COURSE NOTES FOR ELLIPTIC COHOMOLOGY

MIKE HOPKINS

Contents

Introduction 1

Part 1. Preliminaries 1 1. Model categories 1

2. May’s category of spectra 4

3. Elements of enriched category theory 9

4. Operads and monads 13

5. Smash products and stable homotopy 17

6. Geometric realization of simplicial T-algebras 24

7. Lazard’s theorem 25

8. Characteristic classes in generalized cohomology 29

9. Complex oriented cohomology and the MU-spectrum 33

10. More about formal group laws 35

11. Elliptic curves 40

12. The 2-category of Hopf algebroids 42

13. Sheaves and Grothendieck topologies 45

14. Groupoids 49

15. Representable groupoids and morphisms 54

16. Stacks, finally 60

17. The flat topology 62

18. Hopf algebroids and stacks 66

Part 2. The dénouement 82 19. Computations 82 References 87

Introduction

This intro should be

An elliptic spectrum is roughly a ring spectrum E and an elliptic curve over expanded. π_{0}E such that the multiplication on E and the group structure on the elliptic curve are suitably related. The theory of elliptic curves gives an algebraic model for the category of elliptic spectra and we’ll develop the machinery necessary for the study of elliptic spectra. One goal will be to produce a functor from the category of elliptic curves to the category of elliptic spectra.

Date: March 16, 1999.

These notes were typed up by the students in Mike’s MIT course in the fall of 1995 and are in an incomplete and preliminary state. Mike hasn’t yet had a chance to go over them carefully, so please don’t distribute them without his permission. Please give all corrections and comments to Mike Hopkins or Dan Christensen.

Part 1. Preliminaries

Yuck, I don’t like the Part header style. In the first third of these notes, we visit topics that entered algebraic topology

in the late 60’s or early 70’s, and some from the 80’s. Our treatment will be correspondingly brisk, with occasional exercises; we supply references for those parts that have become standard in the literature.

1. Model categories

Give good ref here, since there are.

Definition 1.1. A model category is a category 𝒞 equipped with three classes of maps, called weak equivalences, cofibrations and fibrations, subject to the axioms below. A map displayed as ⟶^{∼} will always be a weak equivalence, a map displayed as ↣ will always be a cofibration, and a map displayed as ↠ will always be a fibration. A map which is both a (co)fibration and a weak equivalence will be called an acyclic (co)fibration. CM1: 𝒞 is closed under finite limits and colimits. CM2: If f and g are maps in 𝒞 such that fg makes sense, and if two of f, g and

fg are weak equivalences, then so is the third. This is referred to as the “two out of three” axiom.

CM3: A retract of a fibration, cofibration or weak equivalence is respectively a fibra-

tion, cofibration or weak equivalence. CM4: Given the commuting solid arrow diagram

A_{i}  X_{p} B _{l} Y

with i a cofibration and p a fibration, if either i or p is a weak equivalence then a lifting l exists making both triangles commute. When such a lift exists for each pair of horizontal maps making the square commute, we say that i has the left lifting property (LLP) with respect to p and that p has the right lifting property (RLP) with respect to i.

CM5: Each map f can be factored into pi with p a fibration and i an acyclic cofibration, and into qj with q an acyclic fibration and j a cofibration.

Exercise 1.2. Prove that a map is a cofibration if and only if it has the left lifting property with respect to all acyclic fibrations, and is an acyclic cofibration if and only if it has the LLP with respect to all fibrations.

Exercise 1.3. Prove the dual to the previous exercise.

Exercise 1.4. Prove that fibrations and cofibrations are preserved under base and cobase change, respectively.

The small object argument. Let 𝒞 be any category closed under countable directed colimits and let I be a collection of maps in 𝒞 preserved under cobase change.

Definition 1.5. An object A of 𝒞 is I-small (sometimes abbreviated to small) if for each countable directed system X_{α} all maps of which are in I, we have that the natural map colim 𝒞(A, X_{α}) → 𝒞(A, colim X_{α}) is an isomorphism.

Definition 1.6. Let 𝒞 be a cocomplete model category (cocomplete means that 𝒞 has all set-indexed colimits) and let I be a collection of maps in 𝒞, stable under cobase change. We say that 𝒞 is generated by I-small objects if there exist two sets {U_{α} → V_{α}} and {S_{α} → D_{α}} of maps which are in I and whose domains are I-small, and having the property that a map f is a fibration if and only if it has the RLP with respect to the maps U_{α} → V_{α} and is an acyclic fibration if and only if it has the RLP with respect to the maps S_{α} → D_{α}.

Example 1.7. Consider the category s𝒮et of simplicial sets and let I contain all of the maps in s𝒮et. Then the sets of maps {V_{n,k} → Δ_{n}} and {∂Δ_{n} → Δ_{n}} show that s𝒮et is generated by I-small objects. We remind the reader that the cofibrations in this model category are the injective maps, the fibrations are the maps having the right lifting property with respect to the maps V_{n,k} → Δ_{n}, and the weak equivalences are the maps whose geometric realizations are weak homotopy equivalences. Say what the horns

are.

Example 1.8. Let 𝒯op be the category of weak Hausdorff compactly generated spaces and let I be the class of maps which are inclusions of closed subspaces. Then the sets {|V_{n,k}| → |Δ_{n}|} and {S^{n−1} → D^{n}} show that 𝒯op is generated by I-small objects. Again we recall the model category structure in this example. A map is a weak equivalence if it is a weak homotopy equivalence, it is a fibration if it is a Serre fibration, and it is a cofibration if it is a retract of a “cellular map”.

A couple of words about point set topology may be in order. Possibly the best way to phrase the condition that a space X be weak Hausdorff is to say that the image of any map from a compact Hausdorff space to X is closed in X. If X is compactly generated, this is equivalent to requiring that the diagonal in the compactly generated product X × X be closed.

Exercise 1.9. Check that 𝒯op is generated by I-small objects.

Similarly, the category 𝒯op_{∗} of pointed spaces is generated by the sets {|V_{n,k}|_{+} → |Δ_{n}|_{+}} and {S_{+}^{n−1} → D_{+}^{n}}, where the subscript indicates the adjunction of a disjoint basepoint.

The technique used to prove the following proposition is a typical example of what is called the “small object argument”. And suppose there

were a separate section entitled ‘the small object argument and cofibrantly generated model categories’, spelling out just what is meant by them? they’ll be referenced three times more!

Proposition 1.10. If 𝒞 is generated by I-small objects, then the factorizations in CM5 can be made functorial.

Proof. Given f : X → Y, we associate to it functorial diagrams

U(f)  V (f) X _{f} Y  and  S(f)  D(f) X _{f} Y

in the following way. We define U(f) to be ∐ U_{α}, where the coproduct is taken over all diagrams

U_{α}  V_{α} X _{f} Y .

The other three functors are defined similarly, and the maps are the obvious ones.

We’ll construct the factorization of f into qj where q is an acyclic fibration and j is a cofibration. We set X_{0} = X and q_{0} = f : X_{0} → Y. Having defined the objects and maps

X_{n−2} ^{j_{n−2}}_{q_{n−2}} X_{n−1}_{q_{n−1}} Y

in such a way that the triangle commutes, we define X_{n}, j_{n−1} : X_{n−1} → X_{n} and q_{n} : X_{n} → Y via the pushout square

S(q_{n−1})  D(q_{n−1}) X_{n−1} _{j_{n−1}} _{q_{n−1}}X_{n}  _{q_{n}} Y .

Let X_{∞} = colim X_{n} and denote by j the inclusion X → X_{∞} and by q the map X_{∞} → Y induced by the q_{n}’s. It’s easy to check the lifting property of j with respect to all acyclic fibrations and of q with respect to the maps S_{α} → D_{α}.

2. May’s category of spectra

We now describe May’s category of “coordinate free” spectra. A basic reference for this material is [16].

Definition 2.1. A universe 𝒰 is a countably infinite dimensional inner product space over ℝ topologized as the direct limit of its finite dimensional subspaces. An indexing set is a set 𝒜 of finite dimensional subspaces of 𝒰 such that each finite dimensional V ⊂ 𝒰 is contained in some U in 𝒜. If V is a finite dimensional vector space, let S^{V} be the one-point compactification of V. Write S^{V} X for S^{V} ∧X and Ω^{V} X for 𝒯op_{∗}(S^{V} , X). Here 𝒯op_{∗} is the category of pointed (weak Hausdorff compactly generated) spaces. A prespectrum indexed on 𝒜 is a collection of pointed spaces {X_{V} }_{V ∈𝒜} together with maps S^{W −V} X_{V} → X_{W} for each pair V ⊆ W of ‘indices’ in 𝒜, such that

S^{W −V} S^{V −U}X_{U}  S^{W −V} X_{V} S^{W −U}X_{U}  X_{W}

commutes and such that S^{0}X_{W} → X_{W} is the identity. (If V is contained in W, then W − V denotes the orthogonal complement of V in W.) A spectrum indexed on 𝒜 is a prespectrum {X_{U}} indexed on 𝒜 such that the adjoints X_{V} → Ω^{W −V} X_{W} to the structure maps are homeomorphisms.

Remark 2.2. If 𝒰 = ℝ^{∞} and 𝒜 = {ℝ^{n}}_{n≥0}, then a prespectrum on 𝒜 is a spectrum in the traditional sense.

Definition 2.3. A map f : X → Y of prespectra X and Y is a collection of maps f_{U} : X_{U} → Y_{U} commuting with the structure maps. This gives the category 𝒫𝒜 of spectra indexed on 𝒜. The category 𝒮𝒜 of spectra is defined to be a full subcategory of the category of prespectra.

Using Freyd’s adjoint functor theorem one shows that the inclusion 𝒮𝒜 → 𝒫𝒜 has a left adjoint, denoted L and called spectrification.

Let X be a prespectrum. We call X an inclusion prespectrum if each map X_{V} → Ω^{W −V} X_{V} is the inclusion of a closed subspace.

Remark 2.4. If each S^{W −V} X_{V} → X_{W} is the inclusion of a closed subspace, then so is each X_{V} → Ω^{W −V} X_{V}.

Proposition 2.5. If X is an inclusion prespectrum then

(LX)_{V} = colim_{W ⊇V} Ω^{W −V} X_{W} .

Proof. With LX defined as in the statement, we prove that the natural map X → LX is the initial map from X to a spectrum, and hence that L is left adjoint to the inclusion. That every map from X to a spectrum factors uniquely through X → LX isn’t hard to see, so it remains to show that LX is in fact a spectrum. Well, if W ⊇ V then we have a sequence of natural isomorphisms

Ω^{W −V} (LX)_{W} = Ω^{W −V} colim_{U⊇W} Ω^{U−W} X_{U}
= colim_{U⊇W} Ω^{W −V} Ω^{U−W} X_{U}
= colim_{U⊇W} Ω^{U−V} X_{U}
= colim_{U⊇V} Ω^{U−V} X_{U} = (LX)_{V} .

The second isomorphism is because the sphere is small with respect to closed inclusions, and the fourth follows by cofinality.

Definition 2.6. Suppose Q is a topological space. Then its suspension spectrum Σ^{∞}Q is the spectrum L(V ↦ Σ^{V} Q).

If we have two indexing sets 𝒜 ⊆ ℬ then there’s an obvious functor 𝒫ℬ → 𝒫𝒜. This clearly descends to a functor 𝒮ℬ → 𝒮𝒜, and moreover the following proposition holds (the proof is left to the reader):

Proposition 2.7. The functor 𝒮ℬ → 𝒮𝒜 is an equivalence of categories.

Note 2.8. If ℬ denotes the set of all finite dimensional subspaces of 𝒰, then the above proposition implies that for any indexing set 𝒜 we have an equivalence of categories 𝒮ℬ ⟶^{∼} 𝒮𝒜. The indexing set will therefore tend to drop out of our notation, and we’ll henceforth adopt the practice of writing 𝒮𝒰 (and sometimes 𝒮_{U}, and sometimes just 𝒮) instead of 𝒮ℬ.

Remark 2.9. The hom-objects of 𝒫𝒜 and 𝒮𝒜 have natural topologies. For example, 𝒫𝒜(X, Y) is a subset of ∏_{V ∈A} 𝒯op_{∗}(X_{V} , Y_{V})—in fact, it’s the subobject which is equalized under the two obvious arrows

∏_{V ∈A} 𝒯op_{∗}(X_{V} , Y_{V}) ⇉ ∏_{V ⊆W} 𝒯op_{∗}(S^{W −V} X_{V} , Y_{W}).

And of course each set 𝒯op_{∗}(X_{V} , Y_{V}) has a natural structure as topological space, which then gets inherited by 𝒫𝒜(X, Y). Since 𝒮𝒜 is a full subcategory, this defines a topology on each 𝒮𝒜(X, Y) when X and Y are spectra.

Definition 2.10. For Q ∈ 𝒯op_{∗} and X ∈ 𝒮𝒜, we make the following definitions:

(a) Q ∧ X = L(V ↦ Q ∧ X_{V}). That is to say, Q ∧ X is the spectrification of the prespectrum whose value on V ∈ 𝒜 is Q ∧ X_{V}.

(b) X^{Q} is the spectrum defined by (X^{Q})_{V} = (X_{V})^{Q}, where (X_{V})^{Q} denotes the space 𝒯op_{∗}(Q, X_{V}). One can check that this really does give a spectrum.

Proposition 2.11. There are natural maps

𝒮(Q ∧ X, Y) → 𝒯op_{∗}(Q, 𝒮(X, Y)) ← 𝒮(X, Y^{Q}),

and moreover, these maps are homeomorphisms.

Proof. Left to the reader.

Our goal is to make the category of spectra into a model category, and the way we’ll do this is by finding functors which allow us to pass back and forth between 𝒮 and the model category 𝒯op_{∗}. To this end, we first note that for each V ∈ 𝒜 there’s an obvious functor (−)_{V} : 𝒮 → 𝒯op_{∗} which maps X → X_{V}. It turns out that such functors are actually corepresentable in our category, and in fact the objects corepresenting them are just the ‘de-suspensions’ of a sphere. The key point is that once we know a functor is corepresentable then by the above proposition smashing with the corepresenting object is a left adjoint to that functor—thus, we obtain functors from 𝒯op_{∗} back to 𝒮.

Lemma 2.12. There exists a unique spectrum S^{−V} for which 𝒮(S^{−V} , X) ≅ X_{V} (as topological spaces, in fact). That is to say, S^{−V} corepresents the functor (−)_{V}.

Proof. Let’s use T to denote the prespectrum

W ⟼ \begin{cases}S^{W −V} & if W ⊇ V \\ ∗ & otherwise, \end{cases}

and let S^{−V} be the spectrification of T. The fact that spectrification is a left adjoint of the forgetful functor says that 𝒮(S^{−V} , X) = 𝒫(T, X) for any spectrum X. Moreover, the functor (−)_{V} mentioned above yields a map 𝒫(T, X) → 𝒯op_{∗}(T_{V} , X_{V}), and since T_{V} = S^{0} the target is really 𝒯op_{∗}(S^{0}, X_{V}) ≅ X_{V}. So it will suffice to show that this map is an isomorphism, and to do this we’ll actually construct its inverse.

First note that since X is a spectrum the maps X_{V} → Ω^{W −V} X_{W} are homeomorphisms for W ⊇ V. These can be thought of as maps 𝒯op_{∗}(S^{0}, X_{V}) → 𝒯op_{∗}(S^{W −V} , X_{W}) = 𝒯op_{∗}(T_{W} , X_{W}). On the other hand, if W ⊉ V then T_{W} = ∗ and there’s a unique map 𝒯op_{∗}(S^{0}, X_{V}) → 𝒯op_{∗}(T_{W} , X_{W}) = ∗. These maps all patch together to give

𝒯op_{∗}(S^{0}, X_{V}) → ∏_{W ⊆𝒰} 𝒯op_{∗}(T_{W} , X_{W}).

It’s not hard to check that the image actually lies in 𝒫(T, X), and that this map is the desired inverse.

Theorem 2.13.

(a) 𝒮 becomes a closed model category if we define the fibrations (resp. weak equivalences) to be the maps f : X → Y such that each f_{V} : X_{V} → Y_{V} is a fibration (resp. weak equivalence), and the cofibrations to be precisely those maps which have the LLP with respect to the acyclic fibrations.

(b) If I = {f : X → Y | each X_{V} → Y_{V} is a closed inclusion}, then 𝒮 is cofibrantly generated by the following sets of maps between I-small objects:

1. A = {V_{n,k+} ∧ S^{−V} ↣ Δ_{n+} ∧ S^{−V} }

2. B = {V_{n,k+} ∧ S^{−V} ↣ Δ_{n+} ∧ S^{−V} } ∪ {S_{+}^{n−1} ∧ S^{−V} ↣ D_{+}^{n} ∧ S^{−V} }. (That is to say, the maps in A detect all fibrations and those in B detect all acyclic fibrations.)

Proof. CM1 requires that 𝒮 possess all finite limits and colimits. This is easy once we make the following observations:

1. Limits and colimits in the category of prespectra can be computed spacewise. That is, if the {X^{α}} form a diagram of prespectra then the assignment

V ⟼ \begin{matrix}lim \\ -------→ \end{matrix}_{α} X_{V}^{α}

defines a prespectrum, and this prespectrum is the colimit of the diagram {X^{α}}. Likewise for inverse limits. In particular, limits and colimits of prespectra always exist.

2. If {X^{α}} is a diagram of spectra then its colimit (or limit) can be obtained by first taking the colimit in the category of prespectra and then spectrifying it.

3. Although not necessary for the argument, we should also note that for inverse limits this last step is superfluous. That is, if one takes a prespectra inverse limit of spectra, then one actually does get a spectrum—or, in other words, inverse limits of spectra can be computed spacewise. (The reason is that the functor Ω commutes with \begin{matrix}lim \\ ←------- \end{matrix}.)

4. One last observation along these lines which will be useful later on is that a sequential direct limit of spectra along closed inclusions can also be computed spacewise. (The crucial point here is that the sphere is small with respect to closed inclusions, and so the functor Ω commutes with \begin{matrix}lim \\ -------→ \end{matrix} when such maps are involved.)

We next remark that axioms CM2 and CM3 are quite trivial, following immediately from the fact that they hold in 𝒯op_{∗}. For instance, a retract of a fibration in 𝒮 will necessarily be a level-wise retract of fibrations in 𝒯op_{∗}; therefore by CM5 for 𝒯op_{∗} an actual level-wise fibration; hence, a fibration in 𝒮.

Part of CM4 is trivial, in that we defined the cofibrations as the maps which have the LLP with respect to the acyclic fibrations. What we must show is that the acyclic cofibrations have the LLP with respect to all fibrations. We defer this for a moment, and prove CM5 first—this is where the heart of the proof really lies.

Our main tool will be the small object argument, and in fact we’ll prove the claims in (b) simultaneously. Step 1: A map if a fibration (resp. acyclic fibration) iff it has the RLP with respect to all maps in A (resp. B). In particular, all the maps in A (and B) are cofibrations.

This follows by virtue of our adjoint functors and by the nature of the way we defined fibrations and weak equivalences. For example, suppose that f : X → Y is a fibration and that we have a diagram

V_{n,k+} ∧ S^{−V}  X_{f} Δ_{n+} ∧ S^{−V}  Y.

Applying the functor (−)_{V} gives a diagram

V_{n,k+}  (V_{n,k+} ∧ S^{−V})_{V}  X_{V}_{f_{V}} Δ_{n+}  (Δ_{n+} ∧ S^{−V})_{V}  Y_{V}

(the left square exists because anytime one has adjoint functors L and R one gets a natural transformation Id → RL). The fact that f is a fibration implies that f_{V} must also be one, and so we get a lifting Δ_{n+} → X_{V} in the above diagram. But now if we go back and apply the functor (−) ∧ S^{−V} then we get

V_{n,k+} ∧ S^{−V}  X_{V} ∧ S^{−V}  X_{f} Δ_{n+} ∧ S^{−V}  Y_{V} ∧ S^{−V}  Y

(and here the right square exists because adjoint functors give a natural transformation LR → Id.) The above diagram shows how to get our desired lifting Δ_{n+} ∧ S^{−V} → X, and so we’ve shown that all fibrations belong to RLP(A). Showing that everything in RLP(A) is in fact a fibration proceeds in a similar manner. Step 2: The objects V_{n,k+} ∧ S^{−V}, Δ_{n+} ∧ S^{−V}, etc. are I-small.

To see this, observe that

𝒮(Δ_{n+} ∧ S^{−V} ,\begin{matrix}lim \\ -------→ \end{matrix} X_{n}) = 𝒯op_{∗}(Δ_{n+}, 𝒮(S^{−V} ,\begin{matrix}lim \\ -------→ \end{matrix} X_{n}))

= 𝒯op_{∗}(Δ_{n+}, (\begin{matrix}lim \\ -------→ \end{matrix} X_{n})_{V})

= 𝒯op_{∗}(Δ_{n+},\begin{matrix}lim \\ -------→ \end{matrix}(X_{n})_{V}) (∗)

= \begin{matrix}lim \\ -------→ \end{matrix} 𝒯op_{∗}(Δ_{n+}, (X_{n})_{V})

= \begin{matrix}lim \\ -------→ \end{matrix} 𝒮(Δ_{n+} ∧ S^{−V} , X_{n}).

((∗)—this is because direct limits of spectra along closed inclusions can be computed spacewise; cf. remark (4) made in the course of proving CM1.)

The above steps give us the grounding to now apply the small object argument. Before proceeding, we wish to make two basic observations about what that argument really gives us (which are really just restatements of things which came up in section 1):

(i) If U is a set of maps, write LLP(U) for the collection of maps which have the LLP with respect to all maps in U. It’s quite easy to check that LLP(U) is closed under coproducts, cobase changes, and sequential direct limits.

(ii) (The Small Object Argument) Let S be a set of maps and let T consist of all maps that can be obtained from elements in S by applying the operations mentioned in (i). If the targets of all the maps in S are T -small, then any map X → Y can be factored into X ⟶^{i} X_{∞} ⟶^{p} Y where p ∈ RLP(S) and i ∈ T.

Step 3: Producing the factorizations.

So suppose that we have a map X ⟶^{f} Y. Applying (ii) to the set of maps B yields a factorization X ⟶^{i} X_{∞} ⟶^{p} Y where p has the RLP with respect to maps in B (and is therefore an acyclic fibration) and i is obtained from maps in B by making coproducts, cobase changes, and taking sequential direct limits. Note that all the maps in B are cofibrations, meaning that they have the LLP with respect to acyclic fibrations, and by (i) such maps are closed under all the operations just mentioned—so i is a cofibration. This gives us one of our factorizations, and so we’re half done.

To continue, apply (ii) to the set of maps A. Then just as above we obtain a factorization where p is a fibration and i is a cofibration. Indeed, we know more, for the maps in A have the LLP with respect to all fibrations, and so i will also have this property (strictly speaking, this is not needed for CM5—but it’s a crucial point, and will come up later on). The only thing that remains to be seen is that i is acyclic—this is the tricky part.

If we let X_{n} denote the space obtained at the nth stage of the small object argument (so that X_{∞} = \begin{matrix}lim \\ -------→ \end{matrix} X_{n}), it will suffice to show that each map X_{n−1} → X_{n} is a levelwise homotopy equivalence. But this map is obtained by pushing out by maps in A, and in general if we have the pushout diagram Something more

needs to be said in this last part of CM5. To be truthful, I don’t really understand how the argument is going.

V_{n,k+} ∧ S^{−V}  C Δ_{n+} ∧ S^{−V}  D,

then the map on the left is a levelwise homotopy equivalence and the levelwise retractions induce levelwise retractions D → C.

Our final task is to return and prove the missing half of CM4—that acyclic cofibrations have the LLP with respect to fibrations. This is actually quite easy now, for if A ⟶^{f} B is an acyclic cofibration then we may use the factorization produced above and write it as a composition A ⟶^{i} A_{∞} ⟶^{p} B where p is a fibration and i is a weak equivalence with the LLP with respect to all fibrations. The two-out-of-three axiom yields that p is a weak equivalence (therefore an acyclic fibration), and then because f is a cofibration we know that a lifting exists in the diagram below:

A ^{i}_{f}  A′_{p} B _{l} B

This diagram actually expresses f as a retract of A ⟶^{i} A′, for we may redraw it as:

A_{f}  A_{i}  A_{f} B _{l} A′ _{p} B .

But the map i had the LLP with respect to fibrations, and such things are closed under retracts as we’ve remarked before—so this shows that f also has this property, which is what we wanted.

3. Elements of enriched category theory

We are about to take a detour in this section, where we give a long string of definitions from category theory. In the previous section we developed quite a bit of structure which connected the categories 𝒯op_{∗} and 𝒮, and there is even more structure yet to come. But here we step aside and introduce a uniform language to describe it all. Good references include Kelly [15] and Borceux’s handbook [4].

Definition 3.1. A monoidal category consists of the data (𝒱, ⊗, I, r, l, a), where 𝒱 is a category, ⊗ : 𝒱 × 𝒱 is a bifunctor, I is a distinguished object in 𝒱, and r,l, and a are natural isomorphisms as follows:

(a) l : I ⊗ X ⟶^{∼} X,

(b) r : X ⊗ I ⟶^{∼} X,

(c) a : (X ⊗ Y) ⊗ Z ⟶^{∼} X ⊗ (Y ⊗ Z).

We ask further that the map l_{I} : I ⊗ I → I coincide with the map r_{I}, and also require axioms stating that the following diagrams commute: MC1: (The Pentagonal axiom)

((W ⊗ X) ⊗ Y) ⊗ Z  (W ⊗ X) ⊗ (Y ⊗ Z) (W ⊗ (X ⊗ Y)) ⊗ Z  W ⊗ (X ⊗ (Y ⊗ Z)) W ⊗ ((X ⊗ Y) ⊗ Z)

MC2:

(X ⊗ I) ⊗ Y  X ⊗ Y .  X ⊗ (I ⊗ Y)

Remark 3.2. The theory of “coherence” is sometimes summarized by saying that if the diagrams in MC1 and MC2 commute then any diagram one expects to also commutes. Mac Lane has a precise syntactical theorem; see [17]. The essence of the statement is that the natural transformations that exhibit (functorially) the isomorphisms between objects one knows to be isomorphic form a suitable algebraic structure under composition.

Definition 3.3. A symmetric monoidal category is a monoidal category equipped with a natural isomorphism s : X ⊗ Y → Y ⊗ X such that s^{2} = id and making the following diagrams commute: SM3:

I ⊗ X  ^{s}_{l}  X ⊗ I_{r} X

SM4:

X ⊗ (Y ⊗ Z)_{1⊗s}  ^{a} (X ⊗ Y) ⊗ Z ^{s} Z ⊗ (X ⊗ Y)_{a} X ⊗ (Z ⊗ Y) _{a} (X ⊗ Z) ⊗ Y _{s⊗1} (Z ⊗ X) ⊗ Y .

Note 3.4. We again make the point that commutativity of the diagrams in SM3 and SM4 provides for coherence: diagrams which should commute actually do.

Example 3.5.

(a) (𝒮et, ×) is a fundamental example of a symmetric monoidal category. The unit is of course the one point set.

(b) The category (𝒜b, ⊗) is a symmetric monoidal category, where the object I is taken to be ℤ and the natural isomorphisms are the obvious ones.

(c) More generally, if R is any commutative ring then the category (ℳod_{R}, ⊗) is symmetric monoidal.

(d) (𝒯op_{∗}, ∧) is symmetric monoidal, and here the unit object I is just S^{0}.

Definition 3.6. A category enriched over a monoidal category 𝒱 is a collection of objects 𝒞 together with “hom-sets” 𝒞(X, Y) in 𝒱 for each pair of objects X and Y, and with 𝒱-maps 𝒞(Y, Z) ⊗ 𝒞(X, Y) → 𝒞(X, Z) and I → 𝒞(X, X) satisfying the usual associativity and unit axioms. (One should of course think of these maps as a “composition law” and an “identity map”, respectively).

Note that in the above definition 𝒞 is not in the strictest sense a category at all (e.g., the “hom-sets” are not really sets, etc.) However, we can always obtain a category from 𝒞 which we’ll call 𝒞_{0}, the underlying category, by taking the same objects as in 𝒞 and defining 𝒞_{0}(X, Y) = 𝒱(I, 𝒞(X, Y)).

Examples 3.7.

(a) A 𝒮et-category 𝒞 is a category in the usual sense. There is an obvious isomorphism 𝒞 ≅ 𝒞_{0} of categories.

(b) An 𝒜b-category is precisely what is usually called a pre-additive category (that is, a category in which the hom-sets are all Abelian groups and in which composition is bilinear.)

(c) The category 𝒮 of spectra is a 𝒯op_{∗}-category.

(d) If 𝒱 is the category of small categories, a 𝒱-category is what is often called a 2-category.

We did not need the full power of a symmetric monoidal category to express the fact that there are composable “hom-objects” with units etc.; in fact, the symmetry never figured. It would, if we wanted to formulate what the opposite of an enriched category is. And in order to imitate other useful features of 𝒮et-categories, it turns out useful to demand that the enriching category 𝒱 satisfy a certain property. All our examples above enjoy this property.

Definition 3.8. A symmetric monoidal category is said to be closed if each functor (−) ⊗ X : 𝒱 → 𝒱 has a distinguished right adjoint (−)^{X} : 𝒱 → 𝒱.

Proposition 3.9. A closed symmetric monoidal category is (canonically) enriched over itself.

Proof. See [15] or [4].

To get a feel for the notion of (enriched) tensor and cotensor, consider the example from 𝒮et-categories: given a bicomplete (complete and cocomplete) category 𝒞, an X ∈ ob 𝒞 and a set U, let U ⊗ X mean the coproduct of a family of copies of X indexed on (members of) U; and let X^{U} mean the product of such a family. Examining the adjunctions these objects satisfy, one arrives at the following notion:

Definition 3.10. Let 𝒱 be a monoidal category enriched over itself (e.g. a closed symmetric monoidal category), and 𝒞 a 𝒱-category. 𝒞 is called a tensored 𝒱category if for every U ∈ 𝒱, X ∈ 𝒞 there exists an U ⊗ X ∈ ob 𝒞 with a 𝒱-natural isomorphism 𝒞(U ⊗ X, Y) ≅ 𝒱(U, 𝒞(X, Y)). The notion of cotensored is dual: for every U ∈ 𝒱, X ∈ 𝒞 there exists an X^{U} ∈ ob 𝒞 with a 𝒱-natural isomorphism 𝒞(Y, X^{U}) ≅ 𝒱(U, 𝒞(Y, X)).

It follows that U ⊗ (−), (−)^{U} : 𝒞 → 𝒞 are what are called 𝒱-functors. A closed symmetric monoidal category is always tensored and cotensored (as a category over itself). See the references for an elaboration of these points.

As we shall have much occasion to dwell in one particular situation, let us agree on a piece of terminology:

Note 3.11. Suppose that 𝒱 is a closed symmetric monoidal category, and 𝒞 a tensored and cotensored 𝒱-category. Then we simply say 𝒞 is closed over 𝒱.

Indexed limits. All of the essentials of 𝒮et-category theory can be extended to the enriched setting. Enriched, or indexed limits are sometimes also called weighted limits; the notion of coend is closely related.

Definition 3.12. Let 𝒞 be a category enriched over the monoidal category 𝒱. Let K : I → 𝒱 and D : I → 𝒞 be two functors; here I is an indexing category, i.e. a usual category over 𝒮et (so that K and D are diagrams). The indexed limit lim_{K} D is an object of 𝒞 with the property that there is an isomorphism in 𝒱, 𝒞(X, lim_{K} D) ≅ 𝒱^{I}(K, D^{X}), natural in X. Here D^{X} : I → 𝒱 is the functor D^{X}(−) = 𝒞(X, D(−)).

Examples 3.13. The first three are familiar from other contexts:

(1) Take I to be Δ, the cosimplicial indexing category, and 𝒞 = 𝒯op, which is cartesian closed. (I.e. the symmetric monoidal structure is that coming from the categorical product.) D : I → 𝒞 is then some cosimplicial space. Define K : I → 𝒞 by K([n]) = Δ_{n}, the n-simplex, with the usual simplicial maps. Then lim_{K} D = Tot D.

(2) Let 𝒞 still be 𝒯op, and I = {• → • ← •}, so D is a diagram E_{2} → B ← E_{1} of spaces. If one takes K to be ∗ → ∗ ← ∗, where ∗ is the one-point space, then lim_{K} D = E_{2} ×_{B} E_{1}, the usual pullback. But by taking K = {0} → [0, 1] ← {1}, one gets lim_{K} D = homotopy fibre product ⊆ E_{2} × B^{[0,1]} × E_{1}.

(3) Take now 𝒞 to be 𝒞at, again a closed category, I as in (2), so D is a diagram A_{1} ⟶^{p_{1}} B ⟵^{p_{2}} A_{2}, and let K be {•} → {•} ← {•}, a diagram of trivial categories. lim_{K} D is a category with objects {(a_{1}, a_{2})| p_{1}(a_{1}) = p_{2}(a_{2}) ∈ ob B}; in short, the usual pullback.

(4) Let everything be as in (3), save take K = {•} ⟶ {• → ⋆} ⟵ {⋆}. Then

lim_{K} D will have objects {(a_{1}, a_{2}, f)| a_{i} ∈ ob A_{i}, p_{1}(a_{1}) ⟶^{f} p_{2}(a_{2}) ∈ mor B}.

Proposition 3.14. If 𝒞 has enough limits, lim_{K} D can be constructed as the following equalizer:

lim_{K} D ⟶ ∏_{i_{0}∈ob I} D(i_{0})^{K(i_{0})} ⇉ ∏_{i_{0}→i_{1}∈mor I} D(i_{1})^{K(i_{0})}.

Quillen’s “corner axiom” for enriched model categories. We want to bring this section to an end by saying a little about the situation when 𝒱 is a model category with a monoidal structure which is enriched over itself and where 𝒞 is a model category which is enriched and closed over 𝒱. It turns out that there’s an additional axiom formulated by Quillen which we’ll want to assume for this setup.

Suppose that A ⟶^{f} B is a morphism in 𝒱 and that X ⟶^{p} Y is a morphism in 𝒞. We define objects f ⊗ p and Hom(f, p) by the following pushout and pullback diagrams:

A ⊗ X  A ⊗ Y  Hom(f, p)  X^{A} B ⊗ X  f ⊗ p  Y^{B}  Y^{A} .

Note that there exist obvious maps f ⊗ p → B ⊗ Y and X^{B} → Hom(f, p).

The following axiom was introduced in [24]: SM7: If f and p are cofibrations then so is f ⊗ p → B ⊗ Y, and this map is a weak

equivalence provided that either f or p is. Likewise, if f is a cofibration and p is a fibration then X^{B} → Hom(f, p) is a fibration, and it’s a weak equivalence if either f or p is one.

Remark 3.15. Let us pretend for a moment that we are in 𝒮et, and pullbacks are computed as usual. Hom(f, p) can then be identified with the set of all commutative diagrams Check on this and

make it better.

A_{f}  X_{p} B  Y

and the image of the map X^{B} → Hom(f, p) consists precisely of those diagrams in which a lifting exists. See what Quillen has

to say ’bout SM7.

Exercise 3.16. Show that 𝒮 satisfies SM7 as a category enriched over 𝒯op_{∗}. Hint: one of the two parts of SM7 usually follows from the other.

4. Operads and monads

Note 4.1. If X is an object in some category 𝒞, a Σ_{n}-action on X is simply a map of monoids Σ_{n} → 𝒞(X, X). From this we can obtain a diagram with the single object X and maps X → X representing the Σ_{n}-action. The object X/Σ_{n} is defined to be the colimit of this diagram, if it exists.

Definition 4.2. Let 𝒱 be a symmetric monoidal category. An operad in 𝒱 is a collection of objects T_{0}, T_{1}, … ∈ 𝒱, with a Σ_{n}-action on T_{n}, together with Σ_{n} × Σ_{i_{1}} × ⋯ × Σ_{i_{n}}-equivariant maps T_{n} ⊗ T_{i_{1}} ⊗ … ⊗ T_{i_{n}} → T_{i_{1}+···+i_{n}} satisfying a certain associativity condition. Although this condition is too cumbersome to write out in full generality, an alert reader will get the idea from the following diagram:

T_{2} ⊗ T_{n} ⊗ T_{m} ⊗ T_{i_{1}} ⊗ … ⊗ T_{i_{n}} ⊗ T_{j_{1}} ⊗ … ⊗ T_{j_{m}} T_{n+m} ⊗ T_{i_{1}} ⊗ … ⊗ T_{j_{1}} ⊗ … ⊗ T_{j_{m}}  T_{2} ⊗ T_{i_{1}+···+i_{n}} ⊗ T_{j_{1}+···+j_{m}} T_{i_{1}+···+j_{1}+···+j_{m}} .

In moving down along the left path in the above diagram, one uses T_{2} to ‘operate’ on T_{n} and T_{m}, and then uses the resulting T_{n+m} to ‘operate’ on the T_{i}’s and the T_{j}’s together. The right path, however, says to use T_{n} to operate on the T_{i}’s, use T_{m} to operate on the T_{j}’s, and then to use T_{2} to operate on the two resultants. We’re requiring that these two ways of ‘operating’ give the same answer.

We have been quite cavalier above, since in a monoidal category that is not strict monoidal, iterated ⊗ expressions lack a canonical meaning. The reader is advised to peruse May’s original definition of topological operads in [19], where this issue does not surface (the reason being that in 𝒯op with cartesian product, one tends to ignore the canonical homeomorphisms between X × Y and Y × X, X × (Y × Z) and (X × Y) × Z etc).

In the general setting, one can use an indexing set larger than ℕ— for example, the set of all finite sets—to keep track of all the canonical isomorphisms. This point

HELP!!! I learned of view is presented in e.g. it from Ezra, and Charles learned it from him too, and he learned it from some Russian.

Example 4.3. If X ∈ ob 𝒱 then the endomorphism operad End(X) of X is the operad of sets defined by End(X)_{n} = 𝒱(X^{⊗n}, X). The structure maps are the obvious ones: if f ∈ 𝒱(X^{⊗n}, X) and g_{i_{k}} ∈ 𝒱(X^{⊗i_{k}}, X), our map is induced by the assignment

(f, g_{i_{1}}, …, g_{i_{n}}) ⟼ f ∘ (g_{i_{1}} ⊗ … ⊗ g_{i_{n}}).

If 𝒱 is enriched over another symmetric monoidal category 𝒱′, then End(X) is an operad in 𝒱′.

Units. End(X) has a distinguished element in degree 1, id_{X}, that behaves as a two-sided unit under composition. We might demand that other operads have this property; one of the several ways of phrasing this is that there is a map I ⟶^{u} T_{1} satisfying

T_{1} ⊗ T_{k}  ^{ξ_{1;k}}  T_{k} I ⊗ T_{k}_{u⊗id}  _{l}

and

T_{n} ⊗ T_{1} ⊗ T_{1} … ⊗ T_{1}  ^{ξ_{n;1,1,...,1}}  T_{n} T_{n} ⊗ I ⊗ I … ⊗ I_{id ⊗u⊗...⊗u}

equivariantly, where ξ_{1;k} and ξ_{n;1,1,...,1} are structure maps, and the unit isomorphisms for ⊗ have been unlabeled.

Henceforth, all our operads will be equipped with units.

Definition 4.4. An algebra over an operad {T_{n}} (or a T -algebra) is an object A ∈ 𝒱 together with Σ_{n}-equivariant maps T_{n} ⊗ A^{⊗n} → A satisfying an associativity condition:

T_{n} ⊗ T_{i_{1}} ⊗ … ⊗ T_{i_{n}} ⊗ A^{⊗i_{1}} ⊗ … ⊗ A^{⊗i_{n}}  T_{n} ⊗ A^{⊗n} T_{i_{1}+···+i_{n}} ⊗ A^{⊗i_{1}+···+i_{n}}  A .

The unit condition for the algebra is the commutativity of

I ⊗ A _{l}  A .  T_{1} ⊗ A

Examples 4.5. We’ll work in the monoidal category (𝒜b, ⊗).

1. If T is the operad with T_{n} = ℤ for all n, then a T -algebra is simply a commutative ring with unit.

2. If W_{n} = ℤ[Σ_{n}] (the group ring of Σ_{n} over ℤ) then a W -algebra is just an associative ring with unit.

In the above examples, the algebra for the operad came from the same category as the coefficients of the operad, in fact the monoidal category (𝒜b, ⊗) is closed, but this is not necessary. For this and other reasons — among them the simplification of the operad structure maps — we rebuild our notions another way, using monads.

Definition 4.6. A triple (or “monad”) in a category 𝒞 consists of a functor T : 𝒞 → 𝒞, together with natural transformations η : Id → T and μ : T^{2} → T (a “unit” and a “product”) making the following diagrams commute:

T^{3} ^{T μ}_{μT}  T^{2}_{μ}  Id ∘T ^{ηT} T^{2}_{μ}  T ∘ Id^{T η} T^{2} _{μ} T  T .

Alternatively, if we observe that the category of functors 𝒞 → 𝒞 has a (strict) monoidal structure given by composition, then a triple is nothing other than a monoid in this category.

Definition 4.7. An algebra over a monad T in a category 𝒞 consists of an X ∈ ob 𝒞 and a morphism T X ⟶^{ξ} X making the following diagrams commute:

T^{2}X ^{T ξ}_{μ_{X}}  T X_{ξ}  X ^{η_{X}} T X_{ξ} T X _{ξ} X  X .

It was realized about thirty years ago that many constructions from universal algebra can be compressed into these very simple structure maps. Take the example of (non-unital) monoids over the category of sets, where we have adjoint functors

𝒮et ^{F} ℳonoid_{U} .

Here U is once again the forgetful functor, and F takes a set X to the free monoid generated by X, which is just ∐_{n} X^{n} with the obvious multiplication.

Notice that making a set X into a monoid is the same as giving a map of sets F X → X which makes certain diagrams commute (the map tells us what all the n-fold products should be, but we need to force associativity).

This situation obtains in general, in fact.

Definition 4.8. An algebra map between two T -algebras X and Y is a map X ⟶^{f} Y ∈ mor 𝒞 making

T X_{ξ_{X}} ^{T f} T Y_{ξ_{Y}} X _{f} Y

commute.

Let 𝒞^{T} be the category of T -algebras. There is a forgetful functor U from T - algebras to 𝒞, and it turns out that this functor has a left adjoint which we will also call T:

Proposition 4.9. Equip T X with the structure map coming from the monad: T T X ⟶^{μ_{X}} T X. This makes T X into a T -algebra and in fact defines a functor

𝒞 ^{T} 𝒞^{T}_{U} .

Remark 4.10. T X should of course be thought of as the “free T-algebra on the object X”.

Let us now return to operads. Suppose 𝒱 has countable colimits; then any operad {T_{i}} gives rise to an endofunctor T (X) = ∐_{n} T_{n} ⊗_{Σ_{n}} X^{⊗n}. Σ_{n} acts on X^{⊗n}

by “permuting the objects” and diagonally on T_{n} ⊗ X^{⊗n}; ⊗_{Σ_{n}} stands for the colimit,

as mentioned at the beginning of this section.

This endofunctor is part of a triple. We begin the verification by making the following conventions: if \underline{i} = (i_{1}, …, i_{n}) then write l(\underline{i}) = n, | \underline{i} |= i_{1} + ⋯ + i_{n}, and

T\underline{i} = T_{i_{1}} × ⋯ × T_{i_{n}}, X^{(\underline{i})} = X^{(i_{1})} ⊗ ⋯ ⊗ X^{(i_{n})},

Σ\underline{i} = Σ_{i_{1}} × ⋯ × Σ_{i_{n}}, T\underline{i} ⊗_{Σ\underline{i}} X^{(\underline{i})} = (T_{i_{1}} ⊗_{Σ_{i_{1}}} X^{(i_{1})}) ⊗ ⋯ ⊗ (T_{i_{n}} ⊗_{Σ_{in}} X^{(i_{n})}).

In this notation we may write

T T X = ∐_{m} T_{m} ⊗_{Σ_{m}} (∐_{n} T_{n} ⊗_{Σ_{n}} X^{(n)})^{(m)} = ∐_{m} T_{m} ⊗_{Σ_{m}} ( ∐_{l(\underline{i})=m} T\underline{i} ⊗_{Σ\underline{i}} X^{(\underline{i})}).

Now the structure maps for the operad T_{m} ⊗ T\underline{i} → T_{|\underline{i}|} induce maps

T_{m} ⊗_{Σ_{m}} (T\underline{i} ⊗_{Σ\underline{i}} X^{(\underline{i})}) = (T_{m} ⊗ T\underline{i}) ⋊_{Σ_{m}×Σ\underline{i}} X^{(\underline{i})} → T_{|\underline{i}|} ⊗_{Σ_{|\underline{i}|}} X^{|\underline{i}|}

and these then give us a map T T X → T X.

Also note that our distinguished unit I → T_{1} yields a map

X = I ⊗ X → T_{1} ⊗ X → ∐_{n} T_{n} ⊗ X^{⊗n} = T X.

One may check that these two maps satisfy the axioms for a triple. The calculation needs that X ⊗ − commute with colimits. (Also, for full rigor regarding coherence, there is the aforementioned trick of using all finite sets to index the coefficients of an operad.)

One can then define an algebra for an operad to be an algebra for this triple; this extends our previous definition.

Remark 4.11. If T is an operad, then the set of T -algebra structures on an object X can be identified with the set of operad maps T → End(X). (And an operad map T → U, by the way, is the obvious thing: a collection of Σ_{n}-equivariant maps T_{n} → U_{n} which commute with the structure maps. It corresponds to a notion of map of triples.)

5. Smash products and stable homotopy

Suppose that 𝒰 and 𝒱 are universes and that f : 𝒰 → 𝒱 is a linear isometric embedding (that is, a linear map which is injective and preserves the metric—the map need not be surjective.) We’ll write LIE(𝒰, 𝒱) for the space of all such maps. Since 𝒰 is topologized as the direct limit of its finite dimensional subspaces, we find that LIE(𝒰, 𝒱) can be topologized as an inverse limit of the sets LIE(W, 𝒱) where W is a finite dimensional subspace of 𝒰. (Note: there’s a better way of looking at the topology on LIE(𝒰, 𝒱), which we’ll see later).

Our first goal is to explain how we can use the map f to “change universes”—i.e, to push forward a spectrum on 𝒰 to a spectrum on 𝒱 and to pull back a spectrum on 𝒱 to one on 𝒰. For Y ∈ 𝒮_{V}, let f^{∗}Y ∈ 𝒮_{U} be given by (f^{∗}Y)_{W} = Y_{f(W)}. As usual, the reader should verify that this construction really does give a spectrum. Conversely, for X ∈ 𝒮_{U} we can define f_{∗}X ∈ 𝒮_{V} by f_{∗}X = L(W ↦ S^{W −f(U)}X_{U}), where U = f^{−1}W.

Exercise 5.1. Show that f^{∗} and f_{∗} are adjoint—i.e., that 𝒮_{V} (f_{∗}X, Y) = 𝒮_{U}(X, f^{∗}Y) for all X ∈ 𝒮_{U} and Y ∈ 𝒮_{V}.

Remark 5.2. f^{∗} carries fibrations to fibrations, acyclic fibrations to acyclic fibrations, etc. In fact, these adjoint functors can be shown to satisfy Quillen’s conditions for an equivalence of model categories.

Now we want to generalize the above construction, so that for any topological space Q, each map Q → LIE(𝒰, 𝒱), and each X ∈ 𝒮_{U}, Y ∈ 𝒮_{V} we get objects Q ⋊ X ∈ 𝒮_{V} (read ‘half-smash’) and Y^{Q} ∈ 𝒮_{U}. We’ll want the functors Q ⋊ (−) and (−)^{Q} to form an adjoint pair, and to coincide with the above constructions in the case where Q is a point.

We first note that it suffices to do this for K compact, since in general we may define

Q ⋊ X := \begin{matrix}lim \\ -------→ \end{matrix}_{K} K ⋊ X

and

Y^{Q} := \begin{matrix}lim \\ ←------- \end{matrix}_{K}Y^{K},

We should choose where in both equations K runs over the compact subspaces of Q. another notation for this Y^{Q}. Is it used anywhere else?

So choose an ascending sequence of finite dimensional subspaces U_{0} ⊆ U_{1} ⊆ ⋯ ⊆ 𝒰 whose union is 𝒰, and likewise choose an ascending sequence of finite dimensional subspaces V_{0} ⊆ ⋯ ⊆ 𝒱 whose union is 𝒱. Our map K → LIE(𝒰, 𝒱) gives a linear map K ×U_{i} → 𝒱 (for each i), and since it’s linear the map is determined completely by its restriction to K × B(U_{i}) where B(U_{i}) is the unit ball in U_{i}. But K × B(U_{i}) is compact, and so its image must lie in one of the V_{j}. We can assume that the V_{j} were chosen so that in fact the image lies in V_{i}—our construction will actually turn out to be independent of the choice of V_{j}’s.

From the map K × U_{i} ⟶^{f} V_{i} we can get a vector bundle over K for which the fibre over a point k will be V_{i} − f(k, U_{i}). Write K^{V_{i}−U_{i}} for the Thom complex of this bundle. We’ll then define

K ⋊ X := L(V_{i} ⟼ K^{V_{i}−U_{i}} ∧ X_{U_{i}}) and Y_{U_{i}}^{K} := 𝒯op_{∗}(K^{V_{i}−U_{i}}, Y_{U_{i}}).

(One should check that Y^{K} is indeed a spectrum.)

Exercise 5.3. Show that 𝒮_{V} (K ⋊ X, Y) ≅ 𝒮_{U}(X, Y^{K}) as topological spaces.

Given spectra X ∈ 𝒮_{𝒰} and Y ∈ 𝒮_{𝒱} , we want to define a space which we’ll denote 𝒮(X, Y) and which we can think of as ‘morally’ being the space of all spectrum maps X → Y. The fact that X and Y are defined on different universes(!) means we have to be a bit crafty in our definition. As a set, we’ll define

𝒮(X, Y) = {(t, f) : t ∈ LIE(𝒰, 𝒱) and f : X → t^{∗}Y is a spectrum map}.

Note that we have an obvious projection map 𝒮(X, Y) → LIE(𝒰, 𝒱). To give the topology, we’ll say that a map p : Q → 𝒮(X, Y) is continuous if and only if

(i) the composite map Q → LIE(𝒰, 𝒱) is continuous, and (ii) there exists a (necessarily unique) map of spectra Q ⋊ X → Y such that for

each q ∈ Q the following diagram commutes:

q ⋊ X _{f}  Y ,  Q ⋊ X

where f is the second coordinate of p(q). To be explicit, a subset of 𝒮(X, Y) is open if and only if its preimage under each such map Q → 𝒮(X, Y) is open.

Note 5.4. What’s being said here is that, given a topological space Q and a map Q → LIE(𝒰, 𝒱), the space 𝒮_{𝒱} (Q⋊X, Y) can be identified with the space of sections Q → 𝒮(X, Y) in the diagram below:

𝒮(X, Y) Q  LIE(𝒰, 𝒱) .

Smash products of spectra.

Definition 5.5. If X, Y ∈ 𝒮_{𝒰} then the data {X_{V} ∧ Y_{W} }_{V,W ⊆𝒰} defines a prespectrum on the set 𝒜 = {V × W ⊆ 𝒰 × 𝒰 : V, W ⊆ 𝒰 are finite dimensional}. We define the smash product X ∧Y to be the spectrification of this prespectrum, which is an element of 𝒮_{𝒰×𝒰} . We denote by X^{(n)} the smash product of X with itself n times.

What we would really like, in the best of all possible worlds, would be to have the smash product of X and Y be a spectrum on 𝒰. We can get such a thing by choosing an isomorphism 𝒰 → 𝒰 × 𝒰 and then pulling back X ∧ Y using this map. Unfortunately, it turns out that no single choice for this map will do everything we really want it to. We therefore choose to stick with the present situation in which smashing also puts one in a new universe.

Now given a spectrum X we want some kind of ‘endomorphism operad’ associated to X. The fact that X^{(n)} and X don’t lie in the same category (i.e., they’re defined on different universes) means that we can’t take the usual route in defining this. Instead, the natural thing to do is to define End(X)_{n} = 𝒮(X^{(n)}, X), where the latter set is the object defined above which sits naturally over LIE(𝒰^{n}, 𝒰).

Note that the collection of spaces {ℒ(n)} = {LIE(𝒰^{n}, 𝒰)} is an operad in a natural way; e.g., the map ℒ(2) × ℒ(n) × ℒ(m) → ℒ(n + m) is the assignment (f, g, h) ⟼ f ∘ (g × h). (And observe that this is just the usual endomorphism operad of 𝒰, if we’re working in the right category.) So what we’ve said is that for a spectrum X ∈ 𝒮_{𝒰} , the endomorphism operad End(X) is naturally an operad over ℒ. We point out that each space ℒ(n) is contractible.

From here on this will always be our setup: we’ll work in the situation where T = {T_{n}} is an operad over the linear isometries operad ℒ, and moreover we’ll assume that T is pointed—that is, that T is equipped with an element ∗ ∈ T_{1} which maps to the identity element of ℒ(1). Note that End(X) is naturally pointed, since there is a distinguished element id_{X} ∈ End(X)_{1} = 𝒮(X, X).

mention that colim have to commute w/ smash prod for operad thingies to work Definition 5.6. For X ∈ 𝒮_{U}, a T -algebra structure on X is a map of pointed operads over ℒ

T ℒ . End(X)

So in particular, for each n we have a map T_{n} ⋊_{Σ_{n}} X^{(n)} → X such that the following

diagram commutes:

T_{n} ⋊_{Σ_{n}} ((T_{i_{1}} ⋊_{Σ_{i_{1}}} X^{(i_{1})}) ∧ ⋯ ∧ (T_{i_{n}} ⋊_{Σ_{in}} X^{(i_{n})}))  T^{n} ⋊_{Σ_{n}} X^{(n)} T_{i_{1}+···+i_{n}} ⋊_{Σ_{i_{1}+···+in}} X^{(i_{1}+···+i_{n})}  X .

Examples 5.7.

(a) When T = ℒ a T -algebra is called an E_{∞}-algebra, the topological analogue of a commutative ring.

(b) When T_{n} = ℒ(n) ⋊ Σ_{n} a T -algebra is called an A_{∞}-algebra, and is the topological analogue of an associative ring.

Our goal is the following fundamental result:

Theorem 5.8. If T is an operad then 𝒮^{T} becomes a closed model category, enriched, tensored and cotensored over 𝒯op_{∗}. A map is defined to be a fibration (resp. weak equivalence) iff Uf is one, and the cofibrations are taken to be precisely those maps which have the LLP with respect to all acyclic fibrations. Moreover, 𝒮^{T} is cofibrantly generated by the following sets of maps:

• A = {T (V_{n,k+} ∧ S^{−V}) → T (Δ_{n+} ∧ S^{−V})}

• B = A ∪ {T (S_{+}^{n−1} ∧ S^{−V}) → T (D_{+}^{n} ∧ S^{−V})}. (As usual, this means that the maps in A detect all fibrations and those in B detect all acyclic fibrations.)

Remark 5.9. The proof of this theorem will be deferred for a bit while we do the necessary groundwork. Take note, however, that just as when we showed that 𝒮 was a model category, most of the work will fall out quite easily from our adjoint functors U and T. In particular, the proof that A detects all fibrations and B detects all acyclic fibrations follows exactly as before.

Exercise 5.10. Suppose that T → ℒ is an operad and that X is a T -algebra (in the category of spectra, of course). Show that:

(a) The two maps T^{2}X ⇉ T X are maps of T -algebras. (One of the maps is the one described above, and the other is obtained by applying T to the map T X → X which makes X into a T -algebra. T X (T^{2}X) is made into a T -algebra using μ_{X} (resp. μ_{T X}).)

(b) The diagram T^{2}X ⇉ T X → X is a coequalizer in 𝒮^{T}. (Hint for both parts: Think about the analogous situation for monoids. In fact, the results hold for any triple.)

Proposition 5.11. 𝒮^{T} is bicomplete, and closed over 𝒯op_{∗}. (Just as a reminder, this means that 𝒮^{T} contains all limits and colimits, and that there exist adjoint functors Q ⊗ (−) and (−)^{Q} : 𝒮^{T} → 𝒮^{T} for any topological space Q. See Note 3.11.)

Proof. The easy part of this is constructing inverse limits and function objects— these are “created by the forgetful functor”. For instance, if the {X_{α}} form a diagram of T -algebras and \begin{matrix}lim \\ ←------- \end{matrix}X_{α} exists, then it’s easy to see that we must have U(\begin{matrix}lim \\ ←------- \end{matrix}X_{α}) = \begin{matrix}lim \\ ←------- \end{matrix}U(X_{α}). The reason is that U is a right adjoint and right adjoints always commute with inverse limits. More explicitly, we can observe that for any spectrum Y we would have

𝒮(Y, U(\begin{matrix}lim \\ ←------- \end{matrix}X_{α})) = 𝒮^{T} (T Y, \begin{matrix}lim \\ ←------- \end{matrix}X_{α}) = \begin{matrix}lim \\ ←------- \end{matrix}𝒮^{T} (T Y, X_{α})

= \begin{matrix}lim \\ ←------- \end{matrix}𝒮(Y, UX_{α}) = 𝒮(Y, \begin{matrix}lim \\ ←------- \end{matrix}UX_{α});

this shows that U(\begin{matrix}lim \\ ←------- \end{matrix}X_{α}) and \begin{matrix}lim \\ ←------- \end{matrix}UX_{α} corepresent the same functor, and therefore they must be isomorphic.

So to construct inverse limits in 𝒮^{T}, our task becomes that of outfitting \begin{matrix}lim \\ ←------- \end{matrix}UX_{α} with a natural structure as a T -algebra. So what we need is a spectrum map T (\begin{matrix}lim \\ ←------- \end{matrix}UX_{α}) → \begin{matrix}lim \\ ←------- \end{matrix}UX_{α} which makes the usual diagrams commute. The fact that each X_{α} is a T -algebra means that we have structure maps (of spectra) T (UX_{α}) → UX_{α}; and the fact that the maps in the diagram {X_{α}} are T -algebra maps means that the diagrams

T UX_{α}  T UX_{β} UX_{α}  UX_{β}

all commute. Thus, we get a map of spectra-diagrams {T UX_{α}} → {UX_{α}}, and this induces a map on the inverse limits: \begin{matrix}lim \\ ←------- \end{matrix}T UX_{α} → \begin{matrix}lim \\ ←------- \end{matrix}UX_{α}. But observe that for any functor F there is always a canonical map F (\begin{matrix}lim \\ ←------- \end{matrix}X_{α}) → \begin{matrix}lim \\ ←------- \end{matrix}F X_{α}, and precomposing with this map for the functor T yields the desired morphism T (\begin{matrix}lim \\ ←------- \end{matrix}UX_{α}) → \begin{matrix}lim \\ ←------- \end{matrix}UX_{α}. (The reader should check that it really does satisfy the right properties.)

Note: It’s useful to organize all this using the following diagram (where the circles indicate “lots of objects” and the double arrows mean “lots of arrows”):

\begin{matrix}lim \\ ←------- \end{matrix}UX_{α}  UX_{α} T (\begin{matrix}lim \\ ←------- \end{matrix}UX_{α})  \begin{matrix}lim \\ ←------- \end{matrix}T UX_{α}  T UX_{α}

We make note that the construction of function objects X^{Q} is quite similar: since (−)^{Q} is a right adjoint, it will necessarily commute with the functor U and so the task becomes that of putting a natural T -algebra structure on (UX)^{Q}. Doing this is not all that different from the above, so we omit the details.

Now it happens that producing colimits and tensor objects Q ⊗ X is a bit more difficult. The easy case is where X has the form T A, for we would need to have

\begin{aligned}𝒮^{T} (Q ⊗ T A, Y) = 𝒯op_{∗}(Q, 𝒮^{T} (T A, Y)) & = 𝒯op_{∗}(Q, 𝒮(A, UY)) \\ & = 𝒮(Q ∧ A, UY) = 𝒮^{T} (T (Q ∧ A), Y). \end{aligned}

Thus, we are forced to define things so that Q ⊗ T A is isomorphic to T (Q ∧ A), for they must represent the same functor.

In a similar manner, we conclude that if the {A_{α}} form a diagram of spectra then we must have \begin{matrix}lim \\ -------→ \end{matrix} T (A_{α}) = T (\begin{matrix}lim \\ -------→ \end{matrix} A_{α}). (To put things more briefly, T is a left adjoint and so must commute with colimits.)

Extending these definitions to all T -algebras requires some machinery. First note that the functor Q ⊗ (−), once it is defined, must preserve coequalizers (since it will be a left adjoint). Applying it to T^{2}X ⇉ T X → X (which was proven to be a coequalizer in the previous exercise), and using the formula for Q ∧ T A derived above, we obtain the following coequalizer diagram of T -algebras:

T (Q ∧ T X) ^{d_{0}}_{d_{1}} T (Q ∧ X)  Q ⊗ X .

Thus, we may define Q ⊗ X as this coequalizer, provided that it actually exists. (The map d_{0} is simply the one induced by the map T X → X, whereas d_{1} comes from the natural transformation T^{2} → T in the following roundabout way: the map T^{2}X → T X yields a natural transformation of functors

𝒯op_{∗}(Q, 𝒮^{T} (T X, −))  𝒯op_{∗}(Q, 𝒮^{T} (T^{2}X, −)) 𝒮^{T} (T (Q ∧ X), −)  𝒮^{T} (T (Q ∧ T X), −)

and the Yoneda lemma guarantees that a natural transformation between representable functors always comes from a map between the objects representing them.)

So the game is to construct the above coequalizer, and it turns out that there’s a trick—it involves the fact that there is some extra information lying around that we have not used as of yet. Recall that the unit for the triple T gives us a map of spectra X → T X, and this will extend to a map T (Q ∧ X) ⟶^{s} T (Q ∧ T X) of T -algebras. One can easily check that the properties of a unit say that this map is a section of both d_{0} and d_{1}; i.e., d_{0}s = d_{1}s = id. Thus, we are led to the following

Definition 5.12. Reflexive coequalizer data is a diagram of the form

X_{1} ^{d_{0}}_{d_{1}} X_{0}_{s}

where d_{0}s = d_{1}s = id. A reflexive coequalizer is a coequalizer whose parallel maps, say d_{0} and d_{1} have a common section s as above.

The following lemma is much more useful than would seem at first glance.

Lemma 5.13. If T : 𝒮 → 𝒮 preserves reflexive coequalizers, then U : 𝒮^{T} → 𝒮 creates reflexive coequalizers.

Remark 5.14. Let us spell out the claim in detail. The premise is the following: if A_{1} ⇉ A_{0} → A is a reflexive coequalizer in 𝒮, then T A_{1} ⇉ T A_{0} → T A is a coequalizer in 𝒮. And the conclusion means: if

X_{1}  X_{0}

is reflexive coequalizer data in the category of T -algebras, then the coequalizer X of UX_{1} ⇉ UX_{0} has a unique T -algebra structure making X_{0} → X a map of T - algebras, and with this T -algebra structure X becomes the coequalizer of X_{1} ⇉ X_{0} in 𝒮^{T}.

Proof. If we are given reflexive coequalizer data X_{1} X_{0} in the category of T -algebras, let X be the coequalizer of UX_{0} ⇉ UX_{1}. Then we can look at the diagram of spectra

UX_{1}  UX_{0}  X T UX_{1}  T UX_{0}  T X_{ρ}

(where the vertical arrows are the maps giving the X_{i} their T -algebra structures). The premise says that the bottom row is a coequalizer in 𝒮, therefore we get the induced map ρ. This map gives X a T -algebra structure, and from there it is easy to see that X is the coequalizer of X_{1} ⇉ X_{0} as T -algebras.

Proof that the hypothesis of Lemma 5.13 is satisfied. The diagram we are interested in is a coproduct of diagrams of the form

T_{n} ⋊ A_{1}^{(n)} ⇉ T_{n} ⋊ A_{0}^{(n)} → T_{n} ⋊ A^{(n)},

and so it will suffice to show that each of these is a coequalizer. But the functor T_{n} ⋊ (−) : 𝒮_{𝒰^{n}} → 𝒮_{𝒰} is a left adjoint (thus, commutes with coequalizers), and so it actually suffices to show that A_{1}^{(n)} ⇉ A_{0}^{(n)} → A^{(n)} is a coequalizer diagram in 𝒮_{𝒰^{n}} . Next we observe that A_{i}^{(n)} = L(V_{1} × ⋯ × V_{n} ⟼ A_{i}(V_{1}) ∧ ⋯ ∧ A_{i}(V_{n})) and that L is a left adjoint, and so it will suffice to show that we have a coequalizer at the prespectrum level. Thus, all we really need to do is to show that if we are given reflexive coequalizers X_{i} ⇉ Y_{i} → Z_{i} in 𝒯op_{∗} (i = 1, …, n), then

X_{1} ∧ ⋯ ∧ X_{n} ⇉ Y_{1} ∧ ⋯ ∧ Y_{n} → Z_{1} ∧ ⋯ ∧ Z_{n}

is also a coequalizer in 𝒯op_{∗}. This is not too hard, and left to the reader.

Remark 5.15. This last — unproved — assertion is a purely formal consequence of the fact that smash product commutes with colimits in the category of compactly generated weak Hausdorff spaces. Let us also remark that it is only at this point that we exploit the specific operad structure.

Remark 5.16. There are variants on Lemma 5.13; for example, if T preserves colimits (coequalizers), then 𝒞^{T} has colimits (coequalizers). See [4] p.198 or [3] p.324. Below we will essentially prove that if 𝒞 is cocomplete and 𝒞^{T} has coequalizers, then 𝒞^{T} is cocomplete. Cf. [4] p.200; here 𝒞 is any category. In all these cases, however, verifying the hypotheses on T may be non-trivial. Indeed, in general a category of algebras need not have coequalizers.

Continuation of the proof that 𝒮^{T} is complete. The above machinery finishes up the construction of tensor objects: if X is a T -algebra and Q is a space then

T (Q ∧ T X)  T (Q ∧ X)

is reflexive coequalizer data in 𝒮^{T}, and by the above its coequalizer exists—we define Q ⊗ X to be this coequalizer.

A similar procedure allows us to construct colimits. If the {X_{α}} form a diagram of T -algebras, then the coequalizers T^{2}X_{α} ⇉ T X_{α} → X_{α} give maps of diagrams {T^{2}X_{α}} ⇉ {T X_{α}} → {X_{α}}, and if colimits exists then (because they necessarily commute with coequalizers) we would have the coequalizer

\begin{matrix}lim \\ -------→ \end{matrix} T^{2}X_{α} ⇉ \begin{matrix}lim \\ -------→ \end{matrix} T X_{α} → \begin{matrix}lim \\ -------→ \end{matrix} X_{α}.

Since T commutes with direct limits, what this says is that \begin{matrix}lim \\ -------→ \end{matrix} X_{α}, if it exists, must be the coequalizer of

T (\begin{matrix}lim \\ -------→ \end{matrix} UT X_{α}) ⇉ T (\begin{matrix}lim \\ -------→ \end{matrix} UX_{α})

(where the U’s have been inserted to clarify that these are direct limits of spectra.)

So again, the task becomes that of showing that this coequalizer actually exists— and once again the unit for T gives it the structure of a reflexive coequalizer, so our machinery shows that indeed it does.

We are now in a position to prove Theorem 5.8.

Proof of Theorem 5.8. Now that we have proved CM1, the rest of the proof is exactly analogous to that of Theorem 2.13. We will only remark on the proof of CM5. First, we will find a factorization of a map f : X → Y into X →^{i} X^{∞} →^{p} Y where i is an acyclic cofibration and p is a fibration. We use the small object argument as before; at each stage, we have the pushout diagram

T (V_{n,k+} ∧ S^{−V}) ^{j} T (Δ_{n} + ∧S^{−V}) X_{n−1}  _{i_{n}}  X_{n} .

Since j is a spacewise homotopy equivalence, i_{n} is a weak equivalence. Furthermore, f is an inclusion and X_{n−1} is a retract, so i_{n} is a spacewise closed inclusion and is thus in the class I. Thus, i = i^{∞} is an acyclic cofibration. As before, p is a fibration since it is in RLP(A).

Now we need only factor a map f : X → Y into X →^{i} X^{∞} →^{p} Y where i is a cofibration and p is an acyclic fibration. The argument goes as above; all that needs to be remarked is that i_{n} is a spacewise closed inclusion in the diagram

T (S_{+}^{n−1} ∧ S^{−V}) ^{j} T (D_{+}^{n} ∧ S^{−V}) X_{n−1}  _{i_{n}}  X_{n} .

This proof to be finished.

6. Geometric realization of simplicial T-algebras

Although we will not use it for some time to come, as long as we are discussing T-algebras, we would like to develop the definition and basic properties of the realization of a simplicial T-algebra.

Suppose [n] ↦ X_{n} is a simplicial T-algebra. Realization is a functor from simplicial T-algebras to T-algebras. The realization |X_{▪}|^{T} of X_{▪} is defined as the

somehow I’d like to coequalizer of see a more substantial blob designate simplicial objects.

∐_{[n]→[m]} Δ_{n} ⊗ X_{m} ⇉_{f_{2}}^{f_{1}} ∐_{n} Δ_{n} ⊗ X_{n} .

There are several things to note about this definition. The coequalizer can be defined either in T-algebras or in spectra, since it is a reflexive coequalizer. The coproduct is taken in the category of T-algebras. The map [n] → [m] gives a map X_{m} → X_{n}, which we can tensor with Δ_{n} to get f_{1}. For f_{2}, tensor the map Δ_{n} → Δ_{m} with X_{m}. Similarly, we can define geometric realization for simplicial spectra or simplicial prespectra, which will be denoted |−|, without the ^{T} decoration.

This definition is the natural generalization of geometric realization in the topological case. For topological spaces, Δ_{n} ⊗ X_{n} simply gives us a copy of Δ_{n} for each n-simplex, including the degenerate simplices. Note that f_{1} : Δ_{p} ⊗ X_{q} → Δ_{p} ⊗ X_{p} and f_{2} : Δ_{p} ⊗X_{q} → Δ_{q} ⊗X_{q}. If q < p, the identification eliminates a degeneracy. If p < q, the identification glues a lower-dimensional simplex to the higher-dimensional simplex of which it is a face.

Also note that the geometric realization corepresents the functor Y ↦ Tot(𝒮^{T} (X_{▪}, Y)), where 𝒮^{T} (X_{▪}, Y) is just the cosimplicial space [n] ↦ 𝒮^{T} (X_{n}, Y).

Proposition 6.1. If X_{▪} is a simplicial T-algebra, the map

|UX_{▪}| ⟶ U|X_{▪}|^{T}

is an isomorphism.

The proof of this proposition is easy once we have the following lemma.

Lemma 6.2. If V_{▪} is a simplicial spectrum, the map

|UT V_{▪}| ⟶ UT |V_{▪}|

is an isomorphism.

Remark 6.3. In the lemma, T is taken to be the free algebra functor 𝒮 → 𝒮^{T}, of course. T V_{▪} is defined by levelwise application of T.

Proof that Lemma 6.2 implies Proposition 6.1. First we prove a special case, when X_{▪} = T V_{▪} for some simplicial spectrum V_{▪}. That is, we want to show that |UT V_{▪}| ≅ U|T V_{▪}|^{T}. We know from the above lemma that |UT V_{▪}| ≅ UT |V_{▪}|, so we need only see that T |V_{▪}| = |T V_{▪}|^{T}. But this is just general nonsense:

𝒮^{T} (T |V_{▪}|, E) = 𝒮(|V_{▪}|, UE)
= Tot(𝒮(V_{▪}, UE))
= Tot(𝒮^{T} (T V_{▪}, E))
= 𝒮^{T} (|T V_{▪}|^{T} , E).

To prove the general case, consider the diagram:

|T^{2}X_{▪}|  |T X_{▪}|  |X_{▪}| |T^{2}X_{▪}|^{T}  |T X_{▪}|^{T}  |X_{▪}| .

By the special case, we know that the down arrows are isomorphisms. But the coequalizer of |T^{2}X_{▪}|^{T} ⇉ |T X_{▪}|^{T} is |X_{▪}|^{T}, since realization is a colimit and therefore commutes with other colimits.

Proof of Lemma 6.2 . Recall that

|T V_{▪}| = |∐ T_{n} ⋊_{Σ_{n}} V_{▪}^{(n)}|,

where V_{▪}^{(n)} is taken levelwise. Since realization is a colimit, it commutes with all left adjoints, so in particular it commutes with L, T_{n}⋊, and division by Σ_{n}. Thus,

|T V_{▪}| = ∐ T_{n} ⋊_{Σ_{n}} L|V_{▪}^{(n)}|,

where V_{▪}^{(n)} is the smash product in prespectra. So the isomorphism reduces to the fact that for simplicial spaces A_{▪} and B_{▪}, the map |diag A_{▪} ∧ B_{▪}| → |A_{▪}| ∧ |B_{▪}| is an

give ref on bisimpli- isomorphism. Thus, cial sets and the di-

agonal thm. |T V_{▪}| = ∐ T_{n} ⋊_{Σ_{n}} L|V_{▪}|^{(n)} = T |V_{▪}|.

Now we will no longer decorate |−|, since it has an unambiguous meaning.

Part of one lecture to be filled in here.

7. Lazard’s theorem

We saw that to a complex oriented E one can associate a formal group law over the ring of coefficients π_{∗}E. Therefore it is important to study in general the formal group laws over an arbitrary ring R. The main result here is Lazard’s theorem.

For this section, good references are [1] and [8].

Theorem 7.1 (Lazard). If 𝕃(R) denotes the functor sending R to the set of formal group laws over R, then 𝕃 is corepresentable by a ring L called the Lazard ring.

Furthermore, L has a natural grading and is isomorphic as a graded ring to a polynomial ring ℤ[a_{1}, a_{2}, …] where the degree of a_{n} is n.

Before proving this, let us state a corollary which is important in the theory of formal groups, and which would be very difficult to prove without Lazard’s theorem.

Corollary 7.2. If R → S is a surjective map of rings, then any formal group law over S lifts to R. (I.e. the canonical map 𝕃(R) → 𝕃(S) is surjective; but that’s easy, since according to Lazard’s theorem L is free, hence ℛing(L, −) sends surjections to surjections.)

Proof. (of Lazard’s theorem) It is obvious that 𝕃 is corepresentable, namely by

L := ℤ[a_{ij} : i, j ≥ 1]/J,

where J is the ideal generated by a_{ij} − a_{ji} and some complicated terms expressing the associativity. (Really, it should be a quotient of ℤ[a_{ij} : i, j ≥ 0], but if we use the condition x +_{F} 0 = x = 0 +_{F} x, we obtain x +_{F} y = x + y + Σ_{i,j≥1}a_{ij}x^{i}y^{j}.)

Remark 7.3. L has a natural grading. To see this, consider the ring of Laurent series ℤ[λ, λ^{−1}], which is a Hopf algebra if we take as comultiplication the map λ ↦ λ ⊗ λ and as counit the map λ ↦ 1. Notice that ℤ[λ^{±1}] corepresents the multiplicative group functor G_{m} : ℛing → 𝒮et, R ↦ R^{∗} = {u ∈ R : u is a unit }.

If A is a Hopf algebra, a coaction of A on an abelian group M is a map ψ : M → A ⊗ M which is coassociative and counital.

Exercise 7.4. Prove that to give an abelian group M a coaction of the coalgebra ℤ[λ^{±1}] is to give a grading on M with

M_{n} = {m ∈ M : ψ(m) = λ^{n} ⊗ m}.

Now, using this exercise, to grade L as a ring we need to find a ring homomorphism

L ⟶ ℤ[λ^{±1}] ⊗ L.

Take this map to be the one classifying the formal group law

F^{λ}(x, y) := λ^{−1} · F (λx, λy).

(Note that this last equality defines a natural transformation 𝕃(R) × R^{×} → 𝕃(R).) The map is a coaction of ℤ[λ^{±1}] on L and hence L a graded ring. To see what the grading is, write

F (x, y) = x + y + Σ_{i,j≥1}a_{ij}x^{i}y^{j}.

Then

F^{λ}(x, y) = x + y + Σ_{i,j≥1}λ^{i+j−1}a_{ij}x^{i}y^{j},

so

a_{ij} ↦ λ^{i+j−1}a_{ij}.

Hence a_{ij} ∈ L_{i+j−1}(i, j ≥ 1) and 1 ∈ L_{0}. So L is a connected graded ring (meaning that L_{0} = ℤ and L_{i} = 0 for i < 0) and is of finite type, i.e. each L_{n} is finitely generated.

Definition 7.5. A formal group law F over a graded ring R is called homogeneous if, writing F (x, y) = x + y + Σ_{i,j≥1}r_{ij}x^{i}y^{j} and considering the degrees of x and y to be −1, the expression is homogeneous of total degree −1. That’s equivalent to saying that deg r_{ij} = i + j − 1 and hence to saying that the ring map classifying F is a map of graded rings.

The idea is to understand L by understanding its indecomposables. Denote by I the augmentation ideal of L, so I = ⊕_{n>0}L_{n}. Let Q be I/I^{2}, the indecomposables in L, and let Q_{n} denote that part of Q in degree n. Then L/I^{2} = ℤ⊕Q_{1} ⊕Q_{2} ⊕⋯ as a ring. Note that Q_{n} has square zero and ℤ ⊕ Q_{n} is a quotient ring of L.

Remark 7.6. The ring ℤ⊕Q_{n} corepresents 𝕃 for graded rings of the form R_{0}⊕R_{n}.

With this rather obvious remark, we’ve come to the main point of the proof. The fact is that we can classify all formal group laws over these kinds of rings, and therefore deduce the structure of ℤ ⊕ Q_{n}. More precisely, if F is a homogeneous formal group law over R_{0} ⊕ R_{n}, then F has a simple form:

F (x, y) = x + y + g(x, y),

with g a homogeneous polynomial of degree n + 1 in x and y satisfying 1. g(x, y) = g(y, x),

2. g(x, 0) = g(0, x) = 0, and

3. g(x + y, z) + g(x, y) = g(x, y + z) + g(y, z) (2-cocycle identity). A g satisfying these three identities is called a homogeneous symmetric 2cocycle. Now, there is result due to Frölich which gives an explicit formula for any such g:

Lemma 7.7 (Symmetric 2-cocycle lemma). Any homogeneous symmetric 2-cocycle of degree n over a ring R has the form

a · C_{n}(x, y),

for some a ∈ R, where

C_{n}(x, y) = \begin{cases}\frac{1}{p}[(x + y)^{n} − x^{n} − y^{n}] & if n = p^{k} for some k, \\ (x + y)^{n} − x^{n} − y^{n} & otherwise. \end{cases}

We are not going to prove this. See [1] or [8].

Using this lemma, the formal group law F we started with should have the form

F (x, y) = x + y + a · C_{n+1}(x, y),

for some a ∈ R_{n}. Therefore the formal group laws over rings of the form R_{0} ⊕ R_{n} are classified by a parameter a such that a^{2} = 0. Combining this with Remark 7.6, we get that

ℤ ⊕ Q_{n} ≅ ℤ[a_{n}]/a_{n}^{2}.

Now define a map f : ℤ[a_{1}, a_{2}, …] → L, by sending a_{n} to any element of L which projects to a generator of Q_{n}. The map f is surjective modulo I and modulo I^{2}, hence surjective, since L is connected graded of finite type. Another way of saying this is: all indecomposables are hit by f, because in each dimension they are generated by a_{n}. ℤ = L_{0} is also hit by f. But the indecomposables together with L_{0} generate the ring. Therefore f is surjective.

We have to show that f is a monomorphism. It will suffice to give a map out of L such that the composite is a monomorphism, and we know how to map out of L. Over ℤ[b_{1}, b_{2}, …] define

exp(x) := x + b_{1}x^{2} + ⋯ + b_{n}x^{n+1} + ⋯,
log(x) := exp^{−1}(x),

the inverse in the power series ring, and

G(x, y) := exp(log(x) + log(y)).

It’s easy to see that G is a formal group law and it’s homogeneous, provided we take deg b_{n} = n. Now look at G(x, y) modulo b_{i} for i < n. Then

exp(x) = x + b_{n}x^{n+1} + ⋯,
log(x) = x − b_{n}x^{n+1} + ⋯,

and

exp(log(x) + log(y)) = (x − b_{n}x^{n+1} + y − b_{n}y^{n+1}) + b_{n}(x + y)^{n+1} + ⋯
= x + y + b_{n}[(x + y)^{n+1} − x^{n+1} − y^{n+1})] + · · · .

So

G(x, y) = x + y + d_{n} · b_{n} · C_{n+1}(x, y) + higher powers of x and y,

where

d_{n} = \begin{cases}p, & if n = p^{k} \\ 1, & otherwise. \end{cases}

Therefore, if we consider the composite

ℤ[a_{1}, a_{2}, …] ⟶ L ⟶ ℤ[b_{1}, b_{2}, …],

where the second map is the one classifying G, we deduce that a_{n} is sent to d_{n} · b_{n} + lower terms.

We thus obtain that the composite

ℤ[a_{1}, a_{2}, …] ⟶ L ⟶ ℤ[b_{1}, b_{2}, …]

is a monomorphism. Therefore ℤ[a_{1}, a_{2}, …] → L is a monomorphism and hence an isomorphism.

Let us now continue the discussion about complex oriented cohomology theories E. We saw previously that to a complex orientation on E corresponds a formal group law over π_{∗}E. This is classified by a map of graded rings

L ⟶ π_{∗}E,

provided we slightly change the grading (deg a_{n} = 2(n − 1)).

On the other hand, we’ll see in the next section that a complex orientation on E translates geometrically into a map of spectra

MU ⟶ E.

If we take now E to be the complex cobordism theory MU, the identity map MU → MU gives a canonical orientation on MU, hence a map of graded rings

L ⟶ π_{∗}MU.

Quillen’s theorem now asserts that this map is an isomorphism.

But, before speaking about these things, let us see how a complex oriented E gives rise to a map of spectra MU → E. For this we need to talk about Thom classes in generalized cohomology.

8. Characteristic classes in generalized cohomology

To define MU, the Thom spectrum of the unitary group, start with the universe

𝒰 = ℂ^{∞} = ⊕_{n=1}^{∞}ℂ.

We give it the standard hermitian inner product and use the real part for its real inner product. Take the indexing set

𝒜 = {V ⊆ 𝒰 : V is a finite dimensional complex subspace}.

Definition 8.1. Let ζ_{V} = (EU(V) ×_{U(V)} V → BU(V)) be the universal complex vector bundle over BU(V), where BU(V) is the classifying space of the unitary group U(V). Then

MU(V) := Thom complex of ζ_{V},
MU := L(V ↦ MU(V)),

i.e. MU = the spectrification of the prespectrum V ↦ MU(V).

To see that V ↦ MU(V) is indeed a prespectrum, take V ⟶^{i} W and get

BU(V) ⟶^{i} BU(W). Then

i^{∗}ζ_{W} = ζ_{V} + (W − V),

so Th(i^{∗}ζ_{W}) = Σ^{W −V} MU(V). (If ξ : V → X is a vector bundle, then Th(ξ) stands for its Thom complex. This is also written X^{V}.) The natural map i : i^{∗}ζ_{W} → ζ_{W} will give us finally, passing to the Thom spaces, the structure map of the prespectrum:

Σ^{W −V} MU(V) ⟶ MU(W).

Notice that V ↦ MU(V) is an inclusion prespectrum, so L is explicitly defined. It is easy to show that MU is a ring spectrum. (It is even an E_{∞}-spectrum).

We now discuss Thom classes and the Thom isomorphism in generalized cohomology. A nice place to read this is in [1]. We begin by recalling some facts about Thom classes in ordinary cohomology, and for this one can consult [21].

Definition 8.2. A system of Thom classes in ordinary cohomology consists of a choice for each complex bundle V → X of rank n of a class u_{V} ∈ H^{2n}(X^{V}; ℝ) = H^{2n}(X^{V}) called the Thom class with the following properties:

1. Naturality: For Y ⟶^{f} X,

f^{∗}u_{V} = u_{f^{∗}V} ∈ H^{2n}(Y^{f^{∗}V});

2. Multiplicativity under the isomorphism (X × Y)^{V ×W} = X^{V} ∧ Y^{W},

u_{V ×W} = u_{V} · u_{W};

3. For V = (ℂ → ∗), u_{V} is the preferred generator of H̃^{2}(∗^{V}) = H̃^{2}(S^{2}).

Once you have such a thing, you have a functorial Thom isomorphism theorem as well, meaning that H̃^{∗}(X^{V}) is a free module over H^{∗}(X), generated by u. Also, because we’ve chosen a preferred generator of H̃^{2}(S^{2}), we have the following result:

Theorem 8.3. There is a unique system of Thom classes for ordinary cohomology.

Proof. The proof is easy if we use ordinary cohomology and the cell structure of the Thom complex, but we take a more complicated approach to allow generalization.

Lemma 8.4. If L is the canonical line bundle over ℂP^{∞}, then a choice of a preferred generator in H̃^{2}(S^{2}) forces the choice of the Thom class

u_{L} ∈ H̃^{2}((ℂP^{∞})^{L}).

To see this, notice that (ℂP^{∞})^{L} and ℂP^{∞} are homotopy equivalent. (This is because (ℂP^{∞})^{L} = Th(L) is the quotient of the disc bundle of L by the sphere bundle of L. But the disc bundle retracts down to ℂP^{∞} and the sphere bundle is contractible since it is an ES^{1}.) Now the assertion follows from the sequence of isomorphisms

H̃^{2}((ℂP^{∞})^{L}) ⟶^{∼} H̃^{2}(ℂP^{∞}) ⟶^{∼} H̃^{2}(ℂP^{1}) ⟶^{∼} H̃^{2}(S^{2})

and the fact that the composite is just the restriction map H̃^{2}((ℂP^{∞})^{L}) → H̃^{2}(∗^{L}).

Remark 8.5. This lemma is not true in generalized cohomology.

Now the theorem will follow after a series of standard calculations:

1. H̃^{∗}((ℂP^{∞})^{n}) = H^{∗}(∗)[[x_{1}, …, x_{n}]] where x_{i} = π_{i}^{∗}x. (Recall that the notation R[[x]] means the inverse limit in graded algebras of the truncated polynomial algebras. So H^{∗}(∗)[[x_{1}, …, x_{n}]] is just a polynomial ring. In generalized cohomology this might not be true.)

2. T^{n}, the n-dimensional torus, is a maximal torus in U(n), if regarded as the subgroup of diagonal matrices. The symmetric group Σ_{n} acts on T^{n} by permuting the diagonal elements. The inclusion map T^{n} → U(n) gives a map BT^{n} → BU(n), i.e. a map (ℂP^{∞})^{n} → BU(n). In cohomology we obtain a monomorphism

H^{∗}BU(n) ⟶ H^{∗}((ℂP^{∞})^{n})

and the image is invariant under the action of Σ_{n}. A calculation will give

H^{∗}BU(n) ⟶^{∼} (H^{∗}(ℂP^{∞})^{n})^{Σ_{n}},

where the right hand side represents the Σ_{n}-invariant elements in H^{∗}((ℂP^{∞})^{n}).

3. So

H^{∗}BU(n) = H^{∗}(∗)[[c_{1}, …, c_{n}]],

where (1+x_{1}t) ⋯ (1+x_{n}t) = 1+c_{1}t+⋯+c_{n}t^{n}. c_{1}, …, c_{n} are the fundamental symmetric polynomials in x_{i} and are called the universal Chern classes. They satisfy: A. The Cartan formula: c_{t}(V ⊕ W) = c_{t}(V) · c_{t}(W), where c_{t} := Σ_{k≥0}c_{k}t^{k}.

Expanding this gives c_{k}(V ⊕ W) = Σ_{i+j=k}c_{i}(V)c_{j}(W). B. c_{1}(L_{1} ⊗ L_{2}) = c_{1}(L_{1}) + c_{1}(L_{2}), for line bundles L_{1} and L_{2}. Remark. These are not unrelated facts.

4. There is the zero section BU(n) → MU(n), where MU(n) is the Thom complex of the universal n-dimensional complex vector bundle ζ_{n}). Calculation shows that

H^{∗}MU(n) ⟶ H^{∗}BU(n)

is a monomorphism and that the image is the ideal generated by c_{n}. Thus we might define u_{n} as the preimage of c_{n}. We think of u_{n} as the universal Thom class for n-bundles. The existence part of the theorem now follows.

With these calculations, the complicated proof of the uniqueness part of the theorem goes like this: Look at the following commutative diagram

ℂP^{∞} × ⋯ × ℂP^{∞}  (ℂP^{∞} × ⋯ × ℂP^{∞})^{L_{1}×···×L_{n}} BU(n)  MU(n).

In cohomology, H^{∗}MU(n) → H^{∗}(ℂP^{∞} × ⋯ × ℂP^{∞}) is a monomorphism, hence the map H^{∗}MU(n) → H^{∗}(ℂP^{∞} × ⋯ × ℂP^{∞})^{L_{1}×···×L_{n}} is a monomorphism. But using Lemma 8.4 and the multiplicativity property, we have that the universal Thom class u is forced to go to the exterior product of the canonical classes in H^{∗}(ℂP^{∞})^{L_{i}} (given by the Lemma), and the uniqueness follows.

Let’s do now the same thing for generalized cohomology. Let E be a ring spectrum.

Definition 8.6 (Compare to Definition 8.2). A system of Thom classes for E consists of a choice for each complex vector bundle (V → X) of rank n of an element u_{V} in E^{2n}(X^{V}) satisfying:

1. Naturality;

2. Multiplicativity;

3. For V = (ℂ → ∗), the class u_{V} is in Ẽ^{2}(∗^{V}) and is sent to the unit 1 ∈ E_{0}(∗) by the following composite isomorphism:

Ẽ^{2}(∗^{V}) ⟶^{∼} Ẽ^{2}(S^{2}) ⟶^{∼} Ẽ^{0}(S^{0}) ⟶^{∼} E_{0}(∗).

(By the preferred generator in Ẽ^{2}(S^{2}), we mean the element that is sent to 1 in E_{0}(∗) by the above sequence of isomorphisms.)

Some authors don’t impose the condition that u is in dimension 2n. However, it seems easier to work with this assumption.

Remark 8.7. Lemma 8.4 is not true anymore, because we don’t have the isomorphism Ẽ^{2}(ℂP^{∞}) → Ẽ^{2}(ℂP^{1}) = Ẽ^{2}(S^{2}). Instead we have a map and, once we fix an element x (a complex orientation) in the preimage of the preferred generator, we can prove a similar theorem about the existence and uniqueness of a system of Thom classes.

Theorem 8.8. Given x ∈ Ẽ^{2}(ℂP^{∞})^{L} such that x is sent to 1 under the composite map

Ẽ^{2}(ℂP^{∞})^{L} ⟶ Ẽ^{2}(∗^{L}) ⟶^{∼} E^{2}(S^{2}) ⟶^{∼} E_{0}(∗),

there is a unique system of Thom classes with u_{L} = x.

Notice that these kind of x’s correspond to complex orientations, via the homotopy equivalence (ℂP^{∞})^{L} ⟶^{∼} ℂP^{∞}.

Proof. Again we need some calculations that we’re not going to work out here. They can be found in [1].

1. For each n, the Atiyah-Hirzebruch Spectral Sequence (AHSS) for E^{∗}(ℂP^{∞})^{n} collapses and gives an isomorphism

E^{∗}((ℂP^{∞})^{n}) = E^{∗}(∗)[[x_{1}, …, x_{n}]] (x_{i} = π_{i}^{∗}x).

2. The map

E^{∗}BU(n) ⟶ (E^{∗}(ℂP^{∞})^{n})^{Σ_{n}}

is an isomorphism. (The AHSS for the left hand side collapses since it injects into the AHSS for the right hand side.)

3. If c_{i} are the Chern classes defined as before, then

E^{∗}BU(n) = E^{∗}(∗)[[c_{1}, …, c_{n}]].

4. The zero section gives rise to a monomorphism

E^{∗}MU(n) ↣ E^{∗}BU(n),

and the image is the ideal generated by c_{n}. This also follows from looking at the AHSS’s, plus knowledge that c_{n} is in the image of E^{∗}MU(n). This last fact comes from looking at the cofibre sequence

BU(n − 1) ⟶ BU(n) ⟶ MU(n).

In cohomology we get a short exact sequence

E^{∗}MU(n) ↣ E^{∗}BU(n) ↠ E^{∗}BU(n − 1)

and c_{n} ∈ E^{∗}BU(n) is sent to 0 ∈ E^{∗}BU(n − 1) (because one of the x_{i}’s is sent to zero).

Now the proof of the theorem goes as before.

We also get a Thom isomorphism:

Proposition 8.9. Given a Thom class u ∈ E^{2n}(X^{V}), Ẽ^{∗}(X^{V}) is free on u as an E^{∗}(X)-module.

Proof. (A little more subtle.) By the usual Thom isomorphism theorem, at the E_{2}-term of the AHSS, multiplication by u induces an isomorphism

E_{2}^{∗,∗}(X) ⟶^{∼} E_{2}^{∗,∗}(X^{V}),

where E_{2}^{∗,∗}(X) := H^{∗}(X, π_{∗}E). This multiplication is also a map of spectral sequences (i.e. d_{r}(ua) = u d_{r}(a)), since d_{r}(ab) = d_{r}(a) · b + a · d_{r}(b), and d_{r}u = 0 for each r because u survives to E_{∞}. Now use the comparison theorem for spectral sequences.

Remark 8.10. The Cartan formula holds:

c_{n}(V ⊕ W) = Σ_{i+j=n}c_{i}(V)c_{j}(W).

The proof follows from the above calculations with BU(n), etc.

Remark 8.11. Formula B. is not true anymore, i.e. if L_{1} and L_{2} are line bundles,

c_{1}(L_{1} ⊗ (L_{2}) ≠ c_{1}(L_{1}) + c_{1}(L_{2})

in general. Nevertheless,

c_{1}(L_{1} ⊗ L_{2}) = F (c_{1}(L_{1}), c_{1}(L_{2})),

with F a formal group law over E^{0}(∗).

Example 8.12. In K-theory, if L is the canonical bundle over ℂP^{∞}, the universal Thom class for line bundles is given by

u_{L} = 1 − L = c_{1}(L).

(This actually happens in K̃^{0}ℂP^{∞} = K̃^{0}(ℂP^{∞})^{L}, but we can go to K̃^{2}(ℂP^{∞})^{L}, by using Bott periodicity.) We get

c_{1}(L_{1} ⊗ L_{2}) = 1 − L_{1}L_{2}
= 1 − L_{1} + 1 − L_{2} − (1 − L_{1})(1 − L_{2})
= c_{1}(L_{1}) + c_{2}(L_{2}) − c_{1}(L_{1})c_{2}(L_{2}).

Thus

F (x, y) = x + y − xy,

which is the multiplicative formal group law.

9. Complex oriented cohomology and the MU-spectrum

We recall the definition of the spectrum MU. Take 𝒰 = ℂ^{∞}, 𝒜 = {ℂ^{n} : n ≥ 1} and

MU := L(ℂ^{n} ↦ MU(n)).

Instead of this definition using spectrification, one can write MU as a colimit of more easily described spectra. Remember that we have the structure map

Σ^{2}MU(n) ⟶ MU(n + 1),

so we get a map in the category of spectra

Σ^{∞}Σ^{2}MU(n) ⟶ Σ^{∞}MU(n + 1),

or

Σ^{∞−2n}MU(n) ⟶ Σ^{∞−2(n+1)}MU(n + 1).

One can check that

MU = colim Σ^{∞−2n}MU(n).

There’s nothing special about MU: if X is any spectrum, then

X = colim Σ^{∞−V} X_{V} .

Now we would like to investigate what it means in terms of the spectrum MU to give a system of Thom classes. We know from the results of the preceding section that it’s enough to look at the universal classes. Recall that ζ_{n} is the universal complex vector bundle over BU(n), and u_{n} := u_{ζ_{n}} ∈ E^{2n}MU(n) is the nth universal Thom class. u_{n} is represented by a map of spectra

Σ^{∞}MU(n) ⟶ Σ^{2n}E,

or, equivalently,

Σ^{∞−2n}MU(n) ⟶ E.

Using the commutative diagram

ζ_{n} ⊕ ℂ  ζ_{n+1} BU(n) _{i} BU(n + 1)

and the three defining properties of the Thom classes, we get

i^{∗}u_{n+1} = u_{(ζ_{n}⊕ℂ)} = u_{ζ_{n}} · u_{ℂ},

where u_{ℂ} corresponds to the preferred generator in Ẽ^{2}(S^{2}). But this translates into the commutativity of the diagram

Σ^{∞−2n}MU(n) ^{u_{n}}  E Σ^{∞−2(n+1)}MU(n + 1)_{u_{n+1}}  .

Thus the system of Thom classes gives an element of

colim_{n} Ho 𝒮(Σ^{∞−2n}MU(n), E),

which, using a Milnor sequence with lim^{1} = 0, equals

Ho 𝒮(MU, E).

(Here Ho 𝒮 stands for the homotopy category of spectra.) Also, the multiplicativity property of the system of Thom classes says that the following diagram commutes:

Σ^{∞−2n−2m}MU(n) ∧ MU(m)  E ∧ E Σ^{∞−2n−2m}MU(n + m)  E.

The upshot of the whole discussion is the following geometric result:

Proposition 9.1. To give a system of Thom classes is the same as to give a map of ring spectra

MU ⟶ E.

Corollary 9.2. The set of maps of ring spectra MU → E is in bijective correspondence with the collection of complex orientations x ∈ E^{2}ℂP^{∞}.

So we’ve come up with the following diagram:

{complex orientations of E}  {maps MU → E of ring spectra} {formal group laws over π_{∗}E}  {ring homomorphisms L → π_{∗}E}.

By taking E = MU and the identity map MU → E, one obtains a ring map L → π_{∗}MU.

Theorem 9.3 (Quillen). The map

L ⟶ π_{∗}MU

thus obtained is an isomorphism.

Proof. (Sketch) We have the following commutative diagram

L  π_{∗}MU ℤ[b_{1}, b_{2}, …] _{∼} H_{∗}MU .

The left vertical arrow is the one given by the proof of Lazard’s theorem and is a monomorphism; the right vertical arrow is the Hurewicz map given π_{∗}MU → π_{∗}(MU ∧ H) = H_{∗}MU; the lower horizontal map is an isomorphism coming from the computation of H_{∗}MU.

Then, using the Adams Spectral Sequence, one sees that the Hurewicz map is a monomorphism, and both vertical maps have the “same” image, hence the map L → π_{∗}MU is an isomorphism. See Adams [1] for details.

Quillen’s theorem allows us now to tackle an important problem. We saw that to any complex oriented cohomology theory, one can associate a formal group law over the ring of coefficients. Question: Is there a way to go back? The answer is yes, but it is not yet completely clear.

Let R be a graded ring and F a homogeneous formal group law over R. Then the map L → R classifying F will be a map of graded rings. On the other hand, we have the map L → MU_{∗}, so it makes sense to speak about MU_{∗}(S) ⊗_{L} R, for S a topological pointed space. Define R_{∗} : 𝒯op_{∗} → 𝒜b by R_{∗}(S) := MU_{∗}(S) ⊗_{L} R.

With this definition, notice that R_{∗}(∗) = R, by Quillen’s theorem. Under which conditions is R_{∗} a homology theory?

Theorem 9.4 (Landweber’s exact functor theorem). The functor R_{∗}(S) is a homology theory if the map L → R is regular (see below).

To define what it means for a map L → R to be regular, we introduce the elements v_{k} in R. Fix a prime p and denote by [p]x the formal sum x +_{F} ⋯ +_{F} x of x with itself p times. Then v_{k} is the coefficient of x^{p^{k}} in the power series [p]x. We say that the map L → R is regular if this sequence (v_{0}, v_{1}, v_{2}, …) is regular in R for each p. This means that multiplication by v_{k} in R/(v_{0}, …, v_{k−1}) is a

I don’t think I’ve monomorphism. phrased this very well. What’s a better way?

Finally, use Brown representability to get a spectrum out of R_{∗}, hence a generalized cohomology theory.

10. More about formal group laws

We would like to think of a formal group as an isomorphism class of formal group laws. Let R be a commutative ring. Let 𝒜dic_{R} be the category of augmented Ralgebras with nilpotent augmentation ideal. Suppose F (x, y) is a formal group law over R. Given a and b in 𝔄, the augmentation ideal, the sum a +_{F} b = F (a, b) = a + b + ⋯ is a finite sum and gives a new abelian group structure on 𝔄 natural in 𝔄. Let A^{n} denote the functor from 𝒜dic_{R} to 𝒮et, given by

A^{n}(R ⊕ 𝔄) = 𝔄^{n} = 𝔄 × 𝔄 × ⋯ × 𝔄.

Definition 10.1. A formal group over R is a functor 𝒜dic_{R} → 𝒜b which when regarded as a functor to 𝒮et is isomorphic to A^{1}. A formal group law is a group structure on A^{1}. In other words, it is an explicit choice of isomorphism between a formal group regarded as a functor to 𝒮et and A^{1}.

Proposition 10.2. The functor A^{n} is pro-represented by R[[x_{1}, …, x_{n}]]. Hence

Map[A^{n}, A^{m}] = {(f_{1}, …, f_{m}) | f_{i} ∈ R[[x_{1}, …, x_{n}]], f_{i}(0) = 0}.

Proof. We need to show

A^{n}(R ⊕ 𝔄) ≅ colim_{k} 𝒜dic_{R}(R[[x_{1}, …, x_{n}]]/(x_{1}, …, x_{n})^{k}, R ⊕ 𝔄).

But this is obvious. The rest follows from the Yoneda lemma.

Now let E be a complex orientable cohomology theory. We know that

E^{∗}(ℂP^{∞}) ≅ \begin{matrix}lim \\ ←------- \end{matrix}_{k} E^{∗}(ℂP^{k}) and each E^{∗}(ℂP^{k}) ∈ 𝒜dic_{E^{∗}}. Define a functor

F : 𝒜dic_{E^{∗}} ⟶ 𝒜b by

F (E^{∗} ⊕ 𝔄) = colim_{k} 𝒜dic_{E^{∗}}(E^{∗}(ℂP^{k}), E^{∗} ⊕ 𝔄).

The action of the multiplication map, μ, in cohomology gives F the structure of an abelian group. Since we know that once we choose an orientation x ∈ E^{2}(ℂP^{∞}), then E^{∗}(ℂP^{k}) ≅ E^{∗}[[x]]/(x^{k}), it follows that F is abstractly isomorphic to A^{1}. This is the formal group attached to the cohomology theory E.

Remark 10.3. Notice that fixing an isomorphism between F and A^{1} is equivalent by the Yoneda lemma to choosing an orientation for the cohomology theory. This gives us a formal group law in the usual sense, namely as a power series in two variables.

Remark 10.4. The definition of F is a bit misleading since E^{∗}[[x]] pro-represents the functor that picks out only the degree 2 part of the augmentation ideal. In any case, the above serves as a good motivation. Next, we fix this problem by making a few assumptions on the cohomology theory E.

In order to fix this problem let us consider complex orientable cohomology theories E for which π_{2}E contains a unit and such that π_{odd}E = 0. Dividing by the unit allows us to assume that the orientation has degree 0. Let us pick a unit u ∈ π_{2}E. Then

E^{∗}(ℂP^{∞}) ≅ E^{0}(ℂP^{∞}) ⊗ ℤ[u^{±1}]

and

E^{0}(ℂP^{∞}) ≅ π_{0}E[[x]].

We have now managed to fix the problem by working with 𝒜dic_{E^{0}} algebras instead.

Remark 10.5. The assumption that π_{odd}E = 0 forces our cohomology theory to be orientable since the AHSS has no room for differentials.

Another way to view this formalism is as follows. Let the functor pro-represented by π_{0}E be called A^{0}. The augmentation represents a unique map (the unit) A^{0} → F. Note that Map(F, A^{1}) ≅ E^{0}(ℂP^{∞}), so we can think of E^{0}(ℂP^{∞}) as the ring of functions on F. It follows that 𝔪 = Ẽ^{0}(ℂP^{∞}) can be identified with the ring of functions which vanish at the unit and that 𝔪/𝔪^{2} ≅ Ẽ^{0}(ℂP^{1}) can be thought of as the cotangent space to F at 1. Since Ẽ^{0}(ℂP^{1}) = π_{2}E, we can identify Ẽ^{0}(ℂP^{∞}) ≅ π_{2}E ⊗_{π_{0}E} E^{0}(ℂP^{∞}) with the module of 1-forms on F. I am a bit shaky on

this stuff; it would be nice if someone could elaborate on it a bit

Let R be a ring and let G : 𝒜lg_{R} → 𝒜b be a functor, where 𝒜lg_{R} denotes the category of R-algebras.

Definition 10.6. The formal completion of G, denoted by

Let’s use either G^{f} or G̃ and stick to it.

G^{f} = G̃ : 𝒜dic_{R} ⟶ 𝒜b,

is given by

G̃(R ⊕ I) = ker(G(R ⊕ I) ⟶ G(R)).

Example 10.7. If G_{m} is defined by G_{m}(T) = T^{×}, the group of units in T, then G_{m}^{f}(R ⊕ I) = (1 ⊕ I)^{×}. This is a formal group and is related to K-theory via the Landweber exact functor theorem.

Let us talk a bit about elliptic curves. We want a “nice” way of attaching a group to a curve. What follows is just a motivation on how we could go about doing this. Consider a curve C defined over a ring R. Suppose each line meets C Drawing goes here in 3 points. We can define a group structure by declaring a + b + c = 0 when a, b and c are colinear points. Let e ∈ C be a unit and let l be a tangent line to C at e. Since l is tangent to C at e, e has multiplicity at least two. Writing l ∩C = {e, e, a} we find that a + 2e = a = e. This means that l ∩ C = {e, e, e} and so there must be some point where the tangent line meets with multiplicity 3.

Choose coordinates so that l is the line at ∞ (i.e. points of the form [x, y, 0] ∈ P (R^{3})) and e is the point at which l intersects the line {x = 0} with multiplicity 3. Let C be given by the set of points (x, y) such that f(x, y) = 0.

Since every line meets f in 3 points, f must have the form of a polynomial with total degree at most 3 and so

f(x, y) = λ_{1}y^{2} + a_{1}xy + a_{3}y − (λ_{2}x^{3} − a_{2}x^{2} − a_{4}x − a_{6}) + dx^{2}y + exy^{2} + gy^{3}.

Each vertical line meets C at e; so it meets f(x, y) = 0 in at most 2 points. That is, f(0, y) has degree 2. This implies that g = 0 and λ_{1} is a unit. Similarly, examining C on the line at infinity yields that λ_{2} is a unit.

Using x and z as coordinates near ∞ (setting y = 1) we get

f(x, z) = (−λ_{2}x^{3} + dx^{2} + ex) + z(λ_{1} + a_{2}x^{2} + a_{1}x) + z^{2}(a_{3} + a_{4}x) + a_{6}z^{3}.

In order for the tangent at ∞ to be a triple tangent we need d = e = 0. So the most general curve is given by

λ_{1}y^{2} + a_{1}xy + a_{3}y = λ_{2}x^{3} + a_{2}x^{2} + a_{4}x + a_{6},

where we could substitute x′ = ηx, y′ = μy. So the case λ_{1} = λ_{2} = 1 is “general” and therefore C is given by the equation:

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}.

Definition 10.8. Suppose that F and G are formal group laws over a ring R. A homomorphism f : F → G is a power series f ∈ xR[[x]] satisfying:

f(x +_{F} y) = f(x) +_{G} f(y).

That is,

f(F (x, y)) = G(f(x), f(y)).

If f is an isomorphism then there exists a homomorphism g such that f ∘ g(x) = g ∘ f(x) = x and this is true if and only if f = λx + higher order terms (where λ is a unit). We say that f is a strict isomorphism if and only if λ = 1.

A homomorphism f is an isogeny if the map f : R[[y]] ⟶ R[[x]] given by y ↦ f(x) is finite. That is, R[[x]] is a finitely generated R[[y]] module.

Example 10.9. Let E be complex orientable and let x_{1} and x_{2} be two complex orientations that are sent to 1 under the composite map

Ẽ^{2}(ℂP^{∞}) ⟶ Ẽ^{2}(S^{2}) ⟶^{∼} Ẽ^{0}(S^{0}).

Let F_{1} and F_{2} be the corresponding formal group laws.

Proposition 10.10. F_{1} and F_{2} are canonically strictly isomorphic.

Proof. Since E^{∗}(ℂP^{∞}) ≅ E^{∗}[[x_{1}]], we can write x_{2} as a power series f(x_{1}). Also since x_{1} and x_{2} have the same image in E^{∗}(ℂP^{1}) ≅ E^{∗}[[x_{1}]]/(x_{1}^{2}), it follows that the leading term in f(x_{1}) is x_{1}. We would be done if we could prove that f(x_{1}+_{F_{1}} x_{2}) = f(x_{1})+_{F_{2}} f(x_{2}). To prove this let us look at the multiplication map in cohomology:

μ^{∗} : E^{∗}(ℂP^{∞}) ⟶ E^{∗}(ℂP^{∞} × ℂP^{∞}) ≅ E^{∗}(ℂP^{∞}) ⊗_{E_{∗}} E^{∗}(ℂP^{∞}).

The image of x_{1} under this map is x_{1} +_{F_{1}} y_{1} and since the map is a ring homomorphism, the image of f(x_{1}) is f(x_{1} +_{F_{1}} y_{1}). By assumption, f(x_{1}) = x_{2}, and by definition its image is x_{2} +_{F_{2}} y_{2} = f(x_{1}) +_{F_{2}} f(y_{1}) and so we are done.

Exercise 10.11. What happens if we weaken the definition of an orientation by requiring that it lie in E^{∗}(ℂP^{∞}) and that it’s image under the obvious map be some unit in E^{∗−2}(pt)?

Now assume that E is some complex oriented cohomology theory. Then the two maps

S^{0} ∧ ℂP^{∞} ⟶ S^{0} ∧ Σ^{2}MU ⟶ E ∧ Σ^{2}MU ≅ Σ^{2}E ∧ MU

and

ℂP^{∞} ∧ S^{0} ⟶ Σ^{2}E ∧ S^{0} ⟶ Σ^{2}E ∧ MU

provide us with two different orientations x_{MU} and x_{E} on E ∧ MU. We saw that the corresponding formal group laws are strictly isomorphic. Now if we have a ring homomorphism from π_{∗}(E ∧MU) = E_{∗}MU to any ring R, then by pushing forward the group laws and the isomorphism, we obtain two formal group laws F and F^{E} on R and a strict isomorphism F^{E} ⟶^{φ} F between them.

Proposition 10.12. The transformation

𝒜lg_{E_{∗}}(E_{∗}MU, R) ⟶ {(F^{E}, F, φ)}

is an isomorphism of functors.

Note 10.13. The data on the RHS is a map π_{∗}E → R giving F^{E}, a formal group law F over R, and a strict isomorphism φ : F^{E} → F.

Before we prove the Proposition, we need some preliminary results.

In the following, V will denote a vector bundle over X and X^{V} will denote the suspension spectrum of the Thom complex. This notation can be extended to the case of virtual bundles so that the usual properties still hold:

(X × Y)^{V ×W} = X^{V} ∧ Y^{W} .

In particular,

X^{V ×ℝ^{n}} = Σ^{n}X^{V} .

Definition 10.14. Let X be a finite CW complex. Let V and W be vector bundles over X. We can find a vector bundle U over X such that

U ⊕ W = ℝ^{n} × X.

We define X^{V −W} by

X^{V −W} = Σ^{−n}X^{V ⊕U}.

There is a unique way to do this when X is not a finite complex. One simply A unique way if we insist that the construction respect colimits? takes the colimit of this construction over the finite subcomplexes of X. Hence (ℂP^{∞})^{L−1} = Σ^{−2}ℂP^{∞}.

Definition 10.15. A complex orientation is a class x ∈ E^{0}(ℂP^{∞})^{L−1} such that x is sent to 1 under the map

E^{0}(ℂP^{∞})^{L−1} ⟶ E^{0}(pt^{L−1}).

Example 10.16. Let V be the universal 0-dimensional virtual bundle over BU. Then BU^{V} = MU. (Why?) The map ℂP^{∞} → BU classifying the canonical virtual bundle, L − 1, gives the orientation, ℂP^{∞}^{L−1} → MU, on “Thomification”.

Now let us make some calculations in E homology. We shall assume the following standard results. Let x ∈ H^{2}(ℂP^{∞}) be a generator, let b_{n} ∈ H_{2n}(ℂP^{∞}) be the element dual to x^{n}. It is well known that

H_{∗}(BU) ≅ Sym H_{∗}(ℂP^{∞})/(b_{0} − 1),

i.e. H_{∗}(BU) is a polynomial algebra on H_{∗}(ℂP^{∞}) with one relation. The map from ℂP^{∞} is the obvious inclusion as BU(1).

Remark 10.17. One proof of the above is as follows. It can be seen via Borel’s theorem or otherwise that H_{∗}(ΩSU(n)) is a polynomial algebra on H_{∗}(ℂP^{n}) modulo the relation b_{0} − 1, the map from ℂP^{n} → ΩSU(n) being the restriction of the Bott map η : BU → ΩSU. Since the Bott map is a homotopy equivalence of Hspaces, the result follows by letting n tend to infinity. One can give another proof using only spectral sequences.

So if E is any complex oriented spectrum (i.e. one that corresponds to a complex oriented cohomology theory), then we know that E^{∗}(ℂP^{∞}) ≅ E^{∗}[[x]]. As before, let b_{i} ∈ E_{2i}(ℂP^{∞}) be the dual elements (a good reference is the section on Universal Coefficient Theorem in Adams’ book [1]). Now by comparing the AHSS for E_{∗}(BU) and E_{∗}(ℂP^{∞}), it follows that

E_{∗}(BU) ≅ Sym_{E_{∗}} E_{∗}(ℂP^{∞})/(b_{0} − 1).

We use the Thom isomorphism to conclude that

E_{∗}(MU) ≅ Sym_{E_{∗}} E_{∗}(ℂP^{L−1})/(β_{0} − 1),

where the β′_{i}s correspond to the b′_{i}s under the Thom isomorphism. Thus

E_{∗}(MU) ≅ E_{∗}[β_{1}, β_{2}, …].

We are now ready to prove Proposition 8.3

Proof. We are trying to show that

𝒜lg_{E_{∗}}(E_{∗}(MU), R) ⟶ {(F, φ, F^{E} ⟶^{φ} F)}

is a bijection. We already know the map in one direction. To construct the inverse map let us first notice that any 𝒜lg_{E_{∗}} map from E_{∗}(MU) to R is uniquely defined by giving its values on the β_{i}, i ≥ 1. Now our data contains the strict isomorphism φ = x + ∑_{i≥1} d_{i}x^{i+1} ∈ R[[x]]. We define the inverse map by demanding that the image of this data take the values d_{i} on β_{i}. One can easily check that this is indeed an inverse. Again, a good reference is Adams’ book [1].

Example 10.18. Let E = H be the ordinary homology with ℤ coefficients. Then F^{H}(x, y) = x + y is the additive formal group law, G_{a}. So H_{∗}MU = π_{∗}(H ∧ MU) represents Iso(F, G_{a}).

11. Elliptic curves

In the previous section (see page 37) we saw that a large class of curves can be described by a Weierstrass equation

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}.

Clearly there is a universal one of these living over the ring ℤ[a_{1}, a_{2}, a_{3}, a_{4}, a_{6}] in the sense that any curve in Weierstrass form can be specified by a ring homomorphism ℤ[a_{1}, a_{2}, a_{3}, a_{4}, a_{6}] → R. Such a curve is said to be non-singular if a certain polynomial Δ in the a_{i} doesn’t vanish. We will specify Δ a few lines below. Hence, Does this conceptual

order really make sense? non-singular curves give rise to a functor

E : 𝒜dic_{A} ⟶ 𝒜b,

where A = ℤ[a_{1}, a_{2}, a_{3}, a_{4}, a_{6}][Δ^{−1}], by using the group structure on the curve as we described above. (Recall that 𝒜dic_{A} denotes the category of augmented A-algebras with nilpotent augmentation ideal.)

Proposition 11.1. The formal completion E^{f} of a curve E is a formal group.

Proof. We need to show that ker(E(A ⊕ 𝔄) → E(A)) is isomorphic to 𝔄 as a set. Recall that the point at infinity serves as unit in the group structure and has homogeneous coordinates ∞ = [0, 1, 0]. Thus every element [x, y, z] in the kernel has 0 augmentation in x, z and y ∈ (A⊕𝔄)^{×}. Hence, we may write the homogeneous Weierstrass equation

y^{2}z + a_{1}xyz + a_{3}yz^{2} = x^{3} + a_{2}x^{2}z + a_{4}xz^{2} + a_{6}z^{3}

in terms of the local parameters u = x/y, v = z/y and get

v + a_{1}uv + a_{3}v^{2} = u^{3} + a_{2}u^{2}v + a_{4}uv^{2} + a_{6}v^{3}. (1)

Now the identity element is (u, v) = (0, 0) and E^{f}(A ⊕ 𝔄) consists of exactly those pairs (u, v) for which (1) holds. We rewrite (1) as

v = −a_{1}uv − a_{3}v^{2} + u^{3} + a_{2}u^{2}v + a_{4}uv^{2} + a_{6}v^{3}

and observe that the right hand side is of higher order. If we substitute in this This argument needs about 100 more words. We’re using that 𝔄 is nilpotent. expression and iterate, v can be expressed as a formal power series of u. Since you may take any arbitrary element in 𝔄 for u, we have shown E^{f}(a ⊕ 𝔄) = 𝔄.

Example 11.2. Consider the (singular) curve y^{2} = x^{3}. In order to write down its formal group law we introduce the local parameter t = x/y near infinity. Then y^{2} = y^{3}t^{3} or (x, y) = (1/t^{2}, 1/t^{3}). Now let t_{1} +_{E} t_{2} = t. Then the slopes of the lines through t_{1}, t_{2} and t_{1}, −_{E}t are identical:

\frac{\frac{1}{t_{2}^{3}} − \frac{1}{t_{1}^{3}}}{\frac{1}{t_{2}^{2}} − \frac{1}{t_{1}^{2}}} = \frac{−\frac{1}{t^{3}} − \frac{1}{t_{1}^{3}}}{\frac{1}{t^{2}} − \frac{1}{t_{1}^{2}}} .

You can clear the denominators and obtain a cubic equation in t for which the two roots t = −t_{1} and t = −t_{2} are known.

Exercise 11.3. Solve for the third root.

The Weierstrass equation is not an invariant of a curve. Any change of variables of the form x′ = x + r, y′ = y + sx + t preserves the Weierstrass form with different coefficients, but we consider them coming from the same curve. Even the unit and the line at infinity are fixed under these transformations. When 2 is a unit in our ring R we can simplify the Weierstrass equation by completing the square in y giving

y^{2} = x^{3} + b_{2}x^{2} + b_{4}x + b_{6}.

If we assume further that 3 is a unit, the transformation x ↦ x − 1/3 b_{2} completes the cube:

y^{2} = x^{3} + c_{4}x + c_{6}.

Over the complex numbers we may even write

y^{2} = (x − λ_{1})(x − λ_{2})(x − λ_{3}).

Here our discriminant Δ takes the easy form

Δ = (λ_{1} − λ_{2})(λ_{1} − λ_{3})(λ_{2} − λ_{3}).

The curve E is what is called a Riemann surface. It comes to us with a twosheeted covering over ℙ^{1} which projects (x, y) to its x-coordinate. The covering is branched at x = λ_{i} for i = 1, 2, 3 and at ∞. We can use it to figure out the Euler characteristic of E. For that triangulate ℙ^{1} so that the four branches are among the vertices. Let F, E, V be the numbers of faces, edges and vertices respectively. Then the Riemann-Hurwicz formula tells us F −V +E = χ(ℙ^{1}) = 2. Now E has 2F faces, 2E edges and in the non-singular case (Δ ≠ 0) it has 2V − 4 vertices. Hence, the Euler characteristic is χ(E) = 2F − 2E + 2V − 4 = 0. Of course, this result can also be deduced from the fact that the group structure furnishes a non-vanishing vector field. E has genus 1 and thus is a torus. Now suppose λ_{1} = λ_{2} ≠ λ_{3}. Then χ(E) = 1 and E is a degenerate torus. Finally, in the case λ_{1} = λ_{2} = λ_{3} we get a topological sphere.

My notes say more. In the singular case let E_{sm} be the set of all smooth points. The “colinear” rule

defines an addition E_{sm} ×E → E. Since the identity element is a smooth point, the

in which sense is it natural transformation E_{sm}^{f} → E^{f} is an isomorphism, so indeed we get a formal an isomorphism? group.

Definition 11.4. An elliptic curve over a ring R is an equivalence class of nonsingular curves of the form

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}.

A generalized elliptic curve is an equivalence class of curves of the same form, allowing a singularity of the form λ_{1} = λ_{2} ≠ λ_{3}. (We describe equivalence below.) Furthermore, our assumption of an embedding in ℙ^{2} might only hold locally.

Definition 11.5. An elliptic spectrum is a multiplicative cohomology theory E such that

(i) π_{odd}E = 0, and (ii) π_{2}E contains a unit, together with a generalized elliptic curve A over π_{0}E and an isomorphism t : A^{f} → F from the formal completion A^{f} of A to the formal group F of E.

We want to be more precise and describe all transformation that yield equivalent curves. We allow

x ↦ λ^{2}x + r, y ↦ λ^{3}y + sλ^{2}x + t

for all r, s, t ∈ R and λ ∈ R^{×}. Then for λ = 1 the curve

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}. (2)

goes over to

\begin{matrix}y^{2} + (2s + a_{1})xy + (2t + ra_{1} + a_{3})y = x^{3} + (3r − s^{2} − sa_{1} + a_{2})x^{2} + (3r^{2} − 2st \\ −rsa_{1} − ta_{1} + 2ra_{2} − sa_{3} + a_{4})x + (r^{3} + a_{2}r^{2} + a_{4}r + a_{6} − t^{2} − rta_{1} − ta_{3}). \end{matrix}

Moreover, if (x, y) is a solution to (2) then (λ^{2}x, λ^{3}y) is a solution to

y^{2} + λa_{1}xy + λ^{3}a_{3}y = x^{3} + λ^{2}a_{2}x^{2} + λ^{4}a_{4}x + λ^{6}a_{6}.

In this way we obtain explicit formulas for the new coefficients ã_{i} under the action of any transformation.

Summarizing, we have described a functor ℛings → 𝒢roupoids sending a ring R to the groupoid G_{R} of elliptic curves over R. An object in G_{R} is a 5-tuple (a_{1}, ⋯, a_{6}) ∈ R^{5} and morphisms are of the form (a_{i}, r, s, t, λ) ∈ R^{5} × R^{3} × R^{×}, where the a_{i} specify the source of the morphism, the target being determined. The functor R ↦ ob G_{R} is represented by the ring A = ℤ[a_{1}, ⋯, a_{6}] and the functor R ↦ mor G_{R} by Γ = A[r, s, t, λ, λ^{−1}]. These data give what is called a Hopf algebroid and what will be the subject of the next section.

12. The 2-category of Hopf algebroids

Definition 12.1. A Hopf algebroid is the data representing a functor ℛing → 𝒢roupoid such that the map representing domain is flat.

In detail, we have rings A and Γ representing ob G_{R} and mor G_{R} respectively and ring homomorphisms η_{L}, η_{R} : A → Γ representing domain and range with η_{L} flat. These maps gives Γ the structure of a left and right A-module respectively. (In the case of elliptic curves η_{L}(a_{i}) = a_{i} and η_{R}(a_{i}) = ã_{i}.) In addition, there Ref for notation ã_{i}? is a ring homomorphism ψ : Γ → Γ ⊗_{A} Γ describing the composition law for “pairs of composable morphisms”. Associativity of composition translates into coassociativity of ψ. There is a map c : Γ → Γ corresponding to the inversion of morphisms. One easily checks the equalities cη_{R} = η_{L}, cη_{L} = η_{R} and c^{2} = id. In particular, η_{R} is flat as well. Finally, there is a map ϵ : Γ → A which associates to each object its identity morphism. One finds that ϵη_{L} = 1 = ϵη_{R} (so ϵ is a map of A-bimodules) and that ϵ is a counit for ψ.

Exercise 12.2. Look at the curve y = x^{2} + bx + c and define the related groupoid G_{R} to have objects (b, c) ∈ R^{2} and morphisms x ↦ x + r. Write down A, Γ, η_{L}, η_{R}, ψ, c and ϵ.

Example 12.3. Any Hopf algebra (commutative, not necessarily cocommutative, with conjugation) is a Hopf algebroid if its unit map is flat. If B is a Hopf algebra over k, then the k-algebra homomorphisms B → R form a group, a groupoid with one object.

Example 12.4. Let E be a ring spectrum such that π_{∗}E^{(k)} is commutative for all k. A while ago we considered the Adams resolution More precise ref.

S^{0}  E  E ∧ E  E ∧ E ∧ E  ⋯

which can be regarded as a cosimplicial object in the homotopy category. It gives We’re using ∧ quite freely. Should we justify this? rise to a spectral sequence with

E_{2}^{s,t} = H^{s}(π_{t}(E ∧ ⋯ ∧ E); Σ(−1)^{i}d^{i})

and converging to π_{t−s}L_{E}S^{0}. Here L_{E} denotes the Bousfield localization functor Is it converging to, with respect to E-homology. We want to give a homological interpretation to the or just abutting to? E_{2}-term, and we’ll need to assume that A = π_{∗}E → π_{∗}(E ∧ E) = Γ is flat. We easily see that

π_{∗}(\underbrace{E ∧ ⋯ ∧ E}_{k times}) = \underbrace{Γ ⊗_{A} ⋯ ⊗_{A} Γ}_{k − 1 times} .

In this situation (A, Γ) is a Hopf algebroid and E_{2}^{s,t} depends only on the Hopf algebroid structure. In fact, the functor

ℛing(π_{∗}(E^{(k)}), −) : ℛings ⟶ s𝒮et

represented by the cosimplicial spectrum above is the ‘nerve’ of the Hopf algebroid

We should refer- (A, Γ). ence Adams’ blue book and Miller’s “Relations between Adams spectral sequences … ”, JPAA, 1981(?).

We want to explain this concept in more detail. There are two obvious maps d_{0}, d_{1} : E ⇉ E ∧ E obtained from writing E as S^{0} ∧ E and E ∧ S^{0} respectively. In homotopy π_{∗}d_{0}, π_{∗}d_{1} : A ⇉ Γ represent range and domain, the multiplication Γ → A represents the map sending an object to its identity morphism, and the twist map π_{∗}τ : Γ → Γ represents the map sending a morphism to its inverse. The composition law is given by factoring the map

π_{∗}(E ∧ E) ⊗_{ℤ} π_{∗}(E ∧ E) ⟶ π_{∗}(E ∧ E ∧ E ∧ E) ⟶ π_{∗}(E ∧ E ∧ E),

through the tensor product over A. The second map in the displayed composite uses the multiplication of the two middle factors. The maps π_{∗}d_{0}, π_{∗}d_{1}, π_{∗}d_{2} : π_{∗}(E ∧ E) → π_{∗}(E ∧ E ∧ E) represent the maps sending (f, g) to g, gf and f respectively. It is clear how to continue to obtain a simplicial object.

We could simply check directly that (A, Γ) is a Hopf algebroid, but we would like to explain this in a more conceptual manner. The ‘nerve’ functor N : 𝒞ategories → s𝒮et is the inclusion of a full subcategory. Its image is the full subcategory of simplicial sets

X_{0}  X_{1}  X_{2}

for which X_{k} → X_{1} ×_{X_{0}} X_{1} ×_{X_{0}} ⋯ ×_{X_{0}} X_{1} is a bijection for each k ≥ 0. Here the map to the ith factor is induced by {0, 1} → {0 < 1 < … < n} : 0 ↦ i − 1, 1 ↦ i and the pullbacks are with respect to the face maps δ_{0} and δ_{1}. One can check that our functor from ℛing to s𝒮et takes values is such simplicial sets, and so factors through 𝒞at. In fact, it is clear then that it lands in 𝒢roupoid. In this sense, the Adams resolution is the nerve of the Hopf algebroid (A, Γ).

Definition 12.5. An (A, Γ)-comodule is a left A-module M, together with a map η : M → Γ ⊗_{A} M of left A-modules such that the two composites

M  Γ ⊗_{A} M  Γ ⊗_{A} Γ ⊗_{A} M γ ⊗ m  \begin{matrix}γ ⊗ η(m) \\ ψ(γ) ⊗ m \end{matrix}

coincide and the composite M ⟶^{η} Γ ⊗_{A} M ⟶^{ϵ⊗1} A ⊗_{A} M ≅ M is the identity.

For any spectrum X there is an Adams resolution

X  E ∧ X  E ∧ E ∧ X  E ∧ E ∧ E ∧ X  ⋯

and a spectral sequence H^{s}(π_{t}(E ∧ … ∧ E ∧ X)) trying to converge to π_{t−s}L_{E}X. Furthermore, M := E_{∗}X = π_{∗}(E ∧ X) is a comodule over (A, Γ) by taking η to be

E_{∗}X ≅ E_{∗}(S^{0} ∧ X) ⟶ E_{∗}(E ∧ X) ≅ E_{∗}E ⊗_{E_{∗}} E_{∗}X.

Exercise 12.6. Verify the algebraic identities in this geometric example.

The collection of (A, Γ)-comodules forms an abelian category with enough injectives. Here a map between comodules M_{1} and M_{2} is a map of left A-modules for which

M_{1}  ^{f}  M_{2} Γ ⊗_{A} M_{1} _{1⊗f} Γ ⊗_{A} M_{2}

commutes. The category has all colimits and they are preserved by the forgetful functor U to the category of A-modules. The E_{2}^{s,t}-term can be identified with Ext_{(A,Γ)}^{s}(A, Σ^{−t}M) where M = E_{∗}X, and Σ^{−t}M = E_{∗}Σ^{−t}X = E_{∗+t}X.

Definition 12.7. A 2-category is a category enriched over the category of small categories. If G is a 2-category and a and b are objects in G, the objects of G(a, b) are called 1-morphisms and the morphisms of G(a, b) are called 2-morphisms.

The category of Hopf algebroids forms a 2-category in the following way. Let (A_{i}, Γ_{i}) represent the functor R ↦ G_{i}(R) for i = 1, 2. Then a 1-morphism is the data representing a functor G_{1}(R) → G_{2}(R), natural in R. A 1-morphism consists of maps A_{1} → A_{2} and Γ_{1} → Γ_{2} satisfying some obvious conditions. A 2-morphism is the data representing a “natural” natural transformation. Two Hopf algebroids are naturally equivalent if the two groupoids are naturally equivalent. This last notion makes sense in any 2-category:

Definition 12.8. Two objects a, b in a 2-category are homotopy equivalent if

there are 1-morphisms a ^{f} b_{g} and 2-morphisms g f ⟹ 1_{a}, f g ⟹ 1_{b}.

Proposition 12.9. Let f : (A_{1}, Γ_{1}) → (A_{2}, Γ_{2}) be a 1-morphism and M be an (A_{1}, Γ_{1})-comodule. Then A_{2} ⊗_{A_{1}} M =: f^{∗}M is an (A_{2}, Γ_{2})-comodule. If f is part Why f^{∗}M and not of a natural equivalence of Hopf algebroids, then f_{∗}M?

Ext_{(A_{1},Γ_{1})}^{s}(A_{1}, M) ⟶^{≅} Ext_{(A_{2},Γ_{2})}^{s}(A_{2}, f^{∗}M).

Proof. The natural equivalence gives rise to a chain homotopy equivalence of standard resolutions. For details the reader is referred to Ravenel’s book. Supply a precise ref.

Example 12.10. We look at the curve y = x^{2}+b+c of Exercise 12.2 with ob G_{R} = R^{2} and morphisms of the form x ↦ x+r. The reader was to check that A = ℤ[b, c], Γ = A[r], η_{L}(b) = b, η_{L}(c) = c, η_{R}(b) = b + 2r and η_{R}(c) = c + br + r. Now we want to calculate the cohomology

H^{∗}(A, Γ) = Ext_{(A,Γ)}^{∗}(A, A).

The nerve of the category is huge, so let’s tackle a simpler example. Consider the Is there a better way to explain the jump to a new Hopf algebroid? related Hopf algebroid

R ⟼ {(b, c, α) ∈ R^{3}| α^{2} + bα + c = 0}.

That is, we look at a curve together with a root. Now the transformation x ↦ x+r takes α to α−r. This Hopf algebroid is represented by A_{1} = ℤ[b, c, α]/(α^{2} +bα+c) and Γ_{1} = A[r] and we call it G_{1}(R). But G_{1}(R) is naturally isomorphic to something much easier. We define the Hopf algebroid G_{2}(R) to be the full subcategory of G_{1}(R) with objects of the form x^{2} + bx and 0 as root. Note that there are only identity morphisms in G_{2}(R), so G_{2} is represented by A_{2} = Γ_{2} = ℤ[b]. Furthermore, any object in G_{1}(R) is isomorphic to an object in G_{2}(R) by the transformation x ↦ x + α. In particular G_{1} is naturally equivalent to G_{2} and we conclude that

H^{∗}(A_{1}, Γ_{1}) = H^{∗}(A_{2}, Γ_{2}) = \{ \begin{matrix}0, & ∗ > 0 \\ A_{2} = ℤ[b], & ∗ = 0. \end{matrix}

13. Sheaves and Grothendieck topologies

The origins of the subject. When modern complex analysis was born in the 19th century, the need arose to effectively manipulate objects that are locally simple (e.g. a complex analytic function is locally given by a power series) but when “patched together” can give rise to complicated behavior (e.g. monodromy). An elaboration of this point of view — and at once a nice motivation for sheaves — can be found in [25] and [7]. Another impetus came from Leray’s early work in algebraic topology.

In these examples, the underlying geometric object was a topological space, even a manifold. With the advent of the more abstract approach to algebraic geometry, the underlying object was replaced by a category; but it also had to support the idea of sheaves.

Consider of the category τ_{X} of open subsets of a topological space X, with inclusions serving as morphisms. One obviously has the notion of a “covering” of an open set by open sets. This notion is transitive; and intersecting any covering by an open set gives a covering of the intersection. Finally, notice that in the category τ_{X} the intersection of two subsets U and V of W is the pullback of U → W along V → W. This motivates the following definition, which has proved enormously fruitful in mathematics.

Definition 13.1. A Grothendieck topology T is given by the data:

1. A category 𝒞 admitting finite limits

2. A collection (not necessarily a set) of sets of morphisms of 𝒞 with the same target. (This is the collection of coverings. {U_{i} → U} will denote a covering, the index set being understood.)

subject to the following axioms:

(i) Isomorphisms: any singleton {ϕ : U_{1} → U_{2}} consisting of an isomorphism is a covering.

(ii) Transitivity: if {U_{i} → U} is a covering and for each i, {V_{ij} → U_{i}} is a

covering, then {V_{ij} ⟶^{composite} U} is a covering. (iii) Base change: if {U_{i} → U} is a covering and V → U a morphism, then

{V ×_{U} U_{i} → V } is also a covering.

A small category with a Grothendieck topology is often called a Grothendieck site. This is but one of several ways of modeling “geometric” notions in category theory; in fact one has an alternative description of Grothendieck topologies via sieves or localizing systems. Locales are lattices that mimic the properties of the lattice of open subsets of a topological space. Borceux’s handbook [4] devotes several chapters to these.

Example 13.2. Let X be a topological space. The category τ_{X} of its open subsets, with the usual notion of cover, is a Grothendieck topology.

Example 13.3. For a topological space X, let ℰt/X be the following category: the objects are maps f : U → X, where f is a covering space of some open subset of X (so in particular f is a local homeomorphism), and morphisms in ℰt/X are maps U → V over X. Let {m_{i} : U_{i} → V } be a covering if V = ⋃ Im m_{i}. This is a Grothendieck site, called the étale topology. Unlike the previous example, here open sets can have automorphisms.

Example 13.4. Let G be a finite group, and 𝒞 the category of finite G-sets and equivariant maps. Declare {f_{i} : U_{i} → U} to be a covering if U = ⋃ Im f_{i}. This is again a Grothendieck topology.

Sheaves. The canonical construction comes again from topological spaces:

Example 13.5. Let τ_{X} be as above. Given a map E ⟶^{p} X, consider the functor F from τ_{X} to 𝒮et defined by

F (U) = \{sections of \begin{matrix}U ×_{X} E \\ U \end{matrix} \} .

(Here U ×_{X} E is just E|_{U} = p^{−1}(U).)

This functor, “sections over U”, satisfies two properties:

(i) F is a contravariant functor τ_{X} → 𝒮et. (ii) If {U_{i} → U} is a covering, then the diagram

F (U)  ∏_{i} F (U_{i})  ∏_{i,j} F (U_{i} ×_{U} U_{j})

is exact, i.e. an equalizer. Here the two parallel arrows are induced by the projections from U_{i} ×_{U} U_{j} to U_{i} and U_{j}.

The second condition is called the patching condition and can be rephrased as follows: if there is a section over every element of a covering, and these sections agree on intersections, then they “glue together” to give a section; that is, they are restrictions of a single section.

By keeping just condition (i), but allowing arbitrary target categories (note that 𝒯op could have served the role of 𝒮et in the example above), we obtain

Definition 13.6. A presheaf with values in a category 𝒟 on a Grothendieck topology 𝒞 is a contravariant functor F : 𝒞 → 𝒟. A presheaf F is a sheaf if, in addition, condition (ii) is satisfied.

F (U) is then called the object of sections over U in the respective category 𝒟. Of course, 𝒟 has to have products for the condition to make sense. A map of presheaves is defined to be a natural transformation. Define the category of sheaves to be a full subcategory of presheaves.

Let us now fix 𝒟 to be 𝒜b, the category of abelian groups, and denote by 𝒫 and 𝒮, respectively, the categories of presheaves and sheaves on 𝒞.

Theorem 13.7. 𝒫 is an abelian category with enough injectives.

In fact, any functor category into an abelian category is also abelian; in 𝒫, a sequence F_{0} → F_{1} → F_{2} is exact iff for every U ∈ ob 𝒞, the sequence F_{0}U → F_{1}U → F_{2}U is exact.

We shall construct the injectives explicitly.

Lemma 13.8. Fix U ∈ ob 𝒞. Define the evaluation functor 𝒫 → 𝒜b by F ↦ F (U). This functor has both a left and a right adjoint.

Remark 13.9. We have here an instance of Kan extensions: given a category 𝒰, a functor f : 𝒜 → ℬ induces a functor f^{∗} : 𝒰^{ℬ} → 𝒰^{𝒜} on the functor categories by composition. If 𝒰 has enough limits and colimits, then f^{∗} has left and right adjoints. See [17] for more on Kan adjoints.

Proof. Here’s a construction of the left adjoint L_{U} for the lemma: given A ∈ 𝒜b, define

L_{U}(A)(V) = ⨁_{V →U} A

for objects V ∈ 𝒞, where the effect of L_{U} on a morphism V_{0} → V_{1} in

⨁_{V_{1}→U} A ⟶ ⨁_{V_{0}→U} A

is induced by taking the V_{1} → U summand to the V_{0} → V_{1} → U summand.

Let us check that this is indeed a left adjoint, i.e. 𝒫(L_{U}(A), G) = 𝒜b(A, G(U)). Given f : L_{U}(A) → G, so f_{V} : L_{U}(A)(V) → G(V), consider the diagram

A_{Id_{U}}  ⨁_{U→U} A  L_{U}(A)(U)_{L_{U} (A)i} ^{f_{U}}  G(U)_{G(i)} A_{i:V →U}  ⨁_{V →U} A  L_{U}(A)(V) _{f_{V}}  G(V) .

One sees that such f are in canonical bijection with the image of A_{Id_{U}} in 𝒜b. The right adjoint R_{U} is constructed dually.

We can now show that 𝒫 has enough injectives:

Proof. If I is an injective abelian group, R_{U}(I) is injective in 𝒫, since right adjoints preserve injectives. So, given a presheaf F, choose for each U ∈ ob 𝒞 an embedding F (U) ↪ I_{U} with I_{U} injective in 𝒜b. Define now

I = ∏_{U∈ob 𝒞} R_{U}(I_{U}) .

There is a map F → I, which is an embedding of F into an injective.

Theorem 13.10. The inclusion 𝒮 ↪ 𝒫 has a left adjoint, F ↦ ass F.

The first results on the existence of the associated sheaf functor go back all the way to Grothendieck’s Tôhoku paper [12]. Gray [10] has a general existence theorem, and the construction we are about to sketch appears in, e.g. [13]. See also Mac Lane–Moerdijk [22] for a wealth of material on sheaves.

The plus construction. For U ∈ 𝒞 define a category 𝒥_{U} as follows:

ob 𝒥_{U} = {coverings of U}

mor 𝒥_{U} = {maps of coverings} .

In more detail, a morphism between {U_{i} → U}_{i∈A} and {V_{j} → U}_{j∈B} is a function

A ⟶^{t} B, and maps U_{i} → V_{t(i)} over U.

If F is a presheaf on 𝒞 with values in 𝒟, we can define a contravariant functor P F : 𝒥_{U} → 𝒟 by

P F {U_{i} → U} = eq (∏ F (U_{i}) ⇉ F (U_{i} ×_{U} U_{j}))

acting appropriately on morphisms. This ought to give back F (U) but, since F is not a sheaf, the best we can do is define

F^{+}(U) = colim_{J_{U}} P F .

Miracle: For most 𝒟, F^{++} is a sheaf.

We are assuming, as always, that 𝒟 has enough (co)limits for the definitions to make sense. Think of the plus construction as giving a “better approximation” to a sheaf. The construction needs to be iterated into the transfinite, in general; however, for categories with good “regularity properties” — 𝒜b and 𝒮et, in particular

— two will suffice. This was conjectured by Gray, and proved by Grillet [11].

Corollary 13.11. 𝒮, the category of sheaves of abelian groups, has enough projectives.

Proof. There are enough projectives in 𝒜b; by a construction dual to the one in Theorem 13.7, there are enough projectives in 𝒜b-valued presheaves; and the associated sheaf functor, which is a left adjoint, preserves projectives and epimorphisms. Moreover, ass F ≅ F when F is already a sheaf.

Theorem 13.12. 𝒮 has enough injectives.

Theorem 13.13. 𝒮 is an abelian category.

These classical theorems are due to Grothendieck. To see that 𝒮 is an abelian category, one shows that the presheaf kernel of a map of sheaves is in fact a sheaf (and so is the kernel in the category of sheaves) and that the sheafification of the presheaf cokernel is the cokernel in the category of sheaves. Similar arguments show that monics are kernels and epis are cokernels. See e.g. Artin [2] for details. Mention briefly why

𝒮 has enough injectives.

Cohomology. Let 𝒞 be a Grothendieck topology; we required 𝒞 to have all finite limits, so in particular 𝒞 has a terminal object X. Let 𝒮 be as above, and F ∈ ob 𝒮, i.e. F is a sheaf of abelian groups on 𝒞.

Definition 13.14. Let H^{0} : 𝒮 → 𝒜b be the functor H^{0}(F) = F (X). This is called the global sections functor.

Proposition 13.15. H^{0} is left exact.

Proof. H^{0} is the composite 𝒮 ↪ 𝒫 ⟶^{H^{0}} 𝒜b. But the inclusion of sheaves into presheaves is a right adjoint, so it is left exact; and the global sections functor is exact in 𝒫. (Recall that exactness is detected “pointwise” in 𝒫.)

Definition 13.16. Let H^{n} be the n^{th} right derived functor of H^{0}.

Examples 13.17. Let X be a topological space and A an abelian group (with the discrete topology). Let Ã be the sheaf of sections of the projection X × A → X. It is called the constant sheaf, and Ã(U) = 𝒯op(U, A). Then H^{n}Ã = H^{n}(X, A). Now suppose that π_{1}X acts on A, and take the projection E = X̃ ×_{π_{1}X} A → X, where X̃ is the universal cover of X. If Ã is the corresponding sheaf of continuous sections, then H^{n}Ã = H^{n}(X, \underline{A}), cohomology with local coefficients.

Exercise 13.18. Let G be a finite group, and 𝒞 the category of finite G-sets as in Example 13.4. Show that there is an equivalence between the categories of sheaves Is 𝒟 any category? on 𝒞 with values in 𝒟 and G-objects in 𝒟.

Exercise 13.19. In the exercise above, take 𝒟 to be the category of abelian groups, M a G-module, and M̃ the corresponding sheaf. Show that H^{n}M̃ = H^{n}(G, M).

14. Groupoids

The references for this part of the material are scattered. Giraud [9] has the fullest discussion; some also appears in Section 4 of the paper by Deligne and Mumford [5]. Another article by Mumford [23], though it does not mention stacks explicitly, has the advantage of being very readable, and close in its philosophy to ours.

Representing objects and pullbacks. We are used to representing a functor F : 𝒞 → 𝒮et by an object A ∈ 𝒞, i.e. F (X) = 𝒞(A, X). Now suppose we wanted to represent isomorphism classes of principal G-bundles on a space X this way. We know these classes biject with homotopy classes of maps X → BG, and the corresponding G-bundle can be obtained by pullback from the universal one. The Yoneda lemma tells us right away that we cannot find a representing object in 𝒯op. If we pass to the homotopy category, we lose the usual pullbacks.

The solution to this dilemma is to consider the entire pullback diagram

E  E_{univ} X  B_{univ}

as data. Also, it turns out to be a good idea to represent objects together with the connecting isomorphisms, rather than just isomorphism classes.

The “classifying space” B_{univ} will no longer be a geometric object; still, many devices of algebraic topology apply to it, in the guise of the machinery we have amassed.

Definition 14.1. A groupoid is a category in which every arrow is an isomor-

We’ve used the word phism. groupoid before, so something is in the wrong order.

A group is then a groupoid with a single object. (There is a purely algebraic notion of groupoid — called a groupoid in the sense of Brandt — which bears the same relation to categorical groupoids as the usual groups in 𝒮et bear to groups in the above sense; a Brandt groupoid is a set together with a partial composition, satisfying suitable associativity and cancellation properties.)

Remark 14.2. The groupoid property can also be formulated in the following way: given a diagram of the form

• • •_{∃!}

there is a unique arrow filling it in.

Exercise 14.3. Recall that via the nerve construction, one can think of a category as a simplicial set for which the map X_{n} ⟶^{f} X_{1} ×_{X_{0}} ⋯ ×_{X_{0}} X_{1} is an isomorphism. The pullback is with respect to δ_{1} and δ_{0}, alternating, and the components of f are simplicial structure maps corresponding to {0 < 1} → {0 < ⋯ < n}, 0 ↦ k − 1, 1 ↦ k.

Show that a groupoid is a simplicial set for which X_{n} ⟶^{f} X_{1} ×_{X_{0}} ⋯ ×_{X_{0}} X_{1} is an isomorphism for each n. Here the components of f are {0 < 1} → {0 < ⋯ < n}, 0 ↦ 0, 1 ↦ k ≠ 0, and the pullbacks are with respect to δ_{0} only.

Let now 𝒞 be some category.

Definition 14.4. A groupoid over 𝒞 is a functor 𝒢 ⟶^{p} 𝒞 such that

(1) There is a fill-in arrow in the diagram:

E_{1} ^{∃g} E_{0} X_{1} _{f} X_{0} .

In words, given X_{1} ⟶^{f} X_{0} ∈ mor 𝒞 and E_{0} ∈ ob 𝒢 such that pE_{0} = X_{0}, there exists E_{1} ⟶^{g} E_{0} ∈ mor 𝒢 such that pE_{1} = X_{1} and p(g) = f.

(2) There is a unique fill-in arrow in the diagram:

E_{1}  ^{∃!β}_{g_{1}}  E_{2}_{g_{2}} E_{0} X_{1}  _{α}_{f_{1}}  X_{2}_{f_{2}} X_{0}  .

That is, given p(g_{1}) = f_{1} and p(g_{2}) = f_{2}, then for each α making the lower triangle commute, there exists exactly one E_{1} ⟶^{β} E_{2} ∈ mor 𝒢 with p(β) = α making the upper triangle commute.

Remark 14.5. Consider the following special case of (2): the diagram

E_{1}_{∃!}  E_{0} E_{2}  _{p} X_{1}  X_{0}

has a unique fill-in, i.e. given two lifts of the arrow X_{1} → X_{0} with identical targets, there exists exactly one E_{1} ⟶^{β} E_{2} ∈ mor 𝒢 making the upper triangle commute such that p(β) = Id_{X_{1}}. In the presence of (1), this special case already implies all of (2), so is convenient when checking that something is a groupoid over 𝒞.

Example 14.6. Let 𝒢 be the category of principal G-bundles over spaces: the objects of 𝒢 are principal G-bundles E → X and the morphisms are pullback

squares _{E_{1}}  _{E_{2}}_{X_{1}}  _{X_{2}} . Composition is by “pasting pullback diagrams”. We have a

groupoid over 𝒞 = 𝒯op by sending E → X to X, { _{E_{1}}  _{E_{2}}_{X_{1}}  _{X_{2}} } to X_{1} → X_{2}.

Definition 14.7. The fibre of a functor p : 𝒢 → 𝒞 at X ∈ ob 𝒞, denoted by 𝒢_{X}, is the category with ob 𝒢_{X} = {E| pE = X} and mor 𝒢_{X} = {g| p(g) = Id_{X}}.

Let 𝒢 ⟶^{p} 𝒞 be a groupoid over 𝒞. Putting Remark 14.2 and the X_{1} ⟶^{id_{X_{1}}} X_{1} case of 14.5 together, one sees that the fibre of p at any object X is a groupoid.

Note that in Example 14.6 we also had pullbacks available, i.e. there exist functors 𝒯op^{{•←•→•}} → 𝒯op computing the pullback. Suppose we wanted to imitate this property in an arbitrary groupoid 𝒢 over 𝒞. The best we can do is pick, using condition (1), for each X_{1} ⟶^{f} X_{0} ∈ mor 𝒞 and E_{0} ∈ ob 𝒢 such that pE_{0} = X_{0}, an appropriate lift E_{1} ⟶^{g} E_{0} ∈ mor 𝒢. In particular, associating E_{1} to E_{0}, we get a map f^{∗} : ob 𝒢_{X_{0}} → ob 𝒢_{X_{1}}. What happens to morphisms of 𝒢_{X_{0}}? The diagram

E_{1}  _{∃!m_{1}}  E_{0}  _{m_{0}} F_{1}  F_{0} X_{1}  _{f}  X_{0}

is an instance of 14.5; hence there is exactly one arrow E_{1} ⟶^{m_{1}} F_{1} in 𝒢_{X_{1}} making the upper parallelogram commute. Set f^{∗}(m_{0}) = m_{1}.

Proposition 14.8. f^{∗} is a functor 𝒢_{X_{0}} → 𝒢_{X_{1}}.

Proof. Observe that f^{∗}(m_{0} ∘ n_{0}) and f^{∗}(m_{0}) ∘ f^{∗}(n_{0}) are both fill-in arrows for a diagram as in 14.5.

So, for each X_{1} ⟶^{f} X_{0} ∈ mor 𝒞, up to certain choices, we have a “pullback” f^{∗} between the respective fibres. Let us, for the time being, call the maps obtainable

I think this dis- this way hand-manufactured pullbacks. cussion of pullbacks needs to be fixed up.

Exercise 14.9. Let f_{1}^{∗}, f_{2}^{∗} both be hand-manufactured pullbacks along f. Check that there is a unique fiberwise natural transformation f_{1}^{∗} → f_{2}^{∗}; it follows that it

What does “fiberwise” mean in this context? is an isomorphism. (Hint: Both existence and uniqueness follow from 14.5 applied at each object of 𝒢_{X_{0}}, which then assemble as in Proposition 14.8.)

Observe that if f^{∗} and g^{∗} are hand-manufactured pullbacks over f and g, two composable arrows of 𝒞, then f^{∗}g^{∗} is a hand-manufactured pullback along gf. It is not necessarily the same as (gf)^{∗}; however, by 14.9, the two are connected by a

unique natural isomorphism (gf)^{∗} ⟶^{θ_{f,g}} f^{∗}g^{∗}.

Proposition 14.10. Let f, g and h be composable arrows of 𝒞, together with handmanufactured pullbacks along them and their composites. The following diagram of natural transformations commutes:

(hgf)^{∗}  ^{θ_{gf,h}}_{θ_{f,hg}}  (gf)^{∗}h^{∗}_{θ_{f,g}∘h^{∗}} f^{∗}(hg)^{∗} _{f^{∗}∘θ_{g,h}} f^{∗}g^{∗}h^{∗} .

Proof. The two ways of going around the diagram exhibit two natural transformations between two hand-manufactured pullbacks over hgf. These natural transformations must coincide by 14.9.

Let us now demand — this has not been forced by the preceding — that Id_{X}^{∗} be the identity functor on 𝒢_{X}, for every X ∈ ob 𝒞. Recall that 𝒢roupoid is a 2-category, with natural transformations as 2-morphisms. While the hand-manufactured pullbacks do not quite assemble to give an honest functor 𝒞 → 𝒢roupoid, the natural transformations θ_{f,g}, subject to the coherence or cocycle condition 14.10 give exactly the data for another notion due to Grothendieck:

Definition 14.11. Let 𝒞 be a category and 𝒟 a 2-category. A pseudo-functor F : 𝒞 ⇝ 𝒟 is given by associating to every X ∈ ob 𝒞 an F X ∈ ob 𝒟, to every f : X → Y ∈ mor 𝒟 a 1-morphism F f : F X → F Y in 𝒟, and to every composable pair X ⟶^{f} Y ⟶^{g} Z ∈ 𝒞 a 2-isomorphism θ_{g,f} of 𝒟 from F (gf) to F (g)F (f), subject to

(i) F (Id_{X}) = Id_{F X} (ii) θ_{f,Id_{Y}} = θ_{Id_{X},f} = the identity 2-morphism on F f

(iii) For every triple X ⟶^{f} Y ⟶^{g} Z ⟶^{h} W ∈ mor 𝒞, the diagram of 2-morphisms

F (hgf)  ^{θ_{h,gf}}_{θ_{hg,f}}  F (h)F (gf)_{F (h)∘θ_{g,f}} F (hg)F (f) _{θ_{h,g}∘F (f)} F (h)F (g)F (h)

commutes.

Remark 14.12. The more usual notion of pseudo-functor does not demand (i), but allows the identities to get “twisted” by a 2-morphism; one then needs to replace (ii) by a coherence condition for units. The even broader notion of lax functor is obtained by not demanding that the requisite 2-morphisms be isomorphisms.

Proposition 14.13. The following two kinds of data are equivalent: What is the precise claim? A bijection of equivalence classes, or an equivalence of (some unspecified) categories?

\begin{cases}a groupoid 𝒢 ⟶^{p} 𝒞 \\ together with \\ hand-manufactured pullbacks \end{cases}\} ⟺ \begin{cases}a pseudo-functor \\ F : 𝒞^{op} ⇝ 𝒢roupoid \end{cases}\}

Proof. We have just seen that given a groupoid 𝒢 → 𝒞 over 𝒞 along with handmanufactured pullbacks, we can construct a pseudo-functor from 𝒞 to 𝒢roupoid. To go the other way, define 𝒢_{X} to be F X and f^{∗} to be F (f^{op}) (here f ∈ mor 𝒞). Now set ob 𝒢 to be {(X, E)| X ∈ ob 𝒞, E ∈ ob 𝒢_{X}} and let a morphism in 𝒢 from

(X_{1}, E_{1}) to (X_{2}, E_{2}) be any pair (f, m) with X_{1} ⟶^{f} X_{2} ∈ mor 𝒞 and f^{∗}E_{2} ⟶^{m}

E_{1} ∈ mor 𝒢_{X_{1}}. Given arrows

(X_{1}, E_{1}) ⟶^{(f,m)} (X_{2}, E_{2}) ⟶^{(g,n)} (X_{3}, E_{3})

in 𝒢, define the composition to be (gf, m ∘ f^{∗}n ∘ θ_{g,f}E_{2}); here θ_{g,f} is shorthand for θ_{f^{op},g^{op}}. By virtue of coherence condition (iii), composition is associative. (Id_{X}, Id_{E}) is a morphism in 𝒢 by virtue of (i), and it is the identity on (X, E) because of (ii). One checks that 𝒢 is a groupoid over 𝒞 via the obvious functor (X, E) ↦ X, (f, m) ↦ f. Finally, by going back and forth along this correspondence, we arrive at a groupoid (or pseudo-functor) naturally isomorphic to what we started out with.

Remark 14.14. Suppose we keep (i), (ii) and (iii) of Definition 14.11 but do allow the θ_{f,g} to be any 2-morphisms. The standard resolution F_{∗}𝒞 of 𝒞 is a simplicial category canonically associated to 𝒞; see [6] for the definition. One can also turn 𝒟 into a simplicial category by applying the nerve functor to each Hom category, and extending composition appropriately. Let us denote that category by s𝒟. The following is not hard to check: there exists a canonical bijection between pseudofunctors 𝒞 ⇝ 𝒟 and simplicial functors F_{∗}𝒞 → s𝒟. One can instead turn 𝒞 into a 2-category 𝒞_{2} canonically and show that the various flavors of lax functors correspond to classes of 2-functors 𝒞_{2} → 𝒟.

Definition 14.15. Let 𝒢roupoid/𝒞, the category of groupoids over 𝒞, be the

What is 𝒞at/𝒞? I full sub-2-category of the over category 𝒞at/𝒞 whose objects are groupoids over 𝒞. think it’s not what is defined below.

Proposition 14.16. 𝒢roupoid/𝒞, which is enriched over 𝒞at, has all indexed limits and colimits.

15. Representable groupoids and morphisms

I’m not sure if this Enriched pullbacks. is the best place for this subsection.

For later use, for example when discussing representable morphisms, we define here the enriched pullback of categories. Suppose we have a diagram C in 𝒞at which looks like

𝒞_{2}_{p_{2}} 𝒞_{1} _{p_{1}} 𝒟 .

Then we will let 𝒞_{1}×_{𝒟} 𝒞_{2} denote the indexed limit of this diagram over the diagram of categories K which looks like

{•} → {• → •} ← {•}

(this indexed limit is just the hom-object of maps 𝒞at(K, C), which is itself a category.) Thus we see that

ob 𝒞_{1}×_{𝒟} 𝒞_{2} = {(C_{1}, C_{2}, t)|C_{i} ∈ ob 𝒞_{i}, t: p_{1}C_{1} → p_{2}C_{2}},

and

𝒞_{1}×_{𝒟} 𝒞_{2}((C_{1}, C_{2}, t), (C′_{1}, C′_{2}, t′))

is the set of pairs of maps f_{i}: C_{i} → C′_{i} ∈ mor 𝒞_{i} for i = 1, 2 such that the diagram

p_{1}𝒞_{1} ^{p_{1}f_{1}}_{t}  p_{1}𝒞′_{1}_{t′} p_{2}𝒞_{2} _{p_{2}f_{2}} p_{2}𝒞′_{2}

commutes. This “enriched pullback” construction should not be confused with the ordinary pullback of categories, which is denoted by 𝒞_{1} ×_{𝒟} 𝒞_{2}.

Using this construction, we define a category 𝒞at/𝒟 with objects Bad notation! Conflicts with usual overcategory notation, which we also use.

ob 𝒞at/𝒟 := {functors 𝒞 → 𝒟},

and maps

𝒞at/𝒟(𝒞_{1} ⟶^{p_{1}} 𝒟, 𝒞_{2} ⟶^{p_{2}} 𝒟) := Sect(𝒞_{1}×_{𝒟} 𝒞_{2} → 𝒞_{1}),

where these are functors 𝒞_{1} → 𝒞_{1}×_{𝒟} 𝒞_{2} such that the composition with the projection is the identity functor on 𝒞_{1}. Thus a map (𝒞_{1} ⟶^{p_{1}} 𝒟) → (𝒞_{2} ⟶^{p_{2}} 𝒟) corresponds to a pair (f, a) consisting of a functor f: 𝒞_{1} → 𝒞_{2} and a natural transformation a: p_{1} → p_{2}f of functors 𝒞_{1} → 𝒟.

Representable groupoids.

Definition 15.1. Given an object Y ∈ ob 𝒞 we define a groupoid over 𝒞, also called Y, as follows. For each X ∈ ob 𝒞 let Y_{X} be the groupoid with

ob Y_{X} := 𝒞(X, Y),

and with only identity morphisms. For each map f: X′ → X ∈ mor 𝒞 define the pullback functor f^{∗}: Y_{X} → Y_{X′} to be the map 𝒞(X, Y) → 𝒞(X′, Y) induced by f.

Note that the pullback functors used in the above definition actually commute; i.e., g^{∗}f^{∗} = (gf)^{∗}. Furthermore, the groupoid Y can be described as the functor 𝒞/Y → 𝒞, and we use this notation as well.

Recall that the category 𝒢roupoid/𝒞 of groupoids over 𝒞, whose objects are groupoids 𝒢 → 𝒞 and whose 1-morphisms (𝒢_{1} → 𝒞) → (𝒢_{2} → 𝒞) are commutative diagrams of functors

𝒢_{1}  𝒞  𝒢_{2},

is itself enriched over categories, and thus forms a 2-category; thus we may speak of an equivalence of groupoids over 𝒞.

Definition 15.2. A groupoid over 𝒞 is representable if it is equivalent to 𝒞/Y → 𝒞 for some Y ∈ ob 𝒞.

The following lemma justifies the terminology “representable groupoid”.

Lemma 15.3. Suppose p: 𝒢 → 𝒞 is a groupoid over 𝒞, and let X ∈ ob 𝒞. Then Why the switch from there is an isomorphism of categories Y to X?

𝒢roupoid/𝒞(𝒞/X, 𝒢) = Sect(𝒢/X → 𝒞/X),

and the functor

Sect(𝒢/X → 𝒞/X) ⟶ 𝒢_{X}

is an equivalence of categories.

Remark 15.4. The functor Sect(𝒢/X → 𝒞/X) → 𝒢_{X} is evaluation at the identity object of 𝒞/X; that is, a section s: 𝒞/X → 𝒢/X is sent to the object s(X ⟶^{id} X).

Proof. The first statement about equality is immediate; it suffices to note that 𝒢/X = 𝒞/X ×_{𝒞} 𝒢. (See Lemma 15.8.)

To prove that Sect(𝒢/X → 𝒞/X) → 𝒢_{X} is an equivalence, we need to introduce a functor

𝒢_{X} ⟶ Sect(𝒢/X → 𝒞/X)

as follows: First, for each E ∈ ob 𝒢_{X} and each f: Y → X ∈ mor 𝒞 we make an (arbitrary) choice

\begin{matrix}E_{Y} ⟶^{g} E \\ p \\ Y ⟶^{f} X \end{matrix}

of E_{Y} ∈ 𝒢 and g: E_{Y} → E such that p(E_{Y}) = Y and p(g) = f, using Axiom 1 for a groupoid over 𝒞. It is convenient to choose the identity map E ⟶^{1} E to cover the identity map X ⟶^{1} X.

Now given a commutative triangle of maps in 𝒞 the second axiom for groupoids over 𝒞 provides a unique fill-in map:

E_{Y_{1}}  ^{∃!}_{g_{1}}  E_{Y_{2}}_{g_{2}} E Y_{1}_{f_{1}}  Y_{2}_{f_{2}} X

The uniqueness of the fill-in shows that we get a well-defined functor s_{E}: 𝒞/X → 𝒢/X which is a section of p.

Given a map E → E′ ∈ mor 𝒢_{X} we again use Axiom 2 to produce a unique natural transformation s_{E} → s_{E′} of sections which thus induce a well-defined functor

𝒢_{X} ⟶ Sect(𝒢/X → 𝒞/X).

Note that the composite

𝒢_{X} ⟶ Sect(𝒢/X → 𝒞/X) ⟶ 𝒢_{X}

is the identity by construction. The other composite

β: Sect(𝒢/X → 𝒞/X) ⟶ 𝒢_{X} ⟶ Sect(𝒢/X → 𝒞/X)

sends a section s to some other section s′ which has the property that s′(X) = s(X). By Axiom 2 there is a unique isomorphism s′ → s which is the identity at X. More generally, given any two sections s_{1} and s_{2} such that s_{1}(X) = E_{1} and s_{2}(X) = E_{2}, and a map g: E_{1} → E_{2} ∈ 𝒢_{X}, there is by Axiom 2 a unique natural transformation s → s′ of sections extending g. This proves that the composite of functors β is naturally isomorphic to the identity functor.

Remark 15.5. Note that the above proof makes essential use of the properties of groupoids; it is false for an arbitrary map of categories 𝒢 → 𝒞.

Remark 15.6. It is not hard to show that the functor

𝒞 ⟶ 𝒢roupoid/𝒞

given by

Y ⟼ (𝒞/Y → 𝒞)

is fully faithful. Thus we shall from now on regard 𝒞 as a subcategory (in fact, a sub-2-category) of 𝒢roupoid/𝒞. This explains our habit of writing Y for the groupoid 𝒞/Y → 𝒞 represented by Y.

We noted earlier that for each groupoid 𝒢 over 𝒞 we could choose (in a noncanonical manner) a pseudo-functor sending X ∈ ob 𝒞 to the groupoid 𝒢_{X}. The following lemma says that we could just as easily have used the assignment X ↦ Sect(𝒢/X → 𝒞/X) which is actually a functor.

Lemma 15.7. Given a groupoid 𝒢 over 𝒞, the assignment

𝒞 ⟶ 𝒢roupoid

given on objects by

X ⟼ Sect(𝒢/X → 𝒞/X)

is a contravariant functor, which is equivalent as a pseudo-functor to the assignment

X ⟼ 𝒢_{X},

equipped with some choice of pullbacks.

Proof. First let’s show that it is a functor. Given f: Y → X ∈ mor 𝒞 we obtain a functor

𝒞/Y ⟶ 𝒞/X

which on objects sends

(Z → Y) ⟼ (Z → Y ⟶^{f} X).

In like manner we obtain a functor 𝒢/Y → 𝒢/X. By Lemma 15.8 the induced commutative square

𝒢/Y  𝒢/X 𝒞/Y  𝒞/X

is a pullback square. Thus, a section of 𝒢/X → 𝒞/X pulls back to a section 𝒢/Y → 𝒞/Y in a canonical manner.

For the rest of the statement, we’ve already defined maps of groupoids 𝒢_{X} → Sect(𝒢/X → 𝒞/X) and Sect(𝒢/X → 𝒞/X) → 𝒢_{X} which at each X ∈ ob 𝒞 give an equivalence of groupoids. The only thing left to show is that these maps are “pseudo-functorial” in the X variable; this is an “easy” consequence of Axiom 2.

Lemma 15.8. Given a functor π : 𝒟 → 𝒞 and a morphism f: X → Y ∈ mor 𝒞 the square diagram of categories

𝒟/X  𝒟/Y 𝒞/X  𝒞/Y

is a pullback square in the category of categories. (An actual pullback, not an enriched pullback.)

Exercise 15.9. Describe the groupoid associated to X ↦ Sect(𝒢/X → 𝒞/X) as a category over 𝒞.

Representable morphisms of groupoids.

Motivation for this Definition 15.10. A morphism of groupoids 𝒢_{1} → 𝒢_{2} over 𝒞 is representable if definition? for each Y ∈ ob 𝒞 and each map Y → 𝒢_{2} of groupoids over 𝒞, the enriched pullback

Y ×_{𝒢_{2}} 𝒢_{1} is a representable groupoid over 𝒞.

Proposition 15.11. Let 𝒢 be a groupoid over 𝒞. Then the diagonal map Δ: 𝒢 → 𝒢 × 𝒢 is representable if and only if for each diagram X → 𝒢 ← Y of groupoids over 𝒞 with X, Y ∈ ob 𝒞 the enriched pullback X ×_{𝒢} Y is a representable groupoid. (Here 𝒢×𝒢 denotes the product in the category of groupoids over 𝒞, so it is actually 𝒢 ×_{𝒞} 𝒢. This convention is used elsewhere as well.)

This proof seems to Proof. Suppose the diagonal map Δ is representable. Thus, for any pair of maps use the “fact” that 𝒢 → 𝒢′ is an equivalence of groupoids if each 𝒢_{X} → 𝒢′_{X} is. Is this true? Yes. Should make argument more explicit. f: X → 𝒢 and g: Y → 𝒢 where X, Y ∈ ob 𝒞, the enriched pullback X × Y ×_{𝒢×𝒢} 𝒢 is representable. That is, there is an enriched pullback square of the form

𝒵  𝒢_{Δ} X × Y _{f×g} 𝒢 × 𝒢

where 𝒵 is representable. Note that the map f classifies some E_{X} ∈ ob 𝒢_{X} and the map g classifies some E_{Y} ∈ ob 𝒢_{Y}. Because this is an enriched pullback square, to give a map W → 𝒵 ∈ mor 𝒢 is to give: an object E_{W} ∈ ob 𝒢_{W}, maps i: W → X and j: W → Y in 𝒞, and isomorphisms i^{∗}E_{X} ⟶^{∼} E_{W} and j^{∗}E_{Y} ⟶^{∼} E_{W} in 𝒢_{W}. Such data are the objects of the groupoid 𝒢roupoid/𝒞(W, 𝒵), but this groupoid is equivalent to the groupoid which has as objects the data: maps i: W → X and j: W → Y in 𝒞, and an isomorphism i^{∗}E_{X} ⟶^{∼} j^{∗}E_{Y} in 𝒢_{W}. But these correspond to maps W → X ×_{𝒢} Y; thus we have an equivalence between X ×_{𝒢} Y and a representable groupoid 𝒵.

Now suppose that the enriched pullback X ×_{𝒢} Y is representable for every X, Y ∈ ob 𝒞. We want to show that for a map t: Z → 𝒢×𝒢 such that Z ∈ ob 𝒞, the enriched pullback 𝒲 is representable.

𝒲  𝒢_{Δ} Z _{t} 𝒢 × 𝒢

By reversing the argument in the first paragraph of the proof, we find that if Z = X × Y and t = f × g then 𝒲 is indeed representable. In general, we can factor the above square into a diagram

𝒲  𝒲̃  𝒢_{Δ} Z _{Δ} Z × Z _{π_{1}t×π_{2}t} _{𝒢 × 𝒢 ,}

where π_{1}, π_{2} : 𝒢×𝒢 → 𝒢 denote the two projections, such that the right-hand square is an enriched pullback, and thus 𝒲̃ is representable, and such that the left-hand square is an ordinary pullback, and thus 𝒲 is representable, as desired.

Example 15.12. Let 𝒞 = 𝒯op and let f: H → G be a map of topological groups. Let ℳ_{H} denote the category of principal H-bundles and let ℳ_{G} denote the category of principal G-bundles. Both ℳ_{H} and ℳ_{G} are groupoids over 𝒞. There is an induced morphism ℳ_{f}: ℳ_{H} → ℳ_{G} of groupoids over 𝒞, given on objects by

(E → X) ↦ (E ×_{H} G → X).

You might ask the question, is ℳ_{f} representable?

Proposition 15.13. A necessary (but not sufficient) condition that a groupoid 𝒢 over 𝒞 be representable, is that for each X ∈ ob 𝒞 and each E ∈ ob 𝒢_{X} we have that 𝒢_{X}(E, E) ≃ {id}.

Consider the enriched pullback diagram

𝒵  ℳ_{H} X _{ξ} _{ℳ_{G} ,}

where X is a space, and ξ classifies a principal G-bundle E → X. Then for W ∈ ob 𝒞 the groupoid

(𝒢roupoid/𝒞)(W, 𝒵)

has the following structure. The objects consist of pairs

( \begin{matrix}E′ \\ W \end{matrix} , E′ ×_{H} G E W X )

where E′ → W is a principal H-bundle and the square is a pullback square of G-bundles. A morphism

( \begin{matrix}E′_{1} \\ W \end{matrix} , E′_{1} ×_{H} G E W  X ) ⟶ ( \begin{matrix}E′_{2} \\ W \end{matrix} , E′_{2} ×_{H} G E W  X )

consists of a bundle isomorphism E′_{1} W E′_{2} making the obvious diagram of

G-bundles commute.

Now we can see that the necessary condition of Proposition 15.13 does not hold, unless H ⊆ G is a subgroup. In the case when H ⊆ G we see that 𝒵 is representable, by Shouldn’t the rep-

resenting object be simply a space? I think X is the representing object.

\begin{matrix}E ×_{G} G/H \\ X \end{matrix}

To sum up, we have “inflated” our original category 𝒞 to a bigger category 𝒢roupoid/𝒞 in which bundle-like constructions are representable.

16. Stacks, finally

Let 𝒞 be a category with a Grothendieck topology. Let 𝒢 be a groupoid over 𝒞 equipped with a choice of pullbacks which makes X ↦ 𝒢_{X} into a functor. We’ll refer to such a groupoid as a strict groupoid. (It’s possible to do everything without this assumption about pullbacks, but it will make things easier to describe if pullbacks are functorial. Recall that we showed that any groupoid over 𝒞 can be replaced by one with functorial pullbacks, so we don’t lose much.)

Notation 16.1. Given f: Y → X ∈ mor 𝒞, we will write the induced pullback functor 𝒢_{X} → 𝒢_{Y} as E ↦ E|_{Y}.

Let J = {U_{i} → U} be a covering in 𝒞.

Definition 16.2. A descent datum relative to J consists of

(1) an object E_{i} ∈ ob 𝒢_{U_{i}} for each element U_{i} → U of J,

(2) a morphism

ϕ_{ij}: E_{i}|_{U_{i}×_{U} U_{j}} ⟶ E_{j}|_{U_{i}×_{U} U_{j}} ∈ mor 𝒢_{U_{i}×_{U} U_{j}}

for every pair of elements of J, satisfying

(3) the cocycle condition

ϕ_{jk}|_{U_{i}×_{U} U_{j}×_{U} U_{k}} ∘ ϕ_{ij}|_{U_{i}×_{U} U_{j}×_{U} U_{k}} = ϕ_{ik}|_{U_{i}×_{U} U_{j}×_{U} U_{k}}.

Definition 16.3. A descent datum {E_{i}, ϕ_{ij}} for a cover J is effective if there exists an E ∈ ob 𝒢_{U} and isomorphisms

s_{i}: E|_{U_{i}} → E_{i} ∈ mor 𝒢_{U_{i}}

such that for each i, j the diagram

E|_{U_{i}×_{U} U_{j}}  ^{s_{i}|_{U_{i}×_{U} U_{j}}} E_{i}|_{U_{i}×_{U} U_{j}}_{ϕ_{ij}} E|_{U_{i}×_{U} U_{j}}  _{s_{j}|_{U_{i}×_{U} U_{j}}} E_{j}|_{U_{i}×_{U} U_{j}}

commutes.

In other words, descent data is effective if you can patch it together.

Definition 16.4. A strict groupoid 𝒢 over 𝒞 is a stack in 𝒞 if (i) every descent datum in 𝒢 is effective, and (ii) for every X ∈ ob 𝒞 and E, F ∈ ob 𝒢_{X} the functor

𝒞/X ⟶ 𝒮et

given by

(f: Y → X) ⟼ 𝒢_{Y} (f^{∗}E, f^{∗}F)

is a sheaf on 𝒞/X.

Remark 16.5. Recall that 𝒞/X has a natural Grothendieck topology in which {(U_{i} → X) → (U → X)} is a cover in 𝒞/X if and only if {U_{i} → U} is a cover in 𝒞.

Condition (ii) gives uniqueness up to isomorphism in (i).

There is a stackification functor which converts a groupoid over 𝒞 into the closest available stack in 𝒞. We won’t discuss stackification in general.

Example 16.6. Let 𝒞 be the category of spaces with the usual topology and let G be a topological group. The groupoid 𝒢 over 𝒞 given on objects by

X ∈ ob 𝒞 ↦ \begin{cases}principal G-bundles E → X \\ which admit a global section \end{cases}\}

and having as morphisms pullback diagrams of G-bundles is not a stack; descent data is certainly not effective here, since you can patch trivializable bundles together and obtain a non-trivial bundle.

On the other hand, the groupoid 𝒢′ over 𝒞 given on objects by

X ∈ ob 𝒞 ⟼ {principal G-bundles E → X}

and having as morphisms pullback diagrams of G-bundles is a stack. In fact, 𝒢′ is the stackification of 𝒢.

Let 𝒞 be a Grothendieck topology, and let ℳ be a stack in 𝒞 such that the I think everything from here to the end of the section was restated in a more precise manner in later lectures. diagonal map Δ: ℳ → ℳ × ℳ is representable.

Definition 16.7. Let 𝒞/ℳ denote the Grothendieck topology with objects

ob 𝒞/ℳ = {Y → ℳ | Y ∈ ob 𝒞},

with morphisms

𝒞/ℳ(X → ℳ, Y → ℳ) = Sect(X ×_{ℳ} Y → X),

and such that a collection of maps {(U_{i} → ℳ) → (U → ℳ)} is a cover if and only if {U_{i} → U} is a cover in 𝒞.

Example 16.8. Let 𝒞 be the category of spaces with the usual topology, and let ℳ be the stack of principal G-bundles. Then

ob 𝒞/ℳ = {principal bundles E → X},

mor 𝒞/ℳ = {pullback squares E F X Y }

and a collection of maps { E_{i} E U_{i} U } is a cover if and only if {U_{i} → U} is a cover

of spaces.

Definition 16.9. A collection of maps of stacks {U_{i} → ℳ} in 𝒞 with U_{i} ∈ 𝒞 is a cover of ℳ if for each Y ∈ ob 𝒞 and map Y → ℳ the collection of maps

{Y ×_{ℳ} U_{i} → Y }

is a cover in 𝒞.

More generally, a collection {ℳ_{i} → 𝒩} of representable morphisms of stacks is a cover if for each Y ∈ ob 𝒞 and map Y → 𝒩 the collection of maps

{Y ×_{𝒩} ℳ_{i} → Y }

is a cover in 𝒞.

Definition 16.10. The category of stacks 𝒮t in 𝒞, with

ob 𝒮t = {stacks ℳ in 𝒞},
𝒮t(ℳ, 𝒩) = Sect(ℳ×_{𝒞} 𝒩 → ℳ),

has a Grothendieck topology with covers as defined above.

Exercise 16.11. Check that a composite of representable morphisms is representable. Check that these covers satisfy the axioms of a Grothendieck topology.

17. The flat topology

Definition 17.1. Let 𝒜ff denote the opposite category of commutative rings. We sometimes write Spec R for the object in 𝒜ff corresponding to the ring R, but keep in mind that the notation Spec R is also used to designate the set of prime ideals in R.

Definition 17.2. A map A → B of rings is flat if (−) ⊗_{A} B: ℳod_{A} → ℳod_{B} is exact. A flat map A → B is faithfully flat if M ⊗_{A} B = 0 implies that M = 0.

It follows that if A → B is faithfully flat, then the induced functor ℳod_{A} → ℳod_{B} is faithful and exact.

Lemma 17.3. Suppose f: A → B is flat. Then the following are equivalent.

(1) f is faithfully flat.

(2) A sequence M′ → M → M′′ of A-modules is exact if and only if B ⊗_{A} M′ → B ⊗_{A} M → B ⊗_{A} M′′ is an exact sequence of B-modules.

(3) The map Spec B → Spec A of prime ideal spectra is surjective.

(4) For all maximal ideals 𝔪 ⊂ A we have that f(𝔪)B ≠ B.

Proof. See [20] or [18, Theorems 7.2 and 7.3].

Example 17.4. Suppose (A, Γ) is a Hopf algebroid. Then both η_{L}, η_{R}: A → Γ are faithfully flat; this follows from the fact that they are flat by definition and the existence of a retraction ϵ: Γ → A.

The flat topology on 𝒜ff .

Definition 17.5. The flat topology or fppf topology on 𝒜ff is a Grothendieck topology defined as follows: a collection of maps {Spec B_{i} → Spec A} in 𝒜ff is a cover if

(1) it is a finite set,

(2) each B_{i} is flat and finitely presented as an A-algebra (finitely presented means that B_{i} ≃ A[x_{1}, …, x_{n}]/(r_{1}, …, r_{m})), and

(3) the map A → ∏_{i} B_{i} is faithfully flat.

Requiring that each B_{i} be flat is equivalent to requiring that the (finite) product ∏ B_{i} be flat.

Remark 17.6. There is a variant of the fppf topology in which we drop the condition that the maps A → B_{i} be finitely presented; this will be useful when we look at the stack of formal group laws.

We need to prove that this satisfies the axioms of a Grothendieck topology.

(i) Isomorphisms: Clearly any isomorphism is faithfully flat and finitely presented.

(ii) Transitivity: Given covers {Spec C_{ij} → Spec B_{i}} and {Spec B_{i} → Spec A} we want to show that {Spec C_{ij} → Spec A} is a cover. This follows from the fact that compositions and finite products of flat (resp. faithfully flat) maps are flat (resp. faithfully flat), and that likewise composition of finitely presented maps is finitely presented.

(iii) Base Change: Suppose {Spec B_{i} → Spec A} is a cover and that Spec C → Spec A is a map. We need to check that {Spec C ×_{Spec A} Spec B_{i} → Spec C} is a cover; i.e., that

(a) each map C → C ⊗_{A} B_{i} is flat and finitely presented, and

(b) the map C → ∏_{i} C ⊗_{A} B_{i} = C ⊗_{A} ∏ B_{i} is faithfully flat.

That C → C ⊗_{A} B_{i} is finitely presented is clear. The rest follows from a lemma.

Lemma 17.7. The collection of flat (resp. faithfully flat) maps is closed under cobase change.

Proof. If A → B is flat, A → C is a map of rings and M′ → M → M′′ is an exact sequence of C-modules, then

M′ ⊗_{C} (C ⊗_{A} B) ⟶ M ⊗_{C} (C ⊗_{A} B) ⟶ M′′ ⊗_{C} (C ⊗_{A} B),

which is isomorphic to

M′ ⊗_{A} B ⟶ M ⊗_{A} B ⟶ M′′ ⊗_{A} B,

is an exact sequence. The proof for A → B faithfully flat is similar.

Descent theory. Let A → B be a map of commutative rings.

Definition 17.8. The category 𝒟esc(B/A) of descent data is the category whose objects are pairs

(M, t: B ⊗_{A} M ⟶^{∼} M ⊗_{A} B),

such that M is a B-module and t is an isomorphism of B ⊗_{A} B-modules which satisfies the cocycle condition, which says that the diagram

B ⊗_{A} B ⊗_{A} M _{1⊗t} B ⊗_{A} M ⊗_{A} B _{t⊗1} M ⊗_{A} B ⊗_{A} B

commutes. (The top arrow in the triangle is a bit hard to describe; you use t to switch the first and third factors in the tensor product.) The morphisms (M, t) → (M′, t′) in 𝒟esc(B/A) consist of maps f: M → M′ of B-modules such that the diagram

B ⊗_{A} M ^{t}_{B⊗f}  M ⊗_{A} B_{f⊗B} B ⊗_{A} M′ _{t′} M′ ⊗_{A} B

commutes.

There is functor ℳod_{A} → 𝒟esc(B/A) which sends N to (B ⊗_{A} N, t), where t: B ⊗_{A} B ⊗_{A} N → B ⊗_{A} N ⊗_{A} B switches the second and third factors.

Theorem 17.9 (Descent Theory). Suppose A → B is a faithfully flat morphism. Then the functor ℳod_{A} → 𝒟esc(B/A) is an equivalence of categories.

Proof. See [20] or [4, Vol. II, Thm. 4.7.8]. Borceux’s version uses a different definition of descent data and has different hypotheses.

Let ℳod denote the category with objects (R, M) where R is a commutative ring and M is an R-module, and such that maps

(R_{1}, M_{1}) ⟶ (R_{2}, M_{2})

consist of:

1. a map R_{2} → R_{1} of rings, and

2. an isomorphism R_{1} ⊗_{R_{2}} M_{2} → M_{1} of R_{1}-modules. There is an obvious functor ℳod → 𝒜ff , and ℳod is a groupoid over 𝒜ff .

Corollary 17.10. The groupoid ℳod over 𝒜ff is a stack.

Proof. This follows almost immediately from Theorem 17.9.

Given any category 𝒞, let 𝒞^{I} denote the category with ob 𝒞^{I} = {X → Y ∈ mor 𝒞}, and with maps (X → Y) → (X′ → Y′) ∈ mor 𝒞^{I} pullback squares. There is a functor 𝒞^{I} → 𝒞 sending (X → Y) to Y making 𝒞^{I} a groupoid over 𝒞.

Corollary 17.11. 𝒜ff^{I} is a stack in 𝒜ff .

Proof. Presumably this gets proved by noting that there is a functor 𝒜ff^{I} → ℳod, and that all the stuff you need to check has been proved in ℳod.

Rank 1 projective modules.

Exercise 17.12. Show that the category of “rank 1 free modules” (a sub-groupoid of ℳod) is not a stack.

Does “locally free” Exercise 17.13. Show that the category 𝒢 of “rank 1 projective modules” (here mean locally in the Zariski topology or in the flat topology?

“rank 1” means “locally free of rank 1”) is a stack.

In the case when 𝒢 is the category of rank 1 projective modules, we would like to know whether the diagonal Δ: 𝒢 → 𝒢×𝒢 is representable. Thus, given X = Spec A and Y = Spec B in 𝒜ff , we consider the enriched pullback

𝒵  Y X  𝒢

and ask if 𝒵 is representable. Note that the map X → 𝒢 classifies an A-module M, and the map Y → 𝒢 classifies a B-module N. Thus for a ring R, the objects of (𝒢roupoid/ 𝒜ff )(Spec R, 𝒵) consist of the data: maps A → R and B → R of rings, and an isomorphism R ⊗_{A} M ⟶^{∼} R ⊗_{B} N. The morphisms of (𝒢roupoid/ 𝒜ff )(Spec R, 𝒵) are all identity maps.

Lemma 17.14. Suppose S is a commutative ring and M and N are two rank 1 projective S-modules. Then the functor

{S-algebras} ⟶ 𝒮et

defined by

R ⟼ Iso(R ⊗_{S} M, R ⊗_{S} N)

is representable.

Proof. Let

M^{∗} := Hom_{S}(M, S),
N^{∗} := Hom_{S}(N, S).

It is not hard to see that Sym(M ⊗_{S} N^{∗}) represents R ↦ Hom_{R}(R ⊗_{S} M, R ⊗_{S} N) and that Sym(M^{∗} ⊗_{S} N) represents R ↦ Hom_{R}(R ⊗_{S} N, R ⊗_{S} M). (Here Sym(P) denotes the free symmetric S-algebra on an S-module P.) Thus Sym(M ⊗_{S} N^{∗} ⊕ M^{∗} ⊗_{S} N) represents Hom_{R}(R ⊗_{S} M, R ⊗_{S} N) × Hom_{R}(R ⊗_{S} N, R ⊗_{S} M).

Define a map c: Sym(M ⊗_{S} M^{∗}) → Sym((M ⊗_{S} N^{∗}) ⊕ (M^{∗} ⊗_{S} N)) as follows. The identity map

M ⊗_{S} M^{∗} ⊗_{S} N ⟶ M ⊗_{S} M^{∗} ⊗_{S} N

is adjoint to a map

i: M ⊗_{S} M^{∗} ⟶ M ⊗_{S} N^{∗} ⊗_{S} M^{∗} ⊗_{S} N.

Now there is a natural inclusion

M ⊗_{S} N^{∗} ⊗_{S} M^{∗} ⊗_{S} N ⟶ Sym((M ⊗_{S} N^{∗}) ⊕ (M^{∗} ⊗_{S} N)),

and composition of this map with i induces the desired map c. One sees that c represents composition:

Hom_{R}(R ⊗_{S} M, R ⊗_{S} N) × Hom_{R}(R ⊗_{S} N, R ⊗_{S} M) ⟶ Hom_{R}(R ⊗_{S} M, R ⊗_{S} M).

Now the push-out of c along the map e: Sym(M ⊗_{S} M^{∗}) → S, which represents the identity map of R ⊗_{S} M, is an S-algebra which represents R ↦ Iso_{R}(R ⊗_{S} M, R ⊗_{S} N). We call this representing object Iso(M, N); it represents the groupoid 𝒵.

Exercise 17.15. Think through the above argument for the category of “rank n locally free modules”.

Let 𝒢 as above denote the category of rank 1 projective modules; it is a stack in 𝒜ff . There is a map Spec ℤ → 𝒢 which classifies the free rank 1 ℤ-module.

Claim 17.16. The map Spec ℤ → 𝒢 is a covering of the stack 𝒢; that is, for each map Spec R → 𝒢 classifying some R-module M there is an enriched pullback diagram of the form

Spec S  Spec ℤ Spec R  𝒢

such that the map of rings R → S is faithfully flat and finitely presented (fppf).

Proof. It suffices to construct an fppf map R → B for which the induced map B → B ⊗_{R} S is fppf. First note that we can always choose an fppf map R → B such that B ⊗_{R} M is free of rank 1. Thus it suffices to prove the claim for maps Spec R → 𝒢 which factor through the map Spec ℤ → 𝒢. In this case the enriched pullback square factors as

Spec S  Spec Γ  Spec ℤ Spec R  Spec ℤ  𝒢

where the right-hand square is an enriched pullback and the left-hand square is a pullback. Thus it suffices to show that ℤ → Γ is fppf. We have shown that Spec Γ must represent the functor

R ↦ Iso(ℤ, ℤ)(R) ≃ Iso_{R}(R, R) ≃ R^{×}.

Thus Γ = ℤ[λ, λ^{−1}], and the map Γ → Z is clearly fppf.

Exercise 17.17. Carry out the analogue of this proof for rank n projective modules.

Note that in the above proof Γ = ℤ[λ, λ^{−1}] is a Hopf algebra over ℤ. We shall soon see that any Hopf algebroid (A, Γ) induces a stack along with a covering by a representable object.

Lemma 17.18. Suppose 𝒢 is a groupoid over 𝒞 for which the diagonal map Δ: 𝒢 → 𝒢 × 𝒢 is representable. Given X_{0} ∈ ob 𝒞 and a map X_{0} → 𝒢, let X_{1} = X_{0}×_{𝒢} X_{0}. Then

X_{0}  X_{1}  X_{1} ×_{X_{0}} X_{1}

is a groupoid object in 𝒞. (Note that X_{1} ×_{X_{0}} X_{1} ≃ X_{0}×_{𝒢} X_{0}×_{𝒢} X_{0}.)

Consider a stack 𝒢 in 𝒜ff such that the diagonal is representable. Given a map Spec A → 𝒢 let

Spec Γ = Spec A×_{𝒢} Spec A.

By Lemma 17.18 the pair of rings (A, Γ) represents a functor ℛing → 𝒢roupoid. If we furthermore suppose that Spec A → 𝒢 is a cover then the right unit map η_{R}: A → Γ will be faithfully flat. (It need not be finitely presented, so now is a good time to quietly drop the finite presentedness condition.)

In the next section, we will show that there is an equivalence of 2-categories:

\{Hopf algebroids (A, Γ)\} ≃ \begin{cases}stacks ℳ with \\ cover Spec A → ℳ \end{cases}\} .

18. Hopf algebroids and stacks

There is a strong connection between Hopf algebroids and stacks. More precisely, to any Hopf algebroid (A, Γ) we can associate a stack over 𝒜ff , denoted by ℳ(A, Γ). (Also, as we are going to see in the next section, one can describe the comodules over a Hopf algebroid in these terms.) Conversely, for a certain type of stack over 𝒜ff (we’ll call it a rigidified stack, for lack of a better name), there is a Hopf algebroid associated to it. The relationship between Hopf algebroids and stacks is part of a more general theory, but, except for Lemma 18.10, we are not going to talk about this.

We are working here with the category 𝒞 = 𝒜ff endowed with the faithfully flat Grothendieck topology. We don’t want to restrict ourselves to the finitely presented case, i.e. to the f.p.p.f. topology, because then we would exclude the stack of formal groups, where one doesn’t have finite presentation.

Definition 18.1. Consider the following 2-category, which we are going to call ℛigidified𝒮tacks.

a) Objects: covers (i.e. faithfully flat morphisms of stacks)

\begin{matrix}Spec A \\ ℳ \end{matrix}

with ℳ a stack over 𝒜ff for which the diagonal is representable;

b) 1-morphisms: commutative diagrams

Spec A_{1}  Spec A_{2} ℳ_{1}  ℳ_{2}

between objects;

c) 2-morphisms: natural transformations between 1-morphisms (denoted as usual by the sign “⟹”)

Spec A_{1}  Spec A_{2} ℳ_{1}  _{⇑}  ℳ_{2}

The reason for defining the 2-category of rigidified stacks in this way is made transparent by the following straightforward lemma, which gives a functor between ℛigidified𝒮tacks and the already familiar 2-category of ℋopfAlgebroids. (In the end, these two categories will be equivalent).

Lemma 18.2. Suppose Spec A → ℳ is a rigidified stack, and let

Spec Γ := Spec A ×_{ℳ} Spec A .

Then (A, Γ) is a Hopf algebroid.

Proof. According to the definition of a Hopf algebroid, we need to show that (A, Γ) represents a functor ℛings → 𝒢roupoids, and also that the domain map η_{L} : A → Γ is flat (which is equivalent to η_{R} being flat). To prove the first assertion, take for each ring R the groupoid given by

- Objects: Hom(A, R) = Hom(Spec R, Spec A) ;

- Morphisms: Hom(Γ, R) = Hom(Spec R, Spec A ×_{ℳ} Spec A) . Because we know the stack maps between a representable and a fibre product, we can rewrite the groupoid as

- Objects: maps

Spec R → Spec A ; - Morphisms: diagrams The arrow inside the

diagram will appear slanted only when printed

Spec R ⟹ Spec A Spec A  ℳ .

Now, this is an honest groupoid (because ℳ_{Spec R} is).

For the second assertion, notice that in the homotopy pullback diagram

Spec Γ^{Spec η_{L}}_{Spec η_{R}}  ⟹ Spec A Spec A  ℳ ,

since Spec A → ℳ is a cover, it follows from the definition of a cover that η_{R} is faithfully flat.

Theorem 18.3. The 2-functor

ℛigidified𝒮tacks ⟶^{Φ} ℋopfAlgebroids
(Spec A → ℳ) ⟼ (A, Γ)

given by the lemma is part of an equivalence of 2-categories.

Exercise 18.4. Check that this sends 2-morphisms to 2-morphisms.

Remark 18.5. The way the functor in the statement of the theorem is constructed doesn’t require ℳ to be a stack or to work over 𝒜ff . It is enough that ℳ be groupoid over 𝒞 with representable diagonal.

Proof. We construct a functor

ℋopfAlgebroids ⟶^{Ψ} ℛigidified𝒮tacks
(A, Γ) ⟼ (Spec A → ℳ(A, Γ))

so that:

1. ℳ(A, Γ) is a stack over 𝒜ff .

2. ℳ(A, Γ) has representable diagonal.

3. Spec A → ℳ(A, Γ) is a cover. (The first three properties ensure that Spec A → ℳ(A, Γ) is a rigidified stack, i.e. that Ψ is a well-defined functor.)

4. ΦΨ = id.

5. ΨΦ ≃ id (naturally equivalent to the identity). We start by constructing for a Hopf algebroid (A, Γ) a groupoid over 𝒜ff denoted by ℳ(A, Γ). Namely, the fibre over Spec R is

ℳ(A, Γ)_{Spec R} ,

which consists of:

- Objects:

(a) A faithfully flat map

R → E

(b) A map of Hopf algebroids

(A, Γ) → (E, E ⊗_{R} E) .

(From now on, to denote the objects, we shall use just the map from (b).)

- Morphisms: A 2-morphism of Hopf algebroids in the diagram

(A, Γ)  ⟹ (E_{1}, E_{1} ⊗_{R} E_{1}) (E_{2}, E_{2} ⊗_{R} E_{2})  (E, E ⊗_{R} E) ,

which goes from ((A, Γ) → (E_{1}, E_{1} ⊗_{R} E_{1})) to ((A, Γ) → (E_{2}, E_{2} ⊗_{R} E_{2})). Here E = E_{1} ⊗_{R} E_{2}, and it’s easy to see, by looking at the pullback diagram

R  E_{1} E_{2}  E_{1} ⊗_{R} E_{2} ,

that the map R → E is faithfully flat. Now what’s the pullback functor in ℳ(A, Γ)? Suppose we have a map

Spec R_{2} ⟶^{ϕ} Spec R_{1}

(i.e. R_{1} → R_{2}). Start with an object over Spec R_{1}, e.g.

(A, Γ) → (E, E ⊗_{R_{1}} E) Spec R_{2}  Spec R_{1} .

Call it x. Then the pullback ϕ^{∗}x will be the object Maybe you’d like to comment more on this strictness

(A, Γ) → (R_{2} ⊗_{R_{1}} E, R_{2} ⊗_{R_{1}} E ⊗_{R_{1}} E) .

That’s a strict pullback (as strict as the tensor product is associative).

Remark 18.6. Although the meaning of the Hopf algebroid (E, E ⊗_{R} E) will be clarified later, observe for now that a comodule structure over it is the same as a descent datum in the stack ℳod associated to the covering {Spec E → Spec R}.

To continue with the definition of Ψ, we need to define a map

Spec A → ℳ(A, Γ) .

But that’s equivalent to giving an object in ℳ(A, Γ)_{Spec A}. Choose this object to be the Hopf algebroid map

(A, Γ) ⟶^{(id,ϵ)} (A, A) .

Here, and in all that follows, the Hopf algebroid (R, R) is supposed to have all its structure maps the identity.

1. The proof that ℳ(A, Γ) is a stack is relegated to Lemma 18.10, because it involves some more general concepts.

2. To prove that ℳ(A, Γ) has representable diagonal, we know it’s enough to see that in the pullback diagram

𝒵  _{p⃗b} Spec S Spec R  ℳ(A, Γ)

(∗)

𝒵 = Spec R ×_{ℳ(A,Γ)} Spec S is representable. (Notice the new notation for homotopy pullback: “p⃗b” instead of “⟹”, which now stands for a 2-morphism.)

Is this notation con- Suppose the horizontal map represents the object x : (A, Γ) → (E, E ⊗_{R} E) in sistent with the previous notes?

ℳ(A, Γ)_{Spec R}. (We are given implicitly a map Spec E ⟶^{ϕ} Spec R, which is a cover.)

Then the composite Spec E ⟶^{ϕ} Spec R ⟶ ℳ(A, Γ) represents the pullback object ϕ^{∗}x : (A, Γ) → (E ⊗_{R} E, E ⊗_{R} E ⊗_{R} E) in ℳ(A, Γ)_{Spec E}. It is not difficult to see that this object is isomorphic to (A, Γ) → (E, E) (because ℳ(A, Γ)_{Spec E} is a groupoid, and there is a map between the two objects, hence an isomorphism). Suppose now that we are able to prove that in the pullback diagram

𝒵̃  _{p⃗b}  𝒵  _{p⃗b} Spec S Spec E _{ϕ} Spec R  ℳ(A, Γ) ,

𝒵̃ is representable. This means 𝒵 is locally representable, hence, according to Lemma 18.9, 𝒵 is representable. It is enough therefore to assume that in the initial diagram (∗), the lower horizontal map represents an object (A, Γ) → (R, R), and, using the same trick, that the vertical arrow represents an object (A, Γ) → (S, S). Thus, we are able to use Lemma 18.7, and get that 𝒵 is representable (by Spec(R ⊗_{A} Γ ⊗_{A} S)).

3. To show that Spec A → ℳ(A, Γ) is a cover, we use the same trick as in the proof of 2., and start with a diagram

𝒵̃ = Spec(E ⊗_{R} W)_{p⃗b} 𝒵 = Spec W  _{p⃗b}  Spec A_{(A,Γ)→(A,A)} Spec E  _{ϕ}  _{(A,Γ)→(E,E)} Spec R _{(A,Γ)→(E,E⊗_{R}E)} ℳ(A, Γ) .

(Here we have already used 2., saying that 𝒵 is representable by a Spec W.) Note that, according to Lemma 18.7, 𝒵̃ is also representable by Spec(E ⊗_{A} Γ ⊗_{A} A) = Spec(E ⊗_{A} Γ). But the map E → E ⊗_{A} Γ is faithfully flat (A ⟶^{η_{L}} Γ is), and it coincides with the map E → E ⊗_{R} W. From a well-known property of faithfully flat maps, we deduce that the map R → W is faithfully flat (because R → E is). This says that Spec W → Spec R is a cover, which is what we wanted.

4. Let us prove now that the composite

ℋopfAlgebroids ^{Ψ} ℛigidified𝒮tacks ^{Φ} ℋopfAlgebroids (A, Γ)  (Spec A → ℳ(A, Γ))  (A, W)

is the identity, where W comes from

Spec W  Spec A_{(A,Γ)→(A,A)} Spec A _{(A,Γ)→(A,A)} ℳ(A, Γ) .

Using Lemma 18.7 again, we know that W is isomorphic to A ⊗_{A} Γ ⊗_{A} A, i.e. to Γ, and the structure maps are the desired ones.

5. The other composite

ℛigidified𝒮tacks ^{Φ} ℋopfAlgebroids ^{Ψ} ℛigidified𝒮tacks (Spec A → ℳ)  (A, Γ)  (Spec A → ℳ(A, Γ)) ,

where Γ = A×_{ℳ} A, is not the identity, but naturally equivalent to it. For this, we need to show that ℳ(A, Γ) and ℳ are equivalent stacks and that the equivalence functors make the diagram

ℳSpec Aℳ(A, Γ)

commutative (enough to prove just for one equivalence functor).

We start by constructing a functor ℳ(A, Γ) → ℳ, or, taking an arbitrary Spec R ∈ 𝒜ff , a functor ℳ(A, Γ)_{Spec R} → ℳ_{Spec R}. A generic object in ℳ(A, Γ)_{Spec R} is a Hopf algebroid map (A, Γ) → (E, E ⊗_{R} E). Now, looking at the diagram

Spec E_{p⃗b}  Spec A_{p⃗b} Spec(E ⊗_{R} E)  Spec Γ Spec R  ℳ , Spec E  Spec A

we see that {maps (A, Γ) → (E, E ⊗_{R} E)} = {descent data associated to Spec E → Spec R in ℳ, factored through Spec E → Spec A → ℳ}. Because ℳ is a stack, any descent datum is effective, so we get an object in ℳ_{Spec R} (which is the same as a map Spec R → ℳ).

To obtain a functor back, given a map Spec R → ℳ, take in the above diagram Spec E = Spec R ×_{ℳ} Spec A, and get a map Spec(E ⊗_{R} E) = Spec E ×_{Spec R} Spec E = Spec R ×_{ℳ} Spec Γ → Spec Γ, and thus a map (A, Γ) → (E, E ⊗_{R} E).

That these functors are equivalences of categories comes essentially from the fact that a map between two objects (A, Γ) → (E_{i}, E_{i} ⊗_{R} E_{i}) (i = 1, 2) descends to a map between the corresponding descended objects in ℳ_{Spec R} (here one uses the other stack axiom for ℳ, namely that morphisms form a sheaf).

It only remains to show that the diagram

Spec A _{(A,Γ)⟶^{(id,ϵ)}(A,A)} ℳ  ℳ(A, Γ)

is commutative. But that’s easy, since, if we take the above cubical diagram for R = E = A and Spec R = Spec E ⟶^{id} Spec A, then there is a trivial descent to the given map Spec A → ℳ.

To really complete the proof of the theorem, there are some lemmas which we have already used, and we need to prove. But first, we want to say something more about the trivial Hopf algebroid (R, R). Remarks.

(a) As we already pointed out, the trivial Hopf algebroid (R, R) has each structure map the identity.

(b) Hopf algebroid maps (A, Γ) ⟶^{(f_{0},f_{1})} (R, R) are in bijective correspondence with maps A ⟶^{f_{0}} R (in which case f_{1} = f_{0}ϵ), where Γ ⟶^{ϵ} A is the counit map).

(c) Natural transformations

(A, Γ) ^{(f_{0},f_{1})}_{(g_{0},g_{1})}^{⇓} (R, R)

are in bijective correspondence with maps Γ ⟶^{α} R such that αη_{L} = f_{0} and αη_{R} = g_{0}.

We are not going to prove (b) or (c). The only skills required for this are the knowledge of the abstract definition of a natural transformation, plus some diagram chasing, with the observation that ϵ is a counit for Δ (Δ is the coproduct map on Γ), which helps in getting rid of the extra conditions which involve Δ.

Lemma 18.7. In the following pullback diagram in the category of groupoids over 𝒜ff

𝒵  _{p⃗b}  Spec S_{(A,Γ)⟶^{g_{0}}(S,S)} Spec R _{(A,Γ)⟶^{f_{0}}(R,R)}ℳ(A, Γ)

𝒵 is representable. Namely, it is represented by Spec(R ⊗_{A} Γ ⊗_{A} S). (The maps which appear in the tensor product are R _{f_{0}} A _{η_{L}} Γ_{η_{R}} A _{g_{0}} S .)

Proof. As usual, we analyze 𝒵 by looking at the maps from a representable groupoid Spec W into it. Thus we have

Spec W  _{Spec ψ}_{Spec ϕ}  𝒵  _{p⃗b} Spec S Spec R  ℳ(A, Γ) .

Then

Hom(Spec W, 𝒵) = Set of \{ \begin{matrix}Spec W ⟶^{Spec ϕ} Spec R \\ Spec W ⟶^{Spec ψ} Spec S \\ A natural transformation & (A, Γ) ^{ϕf_{0}}_{ψg_{0}}^{⇓} (W, W) \end{matrix}

= Set of \{ \begin{matrix}R ⟶^{ϕ} W \\ S ⟶^{ψ} W \\ Γ ⟶^{α} W such that αη_{L} = ϕf_{0}, αη_{R} = ψg_{0}. \end{matrix}

= Set of {  R  _{ϕ}_{≡} A _{f_{0}}_{g_{0}}_{η_{L}}_{η_{R}} Γ _{α} W S  _{ψ}_{≡}

= Hom(R ⊗_{A} Γ ⊗_{A} S, W)

= Hom(Spec W, Spec(R ⊗_{A} Γ ⊗_{A} S)) .

Therefore 𝒵 is representable by Spec(R ⊗_{A} Γ ⊗_{A} S).

The next two lemmas show the local character of equivalence and representability, when dealing with stacks over representable groupoids.

Lemma 18.8. Given a Grothendieck topology 𝒞, a cover {U_{i} → Z} in 𝒞, and a map f of stacks over 𝒞 as in the diagram

ℳ_{1} ^{f} ℳ_{2} U_{i}  Z ,

then the map f is an equivalence if and only if U_{i} ×_{Z} f (or U_{i} ×_{Z} f: for representables, “×” and “×” are the same) is an equivalence for each i.

Proof. Because of the Yoneda-like lemma in the case of 𝒢roupoids/𝒞, it is enough to show that, for each X → Z, the obvious functor

G_{1} = Hom_{Z}(X, ℳ_{1}) ⟶^{F} G_{2} = Hom_{Z}(X, ℳ_{2})

is an equivalence of categories naturally in X. Thus we’d like to prove that F is fully faithful and essentially surjective.

1. F is fully faithful: Given X ^{a}_{b} ℳ_{1} we need to show that

F : G_{1}(a, b) → G_{2}(F a, F b)

is an isomorphism. Regard now G_{1}(a, b) and G_{2}(F a, F b) as sheaves on 𝒞/X. For example, G_{1}(a, b) : 𝒞/X → 𝒮et is given by

(Y ⟶^{t} X) ↦ G_{1}(at, bt) = G_{1}(t^{∗}a, t^{∗}b) .

G_{1}(a, b) is a sheaf, because, if we consider a and b as representing two objects in (ℳ_{1})_{X}, an element in G_{1}(a, b) is a morphism between those two objects, and we can use the stack axiom for ℳ_{1} which says that morphisms form a sheaf. Similarly for G_{2}(F a, F b).

Let V_{i} = U_{i} ×_{Z} X. Then {V_{i} → X} is a cover. Suppose we prove that

F : G_{1}(t_{i}^{∗}a, t_{i}^{∗}b) → G_{2}(t_{i}^{∗}F a, t_{i}^{∗}F b)

is an isomorphism. Then we are done, because this means that F is an isomorphism locally, but, being a map of sheaves, F will be an isomorphism globally, i.e. for t = X ⟶^{id} X. Now, to show that F : G_{1}(t_{i}^{∗}a, t_{i}^{∗}b) → G_{2}(t_{i}^{∗}F a, t_{i}^{∗}F b) is an isomorphism,

we need to analyze G_{1}(t_{i}^{∗}a, t_{i}^{∗}b) = { V_{i} ^{at_{i}}^{⇓}_{bt_{i}} ℳ_{1} }. It would be enough to have an

isomorphism

{ V_{i} ^{at_{i}}^{⇓}_{bt_{i}} ℳ_{1} } ≅ { V_{i} ^{U_{i}×_{Z}a}^{⇓}_{U_{i}×_{Z}b} U_{i} ×_{Z} ℳ_{1} } ,

because then we could use the hypothesis, which says U_{i}×_{Z} f is an equivalence. But the objects in (U_{i} ×_{Z} ℳ_{1})_{V_{i}} can be described easily, and so can the maps between two such objects. Using the particular structure of the representable groupoids (the only maps there are the identities), we obtain the desired isomorphism.

2. F is essentially surjective (i.e. surjective up to isomorphism): Let b ∈ ob G_{2}, which means b ∈ (ℳ_{2})_{X}. Then t_{i}^{∗}b ∈ (ℳ_{2})_{V_{i}}. Because of the local equivalence given by the hypothesis, there are objects a_{i} ∈ (ℳ_{1})_{V_{i}} for all i, such that

F a_{i} _{ϕ_{i}}^{∼} t_{i}^{∗}b .

We claim that (a_{i})_{i} form a descent datum in ℳ_{1} associated to the cover {V_{i} → X}. First, (t_{i}^{∗}b)_{i} obviously form a descent datum, so (F a_{i})_{i} form a descent datum. But, using the previous result (F iso on morphisms), (a_{i})_{i} form a descent datum.

Thus (a_{i})_{i} descend to an object a ∈ (ℳ_{1})_{X}, with t_{i}^{∗}a = a_{i} (really, isomorphic). By a standard argument using the axiom that morphisms form a sheaf, the isomorphisms F a_{i} ⟶^{∼} t_{i}^{∗}b in (ℳ_{2})_{V_{i}} glue together to give an isomorphism F a ⟶^{∼} b.

Lemma 18.9. We are in the conditions of the previous lemma, with the additional assumption that 𝒞^{I} → 𝒞 is a stack (true for 𝒞 = 𝒯op, 𝒜ff ). If for the diagrams

ℳ U_{i}  Z

each U_{i} ×_{Z} ℳ is representable, then ℳ is.

Reminder. In the groupoid 𝒞^{I} over 𝒞, the fibre over a Z ∈ 𝒞 is Hom(−, Z) Is this reminder nec- (objects over Z), with pullback diagrams as maps. essary?

For 𝒞 = 𝒯op, 𝒞^{I} is a stack, because objects defined on covers glue the same way bundles glue.

For 𝒞 = 𝒜ff , 𝒞^{I} is a stack, because of the descent theory.

Proof. Suppose U_{i} ×_{Z} ℳ is represented by M_{i}, so we have maps M_{i} → U_{i}. This means M_{i} ∈ (𝒞^{I})_{U_{i}}, and it’s easy to see that (M_{i})_{i} form a descent datum associated to the cover {U_{i} → Z} in the stack 𝒞^{I}. So there is an object M → Z, and we have isomorphisms U_{i} ×_{Z} M ⟶^{∼} M_{i} compatible with the descent data. Also, the maps U_{i} ×_{Z} ℳ → ℳ, or M_{i} → ℳ, descend to a map

M  ^{f}  ℳ Z

(because {M_{i} → M} is a cover and ℳ is a stack). But the localized map

M_{i} ≅ U_{i} ×_{Z} M ⟶^{U_{i}×_{Z}f} U_{i} ×_{Z} ℳ

is an equivalence, so, by the previous lemma, f is an equivalence. Therefore ℳ is represented by M.

Lemma 18.10. For any Hopf algebroid (A, Γ), ℳ(A, Γ) is a stack.

Proof. Using some ideas from part 5. of the previous theorem, we could find an abstract proof for the lemma, if we introduce the notion of stackification and realize that ℳ(A, Γ) is the stackification of the groupoid whose fibre over Spec R is {(A, Γ) → (R, R) Hopf algebroid map}.

However, the following example hints at an alternative proof. Consider 𝒞 = 𝒯op, G a group object in 𝒞 regarded as a groupoid (ob = ∗, mor = G). Then we can define a groupoid ℳ(∗, G) over 𝒞, by taking its objects to be:

- a cover U → X;

- a groupoid map (U, U ×_{X} U) → (∗, G).

The maps are defined in the same way as those in ℳ(A, Γ). It is easy to show that ℳ(∗, G) is equivalent to the stack of principal G-bundles over 𝒯op, therefore ℳ(∗, G) itself is a stack.

Notice that (∗, G) is a groupoid object in 𝒯op, and also that to any Hopf algebroid (A, Γ) is associated a groupoid object (Spec A, Spec Γ) in 𝒜ff . So let us consider a Grothendieck topology 𝒞 such that 𝒞^{I} → 𝒞 is a stack. We also want 𝒞 to have coequalizers, and other good properties, which we might collect under the name faithfully flat Grothendieck topology. More precisely, we require first that every cover {U_{i} → X} be replaceable by a cover {U → X} with a single element U (usually, this is done by taking U = ∐_{i} U_{i}). Also, we want any cover U → X to be faithfully flat, i.e. the functor U ×_{X} (−) : 𝒞/X → 𝒞 has the property that, for any objects A, B and C in 𝒞/X,

U ×_{X} A  U ×_{X} B  U ×_{X} C

is an coequalizer if and only if

A  B  C

is a coequalizer. Let 𝒢 = (G_{0}, G_{1}) be a groupoid object in 𝒞 (this means we have

maps G_{0} _{u} G_{1}_{d}^{r} ^{i} and G_{1}×_{G_{0}} G_{1} ⟶^{c} G_{1} which satisfy the obvious properties),

such that the map d is faithfully flat. Let ℳ(G_{0}, G_{1}) be the groupoid over 𝒞 with the fibre ℳ(G_{0}, G_{1})_{X} given by objects:

- a cover U → X (this is usually omitted);

- a groupoid map (U, U ×_{X} U) → (G_{0}, G_{1}), and maps (from (U_{1}, U_{1} ×_{X} U_{1}) → (G_{0}, G_{1}) to (U_{2}, U_{2} ×_{X} U_{2}) → (G_{0}, G_{1})):

(Ũ, Ũ ×_{X} Ũ) ⟹ (U_{1}, U_{1} ×_{X} U_{1}) (U_{2}, U_{2} ×_{X} U_{2})  (G_{0}, G_{1}) ,

where Ũ = U_{1} ×_{X} U_{2}. The pullback is defined in the same way as that for ℳ(A, Γ).

Definition 18.11. A (G_{0}, G_{1})-space, or 𝒢-space, is an object over G_{0}, Q → G_{0}, together with a map Q ×_{G_{0}} G_{1} → Q (in the fibre product we use G_{1} ⟶^{d} G_{0}) called the 𝒢-action, such that the two composites

Q ×_{G_{0}} G_{1} ×_{G_{0}} G_{1}  Q ×_{G_{0}} G_{1}  Q

are equal.

As an example of a 𝒢-space, take X ×_{G_{0}} G_{1} (where X is any object over G_{0}) with the obvious 𝒢-action.

Definition 18.12. A principal (G_{0}, G_{1})-bundle, or simply 𝒢-bundle, over X ∈ ob 𝒞, is a 𝒢-space P together with a faithfully flat map P → X, such that there exist a cover U → X, a map U → G_{0}, and an isomorphism

U ×_{X} P ^{∼} U ×_{G_{0}} G_{1} U

of 𝒢-spaces.

As a first example of a 𝒢-bundle over X, take X ×_{G_{0}} G_{1} → X (implicitly we are given here X → G_{0}). We may call this a trivial 𝒢-bundle. Note its resemblance to the notion of relatively free (A, Γ)-comodule, which is an M ⊗_{A} Γ, with M any A-module.

Definition 18.13. We define 𝒫(G_{0}, G_{1}), the category over 𝒞 of 𝒢-bundles, by saying that its fibre over X ∈ ob 𝒞 has objects 𝒢-bundles over X and morphisms 𝒢-space maps over X, and the pullback is defined as in the case of the stack 𝒞^{I}.

Before we prove anything else, we want to show that 𝒫(G_{0}, G_{1}) is a groupoid over 𝒞. First, note that for any cover U → X (which we know is faithfully flat), the sequence

U ×_{X} U  U  X

is a coequalizer. To see this, apply the functor U ×_{X} (−) and observe that

U ×_{X} U ×_{X} U  U ×_{X} U ^{Δ} U

is a coequalizer, because of the splitting map Δ. Now take two 𝒢-bundles P and P′ over X, and a map P ⟶^{f} P′ between them. Suppose both P and P′ are trivial over U. Then we have the diagram

U ×_{X} U ×_{X} P  U ×_{X} P_{U×_{X}f}  P_{f} U ×_{X} U ×_{X} P′  U ×_{X} P′  P′ .

Because P and P′ cover X, both rows are coequalizers, so we need to show that U ×_{X} f is an isomorphism. But it’s easy to show that any map between trivial 𝒢-bundles is an isomorphism, because 𝒢-maps

U ×_{G_{0}} G_{1} ^{Φ} U ×_{G_{0}} G_{1} U

are completely determined by maps

U ^{ϕ} G_{1}_{r}_{d} G_{0}

and we can use the inverse structure map G_{1} ⟶^{i} G_{1}. This proves that 𝒫(G_{0}, G_{1}) is a groupoid over 𝒞. Now we want to show that

1. ℳ(G_{0}, G_{1}) and 𝒫(G_{0}, G_{1}) are equivalent, and

2. 𝒫(G_{0}, G_{1}) is a stack. These will imply that ℳ(G_{0}, G_{1}) is a stack, which concludes the lemma.

1. We show that the skeleta of ℳ(G_{0}, G_{1}) and 𝒫(G_{0}, G_{1}) are isomorphic. Identify them with ℳ̃(G_{0}, G_{1}), 𝒫̃(G_{0}, G_{1}), which are the categories of isomorphism classes of objects in the respective initial categories. Of course, these are still groupoids

over 𝒞. Construct now two functors ℳ̃(G_{0}, G_{1}) ^{F} 𝒫̃(G_{0}, G_{1})_{F′} .

To construct F, start with a groupoid map (U, U ×_{X} U) ⟶^{(ϕ_{0},ϕ_{1})} (G_{0}, G_{1}). It’s not difficult to show (we’ve already noticed this in the course of the proof of the theorem, part 5.) that this corresponds to a descent datum associated to the cover U → X in the stack 𝒞^{I}, namely

U ×_{X} U ×_{G_{0}} G_{1}_{Φ}  U ×_{G_{0}} G_{1}  P U ×_{X} U ×_{G_{0}} G_{1} U ×_{X} U  U  X , (∗∗)

with Φ given by (x, y, g) ↦ (x, y, ϕ_{1}(x, y)g). (Here, of course, ϕ_{1}(x, y)g means c(ϕ_{1}(x, y), g), using the composition map c : G_{1} ×_{G_{0}} G_{1} → G_{1}.) So it descends to a space P over X. Because the map U → X is faithfully flat, P → X is faithfully flat if and only if U ×_{G_{0}} G_{1} → U is. But U ×_{G_{0}} G_{1} → U is faithfully flat, because it’s a base change of d : G_{1} → G_{0}. So P → X is faithfully flat. Now look at the following commutative diagram

U ×_{X} (U ×_{X} P)  U ×_{X} P  P U ×_{X} (U ×_{G_{0}} G_{1}) U ×_{G_{0}} G_{1} P U ×_{X} U  U  X .

In the first row the double arrows are (x, y, u) ↦ (x, y) and (y, u), and in the second row they are (x, y, g) ↦ (x, ϕ(x, y)g) and (y, g), respectively. The upper middle vertical arrow is an isomorphism, written as (x, u) ↦ (x, g), coming from the fact that on the right we have two pullback diagrams. We know U ×_{X} U U X is a coequalizer, so the first row is also a coequalizer, because P → X is flat. This implies that the second row is a coequalizer. We want to make P canonically a 𝒢-space. For this, we need a map P → G_{0}, which we get from the diagram of coequalizers:

U ×_{X} U ×_{G_{0}} G_{1}  U ×_{G_{0}} G_{1}  P G_{1} ×_{G_{0}} G_{1}  _{c}_{π_{2}}  G_{1}  G_{0} .

Next, look at

U ×_{X} U ×_{G_{0}} G_{1} ×_{G_{0}} G_{1}  U ×_{G_{0}} G_{1} ×_{G_{0}} G_{1}  P ×_{G_{0}} G_{1} U ×_{X} U ×_{G_{0}} G_{1}  U ×_{G_{0}} G_{1}  P .

It’s easy to show that the dotted map is a 𝒢-action. Now obviously the map U ×_{X} P → U ×_{G_{0}} G_{1} is an isomorphism of 𝒢-spaces. So P is a 𝒢-bundle. This constructs the functor F, which is well-defined, because a map between objects (U, U ×_{X} U) → (G_{0}, G_{1}) in ℳ(G_{0}, G_{1}) gives a map of descent data, and this descends to a map of the resulting 𝒢-bundles (and we use that 𝒫(G_{0}, G_{1}) is a groupoid/𝒞).

On the other hand, given a 𝒢-bundle P over X which is trivial over U, we obtain a descent datum associated to the cover U → X, therefore a map like Φ in the diagram (∗∗). This gives a map U ×_{X} U ⟶^{ϕ_{1}} G_{1} with certain properties, which turn out after a short computation to be that

(U, U ×_{X} U) ⟶^{(ϕ_{0},ϕ_{1})} (G_{0}, G_{1})

is a groupoid map, where U ⟶^{ϕ_{0}} G_{0} comes from the definition of P. We obtain a functor F′, which is well-defined because a 𝒢-bundle map P ⟶^{f} Q gives a map U ×_{X} P ⟶^{U×_{X}f} U ×_{X} Q between the trivializations (suppose both bundles are trivial over U), which in turn defines a map between the corresponding descent data (which are given as in the diagram (∗∗)). This finally gives a map between the corresponding objects in ℳ(G_{0}, G_{1}). It’s trivial to see now that F F′ and F′F are the identities.

2. We want to prove that 𝒫(G_{0}, G_{1}) is a stack. Take a descent datum

Q U _{i} X ,

so Q descends to an object P → X. We want P to be a 𝒢-bundle, such that i^{∗}P is isomorphic to Q as 𝒢-spaces. Assume Q is trivializable over V, where V ⟶^{j} U is a cover. We obtain the composite pullback diagram

j^{∗}Q  Q  P V  U  X .

j^{∗}Q gives a descent datum associated to V → X, and it descends to P. According to the proof of 1., this makes P into a 𝒢-bundle. Also, the map i^{∗}P → Q, which comes from coequalizers over V ×_{U} V V U, is a 𝒢-space map.

The other stack axiom, that morphisms form a sheaf, is satisfied for 𝒫(G_{0}, G_{1}) for similar reasons (the only thing to prove here is that the descended map is a 𝒢-space map).

Remark 18.14. The condition that 𝒞^{I} be a stack is superfluous. All we needed in the proof was that a descent datum

N U  X ,

with N → U faithfully flat, descends to M → X by a certain coequalizer, i.e. in the diagram of coequalizers

U ×_{X} N  N  M U ×_{X} U  U  X ,

N is isomorphic to U ×_{X} M (meaning that N descends to M). But applying U ×_{X} (−) to the upper row, and (−)×_{X} N to the lower row, we get two coequalizers

N U ×_{X} U ×_{X} N  U ×_{X} N  U ×_{X} M ,

which give the desired isomorphism.

Also, note that in the stack 𝒞^{I} we have uniqueness of the descended object up to isomorphism, provided we assume, as in the lemma, that covers are faithfully flat.

Remark 18.15. The theorem says that isomorphic Hopf algebroids give rise to isomorphic rigidified stacks. One might ask under which conditions two Hopf al-

Is this statement gebroids give rise to equivalent stacks (not necessarily as rigidified stacks). The about cohomology right? question is of interest, because in this case one might show that the two Hopf algebroids have the same cohomology. A partial answer is given by the following lemma, which we won’t prove.

Lemma 18.16. Let (A, Γ) be a Hopf algebroid, A → B a ring map, and (B, Γ_{B}) the Hopf algebroid given by Γ_{B} = B ⊗_{A} Γ ⊗_{A} B. If there is a faithfully flat map A → Ã such that

A  ^{η_{L}} Γ ⊗_{A} B Ã

commutes, then ℳ(B, Γ_{B}) → ℳ(A, Γ) is an equivalence of stacks.

Example 18.17. Consider A = ℤ[a, b], Γ = ℤ[a, b, r] the Hopf algebroid representing the functor ℛings → 𝒢roupoids

R ↦ 𝒢_{R} = {x^{2} + ax + b | a, b ∈ R} ,

where the maps in 𝒢_{R} are given by translations {x ↦ x + r}. The structure maps are: η_{L} is the identity on the generators a, b, and η_{R} is (a, b) ↦ (a + 2r, b + ar + r^{2}). Take B = A/(b) = ℤ[a]. Then, after a short computation,

Γ_{B} = ℤ[a, r]/(ar + r^{2}) .

Also,

Γ ⊗_{A} B = ℤ[a, b, r]/(b + ar + r^{2}) .

Γ ⊗_{A} B is obtained from A by adjoining a root of a second-degree polynomial. Therefore A ⟶^{η_{L}} Γ⊗_{A}B is faithfully flat, which means we can take Ã = Γ⊗_{A}B. The lemma now says that (A, Γ) and (B, Γ_{B}) have the same cohomology. But (B, Γ_{B}) is easier to work with, because its cohomology is given by a periodic resolution.

Definition 18.18. The stack of elliptic curves ℳ_{Ell} is the stack ℳ(A, Γ) associated to the Hopf algebroid

\{ \begin{matrix}A = ℤ[a_{1}, a_{2}, a_{3}, a_{4}, a_{6}][Δ^{−1}] \\ Γ = A[r, s, t, λ^{±1}] , \end{matrix}

with structure maps: η_{L} identity on generators, and η_{R} given by the equation

y^{2} + η_{R}(a_{1})xy + η_{R}(a_{3})y = x^{3} + η_{R}(a_{2})x^{2} + η_{R}(a_{4})x + η_{R}(a_{6}) ,

which has to be the same as the equation

λ^{−6} { Y^{2} + a_{1}XY + a_{3}Y = X^{3} + a_{2}X^{2} + a_{4}X + a_{6} } ,

with the substitutions

\{ \begin{matrix}X = λ^{2}x + r \\ Y = λ^{3} + sλ^{2}x + t . \end{matrix}

[There are roughly five lectures missing at this point.] Part 2. The dénouement

19. Computations

In the “previous” lecture, we produced an essentially unique functor

ℰt′/ℳ_{Ell} ⟶^{E} A_{∞}^{Ell}.

Given any quasi-étale 1-morphism 𝒩 ⟶ ℳ_{Ell}, we define E(𝒩) to be

E(𝒩) = Tot(U^{▪}),

where U^{▪} is a fibrant cosimplicial A_{∞}-ring spectrum, constructed as follows: we choose a quasi-étale covering

Spec R _{f}  𝒩 ℳ_{Ell}

(note that the map f is in ob ℰt′/ℳ_{Ell}), and set U^{n} = E(\overbrace{Spec R ×_{𝒩} … ×_{𝒩} Spec R}^{n}).

E(𝒩) is independent of the choice of cover.

I don’t know if this Now, E(𝒩) comes equipped with a spectral sequence is an E_{1}-term or an E_{2}-term. I’m going to pretend that it is an E_{1}-term; I only mention it one more time, in 15 more lines.

H^{s}(𝒩; ω^{t}) ⇒ π_{2t−s}E(𝒩).

It is interesting to consider this in the case 𝒩 = ℳ_{Ell}. Then H^{0}(ℳ_{Ell}; ω^{t}) is the modular forms of weight t, so

• E(ℳ_{Ell}) is a topological model for the moduli space of elliptic curves, and • π_{∗}E(ℳ_{Ell}) is a topological model for the ring of modular forms.

We call π_{∗}E(ℳ_{Ell}) the ring of topological modular forms; it has been computed completely. Here, we indicate how to compute the E_{1}-term, at least after reducing

This is the one time mod 3. I mention the E_{1}- term.

How to compute H^{s}(ℳ_{Ell}; ω^{t}). Consider the Hopf algebroid

A = ℤ[a_{1}, a_{2}, a_{3}, a_{4}, a_{6}, Δ^{−1}],
Γ = A[r, s, t, λ^{±1}].

Recall that we have a pullback diagram

Spec Γ  Spec A Spec A  ℳ_{Ell} .

The covering Spec A ⟶ ℳ_{Ell} is not quasi-étale, so we can’t just apply our theory to that; however, we can use this covering to compute cohomology (à la ordinary homological algebra).

Consider the spectral sequence

E_{1} = H^{s}(\overbrace{Spec A ×_{ℳ_{Ell}} … ×_{ℳ_{Ell}} Spec A}^{n}; ω^{t}) ⇒ H^{s+n}(ℳ_{Ell}; ω^{t}).

Note that

\overbrace{Spec A ×_{ℳ_{Ell}} … ×_{ℳ_{Ell}} Spec A}^{n} = Spec(\overbrace{Γ ⊗_{A} … ⊗_{A} Γ}^{n−1}),

so H^{s} = 0 for s > 0, and hence the spectral sequence degenerates at E_{2}. Since the E_{1}-term of the spectral sequence, together with d_{1}, is the cobar complex for (A, Γ), we have the following.

Lemma 19.1. H^{s}(ℳ_{Ell}; ω^{t}) is the cohomology of the Hopf algebroid (A, Γ) with coefficients in the comodule associated to ω^{t}.

To make life easy, we reduce everything mod 3 in what follows.

Let B = ℤ/3[b_{2}, b_{4}, b_{4}^{−1}, (b_{2}^{2} − b_{4})^{−1}], and consider the elliptic curve

y^{2} = x^{3} + b_{2}x^{2} + b_{4}x.

For the definition of B, we need to invert the discriminant, which is the product of the differences of the roots of the right-hand side of this equation; this is where the two inverted terms come from. This curve gives us a 1-morphism Spec B ⟶ ℳ_{Ell}.

Claim 19.2. Spec B ⟶ ℳ_{Ell} is a flat cover.

Proof. Consider the pullback diagram

Spec R  Spec A Spec B  ℳ_{Ell} .

Since Spec A ⟶ ℳ_{Ell} is a cover, we need to show that A ⟶ R is faithfully flat. We have a lift Spec B ⟶ Spec A because B was defined by giving an elliptic curve in the standard form; if you prefer, there is clearly a surjection A ↠ B which induces the elliptic curve over B from the standard one over A. In any case, from this diagram, we have an explicit description of R: we have R = B ⊗_{A} Γ. You can see or is it Γ ⊗_{A} B, or this by looking at the following diagram, in which both squares are pullbacks: does it matter?

Spec R  Spec Γ  Spec A Spec B  Spec A  ℳ_{Ell} .

Is A ⟶ B ⊗_{A} Γ = B[r, s, t, λ^{±1}] faithfully flat? We take the standard elliptic curve over A and map it to B by the rules

a_{i} ⟼ b_{i},
y ⟼ λ^{3}y + sx + t,
x ⟼ λ^{2}x + r.

This will tell us how our map A ⟶ B ⊗_{A} Γ is defined: after dividing by λ^{6}, the coefficient of xy will be the image of a_{1}, the coefficient of y will be the image of a_{2}, etc. Here are the details: start with the equation

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}, (∗)

make the above substitutions, and multiply out: I haven’t gone over this carefully—I’m just relying on the calculation in class. λ^{6}y^{2} − λ^{3}sxy − λ^{3}ty

= λ^{6}x^{3} + (λ^{4}b_{2} − s^{2})x^{2} + (−λ^{2}b_{2}r + λ^{2}b_{4} + st)x + (r^{3} + b_{2}r^{2} + b_{4}r − t^{2}).

Now divide by λ^{6}. Then our map is given by

ℤ/3[a_{1}, a_{2}, a_{3}, a_{4}, a_{6}, Δ^{−1}] ⟶ ℤ/3[b_{2}, b_{4}, b_{4}^{−1}, (b_{2}^{2} − b_{4})^{−1}, r, s, t, λ^{±1}]
a_{1} ⟼ −λ^{−3}s
a_{3} ⟼ −λ^{−3}t
a_{2} ⟼ λ^{−2}b_{2} − λ^{−6}s^{2}
a_{4} ⟼ −λ^{−4}b_{2}r + λ^{−4}b_{4} + λ^{−6}st
a_{6} ⟼ λ^{−6}(b_{2}r^{2} + b_{4}r − t^{2} + r^{3}).

This is clearly injective (a_{1} essentially hits s, a_{3} essentially hits t, etc.—you can see that the images of the a_{i}’s are algebraically independent). So

B ⊗_{A} Γ = A[λ^{±1}, r]/(r^{3} + b_{2}r^{2} + b^{4}r − t^{2} = λ^{6}a_{6}).

Hence B ⊗_{A} Γ is free of rank 3 over A[λ^{±1}], and hence is faithfully flat over A.

[Motivation for this: if you take the standard formula (∗) for an elliptic curve, and if 2 is a unit, then you can complete the square on the left-hand side by the substitution y ↦ y − \frac{1}{2}(a_{1}x + a_{3}). This gives

y^{2} = x^{3} + b_{2}x^{2} + b_{4}x + b_{6}.

Now make the substitution x ↦ x + r where r is a root of the right-hand side—a root will exist after a faithfully flat change of base. Then we get

y^{2} = x^{3} + b_{2}x^{2} + b_{4}x.

This shows that the above map is faithful. The argument in the claim is necessary to show flatness.]

Now consider the pullback diagram

Spec W  Spec B Spec B  ℳ_{Ell} .

Then

W = B ⊗_{A} Γ ⊗_{A} B
= B[λ^{±1}, r, s, t]/(s, t, r^{3} + b_{2}r^{2} + b_{4}r)
= B[λ^{±1}, r]/(r^{3} + b_{2}r^{2} + b_{4}r).

(This corresponds to the substitution x ↦ λ^{2}x + r, y ↦ λ^{3}y.) This defines a new Hopf algebroid, (B, W), with structure maps as follows (left unit on top, right unit

(I’m pretty sure) on the bottom):

b_{2} ↦ b_{2},
b_{4} ↦ b_{4},
B ⇉ W,
b_{2} ↦ λ^{−2}b_{2},
b_{4} ↦ λ^{−4}(b_{4} − b_{2}r).

(The bottom two formulas are derived by substituting x ↦ λ^{2}x + r, y ↦ λ^{3}y, and solving for the coefficients of x^{2} and x.)

Note that the comodule ω is free of rank one over B, generated by e, where e ↦ λ ⊗ e. So λ and the powers of ω define a grading on B and W. Thus we end up computing the cohomology of the Hopf algebroid

B_{∗} = ℤ/3[b_{2}, b_{4}, Δ^{−1}], |b_{2}| = 2, |b_{4}| = 4,
W_{∗} = B[r]/(r^{3} + b_{2}r^{2} + b_{4}r), |r| = 2.

Conclusion. The cohomology of (B_{∗}, W_{∗}) in degree i is isomorphic to H^{∗}(ℳ_{Ell}; ω^{i} ⊗ ℤ/3).

We construct a complex for computing this. Consider the map

W_{∗} ⟶^{\frac{d}{dr}} Σ^{2}W_{∗}.

For convenience of notation, we also denote this map by D.

Claim 19.3. This is a map of W_{∗}-comodules.

Proof. We have to show that the diagram

W_{∗}  ^{D}  Σ^{2}W_{∗} W_{∗} ⊗_{B_{∗}} W_{∗} _{1⊗D} W_{∗} ⊗_{B_{∗}} Σ^{2}W_{∗}

commutes, where the vertical maps are given by the coproduct of W_{∗},

W_{∗} ⟶ W_{∗} ⊗_{B_{∗}} W_{∗},
r ⟼ r ⊗ 1 + 1 ⊗ r.

Note that this is a map of left B_{∗}-modules, so it suffices to check the commutativity of our diagram on the B_{∗}-module generators of W_{∗}—that is, on 1, r, and r^{2}. Here are the relevant pictures:

1  0 1 ⊗ 1  0 ,  r  1 \begin{matrix}r ⊗ 1 \\ +1 ⊗ r \end{matrix}  1 ⊗ 1 ,  r^{2}  −r \begin{matrix}r^{2} ⊗ 1 \\ +2r ⊗ r \\ +1 ⊗ r^{2} \end{matrix}  \begin{matrix}2(r ⊗ 1 \\ +1 ⊗ r) \end{matrix}

Note that ker D = B^{∗}, and coker D is free of rank 1 over B_{∗}, generated by r^{2} (which has coaction r^{2} ↦ 1 ⊗ r^{2}, modulo (1 ⊗ DW_{∗})). This gives an exact sequence of W_{∗}-comodules:

0 ⟶ B_{∗} ⟶ W_{∗} ⟶ Σ^{2}W_{∗} ⟶ Σ^{6}B_{∗} ⟶ 0.

We can splice this with (suspensions of) itself to get a periodic resolution of B_{∗} by relative injectives:

0  B_{∗}  W_{∗} ^{D} Σ^{2}W_{∗} Σ^{6}B_{∗} Σ^{6}W_{∗} ^{D} Σ^{8}W_{∗}  ….

We compute the Hopf algebroid cohomology by finding the primitives at each stage, and then taking the cohomology of the resulting chain complex.

The set of primitives of W_{∗} is just B_{∗}, generated by 1, but it’s a bit tricky to work out what the induced maps on the primitives are; Mike made an attempt at it, but mistakes were made. Here are a few calculations, anyway. Consider

W_{∗}  ^{D} Σ^{2}W_{∗} B_{∗}  Σ^{2}B_{∗} .

We have

b_{2}  0 b_{2}  , b_{4} − b_{2}r  −b_{2} b_{4}  ,

so in the first stage of the resolution, b_{2} ↦ 0 and b_{4} ↦ −b_{2}. Similarly, one can show that in the map in the next homological degree, Σ^{2}B_{∗} ⟶ Σ^{6}B_{∗}, we have b_{2} ↦ 0, b_{4} ↦ 0, and b_{4} ↦ b_{2}^{2}. For example, the last one of these comes from the diagram

Σ^{2}W_{∗} ^{D} Σ^{6}W_{∗} Σ^{2}B_{∗}  Σ^{6}B_{∗} , b_{4}^{2} − 2b_{2}b_{4}r + b_{2}r^{2}  b_{2}^{2} b_{4}^{2}  .

(Remember that the map Σ^{2}W_{∗} ⟶ Σ^{6}W_{∗} factors through the map Σ^{2}W_{∗} ⟶ Σ^{6}B_{∗} which sends 1 and r to zero, and sends r^{2} to 1.)

Unfortunately, while the maps on the primitives in the resolution are b_{2}-linear, they are not derivations, so one cannot immediately conclude anything about what happens to a general element of the primitives. One can calculate everything, though, if one knows enough; here’s the final answer.

Proposition 19.4.

H^{∗}(ℳ_{Ell}; ω^{t}) = ℤ/3[b_{2}, Δ^{±1}] ⊗ E[α] ⊗ P [β]/(b_{2}α, b_{2}β),

where Δ = b_{4}^{2}(b_{2}^{2} − b_{4}), α ∈ H^{1}(ℳ_{Ell}; ω^{2}), and β ∈ H^{2}(ℳ_{Ell}; ω^{6}). Also note that we have an isomorphism

Δ : H^{∗}(ℳ_{Ell}; ω^{n}) ⟶^{≅} H^{∗}(ℳ_{Ell}; ω^{n+6}).

Note that there are differentials in the spectral sequence

H^{∗}(ℳ_{Ell}; ω^{n}) ⇒ π_{∗}E(ℳ_{Ell}).

Mike knows what they are. In any case, one can compute π_{∗}E(ℳ_{Ell}), the ring of but he mentioned them as an aside, and I wasn’t listening closely enough topological modular forms.

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Department of Mathematics, M.I.T., Cambridge, MA 02139