- Tyler Lawson
- Hovey's problem list
- Model categories
This is part of an algebraic topology problem
list, maintained by Mark Hovey.
I am not sure working on model categories is a safe thing to do. It
is too abstract for many people, including many of the people who will
be deciding whether to hire or promote you. So maybe you should save
these until you have tenure.
- The safest sort of problem to work on with model categories is
building one of interest in applications. The essential idea is:
whenever someone uses the word homology, there ought to be a model
category around. I like this idea a great deal, and it might lead to
expansion of algebraic topology into many different areas. The simplest
example that I personally do not understand is complexes of
(quasi-coherent?) sheaves over a scheme. There is certainly a model
structure here, and it is probably even known. But I think it would be
good to find this out, and find out how the model structure is built.
I believe this should be a symmetric monoidal model category. I also
have the impression that one can not generalize the usual model
structure on chain complexes over a ring, because you won't have
projectives. But these two impressions sort of contradict each other,
since the second one would lead you to generalize the injective model
structure on chain complexes over a ring, but this model structure is
not symmetric monoidal. So there is something for me at least to learn
- A scheme is a generalization of a ring, in the same way that a
manfold is a generalization of R^n. So maybe there is some kind of
model structure on sheaves over a manifold? Presumably this is where de
Rham cohomology comes from, but I don't know. It doesn't seem like
homotopy theory has made much of a dent in analysis, but I think this is
partly due to our lack of trying. Floer homology, quantum
cohomology--do these things come from model structures?
- Every stable homotopy category I know of comes from a model
category. Well, that used to be true, but it is no longer. Given a
flat Hopf algebroid, Strickland and I have constructed a stable homotopy
category of comodules over it. This clearly ought to be the homotopy
category of a model structure on the category of chain complexes of
comodules, but we have been unable to build such a model structure. My
work with Strickland is still in progress, so you will have to contact
me for details.
- Given a symmetric monoidal model category C, Schwede and Shipley have
given conditions under which the category of monoids in C is again a
model category (with underlying fibrations and weak equivalences). On
the other hand, the category of commutative monoids seems to be much
more subtle. It is well-known that the category of commutative
differential graded algebras over Z can not be a model category with
uinderlying fibrations and weak equivalences (= homology isos). On the
other hand, the solution to this is also pretty well-known--you are
supposed to be using E-infinity DGAs, not commutative ones. Find a
generalization of this statement. Here is how I think this should go,
broken down into steps. The first step: find a model structure on the
category of operads on a given model category. (Has this already been
done? Charles Rezk is the person I would ask). We probably have to
assume the model category is cofibrantly generated.
- The second step: show that the category of algebras over a
cofibrant operad admits a model structure, where the fibrations and weak
equivalences are the underlying ones. Show that a weak equivalence of
cofibrant operads induces a Quillen equivalence of the categories of
algebras. Show that an E-infinity operad
is just a cofibrant approximation to the commutative ring operad. (This
latter statement is probably known, since to me it seems to be the whole
point of E-infinity).
- Find conditions under which algebras over a noncofibrant operad
admit a model structure that generalize the monoid axiom of
Schwede-Shipley. This would include the case where everything is
fibrant, for example. Show that, under some more conditions, a weak
equivalence of operads induces a Quillen equivalence of the algebra
categories. Thus, sometimes you can use commutative, sometimes you
can't, but you can always use E-infinity. And using E-infinity will not
hurt you when you can use commutative.
- Let A be a cofibrant operad as above. Use the above results to
construct spectral sequences that converge to the homotopy groups of the
space of A-algebra structures on a given object X, and to the homotopy
groups of the mapping space of A-algebra maps between two given
A-algebras. These spectral sequences for the A-infinity operad are the
key formal ingredients to the Hopkins-Miller proof that Morava E-theory
admits an action by the stabilizer group.
- My general theory is that the category of model categories is not
itself a model category, but a 2-model category. Weak equivalences of
model categories are Quillen equivalences, and weak equivalences of
Quillen functors are natural weak equivalences. Define a 2-model
category and show the 2-category of model categories is one. Note that
the homotopy 2-category at least makes sense (in a higher universe): we
can just invert the Quillen equivalences and the natural weak
equivalences. This localization process for an n-category has been
studied by Andre Hirschowitz and Carlos Simpson in descent pour les
n-champs, on xxx.
- The 2-category of simplicial model categories is supposed to be
(according to me) 2-Quillen equivalent to the 2-category of model
categories. Even without having all the definitions one can try to find
out if this is true. For example, Dan Dugger has shown that every model
category (with some hypotheses--surely cofibrantly generated at least)
is Quillen equivalent to a simplicial model category. Understand his
result in the context of the preceding two problems. That is, does Dugger's
construction in fact give a 2-functor from model categories to
simplicial model categories? Does it preserve enough structure to make
it clear that it will induce some kind of equivalences on the homotopy
- Is every monoidal model category Quillen equivalent to a simplicial
monoidal model category? This would remove the loose end in my book on
model categories, where I am unable to show that the homotopy category
of a monoidal model category is a central algebra over the homotopy
category of simplicial sets. The centrality is the problem, and I can
cope with this problem for simplicial monoidal model categories.
- Charles Rezk has a homotopy theory of homotopy theories.
This is just a category, though it is large. The objects are
generalizations of categories where composition is not associative on
the nose--that is, they are some kind of simplicial spaces. Understand
the relationship between Rezk's point of view and mine on the 2-category
of model categories. They should be equivalent in some sense.
- In the appendix to my book on model categories, I said maybe what
we are doing in associating to a model category its homotopy category is
the wrong thing. Maybe we should be associating to a model category C
the homotopy categories of all the diagram categories C^I, together
with all the adjunctions induced by functors I --> J. This would make
homotopy limits and colimits part of the structure. Does this viewpoint
have any value?
- Find a model category you can prove is not cofibrantly generated.
This is just an annoyance, not a very significant problem, but it has
been bugging me for a while. The obvious candidate for this is the
simplest nontrivial model category, the one on chain complexes where
weak equivalences are chain homotopy equivalences. Mike Cole is, so far
as I know, the first to write down a desciption of this model category,
though one certainly has the feeling that Quillen must have known about
it. But how do you prove something is not cofibrantly generated?
Department of Mathematics