# 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.

1. 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 here.

2. 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?

3. 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.

4. 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.

5. 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).

6. 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.

7. 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.

8. 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.

9. 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 2-categories?

10. 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.

11. 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.

12. 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?

13. 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?

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Mark Hovey
Department of Mathematics
Wesleyan University
mhovey@wesleyan.edu