Mark Hovey's open problems in the theory of model categories

Question

Mark Hovey maintains a list of open problems in model category theory. I think this list is quite old, and I don't know if Hovey is still updating it or not.

My question is:

i) which of the 13 problems in that list are still considered open and still important in homotopy theory?

ii) or dually, which of the problems are considered settled or are perceived as no longer relevant given the current status of homotopy theory?

Answer 1

I am a former student of Mark Hovey's, and during grad school, I wrote a document giving an update on the status of the 13 problems (as of 2012 or 2013, I guess). I just briefly went through it a moment ago to give some updates, but it's still in rough shape. I apologize for what I'm sure will be many omissions, as so much work has been done in this area over the past 20 years. Nevertheless, here you go (using Hovey's words to set up each of the problems in case his website goes down again):

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.

The machinery to resolve this question was developed by Mark Hovey (2007) in “Cotorsion pairs, model category structures, and representation theory,” which provides a method for building an abelian model structure on a bicomplete abelian category with prescribed classes of acyclic, cofibrant, and fibrant objects $(W,C,F)$ such that $W$ is thick and the pairs $(C,F \cap W)$ and $(C\cap W, F)$ are complete cotorsion pairs. An excellent survey is Hovey’s “Cotorsion Pairs and Model Categories,” where this appears as Theorem 2.5. Conditions are also given to make this model structure monoidal (Theorem 4.2). Hovey used this theory to build a model structure on unbounded chain complexes of quasi-coherent sheaves over a (nice) scheme where weak equivalences are quasi-isomorphisms and fibrations are dimensionwise split surjections with dimensionwise injective kernel (see Hovey “Model Category Structures on Chain Complexes of Sheaves” Theorem 4.4). He also built a model structure on chain complexes of modules over a nice ringed space (Theorem 5.2) and found conditions to make this model structure satisfy the pushout product and monoid axioms (5.7, 5.9, 5.10). Jim Gillespie then proved a theorem which allows one to build a model structure on Ch(A) where A is an abelian model category with a nice cotorsion pair (Theorem 7.9). Gillespie’s work generalizes the above and can be found in “The Flat Model Structure on Complexes of Sheaves” and “Cotorsion Pairs and Degreewise Homological Model Structures.” Gillespie continued to generalize and hone this theory, resulting in more than 30 papers on the topic. A great survey is his paper "Hereditary abelian model categories." These kinds of model structures, that come from cotorsion pairs, are called abelian model structures and nowadays a compatible pair of cotorsion pairs (the type that give rise to an abelian model structure) is known as a Hovey triple. Many other authors have also worked in this area, including Sergio Estrada, Daniel Bravo, Jan Stovicek, Sinem Odabasi, Hanno Becker, and probably many more people I should mention.

2. A scheme is a generalization of a ring, in the same way that a manifold 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?

These problems appear to still be open as stated. There are good reasons why one cannot have a model category of manifolds, and these are discussed in my expository paper “On Colimits in Various Categories of Manifolds” among other places. I don’t know whether or not there are similar obstructions for sheaves over a manifold. If so then the methods of getting around this obstruction mentioned in the expository paper may apply in those settings as well (i.e., using Voevodsky style enlargement or using Diffeologic Spaces). The note advertises Dan Dugger's nice paper "Sheaves and Homotopy Theory." I should disclaim that I wrote this paper early in my grad student career, so it's probably badly written, naive, and might even have errors.

The years since Hovey originally wrote his problem list have seen a massive development in the theory of infinity categories, and a partial answer can be given in that language. Consider the category of smooth $\infty$-groupoids, which contains the categories of smooth manifolds and Lie groupoids. Making use of the global model structure on simplicial presheaves and the fact that BG is a fibrant object therein, one can recover the de Rham cohomology of a smooth manifold as $\pi_0$ of a mapping space in the category of smooth $\infty$-groupoids. Similarly one can recover Cech hypercohomology. A nice survey of these results can be found at the nLab page on “smooth infinity-groupoid structures.” So, although it remains unknown whether there is a model category of sheaves over a manifold, at least de Rham cohomology can be recovered from model category theoretic considerations. There does not appear to have been any work done to recover Floer homology or quantum cohomology from a model category.

UPDATE: Tyler Lawson has done nice work on homotopy theory for Floer homology. And I still think a model structure in this context is possible. It’s something I’d like to work on someday. Mark Hovey and I used to talk about this and we had some ideas I hope to work out one day.

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.

This was solved by Mark Hovey in “Homotopy Theory of Comodules over a Hopf Algebroid” (published in contemp. math.) where he constructs for a given Hopf algebroid $(A,\Gamma)$ a model structure on the category of unbounded chain complexes of $\Gamma$-comodules with weak equivalences the homotopy isomorphisms (not the homology isomorphisms) and the cofibrations are the degreewise split monomorphisms whose cokernel is a complex of relative projectives with no differential. See Theorems 2.1.1, 2.1.3, 5.1.4, and (for the monoidal structure) 5.1.5.

Regarding the general philosophy behind this question, it is interesting to note that examples have been given for triangulated categories which do not arise as the homotopy category of a model category. The most well-known appears in “Triangulated Categories without Models” by Muro, Schwede, and Strickland. Note however that the definition of triangulated category used in the book Model Categories is different from the standard definition. Hovey’s definition of triangulated category T requires T to come with an action of Ho(sSet). Such triangulated categories are the ones which come up in homotopy theory, but they have not been studied systematically other than in Hovey’s book. So, technically, you could ask if every triangulated category in Hovey's sense comes from a model category. And the answer is probably "no." Since the triangulated category community kinda rejected Hovey's definition of "triangulated category", I don't think an explicit counterexample would generate much interest.

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 cannot be a model category with underlying 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.

This has pretty much been solved. My thesis gave conditions under which a category of commutative monoids has a transferred model structure, and also under which it’s equivalent to E-infinity algebras. Also, there is a model structure on categories of operads, due to Berger and Moerdijk (2003), if M satisfies some conditions, like having a nicely behaved interval object. For more general M, say just cofibrantly generated, there’s a semi-model structure on operads due to Spitzweck (a published reference is Fresse’s book on operads and modules).

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).

Let M be a monoidal model category. If M is nice (like, simplicial sets, equivariant topological spaces, chain complexes over a field of characteristic zero, symmetric spectra, equivariant orthogonal spectra, etc) then algebras over any operad have a transferred model structure. If $M$ is less nice, you have a transferred semi-model structure. Many authors did work in this direction, including Spitzweck, Berger–Moerdijk, Elmendorf–Mandell (for spectra), Fresse, Harper, Hess, Casacuberta, Gutierrez, Moerdijk, Vogt, Caviglia, Harper, Hornbostel (in a motivic setting), Muro, and Pavlov-Scholbach. I give some history in my papers with Donald Yau, and also what we consider to be the most general approach with the weakest hypotheses on $M$. Let's focus on full model structures instead of semi-model structures. Our first paper proves that all operads are admissible (have a transferred full model structure) in chain complexes, spaces, and symmetric spectra, and our second paper covered equivariant spaces and spectra, simplicial abelian groups, the category of small categories, the stable module category, etc. If M is only cofibrantly generated, then you have a semi-model structure on algebras over a Sigma-cofibrant operad, again by Spitzweck and Fresse. My first paper with Donald Yau generalizes this to a wider class of operads (just entrywise cofibrant), and our third paper handles rectification (a weak equivalence of operads inducing a Quillen equivalence of categories of algebras) in greater generality. I should point out that many authors had rectification results for $\Sigma$-cofibrant operads (including Berger-Moerdijk and Fresse), and Pavlov-Scholbach had admissibility and rectification results under different assumptions on $M$.

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.

I did this in my thesis, inventing the Commutative Monoid Axiom to get a model structure on commutative monoids. This also included the situation of rectification with E-infinity (indeed, under some more conditions, because it’s not true in the monoidal model category of compactly generated topological spaces). I generalized this in my work with Donald Yau. In our first paper, we get conditions on a model category M so that all operads are admissible (meaning: have a transferred model structure), or weaker conditions so that entrywise cofibrant operads are semi-admissible (have a transferred semi-model structure), and doing rectification in our third paper. To respond to Dmitri Pavlov's answer, let me remark that Jacob Lurie also had a result that ends up with a model structure on commutative monoids, but under much stronger conditions on $M$. Mark and I were not aware of Lurie's approach until after mine was done, and back then there was actually an error in Lurie's approach that wiped out the applications (it was only applicable to chain complexes over a field of characteristic zero). Since my approach worked for all known examples (where commutative monoids have a transferred model structure) plus some new ones, we decided to call my condition the "commutative monoid axiom" instead of giving Lurie's condition that name.

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.

This has been done. Rezk got the program started in part 2 of his thesis, setting up a spectral sequence to compute the homotopy groups of the moduli space of $A$-algebra structures on $X$, when $A$ is a cofibrant operad (in simplicial sets). Vigleik Angeltveit did it in the context of spectra. A great paper by Niles Johnson and Justin Noel "Lifting homotopy T-algebra maps to strict maps" sets up a Bousfield-Kan spectral sequence that converges to the homotopy groups of the space of $A$-algebra maps between two spaces. Here $A$ is a simplicial monad and $M$ is a simplicial model category. Possibly there is room here for generalization to non-simplicial cases, but I doubt that Hovey was asking for an answer in that level of generality.

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.

It will be debatable whether or not this problem has been satisfactorily solved. I’ve got a recent paper with Boris Chorny that can be thought of as seeking the internal hom of the “2-model category of model categories” (meaning, a model structure on a category of functors between two model categories). Boris has done a lot of work in the direction of this problem and would not think it’s solved as of now. However, I think there’s a strong argument that Reid Barton’s thesis essentially solves this problem, or rather a slight but necessary weakening of what Hovey was asking for. Barton makes the argument himself, in section 1 of his thesis, laying out why he thinks this is a solution to what Hovey was asking for, and why what Hovey was asking for was literally impossible.

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?

Again, people will debate if this has been solved. But if you restrict attention to combinatorial model categories then essentially it has been. Dugger proved that every combinatorial model category is Quillen equivalent to a simplicial combinatorial model category (Batanin and I recently proved the same for combinatorial semi-model categories). Back when Clark Barwick was writing a paper about “partial model categories” I convinced myself that the collection of them was equivalent to partial simplicial model categories. I have a hazy recollection that Lennart Meier did some work related to this in 2015. Barton’s thesis also has Theorem 1.3.4 which is like a 2-model category of simplicial combinatorial premodel 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.

Essentially yes. The relevant paper here is "Admissible replacements for simplicial monoidal model categories" by Bayinder and Chorny. Also, the centrality problem was resolved. I learned one solution from Jerome Scherer, who told me it's in his 2008 paper (with Chacholski) Representations of Spaces. Denis-Charles Cisinski points out that he also solved it in 2002.

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.

Again, modulo the fact that what Hovey had in mind doesn’t exactly exist, this problem has been solved. There are many, many models for the "homotopy theory of homotopy theories", including quasi-categories, relative categories, simplicial categories, topological categories, Segal categories, Segal spaces, complete Segal spaces, $A_\infty$-categories, etc, etc. Each has a model structure and these model structures are all Quillen equivalent, even in a coherent way. See work of Toen and Barwick and Schommer-Pries. There are also models for $\infty$-categories with extra structure, like the category of cofibration categories (whose homotopy theory was worked out beautifully by Karol Szumilo) or the category of fibration categories. There's also a model-free approach due to Riehl and Verity, called $\infty$-cosmoi.

However, the theory of model categories is not equivalent to the theory of $(\infty,1)$-categories, because not every infinity category comes from a model category. If an infinity category comes from a model category, it must have all limits and colimits.

There are two ways to proceed. You can weaken what you mean by "model category" or you can look for an equivalence with $(\infty,1)$-categories with extra structure. Both approaches work. For example, if you weaken from "model category" to, say, "partial model category", then Barwick and Kan’s paper on the subject (linked above) shows how the theory of partial model categories is equivalent to the theory of $\infty$-categories. The same is true if you weaken from "model category" to "relative category". As for $(\infty,1)$-categories with extra structure, the right notion is "presentable $(\infty,1)$-categories." The theory of combinatorial model categories is equivalent in a strong way to the theory of presentable infinity categories.

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?

I think what Mark had in mind here was basically the theory of derivators, first conceived by Grothendieck in 1983, and worked out by many authors over the past fifteen years or so. I would say it’s widely accepted that the viewpoint of derivators does have value and is an acceptable way to do homotopy theory. I don’t claim that it's equivalent in a formal way to the theory of model categories or infinity categories.

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?

Many examples have now been given. I’ll list all the ones I know:

• George Raptis proves in Remark 4.7 of “Homotopy Theory for Posets” that Strom’s model structure on Top is not cofibrantly generated. This answers a question which was implicit in Hovey’s book when he discusses this model structure, but which he never explicitly asked.
• Christensen and Hovey's paper "Quillen model structures for relative homological algebra" prove that the absolute model structure on relative chain complexes is not cofibrantly generated (see section 5).
• Isaksen's model structure on pro-simplicial sets is not cofibrantly generated, and the category of simplicial sets is not fibrantly generated. For many more examples see Scott Balchin's recent book A Handbook of Model Categories
• Chorny showed that the model category of maps of spaces or simplicial sets (i.e. the arrow category) is not cofibrantly generated. This might have been the first example known.
• Adamek, Herrlich, Rosicky, and Tholen produce a model structure on the category of small categories that is not cofibrantly generated.
• Chorny-Rosicky “class combinatorial model categories” includes examples of Pro spaces and Ind spaces, and about Fun(M,N), that are not confibrantly generated in the classical sense for set-theoretic reasons.
• There are several examples in Emily Riehl’s book. Chapters 12 and 13 discuss “category cofibrantly generated” when the left class is a category not a set. The left class must be defined as a retract-closure, but the right is automatically closed under retracts. Examples include Hurewicz models. She also covers "enriched cofibrantly generated" more generally.
• Barthel and Riehl: Cole’s mixed model structure – this is an example like the above for topological model categories.
• Barthel May Riehl – this is an example like the above for chain enriched.
• Lack’s 2007 paper has a non-cofibrantly generated model structure on Arr(Cat), but it is cofibrantly generated in Riehl's sense.
• Bourke and Garner have notion of "cofibrantly generated by a double category", plus an example that is not cofibrantly generated as a category (because the right class is not closed under retracts).

UPDATE: In my paper with Donald Yau about arrow categories, we list several examples of monoidal model categories that are not cofibrantly generated.

Answer 2

4. On the other hand, the category of commutative monoids seems to be much more subtle.

The conditions for the existence of a model structure on commutative monoids were worked out by Jacob Lurie around 2009, see Proposition 4.5.4.6 in Higher Algebra, which establishes an analogue of the monoid axiom for commutative monoids. This proposition proves both the existence of a model structure and a rectification theorem for it, showing it is equivalent to the corresponding ∞-category. The assumptions of Proposition 4.5.4.6 are mostly necessary for the second part, the rectification statement, whereas the assumptions for the first part (existence of a model structure) are rather minimal. These assumptions were further weakened in arXiv:1410.5675 (see Theorem 5.11), where the class of cofibrations is replaced by the considerably wider class of h-cofibrations, which yields the most general known result of this class.

4. Find a model structure on the category of operads on a given model category.

This has been done at various levels of generality in the work Berger–Moerdijk, Gutiérrez–Vogt, Spitzweck, Muro (in the nonsymmetric case), Kro. A version that subsumes the examples constructed in the above papers can be found in arXiv:1410.5675 (see Corollary 9.4.1), which establishes a model structure on colored symmetric operads enriched in a symmetric h-monoidal symmetric monoidal model category.

5. Show that the category of algebras over a cofibrant operad admits a model structure

5. Show that a weak equivalence of cofibrant operads induces a Quillen equivalence of the categories of algebras.

6. Find conditions under which algebras over a noncofibrant operad admit a model structure that generalize the monoid axiom of Schwede-Shipley.

Work in this direction has been done by Spitzweck (who resolved Problem 5 in his PhD thesis), Elmendorf–Mandell, Berger–Moerdijk, Caviglia, Harper, Hornbostel, Muro (in the nonsymmetric case), Hinich, Lurie. A version that subsumes the examples constructed in the above papers can be found in arXiv:1410.5675 (see Theorems 5.11, 7.5, 7.11, 8.10), which establishes these results in a very general setting of operads enriched in symmetric monoidal model categories, and provides an appropriate analogue of the monoid axiom.

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.

Some adjustments to the statement are necessary because limits and colimits of simplicial model categories typically do not exist. A proof of one rigorous formulation of this conjecture is given as Theorem 1.2 of arXiv:2110.04679.

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.

Theorem 1.1 of arXiv:2110.04679 establishes a Dwyer–Kan equivalence between combinatorial model categories and presentable quasicategories, which can be seen as a rigorous formulation of Hovey's conjecture.