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I've read about model categories from an Appendix to one of Lurie's papers.

What are the examples of model categories? What should be my intuition about them?

E.g. I understand the typical examples come from taking homotopy of something — but are all model categories homotopy categories?

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I edited the question a bit so the citations below are slightly different. –  Ilya Nikokoshev Oct 25 '09 at 0:53
    
I want to say something about my choice of an answer. There are at least three extremely interesting other answers, and I recommend all of them: thanks to Benjamin Antieau, Timo Schürg and A.J. Tolland and others. Ordinarily it would be very hard to choose an answer. But Reid Barton helped me with posting a very simple and I believe deep reason for why model categories are important. It's certainly subjective, but it hit me as a correct answer to my question about "What should be my intuition about them?" –  Ilya Nikokoshev Oct 26 '09 at 22:30
    
But you feel like another answer deserves to be accepted, then please point that out by voting and leaving comments! –  Ilya Nikokoshev Oct 27 '09 at 19:47
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6 Answers 6

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Model categories are 1-categorical presentations of (∞,1)-categories, which you can just think of as categories enriched in topological spaces, such as the category of spaces itself. (Actually, there are conditions on (∞,1)-categories that come from model categories--most importantly they must have all homotopy limits and colimits.) They're particularly convenient for computations, as with any kind of presentation.

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What is the precise statement that establishes a connection between model categories and (∞,1)-categories? Do you have a reference for it? –  Dmitri Pavlov Oct 28 '09 at 19:06
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The ideal statement would involve something like a "2-model category of combinatorial model categories" and give an equivalence between its homotopy 2-category (combinatorial model categories, Quillen pairs with Quillen equivalences inverted, natural transformations) and the homotopy 2-category of presentable (∞,1)-categories. I don't know of a reference or even a precise statement of this full equivalence. However, Proposition A.3.7.6 of Higher Topos Theory and the following Remark establish a bijection on the level of equivalence classes of objects. –  Reid Barton Oct 28 '09 at 19:31
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In that Proposition, the word "simplicial" is not essential, if you replace N with the simplicial localization. The word "combinatorial" is essential (a non-combinatorial model category can easily yield a non-presentable (∞,1)-category). –  Reid Barton Oct 28 '09 at 19:35
    
Do you know any similar statement for triangulated categories, i.e., every triangulated category satisfying some additional properties is the homotopy category of some stable (∞,1)-category? I was only able to find an informal Remark 3.2 in DAG-I, which says “The theory of triangulated categories is an attempt to capture those features of stable ∞- categories which are visible at the level of homotopy categories. Triangulated categories which appear naturally in mathematics are usually equivalent to the homotopy categories of suitable stable ∞-categories.” –  Dmitri Pavlov Dec 15 '09 at 18:15
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If you have a stable ∞-category C with sufficient colimits then you can form a system of categories Ho (C^I) as I varies over small categories, with restriction and left Kan extension functors between them. Such a system with certain properties (automatically satisfied in this case) is called a derivateur. I have heard that the octahedral axiom for a triangulated category T follows from the existence of a derivateur D with D(*) = T, and that there are similar higher "octahedral" axioms which do not follow from the usual one. So I think any statement of this kind would be quite complicated. –  Reid Barton Dec 15 '09 at 19:03
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Here's a bit of the historical reason why model categories came up. If you have a functor on an abelian category which doesn't quite behave as you would like it to, e.g. is not left exact, there is a simple recipe to derive it: 1) Embed your category into the bigger category of complexes 2) Find a qausi-isomorphic replacement for the object you want to plug into the functor. This was so easy because the starting category was abelian. One important example of a category that is not abelian is the category of commutative rings, and there is one important functor on this category, namely the functor of Kaehler differentials, which is not quite exact. (Remember the conormal sequence from Hartshorne, it's only got a zero on one side). So what to do? Since commutative rings are not abelian, there's no way that you can work with complexes. Quillen's great move was to use simplicial objects instead of complexes. This takes care of step 1. You've just succesfully embedded the category of commutative rings into the category of simplicial commutative rings. Now you have to care of step 2. This is possible if the larger category you just moved into is a model category. The replacement is either called cofibrant or fibrant replacement, depending on your functor. The existence of such a replacement follows immediately from the axioms of a model category, since you can factor the map from the initial object to any object into a cofibration and a trivial fibration. Vice versa with final object for fibrant replacement. So we can also do step 2 now! Coming back to the example of the Kaehler differentials, we can derive them now: 1) View your commutative ring as a trivial simplicial ring. 2) Replace it cofibrantly 3)Apply \Omega levelwise. What you get is a simplicial module. If you apply the Dold-Kan correspondence you get complex, and this is the cotangent complex.

Of course there's much more to Model categories, but deriving functors from non-abelian categories is already quite something.

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Besides the comments above, it might be useful to have a list of categories that have natural model structures:

  • simplicial sets,
  • CW-complexes (Serre fibrations),
  • topological spaces (Hurewicz fibrations),
  • simplicial objects in other categories, like abelian groups, rings, etc,
  • diagrams of simplicial sets, especially,
  • cosimplicial simplicial sets, and
  • presheaves of simplicial sets,
  • quasi-categories (a category whose objects are a special type of simplicial set),
  • the category of small categories (see Thomason),
  • A1-homotopy invariant Nisnevich presheaves with transfers.

It is important to emphasize that these categories often carry several different model structures giving non-equivalent homotopy categories. Another point is that most of these are in some sense built out of simplicial sets. Grothendieck in letters (to Brown?) suggested the idea of a test category, categories which could play the same role as the category Delta. In addition these may come with extra structure. I believe this is the case for cubical objects and similar constructions.

There are also two common operations to do on such model categories: localization and stabilization. The first produces a new model structure on a fixed model category, while the second produces a new category with a model structure.

The idea of localization is to allow us to make certain types of morphisms into isomorphisms in the homotopy category. For instance, in the category of simplicial sets, we may want to consider p-local equivalence. There is a model category structure on simplicial sets such that X -> Y is an isomorphism in the homotopy category if and only if it induces an isomorphism on p-localized homotopy groups. Here, p-localized usually means p-adic. This was first systematically laid out in the book Homotopy limits, completions, and localizations by Bousfield and Kan. A modern account is in Hirschhorn's book Model categories and their localizations.

As for stabilization, this is the process of making the loop space functor invertible. In all of the examples above there exist small homotopy limits, and we can thus form the loop space of any object X as the homotopy fiber product of two maps sending a point to the base-point of X. In this setting we can apply stabilization. For instance, using this on simplicial sets gives us the category of spectra (or, pre-spectra), and the fibrant objects in this category are the Omega-spectra. This category has the incredibly important feature that fiber and cofiber sequences coincide. See the paper Homotopy theory of Gamma-spaces, spectra, and bisimplicial sets by Bousfield and Friedlander.

The more modern versions of spectra are S-modules and symmetric spectra. For the latter, see the paper by Hovey, Shipley, and Smith. These new versions address the following problem. On the normal category of spectra, there is a natural commutative tensor product (the smash product) on the homotopy category of spectra. But, this does not 'lift' to a commutative tensor structure on spectra. The category of symmetric spectra solves this problem by rigidifying, taking into account the action of the symmetric groups on tensors.

Finally, let me note that you don't get anything for free by using model categories. Specifically, while it is often easy to understand what fibrant objects are, the process of fibrant replacement is very very difficult to understand. This is entirely analogous to the point of view of chain complexes: injective objects are easy to understand. But, the injective resolution of some given object is very difficult.

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Arggg. Why doesn't the bulleted list work? –  Benjamin Antieau Oct 24 '09 at 15:36
    
I think you just need to add a newline before the first entry of the list. –  Reid Barton Oct 24 '09 at 16:29
    
@Benjamin, I took the liberty of changing the formatting in your question to the one you meant. –  Ilya Nikokoshev Oct 28 '09 at 21:02
    
@Ilya. Thank you. –  Benjamin Antieau Oct 28 '09 at 22:37
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you might want to add various DG type categories (chain complexes) and spectra to your list, just for posterity. Dold-kan shows that the non-negatively graded chain complexes in A and simplicial objects in A are equivalent. –  Sean Tilson Jul 29 '10 at 21:06
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E.g. I understand the typical examples are homotopy categories of something — but are all model categories homotopy categories?

This isn't quite right. A model category is a category from which you can construct a nicely behaved homotopy category, by inverting weak equivalences and then passing to equivalence classes of maps. So, for example, the category of (reasonable) topological spaces has a nice homotopy category, and the category of complexes of coherent sheaves on a scheme has a derived category. (You might that say that what you really want is the homotopy category, and that the model category is merely a good way of representing the homotopy category. Certainly in Lurie's stuff, the homotopy type is what counts.)

Dwyer & Spalinski's review (http://hopf.math.purdue.edu/Dwyer-Spalinski/theories.pdf) is a reasonable explanation of the basics of the theory of model categories. But I've found that, to get anything out of that theory, I needed to have an example very firmly in mind. One good example is Pete May's Concise Course in Algebraic Topology (http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf), which presents the basics of algebraic topology in manner that reflects the model structure.

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Yes! That was my misunderstanding. I had it written when I read the thing, but forgot. –  Ilya Nikokoshev Oct 23 '09 at 21:50
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I would say what you really want is the (∞,1)-category, and that the model category is merely a good way of representing the (∞,1)-category. –  Reid Barton Oct 24 '09 at 16:30
    
Yes! That makes sense. \infty-categories (that's how Lurie calls them) are very nice and model categories are approximations. –  Ilya Nikokoshev Oct 26 '09 at 22:25
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One can also view model structures as a solution to the problem of when is a localization of a category locally small. In other words when one wants to invert a collection of morphisms the morphisms in the resulting category are equivalence classes of diagrams and there is no guarantee that there will only be a small set of equivalence classes. If the localization can be performed via taking the homotopy category of a model category then one knows that the hom-sets are fine.

Another thing, which is probably more useful in applications, is that a model structure allows one to perform certain constructions such as taking homotopy limits which can't be done just at the level of homotopy categories.

Maybe this isn't what you intended to ask in the last part of your question but it might be helpful. It is not true that every triangulated category comes from a model structure on some category. In the paper Triangulated Categories Without Models (Muro, Schwede, and Strickland) an example of a triangulated category that does not arise via a model structure is given.

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It's not really my thing, but my understanding is that the standard model category is CW-complexes with homotopy classes of maps, and that all the definitions (fibration, cofibration, weak equivalence, etc) are trying to abstract the concepts with the same names in this category. Moving a bit more algebraic, I believe that Derived Categories of Abelian Categories are a type of model category, as are categories of simplicial X where X is pretty much anything. What do all of these have in common? We want to try to do homotopy in them, in some sense. So the intuition I've taken away is that something like "homotopy" should work in these categories.

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Indeed, those are the typical examples, but I'm sure there would be no need for the notion if these were the only two. :) –  Ilya Nikokoshev Oct 23 '09 at 21:09
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Derived categories themselves are not model categories; they are the homotopy categories of a model structure on the category of chain complexes (the same way there is a homotopy category of spaces). –  Eric Wofsey Oct 23 '09 at 21:09
    
Ahh. I've not been clear on that point. Thanks Eric. And Ilya, my understanding is that there are quite a few model categories, but those are the paradigm examples, and the ones that people care most about, but people also want to see which properties of homotopy follow completely algebraically, sort of like how they want to understand what properties of (co)homology follow algebraically, so we look at things like abelian categories, exact and derived functors, and the Eilenberg-Steenrod axioms. –  Charles Siegel Oct 23 '09 at 21:12
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Incidentally, the case of chain complexes and derived categories is actually a special case of "simplicial X". By the Dold-Kan theorem, simplicial objects in an abelian category are equivalent to (nonnegatively graded) chain complexes, and the standard model structure on simplicial stuff is then the standard projective model structure on chain complexes. In this way you can think of more general simplicial objects as versions of chain complexes and derived categories for nonabelian categories. –  Eric Wofsey Oct 23 '09 at 21:45
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There are several answers and comments here that suggest that "CW-complexes" is a model category. This isn't quite right (model categories need to be closed under (at least finite) limits and colimits). The correct statement is: topological spaces have a model category where weak equivalences are weak equivalences, fibrations are Serre fibrations, and the class of cofibrant objects includes CW-complexes. –  Charles Rezk Oct 24 '09 at 16:35
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