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One modern POV on model categories is that they are presentations of $(\infty, 1)$-categories (namely, given a model category, you obtain an $\infty$-category by localizing at the category of weak equivalences; better, if you have a simplicial model category, is to take the homotopy coherent nerve of the fibrant-cofibrant objects).

What other functions, then, do model categories serve today? I understand that getting the theory of $\infty$-categories off the ground (as in HTT, for instance) requires a significant use of a plethora of model structures. However, if we assume that there exists a good model of $(\infty, 1)$-categories that satisfies the properties we want (e.g. that mapping sets are replaced with mapping spaces, limits and colimits are determined by homotopy limits of spaces), how are model categories useful?

I suppose one example would be computing hom-spaces: a simplicial model category gives you a nice way of finding the mapping space between two objects in the associated localization. However, in practice one only considers cofibrant or fibrant objects in the $\infty$-category in the first place, as in Lurie's construction of the derived $\infty$-category (basically, one considers the category of projective complexes -- for the bounded-above case, anyway -- and makes that into a simplicial category, and then takes the homotopy coherent nerve).

One example where having a model structure seems to buy something is the theorem that $E_\infty$ ring spectra can be modeled by 1-categorical commutative algebras in an appropriate monoidal model category of spectra (in DAG 2 there is a general result to this effect), and that you can straighten things out to avoid coherence homotopies. I don't really know anything about $E_\infty$-ring spectra, but I'm not sure how helpful this is when one has a good theory of monoidal objects in $\infty$-categories.

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Here's a related question. Let's say we have two model categories (possibly simplicial). What additional information is it to say that they are Quillen equivalent than it is to say that the associated $\infty$-categories are equivalent? –  Akhil Mathew Oct 18 '11 at 1:24
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If they are combinatorial model categories, then (cf Higher Topos Theory A.3.7.7) there is only a small difference. If the $\infty$-cats are equivalent, then the model cats are connected by a chain of Quillen equivalences. If you have a single direct Quillen equivalence, then what you learn is that certain computations done with one model structure can be closely related to analogous computations with the other. –  Chris Schommer-Pries Oct 18 '11 at 1:37
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I'll leave it to someone with stronger opinions to answer your main questions, but in response to your comment-question, I can say that two combinatorial model categories are connected by Quillen equivalences (two of them, in fact) if and only if their underlying $\infty$-categories are equivalent. –  Clark Barwick Oct 18 '11 at 1:40
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Unsurprisingly, Chris is faster than me. –  Clark Barwick Oct 18 '11 at 1:41
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You seem to think that model categories and $(\infty,1)$-categories are so closed to each other that one of them should perhapes stop being useful, but why model categories? Can you translate everything to the new setting in a simpler way? I'd doubt it. –  Fernando Muro Oct 21 '11 at 20:22

4 Answers 4

up vote 79 down vote accepted

I find some of this exchange truly depressing. There is a subject of ``brave new algebra''and there are myriads of past and present constructions and calculations that depend on having concrete and specific constructions. People who actually compute anything do not use $(\infty,1)$ categories when doing so. To lay down a challenge, they would be of little or no use there. One can sometimes use $(\infty,1)$ categories to construct things not easily constructed otherwise, and then one can compute things about them (e.g. work of Behrens and Lawson). But the tools of computation are not $(\infty,1)$ categorical, and often not even model categorical. People should learn some serious computations, do some themselves, before totally immersing themselves in the formal theory. Note that $(\infty,1)$ categories are in principle intermediate between the point-set level and the homotopy category level. It is easy to translate into $(\infty,1)$ categories from the point-set level, whether from model categories or from something weaker. Then one can work in $(\infty,1)$ categories. But the translation back out to the "old-fashioned'' world that some writers seem to imagine expendable lands in homotopy categories. That is fine if that is all that one needs, but one often needs a good deal more. One must be eclectic. Just one old man's view.

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+1000 for «People should learn some serious computations, do some themselves, before totally immersing themselves in the formal theory» :) –  Mariano Suárez-Alvarez Dec 13 '11 at 2:06
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Thanks for this answer. I guess I need to learn some computations! –  Akhil Mathew Dec 13 '11 at 2:28
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Bravo, sir. Seconding Mariano (with my mind on other areas of mathematics) –  Yemon Choi Dec 13 '11 at 2:33
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I agree that it is better to be eclectic, but for a young person who is first encountering a huge subject and trying to grasp the essence of it, it can be quite useful to be conceptual. Maybe even before getting lost in computations. –  Michael Bächtold Dec 13 '11 at 7:54
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Michael, if you know my work, then you know that in recent years I have erred on the side of the conceptual (I find it easier). But if you want to learn a subject, you have to learn its actual content before you learn how to abstract away from that content in order to prove things that are not as accessible as you might hope within the traditional methodology. I do not think $\infty$ categories can be appreciated without solid prior grounding in algebraic topology or algebraic geometry or at least homological algebra, preferably all three. –  Peter May Dec 14 '11 at 19:18

I confess to being confused by all this $(\infty,1)$ category business, and the way $\Pi X$ is used as another name for the singular simplicial set of $X$. This is related to Peter's question on computations.

I thought one reason for moving from loops or paths to fundamental groups or fundamental groupoids, i.e. taking homotopy classes, was that one could do specific computations in groups, and also groupoids. So I began in the 1960s to look for higher dimensional versions of these groupoid methods, again with the aim, or hope, of higher dimensional nonabelian calculations. Of course we were well aware of all the laws on paths, or singular simplices, up to homotopies, e.g. Kan extension conditions, but it seemed difficult to get computational information directly at the path space or singular complex level.

What was surprising, and took a long time to realise, was that we could do these higher dimensional strict groupoid methods, using certain homotopy classes, for certain structured spaces, particularly filtered spaces (11 years), and later $n$-cubes of spaces (Loday) (17 years). In the filtered space work, the insights of model theory have also proved very useful - I think they have not yet been used in the $n$-cube situation. Grothendieck was amazed when I told him in 1985 (6?) that $n$-fold groupoids model homotopy $n$-types (Loday's theorem). Since we can use this idea for specific nonabelian colimit calculations in homotopy theory with the aid of a Higher van Kampen type theorem, I am happy as an old man to rest with the use of strict multiple groupoids of various kinds suitable for the problem at hand. Of course in the proofs, the relations between the weak (spaces of maps) situation and the strict one is crucial.

I see these ideas as another contribution to the tool kit of algebraic topology, and some younger people are using them.

It seems a useful, but not obligatory, test of a theory to ask if it can in some cases produce some numbers not previously available.

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One nice feature of model categories is that you can speak also of the non-bifibrant objects (which is not longer possible, once you passed to the corresponding infinity-category). A few examples where this is useful:

  • Simplicial sets: non-Kan simplicial sets appear again and again. For example, the n-simplex itself.
  • Spectra: in most models for spectra all fibrant objects are $\Omega$-spectra. One often wants to consider non-$\Omega$-spectra like Thom spectra or suspension spectra.
  • Diagram categories: The process of replacing a map (say, between topological spaces) by a fibration can be seen as a fibrant replacement in the arrow category.
  • Chain complexes: not all modules are projective...

For proving abstract theorems, the framework of $\infty$-categories seems to be in many senses very convenient. But model categories are (in my opinion) often nicer if you want to deal with concrete examples (which are often not bifibrant) and want to see how to compute derived functors of them. Also, concrete models of spectra (like symmetric spectra) where $E_\infty$-rings are modeled by strictly commutative monoids are really nice to write down concrete examples.

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This is interesting. I guess maybe this is why in DAG I, Lurie constructs the $\infty$-category of spectra only by looking at $\Omega$-spectra. –  Akhil Mathew Oct 21 '11 at 14:16

Here are some rough analogies:

  • Model Category :: $(\infty, 1)$-category
  • Basis :: Vector space
  • Local coordinates :: Manifold

I especially like the last one. When you do, say, differential geometry in a coordinate free way you often end up with more conceptual, cleaner, and more beautiful proofs. But sometimes you just have to roll up your sleeves, get your hands dirty, and compute in local coordinates. Sometimes different local coordinates are useful for different calculations.

One modern point of view is that Quillen Model categories are the local coordinates of a homotopy theory (i.e. presentable $(\infty,1)$-category, c.f. this MO answer). When you can compute without choosing local coordinate (a model category structure), that is great. But sometimes it can be very helpful (maybe even necessary) to pick a Model structure to do a specific computation. Sometimes it is helpful to jump between different Model structures which describe the same underlying $(\infty,1)$-category.

Model structures are still extremely useful, even essential to the general theory. But they are a tool, just like local coordinates in differential geometry.

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Thanks. I guess I'm still curious: do you have a specific example of where you need the "local coordinates" to prove theorems? (For instance, at least in elementary differential geometry, they are (to my knowledge) unnecessary in most cases.) Though I suppose Quillen's rational homotopy theory was one example of a result that was stated as a Quillen equivalence between model categories. Hm. –  Akhil Mathew Oct 18 '11 at 2:11
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I think Chris uses the word "need" in the sense of "it is substantially more practical for mathematicians (with their educational backgrounds) today" rather than in a strict, reverse-mathematical sense of "it is impossible to prove otherwise". –  S. Carnahan Oct 18 '11 at 2:21
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@Akhil: Scott is right about my use of "need". For the stricter sense, you can cheat and ask a question which is not about the $(\infty,1)$-category but about the model structure itself (An analog would be: "In normal coordinates at p, the metric simplifies to the Kronecker delta"). Barring this, I guess in principle any calculation you would preform with the model category structure could theoretically be computed from the Hammock localization (which only uses the weak equivalences, not the whole model structure). In practice, though, this is not really feasible with our current technology. –  Chris Schommer-Pries Oct 18 '11 at 13:24
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I've heard these rough analogies before, too, and they seem useful as rough guides go, but this answer seems a bit too philosophical to be really satisfactory. A stronger answer would give concrete instances where it seems very hard to do without model-theoretic techniques in performing some calculation, somewhat along the same lines Ira L gave in his comments for the case of differential geometry. Do you have some such examples? –  Todd Trimble Oct 18 '11 at 15:34
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For the past 30 years or so, abstract homotopy theory has utilized the language of model categories to prove results and theorems. That hasn't just evaporated. I think the burden and challenge is really on the OP, Todd, and other skeptics of the utility of model cats. Can you reproduce even a fraction of these results without ever using model categories? If you can, SUPER! Wirte it up! It would be a tremendous help to the field. Personally, I would like to see the Hill-Hopkins-Ravenel work on the Kervaire invariant done without model categories. –  Chris Schommer-Pries Oct 18 '11 at 16:52

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