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May 18, 2010

Squeezing Higher Categories out of Lower Categories

Posted by Mike Shulman

For a long time, homotopy theory involved the study of homotopy categories. More recently, people have started preferring to use (,1)(\infty,1)-categories, since homotopy categories lose a lot of structure. But it’s worth remembering that homotopy categories do still contain a lot of information, and moreover they encapsulate it all in terms of the familar and easy-to-handle notions of 1-category theory. Want to know if two spaces are (weak) homotopy equivalent? Check whether they’re isomorphic in the homotopy category. Want to know the homotopy groups of a (pointed connected) space? Look at maps out of spheres in the homotopy category. Want a representing object for cohomology theory? Apply Brown’s representability theorem in the homotopy category. Want to know the homotopy type of a mapping space? You’re in luck: the homotopy category is closed symmetric monoidal.

Today I want to talk about ways we can use lower-categorical “homotopy categories” to squeeze even more information out of a higher category, without ever having to construct and work with that higher category directly (thereby potentially avoiding a lot of combinatorial complexity). In fact, in good situations, we can actually squeeze out all the information in this way! (At least for a suitable definition of “all.”)

Let’s go back to classical homotopy theory for a moment, and consider all the different ways we have to work with a particular (,1)(\infty,1)-category, such as the (,1)(\infty,1)-category of \infty-groupoids (i.e. “spaces”).

  1. We can work with particular models, such as topological spaces, simplicial sets, cubical sets, etc. Here we can use the tools of ordinary 1-category theory, but we have to keep track of extra information, such as a class of “weak equivalences” and perhaps a whole model category structure. We also have to be careful about how we pass from one model to another “equivalent” one. We can perform constructions such as homotopy limits and colimits by using weighted limits with suitably “fattened up” weights—this is the classical notion of “homotopy limit.”

  2. We can work with the (,1)(\infty,1)-category directly, using the (fairly complicated) tools of (,1)(\infty,1)-category theory. This (,1)(\infty,1)-category can be constructed from any of the models above, with results that are equivalent in the (,1)(\infty,1)-categorical sense. Thus, passage to this level loses information about which model we started from, but if all we care about is the (,1)(\infty,1)-categorical structure, the information we lost was superfluous. At this level, homotopy limits and colimits can be characterized directly via an (,1)(\infty,1)-categorical universal property.

  3. We can pass to the homotopy category, either by forcibly turning weak equivalences into isomorphisms in a model category, or by identifying equivalent 1-morphisms in an (,1)(\infty,1)-category. Once again we are back in the easy world of 1-category theory, but now important information has definitely been lost. In particular, we can no longer characterize homotopy limits and colimits categorically: they can sometimes be identified with particular weak limits in the homotopy category, but even when this is true, it doesn’t suffice to characterize them among all the other less-well-behaved weak limits that may exist. (This is essentially the reason why triangulated categories are frustrating.)

I think it’s not too far off to say that “classical” homotopy theory is the “indirect” study of (,1)(\infty,1)-categories, by way of “squeezing” them in between the first and third approaches. For a lot of purposes, this is good enough, but in some cases we need more. In particular, this “needing of more” usually tends to happen when we start wanting to think about the collection of (,1)(\infty,1)-categories, rather than merely studying one of them at a time. One solution to this is to pick up the second approach, but another potential solution would be to try to apply the “classical” ideas one categorical level up.

In other words, we want to study some particular (,2)(\infty,2)-category. Specifically, we are interested in the (,2)(\infty,2)-category of (,1)(\infty,1)-categories. By analogy, we can imagine the following approaches:

  1. Work with a particular model or models, such as quasicategories, complete Segal spaces, simplicial categories, etc. Again we can use the tools of 1-category theory, keeping track of extra information, such as weak equivalences, model category structures, and so on, and be careful about passing between “equivalent” models.

  2. Work with the (,2)(\infty,2)-category in question directly, choosing a model and using the tools of (,2)(\infty,2)-category theory (which are less well-developed to date than those of (,1)(\infty,1)-category theory).

  3. Pass to the homotopy 2-category of the (,2)(\infty,2)-category in question, either by 2-categorically inverting the weak equivalences in a model of the first type, or by identifying equivalent 2-cells in a direct model for an (,2)(\infty,2)-category.

Here’s a quick and easy construction of the homotopy 2-category of (,1)Cat(\infty,1)Cat. The category QCatQCat of quasicategories is cartesian closed and thus enriched over itself, so if XX and YY are quasicategories we have a mapping-space quasicategory Y XY^X. Taking the homotopy category of a quasicategory is a product-preserving functor ho:QCatCatho\colon QCat \to Cat, so applying this homwise to the QCatQCat-enriched category QCatQCat, we obtain a CatCat-enriched category QCat 2QCat_2, i.e. a 2-category. Up to equivalence, this is the homotopy 2-category of (,1)Cat(\infty,1)Cat; I’ll denote it by (,1)Cat 2(\infty,1)Cat_2 and call it the homotopy 2-category, by analogy with how the homotopy category of Gpd\infty Gpd is traditionally called the homotopy category.

The homotopy 2-category actually contains a lot of information about (,1)(\infty,1)-categories. The following are all pretty obvious.

  • Two (,1)(\infty,1)-categories are equivalent iff they are internally equivalent in (,1)Cat 2(\infty,1)Cat_2.

  • (,1)Cat 2(\infty,1)Cat_2 is cartesian closed as a 2-category, and its internal-homs are the correct (,1)(\infty,1)-categorical functor categories.

  • If XX is an (,1)(\infty,1)-category, then the hom-category (,1)Cat 2(1,X)(\infty,1)Cat_2(1,X) of “global sections” of XX in (,1)Cat 2(\infty,1)Cat_2 is equivalent to the ordinary homotopy category ho(X)ho(X) of XX. In other words, the functor ho:(,1)Cat 2Catho\colon (\infty,1)Cat_2 \to Cat is represented by the terminal object 11. More generally, for (,1)(\infty,1)-categories AA and XX, the category (,1)Cat 2(A,X)(\infty,1)Cat_2(A,X) is the “homotopy category of AA-shaped diagrams in XX,” which is equivalently ho(X A)ho(X^A) (using the cartesian closed structure).

Moreover, the following facts are also (believed to be) true. These are asserted in various versions of Joyal’s notes on quasicategories, but perhaps (?) complete proofs of them are not yet to be found there.

  • An (,1)(\infty,1)-functor has an adjoint, in the (,1)(\infty,1)-categorical sense, iff its image in (,1)Cat 2(\infty,1)Cat_2 does, in the internal 2-categorical sense. (Obviously, this depends on fixing a notion of (,1)(\infty,1)-adjunction.)

  • An (,1)(\infty,1)-category XX has (homotopy) limits of diagrams of shape AA iff the diagonal functor XX AX\to X^A has a right adjoint (in (,1)Cat 2(\infty,1)Cat_2). In particular, XX is complete if this is the case for all small AA.

  • The presheaf category X^=𝒮 X op\widehat{X} = \mathcal{S}^{X^{op}}, where 𝒮\mathcal{S} is the (,1)(\infty,1)-category of \infty-groupoids, is a free cocompletion of XX in (,1)Cat 2(\infty,1)Cat_2, in the appropriate internal 2-categorical sense.

Finally, Emily Riehl, Daniel Schaeppi, Claire Tomesch, and I (and Richard Garner when he was here last week) have been working on showing that

  • (,1)Cat 2(\infty,1)Cat_2 can be equipped with proarrows and/or a Yoneda structure. We hope that in this way, the long-known tools of limits and colimits in such structures can be applied directly to the study of (weighted, homotopy) limits and colimits in (,1)(\infty,1)-categories. This would make precise the observation that lots of (,1)(\infty,1)-categorical facts “follow as usual” in category theory once you have versions of the Yoneda lemma and so on.

Assuming this is all true, it means that much of (,1)(\infty,1)-category theory—in particular, that which deals only with limits and colimits—could be done purely at the level of (,1)Cat 2(\infty,1)Cat_2, without needing to invoke much of the theory of (,1)(\infty,1)-categories directly. (Of course, one has to beware of various things; for instance, just as the homotopy category lacks most limits and colimits, so the homotopy 2-category lacks most 2-limits and 2-colimits.)

Thinking about (,1)Cat 2(\infty,1)Cat_2 also supplies a “central comparison point” for model-independence results. Namely, given any model for (,1)(\infty,1)-categories, if we can show that its homotopy 2-category is (,1)Cat 2(\infty,1)Cat_2 and that its natural notions of limit and colimit agree with those of (,1)Cat 2(\infty,1)Cat_2, then we know immediately that they also agree with those in any other model with those properties. As we’ve discussed elsewhere, work in the past few years has shown that that the various definitions of (,1)(\infty,1)-category form model categories that are Quillen equivalent, but that by itself doesn’t yet imply that the resulting theories of (,1)(\infty,1)-categories are the same, including such important aspects as limits and colimits.

There’s one final simplification we can perform. Observe that CatCat is a full sub-2-category of (,1)Cat 2(\infty,1)Cat_2, and thus we have a restricted Yoneda embedding (,1)Cat 2[Cat op,Cat](\infty,1)Cat_2 \to [Cat^{op},Cat]. (For size reasons, we usually consider the version of this that looks like (,1)CAT 2[Cat op,CAT](\infty,1)CAT_2 \to [Cat^{op},CAT], where CatCat denotes the 2-category of small categories, while the other two can include large ones.) Thus, any (,1)(\infty,1)-category XX gives rise to a 2-functor D X:Cat opCATD_X\colon Cat^{op}\to CAT, where D X(A)=(,1)Cat 2(A,X)D_X(A) = (\infty,1)Cat_2(A,X) is the homotopy category of the (,1)(\infty,1)-functor category X AX^A.

In particular, D X(1)=(,1)Cat 2(1,X)=ho(X)D_X(1) = (\infty,1)Cat_2(1,X) = ho(X) is the homotopy category of XX, so we can regard the 2-functor D XD_X as “the homotopy category of XX equipped with extra information about homotopy-commutative diagrams in XX of all shapes.” The wonderful thing is that this structure is still sufficient to characterize limits and colimits in XX: they are given by left and adjoints to the functors ho(X)=D X(1)D X(A)ho(X) = D_X(1) \to D_X(A), induced by contravariant functoriality from the unique map A1A\to 1. But now we are completely in the realm of ordinary (2-)category theory: talking about 2-functors Cat opCATCat^{op}\to CAT doesn’t require us even to know anything about any definition of (,1)(\infty,1)-categories.

A 2-functor Cat opCATCat^{op}\to CAT is called a prederivator, and if it has all “homotopy Kan extensions” and is otherwise well-behaved, it is called a derivator. These structures were invented by Grothendieck and Heller, and have since been studied extensively by Cisinski and others. Amazingly (to me), they seem to suffice for all sorts of things that one might want to do in an (,1)(\infty,1)-category. For example, just as the poorly behaved structure of a triangulated category can be lifted to the well-behaved structure of a stable (infinity,1)-category, it can similarly be lifted to a stable derivator, with many of the same properties. I expect something similar is also true for a “derivator topos,” but I don’t know whether anyone has written that down. Thus, by equipping ho(X)ho(X) with this extra level of structure, we’ve managed to squeeze a lot more information out of the (,1)(\infty,1)-category XX using only categories and 2-categories.

(Unfortunately for us English-speakers, a lot of the literature on derivators seems to be in French, but I’m gradually working on putting some of the basic facts on the nLab as I learn them. I could use plenty of help, though, especially from anyone out there who speaks French more fluently and would be interested in making derivators more widely known to English-speakers.)

In fact, in good situations, it seems that we can actually squeeze all the information out of XX this way! This paper by Olivier Renaudin proves that the following 2-categories are equivalent:

  • The 2-category PresDer adjPresDer_{adj} of “locally presentable” derivators, and adjunctions between them.

  • The 2-category obtained from the 2-category CombModel adjCombModel_{adj} of combinatorial model categories and Quillen adjunctions by formally inverting the Quillen equivalences.

It seems to me that the proof is very formal: the idea hinges on the facts that combinatorial model categories and locally presentable derivators can both be constructed by (1) forming the free cocompletion of a small category and then (2) formally inverting a set of morphisms. For combinatorial model categories, this is a theorem of Dugger, while for locally presentable derivators, it can be taken as the definition. Given that this is how the proof goes, it seems like one should similarly be able to show that PresDer adjPresDer_{adj} is also equivalent to the homotopy 2-category obtained from the (,2)(\infty,2)-category of locally presentable (,1)(\infty,1)-categories and adjunctions by identifying equivalent 2-cells.

In particular, this means that two locally presentable (,1)(\infty,1)-categories are equivalent if and only if their corresponding derivators are equivalent; and if a derivator comes from a locally presentable (,1)(\infty,1)-category, then that (,1)(\infty,1)-category is determined essentially uniquely. This is the sense in which I meant that the derivator recovers “all” of the structure of the (,1)(\infty,1)-category: we can reconstruct the latter from the former uniquely up to equivalence. (However, some information is certainly lost in passing from the (,2)(\infty,2)-category (,1)Cat(\infty,1)Cat to the 2-category of derivators.)

When I first encountered derivators, my reaction was “gee, that’s artificial: surely we should be working with (,1)(\infty,1)-categories, which are a much more natural structure.” But then Kate Ponto and I were trying to prove something about “triangulated bicategories,” and (a) getting frustrated at the inadequacy of triangulated categories but (b) not wanting to have to define and understand “locally stable” (,2)(\infty,2)-categories in sufficient detail to actually prove something nontrivial about them. Then I realized that derivators may be exactly what we need. I think of a derivator as a “user-friendly interface” to an (,1)(\infty,1)-category, kind of like Ubuntu and Gnome sitting on top of Linux (for the computer nerds in the audience). Sure, sometimes what the GUI offers doesn’t cut it, and you have to go down deeper into the inner workings of the system. And the experts may prefer to always use the low-level structure. But for a lot of day-to-day tasks, the GUI is perfectly adequate, and much easier to understand, so that a lot of people can simply use the GUI and not worry about what lies underneath.

(In closing, I should mention that there is a different way of thinking about derivators, as espoused by Cisinski in this blog comment. If you already believe in (,1)(\infty,1)-categories, as I expect many of our readers do, then I think the approach to them that I sketched above is very natural. But Cisinski argues persuasively that if you start instead from the more classical perspective that homotopy theory is about “what you get when you invert weak equivalences,” then you are led naturally to derivators. You can then prove that every derivator is canonically enriched over the usual homotopy category of \infty-groupoids, and moreover that for any small category AA, the derivator coming from simplicial presheaves on AA is the free cocompletion of AA as a derivator. This then forces you to believe that \infty-groupoids are important.)

Posted at May 18, 2010 2:19 AM UTC

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4 Comments & 0 Trackbacks

Re: Squeezing Higher Categories out of Lower Categories

Thanks, Mike. Very nice story.

I am wondering about this bit here:

[…] it seems like one should similarly be able to show that PresDer adjPresDer_{adj} is also equivalent to the homotopy 2-category obtained from the (,2)(\infty,2)-category of locally presentable (,1)(\infty,1)-categories and adjunctions by identifying equivalent 2-cells.

What exactly do you think is and what should be true?

You say that the 2-category PresDer adjPresDer_{adj} is known to be equivalent to the 2-localization of the 2-category of combinatorial model categories at Quillen adjunctions. So the above sentence should equivalently read

[…] it seems like one should similarly be able to show that CombModel adjCombModel_{adj} is also equivalent to the homotopy 2-category obtained from the (,2)(\infty,2)-category of locally presentable (,1)(\infty,1)-categories and adjunctions by identifying equivalent 2-cells.

Do I understand you correctly here?

So what do we know about this last statement? We know that a morphism of locally presentable (,1)(\infty,1)-categories is an equivalence precisely if it is modeled as a zig-zag of Quillen equivalences of combinatorial simplicial model categories.

Now, what exactly is CombModel adjCombModel_{adj} for you? What are the 2-morphisms? Unrestricted natural transformations between Quillen adjunctions?

Posted by: Urs Schreiber on May 18, 2010 9:34 AM | Permalink | Reply to this

Re: Squeezing Higher Categories out of Lower Categories

…one should similarly be able to show that CombModel adjCombModel_{adj} is also equivalent to…

I presume you mean “the homotopy 2-category obtained from CombModel adjCombModel_{adj} by inverting Quillen equivalences” (i.e. forcibly turning them into internal equivalences). With that change, what you say would be equivalent to the conjecture I stated.

And yes, the morphisms in CombModel adjCombModel_{adj} are arbitrary natural transformations between (say left, for definiteness) Quillen functors. If I understand him correctly, one of Renaudin’s observations is that because CombModel adjCombModel_{adj} has enough “cylinder objects,” turning Quillen equivalences into equivalences is equivalent to inverting natural transformations that are pointwise weak equivalences on cofibrant objects, via essentially the same argument for why inverting homotopy equivalences is the same as quotienting by homotopy.

Posted by: Mike Shulman on May 18, 2010 4:18 PM | Permalink | Reply to this

Re: Squeezing Higher Categories out of Lower Categories

Is this essentially the same philosophy as using Derivators instead of (infinity, 1)-categories?

http://ncatlab.org/nlab/show/derivator

Posted by: Chris Schommer-Pries on May 21, 2010 10:39 PM | Permalink | Reply to this

Re: Squeezing Higher Categories out of Lower Categories

I did talk about derivators in the blog post. Towards the end (sorry it’s so long). So I’m not sure what more you’re looking for.

Posted by: Mike Shulman on May 22, 2010 3:42 AM | Permalink | Reply to this

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