The n-Category Café

Let’s go back to classical homotopy theory for a moment, and consider all the different ways we have to work with a particular $(\infty,1)$-category, such as the $(\infty,1)$-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 $(\infty,1)$-category directly, using the (fairly complicated) tools of $(\infty,1)$-category theory. This $(\infty,1)$-category can be constructed from any of the models above, with results that are equivalent in the $(\infty,1)$-categorical sense. Thus, passage to this level loses information about which model we started from, but if all we care about is the $(\infty,1)$-categorical structure, the information we lost was superfluous. At this level, homotopy limits and colimits can be characterized directly via an $(\infty,1)$-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 $(\infty,1)$-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 $(\infty,1)$-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 $(\infty,1)$-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 $(\infty,2)$-category. Specifically, we are interested in the $(\infty,2)$-category of $(\infty,1)$-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 $(\infty,2)$-category in question directly, choosing a model and using the tools of $(\infty,2)$-category theory (which are less well-developed to date than those of $(\infty,1)$-category theory).

3. Pass to the homotopy 2-category of the $(\infty,2)$-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 $(\infty,2)$-category.

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

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

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

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

• If $X$ is an $(\infty,1)$-category, then the hom-category $(\infty,1)Cat_2(1,X)$ of “global sections” of $X$ in $(\infty,1)Cat_2$ is equivalent to the ordinary homotopy category $ho(X)$ of $X$. In other words, the functor $ho\colon (\infty,1)Cat_2 \to Cat$ is represented by the terminal object $1$. More generally, for $(\infty,1)$-categories $A$ and $X$, the category $(\infty,1)Cat_2(A,X)$ is the “homotopy category of $A$-shaped diagrams in $X$,” which is equivalently $ho(X^A)$ (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 $(\infty,1)$-functor has an adjoint, in the $(\infty,1)$-categorical sense, iff its image in $(\infty,1)Cat_2$ does, in the internal 2-categorical sense. (Obviously, this depends on fixing a notion of $(\infty,1)$-adjunction.)

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

• The presheaf category $\widehat{X} = \mathcal{S}^{X^{op}}$, where $\mathcal{S}$ is the $(\infty,1)$-category of $\infty$-groupoids, is a free cocompletion of $X$ in $(\infty,1)Cat_2$, 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

• $(\infty,1)Cat_2$ 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 $(\infty,1)$-categories. This would make precise the observation that lots of $(\infty,1)$-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 $(\infty,1)$-category theory—in particular, that which deals only with limits and colimits—could be done purely at the level of $(\infty,1)Cat_2$, without needing to invoke much of the theory of $(\infty,1)$-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 $(\infty,1)Cat_2$ also supplies a “central comparison point” for model-independence results. Namely, given any model for $(\infty,1)$-categories, if we can show that its homotopy 2-category is $(\infty,1)Cat_2$ and that its natural notions of limit and colimit agree with those of $(\infty,1)Cat_2$, 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 $(\infty,1)$-category form model categories that are Quillen equivalent, but that by itself doesn’t yet imply that the resulting theories of $(\infty,1)$-categories are the same, including such important aspects as limits and colimits.

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

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

A 2-functor $Cat^{op}\to CAT$ 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 $(\infty,1)$-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 $ho(X)$ with this extra level of structure, we’ve managed to squeeze a lot more information out of the $(\infty,1)$-category $X$ 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 $X$ this way! This paper by Olivier Renaudin proves that the following 2-categories are equivalent:

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

• The 2-category obtained from the 2-category $CombModel_{adj}$ 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 $PresDer_{adj}$ is also equivalent to the homotopy 2-category obtained from the $(\infty,2)$-category of locally presentable $(\infty,1)$-categories and adjunctions by identifying equivalent 2-cells.

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

When I first encountered derivators, my reaction was “gee, that’s artificial: surely we should be working with $(\infty,1)$-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” $(\infty,2)$-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 $(\infty,1)$-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 $(\infty,1)$-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 $A$, the derivator coming from simplicial presheaves on $A$ is the free cocompletion of $A$ as a derivator. This then forces you to believe that $\infty$-groupoids are important.)