Computations in $\infty$-categories

Question

Direct to the point. Since now I've looked a lot of presentations of $\infty$-categories, but it seems that the only way to do explicit computations on these objects is via model categories. Is that so, or maybe there are other way of doing computations in $\infty$-categories, and if there such different ways what are they?

Edit: I really want to thank all people that answered this question, unfortunately I can choose just one answer. Btw because here is yet midnight I wish everybody a happy new year.

Answer 1

(This is an answer to a question below from Akhil Mathew; he wanted examples of ``explicit computations'' since all he knew were classical 1950s calculations and abstract theory. My answer is too long for a comment and too short to do justice to the question.)

That is terrible!!! I don't know where to begin, since there are huge masses of explicit calculations in the past half century. You mention infinite loop spaces. There is a theorem that says that the ordinary mod $p$ homology of $\Omega^n\Sigma^n X$ is an explicit functor of the ordinary mod $p$ homology of $X$, with all information (product, Dyer-Lashof operations, coproduct, Steenrod operations) determined. That surely sounds conceptual, but try to find a conceptual proof! We have tons of explicit calculations in the classical and Novikov Adams spectral sequences, without which the solution of the Kervaire invariant problem would be unthinkable. Chromatic theory as a whole is informed by explicit calculations. In unstable homotopy theory, exponent theorems for the homotopy groups of spheres use a remarkable blend of theoretic and calculational techniques. There are tons of explicit calculations in ordinary and generalized cohomology in the past half century. Spin cobordism and its applications to curvature questions is an example of a blend of algebraic topology and differential geometry. I could go on for 100s of pages without pausing for breath, and none of these results use any model category theory, let alone $\infty$ categories. There is a subject out there, with real content.

Answer 2

It seems to me that $\infty$-categories are perfectly suited for a particular type of argument: namely to establish results in a homotopical context that would follow for purely formal reasons in a discrete context. There is quite a wide range of statements in algebraic geometry, representation theory etc which you know in a classical setting for purely abstract nonsense reasons, typically having to do with adjunctions, monadicity, commutation of limits with limits or colimits with colimits etc. and $\infty$-categories allow one to take such results and prove them in a derived setting. To put it in other words, if your "computation" is such that you can characterize the result by a universal property, then you can carry it out in the $\infty$-setting. On the other hand it seems to me that in this kind of mathematics many of the nontautological statements and real content tend to boil down to things like trying to pass a limit through a colimit. In this case you typically need some nonformal arguments and having models is likely inescapable.

These comments of course only reflect my limited experience with $\infty$-categories - luckily in my work with David Nadler in applying homotopical techniques to representation theoretic questions we have only needed formal statements, and $\infty$-categorical language made it quite easy to carry out the arguments, which would have been certainly impossible for me without it. The closest to a "calculation" we made in the $\infty$-context is in the paper Loop Spaces and Connections. We were able to prove the Hochschild-Kostant-Rosenberg theorem (identifying Hochschild homology of a commutative ring with symmetric algebra of differential forms) for any derived scheme (in characteristic zero) by this kind of abstract nonsense, but in order to show that the circle action on the former is "the same as" the de Rham differential (in the smooth case) was trickier - it's formal to see that the action map for the circle agrees with the de Rham differential but you still need to see all the higher structure of a circle action is uniquely determined (the action map only determines an action of the "free group" or James construction on the circle).. For this it appeared we'd need models until we saw that we could use a formal/structural argument (keeping track of the compatibility of everything with the action of the multiplicative group one finds there is no room for higher obstruction data). But I think this is a case of getting lucky in a sufficiently structured setting.. beyond that I doubt one can get away without models (forcing homotopical charlatans such as myself out of the business).

Answer 3

What a fun question!

I'd like to first mention a speculation on my part: When most people think of $\infty$-categories, the categories they think of probably exist in another formulation. For instance, the category of spaces (or Kan complexes), of chain complexes, of commutative DGAs, of spectra, et cetera, all have well-developed theory. And we're used to doing computations in these categories based on approaches that predate quasi-categories or Lurie's HTT or Higher Algebra. For such well-studied categories, I think you'll almost always find some pre-quasi-category (e.g., model categorical) "computation" that'll get the job done. There's no need to think of them as weak Kan complexes first.

(To compute a homotopy fiber of a map, for instance, we'll probably just replace the map by a standard fibration and compute its fiber thereafter. This is a silly example, but one in which I think working with a model-categorical framework is easier. This is much faster than proving that some over-category has a terminal object.)

But it would be a lie to say that most "computations" one needs to perform in an $\infty$-category can already be done in a non-quasi-category world. Here are two examples:

(1) If you're working with a new $\infty$-category for which someone hasn't done the model-categorical prep work for you. Depending completely on your math-path, you might come across (or define) a category which is most naturally defined as a weak Kan complex. One degree of separation away, you might define a category which is most naturally enriched over Kan complexes, and apply the simplicial nerve construction to obtain a quasi-category. But if you're just a tramp like me, suddenly face-to-face with a new category with some homotopical flavor, you may not have a natural candidate for a model structure, nor have any intuition for how to prove that something really is a "homotopy fiber" for some map, or more generally a homotopy (co)limit for a homotopy coherent diagram. (Whatever a "homotopy (co)limit" means for your category.)

With such a new category, all you a priori have is whatever led you to define this combinatorial gadget (a simplicially enriched category, or a quasi-category) and maybe some interpretations of your morphisms depending on what motivated your definition. So when trying to prove that some object is the homotopy fiber for some morphism in your quasi-category, it might be easiest for you to simply prove that an over-category has a terminal vertex.

I should admit that, while I imagine that examples like this will come up more and more, the only example I have in mind comes from joint work with David Nadler, where we compute kernels in a category we define, which happens to be most naturally a quasi-category.

(By the way, if someone has techniques that make it very easy to prove something is a limit of a diagram in a fibrant simplicial category, please post it as a comment to this post! I'd love to know more techniques.)

(2) If the diagrams you're working with are homotopy coherent but hard to make sense of at a level of strict commutativity. Another example I've come across is to prove that two homotopy coherent diagrams have the same homotopy colimit. Again because there is a concrete model for over/under categories in the $\infty$-categorical model, I could write down an equivalence between the over-categories associated to the two diagrams.

Summary:

If the non-$\infty$-categorical computations are made possible by model-categorical ideas (like knowing how to replace morphisms), then $\infty$-categorical computations are possible because you reduce your computations to simplicial-set ideas, often proving that a simplicial set is contractible or that it has a terminal vertex. In some mathematical universe both paths may amount to the same thing, but it's my impression that the former approach is hard to follow if you're working in a new category without model structure, and is also not as easy to work with when the higher homotopy coherences of your diagrams are subtle. (Please feel free to let me know if I'm mistaken on this point, I'd love to hear more views.) The quasi-category framework allows you to avoid much of that difficulty by passing the buck to the geometry of simplicial sets.

By the way, you also mentioned something about computing spaces of $k$-morphisms. I have no idea how to do such a thing in the $\infty$-category world, or in either world, really. (Please educate me if someone does know how.)

Answer 4

See my answer to the question "Do we still need model categories?" here.

Answer 5

I believe that many computations could be done in the context of $\infty$-categories given enough outside input. For example, I would expect that most of the spectral sequences arising in topology could be constructed in the context of $\infty$-categories (indeed there are some basic spectral sequences constructed in Lurie's Higher Algebra). However, knowing that they converge to the right thing, might require you to know very specific details about limits and colimits in your $\infty$-category. Since the existence of these limits and colimits is typically deduced from the existence of homotopy colimits and limits in some model category we are often forced to consider the constructions in some model. For the issues of convergence, I am thinking of the Eilenberg-Moore and unstable Adams spectral sequences.

Even if we can overcome these problems, we usually need a small stable of existing computations to get us started (e.g. $\pi_1 S^1\cong \mathbb{Z}$, the ring structure on $K(S^2)$, the group cohomology of cyclic groups,...). The $\infty$-category context is an unnatural place to do such computations. It is more natural to work in an actual category with actual point-set models for our objects that let us compute something from first principles.

When using $\infty$-categories that arise from taking the nerves of the subcategories of bifibrant objects in some model categories, we find, after unwinding the definitions, that many of our $\infty$-category constructions involve taking lots of cofibrant/fibrant replacements. Sometimes we don't need to take a bifibrant model for a computation and we could get by with a cofibrant model. Sometimes we have no concrete idea what the fibrant replacement does to our objects. This is particularly true when we have taken a left Bousfield localization of a model category (in the combinatorial case, this corresponds to presentable $\infty$-categories). In such a case, the $\infty$-category context (unnecessarily) forces us to lose control of our point-set models.