The category theory of $(\infty, 1)$-categories

Question

There are many proposed models for the theory of $(\infty, 1)$-categories and it has now been shown that many of these theories have Quillen-equivalent model categories, i.e. that they are equivalent as homotopy theories. But what about as category theories?

For example, suppose I have a homotopical category $\mathcal{C}$ and a homotopically initial object $A$ in $\mathcal{C}$, in the sense of [Dwyer, Kan, Hirschhorn, and Smith, Homotopy limit functors on model categories and homotopical categories]. That means, there are endofunctors $F$ and $G$ on $\mathcal{C}$ such that $\Delta A$ is weakly equivalent to $F$, $G$ is weakly equivalent to $\textrm{id}_{\mathcal{C}}$, and there is a natural transformation $\alpha : F \Rightarrow G$ such that $\alpha_A : F A \to G A$ is a weak equivalence. It is straightforward to show that $A$ descends to an initial object in $\operatorname{Ho} \mathcal{C}$, but what about in, say, the hammock localisation $L^H \mathcal{C}$ – is the hom-space $L^H \mathcal{C}(A, Z)$ contractible for every object $Z$ in $\mathcal{C}$?

More generally, how are the notions of homotopy limit/colimit, adjunction, Kan extension etc. in the different theories (where defined) related? Pointers to relevant literature would be much appreciated.

Answer 1

Browsing through the old unanswered questions, I've come across this one, which happily can be partially answered now by the work of Riehl and Verity (Zhen Lin will be aware of this, which is why I'll CW it, but the average mathematician using the site might not be).

The first step is to identify what "a category theory" is. In ordinary category theory, we have categories, functors, bimodules, and natural transformations. Perhaps the term "bimodule" (or "profunctor" or "distributor") is unfamiliar in this context, but a bimodule from $C$ to $D$ is just a functor $C^\mathrm{op} \times D \to \mathsf{Set}$, and they compose via a coend, analogously to tensoring ordinary bimodules between rings. Categories and functors form a 2-category; so do categories and bimodules. The relationship is that functors are included into bimodules, by sending the functor $F: C \to D$ to the representable bimodule $D(1,F)$, and moreover this bimodule has a right adjoint given by the corepresentable bimodule $D(F,1)$. Such a structure is called a proarrow equipment. It's essential to have the bimodules in addition to the functors in order to express things like what it means to be a pointwise Kan extension.

This much was known. The next step is to try to extract a proarrow equipment from an arbitrary model of $\infty$-categories, and then to check whether various notions such as (co)limits and Kan extensions agree with the notions that have already been defined in ad hoc ways in various models. The idea of course is that the role of "functors" should be played by appropriate $\infty$-functors and the role of "bimodules" should be played by appropriate 2-sided $\infty$-presheaves. Riehl and Verity do this in a uniform manner for some (but not all) models by setting up an axiomatic framework of $\infty$-cosmoi. From an $\infty$-cosmos, one can extract a homotopy 2-category of $\infty$-categories and $\infty$-functors, and then the bimodules are defined using certain weak comma objects which are guaranteed to exist by the $\infty$-cosmos axioms. This is similar to a familiar construction using comma objects, but Riehl and Verity had to weaken the universal property of the comma object in a novel way in order for it to exist in this setting. And it turns out that all the categorical notions formulated this way agree with those which had already been formulated in more ad hoc ways.

Since the proarrow equipment associated to an $\infty$-cosmos is defined in a uniform way, Riehl and Verity are then able to formulate simple criteria for equivalence of these equipments. And they find that all of the models of $\infty$-categories which actually form $\infty$-cosmoi are indeed equivalent. This includes those models which form model categories which are simplicial with respect to the Joyal model structure, including quasicategories, complete Segal spaces, naturally marked simplicial sets, and Segal categories. But, for example, simplical categories and homotopical categories are not among the models which can be compared using the $\infty$-cosmos framework. But it is still natural to approach these comparisons by extracting proarrow equipments from these models.

I've perhaps overstated the role of the proarrow equipment in this story. I think Riehl and Verity regard it as an open question to determine just how much "category theory" is correctly encoded in terms of the proarrow equipment. (I've also glossed over the fact that Riehl and Verity use a slightly weaker, and more-complicated-to-define, but to my mind no less natural structure called a virtual equipment.) But I suspect that even concepts which can't be expressed in terms of proarrow equipments will be expressible in terms of $\infty$-cosmoi, using the addtional structure available which I haven't gone into, and Riehl and Verity have actually constructed equivalences at the $\infty$-cosmos level, so the models they consider should be equivalent in a strong sense.

References: Notes and video from Riehl's YTM lectures. The notes contain fuller references, laid out in the introduction.