Category Theory – What Does it Mean for a Model Category to Present a Higher Category

Question

A model category serves as an abstract/ axiomatic framework for homotopy theory. A higher category, in particular an $(\infty,1)$ category, I'm using the model of quasicategories, is a category with higher morphisms that satisfy all known equations in only a weaker sense, i.e. up to homotopy.

I've seen the term "a model category that presents an infinity category" or "this infinity category, or topos, is presented by a such and such model category" thrown around quite a bit but I am yet to find a precise definition.

My question is: What does it mean for a model category to present a higher category

The closest thing to an explicit definition I've found is in Emily Rhiel's paper From Model Categories to (Infinity,1)-categories. There one sees a known theorem that if a model category $\mathcal{M}$ is, not only simplicially enriched but, Kan enriched then its image under the homotopy coherent nerve $\mathfrak{N}$ is a quasicategory.

That's all well and good and it's probably one of the meanings of the phrase "a model category presents a higher category". But I am suspecting it's not the only one. For example when we say that "Rezk's model topoi present $(\infty,1)$-topoi" do we imply that model topoi are Kan enriched? Or that $(\infty,1)$-topoi lie in the essential image of $\mathfrak{N}$?

On the other hand, a Type Theoretic Model Topos is, in particular, a simplicially enriched model category which would be a step in the right direction to my understanding of the meaning of the aforementioned phrase.

Any help/ references are more than welcome. Thanks in advance!

Answer 1

The $\infty$-category presented by a model category is also called the underlying $\infty$-category of the model category, and there are several equivalent ways to describe it. The idea is that ''a homotopy theory'' is an $\infty$-category, and that every model category encodes a homotopy theory. Therefore, every model category $\mathcal{M}$ should give rise to an $\infty$-category $\mathcal{M}_\infty$ that ''is'' the homotopy theory which $\mathcal{M}$ encodes, and which as such only retains the homotopical data of the model category, throwing away non-homotopical notions such as cofibrations and fibrations.

To actually construct (a model of) $\mathcal{M}_\infty$, therefore, we only want to remember the weak equivalences of the model category. As a first step, then, we map a model category $\mathcal{M}$ to its underlying relative category (also called ''category with weak equivalences'', but there are several meanings to that term) $(\mathcal{M},\mathcal{W})$, where $\mathcal{W}$ denotes the class of weak equivalences in $\mathcal{M}$. Now, you can for instance proceed in the following two ways:

1. Via the hammock localization functor $L^H\colon\mathsf{RelCat}\to\mathsf{sCat}$ we can turn $(\mathcal{M},\mathcal{W})$ into a simplicial category $L^H(\mathcal{M},\mathcal{W})$. We apply the derived homotopy coherent nerve functor $\mathbf{R}N^\mathrm{coh}\colon\mathsf{sCat}\to\mathsf{sSet}_\mathrm{Joyal}$, using the Bergner model structure on $\mathsf{sCat}$. The resulting quasicategory $\mathbf{R}N^\mathrm{coh}L^H(\mathcal{M},\mathcal{W})$ is a particular model for the underlying $\infty$-category of $\mathcal{M}$.
2. There is a right Quillen equivalence $N_\xi\colon\mathsf{RelCat}\to\mathsf{CSS}$, where $\mathsf{CSS}$ is the complete Segal space model structure on simplicial spaces (see Barwick--Kan, Relative categories: Another model for the homotopy theory of homotopy theories). (Edit: I made a mistake here with which functor exactly to use, but I fixed it now.) There is a further right Quillen equivalence $\mathrm{ev}_{(-)}(0)\colon\mathsf{CSS}\to\mathsf{sSet}_\mathrm{Joyal}$ (which is evaluation of the sequence of spaces at their 0-simplices). So, the right derived functor $\mathbf{R}(\mathrm{ev}_{(-)}(0)\circ N_\xi)$ maps $(\mathcal{M},\mathcal{W})$ to a quasicategory $\mathbf{R}(\mathrm{ev}_{(-)}(0)\circ N_\xi)(\mathcal{M},\mathcal{W})$, which also models the underlying $\infty$-category of $\mathcal{M}$.

These two constructions yield equivalent $\infty$-categories, and in fact the constructions themselves are naturally equivalent in the sense that their underlying $\infty$-functors $\mathsf{RelCat}_\infty\to(\mathsf{sSet}_\mathrm{Joyal})_\infty$ are naturally equivalent. This follows from Toën's result on automorphisms of $\mathsf{Cat}_\infty$. If $\mathcal{M}$ is a simplicial model category, then a third (equivalent) way to describe $\mathcal{M}_\infty$ is as the homotopy coherent nerve $N^\mathrm{coh}(\mathcal{M}^\circ)$ of the simplicial category $\mathcal{M}^{\circ}$ of fibrant-cofibrant objects in $\mathcal{M}$. This is what Lurie usually works with in Higher Topos Theory.

So, you can now put as definition the following: a model category $\mathcal{M}$ presents an $\infty$-category $\mathcal{C}$ if you supply an equivalence $\mathcal{M}_\infty\simeq\mathcal{C}$ of $\infty$-categories, where $\mathcal{M}_\infty$ may be taken to be any of the equivalent $\infty$-categories above. If $\mathcal{M}$ is a monoidal model category, and $\mathcal{C}$ is a monoidal $\infty$-category, you probably want to add the requirement that the equivalence $\mathcal{M}_\infty\simeq\mathcal{C}$ is a monoidal equivalence.

There is a long list of results that tell you which extra structure on model categories carries over to extra structure on $\infty$-categories presented by the model categories. You can therefore ask, given an $\infty$-category $\mathcal{C}$ with some extra interesting structure, for this structure to also exist in a model category $\mathcal{M}$ that presents $\mathcal{C}$ (assuming such an $\mathcal{M}$ exists), and you can ask for this structure to be preserved by the equivalence $\mathcal{M}_\infty\simeq\mathcal{C}$. This is often (implicitly) done.

We like presentations for multiple reasons, among which that you can sometimes explicitly compute things in $\infty$-categories by looking at a convenient model category that presents them. In some sense, having a model-categorical presentation is similar to working in a basis for a vector space: it is not intrinsic to the underlying object you are interested in, but can be very helpful. However, note that an $\infty$-category automatically satisfies quite strong properties if it can be presented by a model category (such as being complete and cocomplete), so in particular not all $\infty$-categories admit such a presentation.