# What does it mean for a model category to present a higher category

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!

• @MarianoSuárez-Álvarez Apologies but I don't see why that is.. I guess my reddit habits got the better of me.. what about How to choose a good title Nov 17, 2023 at 9:53
• @MarianoSuárez-Álvarez Right. I've edited my question. Hopefully this is clearer. Thanks for your advice. Consider giving it in a more constructive manner in the future. Nov 17, 2023 at 11:36

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.

• Hi, suppose that I have background in model categories but know just a little about $\infty$-categories, is there any introductory work (maybe notes) rather than Lurie's book so that I can quickly grasp some ideas of this concept? Nov 17, 2023 at 14:51
• If you're only interested in statements and the big picture, and not in proofs, maybe the first chapter of Fabian Hebestreit's lecture notes on algebraic and Hermitian K-theory (typed by Ferdinand Wagner) work for you. It doesn't really need model categories at all. Unfortunately, I don't know any books or notes that quickly allow you to be comfortable with all the technicalities and proofs. But it should be possible to learn to manipulate $\infty$-categories without knowing all the technicalities. (cont.) Nov 17, 2023 at 16:05
• (cont.) As for other sources for the actual theory, you can always read parts of Lurie's book, but there are also good notes by Charles Rezk, a book by Markus Land (which is far more elementary than Lurie, and has exercises), lecture notes by Hebestreit, and notes by Mazel-Gee. My personal preference goes to a combination of Rezk's notes with interesting (not too technical) parts of HTT. Nov 17, 2023 at 16:08
• Hi Daniël, I am familiar with model categories and I have a feeling that most of the statements of model categories carry over to their underlying $\infty$-categories and nowadays, people prefer $\infty$-categories so it urges me to learn it; anyway, there maybe no royal road but still thanks for your recommendations. Nov 18, 2023 at 15:01
• Hi, can you provide a reference for the equivalence between the three definitions? In particular, I wish to compare some constructions that mainly involve 1) and 3). Mar 4 at 9:21