What is a model category from an $\infty$ point of view?

Question

A number of different models for $\infty$ categories can seen to have analogs in $\infty$-category theory. For example:

• Quasicategories: $\Delta \subseteq \mathrm{Cat}_{(\infty, 1)}$ is (?) a dense generator, so we can see $\infty$-categories as presheaves on $\Delta$
• Simplicially enriched categories: Simplicial enrichment is a proxy for enrichment in $\mathrm{Cat}_{(\infty, 0)}$
• Simplicial categories: These can be seen as the objects in an intermediate step towards adjoining simplicially-indexed colimits to $\mathrm{Cat}_{(1,1)}$
• Complete segal spaces: These look to be inspired by models in $\mathrm{Cat}_{(\infty, 0)}$ of the finite limit sketch defining categories
• Relative categories: A pair $(C,W)$ is a proxy for the localization $C[W^{-1}]$.

However, I don't know of a suitable interpretation of a model structure.

Every model category is in particular a relative category, so that's how they correspond to $\infty$-categories. But that doesn't involve the fibrations and cofibrations.

My question is if there is some notion in $\infty$-category theory that is analogous to equipping a relative category with a choice of classes of fibrations and cofibrations.

Answer 1

This is just a long comment, which does not really address the main question. Instead, I give few examples where fibrations are an additional rigidity structure allowing "homotopical commutativity".

In all of the simplicial models for $(\infty, 1)$-categories that you listed, only the fibrant objects of the respective model category structure behave as "categories weakly enriched in $\infty$-groupoids". Inside this categorical framework (if you want, formal 2-categorical), there is a notion of localisation $C[W^{-1}]$ of an $\infty$-category $C$ with respect to a sub-$\infty$-category $W$. Given a relative category $(C, W)$, it is a theorem that the $\infty$-localisation $N(C)[N(W)^{-1}]$ models the $\infty$-category associated to $(C, W)$ (see for instance the appendix of Brown categories and bicategories of Geoffroy Horel). However if $(C, W)$ is not fibrant, nor will be its Rezk's relative nerve, i.e. $N^R(C, W)$ is not really an $\infty$-category.

Suppose now your relative category $(C, W)$ has an additional structure of fibrantions that interacts gently with the weak equivalences (for instance, a model category), let's call such a structure a fibration category. Lennart Meier showed in Fibration categories are fibrant relative categories that any such relative category is fibrant in the Barwick-Kan model category structure and thus $N^R(C, W)$ is an $\infty$-category modelling the $\infty$-categorical localisation $N(C)[N(W)^{-1}]$.

This is just the reflection of something happening entirely in the $\infty$-categorical world. Indeed, in Chapter 7 of Higher categories and homotopical algebra Denis-Charles Cisinski studies the $\infty$-localisation theory of fibration $\infty$-categories. Within this framework, we have the beautiful feature that the $\infty$-localisation of a fibration $\infty$-category has finite $\infty$-limits (see Section 7.5). Moreover the identification of homotopy limits and $\infty$-limits can be stated as an elegant theorem about an equivalence of $\infty$-categories (see Theorem 7.9.8 and Remark 7.9.10).

The longer story tells that fibration $\infty$-categories are closed by exponentiation, that is if $(C, W)$ is a fibration $\infty$-category and $I$ is an $\infty$-category, then $\underline{\text{Hom}}(I, C)$ is still a fibration $\infty$-category, with fiber-wise weak equivalences $W_I$. If $(C, W)$ is just (the nerve of) a regular fibration category and $I$ (the nerve of) a small category, then the $\infty$-localisation $\text{Fun}(I, C)[W_I^{-1}]$ is equivalent to the $\infty$-functor category $\underline{\text{Hom}}(I, C[W^{-1}])$. (Notice that $(\text{Fun}(I, C), W_I)$ is a fibrant relative category!)

In the case where the fibration $\infty$-category is actually a model $\infty$-category, the theory has been studied by Aaron Mazel-Gee, as Dmitri Pavlov says in his nice answer.

Answer 2

One can make a reasonable claim that the analog of model categories in the realm of ∞-categories are model ∞-categories, see, for instance, http://arxiv.org/abs/1412.8411 and other papers by Aaron Mazel-Gee.

Relative categories encode an ∞-categories, whereas the additional data of a model structure can be seen as encoding (say) choices of limit-colimit diagrams for which limits and colimits commute, or (more generally) choices of left/right adjoint functors that preserve some specific limits/colimits. Different choices of model structures produce different commuting limits/colimits. This is explained in more detail here: Why do we need model categories?

In the world of ∞-categories one still needs to commute ∞-limits and ∞-colimits that need not commute in general, or commute an ∞-limit past a left adjoint functor, etc. This is what model ∞-structures accomplish. Thus model ∞-categories perform the same role in the realm of ∞-categories as model categories do in the realm of 1-categories.