Formal $\mathrm{\infty }$-category theory of relative and simplicial categories, and more general enriched categories.

Several models of $(\mathrm{\infty },1)$-categories exist [3], making precise the idea of a category with a homotopy-coherent weak enrichment in spaces (considered up to weak homotopy equivalence). Two different axiomatizations of the homotopy theory of $(\mathrm{\infty },1)$-categories have been given by Toën in [5], and by Barwick and Schommer-Pries in [2]. Knowing that two models are equivalent in the sense that they present the same homotopy theory is not sufficient, however, to conclude that they produce “the same category theory” and that categorical results proven using one particular model can be assumed to apply to all the others. A breakthrough towards model independence has been achieved by Riehl and Verity in a series of works culminating in [4]. Their framework applies to various models of $(\mathrm{\infty },1)$-categories, namely quasicategories, complete Segal spaces, Segal categories, and saturated 1-trivial weak 1-complicial sets. There are two notable exceptions: simplicially enriched categories and relative categories [1]. In this talk, I will show how to fit these into the picture. These two models are important for a variety of reasons: many fundamental examples of $(\mathrm{\infty },1)$-categories arise most naturally as relative categories (for example, those underlying a model category) or as simplicially enriched categories (for example, the $\mathrm{\infty }$-category of spaces itself); moreover, they are a convenient setting for explicit computations via homotopy (co)limits. What matters for the purpose of developing $(\mathrm{\infty },1)$-category theory synthetically as in [4] is the existence of a structure called a proarrow equipment [6] on a suitably defined homotopy $2$-category obtained from a model; if two models produce equivalent proarrow equipments, then, in essence, they can be used interchangeably for doing $(\mathrm{\infty },1)$-category theory. The methods used in [4] to establish the existence and equivalence of the various proarrow equipments do not apply to simplicial categories and relative categories, hence one must proceed in an ad hoc way. In the final part of the talk I will outline how to generalize this result from simplicially enriched categories to categories enriched in a monoidal model category $(𝒱,\mu )$, thus producing a way to encode the theory of $\mathrm{\infty }$-categories enriched in (the underlying $\mathrm{\infty }$-category of) the model category $(𝒱,\mu )$.