Weak composition rule for simplicial categories

Question

Informally, an $\infty$-category should be the following data:

• A collection of objects
• A space of morphisms between any two objects
• Weak associativity rules: Coherent homotopies between all of the different ways to compose morphisms

As far as I understand, simplicially enriched categories can be used as a somewhat strict version model for that: The mapping simplicial-sets are indeed models for mapping spaces (by that I mean a simplicial category is fibrant if the mapping simplicial-sets are Kan complexes), but the composition rule should still be strict.

Is there a model that spells out all the details required for defining an $\infty$-category using the same ingredients as in a simplicial category but with weak composition rules instead?

Answer 1

The most obvious approach is to consider simplicial $\def\Ai{{\sf A}_∞}\Ai$-categories, where $\Ai$ denotes a nonsymmetric operad in simplicial sets that is weakly equivalent to the terminal operad, i.e., the associative operad.

In such a structure, instead of the usual compositions we have maps $$\Ai(n)⨯C(X_{n-1},X_n)⨯⋯⨯C(X_0,X_1)→C(X_0,X_n)$$ that satisfy the usual identities for algebras over operads.

As it turns out, the resulting notion is in some sense equivalent to the usual simplicial categories: every simplicial $\Ai$-category can be rectified to a simplicial category by Proposition 9.2.4 in arXiv:1410.5675v3.

Other generalizations that extend simplicial categories using less obvious notions of coherence for composition include Segal categories and complete Segal spaces. All of these are also equivalent to simplicial categories. The book by Bergner (The Homotopy Theory of (∞,1)-Categories) contains an exposition of these equivalences.