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?



Weak composition rule for simplicial categories
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](https://arxiv.org/abs/1410.5675v3).


Other generalizations that extend simplicial categories using less obvious notions of coherence for composition
include [Segal categories](https://ncatlab.org/nlab/show/Segal+category) and [complete Segal spaces](https://ncatlab.org/nlab/show/complete+Segal+space).
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.


