One may think of the 1-cells in a hom-simplicial set as a 2-morphism, the 2-cells as a 3-morphism and generally a -cell as a k-morphism. Therefore simplicially enriched categories may serves as models for ∞-categories. Precisely which notion of -category depends on which extra structure and property one imposes.
This is discussed in more detail at relation between quasi-categories and simplicial categories.
These -enriched categories are sometimes, somewhat imprecisely, called just simplicial categories.
Since simplicial sets that are Kan complexes are an incarnation of ∞-groupoids, an -category all whose hom-objects happen to be Kan complexes may be regarded as a category enriched in ∞-groupoids. By the logic of (n,r)-category theory this should be a model for an (∞,1)-category.
Treating simplicial categories this way as models for -categories is one of the central tools in homotopy coherent category theory.
To every category with weak equivalences is associated its simplicial localization , which is an -category with the property that its homotopy category of an (∞,1)-category coincides with the homotopy category .
The notion of homotopy Kan extension and hence in particular that of homotopy limit and homotopy colimit has a direct formulation in terms of Kan-complex-enriched categories. See homotopy Kan extension for more..
All of (∞,1)-topos theory can be modeled in terms of -categories. (ToënVezzosi). There is a notion of sSet-site that models the notion of (∞,1)-site and a model structure on sSet-enriched presheaves on -sites that is a presentation for the ∞-stack (∞,1)-toposes on .
See (∞,2)-category for the moment.
simplicially enriched category
The original references on homotopy theory in terms of -categories are
William Dwyer, Dan Kan, Equivalences between homotopy theories of diagrams , in Algebraic topology and algebraic K-theory, Annals of Math. Studies 113, Princeton University Press, Princeton, 1987, 180-205.
Simplicially enriched categories as models for -categories are discussed in some detail in section A.3 of
as well as in section 2 of
Homotopy coherent category theory on -categories is discussed in
which describes resolutions of the simplicial functor categories between two simplicial categories and
which shows that these resolved functor categories are in fact -A-∞ categories.