equivalences in/of $(\infty,1)$-categories
A simplicially enriched category is a category with a simplicial set of morphisms between any two objects.
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 $(k-1)$-cell as a k-morphism. Therefore simplicially enriched categories may serves as models for ∞-categories. Precisely which notion of $\infty$-category depends on which extra structure and property one imposes.
For instance
requiring the hom-simplicial sets to be Kan complexes makes simplicially enriched categories a model for (∞,1)-categories;
similary, equipping the $sSet$-enriched category with the structure of a $sSet_{Quillen}$-enriched model category – a simplicial model category – makes it a model for an $(\infty,1)$-category.
This is discussed in more detail at relation between quasi-categories and simplicial categories.
on the other hand, eqipping the $sSet$-enriched category with the structure of an $sSet_{Joyal}$-enriched model category over the Joyal-model structure for quasi-categories makes it a model for an (∞,2)-category.
A simplicially enriched category is a category enriched over the cartesian monoidal category sSet of simplicial sets.
These $sSet$-enriched categories are sometimes, somewhat imprecisely, called just simplicial categories.
There is a related notion of simplicial groupoid with the added requirement that all arrows in the categories concerned are isomorphisms.
Since simplicial sets that are Kan complexes are an incarnation of ∞-groupoids, an $sSet$-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 $(\infty,1)$-categories is one of the central tools in homotopy coherent category theory.
Indeed, there is a model structure on simplicial categories whose fibrant objects are Kan-complex-enriched categories, and which is one model for the (∞,1)-category of (∞,1)-categories.
By a web of Quillen equivalences this is related to the other incarnations of $(\infty,1)$-categories. Notably to quasi-categories and complete Segal spaces. For more on this see
To every category with weak equivalences $(C,W)$ is associated its simplicial localization $L_W C$, which is an $sSet$-category with the property that its homotopy category of an (∞,1)-category coincides with the homotopy category $Ho_W(C)$.
There is a model structure on sSet-categories that presents the (∞,1)-category (∞,1)Cat.
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 $sSet$-categories. (ToënVezzosi). There is a notion of sSet-site $C$ that models the notion of (∞,1)-site and a model structure on sSet-enriched presheaves on $sSet$-sites that is a presentation for the ∞-stack (∞,1)-toposes on $C$.
See (∞,2)-category for the moment.
simplicially enriched category
In the context of model category theory, simplicially enriched categories (simplicial model categories) appear in
The original references on homotopy theory in terms of $sSet$-categories are
William Dwyer, Dan Kan, Simplicial localization of categories, J. Pure and Appl. Algebra 17 (1980), 267-284.
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 $(\infty,1)$-categories are discussed in some detail in section A.3 of
as well as in section 2 of
Homotopy coherent category theory on $sSet$-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 $sSet$-A-∞ categories.