homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
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, equipping 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.
simplicially enriched category
Discussion the context of (simplicial) model category theory :
Dan Quillen, chapter II, section 1 of Homotopical algebra, Lecture Notes in Mathematics 43, Springer-Verlag 1967, iv+156 pp.
Philip Hirschhorn, Section 9.1.1 of: Model Categories and Their Localizations, AMS Math. Survey and Monographs Vol 99 (2002) (ISBN:978-0-8218-4917-0, pdf toc, pdf)
Jacob Lurie, Section A.3 in: Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press 2009 (pup:8957, pdf)
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 also discussed in
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.