nLab simplicially enriched category

Idea

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.

Definition

Definition

A simplicially enriched category is a category enriched over the cartesian monoidal category sSet of simplicial sets.

Remark

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.

As models for $(\infty,1)$-categories

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

• relation between quasi-categories and simplicial categories.

Properties

Simplicial localization

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 $Ho_W(C)$.

Model structure

There is a model structure on sSet-categories that presents the (∞,1)-category (∞,1)Cat.

Homotopy Kan extension

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.

Presentation of $(\infty,1)$-topos theory

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 $C$.

As models for $(\infty,2)$-categories

See (∞,2)-category.

Related concepts

• simplicial category

  • simplicially enriched category

  • simplicial object in Cat

• simplicial groupoid

• relation between quasi-categories and simplicial categories

• topologically enriched category

References

See also:

• Wikipedia, Simplicially enriched category

Discussion the context of (simplicial) model category theory:

• Daniel Quillen, II §1 of: Homotopical Algebra, Lecture Notes in Mathematics 43, Springer (1967) [doi:10.1007/BFb0097438]

• 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)

• Bertrand Toën, Gabriele Vezzosi, Section 2 of: Topos theory, Advances in Mathematics 193 2 (2005) 257-372 [arXiv:0207028, doi:10.1016/j.aim.2004.05.004]

• Jacob Lurie, Section A.3 in: Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press 2009 (pup:8957, pdf)

• Emily Riehl, §3.6 in: Categorical Homotopy Theory, Cambridge University Press (2014) [doi:10.1017/CBO9781107261457, pdf]

The original references in the context of simplicial localization:

• William Dwyer, Daniel Kan, Simplicial localizations of categories, J. Pure Appl. Algebra 17 3 (1980), 267-284 [doi:10.1016/0022-4049(80)90049-3]

• William Dwyer, Daniel Kan, Equivalences between homotopy theories of diagrams, in: Algebraic topology and algebraic K-theory, Ann. of Math. Stud. 113, Princeton University Pr

  Simplicially enriched categories as models for $(\infty,1)$-categories are also discussed in

Homotopy coherent category theory on $sSet$-categories is discussed in

• Jean-Marc Cordier and Timothy Porter, Homotopy Coherent Category Theory, Trans. Amer. Math. Soc. 349 (1997) 1-54, (pdf)2-2.pdf))

which describes resolutions of the simplicial functor categories between two simplicial categories and

• Michael Batanin, Homotopy coherent category theory and $A_\infty$-structures in monoidal categories (pdf)

which shows that these resolved functor categories are in fact $sSet$-A-∞ categories.