mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a simplicially enriched category, is a

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

enriched over the category of

simplicial set In mathematics, a simplicial set is a sequence of sets with internal order structure ( abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs. Every simplicial set gives rise to a "n ...

s. Simplicially enriched categories are often also called, more ambiguously, simplicial categories; the latter term however also applies to simplicial objects in Cat (the category of

small Small means of insignificant size Size in general is the Magnitude (mathematics), magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to three geometrical measures: length, area, or ...

categories). Simplicially enriched categories can, however, be identified with simplicial objects in Cat whose object part is constant, or more precisely, all face and degeneracy maps are

bijective In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...

on objects. Simplicially enriched categories can model (∞, 1)-categories, but the dictionary has to be carefully built. Namely, many notions (limits, for example) are different from the limits in the sense of enriched category theory. The homotopy coherent nerve of a simplicially enriched category is a simplicial set that generalizes the nerve of a category.

References

* *

External links

* {{Category theory Category theory