Editing Simplex category (section)

==Formal definition==

The '''simplex category''' is usually denoted by <math>\Delta</math>. There are several equivalent descriptions of this category. <math>\Delta</math> can be described as the category of ''non-empty finite ordinals'' as objects, thought of as totally ordered sets, and ''(non-strictly) order-preserving functions'' as [[morphisms]]. The objects are commonly denoted <math> [n] = \{0, 1, \dots, n\} </math> (so that <math> [n] </math> is the ordinal <math> n+1 </math>). The category is generated by coface and codegeneracy  maps, which amount to inserting or deleting elements of the orderings. (See [[simplicial set]] for relations of these maps.)

A [[simplicial object]] is a [[Presheaf (category theory)|presheaf]] on <math>\Delta</math>, that is a contravariant functor from <math>\Delta</math> to another category. For instance, [[simplicial set]]s are contravariant with the codomain category being the category of sets. A '''cosimplicial object''' is defined similarly as a covariant functor originating from <math>\Delta</math>.