Definition Simplicial Category [efr-0016]

The simplicial category is the category of finite, nonempty, totally ordered sets, and order-preserving functions, denoted \mathbb {\Delta }. We denote by [n] for n \geq 0 the ordered set \{0 < 1 < \dots < n\} of n+1 elements---every object on \mathbb {\Delta } is isomorphic to one of the form [n].

A simplicial object in a category \mathcal {C} is a functor \Delta ^\mathrm {op} \to \mathcal {C}.