Simplicial Set

A simplicial set is a contravariant functor from the category of nonempty finite sets equipped with a total order and order-preserving maps to the category of sets. Equivalently, it consists of sets of -simplices with face and degeneracy maps satisfying the simplicial identities. Degeneracies permit repeated vertices, and different simplices can have the same boundary, unlike the more restrictive combinatorial data of a simplicial complex. The nerve of a category is an example, with -simplices given by strings of composable morphisms. Geometric realization associates a topological space to a simplicial set. The Kan-Quillen model category of simplicial sets models the homotopy theory of spaces.