Simplicial set

Summary

A simplicial set is a presheaf on the simplex category, that is, a simplicial object in the category of sets. It records a set $X_n$ of $n$-simplices for each $n$, together with face and degeneracy maps. Simplicial sets are a combinatorial model for topological spaces up to homotopy, and they extend simplicial complexes by allowing degenerate and repeated simplices. Through the nerve they also contain all small categories, which makes them the basis of quasi-categories.