Simplicial Set - The Standard n-simplex and The Simplex Category

The Standard n-simplex and The Simplex Category

Categorically, the standard n-simplex, denoted Δn, is the functor hom(-, n) where n denotes the string 0 → 1 → ... → n of the first (n + 1) nonnegative integers and the homset is taken in the category Δ. In many texts, it is written instead as hom(n,-) where the homset is understood to be in the opposite category Δop.

The geometric realization |Δn| is just defined to be the standard topological n-simplex in general position given by

By the Yoneda lemma, the n-simplices of a simplicial set X are classified by natural transformations in hom(Δn, X). The n-simplices of X are then collectively denoted by X_n. Furthermore, there is a simplex category, denoted by whose objects are maps (i.e. natural transformations) Δn → X and whose morphisms are natural transformations Δn → Δm over X arising from maps n → m in Δ. The following isomorphism shows that a simplicial set X is a colimit of its simplices:

where the colimit is taken over the simplex category of X.