Simplicial object

Summary

A simplicial object in a category $\mathcal{C}$ is a contravariant functor from the simplex category to $\mathcal{C}$. It is a sequence of objects $X_0, X_1, X_2, \dots$ of $\mathcal{C}$ with face and degeneracy maps between them that obey the simplicial identities. In $\mathcal{C} = \mathbf{Set}$ these are the simplicial sets, and in abelian groups they are the simplicial abelian groups of the Dold–Kan correspondence. They let one do homotopy theory with objects of $\mathcal{C}$.