Simplicial object

Remarks

Equivalently, a simplicial object is a family of objects $X_n$, $n \ge 0$, with face maps $d_i \colon X_n \to X_{n-1}$ and degeneracy maps $s_i \colon X_n \to X_{n+1}$ for $0 \le i \le n$, subject to the simplicial identities: $$d_i d_j = d_{j-1} d_i \ (i < j), \qquad s_i s_j = s_{j+1} s_i \ (i \le j),$$ $$d_i s_j = s_{j-1} d_i \ (i < j), \qquad d_j s_j = d_{j+1} s_j = \id, \qquad d_i s_j = s_j d_{i-1} \ (i > j+1).$$ Here $d_i = X(\delta_i)$ and $s_i = X(\sigma_i)$ for the face and degeneracy maps $\delta_i$, $\sigma_i$ of $\Delta$. Example: for an object $A$, the constant functor with $X_n = A$ and every $d_i$, $s_i$ equal to $\id_A$ satisfies all identities, so it is a simplicial object. Non-example: a covariant functor $\Delta \to \mathcal{C}$ is not a simplicial object, because its maps go from $X_{n-1}$ to $X_n$ for faces; such functors are called cosimplicial objects. Simplicial objects in $\mathbf{Set}$ are simplicial sets.