The simplicial identities encode the relationships between the face and degeneracy maps in a simplicial object, in particular, in a simplicial set.
The simplicial identities are the duals to the simplicial relations of coface and codegeneracy maps described at simplex category:
Let $S \in$ sSet with
face maps $\partial_i : S_n \to S_{n-1}$ obtained by omitting the $i$th vertex;
degeneracy maps $s_i : S_n \to S_{n+1}$ obtained by repeating the $i$th vertex.
The simplicial identities satisfied by face and degeneracy maps as above are (whenever these maps are composable as indicated):
$\partial_i \circ \partial_j = \partial_{j-1} \circ \partial_i$ if $i \lt j$,
$s_i \circ s_j = s_j \circ s_{i-1}$ if $i \gt j$.
$\partial_i \circ s_j = \left\{ \array{ s_{j-1} \circ \partial_i & if \; i \lt j \\ id & if \; i = j \; or \; i = j+1 \\ s_j \circ \partial_{i-1} & if\; i \gt j+1 } \right.$
The simplicial identities of def. 1 can be understood as a non-abelian or “unstable” generalization of the identity
satisfied by differentials in chain complexes (in homological algebra).
Write $\mathbb{Z}[S]$ be the simplicial abelian group obtained form $S$ by forming degreewise the free abelian group on the set of $n$-simplices, as discussed at chains on a simplicial set.
Then using these formal linear combinations we can sum up all the $(n+1)$ face maps $\partial_i : S_n \to S_{n-1}$ into a single map:
The alternating face map differential in degree $n$ of the simplicial set $S$ is the linear map
defined on basis elements $\sigma \in S_n$ to be the alternating sum of the simplicial face maps:
This is the differential of the alternating face map complex of $S$:
The simplicial identity def. 1 (1) implies that def. 2 indeed defines a differential in that $\partial \circ \partial = 0$.
By linearity, it is sufficient to check this on a basis element $\sigma \in S_n$. There we compute as follows:
Here
the first equality is (1);
the second is (1) together with the linearity of $d$;
the third is obtained by decomposing the sum into two summands;
the fourth finally uses the simplicial identity def. 1 (1) in the first summand;
the fifth relabels the summation index $j$ by $j +1$;
the last one observes that the resulting two summands are negatives of each other.
For instance definition 1.1 in