Paths and cylinders
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 sSet with
The simplicial identities satisfied by face and degeneracy maps as above are (whenever these maps are composable as indicated):
Relation to nilpotency of differentials
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 be the simplicial abelian group obtained form by forming degreewise the free abelian group on the set of -simplices, as discussed at chains on a simplicial set.
Then using these formal linear combinations we can sum up all the face maps into a single map:
The alternating face map differential in degree of the simplicial set is the linear map
defined on basis elements to be the alternating sum of the simplicial face maps:
This is the differential of the alternating face map complex of :
The simplicial identity def. 1 (1) implies that def. 2 indeed defines a differential in that .
By linearity, it is sufficient to check this on a basis element . There we compute as follows:
the first equality is (1);
the second is (1) together with the linearity of ;
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 by ;
the last one observes that the resulting two summands are negatives of each other.
For instance definition 1.1 in