For a simplicial set and an abelian group, the simplicial homology of is the chain homology of the chain complex corresponding under the Dold-Kan correspondence to the simplicial abelian group of -chains on : formal linear combinations of simplices in with coefficients in .
These abelian groups arrange to a simplicial abelian group
The alternating face map complex of this groups is called the complex of simplicial chains on
The simplicial homology of is the chain homology of the complex of simplicial chains:
where are the face maps of .
Since this has
4 non-degenerate vertices
6 non-degenerate edges
4 non-degenerate faces
the normalized chain complex of is of the form
By writing out the two non-trivial differentials, one can deduce explicitly that
The term simplicial homology is also used in the literature for the homology of polyhedral spaces, based on the theory of simplicial complexes. That homology is defined by first looking at a chain complex of simplicial chains on, say, a triangulation of a space, and then passing to the corresponding homology. The theory then proceeds by proving that the end result is independent of the triangulation used. The resulting homology theory is isomorphic to singular homology, but historically was the earlier theory.
A basic discussion is for instance around application 1.1.3 of
Homology for spaces is discussed in chapter 2 of
and this includes a discussion of the homology of simplicial complexes.