Under the Dold-Kan correspondence, ∞-groupoids with strict abelian group structure (modeled by Kan complexes that are simplicial abelian groups) are identified with non-negatively graded chain complexes of abelian groups
The homology groups of a chain complex of abelian groups are the image under this identification of the homotopy groups of the corresponding ∞-groupoids. More details on this are at chain homology and cohomology.
So at least for the case of chain complexes of abelian groups we have the slogan
Of course historically the development of concepts was precisely the oppposite: chain homology is an old fundamental concept in homological algebra that is simpler to deal with than simplicial homotopy groups. The computational simplification for chain complexes is what makes the Dold-Kan correspondence useful after all.
But conceptually it is useful to understand homology as a special kind of homotopy. This is maybe most vivid in the dual picture: cohomology derives its name from that fact that chain homology and cohomology are dual concepts. But later generalizations of cohomology to generalized (Eilenberg-Steenrod) cohomology and further to nonabelian cohomology showed that the restricted notion of homology is an insufficient dual model for cohomology: what cohomology is really dual to is the more general concept of homotopy. More on this is at cohomotopy and Eckmann-Hilton duality.
Let be an abelian category and let
the homology of in degree is the object
In the special case that is the category of abelian groups, or of vector spaces, this definition reduces to the more familiar simpler statement:
the -th homology group of the chain complex is the quotient group
For more see generalized homology.
The relation between homology, cohomology and homotopy:
|category theory||covariant hom||contravariant hom||tensor product|
|enriched category theory||end||end||coend|
|homotopy theory||derived hom space||cocycles||derived tensor product|
The ingredients of homology and cohomology: