Dold–Kan correspondence
In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence states that there is an equivalence between the category of chain complexes and the category of simplicial abelian groups. Moreover, under the equivalence, the th homology group of a chain complex is the th homotopy group of the corresponding simplicial abelian group, and a chain homotopy corresponds to a simplicial homotopy.
Example: Let C be a chain complex that has an abelian group A in degree n and zero in other degrees. Then the corresponding simplicial group is the Eilenberg–MacLane space.
There is also an ∞-category-version of a Dold–Kan correspondence.
The book "Nonabelian Algebraic Topology" cited below has a Section 14.8 on cubical versions of the Dold–Kan theorem, and relates them to a previous equivalence of categories between cubical omega-groupoids and crossed complexes, which is fundamental to the work of that book.
