Dold–Kan correspondence
In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence (named after Albrecht Dold and Daniel Kan) states that there is an equivalence between the category of (nonnegatively graded) chain complexes and the category of simplicial abelian groups. Moreover, under the equivalence, the formula_1th homology group of a chain complex is the formula_1th homotopy group of the corresponding simplicial abelian group, and a chain homotopy corresponds to a simplicial homotopy. (In fact, the correspondence preserves the respective standard model structures.) 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 formula_3. 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. Detailed construction. The Dold-Kan correspondence between simplicial abelian groups and chain complexes can be constructed explicitly through an adjunction of functors. The first functor is  the normalized chain complex functorformula_4and the second functor is the "simplicialization" functorformula_5constructing a simplicial abelian group from a chain complex. Normalized chain complex. Given a simplicial abelian group formula_6 there is a chain complex formula_7 called the normalized chain complex with termsformula_8and differentials given byformula_9These differentials are well defined because of the simplicial identityformula_10showing the image of formula_11 is in the kernel of each formula_12. This is because the definition of formula_13 gives formula_14. Now, composing these differentials gives a commutative diagramformula_15and the composition map formula_16. This composition is the zero map because of the simplicial identityformula_17and the inclusion formula_18 , hence the normalized chain complex is a chain complex in formula_19. Because a simplicial abelian group is a functorformula_20and morphisms formula_21 are given by natural transformations, meaning the maps of the simplicial identities still hold, the normalized chain complex construction is functorial.