Equivalence between the categories of chain complexes and simplicial abelian groups
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.) The correspondence is an example of the . 
There is also an ∞-category-version of the 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.
Examples.
For a chain complex "C" that has an abelian group "A" in degree "n" and zero in all other degrees, the corresponding simplicial group is the Eilenberg–MacLane space formula_3.
Detailed construction.
The Dold-Kan correspondence between the category sAb of simplicial abelian groups and the category Ch≥0(Ab) of nonnegatively graded chain complexes can be constructed explicitly through a pair of functorspg 149 so that these functors form an equivalence of categories. 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. The formed equivalence is an instance of a special type of adjunction, called the nerve-realization paradigm (also called a nerve-realization context) where the data of this adjunction is determined by what's called a cosimplicial object formula_6, and the adjunction then takes the form
formula_7
where we take the left Kan extension and formula_8 is the Yoneda embedding.
Normalized chain complex.
Given a simplicial abelian group formula_9 there is a chain complex formula_10 called the normalized chain complex (also called the Moore complex) with termsformula_11and differentials given byformula_12These differentials are well defined because of the simplicial identityformula_13showing the image of formula_14 is in the kernel of each formula_15. This is because the definition of formula_16 gives formula_17.
Now, composing these differentials gives a commutative diagramformula_18and the composition map formula_19. This composition is the zero map because of the simplicial identityformula_20and the inclusion formula_21, hence the normalized chain complex is a chain complex in formula_22. Because a simplicial abelian group is a functorformula_23and morphisms formula_24 are given by natural transformations, meaning the maps of the simplicial identities still hold, the normalized chain complex construction is functorial.
References.
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