In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence (named after Albrecht Dold and Daniel Kan) states [1] 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 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. (In fact, the correspondence preserves the respective standard model structures.) The correspondence is an example of the nerve and realization paradigm.
There is also an ∞-category-version of the Dold–Kan correspondence. [2]
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.
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 .
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 functors [1] pg 149 so that these functors form an equivalence of categories. The first functor is the normalized chain complex functor
and the second functor is the "simplicialization" functor
constructing 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 [3] (also called a nerve-realization context) where the data of this adjunction is determined by what's called a cosimplicial object , and the adjunction then takes the form
where we take the left Kan extension and is the Yoneda embedding.
Given a simplicial abelian group there is a chain complex called the normalized chain complex (also called the Moore complex) with terms
and differentials given by
These differentials are well defined because of the simplicial identity
showing the image of is in the kernel of each . This is because the definition of gives . Now, composing these differentials gives a commutative diagram
and the composition map . This composition is the zero map because of the simplicial identity
and the inclusion , hence the normalized chain complex is a chain complex in . Because a simplicial abelian group is a functor
and morphisms are given by natural transformations, meaning the maps of the simplicial identities still hold, the normalized chain complex construction is functorial.
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In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of the next. Associated to a chain complex is its homology, which describes how the images are included in the kernels.
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In mathematics, more precisely, in the theory of simplicial sets, a simplicial group is a simplicial object in the category of groups. Similarly, a simplicial abelian group is a simplicial object in the category of abelian groups. A simplicial group is a Kan complex. The Dold–Kan correspondence says that a simplicial abelian group may be identified with a chain complex. In fact it can be shown that any simplicial abelian group is non-canonically homotopy equivalent to a product of Eilenberg–MacLane spaces,
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