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.
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.
[3] The book "Nonabelian Algebraic Topology"[4] 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
{\displaystyle {\text{Ch}}_{\geq 0}({\textbf {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
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[5] (also called a nerve-realization context) where the data of this adjunction is determined by what's called a cosimplicial object[6]
{\displaystyle dk\colon \Delta ^{\text{op}}\to {\text{Ch}}_{\geq 0}({\textbf {Ab}})}
, and the adjunction then takes the form
{\displaystyle \Gamma =\mathrm {Lan} _{y}dk:{\text{Ch}}_{\geq 0}({\textbf {Ab}})\dashv s{\textbf {Ab}}:\mathrm {Lan} _{dk}y=N}
where we take the left Kan extension and
is the Yoneda embedding.
Given a simplicial abelian group
{\displaystyle A_{\bullet }\in {\text{Ob}}({\text{s}}{\textbf {Ab}})}
called the normalized chain complex (also called the Moore complex) with terms
These differentials are well defined because of the simplicial identity
Now, composing these differentials gives a commutative diagram
This composition is the zero map because of the simplicial identity
, hence the normalized chain complex is a chain complex in
{\displaystyle {\text{Ch}}_{\geq 0}({\textbf {Ab}})}
Because a simplicial abelian group is a functor
{\displaystyle A_{\bullet }\colon {\text{Ord}}\to {\textbf {Ab}}}
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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