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
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
there is a chain complex
called the normalized chain complex (also called the Moore complex) with terms
These differentials are well defined because of the simplicial identity
showing the image of
Now, composing these differentials gives a commutative diagram
and the composition map
This composition is the zero map because of the simplicial identity
, hence the normalized chain complex is a chain complex in
Because a simplicial abelian group is a functor
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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