Dold–Kan correspondence

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Short description: 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[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 nth homology group of a chain complex is the nth 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 all other degrees. Then the corresponding simplicial group is the Eilenberg–MacLane space K(A,n).

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.

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 functors[1]pg 149 so that the compositions of these functors are naturally isomorphic to the respective identity functors. The first functor is the normalized chain complex functor

N:sAbCh0(Ab)

and the second functor is the "simplicialization" functor

Γ:Ch0(Ab)sAb

constructing a simplicial abelian group from a chain complex.

Normalized chain complex

Given a simplicial abelian group

AOb(sAb)

there is a chain complex

NA

called the normalized chain complex with terms

NAn=i=0n1ker(di)An

and differentials given by

NAn(1)ndnNAn1

These differentials are well defined because of the simplicial identity

didn=dn1di:AnAn2

showing the image of

dn:NAnAn1

is in the kernel of each

di:NAn1NAn2

. This is because the definition of

NAn

gives

di(NAn)=0

. Now, composing these differentials gives a commutative diagram

NAn(1)ndnNAn1(1)n1dn1NAn2

and the composition map

(1)n(1)n1dn1dn

. This composition is the zero map because of the simplicial identity

dn1dn=dn1dn1

and the inclusion

Im(dn)NAn1

, hence the normalized chain complex is a chain complex in

Ch0(Ab)

. Because a simplicial abelian group is a functor

A:OrdAb

and morphisms

AB

are given by natural transformations, meaning the maps of the simplicial identities still hold, the normalized chain complex construction is functorial.

References

  1. 1.0 1.1 Goerss & Jardine (1999), Ch 3. Corollary 2.3
  2. Lurie, § 1.2.4.

Further reading