Dold–Kan correspondence

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 ${\displaystyle n}$th homology group of a chain complex is the ${\displaystyle n}$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.)

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 ${\displaystyle K(A,n)}$.

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

${\displaystyle N:s{\textbf {Ab}}\to {\text{Ch}}_{\geq 0}({\textbf {Ab}})}$

and the second functor is the "simplicialization" functor

${\displaystyle \Gamma :{\text{Ch}}_{\geq 0}({\textbf {Ab}})\to s{\textbf {Ab}}}$

constructing a simplicial abelian group from a chain complex.

Normalized chain complex

Given a simplicial abelian group ${\displaystyle A_{\bullet }\in {\text{Ob}}({\text{s}}{\textbf {Ab}})}$ there is a chain complex ${\displaystyle NA_{\bullet }}$ called the normalized chain complex with terms

${\displaystyle NA_{n}=\bigcap _{i=0}^{n-1}\ker(d_{i})\subset A_{n}}$

and differentials given by

${\displaystyle NA_{n}\xrightarrow {(-1)^{n}d_{n}} NA_{n-1}}$

These differentials are well defined because of the simplicial identity

${\displaystyle d_{i}\circ d_{n}=d_{n-1}\circ d_{i}:A_{n}\to A_{n-2}}$

showing the image of ${\displaystyle d_{n}:NA_{n}\to A_{n-1}}$ is in the kernel of each ${\displaystyle d_{i}:NA_{n-1}\to NA_{n-2}}$. This is because the definition of ${\displaystyle NA_{n}}$ gives ${\displaystyle d_{i}(NA_{n})=0}$. Now, composing these differentials gives a commutative diagram

${\displaystyle NA_{n}\xrightarrow {(-1)^{n}d_{n}} NA_{n-1}\xrightarrow {(-1)^{n-1}d_{n-1}} NA_{n-2}}$

and the composition map ${\displaystyle (-1)^{n}(-1)^{n-1}d_{n-1}\circ d_{n}}$. This composition is the zero map because of the simplicial identity

${\displaystyle d_{n-1}\circ d_{n}=d_{n-1}\circ d_{n-1}}$

and the inclusion ${\displaystyle {\text{Im}}(d_{n})\subset NA_{n-1}}$, 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

${\displaystyle A_{\bullet }:{\text{Ord}}\to {\textbf {Ab}}}$

and morphisms ${\displaystyle A_{\bullet }\to B_{\bullet }}$ are given by natural transformations, meaning the maps of the simplicial identities still hold, the normalized chain complex construction is functorial.

References

• Goerss, Paul G.; Jardine, John F. (1999). Simplicial Homotopy Theory. Progress in Mathematics. Vol. 174. Basel, Boston, Berlin: Birkhäuser. ISBN 978-3-7643-6064-1.
• Lurie, J. "Higher Algebra" (PDF). last updated August 2017
• Mathew, Akhil. "The Dold–Kan correspondence" (PDF). Archived from the original (PDF) on 2016-09-13.
• Brown, Ronald; Higgins, Philip J.; Sivera, Rafael (2011). Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids. Tracts in Mathematics. Vol. 15. Zurich: European Mathematical Society. ISBN 978-3-03719-083-8.

Further reading

• Jacob Lurie, DAG-I

External links

• Dold-Kan correspondence at the nLab