Dold–Kan correspondence

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 ${\displaystyle K(A,n)}$ .

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

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

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. 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 ${\displaystyle dk:\Delta ^{\text{op}}\to {\text{Ch}}_{\geq 0}({\textbf {Ab}})}$ , 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 ${\displaystyle y}$  is the Yoneda embedding.

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 (also called the Moore 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.

• 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.

• Jacob Lurie, DAG-I

• Dold-Kan correspondence at the nLab

This category theory-related article is a stub. You can help Wikipedia by expanding it.

• v
• t
• e