Dold–Kan correspondence

In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence 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 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.)

Example: Let C be a chain complex that has an abelian group A in degree n and zero in other degrees. Then the corresponding simplicial group is the Eilenberg–MacLane space .

There is also an ∞-category-version of a 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.

References

• Goerss, P. G.; Jardine, J. F. (1999). Simplicial Homotopy Theory. Progress in Mathematics. 174. Basel, Boston, Berlin: Birkhäuser. ISBN 978-3-7643-6064-1. 
• J. Lurie, Higher Algebra, last updated August 2012
• A. Mathew, The Dold-Kan correspondence
• Brown, R.; Higgins, P.J.; Sivera, R. (2011). Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids. Tracts in Mathematics. 15. Zurich: European Mathematical Society. ISBN 978-3-03719-083-8. 

Further reading

• J. Lurie, DAG-I

External links

• Dold-Kan correspondence in nLab