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 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 $n$th homology group of a chain complex is the $n$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 other degrees. Then the corresponding simplicial group is the Eilenberg–MacLane space $K(A,n)$.

There is also an ∞-category-version of a Dold–Kan correspondence.

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, 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).
• 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