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 homology group of a chain complex is the 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.^[2]

References

1. Goerss–Jardine 1999, Ch 3. Corollary 2.3
2. Lurie 2012, § 1.2.4.

• 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

Further reading

• J. Lurie, DAG-I

External links

• Dold-Kan correspondence in nLab