# 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 .

There is also an ∞-category-version of a Dold–Kan correspondence.^{[2]}

## References

- ↑ Goerss–Jardine 1999, Ch 3. Corollary 2.3
- ↑ 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