Dold–Kan correspondence

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

References

• 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