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.

Further reading

External links


This article is issued from Wikipedia - version of the Sunday, January 18, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.