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[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 th homology group of a chain complex is the 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 .

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

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
  1. Paul Goerss and Rick Jardine (1999, Ch 3. Corollary 2.3)
  2. Lurie 2012, § 1.2.4.
  • Goerss, Paul G.; Jardine, John 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 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. 15. Zurich: European Mathematical Society. ISBN 978-3-03719-083-8.

Further reading
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