Dold–Kan correspondence

Short description: Equivalence between the categories of chain complexes and simplicial abelian groups

In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence (named after Albrecht Dold and Daniel Kan) 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 [math]\displaystyle{ n }[/math]th homology group of a chain complex is the [math]\displaystyle{ n }[/math]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 all other degrees. Then the corresponding simplicial group is the Eilenberg–MacLane space [math]\displaystyle{ K(A, n) }[/math].

There is also an ∞-category-version of the Dold–Kan correspondence.

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.

Detailed construction

The Dold-Kan correspondence between the category sAb of simplicial abelian groups and the category Ch_≥0(Ab) of nonnegatively graded chain complexes can be constructed explicitly through a pair of functors^{pg 149} so that the compositions of these functors are naturally isomorphic to the respective identity functors. The first functor is the normalized chain complex functor

[math]\displaystyle{ N:s\textbf{Ab} \to \text{Ch}_{\geq 0}(\textbf{Ab}) }[/math]

and the second functor is the "simplicialization" functor

[math]\displaystyle{ \Gamma:\text{Ch}_{\geq 0}(\textbf{Ab}) \to s\textbf{Ab} }[/math]

constructing a simplicial abelian group from a chain complex.

Normalized chain complex

Given a simplicial abelian group [math]\displaystyle{ A_\bullet \in \text{Ob}(\text{s}\textbf{Ab}) }[/math] there is a chain complex [math]\displaystyle{ NA_\bullet }[/math] called the normalized chain complex with terms

[math]\displaystyle{ NA_n = \bigcap^{n-1}_{i=0}\ker(d_i) \subset A_n }[/math]

and differentials given by

[math]\displaystyle{ NA_n \xrightarrow{(-1)^nd_n} NA_{n-1} }[/math]

These differentials are well defined because of the simplicial identity

[math]\displaystyle{ d_i \circ d_n = d_{n-1}\circ d_i : A_n \to A_{n-2} }[/math]

showing the image of [math]\displaystyle{ d_n : NA_n \to A_{n-1} }[/math] is in the kernel of each [math]\displaystyle{ d_i:NA_{n-1} \to NA_{n-2} }[/math]. This is because the definition of [math]\displaystyle{ NA_n }[/math] gives [math]\displaystyle{ d_i(NA_n) = 0 }[/math]. Now, composing these differentials gives a commutative diagram

[math]\displaystyle{ NA_n \xrightarrow{(-1)^nd_n} NA_{n-1} \xrightarrow{(-1)^{n-1}d_{n-1}} NA_{n-2} }[/math]

and the composition map [math]\displaystyle{ (-1)^n(-1)^{n-1}d_{n-1}\circ d_n }[/math]. This composition is the zero map because of the simplicial identity

[math]\displaystyle{ d_{n-1}\circ d_n = d_{n-1}\circ d_{n-1} }[/math]

and the inclusion [math]\displaystyle{ \text{Im}(d_n) \subset NA_{n-1} }[/math], hence the normalized chain complex is a chain complex in [math]\displaystyle{ \text{Ch}_{\geq 0 }(\textbf{Ab}) }[/math]. Because a simplicial abelian group is a functor

[math]\displaystyle{ A_\bullet : \text{Ord} \to \textbf{Ab} }[/math]

and morphisms [math]\displaystyle{ A_\bullet \to B_\bullet }[/math] are given by natural transformations, meaning the maps of the simplicial identities still hold, the normalized chain complex construction is functorial.

References

• 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". http://people.fas.harvard.edu/~amathew/doldkan.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

• Jacob Lurie, DAG-I

External links

• Dold-Kan correspondence in nLab