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{{harvp|Goerss|Jardine|1999|loc=Ch 3. Corollary 2.3}} 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 $n$ th homology group of a chain complex is the $n$ 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.) The correspondence is an example of the nerve and realization paradigm.{{nLab|id=nerve+and+realization#doldkan_correspondence|title=nerve and realization}}

There is also an ∞-category-version of the Dold–Kan correspondence.{{harvnb|Lurie|loc=§ 1.2.4.}} The book "Nonabelian Algebraic Topology"{{Cite book | last1=Brown | first1=Ronald|author1-link=Ronald Brown (mathematician)| last2=Higgins | first2=Philip J. | last3=Sivera | first3=Rafael |title=Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids | publisher=European Mathematical Society | location=Zurich | series=Tracts in Mathematics | isbn= 978-3-03719-083-8 | year=2011 | volume=15}} 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.

Examples

For a chain complex C that has an abelian group A in degree n and zero in all other degrees, the corresponding simplicial group is the Eilenberg–MacLane space $K(A, n)$ .

Detailed construction

The Dold–Kan correspondence between the category sAb of simplicial abelian groups and the category $\text{Ch}_{\geq 0}(\textbf{Ab})$ of nonnegatively graded chain complexes can be constructed explicitly through a pair of functors^{pg 149} so that these functors form an equivalence of categories. The first functor is the normalized chain complex functor

$N\colon s\textbf{Ab} \to \text{Ch}_{\geq 0}(\textbf{Ab})$

and the second functor is the "simplicialization" functor

$\Gamma\colon \text{Ch}_{\geq 0}(\textbf{Ab}) \to s\textbf{Ab}$

constructing a simplicial abelian group from a chain complex. The formed equivalence is an instance of a special type of adjunction, called the nerve-realization paradigm{{cite book |last=Loregian |first=Fosco |title=(Co)end Calculus |chapter= |year=2021 |page=85 |publisher=

Cambridge University Press|series=London Mathematical Society Lecture Note Series |volume=468|doi=10.1017/9781108778657 |arxiv=1501.02503 |isbn=978-1-108-77865-7 }} (also called a nerve-realization context) where the data of this adjunction is determined by what's called a cosimplicial object{{nLab|id=cosimplicial+object|title=cosimplicial object}} $dk\colon \Delta^{\text{op}}\to \text{Ch}_{\geq 0}(\textbf{Ab})$ , and the adjunction then takes the form

$\Gamma = \mathrm{Lan}_y dk : \text{Ch}_{\geq 0}(\textbf{Ab}) \dashv s\textbf{Ab} : \mathrm{Lan}_{dk} y = N$

where we take the left Kan extension and $y$ is the Yoneda embedding.

= Normalized chain complex =

Given a simplicial abelian group $A_\bullet \in \text{Ob}(\text{s}\textbf{Ab})$ there is a chain complex $NA_\bullet$ called the normalized chain complex (also called the Moore complex) with terms

$NA_n = \bigcap^{n-1}_{i=0}\ker(d_i) \subset A_n$

and differentials given by

$NA_n \xrightarrow{(-1)^nd_n} NA_{n-1}$

These differentials are well defined because of the simplicial identity

$d_i \circ d_n = d_{n-1}\circ d_i : A_n \to A_{n-2}$

showing the image of $d_n \colon NA_n \to A_{n-1}$ is in the kernel of each $d_i\colon NA_{n-1} \to NA_{n-2}$ . This is because the definition of $NA_n$ gives $d_i(NA_n) = 0$ .

Now, composing these differentials gives a commutative diagram

$NA_n \xrightarrow{(-1)^nd_n} NA_{n-1} \xrightarrow{(-1)^{n-1}d_{n-1}} NA_{n-2}$

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

$d_{n-1}\circ d_n = d_{n-1}\circ d_{n-1}$

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

$A_\bullet \colon \text{Ord} \to \textbf{Ab}$

and morphisms $A_\bullet \to B_\bullet$ are given by natural transformations, meaning the maps of the simplicial identities still hold, the normalized chain complex construction is functorial.

References

{{Cite book | last1=Goerss | first1=Paul G. | last2=Jardine | first2=John F. | authorlink2=Rick Jardine | title=Simplicial Homotopy Theory | publisher=Birkhäuser | location=Basel, Boston, Berlin | series=Progress in Mathematics | isbn=978-3-7643-6064-1 | year=1999 | volume=174}}
{{Lurie-HA}}
{{cite web|first=Akhil|last= Mathew|url= https://math.uchicago.edu/~amathew/doldkan.pdf |title=The Dold–Kan correspondence|archive-url= https://web.archive.org/web/20160913201635/http://people.fas.harvard.edu/~amathew/doldkan.pdf|archive-date= 2016-09-13}}

External links

{{nlab|id=Dold-Kan+correspondence|title=Dold–Kan correspondence}}

Category:Simplicial sets

Category:Theorems in abstract algebra