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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Dold-Kan correspondence} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra} [[!include homological algebra - contents]] \hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory} [[!include higher category theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{statement_abelian_case}{Statement (abelian case)}\dotfill \pageref*{statement_abelian_case} \linebreak \noindent\hyperlink{EquivalenceOfCategories}{Equivalence of categories}\dotfill \pageref*{EquivalenceOfCategories} \linebreak \noindent\hyperlink{ModelCatVersion}{Quillen equivalence of model categories}\dotfill \pageref*{ModelCatVersion} \linebreak \noindent\hyperlink{StatementGeneral}{Statement (general nonabelian case)}\dotfill \pageref*{StatementGeneral} \linebreak \noindent\hyperlink{GlobularAndCubical}{Globular and cubical version}\dotfill \pageref*{GlobularAndCubical} \linebreak \noindent\hyperlink{presentation_of_strict_groupal_groupoids}{Presentation of strict groupal $\infty$-groupoids}\dotfill \pageref*{presentation_of_strict_groupal_groupoids} \linebreak \noindent\hyperlink{nonabelian_forms_of_the_doldkan_correspondence}{Non-abelian forms of the Dold-Kan correspondence.}\dotfill \pageref*{nonabelian_forms_of_the_doldkan_correspondence} \linebreak \noindent\hyperlink{StableDoldKanCorrespondence}{Stable Dold-Kan correspondence}\dotfill \pageref*{StableDoldKanCorrespondence} \linebreak \noindent\hyperlink{Variants}{Generalizations and variants}\dotfill \pageref*{Variants} \linebreak \noindent\hyperlink{Applications}{Applications}\dotfill \pageref*{Applications} \linebreak \noindent\hyperlink{EilenbergMacLaneObjects}{Eilenberg-MacLane objects}\dotfill \pageref*{EilenbergMacLaneObjects} \linebreak \noindent\hyperlink{LoopingAndDelooping}{Looping and delooping}\dotfill \pageref*{LoopingAndDelooping} \linebreak \noindent\hyperlink{abelian_sheaf_cohomology_in_nonabelian_cohomology}{Abelian sheaf cohomology in nonabelian cohomology}\dotfill \pageref*{abelian_sheaf_cohomology_in_nonabelian_cohomology} \linebreak \noindent\hyperlink{computational_aspects}{Computational aspects}\dotfill \pageref*{computational_aspects} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The \emph{Dold--Kan correspondence} asserts there is an [[equivalence of categories]] between [[abelian group|abelian]] [[simplicial groups]] and [[connective chain complex|connective]] [[chain complexes]] of [[abelian groups]]. Since every [[simplicial group]] is in particular a [[Kan complex]] with [[group object|group structure]], hence an [[∞-groupoid]] with group structure, hence an [[∞-group]], the Dold-Kan correspondence says that connective chain complexes are a model for certain [[abelian ∞-groups]], thus the correspondence interpolates between [[homological algebra]] and general [[simplicial homotopy theory]]. (This is part of the [[cosmic cube]] of [[higher category theory]]). The relevance of this is that chain complexes are typically easier to handle: all the tools of [[homological algebra]] apply. In fact, the [[functor]] that identifies simplicial abelian groups with their corresponding chain complexes -- the [[Moore complex|normalized chains functor]] -- does precisely this: it \emph{normalizes} an abelian group by discarding irrelevant information and constructing a smaller and less redundant model for it. There are various variants and generalizations of the Dold-Kan correspondence. These are discussed further \hyperlink{Variants}{below}. \hypertarget{statement_abelian_case}{}\subsection*{{Statement (abelian case)}}\label{statement_abelian_case} Let $A$ be an [[abelian category]]. We say a [[chain complex]] in $A$ is \emph{connective} if it is concentrated in non-negative degree. The [[full subcategory]] \begin{displaymath} Ch^+_\bullet(A) \hookrightarrow Ch_\bullet(A) \end{displaymath} of connective chain complexes is naturally identified with the category of $\mathbb{N}$-graded chain complexes. \hypertarget{EquivalenceOfCategories}{}\subsubsection*{{Equivalence of categories}}\label{EquivalenceOfCategories} \begin{theorem} \label{}\hypertarget{}{} For $A$ an [[abelian category]] there is an [[equivalence of categories]] \begin{displaymath} N \;\colon\; A^{\Delta^{op}} \stackrel{\leftarrow}{\to} Ch_\bullet^+(A) \;\colon\; \Gamma \end{displaymath} between \begin{itemize}% \item the [[category of simplicial objects]] in $A$; \item the [[category of chain complexes|category of connective chain complexes]] in $A$; \end{itemize} where \begin{itemize}% \item $N$ is the [[normalized chains complex]]/normalized [[Moore complex]] functor. \end{itemize} \end{theorem} (\hyperlink{Dold58}{Dold 58}, \hyperlink{Kan58}{Kan 58}, \hyperlink{DoldPuppe61}{Dold-Puppe 61}). \begin{theorem} \label{}\hypertarget{}{} For the case that $A$ is the category [[Ab]] of [[abelian group]]s, the functors $N$ and $\Gamma$ are [[nerve and realization]] with respect to the cosimplicial chain complex \begin{displaymath} \mathbb{Z}[-]: \Delta \to Ch_+(Ab) \end{displaymath} that sends the standard $n$-[[simplex]] to the normalized [[Moore complex]] of the free simplicial abelian group $F_{\mathbb{Z}}(\Delta^n)$ on the [[simplicial set]] $\Delta^n$, i.e. \begin{displaymath} \Gamma(V) : [k] \mapsto Hom_{Ch_\bullet^+(Ab)}(N(\mathbb{Z}(\Delta[k])), V) \,. \end{displaymath} \end{theorem} This is due to (\hyperlink{Kan58}{Kan 58}). More explicitly we have the following \begin{prop} \label{ExplicitUnitAndCounit}\hypertarget{ExplicitUnitAndCounit}{} \begin{itemize}% \item For $V \in Ch_\bullet^+$ the simplicial abelian group $\Gamma(V)$ is in degree $n$ given by \begin{displaymath} \Gamma(V)_n = \bigoplus_{[n] \underset{surj}{\to} [k]} V_k \end{displaymath} and for $\theta : [m] \to [n]$ a morphism in $\Delta$ the corresponding map $\Gamma(V)_n \to \Gamma(V)_m$ \begin{displaymath} \theta^* : \bigoplus_{[n] \underset{surj}{\to} [k]} V_k \to \bigoplus_{[m] \underset{surj}{\to} [r]} V_r \end{displaymath} is given on the summand indexed by some $\sigma : [n] \to [k]$ by the composite \begin{displaymath} V_k \stackrel{d^*}{\to} V_s \hookrightarrow \bigoplus_{[m] \underset{surj}{\to} [r]} V_r \end{displaymath} where \begin{displaymath} [m] \stackrel{t}{\to} [s] \stackrel{d}{\to} [k] \end{displaymath} is the [[weak factorization system|epi-mono factorization]] of the composite $[m] \stackrel{\theta}{\to} [n] \stackrel{\sigma}{\to} [k]$. \item The [[natural isomorphism]] $\Gamma N \to Id$ is given on $A \in sAb^{\Delta^{op}}$ by the map \begin{displaymath} \bigoplus_{[n] \underset{surj}{\to} [k]} (N A)_k \to A_n \end{displaymath} which on the [[direct sum]]mand indexed by $\sigma : [n] \to [k]$ is the composite \begin{displaymath} N A_k \hookrightarrow A_k \stackrel{\sigma^*}{\to} A_n \,. \end{displaymath} \item The [[natural isomorphism]] $Id \to N \Gamma$ is on a chain complex $V$ given by the composite of the projection \begin{displaymath} V \to C(\Gamma(V)) \to C(\Gamma(C))/D(\Gamma(V)) \end{displaymath} with the inverse \begin{displaymath} C(\Gamma(V))/D(\Gamma(V)) \to N \Gamma(V) \end{displaymath} of \begin{displaymath} N \Gamma(V) \hookrightarrow C(\Gamma(V)) \to C(\Gamma(V))/D(\Gamma(V)) \end{displaymath} (which is indeed an [[isomorphism]], as discussed at [[Moore complex]]). \end{itemize} \end{prop} This is spelled out in (\hyperlink{Goerssjardine}{Goerss-Jardine, prop. 2.2 in section III.2}). \begin{prop} \label{}\hypertarget{}{} With the explicit choice for $\Gamma N \stackrel{\simeq}{\to} Id$ as \hyperlink{ExplicitUnitAndCounit}{above} we have that $\Gamma$ and $N$ form an [[adjoint equivalence]] $(\Gamma \dashv N)$ \end{prop} This is for instance (\hyperlink{Weilbel}{Weibel, exercise 8.4.2}). \begin{remark} \label{}\hypertarget{}{} It follows that with the inverse structure maps, we also have an [[adjunction]] the other way round: $(N \dashv \Gamma)$. \end{remark} \hypertarget{ModelCatVersion}{}\subsubsection*{{Quillen equivalence of model categories}}\label{ModelCatVersion} Both $Ch_\bullet^+(A)$ and $A^{\Delta^{op}}$ are [[categories with weak equivalences]] in an standard way: \begin{itemize}% \item the weak equivalences of simplicial abelian groups are the [[weak homotopy equivalence]]s of the underlying [[Kan complex]]es, hence morphisms that induces [[isomorphism]]s on all [[simplicial homotopy group]]; \item the weak equivalences of chain complexes are the [[quasi-isomorphisms]]: the morphisms that induces isomorphisms on all [[chain homology]] groups. \end{itemize} \begin{prop} \label{}\hypertarget{}{} These functors $N$ and $\Gamma$ both respect all weak equivalences with respect to the standard [[model structure on simplicial sets]] [[model structure on chain complexes|and on chain complexes]] in that they induce isomorphisms between [[simplicial homotopy groups]] and [[homology group]]s. \end{prop} The structures of categories with weak equivalences have standard refinements to [[model category]] structures: \begin{itemize}% \item the \emph{projective} [[model structure on chain complexes]] $Ch_\bullet$ has as fibrations the chain maps that are surjections in each positive degree; \item the \emph{[[model structure on simplicial T-algebras|model structure on simplicial abelian groups]]} has as fibrations those whose underlying morphisms in [[sSet]] are fibrations ([[Kan fibrations]]) with respect to the standard [[model structure on simplicial sets]]. \end{itemize} \begin{prop} \label{QuillenEquivalenceBetweensAbAndCh}\hypertarget{QuillenEquivalenceBetweensAbAndCh}{} Both \begin{displaymath} (N \dashv \Gamma) : Ch_\bullet^+ \stackrel{\overset{N}{\leftarrow}}{\underset{\Gamma}{\to}} sAb \end{displaymath} as well as \begin{displaymath} (\Gamma \dashv N) : sAb \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{N}{\to}} Ch_\bullet^+ \end{displaymath} are [[Quillen equivalence]]s with respect to these model structures. \end{prop} This is discussed for instance in (\hyperlink{SchwedeShipley}{Schwede-Shipley, section 4.1}, \href{http://www.math.uic.edu/~bshipley/monoidalequi.final.pdf#page=17}{p.17}). \begin{remark} \label{}\hypertarget{}{} The category [[sAb]] $= Ab^{\Delta^{op}}$ is -- being a [[category of simplicial objects]] of a category with [[colimits]] -- is naturally an [[sSet]]-[[enriched category]] and with the model structure this makes it a [[simplicial model category]]. Since the DK-correspondence is even an [[equivalence of categories]], this induces accordingly the structure of a simplicial model category also on $Ch_\bullet^+$. Therefore the above Quillen equivalence is even a [[simplicial Quillen adjunction]]. \end{remark} \begin{remark} \label{}\hypertarget{}{} The [[free functor|free]]/[[stuff, structure, property|forgetful]] [[adjunction]] $(F \dashv U) : Ab \stackrel{\leftarrow}{\to} Set$ prolongs to [[simplicial object]]s \begin{displaymath} (F\dashv U) : sAb \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} sSet \end{displaymath} as an [[sSet]]-[[enriched functor|enriched]] adjunction. Moreover, by the above the right adjoint $U$ is a right Quillen functor to the standard [[model structure on simplicial sets]]. This means we have a [[simplicial Quillen adjunction]] \begin{displaymath} ( \Gamma F \dashv U N) : Ch_\bullet^+ \stackrel{\overset{}{\leftarrow}}{\underset{U N}{\to}} sSet \,. \end{displaymath} This manifestly [[presentable (infinity,1)-category|presents]] connective chain complexes as models for certain [[∞-groupoid]]s. \end{remark} \hypertarget{StatementGeneral}{}\subsection*{{Statement (general nonabelian case)}}\label{StatementGeneral} \hypertarget{GlobularAndCubical}{}\subsubsection*{{Globular and cubical version}}\label{GlobularAndCubical} There are versions of the Dold-Kan correspondence for other [[geometric shapes for higher structures]] than the [[simplex]], also for the [[globe]] and the [[cube]]. \begin{theorem} \label{}\hypertarget{}{} Write [[Ab]] for the category of [[abelian group]]s. (Could be any [[additive category]] with [[kernel]]s for the following to be true). Then the following categories of structures [[internalization|internal to]] $Ab$ are equivalent. \begin{enumerate}% \item The category of [[chain complex]]es (in non-negative degree). \item The category of [[crossed complex]]es. \item The category of [[cubical set]]s with [[connection on a cubical set]]. \item The category of cubical [[strict ∞-groupoid]]s. \item The category of [[globe|globular]] [[strict ∞-groupoid]]s. \end{enumerate} \end{theorem} A proof with references to the rich literature can be found for instance in \begin{itemize}% \item [[Ronnie Brown]], [[Philip Higgins]], [[Rafael Sivera]], \emph{[[Nonabelian Algebraic Topology]]} \end{itemize} see the section \href{http://ncatlab.org/nlab/show/Nonabelian+Algebraic+Topology#CubicalDoldKan}{Cubical Dold-Kan theorem}. This version of the Dold-Kan theorem reproduces the simplicial Dold-Kan theorem after application of the [[omega-nerve]], i.e. the simplicial Dold-Kan correspondence factors through the globular one via the $\omega$-nerve. \hypertarget{presentation_of_strict_groupal_groupoids}{}\subsubsection*{{Presentation of strict groupal $\infty$-groupoids}}\label{presentation_of_strict_groupal_groupoids} It was mentioned above that the standard simplicial Dold-Kan correspondence $Ch_\bullet(Ab) \stackrel{\leftarrow}{\to} sAb$ may be understood as identifying strictly abelian [[strict ∞-groupoid]]s among all [[∞-groupoid]]s. This statement is also surveyed and put into a larger context at [[cosmic cube of higher category theory]]. We now give a formal version of this statement, following an observation by [[Richard Garner]]. A different but closely analogous sequence of arguments to the same extent is also in the book \begin{itemize}% \item [[Ronnie Brown]], [[Philip Higgins]], [[Rafael Sivera]], \emph{[[Nonabelian Algebraic Topology]]}, European Math. Soc. Tracts in Mathematics 15, 2011. \end{itemize} see . \begin{defn} \label{}\hypertarget{}{} Write \begin{displaymath} (L \dashv R) : Ch_\bullet(Ab)^+ \stackrel{\leftarrow}{\to} Str \infty Cat(Ab) \stackrel{\leftarrow}{\to} Str \infty Cat(Set) \end{displaymath} for the [[adjunction]] obtained by composing the \hyperlink{GlobularAndCubical}{globular Dold-Kan correspondence} with the forgetful functor which forgets the abelian group structure on a strict $\infty$-category in the image of the globular/cubical Dold-Kan map. \end{defn} \begin{prop} \label{}\hypertarget{}{} The functor \begin{displaymath} C_\bullet : \Delta \to Ch_\bullet^+ \end{displaymath} which sends a [[simplex]] to its (normalized) chain complex factors as \begin{displaymath} C_\bullet : \Delta \stackrel{\mathcal{O}}{\to} Str \infty Cat \stackrel{L}{\to} Ch_\bullet^+ \,, \end{displaymath} where the cosimplicial strict $\infty$-category $\mathcal{O}$ is the [[oriental]] functor. \end{prop} This is a remark by [[Richard Garner]] posted . \begin{proof} Use that $\mathcal{O}(n)$ is the [[free construction|free]] strict $\infty$-category on a [[computad]]. Observe that $L$ sends a strict $\omega$-category $X$ to the chain complex obtained from the abelian reflexive globular set $X \times \mathb{Z}$. In particular the value on the $n$-[[globe]] is the chain complex \begin{displaymath} \mathbb{Z} \to \mathbb{Z}\oplus\mathbb{Z} \to \mathbb{Z}\oplus\mathbb{Z} \to \cdots \to \mathbb{Z}\oplus\mathbb{Z} \end{displaymath} with $(n+1)$ terms and differential given by $x \mapsto (x, -x)$ in each dimension. Moreover, the value of $L$ on the boundary of the $n$-globe is the chain complex obtained from this by removing the uppermost copy of $\mathbb{Z}$. Given a [[computad]] $C$, the associated abelian chain complex $L C$ has for $(L C)_n$ the free abelian group on the set of generating $n$-cells of $C$, and differential given by $\partial x = \sum_j t_j - \sum_i s_i$, where $\{s_i\}$ and $\{t_i\}$ are the sets of source- and or target-cells, respectively. A glance at [[Ross Street]]`s presentation of the [[oriental]]s shows that $L(\mathcal{O}(n)) = C_\bullet(\Delta[n])$. \end{proof} \begin{cor} \label{}\hypertarget{}{} The simplicial Dold-Kan map \begin{displaymath} Hom(C_\bullet \Delta[n], -) : Ch_\bullet + \to sSet \end{displaymath} factors as the identification of chain complexes with strictly abelian strict $\infty$-groupoids, followed by the functor that forgets the abelian structure and then followed by the [[omega-nerve]] operation that embeds strict $\infty$-groupoids into all $\infty$-groupoids. \end{cor} \begin{proof} Use the above adjunction and proposition to write for $K_\bullet$ a chain complex \begin{displaymath} Hom_{Ch_\bullet}(C_\bullet \Delta[n], K) = Hom_{Ch_\bullet}(L \mathcal{O}(n), K) = Hom_{Str \infty Cat}(\mathcal{O}(n), R K) = N (R K)_n \,. \end{displaymath} \end{proof} \textbf{Remark} The alternative construction in [[Nonabelian Algebraic Topology]] factors also versions of the nonabelian Dold-Kan correspondence through the $\omega$-nerve. \hypertarget{nonabelian_forms_of_the_doldkan_correspondence}{}\subsection*{{Non-abelian forms of the Dold-Kan correspondence.}}\label{nonabelian_forms_of_the_doldkan_correspondence} Perhaps the `ultimate' form of a `classical' Dold--Kan result is by Pilar Carrasco, who identified the extra structure on chain complexes of groups in order that they be [[Moore complex|Moore complexes]] of simplicial groups. Dominique Bourn has a general form of this result for his [[semi-abelian category|semi-abelian categories]]. His results provide a neat categorical gloss on the theorem. Dominique Bourn's formulation is very pretty. The Moore complex functor is [[monad|monadic]] when the basic category is semi-Abelian (Th. 1.4. p.113 in \emph{Bourn2007} below). Of course for simplicial \emph{groups}, the monad on chain complexes of groups gives the [[hypercrossed complex]]es of Carrasco and Cegarra, but here they fall out from the theory. On the down side there is apparently no full analysis as yet of the actual form of this monad. \hypertarget{StableDoldKanCorrespondence}{}\subsection*{{Stable Dold-Kan correspondence}}\label{StableDoldKanCorrespondence} The Dold-Kan correspondence [[stabilization|stabilizes]] to identify \emph{unbounded} [[chain complexes]] with the category of stably simplicial abelian groups. The latter are closely related to [[combinatorial spectrum|combinatorial spectra]] of [[Daniel Kan]] and can be defined as stably simplicial objects in the category of abelian groups. More precisely, we have the following definitions. \begin{defn} \label{}\hypertarget{}{} The category of \emph{stable simplices} has integer numbers as objects. Given two objects $k$ and $l$, the set of morphisms from $k$ to $l$ is the set of order-preserving maps $h$ from the set of natural numbers to itself such that $h(n)=n+l-k$ for all but a finite number of $n$. Morphisms are composed by composing the corresponding maps. \end{defn} \begin{defn} \label{}\hypertarget{}{} A \emph{stably simplicial abelian group} is a presheaf $F$ of abelian groups on the category of stable simplices such that for any integer $k$ every element $x$ of $F(k)$ belongs to the kernels of all but a finite number of degeneracy maps. Morphisms of stably simplicial abelian groups are morphisms of presheaves. \end{defn} The following theorem was established in 1963 by [[Daniel Kan]] in his paper ``Semisimplicial spectra'' (see Proposition 5.8): \begin{theorem} \label{}\hypertarget{}{} The [[category of unbounded chain complexes]] is [[equivalence of categories|equivalent]] to the category of stably simplicial abelian groups, with equivalences being given by the same functors as in the unstable Dold-Kan correspondence, but appropriately extended to the above categories. \end{theorem} Similarly to the unstable case, the above categories, when interpreted as ∞-categories, are also equivalent to the ∞-category of module spectra over the Eilenberg-MacLane ring spectrum of the integers. For more see at \emph{[[stable Dold-Kan correspondence]]}. \hypertarget{Variants}{}\subsection*{{Generalizations and variants}}\label{Variants} There are various variants, generalizations and enhancements of the Dold--Kan correspondence. \begin{itemize}% \item The [[monoidal Dold-Kan correspondence]] relates [[simplicial algebra]]s with [[dg-algebra]]s. \item In [[rational homotopy theory]], Quillen proved and used an analogous statement for [[Lie algebra]]s: a Quillen equivalence between the reduced rational [[dg-Lie algebra]]s and reduced rational simplicial Lie algebras: D. Quillen, \emph{Rational homotopy theory} , Ann. Math. 90 (1969), 204--265. \item The statement of the Dold--Kan correspondence generalizes to [[sheaf|sheaves]] with values in the respective categories and this way from [[? Grpd]] to more general $(\infty,1)$-[[(infinity,1)-topos|topoi]]: For $X$ be a [[site]], let $Sh(X, sAb)$ be the category of \emph{simplicial abelian sheaves} -- i.e. [[simplicial presheaf|simplicial sheaves]] which take values in simplicial abelian groups -- and let $Sh(X, Ch_+(Ab))$ be the category of [[sheaf|sheaves]] on $S$ with values in non-negatively graded [[chain complexes]] of abelian groups. The normalized chain complex extends objectwise to a functor \begin{displaymath} Sh(X,sAb) \stackrel{\simeq}{\to} Sh(X, Ch_+(Ab)) \end{displaymath} which is an [[equivalence]] of categories. Moreover, both these categories are naturally [[category with weak equivalences|categories with weak equivalences]]: the weak equivalences in $Sh(X, sAb)$ are the stalkwise [[model structure on simplicial sets|weak equivalences of simplicial sets]] and the weak equivalences in $Sh(X, Ch_+(Ab))$ are the [[quasi-isomorphisms]]. The normalized chain complex functor preserves these weak equivalences. This sheaf version of the Dold--Kan correspondence allows to understand [[abelian sheaf cohomology]] as a special case of [[nonabelian cohomology]]. See page 9,10 of \begin{itemize}% \item K. Brown, [[BrownAHT|Abstract Homotopy Theory and Generalized Sheaf Cohomology]] \end{itemize} \item There is a version of the Dold--Kan correspondence in the context of $(\infty,1)$-[[(infinity,1)-category|categories]]: let $C$ be a [[stable (∞,1)-category]]. Then the $(\infty,1)$-categories of non-negatively graded [[complexes]] in $C$ is equivalent to the $(\infty,1)$-category of [[simplicial objects]] in $C$ \begin{displaymath} Fun(N(\mathbb{Z}_{\geq 0}), C) \simeq Fun(N(\Delta)^{op}, C) \,. \end{displaymath} This is [[infinity-Dold-Kan correspondence]] is \href{http://www.math.harvard.edu/~lurie/topoibook/DAGI.pdf}{theorem 12.8, p. 50} of \begin{itemize}% \item [[Jacob Lurie]], [[Stable ∞-Categories]] \end{itemize} \item There is a version of the Dold--Kan correspondence with [[simplicial sets]] generalized to [[dendroidal sets]]. This is described in \begin{itemize}% \item [[Javier Gutierrez|Javier Gutiérrez]], [[Andor Lukacs]], [[Ittay Weiss]], \emph{Dold-Kan correspondence for dendroidal abelian groups} (\href{http://arxiv.org/abs/0909.3995}{arXiv}) \end{itemize} \item Various functor categories of interest in stable homotopy theory and homological stability are involved in generalized Dold-Kan equivalences. These equivalences have been studied independently by several authors, including \hyperlink{Piraashvili}{Pirashvili} , \hyperlink{S?omi?ska}{Słomińska}, \hyperlink{Helmstutler}{Helmstutler}, and \hyperlink{LackStreet}{Lack and Street} \item There is a [[categorification]] of the correspondence, [[categorified Dold-Kan correspondence]] (\hyperlink{Dyckerhoff17}{Dyckerhoff17}) \end{itemize} \hypertarget{Applications}{}\subsection*{{Applications}}\label{Applications} \hypertarget{EilenbergMacLaneObjects}{}\subsubsection*{{Eilenberg-MacLane objects}}\label{EilenbergMacLaneObjects} The Dold-Kan correspondence gives a convenient construction of [[Eilenberg-MacLane object]]s in [[simplicial group]]s. \begin{prop} \label{ExplicitUnitAndCounit}\hypertarget{ExplicitUnitAndCounit}{} For $A$ an [[abelian group]] write $A[-n]$ for the [[chain complex]] concentrated on $A$ in degree $n$. The simplicial abelian group $\Gamma (A[-n])$ is an [[Eilenberg-MacLane object]] $K(A,n)$. And conversely, every such Eilenberg-MacLane object in simplicial abelian groups is related by an [[∞-anafunctor]]-equivalence to a $\Gamma(A[-n])$. \end{prop} \hypertarget{LoopingAndDelooping}{}\subsubsection*{{Looping and delooping}}\label{LoopingAndDelooping} The Dold-Kan correspondence provides a convenient way to describe formation of [[loop space object]]s and [[delooping]] for anything in the image of $\Xi : Ch_\bullet \to sSet$: by the basic fact that the homotopy groups of $\Xi(V_\bullet)$ are the homology groups of $V_\bullet$, looping and delooping simply corresponds to shifting chain complexes up or down in degree. But the relation is also strongly coherent: it respects the standard delooping functor $\bar W : sGrp \to sSet$ for [[simplicial group]]s (see there and at [[looping and delooping]]) (notice that restricted to simplicial abelian groups this produces simplicial abelian groups $\bar W : sAbGrp \to sAbGrp$): \begin{prop} \label{}\hypertarget{}{} There is a [[natural isomorphism]] \begin{displaymath} N \bar W G \simeq (N G)[-1] \end{displaymath} natural in $G \in sAbGrpd$. \end{prop} This appears for instance as (\hyperlink{GoerssJardine}{GoerssJardine, remark III.5.6}) or around (\hyperlink{Jardine}{Jardine, theorem 4.57}). \hypertarget{abelian_sheaf_cohomology_in_nonabelian_cohomology}{}\subsubsection*{{Abelian sheaf cohomology in nonabelian cohomology}}\label{abelian_sheaf_cohomology_in_nonabelian_cohomology} Composed with the [[forgetful functor]] $sAb \to sSet$ the Dold-Kan correspondence presents certain [[simplicial set]]s by chain complexes. Since this is entirely functorial, it prolongs to a functor from chain complexes of (pre)[[sheaves]] on any [[site]] $S$, to [[simplicial presheaves]] \begin{displaymath} \Gamma : [S^{op}, Ch_\bullet^+(ab)] \to [S^{op}, sSet] \,. \end{displaymath} If $[S^{op}, sSet]$ is equipped with the projective [[model structure on simplicial presheaves]] it models the [[(∞,1)-sheaf (∞,1)-topos]] on $S$. The [[derived hom-space]]s compute general [[nonabelian cohomology]]. If the coefficient objects come from sheaves of chain complexes along $\Gamma$, this cohomology restricts to ordinary [[abelian sheaf cohomology]]. See there for more details. \hypertarget{computational_aspects}{}\subsubsection*{{Computational aspects}}\label{computational_aspects} One may view the (monoidal) Dold-Kan correspondence as a relation between a well-behaved theory (simplicial/higher methods) that work in any characteristic but is very abstract and mainly suited to the proof of abstract theorems, and a more computational theory (strict structures in dg-modules) that are particularly well adapted to computations. The relation between these two (symmetric monoidal) theories may only be properly used with characteristic 0 coefficients. This remark is very naive and basic, but certainly at the center of computational implementations of abstract homotopical methods. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item \textbf{Dold-Kan correspondence} \item [[monoidal Dold-Kan correspondence]] \item [[operadic Dold-Kan correspondence]] \item [[infinity-Dold-Kan correspondence]] \item [[stable Dold-Kan correspondence]] \item [[categorified Dold-Kan correspondence]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} Historical references for the Dold--Kan correspondence are \begin{itemize}% \item [[Albrecht Dold]], \emph{Homology of symmetric products and other functors of complexes}, Annals of Mathematics Second Series, Vol. 68, No. 1 (Jul., 1958), pp. 54-80 (\href{http://www.jstor.org/stable/1970043}{jstor}) \end{itemize} which considers the correspondence for categories of [[modules]], and \begin{itemize}% \item [[Albrecht Dold]], [[Dieter Puppe]], \emph{Homologie nicht-additiver Funktoren. Anwendungen}, Annales de l'institut Fourier, 11 (1961), p. 201-312 (\href{http://www.numdam.org/item?id=aif_1961__11__201_0}{numdam}) \end{itemize} that generalizes the result to arbitrary [[abelian category|abelian categories]]. The expression of the correspondence in terms of [[nerve and realization]] is due to \begin{itemize}% \item [[Daniel Kan]], \emph{Functors involving c.s.s complexes}, Transactions of the American Mathematical Society, Vol. 87, No. 2 (Mar., 1958), pp. 330--346 (\href{http://www.jstor.org/stable/1993103}{jstor}). \end{itemize} This remarkable article, which appeared shortly after the work by Dold and Puppe but was apparently not influenced by that, introduces not just the abstract [[nerve and realization]] form of the Dold-Kan correspondence, but introduces the general notion of nerve and realization and in fact the general notion of what is now called [[Kan extension]]. A standard modern textbook reference for the ordinary Dold-Kan correspondence is \href{http://www.maths.abdn.ac.uk/~bensondj/papers/g/goerss-jardine/ch-3.dvi}{chapter III.2} of \begin{itemize}% \item [[Paul Goerss]], [[Rick Jardine]], \emph{[[Simplicial homotopy theory]]} (\href{http://www.maths.abdn.ac.uk/~bensondj/html/archive/goerss-jardine.html}{web}) \end{itemize} Similar material is in section 4.6 of \begin{itemize}% \item [[Rick Jardine]], \emph{Generalized etale cohomology theories} Modern Birkh\"a{}user Classics, (1991) \end{itemize} Remarks about the interpretation in terms of model categories are in \begin{itemize}% \item [[Stefan Schwede]], [[Brooke Shipley]], \emph{Equivalences of monoidal model categories} , Algebr. Geom. Topol. 3 (2003), 287--334 (\href{http://arxiv.org/abs/math.AT/0209342}{arXiv}) \end{itemize} Discussion in the generality of [[idempotent complete category|idempotent complete]] [[additive categories]] is in \begin{itemize}% \item [[Jacob Lurie]], section 1.2.3 of \emph{[[Higher Algebra]]} \end{itemize} The relation between [[strict ∞-groupoids]] and [[crossed complexes]] is in \begin{itemize}% \item [[Ronnie Brown|R. Brown]] and P.J. Higgins, \emph{The equivalence of $\infty$-groupoids and crossed complexes}, Cahiers de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}goriques, 22 no. 4 (1981), p. 371--386 (\href{http://archive.numdam.org/article/CTGDC_1981__22_4_371_0.pdf}{pdf}) \end{itemize} P. 59 of \begin{itemize}% \item R. Brown, \emph{Groupoids and crossed objects in algebraic topology}, Homology, Homotopy and Applications, 1 (1999) 1-78. \end{itemize} gives seven equivalent categories with the equivalences all expressing nonabelian versions of the Dold--Kan correspondence. One of these is given in \begin{itemize}% \item Ashley, N., \emph{Simplicial T-complexes and crossed complexes: a nonabelian version of a theorem of Dold and Kan}. University of Wales PhD Thesis, (1978); Dissertationes Math. (Rozprawy Mat.) 265 (1988) 1--61. \end{itemize} The relation of these with the abelian version is given in \begin{itemize}% \item Brown, R. and Higgins, P. J., \emph{Cubical abelian groups with connections are equivalent to chain complexes}. Homology Homotopy Appl. 5 (1) (2003) 49--52. \end{itemize} The paper \begin{itemize}% \item Ellis, G.J. and Steiner, R. \emph{Higher-dimensional crossed modules and the homotopy groups of $(n+1)$\}-ads.}J. Pure Appl. Algebra\_ \textbf{46} (2-3) (1987) 117--136. \end{itemize} should also be seen as of Dold-Kan type. The homotopical applications considerably generalise results on the [[Blakers-Massey theorem]]. See also \begin{itemize}% \item Brown, R. \emph{Modelling and computing homotopy types: I}, Indagationes Math: Special Issue in honor of L.E.J. Brouwer, (2017) (\href{http://groupoids.org.uk/pdffiles/brouwer-honor.pdf}{pdf}) \end{itemize} The discussion of Dold--Kan in the general context of [[semi-abelian category|semi-abelian categories]] is in \begin{itemize}% \item [[Dominique Bourn]], \emph{Moore normalisation and Dold--Kan theorem for semi-Abelian categories}, in Categories in algebra, geometry and mathematical physics , volume 431 of Contemp. Math., 105--124, Amer. Math. Soc., Providence, RI. (2007) \end{itemize} The classical Dold-Kan theorem occurs as a special case among others from [[combinatorics]] and [[representation theory]], and in particular from homological stability, in: \begin{itemize}% \item Jolanta Słomińska, \emph{Dold?Kan type theorems and Morita equivalences of functor categories}, Journal of Algebra 274.1 (2004): 118-137. (\href{https://doi.org/10.1016/j.jalgebra.2003.10.025}{link}) \end{itemize} A similar framework was independently rediscovered in: \begin{itemize}% \item [[Stephen Lack]], [[Ross Street]], \emph{Combinatorial Categorical Equivalences}, arxiv:1402.7151 (2014). (\href{http://arxiv.org/abs/1402.7151}{link}) \end{itemize} A stable homotopical version of these general correspondences was developed in: \begin{itemize}% \item Randall Helmstutler, \emph{Model category extensions of the Pirashvili-S?omi?ska theorems}, arxiv:0806.1540 (\href{https://arxiv.org/abs/0806.1540}{link}). \end{itemize} Among the correspondences ``of Dold-Kan type'' included in this theory are an equivalence between FI-modules and linear combinatorial species: \begin{itemize}% \item Thomas Church, Jordan S. Ellenberg, Benson Farb, \emph{FI-modules and stability for representations of symmetric groups}, arxiv:1204.4533 (\href{https://arxiv.org/abs/1204.4533}{link}) \end{itemize} A Dold-Kan theorem for $\Gamma$-groups: \begin{itemize}% \item Teimuraz Pirashvili. \emph{Dold-Kan type theorem for ∞-groups}, Mathematische Annalen 318.2 (2000): 277-298. (\href{https://doi.org/10.1007/s002080000120}{link}) \end{itemize} An equivalence between representations of the category of finite-dimensional $\mathbb{F}_q$-vector spaces and representations of its underlying groupoid: \begin{itemize}% \item Nicholas Kuhn, \emph{Generic representation theory of finite fields in nondescribing characteristic}, arxiv:1405.0318 (\href{https://arxiv.org/abs/1405.0318}{link}) \end{itemize} A [[categorification]] to a [[categorified Dold-Kan correspondence]] is discussed here: \begin{itemize}% \item [[Tobias Dyckerhoff]], \emph{A categorified Dold-Kan correspondence} (\href{https://arxiv.org/abs/1710.08356}{arXiv:1710.08356}) \end{itemize} [[!redirects Dold--Kan correspondence]] [[!redirects Dold?Kan correspondence]] [[!redirects Dold-Kan theorem]] [[!redirects Dold--Kan theorem]] [[!redirects Dold?Kan theorem]] [[!redirects dual Dold--Kan correspondence]] [[!redirects dual Dold?Kan correspondence]] [[!redirects dual Dold-Kan theorem]] [[!redirects dual Dold--Kan theorem]] [[!redirects dual Dold?Kan theorem]] \end{document}