and
nonabelian homological algebra
The Dold–Kan correspondence asserts there is an equivalence of categories between abelian simplicial groups and connective chain complexes of abelian groups.
Since every simplicial group is in particular a Kan complex with 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. (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 normalized chains functor – does precisely this: it 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 below.
Let $A$ be an abelian category.
We say a chain complex in $A$ is connective if it is concentrated in non-negative degree. The full subcategory
of connective chain complexes is naturally identified with the category of $\mathbb{N}$-graded chain complexes.
For $A$ an abelian category there is an equivalence of categories
between
the category of simplicial objects in $A$;
the category of connective chain complexes in $A$;
where
For the case that $A$ is the category Ab of abelian groups, the functors $N$ and $\Gamma$ are nerve and realization with respect to the cosimplicial chain complex
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.
More explicitly we have the following
For $V \in Ch_\bullet^+$ the simplicial abelian group $\Gamma(V)$ is in degree $n$ given by
and for $\theta : [m] \to [n]$ a morphism in $\Delta$ the corresponding map $\Gamma(V)_n \to \Gamma(V)_m$
is given on the summand indexed by some $\sigma : [n] \to [k]$ by the composite
where
is the epi-mono factorization of the composite $[m] \stackrel{\theta}{\to} [n] \stackrel{\sigma}{\to} [k]$.
The natural isomorphism $\Gamma N \to Id$ is given on $A \in sAb^{\Delta^{op}}$ by the map
which on the direct summand indexed by $\sigma : [n] \to [k]$ is the composite
The natural isomorphism $Id \to N \Gamma$ is on a chain complex $V$ given by the composite of the projection
with the inverse
of
(which is indeed an isomorphism, as discussed at Moore complex).
This is spelled out around prop. 2.2 in section III.2 of (GoerssJardine).
With the explicit choice for $\Gamma N \stackrel{\simeq}{\to} Id$ is above we have that $\Gamma$ and $N$ form an adjoint equivalence $(\Gamma \dashv N)$
This is for instance exercise 8.4.2 in Weibel.
It follows that with the inverse structure maps, we also have an adjunction the other way round: $(N \dashv \Gamma)$.
Both $Ch_\bullet^+(A)$ and $A^{\Delta^{op}}$ are categories with weak equivalences in an standard way:
the weak equivalences of simplicial abelian groups are the weak homotopy equivalences of the underlying Kan complexes, hence morphisms that induces isomorphisms on all simplicial homotopy group;
the weak equivalences of chain complexes are the quasi-isomorphisms: the morphisms that induces isomorphisms on all chain homology groups.
These functors $N$ and $\Gamma$ both respect all weak equivalences with respect to the standard model structure on simplicial sets and on chain complexes in that they induce isomorphisms between simplicial homotopy groups and homology groups.
The structures of categories with weak equivalences have standard refinements to model category structures:
the projective model structure on chain complexes $Ch_\bullet$ has as weak equivalences the quasi-isomorphisms and as fibrations the chain maps that are surjections in each positive degree;
the model structure on simplicial abelian groups has as weak equivalences and fibrations those whose underlying morphisms in sSet are fibrations (Kan fibrations) with respect to the standard model structure on simplicial sets.
Both
as well as
are Quillen equivalences with respect to these model structures.
This is discussed for instance in section 4.1 of SchwedeShipley
The category $Ab^{\Delta^{op}}$ is – being a category of simplicial objects of a category with colimits – 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.
The free/forgetful adjunction $(F \dashv U) : Ab \stackrel{\leftarrow}{\to} Set$ prolongs to simplicial objects
as an sSet-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
This manifestly presents connective chain complexes as models for certain ∞-groupoids.
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.
Write Ab for the category of abelian groups. (Could be any additive category with kernels for the following to be true). Then the following categories of structures internal to $Ab$ are equivalent.
The category of chain complexes (in non-negative degree).
The category of crossed complexes.
The category of cubical sets with connection on a cubical set.
The category of cubical strict ∞-groupoids.
The category of globular strict ∞-groupoids.
A proof with references to the rich literature can be found for instance in
see the section 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.
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 ∞-groupoids among all ∞-groupoids. 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
see Dold-Kan map and omega-nerve.
Write
for the adjunction obtained by composing the 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.
The functor
which sends a simplex to its (normalized) chain complex factors as
where the cosimplicial strict $\infty$-category $\mathcal{O}$ is the oriental functor.
This is a remark by Richard Garner posted here.
Use that $\mathcal{O}(n)$ is the 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
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 orientals shows that $L(\mathcal{O}(n)) = C_\bullet(\Delta[n])$.
The simplicial Dold-Kan map
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.
Use the above adjunction and proposition to write for $K_\bullet$ a chain complex
Remark The alternative construction in Nonabelian Algebraic Topology factors also versions of the nonabelian Dold-Kan correspondence through the $\omega$-nerve.
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 complexes of simplicial groups. Dominique Bourn has a general form of this result for his 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 monadic when the basic category is semi-Abelian (Th. 1.4. p.113 in Bourn2007 below). Of course for simplicial groups, the monad on chain complexes of groups gives the hypercrossed complexes 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.
The Dold-Kan correspondence stabilizes to identify unbounded chain complexes with the category of stably simplicial abelian groups. The latter are closely related to 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.
The category of 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.
A 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.
The following theorem was established in 1963 by Daniel Kan in his paper “Semisimplicial spectra” (see Proposition 5.8):
The category of unbounded chain complexes is 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.
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 stable Dold-Kan correspondence.
There are various variants, generalizations and enhancements of the Dold–Kan correspondence.
The monoidal Dold-Kan correspondence relates simplicial algebras with dg-algebras.
In rational homotopy theory, Quillen proved and used an analogous statement for Lie algebras: a Quillen equivalence between the reduced rational dg-Lie algebras and reduced rational simplicial Lie algebras:
D. Quillen, Rational homotopy theory , Ann. Math. 90 (1969), 204–265.
The statement of the Dold–Kan correspondence generalizes to sheaves with values in the respective categories and this way from ∞ Grpd to more general $(\infty,1)$-topoi:
For $X$ be a site, let $Sh(X, sAb)$ be the category of simplicial abelian sheaves – i.e. simplicial sheaves which take values in simplicial abelian groups – and let $Sh(X, Ch_+(Ab))$ be the category of sheaves on $S$ with values in non-negatively graded chain complexes of abelian groups. The normalized chain complex extends objectwise to a functor
which is an equivalence of categories. Moreover, both these categories are naturally categories with weak equivalences: the weak equivalences in $Sh(X, sAb)$ are the stalkwise 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
There is a version of the Dold–Kan correspondence in the context of $(\infty,1)$-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 $C$
This is infinity-Dold-Kan correspondence is theorem 12.8, p. 50 of
There is a version of the Dold–Kan correspondence with simplicial sets generalized to dendroidal sets. This is described in
The Dold-Kan correspondence gives a convenient construction of Eilenberg-MacLane objects in simplicial groups.
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 $\Gamma(A[-n])$.
The Dold-Kan correspondence provides a convenient way to describe formation of loop space objects 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 groups (see there and at looping and delooping) (notice that restricted to simplicial abelian groups this produces simplicial abelian groups $\bar W : sAbGrp \to sAbGrp$):
There is a natural isomorphism
natural in $G \in sAbGrpd$.
This appears for instance as (GoerssJardine, remark III.5.6) or around (Jardine, theorem 4.57).
Composed with the forgetful functor $sAb \to sSet$ the Dold-Kan correspondence presents certain simplicial sets 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
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-spaces 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.
Dold-Kan correspondence
Historical references for the Dold–Kan correspondence are
which considers the correspondence for categories of modules, and
that generalizes the result to arbitrary abelian categories.
The expression of the correspondence in terms of nerve and realization is due to
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 chapter III.2 of
Similar material is in section 4.6 of
Remarks about the interpretation in terms of model categories are in
Discussion in the generality of idempotent complete additive categories is in
The relation between strict ω-groupoids and crossed complexes is in
P. 59 of
gives seven equivalent categories with the equivalences all expressing nonabelian versions of the Dold–Kan correspondence. One of these is given in
The relation of these with the abelian version is given in
The discussion of Dold–Kan in the general context of semi-abelian categories is in
The classical Dold-Kan theorem occurs as a special case among others from combinatorics and representation theory here: