nLab
monoidal Dold-Kan correspondence

Idea
General discussion
Simplicial rings and chain dg-algebras
Cosimplicial rings and cochain dg-algebras
References

Idea

The plain Dold-Kan correspondence establishes an equivalence between (co)simplicial groups and (co)chain complexes.

Both these categories carry natural monoidal category structures. It turns out that the Dold-Kan correspondence does respect this monoidal structure, either strictly or in their suitable higher categorical sense.

General discussion

Notice that

monoids in the category of chain complexes are differential graded rings
monoids in the category of simplicial abelian groups are simplicial rings.

If instead we look at chain complexes and simplicial objects not in Ab but in Vect then

monoids in the category of chain complexes of vector spaces are differential graded algebras
monoids in the category of simplicial vector spaces are simplicial algebras.

Analogous statements apply to the dual Dod-Kan correspondence, where the monoids in question are accordingly cosimplicial rings and differential graded algebras with differential of positive degree.

A crucial fact about the Dold-Kan correspondence is that

The Dold-Kan correspondence respects these monoidal structures.

But it doesn’t in general do so strictly, except possibly in one direction. Rather, it does so in the context of higher category theory.

There are two different versions of the monoidal Dold-Kan corespondence, which are almost but apparently not entirely formal duals of each other (at least not in the detailed constructions):

the simplicial version that relates monoids in abelian simplicial groups (simplicial rings) to monoids in chain complexes (chain dg-algebras),
the cosimplicial version that relates monoids in cosimplicial groups (cosimplicial ring) to monoidal in co-chain complexes (cochain dg-algebras).

Simplicial rings and chain dg-algebras

The Dold-Kan correspondence (using the normalized chain complex functor) is in one direction monoidal in the naive (strict, 1-categorical) way, whereas in the other direction it is monoidal only in the expect homotopical sense.

Lemma

The Moore complex functor

C_{•} : sAb \to {Ch}_{+}

as well as the normalized Moore complex functor

N_{•} : sAb \to {Ch}_{+}

is a symmetric lax monoidal as well as lax comonoidal. And in a compatible way: it is actually a Frobenius monoidal functor .

Proof

The proof of symmetric lax monoidalness can be found for instance in

Weibel, An introduction to homological algebra, section 8.5.4

The lax monoidal transformation that exhibits the lax-monoidalness of the Moore chain complex functor is the shuffle map?. Its component

\nabla_{A, B} : (N_{•} A) \otimes (N_{•} B)) \to N_{•} (A \otimes B)

on a pair $A, B$ of simplicial abelian groups is the morphism of chain complexes that sends homogeneous elements $a_{p} \otimes b_{q} \in A_{p} \otimes B_{q} = : C_{p} (A) \otimes C_{q} (B)$ to

\nabla_{A, B} (a \otimes B) = \sum_{(μ, ν)} sign (μ, ν) (s_{ν} a) \otimes s_{μ} (b) \in C_{p + q} (A \otimes B) = A_{p + q} \otimes B_{p + q} .

Here the sum is over all $(p, q)$ -shuffles, i.e. permutation ${μ_{1}, \dots, μ_{p}, ν_{1}, \dots, ν_{q}}$ of the set ${0,1, \dots, p + q - 1}$ that leave the first $p$ and the last $q$ elements in their natural order.

The sign in the above sum is the corresponding sign of this permutation and the degeneracy maps $s_{μ}$ and $s_{ν}$ denote the maps

s_{μ} : = s_{μ_{p}} \dots \circ s_{μ_{1}}

and similarly for $s_{ν}$

(Hm, is that consistent?)

For more on this see also section 2.3 of SchwedeSchipley

The comonoidalness and Frobenius monoidalness of the normalized Moore functor is discussed in

Marcelo Aguiar, Swapneel Mahajan, Coxeter Groups and Hopf Algebras Fields Institute Monographs, vol. 23 pdf

and with more details in

Marcelo Aguiar, Swapneel Mahajan, Monoidal functors, species and Hopf algebras, pdf, over 6 Mb version June 18, 2009, 672 pages.

While the nerve functor $Ξ : {Ch}_{+} \to sAb$ fails to be monoidal in such a direct way, it turns out to be monoidal in a suitable higher categorical, i.e. homotopical way. This is the content of the following statements.

In characteristic zero there is also a Dold–Kan correspondence between simplicial algebras and $ℕ$ -graded chain dg-algebras

Stefan Schwede, Brooke Shipley, Equivalences of monoidal model categories , Algebr. Geom. Topol. 3 (2003), 287–334 (arXiv)

Further work along such lines is

Birgit Richter, Symmetry properties of the Dold-Kan correspondence (pdf)

This shows that the Dold-Kan map from Chain complexes to simplicial abelian groups is, while not a monoidal functor, an E-infinity monoidal functor.

This implies that generalized Eilenberg–Mac Lane spectra on differential graded commutative algebras are E-infinity monoids in the category of $H ℤ$ -module spectra.

Birgit Richter, Homotopy algebras and the inverse of the normalization functor (pdf)

This article shows that the inverse $Ξ$ from chain complexes to simplicial abelian groups sends algebras over arbitrary differential graded E-infinity-operad to E-infinity-algebra in simplicial modules, and is part of a Quillen adjunction for these.

Cosimplicial rings and cochain dg-algebras

The monoidal Dold-Kan correspondence relating cosimplicial algebras to cochain dg-algebras is considered less prominently explicitly in the literature, but does appear implicitly in much classical work. For instance the classical statement that the cochains on simplicial sets form a dg-algebra that is commutative up to coherent higher homotopy, i.e. that is an E-infinity algebra, is really the statement that the Moore cochain complex functor on cosimplicial algebras of functions on simplicial sets is an $∞$ -monoidal functor in a suitable sense.

One article that does make a cosimplicial/cochain monoidal Dold-Kan correspondence explicit is

J.L. Castiglioni, G. Cortiñas, Cosimplicial versus DG-rings: a version of the Dold-Kan correspondence, J. Pure Appl. Algebra 191 (2004), no. 1-2, 119–142, arXiv:math.KT/0306289.

This establishes not quite a Quillen equivalence, but shows that the Dold-Kan correspondence induces an equivalence of homotopy categories for the model structure on cosimplicial rings and the model structure on dg-rings.

However, this article explicitly constructs the (derived) adjoint functor to the Moore cochain complex functor.

Explicit discussion of the Moore co-chain complex functor as inducing an $∞$ -monoidal functor seems not to be in the literature explicitly at time of this writing (?), even though various of its aspects are implicit, partly classical, statements. The following tries to make some aspects explicit.

Alexander–Whitney and shuffle morphisms

A central ingredient in the monoidal Dold-Kan correspondence are the Alexander–Whitney and the shuffle morphisms.

See chapter VI, paragraph 12 of

A. Dold, Lectures on algebraic topology, Grundlehren Math. Wiss. vol 200, Springer-Verlag, New-York-Berlin, 1972

or chapter VIII, paragraph 8 of

Saunders MacLane, Homology , Grundlehren Math. Wiss. vol 114, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963 .

For $A, B$ cosimplicial abelian groups, $C A$ and $C B$ their Moore cochain complexes,

the Alexander–Whitney morphism is the morphism

N A \otimes N B \to N (A \otimes B)

that is given on homogeneous elements $x, y$ of degree $p, q$ , respectively, by

x \otimes y \mapsto δ^{n} \circ \dots \circ δ^{p + 1} (x) \otimes δ^{0} \circ \dots \circ δ^{0} (y) .

Notice that for $A = B$ a cosimplicial algebra, further composing this with the product yields the cup product induced on dg-algebras of cosimplicial algebras. This is spelled out in detail below.

The shuffle morphism goes the other way

C (A \otimes B) \to C A \otimes C B

and is given on homogeneous elements as above by

x \otimes y \mapsto \sum_{(μ, ν)} ϵ (μ, ν) σ^{ν_{1} - 1} \circ \dots \circ σ^{ν_{q} - 1} (x) \otimes σ^{μ_{1} - 1} \circ \dots \circ σ^{μ_{p} - 1} (y)

where the sum is over all $(p, q)$ shuffle?s $(μ, ν)$ and $ϵ (μ, ν)$ is the sign of the shuffle.

Both the Alexander–Whitney morphism as well as the shuffle morphism respect the passage to the normalized Moore complex $N A$ of $C A$ and hence induce also morphisms

AW : N A \otimes N B \to N (A \otimes B)

and

S : N (A \otimes B) \to N A \otimes N B .

In this form they satisfy

S \circ AW = Id .

See for instance theorem 2.1.a in

Samuel Eilenberg, Saunders MacLane, On the groups $H (Π, n)$ II. Methods of computation , Ann. Math. (2) 60 (1954), 49 - 139

A quick summary of all this is in section 7 of

José Burgos Gil, The regulators of Beilinson and Borel (pdf)

Lax monoidalness of the Moore co-chain complex functor

Urs Schreiber:

We claim to show that the Moore cochain complex functor

C : CoS (Ab) \to {Ch}_{+}^{•} (Ab)

from cosimplicial abelian groups to cochain complexes is a lax monoidal functor with respect to the standard monoidal structures on $CoS (Ab)$ and ${Ch}_{+}^{•} (Ab)$ .

This should be old and standard, but somehow explicit statements in the literature to this extent are hard to find.(?) Some central aspects are recalled in section 7 of

José Burgos Gil, The regulators of Beilinson and Borel (pdf)

Every monoid $K$ in $CoS (An)$ – a cosimplicial ring – the Moore cochain complex $C (K)$ is naturally equipped with the structure of a monoid in ${Ch}_{+}^{•} (Ab)$ – a differential graded algebra – by letting the product $⌣ : C (K) \otimes C (K) \to C (K)$ be given by the composite

⌣ : C (K) \otimes C (K) \overset{μ_{K, K}}{\to} C (K \otimes K) \overset{C (- \cdot -)}{\to} C (K)

where

$μ_{K, K}$ is the component of the lax monoidal transformation? that exhibits the lax monoidal structure (the “compositor”);
The operation $- \cdot - : K \otimes K \to K$ is the product on $K$ .

This monoidal structure induced from a cosimplicial ring $K$ on its Moore cochain complex on $C (K)$ is the cup product.

More precisely, for $X$ a topological space and $X^{Δ_{Top}^{n}}$ the set of $n$ -simplices in $X$ , arranging themselves into the simplicial set $Π (X) = X^{Δ_{Top}^{•}}$ – the fundamental ∞-groupoid of $X$ – and for $R$ some ring let $Maps (Π (X), R)$ be the cosimplicial ring of maps $X^{Δ_{Top}^{n}} \to R$ .

Then

the Moore cochain complex $C (Maps (Π (X), R))$ is the cochain complex that computes the singular cohomology of $X$ ;
the monoid structure induced on $C (Maps (Π (X), R))$ by the lax monoidalness of the Moore cochain complex functor is the familiar cup product on singular cohomology.

We now derive this in detail.

Check.

For $A$ some abelian category write $CoS (A)$ for the category of cosimplicial objects in $A$ and ${Ch}_{+}^{•} (Ab)$ for the category of cochain complexes in $A$ concentrated in non-negative degree – called connective or $ℕ$ -graded cochain complexes.

Recall that the Moore cochain complex functor

C : CoS (A) \to {Ch}_{+}^{•} (A)

is an equivalence of categories. This is the Dold-Kan correspondence.

For $A =$ Ab with its standard monoidal category structure $(A, \otimes)$ , there are standard monoidal category structures $(CoS (A), \otimes)$ and $({Ch}_{+}^{•} (X), \otimes)$ :

for $K, L \in CoS (A)$ their tensor product $K \otimes L$ is the degreewise tensor product, i.e. the cosimplicial object with $(K \otimes L)^{n} = K^{n} \otimes L^{n}$ and with cosimplicial maps the tensor product of the cosimplicial maps in $K$ and $L$ .
for $V, W \in {Ch}_{+}^{•} (A)$ their tensor product $V \otimes W$ the graded tensor product, i.e. the cochain complex with $(V \otimes W)^{n} = \oplus_{p + q = n} V^{p} \otimes W^{q}$ whose coboundary map is given on homogeneous elements $ν \otimes ω \in V^{p} \otimes W^{q}$ by

$d (ν \otimes ω) = (d_{V} ν) \otimes ω + (-)^{p} vu \otimes d_{W} ω .$ d (\nu \otimes \omega) = (d_V \nu) \otimes \omega + (-)^{p} \vu \otimes d_W \omega \,.

With respect to these standard monoidal structures the Moore cochain complex functor $C : CoS (A) \to {Ch}_{+}^{•} (A)$ becomes a lax monoidal functor with the following lax monoidal natural transformation map $μ : C (-) \otimes C (-) \to C (- \otimes -) : CoS (A) \times CoS (A) \to {Ch}_{+}^{•} (A)$ .

Definition (lax monoidal structure on Moore cochain complex functor)

For $K, L \in CoS (A)$ define the component map

μ_{K, L} : C (K) \otimes C (L) \to C (K \otimes L)

by defining it on homogeneous elements

$a \otimes b \in C^{p} (K) \otimes C^{q} (L) \subset (C (K) \otimes C (L))^{p + k}$ by

μ_{K, L} : a \otimes b \mapsto (d_{p + 1})^{p} a) \otimes ((d_{0})^{q} b) = : a ⌣ b .

Remark

The iterated application of cosimplicial face maps on the right is to be thought of as producing a $(p + q)$ -cosimplex in $C (K \otimes L)^{p + q} = K^{p + q} \otimes L^{p + q}$ by evaluating the $p$ -cosimplex $a$ on “leftmost” simplicial $p$ -faces and the $q$ -simplex $b$ on “rightmost” simplicial $q$ -faces and then tensoring the resulting group elements.

Proposition

The $μ_{K, L}$ defined this is indeed a cochain map.

Proof

We have to check that $μ$ respect the coboundary maps in that for all $a, b$ as above we have

d \circ μ (a \otimes b) = μ \circ d (a \otimes b) .

By definition of the cup product and the differential on the Moore cochain complex we have

\begin{aligned} d \circ μ (a \otimes b) & = d (a ⌣ b) \\ = d (((d_{p + 1})^{q} a) \otimes ((d_{0})^{p} b)) \\ = \sum_{i = 0}^{p + q + 1} (- 1)^{i} d_{i} ((d_{p + 1})^{q} a) \otimes ((d_{0})^{p} b) \end{aligned}

By definition the face maps on the tensor product $K \otimes L$ are just the tensor products of the face maps of $K$ and $L$ , so that this is

\dots = \sum_{i = 0}^{p + q + 1} (- 1)^{i} (d_{i} (d_{p + 1})^{q} a) \otimes (d_{i} (d_{0})^{p} b) .

Break up this sum in three parts

\dots = \sum_{i = 0}^{p} (- 1)^{i} (d_{i} (d_{p + 1})^{q} a) \otimes (d_{i} (d_{0})^{p} b) + (- 1)^{p + 1} (d_{p + 1} (d_{p + 1})^{q} a) \otimes (d_{p + 1} (d_{0})^{p} b) + \sum_{i = p + 2}^{p + q + 1} (- 1)^{i} (d_{i} (d_{p + 1})^{q} a) \otimes (d_{i} (d_{0})^{p} b)

Now repeatedly use the simplicial identities for face maps

(i < j) \Rightarrow d_{i} \circ d_{j} = d_{j} \circ d_{i - 1}

to pass face maps from the left to the right. This yields

\dots = \sum_{i = 0}^{p} (- 1)^{i} ((d_{p + 2})^{q} d_{i} a) \otimes ((d_{0})^{p + 1} b) + (- 1)^{p + 1} ((d_{p + 1})^{q + 1} a) \otimes ((d_{0})^{p} d_{1} b) + \sum_{i = p + 2}^{p + q + 1} (- 1)^{i} ((d_{p + 1})^{q + 1} a) \otimes ((d_{0})^{p} d_{i - p} b) .

Observe that the second term may now be naturally included in the sum in the third term, leaving two sums

\dots = \sum_{i = 0}^{p} (- 1)^{i} ((d_{p + 2})^{q} d_{i} a) \otimes ((d_{0})^{p + 1} b) + \sum_{i = p + 1}^{p + q + 1} (- 1)^{i} ((d_{p + 1})^{q + 1} a) \otimes ((d_{0})^{p} d_{i - p} b) .

Reduce the summation index on the second sum by $p$ to obtain

\dots = \sum_{i = 0}^{p} (- 1)^{i} ((d_{p + 2})^{q} d_{i} a) \otimes ((d_{0})^{p + 1} b) + (- 1)^{p} \sum_{i = 1}^{q + 1} (- 1)^{i} ((d_{p + 1})^{q + 1} a) \otimes ((d_{0})^{p} d_{i} b) .

Each sum seperately is now almost the alternating sum expression of an application of the Moore complex coboundary map, except for one missing term in each sum. But the two missing terms are equal and of opposite sign, so we can add them in

\begin{aligned} \dots = & \sum_{i = 0}^{p} (- 1)^{i} ((d_{p + 2})^{q} d_{i} a) \otimes ((d_{0})^{p + 1} b) + (- 1)^{p + 1} ((d_{p + 2})^{q} d_{p + 1} a) \otimes ((d_{0})^{p + 1} b) \\ + (- 1)^{p} ((d_{p + 2})^{q} d_{p + 1} a) \otimes ((d_{0})^{p + 1} b) + (- 1)^{p} \sum_{i = 1}^{q + 1} (- 1)^{i} ((d_{p + 1})^{q + 1} a) \otimes ((d_{0})^{p} d_{i} b) \end{aligned} .

One last application of the simplicial identities in the third tirm shows that indeed this is the missing term in the sum in the fourth term. Comparing the result with the definition of coboundary map and cup product we find finally

\begin{aligned} \dots & = (d a) ⌣ b + (- 1)^{p} a ⌣ (d b) \\ = μ \circ d (a \otimes b) \end{aligned} .

Proposition

This $μ$ is indeed natural in $K, L$ . For every $f : K \to K'$ and $g : L \to L'$ we have

\begin{matrix} C (K) \otimes C (L) & \overset{μ_{K, L}}{\to} & C (K \otimes L) \\ ↓^{C (f) \otimes C (g)} & ↓^{C (f \otimes g)} \\ C (K') \otimes C (L') & \overset{μ_{K', L'}}{\to} & C (K' \otimes L') \end{matrix}

Proof

This is immediate from the fact that the morphisms $f, g$ of cosimplicial objects respect the face maps.

Proposition

The natural transformation $μ$ is indeed a lax monoidal transformation in that it is associative and unital in the required sense.

Proof

We need to show that for all $J, K, L \in CoS (A)$ we have

\begin{matrix} C (J) \otimes C (K) \otimes C (L) & \overset{{Id}_{C (J)} \otimes μ_{K, L}}{\to} & C (J) \otimes C (K \otimes L) \\ ↓ & ↓ \\ C (J \otimes K) \otimes C (L) & \overset{μ_{J, K} \otimes {Id}_{C (L)}}{\to} & C (J \otimes K \otimes L) \end{matrix} .

it is sufficient to check this on all homogeneous elements $a \otimes b \otimes c \in C^{o} (J) \otimes C^{p} (K) \otimes C^{q} (L)$ . There it is straightforward to check by using simplicial identities.

Remark

The above statement is obvious when one observes the geometric interpretation of the above remark: the image of $a \otimes b \otimes c$ along both ways around the above square is the $o + p + q$ -dimensional cosimplex that is obtained by evaluating $a$ on the leftmost $o$ -face, $p$ on the middle $p$ -face and $q$ on the rightmost $q$ -face and then tensoring the resulting group elements.

Corollary

The Moore cochain complex functor

C : CoS (Ab) \to {Ch}_{+}^{•} (Ab)

can be equipped with a lax monoidal structure with respect to the standard monoidal structure on cosimplicial abelian groups and on connective cochain complexes of abelian groups.

$E_{∞}$ -cup product on cochains on simplicial sets

At least for those cosimplicial algebras $A$ that are algebras of cochains on simplicial sets $S^{•} \in SSet$ , i.e. $A = C (S^{•}, R)$ it is known that the Moore complex dg-algebra $N^{•} (A)$ equipped with the cup product is an E-∞-algebra. See cochains on simplicial sets for details on this.

References

On top of the references already listed, here are some more.

The Alexander–Whitney/Eilenberg–Zilber equivalences for the normalized chains functor are a special case of the strong deformation retract of chain complexes that was constructed

Samuel Eilenberg, Saunders MacLane, On the groups $H (π, n)$ . II Annals of Mathematics (1954 )

For any commutative ring $R$ , they defined chain equivalences between the tensor product of the normalized chains on two simplicial R-modules and the normalized chains on their levelwise tensor product.

nLab monoidal Dold-Kan correspondence

Contents

Idea

General discussion

Simplicial rings and chain dg-algebras

Lemma

Proof

Cosimplicial rings and cochain dg-algebras

Alexander–Whitney and shuffle morphisms

Lax monoidalness of the Moore co-chain complex functor

Definition (lax monoidal structure on Moore cochain complex functor)

Remark

Proposition

Proof

Proposition

Proof

Proposition

Proof

Remark

Corollary

E ∞-cup product on cochains on simplicial sets

References

nLab
monoidal Dold-Kan correspondence

$E_{∞}$ -cup product on cochains on simplicial sets