There is a plain Dold-Kan correspondence, which 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, to some extent strictly, but generally in the sense of homotopy theory and higher category theory.

This way it extends to a Quillen equivalence between model categories of monoids in simplicial groups – simplicial rings – and monoids in chain complexes – dg-algebras.

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 Dold-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 two functors in the Dold-Kan correspondence individually respect these monoidal structures, in the sense that they are lax monoidal functors (but not strong monoidal functors, see remark 1).

However, the adjunction fails to be a monoidal adjunction of any sort (i.e. the unit and counit are not monoidal natural transformations). (Note that if it were a monoidal adjunction, then by doctrinal adjunction, both functors would necessarily be strong monoidal, and hence an equivalence of monoidal categories.) As a result, the Dold-Kan equivalence of categories does not induce an equivalence of categories or even an adjunction between (co)simplicial rings and (co)chain complexes.

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).

The cosimplicial version is made monoidal by replacing the the Moore complex functor by something else, to obtain a Quillen equivalence. The simplicial version is made monoidal by replacing the other functor by something else.

Summary of Quillen equivalences

Below we discuss the following Quillen equivalences that capture various aspects of the monoidal Dold-Kan correspondence.

(Γ⊣N):sAb⊥⟶N⟵ΓCh • +(\Gamma \dashv N) : sAb \underoverset{\underset{N}{\longrightarrow}}{\overset{\Gamma}{\longleftarrow}}{\bot} Ch_\bullet^+

as well as

(N⊣Γ):Ch • +⊥⟶Γ⟵NsAb(N \dashv \Gamma) : Ch_\bullet^+ \underoverset{\underset{\Gamma}{\longrightarrow}}{\overset{N}{\longleftarrow}}{\bot} sAb

between simplicial abelian groups and connective chain complexes;
(Γ cmon⊣N):CAlg k Δ op⊥⟶N⟵ΓCdgAlg k conn(\Gamma^{cmon} \dashv N) : CAlg_k^{\Delta^{op}} \underoverset{\underset{N}{\longrightarrow}}{\overset{\Gamma}{\longleftarrow}}{\bot} CdgAlg_{k}^{conn}

between connected simplical commutative algebras and connected differential graded-commutative algebras over a field of characteristic zero
(Γ mon⊣N):Alg k Δ op⊥⟶N⟵Γ mondgAlg k(\Gamma^{mon} \dashv N) : Alg_k^{\Delta^{op}} \underoverset{\underset{N}{\longrightarrow}}{\overset{\Gamma^{mon}}{\longleftarrow}}{\bot} dgAlg_{k}

as well as

(N mon⊣Γ):dgAlg k⊥⟶Γ⟵N monAlg k Δ op(N^{mon} \dashv \Gamma) : dgAlg_{k} \underoverset{\underset{\Gamma}{\longrightarrow}}{\overset{N^{mon}}{\longleftarrow}}{\bot} Alg_k^{\Delta^{op}}

between simplicial algebras and connective dg-algebras (neither necessarily commutative) over a commutative ring kk
(Γ mon⊣N):dgRing⊥⟶N⟵Γ monRing Δ (\Gamma^{mon} \dashv N) : dgRing \underoverset{\underset{N}{\longrightarrow}}{\overset{\Gamma^{mon}}{\longleftarrow}}{\bot} Ring^{\Delta}

between connective cochain dg-rings and cosimplicial rings
(Γ mon⊣N):Alg A Δ op⊥⟶N⟵Γ mondgAlg N(A)(\Gamma^{mon} \dashv N) : Alg_A^{\Delta^{op}} \underoverset{\underset{N}{\longrightarrow}}{\overset{\Gamma^{mon}}{\longleftarrow}}{\bot} dgAlg_{N(A)}

between simplicial algebras and connective dg-algebras (neither necessarily commutative) over a commutative simplicial ring AA and its normalized dg-ring N(A)N(A)
(Γ E ∞⊣N):E ∞Alg k Δ op⊥⟶N⟵Γ E ∞E ∞dgAlg k(\Gamma^{E_\infty} \dashv N) : E_\infty Alg_k^{\Delta^{op}} \underoverset{\underset{N}{\longrightarrow}}{\overset{\Gamma^{E_\infty}}{\longleftarrow}}{\bot} E_\infty dgAlg_{k}

between E ∞E_\infty simplicial algebras and connective E ∞E_\infty dg-algebras over a ring kk in any characteristic

Simplicial algebras and chain dg-algebras

We first discuss the extent to which the Moore complex functor is monoidal. Then we use this to discuss various Quillen equivalences on model categories of monoids that it induces.

Bilax monoidal Frobenius structure on normalized chains

Proposition

The Moore complex functor

C •:sAb→Ch • + C_\bullet \colon sAb \to Ch_\bullet^+

as well as the normalized chains/normalized Moore complex functor

N •:sAb→Ch • + N_\bullet \colon sAb \to Ch_\bullet^+

are both

symmetric\;lax monoidal functors
oplax monoidal functors

(but not oplax symmetric monoidal)
bilax monoidal functors
Frobenius monoidal functors.

The lax structure is given by the Eilenberg-Zilber map

∇ A,B:N(A)⊗N(B)→N(A⊗B). \nabla_{A,B} : N(A) \otimes N(B) \to N(A \otimes B) \,.

The oplax structure is given by the Alexander-Whitney map

Δ A,B:N(A⊗B)→N(A)⊗N(B). \Delta_{A,B} : N(A \otimes B ) \to N(A) \otimes N(B) \,.

The composite

N(A)⊗N(B)→∇ A,BN(A⊗B)→Δ A,BN(A)⊗N(B) N(A) \otimes N(B) \stackrel{\nabla_{A,B}}{\to} N(A \otimes B) \stackrel{\Delta_{A,B}}{\to} N(A) \otimes N(B)

is the identity, while the composite

N(A⊗B)→Δ A,BN(A)⊗N(B)→∇ A,BN(A⊗B) N(A \otimes B) \stackrel{\Delta_{A,B}}{\to} N(A) \otimes N(B) \stackrel{\nabla_{A,B}}{\to} N(A \otimes B)

is a chain homotopy equivalence.

Apparently the basic result (without the bilax and Frobenius structure) appears in (MacLaneHomology). The AW/EZ equivalences for the normalized chains functor are a special case of the strong deformation retract of chain complexes that was constructed in (EilenbergMacLane). A review of the proof of symmetric lax monoidalness can be found also in section 8.5.4 of (Weibel). The bilax monoidal and Frobenius structure is described in chapter 5 of (AguiarMahajan). The Frobenius structure has also been observed independently by Kathryn Hess and Steve Lack. See also section 2.3 of (SchwedeShipley).

Proof

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

∇ A,B:(N •A)⊗(N •B)→N •(A⊗B) \nabla_{A,B} : (N_\bullet A) \otimes (N_\bullet B) \to N_\bullet (A \otimes B)

on a pair A,BA,B of simplicial abelian groups is the morphism of chain complexes that sends homogeneous elements a p⊗b q∈A p⊗B q=:C p(A)⊗C q(B)a_p\otimes b_q \in A_p \otimes B_q =: C_p(A) \otimes C_q(B) to

∇ A,B(a⊗b)=∑ (μ,ν)sign(μ,ν)(s νa)⊗s μ(b)∈C p+q(A⊗B)=A p+q⊗B p+q. \nabla_{A,B}(a \otimes b) = \sum_{(\mu,\nu)} sign(\mu,\nu) (s_\nu a) \otimes s_\mu(b) \in C_{p+q}(A \otimes B) = A_{p+q}\otimes B_{p+q} \,.

Here the sum is over all (p,q)(p,q)-shuffles, i.e. permutation {μ 1,…,μ p,ν 1,⋯,ν q}\{\mu_1, \dots, \mu_p, \nu_1, \cdots, \nu_q\} of the set {0,1,⋯,p+q−1}\{0,1,\cdots,p+q-1\} that leave the first pp and the last qq elements in their natural order.

The sign in the above sum is the corresponding sign of this permutation and the degeneracy maps s μs_\mu and s νs_\nu denote the maps

s μ:=s μ p⋯∘s μ 1 s_\mu := s_{\mu_p} \cdots \circ s_{\mu_1}

and similarly for s νs_\nu

(Hm, is that consistent?)

Corollary

Since the normalized Moore complex functor N •N_\bullet is an equivalence of categories, by doctrinal adjunction its inverse nerve functor Γ:Ch +→sAb\Gamma : Ch_+ \to sAb also acquires a lax monoidal and a oplax symmetric monoidal structure.

For more details see oplax monoidal functor.

This implies that generalized Eilenberg?Mac Lane spectra?Eilenberg–Mac Lane spectra on differential graded commutative algebras are E-infinity monoids in the category of HℤH \mathbb{Z}-module spectra.

The article (Richter) shows that the inverse Ξ\Xi 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.

Quillen equivalences

We discuss Quillen equivalences revolving around the monoidal Dold-Kan correspondence. More details on their construction is below.

Write

(Γ⊣N)/(N⊣Γ):Ch • +⊥⟶Γ⟵NsAb (\Gamma \dashv N) / (N \dashv \Gamma) : Ch_\bullet^+ \underoverset{\underset{\Gamma}{\longrightarrow}}{\overset{N}{\longleftarrow}}{\bot} s Ab

for the ordinary Dold-Kan correspondence. Ch • +Ch_\bullet^+ denotes the connective chain complexes, i.e. with differential of degree -1 and concentrated in degrees ≥0\geq 0.

Since NN and Γ\Gamma are strictly inverse to each other, this may be regarded as a pair of adjoint functors in two ways. Moreover, with respect to the standard model category structures (the projective model structure on chain complexes (fibrations the degreewise surjections in positive degree) and the projective model structure on simplicial abelian groups (fibrations the underlying Kan fibration)s ) both adjunctions are Quillen equivalences.

\begin{definition} \label{MonoidalCategoryStructures} (Monoidal category strcutures) \linebreak Let

(Ch • +,⊗)(Ch_\bullet^+, \otimes) be the standard monoidal category structure on the category of chain complexes via the tensor product of chain complexes;
(sAb,⊗)(sAb, \otimes) be the monoidal category structure that is degreewise that of Ab, via the tensor product of abelian groups.

For (C,⊗)(C, \otimes) a monoidal category, write Mon(C)Mon(C) for its category of monoids, hence

Mon(Ch • +)Mon(Ch_\bullet^+) is the category of connective differential graded rings.
Mon(sAb)Mon(sAb) is the category of simplicial rings.

\end{definition}

Lemma

The DK correspondence exhibits connective dg-rings as a full subcategory of simplicial rings

Γ:Mon(Ch • +)↪Mon(sAb). \Gamma : Mon(Ch_\bullet^+) \hookrightarrow Mon(sAb) \,.

The composite functor

NΓ:Mon(Ch • +)↪Mon(Ch • +) N \Gamma : Mon(Ch_\bullet^+) \hookrightarrow Mon(Ch_\bullet^+)

is equivalent to the identity.

Simplicial kk-algebras and connective dg-algebras

This appears in section 4.2 of (SchwedeShipley).

Below in Simplicial algebras and dg-Algebras a generalization of this statement is discussed. But it is worthwhile to spell out the proof just of this special case here.

Proof

Regard the ordinary Dold-Kan correspondence

(Γ⊣N):Mod k Δ op⊥⟶N⟵ΓCh •(k) + (\Gamma \dashv N) : Mod_k^{\Delta^{op}} \underoverset{\underset{N}{\longrightarrow}}{\overset{\Gamma}{\longleftarrow}}{\bot} Ch_\bullet(k)^+

as a Quillen equivalence between the model structure on simplicial k-modules and the projective model structure on chain complexes (fibrations the degreewise surjections in positive degree) with NN regarded as the right adjoint . Notice that both NN and Γ\Gamma preserve all weak equivalences and that the unit and counit of the adjunction are isomorphisms.

Both model categories involved are monoidal model categories. We claim that the above Quillen adjunction is a monoidal Quillen adjunction with respect to this structure.

First of all NN is a lax monoidal functor with the lax monoidal transformation ∇ A,B:N(A)⊗N(B)→N(A⊗B)\nabla_{A,B} : N(A) \otimes N(B) \to N(A \otimes B) given by the Eilenberg-Zilber map. As described at oplax monoidal functor this induces an oplax monoidal structure on Γ\Gamma given by the composite

Δ X,Y:Γ(X⊗Y)→Γ(i X⊗i Y)Γ(NΓX⊗NΓY)→Γ(∇ ΓX,ΓY)ΓN(ΓX⊗ΓY)→ϵ ΓX⊗ΓYΓX⊗ΓY. \Delta_{X,Y} : \Gamma(X \otimes Y) \stackrel{\Gamma(i_X \otimes i_Y)}{\to} \Gamma(N \Gamma X \otimes N \Gamma Y) \stackrel{\Gamma(\nabla_{\Gamma X, \Gamma Y})}{\to} \Gamma N(\Gamma X \otimes \Gamma Y) \stackrel{\epsilon_{\Gamma X \otimes \Gamma Y}}{\to} \Gamma X \otimes \Gamma Y \,.

Now the Eilenberg-Zilber map ∇\nabla is (as discussed there) a chain homotopy equivalence and i ⋅i_\cdot and ϵ ⋅\epsilon_\cdot are even isomorphisms. Since Γ\Gamma preserves all weak equivalences, it follows that Δ X,Y\Delta_{X,Y} is a weak equivalence.

This shows that (Γ⊣N)(\Gamma \dashv N) is a monoidal Quillen equivalence. Moreover, by standard facts the transferred model structures on Mon(Ch •(g) +)=dgAlg k +Mon(Ch_\bullet(g)^+) = dgAlg_k^+ and Mon(Mod k Δ op)Alg k Δ opMon(Mod_k^{\Delta^{op}}) Alg_k^{\Delta^{op}} exist as indicated.

Therefore with the theorem on lifts to monoids described at monoidal Quillen adjunction the claim follows.

But not only is (Γ⊣N)(\Gamma \dashv N) but also (N⊣Γ)(N \dashv \Gamma). This yields:

Proposition

For kk a commutative ring, there is a Quillen equivalence

(N mon⊣Γ):dgAlg k⊥⟶Γ⟵N monAlg k Δ op (N^{mon} \dashv \Gamma) \colon dgAlg_k \underoverset{\underset{\Gamma}{\longrightarrow}}{\overset{N^{mon}}{\longleftarrow}}{\bot} Alg_k^{\Delta^{op}}

for the above model structures.

Simplicial A •A_\bullet-algebras and connective dg-algebras

Theorem

For AA a commutative simplicial ring there is a Quillen equivalence

(Q⊣N):Alg A⊥⟶N⟵QAlg N(A) (Q \dashv N) : Alg_A \underoverset{\underset{N}{\longrightarrow}}{\overset{Q}{\longleftarrow}}{\bot} Alg_{N(A)}

between simplicial AA-algebras and connective differential N(A)N(A)-algebras, where the right adjoint in the normalization functor, but the left adjoint is not Γ\Gamma.

This is the main theorem in (SchwedeShipley).

Commutative and E ∞E_\infty-algebras

Notice that the above statement is not formulated for commutative monoids. But

Theorem

For kk a field of characteristic 0 there is a Quillen equivalence

(Q⊣N):sAlg k C⊥⟶N⟵QdgAlg k C (Q \dashv N) \colon sAlg^C_k \underoverset{\underset{N}{\longrightarrow}}{\overset{Q}{\longleftarrow}}{\bot} dgAlg^C_k

between connected commutative connective dg-algebras over kk and connected commutative simplicial algebras over kk

Here connected means: trivial in degree k≤0k \leq 0 (= “reduced”).

This was originally mentioned in (Quillen 69, remark on p. 223) as following in a way analogous to the Quillen equivalence between L ∞L_\infty-algebras and dg-Lie algebras. Further remarks on this are in Fresse, 6.4.1a, 6.4.5

In arbitrary characteristic we have instead

Theorem

For kk a commutative ring, there is a Quillen equivalence

dgAlg k E ∞→←sAlg k E ∞ dgAlg^{E_\infty}_k \stackrel{\overset{}{\leftarrow}}{\underset{}{\to}} sAlg^{E_\infty}_k

between connective dg-E-∞ algebra over kk and simplicial E ∞E_\infty-algebras over kk.

This is in (Mandell).

Here the model category structures are analogous to those before: for simplicial E ∞E_\infty-algebras the weak equivalences and fibrations are those of the underlying simplicial sets, and for connective dg-E ∞E_\infty-algebras they are the underlying quasi-isomorphisms and the underlying positive degreewise surjections.

HℤH \mathbb{Z}-module spectra and unbounded dg-algebras

An unbounded (“stable”) analog of the monoidal Dold-Kan correspondence is:

there is a Quillen equivalence

HℤAlg≃dgAlg ℤ H \mathbb{Z} Alg \simeq dgAlg_{\mathbb{Z}}

between the model structure on Eilenberg-MacLane spectrum-algebra spectra and the model structure on dg-algebras (unbounded). See algebra spectrum for more on this.

Cosimplicial algebras 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 ∞\infty-monoidal functor in a suitable sense.

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

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 ∞\infty-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.

Lax and oplax monoidalness of conormalized cochains

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,BA, B cosimplicial abelian groups, CAC A and CBC B their Moore cochain complexes,

the Alexander–Whitney morphism is the morphism

NA⊗NB→N(A⊗B) N A \otimes N B \to N(A \otimes B)

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

x⊗y↦δ n∘⋯∘δ p+1(x)⊗δ 0∘⋯∘δ 0(y). x \otimes y \mapsto \delta^n \circ \cdots \circ \delta^{p+1}(x) \;\;\otimes \;\; \delta^0 \circ \cdots \circ \delta^0 (y) \,.

Notice that for A=BA = 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⊗B)→CA⊗CB C(A \otimes B) \to C A \otimes C B

and is given on homogeneous elements as above by

x⊗y↦∑ (μ,ν)ϵ(μ,ν)σ ν 1−1∘⋯∘σ ν q−1(x)⊗σ μ 1−1∘⋯∘σ μ p−1(y) x \otimes y \mapsto \sum_{(\mu,\nu)} \epsilon(\mu,\nu) \sigma^{\nu_1 - 1} \circ \cdots \circ \sigma^{\nu_q - 1} (x) \;\; \otimes \;\; \sigma^{\mu_1 - 1} \circ \cdots \circ \sigma^{\mu_p - 1}(y)

where the sum is over all (p,q)(p,q) shuffles (μ,ν)(\mu,\nu) and ϵ(μ,ν)\epsilon(\mu,\nu) 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 NAN A of CAC A and hence induce also morphisms

AW:NA⊗NB→N(A⊗B) AW : N A \otimes N B \to N(A \otimes B)

and

S:N(A⊗B)→NA⊗NB. S : N(A \otimes B) \to N A \otimes N B \,.

In this form they satisfy

S∘AW=Id. S \circ AW = Id \,.

See for instance theorem 2.1.a in (EilenbergMacLane).

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)→Ch + •(Ab) C : CoS(Ab) \to Ch_+^\bullet(Ab)

from cosimplicial abelian groups to cochain complexes is a lax monoidal functor with respect to the standard monoidal structures on CoS(Ab)CoS(Ab) and Ch + •(Ab)Ch_+^\bullet(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 KK in CoS(An)CoS(An) – a cosimplicial ring – the Moore cochain complex C(K)C(K) is naturally equipped with the structure of a monoid in Ch + •(Ab)Ch_+^\bullet(Ab) – a differential graded algebra – by letting the product ⌣:C(K)⊗C(K)→C(K)\smile : C(K)\otimes C(K) \to C(K) be given by the composite

⌣:C(K)⊗C(K)→μ K,KC(K⊗K)→C(−⋅−)C(K) \smile : C(K)\otimes C(K) \stackrel{\mu_{K,K}}{\to} C(K \otimes K) \stackrel{C(-\cdot -)}{\to} C(K)

where

μ K,K\mu_{K,K} is the component of the lax monoidal transformation that exhibits the lax monoidal structure (the “compositor”);
The operation −⋅−:K⊗K→K-\cdot - : K \otimes K \to K is the product on KK.

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

More precisely, for XX a topological space and X Δ Top nX^{\Delta^n_{Top}} the set of nn-simplices in XX, arranging themselves into the simplicial set Π(X)=X Δ Top •\Pi(X) = X^{\Delta^\bullet_{Top}} – the fundamental ∞-groupoid of XX – and for RR some ring let Maps(Π(X),R)Maps(\Pi(X),R) be the cosimplicial ring of maps X Δ Top n→RX^{\Delta^n_{Top}} \to R.

Then

the Moore cochain complex C(Maps(Π(X),R))C(Maps(\Pi(X),R)) is the cochain complex that computes the singular cohomology of XX;
the monoid structure induced on C(Maps(Π(X),R))C(Maps(\Pi(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 AA some abelian category write CoS(A)CoS(A) for the category of cosimplicial objects in AA and Ch + •(Ab)Ch_+^\bullet(Ab) for the category of cochain complexes in AA concentrated in non-negative degree – called connective or ℕ\mathbb{N}-graded cochain complexes.

Recall that the Moore cochain complex functor

C:CoS(A)→Ch + •(A) C : CoS(A) \to Ch_+^\bullet(A)

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

For A=A = Ab with its standard monoidal category structure (A,⊗)(A,\otimes), there are standard monoidal category structures (CoS(A),⊗)(CoS(A), \otimes) and (Ch + •(X),⊗)(Ch_+^\bullet(X), \otimes):

for K,L∈CoS(A)K,L \in CoS(A) their tensor product K⊗LK \otimes L is the degreewise tensor product, i.e. the cosimplicial object with (K⊗L) n=K n⊗L n(K \otimes L)^n = K^n \otimes L^n and with cosimplicial maps the tensor product of the cosimplicial maps in KK and LL.
for V,W∈Ch + •(A)V,W \in Ch_+^\bullet(A) their tensor product V⊗WV \otimes W the graded tensor product, i.e. the cochain complex with (V⊗W) n=⊕ p+q=nV p⊗W q(V \otimes W)^n = \oplus_{p+q = n} V^p \otimes W^q whose coboundary map is given on homogeneous elements ν⊗ω∈V p⊗W q\nu \otimes \omega\in V^p \otimes W^q by

d(ν⊗ω)=(d Vν)⊗ω+(−) pν⊗d Wω. d (\nu \otimes \omega) = (d_V \nu) \otimes \omega + (-)^{p} \nu \otimes d_W \omega \,.

With respect to these standard monoidal structures the Moore cochain complex functor C:CoS(A)→Ch + •(A)C : CoS(A) \to Ch_+^\bullet(A) becomes a lax monoidal functor with the following lax monoidal natural transformation map μ:C(−)⊗C(−)→C(−⊗−):CoS(A)×CoS(A)→Ch + •(A)\mu : C(-) \otimes C(-) \to C(-\otimes -) : CoS(A) \times CoS(A) \to Ch_+^\bullet(A) .

Definition (lax monoidal structure on Moore cochain complex functor)

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

μ K,L:C(K)⊗C(L)→C(K⊗L) \mu_{K,L} : C(K)\otimes C(L) \to C(K \otimes L)

by defining it on homogeneous elements

a⊗b∈C p(K)⊗C q(L)⊂(C(K)⊗C(L)) p+ka\otimes b \in C^p(K)\otimes C^q(L) \subset (C(K)\otimes C(L))^{p+k} by

μ K,L:a⊗b↦(d p+1) pa)⊗((d 0) qb)=:a⌣b. \mu_{K,L} : a\otimes b \mapsto (d_{p+1})^p a) \otimes ((d_0)^q b) =: a \smile b \,.

Proposition

The μ K,L\mu_{K,L} defined this way is indeed a cochain map.

Proof

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

d∘μ(a⊗b)=μ∘d(a⊗b). d \circ \mu (a \otimes b) = \mu \circ d (a \otimes b) \,.

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

d∘μ(a⊗b) =d(a⌣b) =d(((d p+1) qa)⊗((d 0) pb)) =∑ i=0 p+q+1(−1) id i((d p+1) qa)⊗((d 0) pb) \begin{aligned} d \circ \mu (a \otimes b) &= d (a \smile 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⊗LK \otimes L are just the tensor products of the face maps of KK and LL, so that this is

⋯=∑ i=0 p+q+1(−1) i(d i(d p+1) qa)⊗(d i(d 0) pb). \cdots = \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

⋯=∑ i=0 p(−1) i(d i(d p+1) qa)⊗(d i(d 0) pb)+(−1) p+1(d p+1(d p+1) qa)⊗(d p+1(d 0) pb)+∑ i=p+2 p+q+1(−1) i(d i(d p+1) qa)⊗(d i(d 0) pb) \cdots = \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)⇒d i∘d j=d j∘d i−1 (i \lt 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

⋯=∑ i=0 p(−1) i((d p+2) qd ia)⊗((d 0) p+1b)+(−1) p+1((d p+1) q+1a)⊗((d 0) pd 1b)+∑ i=p+2 p+q+1(−1) i((d p+1) q+1a)⊗((d 0) pd i−pb). \cdots = \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

⋯=∑ i=0 p(−1) i((d p+2) qd ia)⊗((d 0) p+1b)+∑ i=p+1 p+q+1(−1) i((d p+1) q+1a)⊗((d 0) pd i−pb). \cdots = \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 pp to obtain

⋯=∑ i=0 p(−1) i((d p+2) qd ia)⊗((d 0) p+1b)+(−1) p∑ i=1 q+1(−1) i((d p+1) q+1a)⊗((d 0) pd ib). \cdots = \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

⋯= ∑ i=0 p(−1) i((d p+2) qd ia)⊗((d 0) p+1b)+(−1) p+1((d p+2) qd p+1a)⊗((d 0) p+1b) +(−1) p((d p+2) qd p+1a)⊗((d 0) p+1b)+(−1) p∑ i=1 q+1(−1) i((d p+1) q+1a)⊗((d 0) pd ib). \begin{aligned} \cdots =& \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

⋯ =(da)⌣b+(−1) pa⌣(db) =μ∘d(a⊗b). \begin{aligned} \cdots &= (d a) \smile b + (-1)^p a \smile (d b) \\ &= \mu \circ d (a \otimes b) \end{aligned} \,.

Proposition

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

C(K)⊗C(L) →μ K,L C(K⊗L) ↓ C(f)⊗C(g) ↓ C(f⊗g) C(K′)⊗C(L′) →μ K′,L′ C(K′⊗L′) \array{ C(K) \otimes C(L) &\stackrel{\mu_{K,L}}{\to}& C(K \otimes L) \\ \;\;\;\downarrow^{C(f)\otimes C(g)} && \;\;\;\downarrow^{C(f \otimes g)} \\ C(K') \otimes C(L') &\stackrel{\mu_{K',L'}}{\to}& C(K' \otimes L') }

Proof

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

Proposition

The natural transformation μ\mu 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∈CoS(A)J,K,L \in CoS(A) we have

C(J)⊗C(K)⊗C(L) →Id C(J)⊗μ K,L C(J)⊗C(K⊗L) ↓ ↓ C(J⊗K)⊗C(L) →μ J,K⊗Id C(L) C(J⊗K⊗L). \array{ C(J) \otimes C(K) \otimes C(L) &\stackrel{Id_{C(J)} \otimes \mu_{K,L}}{\to}& C(J)\otimes C(K \otimes L) \\ \downarrow && \downarrow \\ C(J \otimes K) \otimes C(L) &\stackrel{\mu_{J,K} \otimes Id_{C(L)}}{\to}& C(J\otimes K \otimes L) } \,.

it is sufficient to check this on all homogeneous elements a⊗b⊗c∈C o(J)⊗C p(K)⊗C q(L)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.

Corollary

The Moore cochain complex functor

C:CoS(Ab)→Ch + •(Ab) C : CoS(Ab) \to Ch_+^\bullet(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 ∞E_\infty-cup product on cochains on simplicial sets

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

Quillen equivalences

Theorem

There is a Quillen equivalence

(Γ mon⊣N):Ring Δ⊥⟶N⟵Γ mondgRing (\Gamma^{mon} \dashv N) : Ring^{\Delta} \underoverset{\underset{N}{\longrightarrow}}{\overset{\Gamma^{mon}}{\longleftarrow}}{\bot} dgRing

between connective cochain dg-rings and cosimplicial rings.

This is the main theorem in CastiglioniCortinas.

A version of this equivalence (refined to an equivalence between dg-dg-algebras and cosimplicial dg-algebras) is in (Pridham, theorem 4.26). There is also discussed how this is a Quillen eqivalence between two realizations of a model structure for L-infinity algebras.

Dold-Kan correspondence
monoidal Dold-Kan corespondence
operadic Dold-Kan correspondence

References

Original results are in

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

Saunders MacLane, Homology

A classical textbook reference is

Charles Weibel, An introduction to homological algebra,

This discusses the symmetric monoidalness of the chains functor in section 8.5.4.

The bilax monoidalness and Frobenius monoidalness of the normalized chains/Moore complex functor is discussed in section 5.4.2 of

Marcelo Aguiar, Swapneel Mahajan,

Coxeter Groups and Hopf Algebras Fields Institute Monographs, vol. 23 (pdf)

Monoidal Functors, Species and Hopf Algebras,

Explicit details on many constructions of the lax/oplax structure are given in section 11 of

Tim Porter, Crossed Menagerie

The Quillen equivalence between connective chain dg-algebras and simplicial algebras is discussed in

Stefan Schwede, Brooke Shipley, Equivalences of monoidal model categories , Algebr. Geom. Topol. 3 (2003), 287–334 (arXiv:math.AT/0209342)

The equivalence between connected cochain dg-algebras and cosimplicial algebras is discussed in

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, doi:10.1016/j.jpaa.2003.11.009)

The equivalence between cosimplicial commutative algebras and commutative cochain dg-algebras in non-negative degree, over a field of char 0 is discussed in

Vladimir Hinich, Vadim Schechtman, Homotopy limits of homotopy algebras, in K-theory: algebra, geometry, arithmetic, Lecture Notes in Mathematics 1289, Springer, Berlin 240 − 264
Jonathan Pridham, Unifying derived deformation theories, Adv. Math. 224 (2010), no.3, 772-826 (arXiv:0705.0344)

and the analogous statement for the simplicial case, with hints for its proof, is also in

Benoit Fresse, Homotopy of Operads and Grothendieck-Teichmüller Groups, chapter 6: Differential graded commutative algebras and cosimplicial algebras (pdf)

The Quillen equivalence between connected simplicial commutative algebras and connected commutative dg-algebras in characteristic 0 is indicated all the way back on p. 223 of

Dan Quillen, Rational homotopy theory The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 (JSTOR)

The Quillen equivalence between E ∞E_\infty dg-algebras and E ∞E_\infty simplicial algebras is in

Michael Mandell, Topological André-Quillen Cohomology and E ∞E_\infty André-Quillen Cohomology Adv. in Math., Adv. Math. 177 (2) (2003) 227–279

and

Birgit Richter

Symmetry properties of the Dold-Kan correspondence (pdf)

Homotopy algebras and the inverse of the normalization functor (pdf)

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)H(\pi, n). II Annals of Mathematics (1954 )

For any commutative ring RR, 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.

monoidal Dold-Kan correspondence (changes) in nLab

Context

Homological algebra

Higher algebra

Algebraic theories

Algebras and modules

Higher algebras

Model category presentations

Geometry on formal duals of algebras

Theorems

Contents

Idea

Summary of Quillen equivalences

Simplicial algebras and chain dg-algebras

Bilax monoidal Frobenius structure on normalized chains

Proposition

Proof

Corollary

Quillen equivalences

Lemma

Simplicial kk-algebras and connective dg-algebras

Proof

Proposition

Simplicial A •A_\bullet-algebras and connective dg-algebras

Theorem

Commutative and E ∞E_\infty-algebras

Theorem

Theorem

HℤH \mathbb{Z}-module spectra and unbounded dg-algebras

Cosimplicial algebras and cochain dg-algebras

Lax and oplax monoidalness of conormalized cochains

Alexander–Whitney and shuffle morphisms

Lax monoidalness of the Moore co-chain complex functor

Definition (lax monoidal structure on Moore cochain complex functor)

Proposition

Proof

Proposition

Proof

Proposition

Proof

Corollary

E ∞E_\infty-cup product on cochains on simplicial sets

Quillen equivalences

Theorem

References