and
nonabelian homological algebra
symmetric monoidal (∞,1)-category of spectra
The monoidal Dold-Kan correspondence relates simplicial algebras with differential graded algebras.
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
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.
Below we discuss the following Quillen equivalences that capture various aspects of the monoidal Dold-Kan correspondence.
$(\Gamma \dashv N) : sAb \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{N}{\to}} Ch_\bullet^+$
as well as
$(N \dashv \Gamma) : Ch_\bullet^+ \stackrel{\overset{N}{\leftarrow}}{\underset{\Gamma}{\to}} sAb$
between simplicial abelian groups and connective chain complexes;
$(\Gamma^{cmon} \dashv N) : CAlg_k^{\Delta^{op}} \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{N}{\to}} CdgAlg_{k}^{conn}$
between connected commutative simplicial algebras and connected commutative dg-algebras over a field of characteristic 0
$(\Gamma^{mon} \dashv N) : Alg_k^{\Delta^{op}} \stackrel{\overset{\Gamma^{mon}}{\leftarrow}}{\underset{N}{\to}} dgAlg_{k}$
as well as
$(N^{mon} \dashv \Gamma) : dgAlg_{k}\stackrel{\overset{N^{mon}}{\leftarrow}}{\underset{\Gamma}{\to}} Alg_k^{\Delta^{op}}$
between simplicial algebras and connective dg-algebras (neither necessarily commutative) over a commutative ring $k$
$(\Gamma^{mon} \dashv N) : dgRing \stackrel{\overset{\Gamma^{mon}}{\leftarrow}}{\underset{N}{\to}} Ring^{\Delta}$
between connective cochain dg-rings and cosimplicial rings
$(\Gamma^{mon} \dashv N) : Alg_A^{\Delta^{op}} \stackrel{\overset{\Gamma^{mon}}{\leftarrow}}{\underset{N}{\to}} dgAlg_{N(A)}$
between simplicial algebras and connective dg-algebras (neither necessarily commutative) over a commutative simplicial ring $A$ and its normalized dg-ring $N(A)$
$(\Gamma^{E_\infty} \dashv N) : E_\infty Alg_k^{\Delta^{op}} \stackrel{\overset{\Gamma^{E_\infty}}{\leftarrow}}{\underset{N}{\to}} E_\infty dgAlg_{k}$
between $E_\infty$ simplicial algebras and connective $E_\infty$ dg-algebras over a ring $k$ in any characteristic
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.
The Moore complex functor
as well as the normalized chains/normalized Moore complex functor
are both
(but not oplax symmetric monoidal)
The lax structure is given by the Alexander-Whitney map
The oplax structure is given by the Eilenberg-Zilber map
The composite
is the identity, while the composite
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).
The lax monoidal transformation that exhibits the lax-monoidalness of the Moore chain complex functor is the shuffle map. Its component
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
Here the sum is over all $(p,q)$-shuffles, i.e. permutation $\{\mu_1, \dots, \mu_p, \nu_1, \cdots, \nu_q\}$ of the set $\{0,1,\cdots,p+q-1\}$ 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 $s_\mu$ and $s_\nu$ denote the maps
and similarly for $s_\nu$
(Hm, is that consistent?)
Since the normalized Moore complex functor $N_\bullet$ is an equivalence of categories, by doctrinal adjunction its inverse nerve functor $\Gamma : Ch_+ \to sAb$ also acquires a lax monoidal and a oplax symmetric monoidal structure.
For more details see oplax monoidal functor.
The upshot is that $N$ and $G$ are both pretty close to being strong monoidal functors, but fail to be so. If they were, the monoidal Dold-Kan correspondence would be a simple corollary of the Dold-Kan correspondence and would hold at the level of 1-categories.
Explicitly, the failure of $N$ to be strong monoidal is in that the Eilenberg-Zilber map is (on normalized chain complexes) a right inverse to the Alexander-Whitney map, but not a left inverse. But it is a homotopy-inverse: because the components of the Alexander-Whitney map are (as discussed there) quasi-isomorphisms. By 2-out-of-3 it follows that also the EZ-maps are quasi-isomorphisms and that these are indeed inverse to the AW map in the homotopy category of chain complexes (the derived category).
Therefore we expect that the monoidal Dold-Kan correspondence holds, while not necessarily at the level of ordinary categories, at least at the level of homotopical categories. This is indeed the case, as discussed below.
Note that
the oplax structure of $N$
and the lax structure of $\Gamma$
are not symmetric monoidal functors, i.e. they do not respect the symmetric monoidal category structure. However, this, too, they do respect up to homotopy, i.e. they are E-infinity monoidal functors in a suitable sense. This is shown in (Richter).
This implies that generalized Eilenberg–Mac Lane spectra on differential graded commutative algebras are E-infinity monoids in the category of $H \mathbb{Z}$-module spectra.
The article (Richter) shows that the inverse $\Xi$ 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.
We discuss Quillen equivalences revolving around the monoidal Dold-Kan correspondence. More details on their construction is below.
Write
for the ordinary Dold-Kan correspondence. $Ch_\bullet^+$ denotes the connective chain complexes: non-negatively graded.
Since $N$ and $\Gamma$ 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.
Let
$(Ch_\bullet^+, \otimes)$ be the standard monoidal category structure on the category of chain complexes;
And let $(sAb, \otimes)$ be the monoidal category structure that is degreewise that of Ab.
For $(C, \otimes)$ a monoidal category, write $Mon(C)$ for its category of monoids. Notice that
$Mon(Ch_\bullet^+)$ is the category of connective differential graded rings.
$Mon(sAb)$ is the category of simplicial rings.
The DK correspondence exhibits connective dg-rings as a full subcategory of simplicial rings
The composite functor
is equivalent to the identity.
For $k$ a commutative ring, there is a Quillen equivalence
between
the model structure on dg-algebras for connective dg-algebras over $k$: weak equivalences are the quasi-isomorphisms, fibrations the degreewise surjections in positive degree
and the model structure on simplicial algebras over $k$: weak equivalences and fibrations are those of the underlying simplicial sets in the standard model structure on simplicial sets.
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.
Regard the ordinary Dold-Kan correspondence
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 $N$ regarded as the right adjoint . Notice that both $N$ and $\Gamma$ 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 $N$ is a lax monoidal functor with the lax monoidal transformation $\nabla_{A,B} : N(A) \otimes N(B) \to N(A \otimes B)$ given by the Eilenberg-Zilber map. As described at oplax monoidal functor this induces an oplax monoidal structure on $\Gamma$ given by the composite
Now the Eilenberg-Zilber map $\nabla$ is (as discussed there) a chain homotopy equivalence and $i_\cdot$ and $\epsilon_\cdot$ are even isomorphisms. Since $\Gamma$ preserves all weak equivalences, it follows that $\Delta_{X,Y}$ is a weak equivalence.
This shows that $(\Gamma \dashv N)$ is a monoidal Quillen equivalence. Moreover, by standard facts the transferred model structures on $Mon(Ch_\bullet(g)^+) = dgAlg_k^+$ and $Mon(Mod_k^{\Delta^{op}}) Alg_k^{\Delta^{op}}$ exist as indicated.
Therefore with the theorem on lifts to monoids described at monoidal Quillen adjunction the claim follows.
But not only is $(\Gamma \dashv N)$ but also $(N \dashv \Gamma)$. This yields:
For $k$ a commutative ring, there is a Quillen equivalence
for the above model structures.
We check that
is a monoidal Quillen adjunction: the oplax monoidal structure on the left adjoint $N$ is given by the Alexander-Whitney map, which is a weak equivalence, as discussed there.
Therefore with the theorem on lifts to monoids described at monoidal Quillen adjunction the claim follows.
For $A$ a commutative simplicial ring there is a Quillen equivalence
between simplicial $A$-algebras and connective differential $N(A)$-algebras, where the right adjoint in the normalization functor, but the left adjoint is not $\Gamma$.
This is the main theorem in (SchwedeShipley).
Notice that the above statement is not formulated for commutative monoids. But
For $k$ a field of characteristic 0 there is a Quillen equivalence
between connected commutative connective dg-algebras over $k$ and connected commutative simplicial algebras over $k$
Here connected means: trivial in degree $k \leq 0$ (= “reduced”).
This is due to the remark on p. 223 of (Quillen).
In arbitrary characteristic we have instead
For $k$ a commutative ring, there is a Quillen equivalence
between connective dg-E-∞ algebra over $k$ and simplicial $E_\infty$-algebras over $k$.
This is in (Mandell).
Here the model category structures are analogous to those before: for simplicial $E_\infty$-algebras the weak equivalences and fibrations are those of the underlying simplicial sets, and for connective dg-$E_\infty$-algebras they are the underlying quasi-isomorphisms and the underlying positive degreewise surjections.
An unbounded (“stable”) analog of the monoidal Dold-Kan correspondence is:
there is a Quillen equivalence
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.
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.
A central ingredient in the monoidal Dold-Kan correspondence are the Alexander–Whitney and the shuffle morphisms.
See chapter VI, paragraph 12 of
or chapter VIII, paragraph 8 of
For $A, B$ cosimplicial abelian groups, $C A$ and $C B$ their Moore cochain complexes,
the Alexander–Whitney morphism is the morphism
that is given on homogeneous elements $x, y$ of degree $p,q$, respectively, by
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
and is given on homogeneous elements as above by
where the sum is over all $(p,q)$ shuffles $(\mu,\nu)$ and $\epsilon(\mu,\nu)$ 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 $N A$ of $C A$ and hence induce also morphisms
and
In this form they satisfy
See for instance theorem 2.1.a in (EilenbergMacLane).
A quick summary of all this is in section 7 of
We claim to show that the Moore cochain complex functor
from cosimplicial abelian groups to cochain complexes is a lax monoidal functor with respect to the standard monoidal structures on $CoS(Ab)$ and $Ch_+^\bullet(Ab)$.
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
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 $Ch_+^\bullet(Ab)$ – a differential graded algebra – by letting the product $\smile : C(K)\otimes C(K) \to C(K)$ be given by the composite
where
$\mu_{K,K}$ 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 $X^{\Delta^n_{Top}}$ the set of $n$-simplices in $X$, arranging themselves into the simplicial set $\Pi(X) = X^{\Delta^\bullet_{Top}}$ – the fundamental ∞-groupoid of $X$ – and for $R$ some ring let $Maps(\Pi(X),R)$ be the cosimplicial ring of maps $X^{\Delta^n_{Top}} \to R$.
Then
the Moore cochain complex $C(Maps(\Pi(X),R))$ is the cochain complex that computes the singular cohomology of $X$;
the monoid structure induced on $C(Maps(\Pi(X),R))$ 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 $A$ some abelian category write $CoS(A)$ for the category of cosimplicial objects in $A$ and $Ch_+^\bullet(Ab)$ for the category of cochain complexes in $A$ concentrated in non-negative degree – called connective or $\mathbb{N}$-graded cochain complexes.
Recall that the Moore cochain complex functor
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 $(Ch_+^\bullet(X), \otimes)$:
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 $V,W \in Ch_+^\bullet(A)$ 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 $\nu \otimes \omega\in V^p \otimes W^q$ by
With respect to these standard monoidal structures the Moore cochain complex functor $C : CoS(A) \to Ch_+^\bullet(A)$ becomes a lax monoidal functor with the following lax monoidal natural transformation map $\mu : C(-) \otimes C(-) \to C(-\otimes -) : CoS(A) \times CoS(A) \to Ch_+^\bullet(A)$ .
For $K,L \in CoS(A)$ define the component map
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
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.
The $\mu_{K,L}$ defined this way is indeed a cochain map.
We have to check that $\mu$ respect the coboundary maps in that for all $a,b$ as above we have
By definition of the cup product and the differential on the Moore cochain complex we have
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
Break up this sum in three parts
Now repeatedly use the simplicial identities for face maps
to pass face maps from the left to the right. This yields
Observe that the second term may now be naturally included in the sum in the third term, leaving two sums
Reduce the summation index on the second sum by $p$ to obtain
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
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
This $\mu$ is indeed natural in $K,L$. For every $f : K \to K'$ and $g : L\to L'$ we have
This is immediate from the fact that the morphisms $f, g$ of cosimplicial objects respect the face maps.
The natural transformation $\mu$ is indeed a lax monoidal transformation in that it is associative and unital in the required sense.
We need to show that for all $J,K,L \in CoS(A)$ we have
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.
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.
The Moore cochain complex functor
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.
At least for those cosimplicial algebras $A$ that are algebras of cochains on simplicial sets $S^\bullet \in SSet$, i.e. $A = C(S^\bullet, R)$ it is known that the Moore complex dg-algebra $N^\bullet(A)$ equipped with the cup product is an E-∞-algebra. See cochains on simplicial sets for details on this.
There is a Quillen equivalence
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.
monoidal Dold-Kan corespondence
Original results are in
A classical textbook reference is
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)
Explicit details on many constructions of the lax/oplax structure are given in section 11 of
The Quillen equivalence between connective chain dg-algebras and simplicial algebras is discussed in.
The Quillen equivalence between connected cochain dg-algebras and cosimplicial algebras is discussed in
Also section 4.4 of
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
The Quillen equivalence between $E_\infty$ dg-algebras and $E_\infty$ simplicial algebras is in
and
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
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.
See also the appendix in
The equivalence between cosimplicial commutative algebras and commutative cochain dg-algebras in non-negative degree, over a field of char 0 is discussed in