monoidal Dold-Kan correspondence

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#### Higher algebra
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#Contents#
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## Idea

The _monoidal Dold-Kan correspondence_ relates [[simplicial algebra]]s with [[differential graded algebra]]s.

There is a plain [[Dold-Kan correspondence]], which establishes an equivalence between (co)[[simplicial group]]s and (co)[[chain complex]]es.
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 ring]]s -- and monoids in chain complexes -- [[dg-algebra]]s.

Notice that

* monoids in the [[category of chain complexes]] are [[differential graded rings]]

* monoids in the category of simplicial abelian groups are [[simplicial object|simplicial]] [[rings]].

If instead we look at chain complexes and [[simplicial object]]s 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 \ref{RemarkOnLaxStrongDistinction}).

However, the [[adjunction]] fails to be a [[monoidal adjunction]] of any sort (i.e. the [[unit of an adjunction|unit]] and [[counit of an adjunction|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 [[monoid]]s in abelian [[simplicial group]]s (simplicial rings)  to monoids in [[chain complex]]es (chain [[dg-algebra]]s),

* the **cosimplicial version** that relates monoids in cosimplicial groups ([[cosimplicial algebra|cosimplicial ring]]) to monoidal in co-chain complexes (cochain [[dg-algebra]]s).

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 {#Summary}

Below we discuss the following [[Quillen equivalence]]s that capture various aspects of the monoidal Dold-Kan correspondence.

* $(\Gamma \dashv N) : sAb \underoverset{\underset{N}{\longrightarrow}}{\overset{\Gamma}{\longleftarrow}}{\bot} Ch_\bullet^+$

  as well as

  $(N \dashv \Gamma) : Ch_\bullet^+ \underoverset{\underset{\Gamma}{\longrightarrow}}{\overset{N}{\longleftarrow}}{\bot} sAb $

  between [[simplicial abelian groups]] and connective [[chain complexes]];

*  $(\Gamma^{cmon} \dashv N) : CAlg_k^{\Delta^{op}} \underoverset{\underset{N}{\longrightarrow}}{\overset{\Gamma}{\longleftarrow}}{\bot} CdgAlg_{k}^{conn}$

   between connected [[simplicial algebra|simplical]] [[commutative algebras]] and connected [[differential graded-commutative algebras]] over a [[field]] of [[characteristic zero]]

*  $(\Gamma^{mon} \dashv N) : Alg_k^{\Delta^{op}} \underoverset{\underset{N}{\longrightarrow}}{\overset{\Gamma^{mon}}{\longleftarrow}}{\bot} dgAlg_{k}$

   as well as

    $(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 $k$

*  $ (\Gamma^{mon} \dashv N) : dgRing \underoverset{\underset{N}{\longrightarrow}}{\overset{\Gamma^{mon}}{\longleftarrow}}{\bot}  Ring^{\Delta}$

   between connective cochain dg-rings and cosimplicial rings

*  $(\Gamma^{mon} \dashv N) : Alg_A^{\Delta^{op}} \underoverset{\underset{N}{\longrightarrow}}{\overset{\Gamma^{mon}}{\longleftarrow}}{\bot} dgAlg_{N(A)}$

   between simplicial algebras and connective dg-algebras (neither necessarily commutative) over a commutative simplicial ring $A$ and its normalized dg-ring $N(A)$

*  $(\Gamma^{E_\infty} \dashv N) : E_\infty Alg_k^{\Delta^{op}} \underoverset{\underset{N}{\longrightarrow}}{\overset{\Gamma^{E_\infty}}{\longleftarrow}}{\bot}  E_\infty dgAlg_{k}$

   between $E_\infty$ simplicial algebras and connective $E_\infty$ dg-algebras over a ring $k$ in any characteristic

## Simplicial algebras and chain dg-algebras {#simplicial}

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 {#BilaxFrobStruc}

+-- {: .num_prop #ChainsIsLaxMonoidal}
###### Proposition

The [[Moore complex]] functor

$$
  C_\bullet \colon sAb \to Ch_\bullet^+
$$

as well as the normalized chains/normalized Moore complex functor

$$
  N_\bullet \colon sAb \to Ch_\bullet^+
$$

are both

* [[symmetric monoidal functor|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]]

$$
  \nabla_{A,B} : N(A) \otimes N(B) \to N(A \otimes B)
  \,.
$$

The oplax structure is given by the [[Alexander-Whitney map]]

$$
  \Delta_{A,B} : N(A \otimes B ) \to N(A) \otimes N(B)
  \,.
$$

The composite

$$
  N(A) \otimes N(B) \stackrel{\nabla_{A,B}}{\to}
  N(A \otimes B)
  \stackrel{\Delta_{A,B}}{\to}
  N(A) \otimes N(B)
$$

is the [[identity]], while the composite

$$
  N(A \otimes B)
  \stackrel{\Delta_{A,B}}{\to}
  N(A) \otimes N(B)
   \stackrel{\nabla_{A,B}}{\to}
  N(A \otimes B)
$$

is a chain [[homotopy equivalence]].

=--

Apparently the basic result (without the bilax and Frobenius structure)  appears in ([MacLaneHomology](#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](#EilenbergMacLane)).  A review of the proof of symmetric lax monoidalness can be found also in section 8.5.4 of ([Weibel](#Weibel)).
The bilax monoidal and Frobenius structure is described in chapter 5 of ([AguiarMahajan](#AguiarMahajan)). The Frobenius structure has also been observed independently by [[Kathryn Hess]] and [[Steve Lack]].
See also section 2.3 of ([SchwedeShipley](#SchwedeShipley)).

+-- {: .proof}
###### Proof

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

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

on a pair $A,B$ of simplicial abelian groups is the morphism of [[chain complex]]es that sends homogeneous elements $a_p\otimes b_q \in A_p \otimes B_q =: C_p(A) \otimes C_q(B)$ to

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

Here the sum is over all $(p,q)$-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

$$
  s_\mu := s_{\mu_p} \cdots \circ s_{\mu_1}
$$

and similarly for $s_\nu$

>(Hm, is that consistent?)

=--

+-- {: .num_cor }
###### Corollary

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]].

+-- {: .num_remark #RemarkOnLaxStrongDistinction}
###### Remark

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-isomorphism]]s. By [[category with weak equivalences|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.

=--

+-- {: .num_remark }
###### Remark

Note that

* the oplax structure of $N$

* and the lax structure of $\Gamma$

are not [[symmetric monoidal functor]]s, 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-k-operad|E-infinity monoidal functors]] in a suitable sense.  This is shown in ([Richter](#Richter)).

=--

This implies that generalized [[Eilenberg?Mac Lane spectra]] on [[differential graded algebra|differential graded commutative algebras]] are [[E-k operad|E-infinity monoids]] in the category of $H \mathbb{Z}$-[[module]] [[spectrum|spectra]].

The article ([Richter](#Richter)) shows that the inverse $\Xi$ from chain complexes to simplicial abelian groups sends algebras over arbitrary differential graded [[E-k operad|E-infinity-operad]] to [[E-k-operad|E-infinity-algebra]] in simplicial modules, and is part of a [[Quillen adjunction]] for these.

### Quillen equivalences

We discuss [[Quillen equivalence]]s revolving around the monoidal Dold-Kan correspondence. More details on their construction is below.

Write

$$
  (\Gamma \dashv N) / (N \dashv \Gamma)
   :
  Ch_\bullet^+
    \underoverset{\underset{\Gamma}{\longrightarrow}}{\overset{N}{\longleftarrow}}{\bot}
  s Ab
$$

for the ordinary [[Dold-Kan correspondence]]. $Ch_\bullet^+$ denotes the [[connective chain complex|connective]] chain complexes, i.e. with [[differential]] of degree -1 and concentrated in degrees $\geq 0$.

Since $N$ and $\Gamma$ are strictly inverse to each other, this may be regarded as a pair of [[adjoint functor]]s 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 T-algebras|model structure on simplicial abelian groups]] (fibrations the underlying [[Kan fibration]])s ) both adjunctions are [[Quillen equivalence]]s.

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

* $(Ch_\bullet^+, \otimes)$ be the standard [[monoidal category]] structure on the [[category of chain complexes]] via the [[tensor product of chain complexes]];

* $(sAb, \otimes)$ be the [[monoidal category]] [[structure]] that is degreewise that of [[Ab]], via the [[tensor product of abelian groups]].

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

* $Mon(Ch_\bullet^+)$ is the category of [[connective chain complex|connective]] [[differential graded rings]].

* $Mon(sAb)$ is the category of [[simplicial rings]].

\end{definition}

+-- {: .num_lemma}
###### Lemma

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

$$
  \Gamma
   :
   Mon(Ch_\bullet^+)
    \hookrightarrow
   Mon(sAb)
  \,.
$$

The composite functor

$$
  N \Gamma
  :
   Mon(Ch_\bullet^+)
    \hookrightarrow
   Mon(Ch_\bullet^+)
$$

is equivalent to the identity.

=--

#### Simplicial $k$-algebras and connective dg-algebras {#QuillenEquSimpkAlgs}

+-- {: .num_prop}
###### Proposition

For $k$ a commutative ring, there is a [[Quillen equivalence]]

$$
  (\Gamma^{mon} \dashv N)
    \colon
  Alg_k^{\Delta^{op}}
    \underoverset{\underset{N}{\longrightarrow}}{\overset{\Gamma^{mon}}{\longleftarrow}}{\bot}
  dgAlg_k
$$

between

* the [[model structure on dg-algebras]] for connective [[dg-algebra]]s over $k$: weak equivalences are the [[quasi-isomorphism]]s, 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](#SchwedeShipley)).

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

+-- {: .proof}
###### Proof

Regard the ordinary [[Dold-Kan correspondence]]

$$
  (\Gamma \dashv N)
  :
  Mod_k^{\Delta^{op}}
   \underoverset{\underset{N}{\longrightarrow}}{\overset{\Gamma}{\longleftarrow}}{\bot}
  Ch_\bullet(k)^+
$$

as a [[Quillen equivalence]] between the [[model structure on simplicial T-algebras|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 [[isomorphism]]s.

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

$$
  \Delta_{X,Y}
  :
  \Gamma(X \otimes Y) \stackrel{\Gamma(i_X \otimes i_Y)}{\to}
  \Gamma(N \Gamma X \otimes N \Gamma Y)
  \stackrel{\Gamma(\nabla_{\Gamma X, \Gamma Y})}{\to}
  \Gamma N(\Gamma X \otimes \Gamma Y)
  \stackrel{\epsilon_{\Gamma X \otimes \Gamma Y}}{\to}
  \Gamma X \otimes \Gamma Y
  \,.
$$

Now the [[Eilenberg-Zilber map]] $\nabla$ is (as discussed there) a chain [[homotopy equivalence]] and $i_\cdot$ 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 structure]]s 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](monoidal+Quillen+adjunction#LiftToMonoids) described at _[[monoidal Quillen adjunction]]_ the claim follows.

=--

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

+-- {: .num_prop}
###### Proposition

For $k$ a commutative ring, there is a [[Quillen equivalence]]

$$
  (N^{mon} \dashv \Gamma)
   \colon
   dgAlg_k
      \underoverset{\underset{\Gamma}{\longrightarrow}}{\overset{N^{mon}}{\longleftarrow}}{\bot}
    Alg_k^{\Delta^{op}}
$$

for the above model structures.

=--

+-- {: .proof}
###### Proof

We check that

$$
  (N \dashv \Gamma)
    \colon
  Ch_\bullet^+(k)
    \underoverset{\underset{\Gamma}{\longrightarrow}}{\overset{N}{\longleftarrow}}{\bot}
  Mod_k^{\Delta^{op}}
$$

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](monoidal+Quillen+adjunction#LiftToMonoids) described at _[[monoidal Quillen adjunction]]_ the claim follows.

=--

#### Simplicial $A_\bullet$-algebras and connective dg-algebras {#AlgebrasOverdgAlgebra}

+-- {: .num_theorem}
###### Theorem

For $A$ a commutative [[simplicial ring]] there is a [[Quillen equivalence]]

$$
  (Q \dashv N) : Alg_A \underoverset{\underset{N}{\longrightarrow}}{\overset{Q}{\longleftarrow}}{\bot} Alg_{N(A)}
$$

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](#SchwedeShipley)).

#### Commutative and $E_\infty$-algebras

Notice that the above statement  is _not_ formulated for commutative monoids. But

+-- {: .num_theorem}
###### Theorem

For $k$ a [[field]] of [[characteristic]] 0 there is a Quillen equivalence

$$
  (Q \dashv N)
    \colon
  sAlg^C_k
    \underoverset{\underset{N}{\longrightarrow}}{\overset{Q}{\longleftarrow}}{\bot}
  dgAlg^C_k
$$

between connected commutative connective [[dg-algebra]]s over $k$ and connected commutative [[simplicial algebra]]s over $k$

Here connected means: trivial in degree $k \leq 0$ (= "reduced").

=--

This was originally mentioned in ([Quillen 69, remark on p. 223](#Quillen69)) as following in a way analogous to the Quillen equivalence between $L_\infty$-algebras and dg-Lie algebras. Further remarks on this are in [Fresse,  6.4.1a, 6.4.5](#Fresse)

In arbitrary characteristic we have instead

+-- {: .num_theorem}
###### Theorem

For $k$ a commutative ring, there is a Quillen equivalence

$$
  dgAlg^{E_\infty}_k \stackrel{\overset{}{\leftarrow}}{\underset{}{\to}}  sAlg^{E_\infty}_k
$$

between connective dg-[[E-∞ algebra]] over $k$ and simplicial $E_\infty$-algebras over $k$.

=--

This is in ([Mandell](#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.

#### $H \mathbb{Z}$-module spectra and unbounded dg-algebras

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

there is a [[Quillen equivalence]]

$$
  H \mathbb{Z} Alg \simeq dgAlg_{\mathbb{Z}}
$$

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 {#cosimplicial}

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-k-operad|E-infinity algebra]], is really the statement that the Moore cochain complex functor on cosimplicial algebras of functions on [[simplicial set]]s is an $\infty$-monoidal functor in a suitable sense.

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

This establishes not quite a [[Quillen equivalence]], but shows that the [[Dold-Kan correspondence]] induces an [[equivalence]] of [[homotopy category|homotopy categories]] for the [[model structure on cosimplicial rings]] and the [[model structure on dg-algebras|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 {#AlexWhitneyShuffle}

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 complex|Moore cochain complex]]es,

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

$$
  x \otimes y \mapsto
  \delta^n \circ \cdots \circ \delta^{p+1}(x)
  \;\;\otimes
  \;\;
  \delta^0 \circ \cdots \circ \delta^0
  (y)
  \,.
$$

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

$$
  x  \otimes y
  \mapsto
  \sum_{(\mu,\nu)}
  \epsilon(\mu,\nu)
  \sigma^{\nu_1 - 1}
  \circ \cdots
  \circ
  \sigma^{\nu_q - 1}
  (x)
  \;\; \otimes
  \;\;
  \sigma^{\mu_1 - 1} \circ \cdots \circ
  \sigma^{\mu_p - 1}(y)
$$

where the sum is over all $(p,q)$ [[shuffle]]s $(\mu,\nu)$ 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

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

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 ([EilenbergMacLane](#EilenbergMacLane)).

A quick summary of all this is in section 7 of

* José Burgos Gil, _The regulators of Beilinson and Borel_ ([pdf](http://atlas.mat.ub.es/personals/burgos/files/brbr.pdf))

#### Lax monoidalness of the Moore co-chain complex functor

+-- {: .standout}

[[Urs Schreiber]]:

We claim to show that the [[Moore complex|Moore cochain complex functor]]

$$
  C : CoS(Ab) \to Ch_+^\bullet(Ab)
$$

from [[cosimplicial abelian groups]] to cochain complexes is a  [[lax monoidal functor]] with respect to the standard monoidal structures on $CoS(Ab)$ 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](http://atlas.mat.ub.es/personals/burgos/files/brbr.pdf#page=63) of

* José Burgos Gil, _The regulators of Beilinson and Borel_ ([pdf](http://atlas.mat.ub.es/personals/burgos/files/brbr.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 $Ch_+^\bullet(Ab)$ -- a [[differential graded algebra]] -- by letting the product $\smile : C(K)\otimes C(K) \to C(K)$ be given by the composite

$$
  \smile : C(K)\otimes C(K) \stackrel{\mu_{K,K}}{\to} C(K \otimes K) \stackrel{C(-\cdot -)}{\to} C(K)
$$

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 $K$.

+-- {: .standout}

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 $n$-[[simplex|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.

+-- {: .standout}

Check.

=--

For $A$ some [[abelian category]] write
$CoS(A)$ for the category of [[simplicial object|cosimplicial object]]s
in $A$ and $Ch_+^\bullet(Ab)$ for the
[[category of chain complexes|category of cochain complexes]] in $A$
concentrated in non-negative degree -- called connective or $\mathbb{N}$-graded
cochain complexes.

Recall that the [[Moore complex|Moore cochain complex]] functor

$$
  C : CoS(A) \to Ch_+^\bullet(A)
$$

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 $(Ch_+^\bullet(X), \otimes)$:

* 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
  $$
    d (\nu \otimes \omega) = (d_V \nu) \otimes \omega + (-)^{p} \nu \otimes d_W \omega
    \,.
  $$

With respect to these standard monoidal  structures
the [[Moore complex|Moore cochain complex]] functor $C : CoS(A) \to Ch_+^\bullet(A)$
becomes a [[lax monoidal functor]] with the following lax monoidal
[[natural transformation]] map
$\mu : C(-) \otimes C(-) \to C(-\otimes -) : CoS(A) \times CoS(A) \to Ch_+^\bullet(A)$ .

+-- {: .num_defn}
###### Definition (lax monoidal structure on Moore cochain complex functor)

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

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

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

$$
  \mu_{K,L} : a\otimes b \mapsto (d_{p+1})^p a) \otimes
   ((d_0)^q b) =: a \smile b
  \,.
$$

=--

+-- {: .num_remark }
###### 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.

=--

+-- {: .num_prop }
###### Proposition

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

=--

+-- {: .proof}
###### Proof

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

$$
  d \circ \mu (a \otimes b) = \mu \circ d (a \otimes b)
  \,.
$$

By definition of the cup product and the differential on the
Moore cochain complex we have
$$
  \begin{aligned}
    d \circ \mu (a \otimes b)
    &=
    d (a \smile b)
    \\
    &= d ( ((d_{p+1})^q a) \otimes ((d_0)^p b) )
    \\
    &=
    \sum_{i=0}^{p+q+1}
    (-1)^i  d_i((d_{p+1})^q a) \otimes ((d_0)^p b)
  \end{aligned}
$$

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

$$
    \cdots
    =
    \sum_{i=0}^{p+q+1}
    (-1)^i  (d_i (d_{p+1})^q a) \otimes (d_i (d_0)^p b)
    \,.
$$

Break up this sum in three parts

$$
    \cdots
    =
    \sum_{i=0}^{p}
    (-1)^i  (d_i (d_{p+1})^q a) \otimes (d_i (d_0)^p b)
    +
    (-1)^{p+1}  (d_{p+1} (d_{p+1})^q a) \otimes (d_{p+1} (d_0)^p b)
    +
    \sum_{i=p+2}^{p+q+1}
    (-1)^i  (d_i (d_{p+1})^q a) \otimes (d_i (d_0)^p b)
$$

Now repeatedly use the [[simplicial identities]] for face maps

$$
   (i \lt j) \;\;\Rightarrow \;\; d_i \circ d_j = d_j \circ d_{i-1}
$$

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

$$
    \cdots
    =
    \sum_{i=0}^{p}
    (-1)^i  ( (d_{p+2})^q d_i a) \otimes ( (d_0)^{p+1} b)
    +
    (-1)^{p+1}  ((d_{p+1})^{q+1} a) \otimes ((d_0)^p d_1 b)
    +
    \sum_{i=p+2}^{p+q+1}
    (-1)^i  ((d_{p+1})^{q+1} a) \otimes ((d_0)^p d_{i-p}b)
    \,.
$$
Observe that the second term may now be naturally included in the
sum in the third term, leaving two sums

$$
  \cdots =
    \sum_{i=0}^{p}
    (-1)^i  ( (d_{p+2})^q d_i a) \otimes ( (d_0)^{p+1} b)
    +
    \sum_{i=p+1}^{p+q+1}
    (-1)^i  ((d_{p+1})^{q+1} a) \otimes ((d_0)^p d_{i-p}b)
    \,.
$$

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

$$
  \cdots =
    \sum_{i=0}^{p}
    (-1)^i  ( (d_{p+2})^q d_i a) \otimes ( (d_0)^{p+1} b)
    +
    (-1)^p
    \sum_{i=1}^{q+1}
    (-1)^i  ((d_{p+1})^{q+1} a) \otimes ((d_0)^p d_{i}b)
    \,.
$$

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

$$
 \begin{aligned}
  \cdots
  =&
    \sum_{i=0}^{p}
    (-1)^i  ( (d_{p+2})^q d_i a) \otimes ( (d_0)^{p+1} b)
    +
    (-1)^{p+1}  ( (d_{p+2})^q d_{p+1} a) \otimes ( (d_0)^{p+1} b)
    \\
    &
    +
    (-1)^{p}  ( (d_{p+2})^q d_{p+1} a) \otimes ( (d_0)^{p+1} b)
    +
    (-1)^p
    \sum_{i=1}^{q+1}
    (-1)^i  ((d_{p+1})^{q+1} a) \otimes ((d_0)^p d_{i}b)
  \end{aligned}
  \,.
$$

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

$$
  \begin{aligned}
    \cdots &= (d a) \smile b + (-1)^p a \smile (d b)
    \\
    &=
    \mu \circ d (a \otimes b)
   \end{aligned}
    \,.
$$

=--

+-- {: .num_prop}
###### Proposition

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

$$
  \array{
    C(K) \otimes C(L) &\stackrel{\mu_{K,L}}{\to}& C(K \otimes L)
    \\
    \;\;\;\downarrow^{C(f)\otimes C(g)} && \;\;\;\downarrow^{C(f \otimes g)}
    \\
    C(K') \otimes C(L') &\stackrel{\mu_{K',L'}}{\to}& C(K' \otimes L')
  }
$$

=--

+-- {: .proof}
###### Proof

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

=--

+-- {: .num_prop}
###### Proposition

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

=--

+-- {: .proof}
###### Proof

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

$$
 \array{
  C(J) \otimes C(K) \otimes C(L) &\stackrel{Id_{C(J)} \otimes \mu_{K,L}}{\to}&
  C(J)\otimes C(K \otimes L)
  \\
  \downarrow && \downarrow
  \\
  C(J \otimes K) \otimes C(L)
  &\stackrel{\mu_{J,K} \otimes Id_{C(L)}}{\to}&
  C(J\otimes K \otimes L)
 }
  \,.
$$

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

=--

+-- {: .num_remark }
###### 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.

=--

+-- {: .num_cor }
###### Corollary

The Moore cochain complex functor

$$
  C : CoS(Ab) \to Ch_+^\bullet(Ab)
$$

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.

=--

#### $E_\infty$-cup product on cochains on simplicial sets {#EooCup}

At least for those [[cosimplicial algebra]]s $A$ 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-k-operad|E-∞]]-[[algebra over an operad|algebra]]. See [[cochains on simplicial sets]] for details on this.

### Quillen equivalences
 {#DualDoldKanQuillenEquivalences}

+-- {: .num_theorem}
###### Theorem

There is a Quillen equivalence

$$
  (\Gamma^{mon} \dashv N) :  Ring^{\Delta}
   \underoverset{\underset{N}{\longrightarrow}}{\overset{\Gamma^{mon}}{\longleftarrow}}{\bot}  dgRing
$$

between connective cochain dg-rings and cosimplicial rings.

=--

This is the main theorem in [CastiglioniCortinas](#CastiglioniCortinas).

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

## Related concepts

* [[Dold-Kan correspondence]]

* **monoidal Dold-Kan corespondence**

* [[operadic Dold-Kan correspondence]]

## References ##

Original results are in

* [[Samuel Eilenberg]], [[Saunders MacLane]], _On the groups $H(\Pi, n)$ II. Methods of computation_ , Ann. Math. (2) 60 (1954), 49 - 139
{#EilenbergMacLane}

* [[Saunders MacLane]], _Homology_
{#MacLaneHomology}

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](http://www.math.tamu.edu/~maguiar/monograph.pdf))

   _[[Monoidal Functors, Species and Hopf Algebras]]_,
{#AguiarMahajan}

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

* {#SchwedeShipley} [[Stefan Schwede]], [[Brooke Shipley]], _Equivalences of monoidal model categories_ , Algebr. Geom. Topol. 3 (2003), 287--334 ([arXiv:math.AT/0209342](http://arxiv.org/abs/math.AT/0209342))

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

* {#CastiglioniCortinas} 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](http://arxiv.org/abs/math/0306289), [doi:10.1016/j.jpaa.2003.11.009](https://doi.org/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

* {#HinichSchechtman} [[Vladimir Hinich]], [[Vadim Schechtman]], _Homotopy limits of homotopy algebras_, in _K-theory: algebra, geometry, arithmetic_, Lecture Notes in Mathematics 1289,  Springer, Berlin 240 − 264

* {#Pridham} [[Jonathan Pridham]], _Unifying derived deformation theories_, Adv. Math. 224 (2010), no.3, 772-826 ([arXiv:0705.0344](http://arxiv.org/abs/0705.0344))

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

* {#Fresse} [[Benoit Fresse]], _[Homotopy of Operads and Grothendieck-Teichmüller Groups](http://math.univ-lille1.fr/~fresse/OperadHomotopyBook/)_, chapter 6: _Differential graded commutative algebras and cosimplicial algebras_ ([pdf](http://math.univ-lille1.fr/~fresse/OperadHomotopyBook/CommutativeAlgebraHomotopy.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

* {#Quillen69} [[Dan Quillen]], _Rational homotopy theory_ The Annals of Mathematics,
Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 ([JSTOR](http://www.jstor.org/stable/1970725))

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

* {#Mandell} [[Michael Mandell]], _Topological André-Quillen Cohomology and $E_\infty$ André-Quillen Cohomology_ Adv. in Math., Adv. Math. 177 (2) (2003) 227–279

and

* {#Richter} [[Birgit Richter]]

  _Symmetry properties of the Dold-Kan correspondence_ ([pdf](http://www.math.uni-hamburg.de/home/richter/doldkan.pdf))

  _Homotopy algebras and the inverse of the normalization functor_ ([pdf](http://www.math.uni-hamburg.de/home/richter/invjpaa.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 $H(\pi, n)$. 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.

See also the appendix in

* [[Pavol Severa]], [[Thomas Willwacher]], _Equivalence of formalities of the little discs operad_, [arXiv:0905.1789](http://arxiv.org/abs/0905.1789)

[[!redirects monoidal Dold–Kan correspondence]]
[[!redirects monoidal Dold--Kan correspondence]]


monoidal double category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea
A *monoidal double category* is a [[double category]] equipped with a [[tensor product]] (cf. *[[monoidal 2-category]]*), compatible with both tight and loose morphisms.

Formally, a monoidal double category is a [[pseudomonoid]] in the ([[product of categories|cartesian]] [[monoidal 2-category|monoidal]]) [[2-category]] $\mathbf{DblCat}$ of double categories, [[double functor|(lax) double functors]] and [[double natural transformation|loose natural transformations]].

## Definition

Recall a double category $\mathbb{D}$ is an [[internal category|internal (pseudo)category]] in [[Cat]], hence is given by two categories $\mathbb{D}_1$ (category of loose arrows) and $\mathbb{D}_0$ (category of objects), plus *source* and *target* functors $s,t:\mathbb{D}_1 \to \mathbb{D}_0$, a *loose identity* functor $U:\mathbb{D}_0 \to \mathbb{D}_1$, and a *loose composition* functor $\odot  : \mathbb{D}_1 {}_s\times_t \mathbb{D}_1 \to \mathbb{D}_0$.
Clearly this data has to satisfy the properties of the composition of a category, thus $U$ provides identities for the $\odot$, which in turn is associative. In [Shulman 2010](#Shulman10) moreover, these properties are only satisfied up to coherent isomorphism, making loose arrows form a [[bicategory]].

\begin{definition}
A monoidal double category is a double category $\mathbb{D}$ equipped with [[double functors]]
$$I: \mathbb{1} \to \mathbb{D}\quad \text{and} \quad \otimes : \mathbb{D} \times \mathbb{D} \to \mathbb{D}$$
and with [[double natural transformations]] $\alpha : (\otimes\otimes)\otimes \cong \otimes(\otimes\otimes)$, $\lambda : (I \times \mathbb{D})\otimes \cong \otimes$, $\rho : (\mathbb{D} \times I)\otimes \cong \otimes$ satisfying the usual coherence laws of a pseudomonoid (pentagon and triangle, see [[monoidal category]]).
\end{definition}

\begin{remark}
As often happens for structure on internal categories, these arise as structures on the categories of loose arrows $\mathbb{D}_1$ and that of objects $\mathbb{D}_0$ when the source, identity and target maps respect it.

Specifically, a monoidal double category arises when both $\mathbb{D}_1$ and $\mathbb{D}_0$ are [[monoidal categories]] and

1.  $s,t:\mathbb{D}_1\to\mathbb{D}_0$ are strict monoidal, meaning if $p:A \nrightarrow B$ and $q:A' \nrightarrow B'$ are loose arrows, $p \otimes q$ is a loose arrow $A \otimes A' \nrightarrow B \otimes B'$,
2. $U:\mathbb{D}_0 \to \mathbb{D}_1$ is strong monoidal, meaning $U_1 : I \nrightarrow I$ is the monoidal unit for $\mathbb{D}_1$ and there is an iso $U_{A \otimes B} \cong U_A \otimes U_B$ for all $A,B:\mathbb{D}_0$.
3. $\odot  : \mathbb{D}_1 {}_s\times_t \mathbb{D}_1 \to \mathbb{D}_0$ is monoidal, meaning there is an iso $(p \otimes q) \odot (p' \otimes q') \cong (p \odot p') \otimes (q \odot q')$ for any suitably composable loose arrows $p,p',q',q' : \mathbb{D}_1$.

and of course these isomorphisms satisfy coherence axioms one can find in [Shulman'10](#Shulman10).
\end{remark}

\begin{definition}
A **braiding** on a monoidal double category is a natural transformation $\tau : \otimes \cong \mathrm{swap}\otimes$ which satisfies the coherence laws of the braiding of a pseudomonoid (see [[braided monoidal category]])
\end{definition}

\begin{definition}
A symmetric monoidal double category is a braided monoidal double category whose braiding $\tau$ satisfies $\tau\tau = 1$.
\end{definition}

## Related concepts

* [[monoidal 2-category]]

## References

* {#Shulman10} [[Mike Shulman]], Def. 2.9 in: _Constructing Symmetric Monoidal Double Categories_ (2010) [[arxiv:1004.0993](https://arxiv.org/abs/1004.0993)]

* Linde Wester Hansen and [[Mike Shulman]]. _Constructing symmetric monoidal bicategories functorially_ (2019) [[arxiv:1910.09240](https://arxiv.org/abs/1910.09240)]

[[!redirects monoidal double categories]]
[[!redirects symmetric monoidal double category]]
[[!redirects symmetric monoidal double categories]]
[[!redirects braided monoidal double category]]
[[!redirects braided monoidal double categories]]


monoidal fibration

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A **monoidal fibration** is a [[functor]] $\Phi\colon E\to B$ such that

* $\Phi$ is a [[Grothendieck fibration]]

* $E$ and $B$ are [[monoidal categories]] and $\Phi$ is a strict [[monoidal functor]], and

* the [[tensor product]] of $E$ preserves [[cartesian arrows]].

If $B$ is [[cartesian monoidal category|cartesian monoidal]], then monoidal fibrations over $B$ are equivalent to [[pseudofunctors]] $B^{op} \to MonCat$, which are called [[indexed monoidal categories]]. In this case the tensor product on $E$ is the [[external tensor product]] of the indexed monoidal category.

## Related concepts

* [[indexed monoidal category]], [[dependent linear type theory]]

* [[monoidal Grothendieck construction]]

## References

* [[Mike Shulman]], _Framed bicategories and monoidal fibrations_, [TAC](http://www.tac.mta.ca/tac/volumes/20/18/20-18abs.html)

[[!redirects monoidal fibrations]]


monoidal functor

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _monoidal functor_ is a [[functor]] between [[monoidal categories]] that preserves the monoidal structure: a [[homomorphism]] of monoidal categories.

## Definition ##

+-- {: .num_defn #LaxMonoidalFunctor}
###### Definition

Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ be two [[monoidal categories]]. A **lax monoidal functor** between them is a [[functor]]:

   $$
     F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}
     \,,
   $$

together with coherence [[maps]]:

1. a [[morphism]]

   $$
     \epsilon \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}})
   $$

1. a [[natural transformation]]

   $$
     \mu_{x,y}
       \;\colon\;
     F(x) \otimes_{\mathcal{D}} F(y)
       \longrightarrow
     F(x \otimes_{\mathcal{C}} y)
   $$

   for all $x,y \in \mathcal{C}$

satisfying the following conditions:

1. **([[associativity]])** For all objects $x,y,z \in \mathcal{C}$ the following [[commuting diagram|diagram commutes]]

   $$
     \array{
       (F(x) \otimes_{\mathcal{D}} F(y)) \otimes_{\mathcal{D}} F(z)
         &\underoverset{\simeq}{a^{\mathcal{D}}_{F(x),F(y),F(z)}}{\longrightarrow}&
       F(x) \otimes_{\mathcal{D}}( F(y)\otimes_{\mathcal{D}} F(z) )
       \\
       {}^{\mathllap{\mu_{x,y} \otimes id}}\downarrow
         &&
       \downarrow^{\mathrlap{id\otimes \mu_{y,z}}}
       \\
       F(x \otimes_{\mathcal{C}} y) \otimes_{\mathcal{D}} F(z)
        &&
       F(x) \otimes_{\mathcal{D}} F(y \otimes_{\mathcal{C}} z)
       \\
       {}^{\mathllap{\mu_{x \otimes_{\mathcal{C}} y , z} } }\downarrow
         &&
       \downarrow^{\mathrlap{\mu_{ x, y \otimes_{\mathcal{C}} z  }}}
       \\
       F( ( x \otimes_{\mathcal{C}} y ) \otimes_{\mathcal{C}} z  )
         &\underset{F(a^{\mathcal{C}}_{x,y,z})}{\longrightarrow}&
       F( x \otimes_{\mathcal{C}} ( y \otimes_{\mathcal{C}} z ) )
     }
   $$

   where $a^{\mathcal{C}}$ and $a^{\mathcal{D}}$ denote the [[associators]] of the monoidal categories;

1. **([[unitality]])** For all $x \in \mathcal{C}$ the following [[commuting diagram|diagrams commute]]

   $$
     \array{
       1_{\mathcal{D}} \otimes_{\mathcal{D}} F(x)
         &\overset{\epsilon \otimes id}{\longrightarrow}&
       F(1_{\mathcal{C}}) \otimes_{\mathcal{D}} F(x)
       \\
       {}^{\mathllap{\ell^{\mathcal{D}}_{F(x)}}}\downarrow
         &&
       \downarrow^{\mathrlap{\mu_{1_{\mathcal{C}}, x }}}
       \\
       F(x)
         &\overset{F(\ell^{\mathcal{C}}_x )}{\longleftarrow}&
       F(1 \otimes_{\mathcal{C}} x  )
     }
   $$

   and

   $$
     \array{
       F(x) \otimes_{\mathcal{D}}  1_{\mathcal{D}}
         &\overset{id \otimes \epsilon }{\longrightarrow}&
       F(x) \otimes_{\mathcal{D}}  F(1_{\mathcal{C}})
       \\
       {}^{\mathllap{r^{\mathcal{D}}_{F(x)}}}\downarrow
         &&
       \downarrow^{\mathrlap{\mu_{x, 1_{\mathcal{C}} }}}
       \\
       F(x)
         &\overset{F(r^{\mathcal{C}}_x )}{\longleftarrow}&
       F(x \otimes_{\mathcal{C}} 1  )
     }
   $$

   where $\ell^{\mathcal{C}}$, $\ell^{\mathcal{D}}$, $r^{\mathcal{C}}$, $r^{\mathcal{D}}$ denote the left and right [[unitors]] of the two monoidal categories, respectively.

{#StrictMonoidalFunctor} If $\epsilon$ and all $\mu_{x,y}$ are [[isomorphisms]], then $F$ is called a **strong monoidal functor**. (Note that 'strong' is also sometimes applied to 'monoidal functor' to indicate possession of a [[tensorial strength]].)   If they are even [[identity morphisms]], then $F$ is called a **strict monoidal functor**.

=--

+-- {: .num_remark}
###### Remark

In the literature often the term "monoidal functor" refers by default to what in def. \ref{LaxMonoidalFunctor} is called a strong monoidal functor.  With that convention then what def. \ref{LaxMonoidalFunctor} calls a lax monoidal functor is called a  **weak monoidal functor**.

=--

+-- {: .num_remark}
###### Remark

Lax monoidal functors are the [[lax morphisms]] for an appropriate [[2-monad]].

=--

+-- {: .num_remark}
###### Definition

An __[[oplax monoidal functor]]__ (with various alternative names including **comonoidal**), is a monoidal functor from the [[opposite categories]] $C^{op}$ to $D^{op}$.

=--

+-- {: .num_remark}
###### Definition

A _[[monoidal transformation]]_ between monoidal functors is a [[natural transformation]] that respects the extra structure in an obvious way.

=--

## Properties

+-- {: .num_prop #MonoidsToMonoidsByLaxMonoidal}
###### Proposition
**(Lax monoidal functors send [[monoid]]s to monoids)**

If $F : (C,\otimes) \to (D,\otimes)$ is a lax monoidal functor and

$$
  (A \in C,\;\; \mu_A : A \otimes A \to A, \; i_A : I \to A)
$$

is a [[monoid object]] in $C$, then the object $F(A)$ is naturally equipped with the structure of a monoid in $D$ by setting

$$
  i_{F(A)} : I_D \stackrel{}{\to} F(I_C) \stackrel{F(i_A)}{\to} F(A)
$$

and

$$
  \mu_{F(A)} :
  F(A) \otimes F(A)
  \stackrel{\nabla_{F(A), F(A)}}{\to}
  F(A \otimes A)
  \stackrel{F(\mu_A)}{\to}
  F(A)
  \,.
$$

This construction defines a functor

$$
  Mon(f) : Mon(C) \to Mon(D)
$$

between the [[categories of monoids]] in $C$ and $D$, respectively.

=--

More generally, lax functors send [[enriched categories]] to enriched categories, an operation known as [[change of enriching category]]. See there for more details.

Similarly:

+-- {: .un_prop #OplaxSendsComonoidsToComonoids}
###### Proposition
**([[oplax monoidal functors]] sends [[comonoids]] to comonoids)**

For $(C,\otimes)$ a [[monoidal category]] write $\mathbf{B}C$ for the corresponding [[delooping]] [[2-category]].

Lax monoidal functor $f : C \to D$ correspond to [[lax 2-functor]]

$$
  \mathbf{B}F : \mathbf{B}C \to \mathbf{B}D
  \,.
$$

If $F$ is strong monoidal then this is an ordinary [[2-functor]]. If it is strict monoidal, then this is a [[strict 2-functor]].

=--

## String diagrams

Just like monoidal categories, monoidal functors have a [[string diagram]] calculus; see [these slides](https://web.archive.org/web/20191021024946/http://web.science.mq.edu.au/~mmccurdy/cms2010talk.pdf) for some examples.

## Related concepts

* **monoidal functor**, **strong monoidal functor**

* [[multifunctor]]

* [[module over a monoidal functor]]

* [[monoidal adjunction]]

* [[indexed monoidal category]]

* [[cartesian functor]]

* **lax monoidal functor**

  * [[functor with smash products]]

* [[oplax monoidal functor]]

* [[bilax monoidal functor]]

* [[Frobenius monoidal functor]]

* [[braided monoidal functor]]

* [[symmetric monoidal functor]]

* [[monoidal (∞,1)-functor]]

* [[monoidal (∞,n)-functor]]

## References

* {#EK65} [[Samuel Eilenberg]], [[G. Max Kelly]], p. 473 in: *Closed Categories*, in:  [[Samuel Eilenberg|S. Eilenberg]], [[D. K. Harrison]], [[S. MacLane]], [[H. Röhrl]] (eds.): *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) 421-562 [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

* [[Saunders MacLane]], §XI.2 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* {#EGNO15} [[Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], §2.4 in: *Tensor Categories*, AMS Mathematical Surveys and Monographs **205** (2015) [[ISBN:978-1-4704-3441-0](https://bookstore.ams.org/surv-205), [pdf](http://www-math.mit.edu/~etingof/egnobookfinal.pdf)]

  > (discussed what we call *[[strong monoidal functors]]*)

* [[Marcelo Aguiar]] and Swapneel Mahajan, _Monoidal functors, species and Hopf algebras_. ([pdf](http://pi.math.cornell.edu/~maguiar/a.pdf))

Exposition of basics of [[monoidal categories]] and [[categorical algebra]]:

* _[[geometry of physics -- categories and toposes]]_, Section 2: _[Basic notions of categorical algebra](geometry+of+physics+--+categories+and+toposes#BasicNotionsOfCategoricalAlgebra)_

[[!redirects lax monoidal functor]]
[[!redirects strict monoidal functor]]
[[!redirects strong monoidal functor]]
[[!redirects weak monoidal functor]]
[[!redirects monoidal functors]]
[[!redirects lax monoidal functors]]
[[!redirects strict monoidal functors]]
[[!redirects strong monoidal functors]]
[[!redirects weak monoidal functors]]
[[!redirects strength]]


This entry is about the book

* {#AguiarMahajan} [[Marcelo Aguiar]], [[Swapneel Mahajan]]:

  \linebreak

  **Monoidal Functors, Species and Hopf Algebras**

  \linebreak

  with forewords by [[Kenneth Brown]], [[Stephen Chase]] and [[André Joyal]]

  CRM Monograph Series __29__

  Amer. Math. Soc. 2010. lii+784 pp.

  [ISBN:978-0-8218-4776-3](https://bookstore.ams.org/crmm-29/)

  [pdf](http://www.math.cornell.edu/~maguiar/a.pdf)

on [[monoidal functors]], [[combinatorial species]] and [[Hopf algebras]].

## Related entries

* [[Hopf monoid]]

* [[partition]]

* [[shuffle]]

* [[duoidal category|Duoidal category]]

category: reference


monoidal groupoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Monoid theory
+-- {: .hide}
[[!include monoid theory - contents]]
=--
#### Categorification
+-- {: .hide}
[[!include categorification - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A [[monoidal category]] whose underlying [[category]] is a [[groupoid]] (hence all whose [[morphisms]] are [[invertible morphism|invertible]]).

Equivalently: an [[A4-spatial groupoid|$A_4$-spatial groupoid]] whose [[unitors]] satisfy the [[triangle identities]];

Equivalently: a [[1-type|1-truncated]] [[A-infinity space|$A_\infty$-space]]/[[En-algebra|$E_1$-space]].

## Definitions

A **monoidal groupoid** is an [[A4-spatial groupoid|$A_4$-spatial groupoid]] $G$ such that the [[triangle identity]] is satisfied for all objects $A:G$ and $B:G$:

$$\tau_{A,B}:(id_A \otimes \lambda_B) \circ \alpha_{A,I,B} = (\rho_A \otimes id_B)$$

## Properties

The [[oidification]] of a monoidal groupoid is a [[(2,1)-category]].

## See also

* [[monoid]]
* [[A-infinity space]]
* [[En-algebra]]
* [[groupoid]]
* [[A4-spatial groupoid]]
* [[2-group]], [[Picard groupoid]]
* [[braided monoidal groupoid]]
* [[k-tuply monoidal n-groupoid]]
* [[ring groupoid]]
* [[2-action]]
* [[monoidal category]], [[monoidal dagger category]]
* [[(2,1)-category]]

[[!include oidification - table]]

[[!redirects monoidal groupoid]]
[[!redirects monoidal groupoids]]


monoidal homotopy category of a monoidal model category

+-- {: .num_prop #MonoidalStructureOnHomotopyCategoryOfMonoidalModelCategory}
###### Proposition

Let $(\mathcal{C}, \otimes, I)$ be a [[monoidal model category]]. Then the [[left derived functor]] of the tensor product exists and makes the [[homotopy category of a model category|homotopy category]] into a [[monoidal category]] $(Ho(\mathcal{C}), \otimes^L, \gamma(I))$.

If in in addition $(\mathcal{C}, \otimes)$ satisfies the [[monoid axiom in a monoidal model category|monoid axiom]], then the [[localization]] functor $\gamma\colon \mathcal{C}\to Ho(\mathcal{C})$ carries the structure of a [[lax monoidal functor]]

$$
  \gamma
  \;\colon\;
  (\mathcal{C}, \otimes, I)
   \longrightarrow
  (Ho(\mathcal{C}), \otimes^L , \gamma(I))
  \,.
$$

=--

+-- {: .proof}
###### Proof

Consider the explicit model of $Ho(\mathcal{C})$ as the category of fibrant-cofibrant objects in $\mathcal{C}$ with left/right-homotopy classes of morphisms between them (discussed at _[[homotopy category of a model category]]_).

A [[derived functor]] exists if its restriction to this subcategory preserves weak equivalences. Now the [[pushout-product axiom]] implies that on the subcategory of cofibrant objects the functor $\otimes$ preserves acyclic cofibrations, and then the preservation of all weak equivalences follows by [[Ken Brown's lemma]].

Hence $\otimes^L$ exists and its [[associativity]] follows simply by restriction. It remains to see its [[unitality]].

To that end, consider the construction of the localization functor $\gamma$ via a fixed but arbitrary choice of (co-)fibrant replacements $Q$ and $R$, assumed to be the identity on (co-)fibrant objects. We fix notation as follows:

$$
\emptyset \underoverset{\in Cof}{i_X}{\longrightarrow} Q X \underoverset{\in W \cap Fib}{p_x}{\longrightarrow} X
\;\;\,,\;\;
X \underoverset{\in W \cap Cof}{j_X}{\longrightarrow} R X \underoverset{\in Fib}{q_x}{\longrightarrow} \ast
  \,.
$$

Now to see that $\gamma(I)$ is the [[tensor unit]] for $\otimes^L$, notice that in the [[zig-zag]]

$$
  (R Q I) \otimes (R Q X)
    \overset{j_{Q I} \otimes (R Q X)}{\longleftarrow}
  (Q I) \otimes (R Q X)
    \overset{(Q I)\otimes j_{Q X}}{\longleftarrow}
  (Q I) \otimes (Q X)
    \overset{p_I \otimes (Q X)}{\longrightarrow}
  I \otimes Q X
    \simeq
  Q X
$$

all morphisms are weak equivalences: For the first two this is due to the [[pushout-product axiom]], for the third this is due to the unit axiom on a monoidal model category. It follows that under $\gamma(-)$ this zig-zig gives an isomorphism

$$
  \gamma(I) \otimes^L \gamma(X)\simeq \gamma(X)
$$

and similarly for tensoring with $\gamma(I)$ from the right.

To exhibit lax monoidal structure on $\gamma$, we need to construct a [[natural transformation]]

$$
  \gamma(X) \otimes^L \gamma(Y) \longrightarrow \gamma(X \otimes Y)
$$

and show that it satisfies the the appropriate [[associativity]] and [[unitality]] condition.

By the definitions at _[[homotopy category of a model category]]_, the morphism in question is to be of the form

$$
  (R Q X) \otimes (R Q Y) \longrightarrow R Q (X\otimes Y)
$$

To this end, consider the [[zig-zag]]

$$
  (R Q X) \otimes (R Q Y)
   \underoverset{\in Cof \cap W}{j_{Q X} \otimes R Q Y}{\longleftarrow}
  (Q X) \otimes (R Q Y)
    \underoverset{\in Cof \cap W}{(Q X) \otimes  j_{Q Y} }{\longleftarrow}
  (Q X) \otimes (Q Y)
    \overset{p_X \otimes (Q Y)}{\longrightarrow}
  X \otimes (Q Y)
    \overset{Y \otimes p_Y}{\longrightarrow}
  X \otimes Y
  \,,
$$

and observe that the two morphisms on the left are weak equivalences, as indicated, by the [[pushout-product axiom]] satisfied by $\otimes$.

Hence applying $\gamma$ to this zig-zag, which is given by the two horizontal part of the following digram

$$
  \array{
    (R Q X) \otimes (R Q Y) &\longleftarrow& R( Q X \otimes Q Y ) &\longrightarrow& R Q (X \otimes Y)
    \\
    \uparrow^{\mathrlap{id}} && \uparrow^{\mathrlap{j_{Q X \otimes Q Y}}} && \uparrow^{\mathrlap{j_{Q(X \otimes Y)}}}
    \\
    \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{p_{X\otimes Y}}}
    \\
    (R Q X) \otimes (R Q Y)
     &\underoverset{\in Cof \cap W}{j_{Q X} \otimes j_{Q Y}}{\longleftarrow}&
    (Q X) \otimes (Q Y)
      &\overset{p_X \otimes p_Y}{\longrightarrow}&
    X \otimes Y
  }
  \,,
$$

and inverting the first two morphisms, this yields a natural transformation as required.

To see that this satisfies associativity if the monoid axiom holds, tensor the entire diagram above on the right with $(R Q Z)$ and consider the following [[pasting]] composite:

$$
  \array{
    (R Q X) \otimes (R Q Y) \otimes (R Q Z) &\longleftarrow& R( Q X \otimes Q Y ) \otimes (R Q Z) &\longrightarrow& (R Q (X \otimes Y)) \otimes (R Q Z)
    \\
    \uparrow^{\mathrlap{id}} && \uparrow^{\mathrlap{j_{Q X \otimes Q Y} \otimes id }} && \uparrow^{\mathrlap{j_{Q(X \otimes Y)}\otimes id }}
    \\
    && && Q(X \otimes Y) \otimes (R Q Z) &\overset{id \otimes j_{Q Z}}{\longleftarrow}& Q(X\otimes Y) \otimes (Q Z)
    \\
       \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{id}}
    &&
      \downarrow^{\mathrlap{p_{(X\otimes Y)} \otimes id }}
    &(\star)&
     \downarrow^{\mathrlap{p_{(X \otimes Y)} \otimes id}}
    \\
    (R Q X) \otimes (R Q Y) \otimes (R Q Z)
     &\underoverset{\in Cof \cap W}{j_{Q X} \otimes j_{Q Y} \otimes id}{\longleftarrow}&
    (Q X) \otimes (Q Y) \otimes (R Q Z)
      &\overset{p_X \otimes p_Y \otimes id}{\longrightarrow}&
    X \otimes Y \otimes (R Q Z)
      &\underset{id \otimes j_{Q Z}}{\longleftarrow}&
    X\otimes Y \otimes Q Z
      &\overset{id \otimes p_Z}{\longrightarrow}&
    X \otimes Y \otimes Z
  }
  \,,
$$

Observe that under $\gamma$ the total top [[zig-zag]] in this diagram gives

$$
  (\gamma(X) \otimes^L \gamma(Y)) \otimes^L \gamma(Z)
  \to \gamma(X\otimes Y)\otimes^L \gamma(Z)
  \,.
$$

Now by the [[monoid axiom in a monoidal model category|monoid axiom]] (but not by the pushout-product axiom!), the horizontal maps in the square in the bottom right (labeled $\star$) are weak equivalences. This implies that the total horizontal part of the diagram is a [[zig-zag]] in the first place, and that under $\gamma$ the total top zig-zag is equal to the image of that total bottom zig-zag. But by functoriality of $\otimes$, that image of the bottom zig-zag is

$$
  \gamma(p_X \otimes p_Y \otimes p_Z)
  \circ
  \gamma(j_{Q X} \otimes j_{Q Y} \otimes j_{Q Z})^{-1}
  \,.
$$

The same argument applies to left tensoring with $R Q Z$ instead of right tensoring, and so in both cases we reduce to the same morphism in the homotopy category, thus showing the associativity condition on the transformation that exhibits $\gamma$ as a lax monoidal functor.

=--


monoidal localization

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The concept of [[localization]] of a [[category]] for the case of [[monoidal categories]] with localization functors being [[monoidal functors]].

## Related concepts

* [[monoidal model category]]

## References

* {#Day73} [[Brian Day]], _Note on monoidal localisation_, Bull. Austr. Math. Soc. Vol 8 (1973) ([[DayMonoidalLocaization73.pdf:file]])

On monoidal [[Bousfield localization of model categories|Bousfield localization]] of [[monoidal model categories]]:

* {#Barwick10} [[Clark Barwick]], Prop. 4.47 in: *On left and right model categories and left and right Bousfield localizations*, Homology Homotopy Appl. **12** 2 (2010) 245-320 [[doi:10.4310/hha.2010.v12.n2.a9](https://doi.org/10.4310/hha.2010.v12.n2.a9), [euclid:1296223884](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-12/issue-2/On-left-and-right-model-categories-and-left-and-right/hha/1296223884.full), subsuming: [arXiv:0708.2067](https://arxiv.org/abs/0708.2067), [arXiv:0708.2832](https://arxiv.org/abs/0708.2832), [arXiv:0708.3435](http://arxiv.org/abs/0708.3435)]

* {#GorchinskiyGuletskii16} [[Sergey Gorchinskiy]], [[Vladimir Guletskii]], Lemma 28 in: *Symmetric powers in abstract homotopy categories*, Adv. Math., **292** (2016) 707-754 [[arXiv:0907.0730](https://arxiv.org/abs/0907.0730), [doi:10.1016/j.aim.2016.01.011](https://doi.org/10.1016/j.aim.2016.01.011)]

* {#White14} [[David White]], *Monoidal Bousfield Localizations and Algebras over Operads*, PhD thesis (2014), published in *[Equivariant Topology and Derived Algebra: A conference in honor of John Greenlees' 60th birthday](https://www.commalg.org/2019/07/29/equivariant-topology-and-derived-algebra-in-honor-of-john-greenlees-60th-birthday/)*, Cambridge University Press (2021) 179-239 [[doi:10.1017/9781108942874.007](https://doi.org/10.1017/9781108942874.007), [digitalcollections:ir-2263](https://digitalcollections.wesleyan.edu/object/ir-2263), [arXiv:1404.5197](https://arxiv.org/abs/1404.5197)]

* {#Lawson2022} [[Tyler Lawson]], Prop. 12.18 in: *An introduction to Bousfield localization*, in: *[[Stable categories and structured ring spectra]]*, MSRI Book Series, Cambridge University Press (2022) [[arXiv:2002.03888](https://arxiv.org/abs/2002.03888)]

* [[Luca Pol]], [[Jordan Williamson]], *The Left Localization Principle, completions, and cofree $G$-spectra*, J. Pure Appl. Algebra **224** 11 (2020) 106408 [[arXiv:1910.01410](https://arxiv.org/abs/1910.01410), [doi:10.1016/j.jpaa.2020.106408](https://doi.org/10.1016/j.jpaa.2020.106408)]

[[!redirects monoidal localizations]]

[[!redirects monoidal localisation]]
[[!redirects monoidal localisations]]


monoidal model category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _monoidal model category_ is a [[model category]] which is also a [[closed monoidal category]] in a compatible way.  In particular, its [[homotopy category]] inherits a closed monoidal structure, as does the [[(infinity,1)-category]] that it presents.

## Definition

\begin{definition}\label{MonoidalModelCategory}
A (symmetric) **monoidal model category** is

* a [[model category]] $\mathcal{C}$

equipped with the further [[structure]] of

* a  *[[closed monoidal category|closed]]* [[symmetric monoidal category]] $(\mathcal{C}, \otimes, I)$

such that the following two compatibility conditions are satisfied:

1. {#PushoutProductAxiom} ([[pushout-product axiom]]) For every [[pair]] of [[cofibrations]] $f \colon X \to Y$ and $f' \colon X' \to Y'$, their [[pushout-product]], hence the induced morphism out of the [[pushout|cofibered]] [[coproduct]] over ways of forming the [[tensor product]] of these objects

   \[
     (X \otimes Y')
       \overset
        {X \otimes X'}
        {\amalg}
     (Y \otimes X')
     \longrightarrow
     Y \otimes Y'
     \,,
   \]

   is itself a [[cofibration]], which, furthermore, is [[acyclic cofibration|acyclic]] (i.e. also a [[weak equivalence]]) if $f$ or $f'$ is so.

   (Equivalently this says that the [[tensor product]] $\otimes \colon C \times C \to C$ is a left [[Quillen bifunctor]].)

1. {#UnitAxiom} (unit axiom) For every [[cofibrant object]] $X$ and every [[cofibrant resolution]] $\emptyset \hookrightarrow Q I \stackrel{p_I}{\longrightarrow} I$ of the [[tensor unit]] $I$, the resulting morphism

   \[
     Q I \otimes X
      \stackrel{p_I \otimes X}{\longrightarrow}
     I\otimes X \stackrel{\simeq}{\longrightarrow} X
   \]

   is a [[weak equivalence]].

\end{definition}

\begin{remark}\label{InternalHomQuillenAdjunction}
**(internal hom Quillen adjunction)**

Let $X$ be a [[cofibrant object]], hence $\varnothing \overset{\exists !}{\to} X$ a [[cofibration]].

In this case the [[pushout-product axiom]] (Def. \ref{MonoidalModelCategory}) says that the [[tensor product]] [[functor]] $X \otimes (-)$ preserves [[cofibrations]] and [[acyclic cofibrations]]. Since the ambient category is assumed to be [[closed monoidal category]], so that this functor has a [[right adjoint]] [[internal hom]] $[X,-]$, this means that it is the [[left Quillen functor]] in a [[Quillen adjunction]]

$$
  \mathcal{C}
  \underoverset
    {\underset{[X,-]}{\longrightarrow}}
    {\overset{X\otimes(-)}{\longleftarrow}}
    {
      \;\;\;\;\;
      \bot_{\mathrlap{Qu}}
      \;\;\;\;\;
    }
  \mathcal{C}
$$

\end{remark}

\begin{remark}\label{MeaningOfPushoutProductAxiom}
**(meaning of the pushout product axiom)**
\linebreak
 The [[pushout-product axiom]] ([above](#PushoutProductAxiom)) is stronger than its implied statement in Exp. \ref{InternalHomQuillenAdjunction}, while the latter might superficially seem to be all that would reasonably be required of a model-category theoretic version of a [[monoidal category]].

It is at this point that the established tradition to just say "monoidal model category" for the above definition is somewhat misleading, in that Def. \ref{MonoidalModelCategory} is really a model-category theoretic version of the stronger concept of *[[closed monoidal categories|closed monoidal]]* categories (*symmetric* closed monoidal really, but the symmetry is not the subtle part). The arguably more accurate terminology "closed monoidal model category" for Def. \ref{MonoidalModelCategory} is probably being avoided because [Quillen (1967)](#Quillen67) originally introduced model categories in general under the name "closed model categories" with the adjective "closed" *not* meant to refer to its use in *[[closed categories]]*...

Namely, the full strength of the pushout-product axiom is needed to imply that not only the [[tensor product]] $\otimes$ but also its [[internal hom]] $[-,-] \,\colon\, C^{op} \times C \to C$ is homotopically well-behaved, to wit that $[-,-]$ satisfies the *[[pullback-power axiom]]*, which in turn implies that not just $[X,-]$ but also $[-,A]$ are [[right Quillen functors]] for all [[cofibrant objects]] $X$ and [[fibrant objects]] $A$.

In short, the [[pushout-product axiom]] is the model-category theoretic way to ensure that not just the [[tensor product]], but also the [[internal hom]] as well as their joint relatin in a [[two-variable adjunction]] are homotopically meaningful.
\end{remark}

\begin{remark}\label{CaseOfCofibrantTensorUnit}
**(cofibrant tensor unit implies unit axiom)**\linebreak
As a special case of Rem. \ref{InternalHomQuillenAdjunction}: If the [[tensor unit]] $I$ happens to be [[cofibrant objects|cofibrant]], then the unit axiom in def. \ref{MonoidalModelCategory} is already implied by the [pushout-product axiom](#PushoutProductAxiom).

(Because then $Q I \to I$ is a [[weak equivalence]] between [[cofibrant objects]] and such are preserved by functors that preserve acyclic cofibrations, by [[Ken Brown's lemma]].)
\end{remark}

+-- {: .num_defn #MonoidAxiom}
###### Definition
**([[monoid axiom]])**

One says that a [[monoidal model category]] (Def. \ref{MonoidalModelCategory}) satisfies the *[[monoid axiom]]* if every [[morphism]] that is obtained as a [[transfinite composition]] of [[pushouts]] of [[tensor products]] $X\otimes f$ of acyclic cofibrations $f$ with any object $X$ is a [[weak equivalence]].

=--

([Schwede-Shipley 00, def. 3.3.](#SchwedeShipley00)).

+-- {: .num_remark}
###### Remark

In particular, the axiom in def. \ref{MonoidAxiom} says that for every object $X$ the functor $X \otimes (-)$ sends acyclic cofibrations to weak equivalences.

=--

## Properties

### Monoidal homotopy category
 {#MonoidalHomotopyCategory}

#### Abstractly

+-- {: .num_prop #LocalizationFunctorIsLaxMonoidal}
###### Proposition

For $(\mathcal{C},\otimes)$ a monoidal model category, def. \ref{MonoidalModelCategory}, then the [[derived functor]] $\otimes^L$ of the tensor product makes the [[homotopy category of a model category|homotopy category of the model category]] itself into a [[monoidal category]], such that the [[localization]] functor

$$
  \gamma
   \;\colon\;
  (\mathcal{C},\otimes)
    \longrightarrow
  (Ho(\mathcal{C}), \otimes^L)
$$

is a [[lax monoidal functor]].

=--

+-- {: .proof}
###### Proof

Let $V$ be a monoidal model category, and consider it as a _[[derivable category]]_ in the sense of ([Shulman 11, section 8](#Shulman11)) with $V_Q$ the subcategory of cofibrant objects and $V_R=V$.  Then $\otimes :V\times V\to V$ is left derivable, i.e. it preserves the $Q$-subcategories and weak equivalences.  Since deriving is [[pseudofunctor|pseudofunctorial]] (and product-preserving) on the [[2-category]] of [[derivable categories]] and [[left derived functor|left derivable functors]], it follows immediately that $Ho(V)$ is monoidal; this is ([Shulman 11, example 8.13](#Shulman11)).

Now let $V_0$ denote the category $V$ with its trivial derivable structure: only isomorphisms are weak equivalences, and all objects are both $Q$ and $R$.  Then of course $Ho(V_0) = V$, and $V_0$ is also a pseudomonoid in derivable categories.  The identity functor $Id : V_0 \to V$ is not left derivable, since it does not preserve $Q$-objects; but it is *right* derivable, since we took all objects in $V$ to be $R$-objects (ignoring the fibrant objects in the model structure on $V$).  Of course $Id$ is [[strong monoidal functor|strong monoidal]], and this monoidality constraint can be expressed as a square in the [[double category]] of ([Shulman 11](#Shulman11)) whose vertical arrows are left derivable functors and whose horizontal arrows are right derivable functors; moreover the axioms on a [[monoidal functor]] may be expressed using products and double-categorical pasting in this double category.  Therefore, it is all preserved by the double pseudofunctor $Ho$; but $Ho(Id) = \gamma : V \to Ho(V)$.  The only thing that is not visible to the double category is the invertibility of the monoidal constraint, and hence this is not preserved by the double pseudofunctor; thus $\gamma$ is only [[lax monoidal functor|lax monoidal]].

=--

#### Explicitly

+-- {: .num_prop #MonoidalStructureOnHomotopyCategoryOfMonoidalModelCategory}
###### Proposition

Let $(\mathcal{C}, \otimes, 1)$ be a [[monoidal model category]] with cofibrant [[tensor unit]]. Then the [[left derived functor]] $\otimes^L$ of the tensor product exists

$$
  \array{
    \mathcal{C}_c\times \mathcal{C}_c
      &\overset{\otimes}{\longrightarrow}&
    \mathcal{C}_c
    \\
    {}^{\mathllap{\gamma \times \gamma}}\downarrow
      &\swArrow_{\mu^{-1}}&
    \downarrow^{\mathrlap{\gamma}}
    \\
    Ho(\mathcal{C})\times Ho(\mathcal{C})
     &\overset{\otimes^L}{\longrightarrow}&
    Ho(\mathcal{C})
  }
$$

and makes the [[homotopy category of a model category|homotopy category]] into a [[monoidal category]] $(Ho(\mathcal{C}), \otimes^L, \gamma(1))$.

Moreover, the [[localization]] functor

$$
  \gamma
  \;\colon\;
  (\mathcal{C}_c, \otimes, 1)
   \longrightarrow
  (Ho(\mathcal{C}), \otimes^L , \gamma(1))
$$

on the [[category of cofibrant objects]] is a [[strong monoidal functor]] with structure morphism the inverse of the above natural isomorpmism

$$
  \mu_{X,Y}
  \;\colon\;
  \gamma(X)\otimes^L \gamma(Y)
    \longrightarrow
  \gamma(X \otimes Y)
  \,.
$$

=--

+-- {: .proof}
###### Proof

For the [[left derived functor]] ([def.](Introduction+to+Stable+homotopy+theory+--+P#LeftAndRightDerivedFunctorsOnModelCategories)) of the tensor product

$$
  \otimes
  \;
  \mathcal{C}\times \mathcal{C}
    \longrightarrow
  \mathcal{C}
$$

to exist, it is sufficient that its restriction to the subcategory

$$
  (\mathcal{C} \times \mathcal{C})_c
  \simeq
  \mathcal{C}_c \times \mathcal{C}_c
$$

of cofibrant objects preserves acyclic cofibrations ([[Ken Brown's lemma]], [here](Introduction+to+Stable+homotopy+theory+--+P#KenBrownLemma)).

Every morphism $(f,g)$ in the [[product category]] $\mathcal{C}_{c}\times \mathcal{C}_{c}$ may be written as a composite of a pairing with an identity morphisms

$$
  (f,g)
    \;\colon\;
  (c_1, d_1)
    \overset{(id_{c_1},g)}{\longrightarrow}
  (c_1,d_2)
    \overset{(f,id_{c_2})}{\longrightarrow}
  (c_2,d_2)
  \,.
$$

Now since the [[pushout product]] (with respect to tensor product) with the initial morphism $(\ast \to c_1)$ is equivalently the tensor product

$$
  (\ast \to c_1) \Box_{\otimes} g
    \;\simeq\;
  id_{c_1} \otimes g
$$

and

$$
  f \Box_{\otimes} (\ast \to c_2)
    \;\simeq\;
  f \otimes id_{c_2}
$$

the [[pushout-product axiom]] (def. \ref{MonoidalModelCategory}) implies that on the subcategory of cofibrant objects the functor $\otimes$ preserves acyclic cofibrations. (This is why one speaks of a _[[Quillen bifunctor]]_, see also [Hovey 99, prop. 4.3.1](#Hovey99)).

Hence $\otimes^L$ exists.

By the same decomposition and using the [[universal property]] of the [[localization]] of a category ([def.](Introduction+to+Stable+homotopy+theory+--+P#HomotopyCategoryOfACategoryWithWeakEquivalences)) one finds that for $\mathcal{C}$ and $\mathcal{D}$ any two [[categories with weak equivalences]] ([def.](Introduction+to+Stable+homotopy+theory+--+P#CategoryWithWeakEquivalences)) then the [[localization]] of their [[product category]] is the product category of their localizations:

$$
  (\mathcal{C} \times \mathcal{D})[(W_{\mathcal{C}} \sqcup W_{\mathcal{D}})^{-1}]
  \simeq
  (\mathcal{C}[W^{-1}_{\mathcal{C}}])
  \times
  (\mathcal{D}[W^{-1}_{\mathcal{D}}])
  \,.
$$

With this, the [[universal property]] as a [[localization]] ([def.](Introduction+to+Stable+homotopy+theory+--+P#HomotopyCategoryOfACategoryWithWeakEquivalences)) of the [[homotopy category of a model category]] ([thm.](Introduction+to+Stable+homotopy+theory+--+P#UniversalPropertyOfHomotopyCategoryOfAModelCategory)) induces [[associators]]: Let

$$
  \array{
    \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c
      &\overset{((-)\otimes(-))\otimes (-)}{\longrightarrow}&
    \mathcal{C}
    \\
    {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C} \times \mathcal{C}}}}\downarrow
      &\swArrow_{\eta}&
    \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}}
    \\
    Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C})
      &\overset{((-)\otimes^L(-))\otimes^L (-)}{\longrightarrow}&
    Ho(\mathcal{C})
  }
  \;\;\;\,,\;\;\;\;\;
  \array{
    \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c
      &\overset{ (-) \otimes ( (-) \otimes (-) ) }{\longrightarrow}&
    \mathcal{C}
    \\
    {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C} \times \mathcal{C}}}}\downarrow
      &\swArrow_{\eta'}&
    \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}}
    \\
    Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C})
      &\overset{ (-) \otimes^L ( (-) \otimes^L (-) )  }{\longrightarrow}&
    Ho(\mathcal{C})
  }
$$

be the [[natural isomorphism]] exhibiting the [[derived functors]] of the two possible tensor products of three objects, as shown at the top. By pasting the second with the [[associator]] natural isomorphism of $\mathcal{C}$ we obtain another such factorization for the first, as shown on the left below,

{#DerivedAssociatorFromUniversalPropertyOfLocalization}
$$
  (\star)
  \;\;\;\;\;\;
  \array{
    \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c
      &\overset{((-)\otimes(-))\otimes (-)}{\longrightarrow}&
    \mathcal{C}
    \\
    {}^{\mathllap{=}}\downarrow
      &\swArrow_{\alpha}&
    \downarrow^{\mathrlap{=}}
    \\
    \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c
      &\overset{ (-) \otimes ( (-) \otimes (-) ) }{\longrightarrow}&
    \mathcal{C}
    \\
    {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C} \times \mathcal{C}}}}\downarrow
      &\swArrow_\eta&
    \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}}
    \\
    Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C})
      &\overset{ (-) \otimes^L ( (-) \otimes^L (-) )  }{\longrightarrow}&
    Ho(\mathcal{C})
  }
  \;\;\;\;\;\;
  \simeq
  \;\;\;\;\;\;
  \array{
    \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c
      &\overset{((-)\otimes(-))\otimes (-)}{\longrightarrow}&
    \mathcal{C}
    \\
    {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C} \times \mathcal{C}}}}\downarrow
      &\swArrow_{\eta'}&
    \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}}
    \\
    Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C})
      &\overset{((-)\otimes^L(-))\otimes^L (-)}{\longrightarrow}&
    Ho(\mathcal{C})
    \\
    {}^{\mathllap{=}}\downarrow
      &\swArrow_{\alpha^L}&
    \downarrow^{\mathrlap{=}}
    \\
    Ho(\mathcal{C})\times Ho(\mathcal{C})\times Ho(\mathcal{C})
      &\underset{(-)\otimes^L((-)\otimes^L (-))}{\longrightarrow}&
    Ho(\mathcal{C})
  }
$$

and hence by the [[universal property]] of the factorization through the derived functor, there exists a unique natural isomorphism $\alpha^L$ such as to make this composite of natural isomorphisms equal to the one shown on the right. Hence the [[pentagon identity]] satisfied by $\alpha$ implies a pentagon identity for $\alpha^L$, and so $\alpha^L$ is an [[associator]] for $\otimes^L$.

The above equation on pasting composites of [[natural isomorphism]] is equivalently just the coherence law for a monoidal functor:

$$
  \array{
    (\gamma(X) \otimes^L \gamma(Y)) \otimes^L \gamma(Z)
     &\overset{\alpha^L_{\gamma(X), \gamma(Y), \gamma(Z)}}{\longrightarrow}&
    \gamma(X) \otimes^L (\gamma(Y) \otimes^L \gamma(Z))
    \\
    {}^{\mathllap{\eta'}}\uparrow
      &&
    \uparrow^{\mathrlap{\eta}}
    \\
    \gamma( (X \otimes Y) \otimes Z )
     &\underset{\gamma(\alpha)}{\longrightarrow}&
   \gamma(X \otimes (Y \otimes Z))
  }
  \,.
$$

=--

## Examples

### Classical homotopy theory

* A [[nice category of spaces|nice category of topological spaces]] with cartesian product and the [[classical model structure on topological spaces]].

* The category of [[simplicial sets]] with cartesian product and the [[classical model structure on simplicial sets]].

* The [[classical model structure on pointed topological spaces]] or pointed simplicial sets with the [[smash product]] of pointed objects.

### Simplicial presheaves
 {#SimplicialPresheaves}

If the underlying [[site]] has [[finite products]], then both the injective and the projective, the global and the local [[model structure on simplicial presheaves]] becomes a [[cartesian monoidal category|cartesian]] [[monoidal model category]] with respect to the standard [[closed monoidal structure on presheaves]].

See at _[[model structure on simplicial presheaves]]_ the section _[Closed monoidal structure](https://ncatlab.org/nlab/show/model+structure+on+simplicial+presheaves#MonoidalStructure)_.

### Monoidal presheaves
 {#MonoidalPresheaves}

More generally, if a [[small category]] $\mathcal{S}$ has [[finite products]] and $\mathcal{M}$ is a [[cofibrantly generated model category|cofibrantly generated]] [[symmetric monoidal model category]], then the [[functor category]] $Func\big(\mathcal{S}^{op}, \mathcal{M}\big)$ with its object-wise [[monoidal category]]-structure and with the [[projective model structure on functors]] is itself a monoidal model category ([Pavlov & Scholbach 2018, inside proof of Prop. 7.9](#PavlovScholbach18)).

### Homological algebra and stable homotopy theory

* The [[category of chain complexes]] with the usual [[tensor product of chain complexes]] carries various monoidal model category structures, see

* the [[model structure on chain complexes]].

With respect to a [[symmetric monoidal smash product of spectra]]:

* the [[model structure on symmetric spectra]]

* the [[model structure on orthogonal spectra]]

* the [[model structure for excisive functors]]

([Schwede-Shipley 00](#SchwedeShipley00), [MMSS 00, theorem 12.1 (iii) with prop. 12.3](#MMSS00))

The standard example of a monoidal model category whose unit is not cofibrant is the category of EKMM S-modules.

### Categorical model structures

* The category [[Cat]] with cartesian product and the [[folk model structure]].

* The category [[Gray]] of [[strict 2-category|strict 2-categories]] with the [[Gray tensor product]] and the [[model structure on 2-categories|Lack model structure]].

### Model structure on $G$-objects

+-- {: .un_def}
###### Assumption

Let $\mathcal{E}$ be a [[category]] equipped with the structure of

* a [[closed monoidal category|closed]] [[symmetric monoidal category]];

* a [[monoidal model category]];

such that

* the model structure is [[cofibrantly generated model category|cofibrantly generated]];

* the tensor unit $I$ is cofibrant.

=--

+-- {: .un_prop}
###### Proposition

Under these conditions there is for each [[discrete group]] $G$ the structure of a [[monoidal model category]] on the [[Borel model structure]] $\mathcal{E}^{\mathbf{B}G}$ of objects in $\mathcal{E}$ equipped with a $G$-[[group action|action]], for which the [[forgetful functor]]

$$
  \mathcal{E}^{\mathbf{B}G} \to \mathcal{E}
$$

preserves and reflects fibrations and weak equivalences.

=--

See [Berger & Moerdijk (2006) 2.5](#BergerMoerdijkResolution).

### Model structure on monoids

See [[model structure on monoids in a monoidal model category]].

## Related concepts

* [[cartesian closed monoidal model category]]

* [[enriched monoidal model category]]

* [[monoidal localization]]

* [[monoidal Quillen adjunction]]

* [[model structure on monoids in a monoidal model category]]

* [[model structure on modules in a monoidal model category]]

## References

Textbook accounts:

* {#Hovey99} [[Mark Hovey]], Chapter 4, Def. 4.2.6, in: *[[Model Categories]]*, Mathematical Surveys and Monographs, **63** AMS (1999) [[ISBN:978-0-8218-4361-1](https://bookstore.ams.org/surv-63-s), [doi:10.1090/surv/063](https://doi.org/http://dx.doi.org/10.1090/surv/063), [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/hovey-model-cats.pdf), [Google books](http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover)]

* [[Jacob Lurie]], Def. A.3.1.2 in: _[[Higher Topos Theory]]_, Annals of Mathematics Studies 170, Princeton University Press 2009 ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf))

{#OriginalReferences} Historically, the first mention of monoidal model categories (without the unit axiom) under this name is in

* [[William G. Dwyer]], [[Philip S. Hirschhorn]], [[Daniel M. Kan]], *[[Model Categories and More General Abstract Homotopy Theory]]* [[pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/dhk.pdf)]

  > (Mentioned in passing in Remark 55.10, with no definition given, but the preceding section discusses the pushout product axiom.)

The term "pushout smash poduct" is used for the case of [[smash products]] in [[pointed category|pointed categories]] (such as of [[pointed simplicial sets]] or [[symmetric spectra]]):

* [[Mark Hovey]], [[Brooke Shipley]], [[Jeff Smith]], _Symmetric spectra_, J. Amer. Math. Soc. **13** (1998) 149-208 [[arXiv:math/9801077](https://arxiv.org/abs/math/9801077), [doi:10.1090/S0894-0347-99-00320-3](https://doi.org/10.1090/S0894-0347-99-00320-3)]

The general notion of monoidal model categories and their pushout-product axiom appears in:

* [[Stefan Schwede]], [[Brooke Shipley]], Def. 2.1 _Algebras and modules in monoidal model categories_, Proc. London Math. Soc. **80** 2  (2000)  491-511  [[arXiv:math/9801082](https://arxiv.org/abs/math/9801082), [doi:10.1112/S002461150001220X](https://doi.org/10.1112/S002461150001220X)]

* {#Hovey98} [[Mark Hovey]], Def. 1.1 in: _Monoidal model categories_ [[arXiv:math/9803002v1](https://arxiv.org/abs/math/9803002)]

Although the earliest mentions of terminology appear to be these sources indicated above, the notion itself is older.
In particular, [Hovey (1999), p. 107](#Hovey99) credits the definition of a [[Quillen bifunctor]] to [[Homotopy Limit Functors on Model Categories and Homotopical Categories|Dwyer, Hirschhorn, Kan & Smith (2004)]] (which had for years earlier been "in preparation"), and the definition is of course just a small variant of the definition [[enriched model categories]], which in its specialization to [[simplicial model categories]] is due to

* {#Quillen67} [[Daniel Quillen]], chapter II, section 2 of: _[[Homotopical Algebra]]_, Lecture Notes in Mathematics __43__, Springer (1967) [[doi:10.1007/BFb0097438](https://link.springer.com/book/10.1007/BFb0097438)]

(The corresponding [[pullback-power axiom]] is axiom "SM7" in [Quillen (1967)](#Quillen67).)

The [[unit axiom]] together with the fact that the [[homotopy category]] is monoidal in this case is due to Hovey.

Conditions for the existence of induced monoidal model structure on [[Reedy model categories]]:

* {#Barwick10} [[Clark Barwick]], p. 282 in: *On left and right model categories and left and right Bousfield localizations*, Homology, Homotopy and Applications **12** 2 (2010) 245–320 [[doi:10.4310/hha.2010.v12.n2.a9](https://doi.org/10.4310/hha.2010.v12.n2.a9), [euclid:1296223884](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-12/issue-2/On-left-and-right-model-categories-and-left-and-right/hha/1296223884.full), subsuming: [arXiv:0708.2067](https://arxiv.org/abs/0708.2067), [arXiv:0708.2832](https://arxiv.org/abs/0708.2832), [arXiv:0708.3435](http://arxiv.org/abs/0708.3435)]

Some relevant homotopy category background:

* {#Shulman11} [[Mike Shulman]], _Comparing composites of left and right derived functors_, New York Journal of Mathematics Volume 17 (2011) 75-125 ([arXiv:0706.2868](http://arxiv.org/abs/0706.2868), [publisher](http://nyjm.albany.edu/j/2011/17-5.html))

The monoidal structures for a [[symmetric monoidal smash product of spectra]] are due to

* {#SchwedeShipley00} [[Stefan Schwede]], [[Brooke Shipley]], _Algebras and modules in monoidal model categories_ Proc. London Math. Soc. (2000) 80(2): 491-511  ([pdf](http://www.math.uic.edu/~bshipley/monoidal.pdf))

* {#MMSS00} [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], part III of _[[Model categories of diagram spectra]]_, Proceedings London Mathematical Society Volume 82, Issue 2, 2000 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf), [publisher](http://plms.oxfordjournals.org/content/82/2/441.short?rss=1&ssource=mfc))

Further variation of the axiomatics is discussed in

* {#BataninBerger13} [[Michael Batanin]], [[Clemens Berger]], _Homotopy theory for algebras over polynomial monads_ ([arXiv:1305.0086](https://arxiv.org/abs/1305.0086))

The monoidal model structure on the [[Borel model structure]] $\mathcal{E}^{\mathbf{B}G}$ is discussed for instance in

* {#BergerMoerdijkResolution} [[Clemens Berger]], [[Ieke Moerdijk]], *The Boardman-Vogt resolution of operads in monoidal model categories*, Topology **45** (2006) 807-849 [[arXiv:math/0502155](https://arxiv.org/abs/math/0502155), [doi:10.1016/j.top.2006.05.001](https://doi.org/10.1016/j.top.2006.05.001)]

Relation to [[symmetric monoidal (infinity,1)-categories]] (in particular, that every locally presentable symmetric monoidal $(\infty,1)$-category arises from a symmetric monoidal model category) is discussed in

* {#NikolausSagave15} [[Thomas Nikolaus]], [[Steffen Sagave]], _Presentably symmetric monoidal infinity-categories are represented by symmetric monoidal model categories_ ([arXiv:1506.01475](http://arxiv.org/abs/1506.01475))

Monoidal Reedy model structures are discussed in

* Moncef Ghazel, Fethi Kadhi, _Reedy diagrams in V-model categories_, [arXiv:1610.06535](https://arxiv.org/abs/1610.06535), [doi:10.1007/s10485-019-09566-w](https://doi.org/10.1007/s10485-019-09566-w).

On enhancement of monoidal model categories up to [[strong monoidal functor|strong]] [[monoidal Quillen adjunction|monoidal]] [[Quillen equivalence]]:

* {#PavlovScholbach18} [[Dmitri Pavlov]], [[Jakob Scholbach]], p. 20 of: *Admissibility and rectification of colored symmetric operads*, Journal of Topology **11** 3 (2018) 559-601 [[doi:10.1112/topo.12008](https://doi.org/10.1112/topo.12008), [arXiv:1410.5675](https://arxiv.org/abs/1410.5675)]

and for monoidal [[simplicial model categories]]:

* [[Haldun Özgür Bayındır]], [[Boris Chorny]], *Admissible replacements for simplicial monoidal model categories*, Algebr. Geom. Topol. **23** (2023) 43-73 [[arXiv:2008.00515](https://arxiv.org/abs/2008.00515), [doi:10.2140/agt.2023.23.43](https://doi.org/10.2140/agt.2023.23.43)]

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