commutative Hopf algebroid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _commutative Hopf algebroid_ is a [[Hopf algebroid]] where all multiplication operations (as opposed to the [[comultiplication]] operation) is required to be commutative. This concept is the [[formal dual]] to [[internal groupoids]] in the [[opposite category]] of [[CRing]].

A commutative Hopf algebroid is in particular a _[[Hopf algebroid over a commutative base]]_, but the latter may be more general.

Commutative Hopf algebroids appear prominently in [[stable homotopy theory]]/[[higher algebra]] as the [[dual Steenrod algebras]] of certain classes of [[multiplicative cohomology theory|multiplicative]] [[generalized homology theories]] $E$. As such they play a key role in the $E$-[[Adams spectral sequence]].

## Definition

### Commutative Hopf algebroids

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

A **[[commutative Hopf algebroid]]** is an [[internal groupoid]] in the [[opposite category]] [[CRing]]${}^{op}$ of [[commutative rings]], regarded with its [[cartesian monoidal category]] structure.

=--

(e.g. [Ravenel 86, def. A1.1.1](#Ravenel86))

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

We unwind def. \ref{CommutativeHopfAlgebroid}.  For $R \in CRing$, write $Spec(R)$ for same same object, but regarded as an object in $CRing^{op}$.

An [[internal category]] in $CRing^{op}$ is a [[diagram]] in $CRing^{op}$ of the form

$$
  \array{
    Spec(\Gamma) \underset{Spec(A)}{\times} Spec(\Gamma)
    \\
    \downarrow^{\mathrlap{\circ}}
    \\
    Spec(\Gamma)
    \\
    {}^{\mathllap{s}}\downarrow \; \uparrow^{\mathrlap{i}} \downarrow^{\mathrlap{t}}
    \\
    Spec(A)
  }
  \,,
$$

(where the [[fiber product]] at the top is over $s$ on the left and $t$ on the right) such that the pairing $\circ$ defines an [[associativity law|associative]] [[composition]] over $Spec(A)$, [[unitality|unital]] with respect to $i$. This is an [[internal groupoid]] if it is furthemore equipped with a morphism

$$
  inv \;\colon\; Spec(\Gamma) \longrightarrow Spec(\Gamma)
$$

acting as assigning [[inverses]] with respect to $\circ$.

The key basic fact to use now is that [[tensor product]] of commutative rings exhibits the [[cartesian monoidal category]] structure on $CRing^{op}$, see at _[CRing -- Properties -- Cocartesian comonoidal structure](CRing#CocartesianComnonoidalStructure)_:

$$
  Spec(R_1) \underset{Spec(R_3)}{\times} Spec(R_2)
  \simeq
  Spec(R_1 \otimes_{R_3} R_2)
  \,.
$$

This means that the above is equivalently a diagram in [[CRing]] of the form

$$
  \array{
    \Gamma \underset{A}{\otimes} \Gamma
    \\
    \uparrow^{\mathrlap{\Psi}}
    \\
    \Gamma
    \\
    {}^{\mathllap{\eta_L}}\uparrow
    \downarrow^{\mathrlap{\epsilon}} \;
    \uparrow^{\mathrlap{\eta_R}}
    \\
    A
  }
$$

as well as

$$
  c \; \colon \; \Gamma \longrightarrow \Gamma
$$

and satisfying [[formal duality|formally dual]] conditions, spelled out as def. \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents} below. Here

* $\eta_L, \eta_R$ are called the left and right _[[unit]] maps_;

* $\epsilon$ is called the _co-unit_;

* $\Psi$ is called the _[[comultiplication]]_;

* $c$ is called the _[[antipode]]_ or _conjugation_

=--

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

Generally, in a commutative Hopf algebroid, def. \ref{CommutativeHopfAlgebroid}, the two morphisms $\eta_L, \eta_R\colon A \to \Gamma$ from remark \ref{CommutativeHopfAlgebroidSpelledOut} need not coincide, they make $\Gamma$ genuinely into a [[bimodule]] over $A$, and it is the [[tensor product]] of [[bimodules]] that appears in remark \ref{CommutativeHopfAlgebroidSpelledOut}. But it may happen that they coincide:

An [[internal groupoid]] $\mathcal{G}_1 \stackrel{\overset{s}{\longrightarrow}}{\underset{t}{\longrightarrow}}$ for which the [[domain]] and [[codomain]] morphisms coincide, $s = t$, is euqivalently a [[group object]] in the [[slice category]] over $\mathcal{G}_0$.

Dually, a [[commutative Hopf algebroid]] $\Gamma \stackrel{\overset{\eta_L}{\longleftarrow}}{\underset{\eta_R}{\longleftarrow}} A$ for which $\eta_L$ and $\eta_R$ happen to coincide is equivalently a **commutative [[Hopf algebra]]** $\Gamma$ over $A$.

=--

Writing out the formally dual axioms of an [[internal groupoid]] as in remark \ref{CommutativeHopfAlgebroidSpelledOut} yields the following equivalent but maybe more explicit definition of commutative Hopf algebroids, def. \ref{CommutativeHopfAlgebroid}

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

A **[[commutative Hopf algebroid]]** is

1. two [[commutative rings]], $A$ and $\Gamma$;

1. ring [[homomorphisms]]

   1. (left/right unit)

      $\eta_L,\eta_R \colon A \longrightarrow \Gamma$;

   1. (comultiplication)

      $\Psi \colon \Gamma \longrightarrow \Gamma \underset{A}{\otimes} \Gamma$;

   1. (counit)

      $\epsilon \colon \Gamma \longrightarrow A$;

   1. (conjugation)

      $c \colon \Gamma \longrightarrow \Gamma$

such that

1. (co-[[unitality]])

   1. (identity morphisms respect source and target)

      $\epsilon \circ \eta_L = \epsilon \circ \eta_R = id_A$;

   1. (identity morphisms are units for composition)

      $(id_\Gamma \otimes_A \epsilon) \circ \Psi  = (\epsilon \otimes_A id_\Gamma) \circ \Psi = id_\Gamma$;

   1. (composition respects source and target)

      1. $\Psi \circ \eta_R = (id \otimes_A \eta_R) \circ \eta_R$;

      1. $\Psi \circ \eta_L = (\eta_L \otimes_A id) \circ \eta_L$

1. (co-[[associativity]]) $(id_\Gamma \otimes_A \Psi) \circ \Psi = (\Psi \otimes_A id_\Gamma) \circ \Psi$;

1. ([[inverses]])

   1. (inverting twice is the identity)

      $c \circ c = id_\Gamma$;

   1. (inversion swaps source and target)

      $c \circ \eta_L = \eta_R$; $c \circ \eta_R = \eta_L$;

   1. (inverse morphisms are indeed left and right inverses for composition)

      the morphisms $\alpha$ and $\beta$ induced via the [[coequalizer]] property of the [[tensor product]] from $(-) \cdot c(-)$ and $c(-)\cdot (-)$, respectively

      $$
        \array{
          \Gamma \otimes A \otimes \Gamma
            &
            \underoverset
              {\longrightarrow}
              {\longrightarrow}
              {}
            &
          \Gamma \otimes \Gamma
             &
             \overset{coeq}{\longrightarrow}
             &
          \Gamma \otimes_A \Gamma
           \\
           &&
           {}_{\mathllap{(-)\cdot c(-)}}\downarrow
           &
            \swarrow_{\mathrlap{\alpha}}
           \\
           && \Gamma
        }
      $$

      and

      $$
        \array{
          \Gamma \otimes A \otimes \Gamma
            &
            \underoverset
              {\longrightarrow}
              {\longrightarrow}
              {}
            &
          \Gamma \otimes \Gamma
             &
             \overset{coeq}{\longrightarrow}
             &
          \Gamma \otimes_A \Gamma
           \\
           &&
           {}_{\mathllap{c(-)\cdot (-)}}\downarrow
           &
            \swarrow_{\mathrlap{\beta}}
           \\
           && \Gamma
        }
      $$

      satisfy

      $\alpha \circ \Psi = \eta_L \circ \epsilon $

      and

      $\beta \circ \Psi = \eta_R \circ \epsilon $.

=--

e.g. ([Ravenel 86, def. A1.1.1](#Ravenel86))

### Comodules

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

Given a [[commutative Hopf algebroid]] $\Gamma$ over $A$ as in def. \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents}, hence an [[internal groupoid]] in $CRing^{op}$, then a **comodule ring** over it is an [[action]] in $CRing^{op}$ of that internal groupoid.

=--

In the same spirit, a **[[comodule]]** over a commutative Hopf algebroid (not necessarily a comodule ring) is a [[quasicoherent sheaf]] on the corresponding [[internal groupoid]] (regarded as a ([[algebraic stack|algebraic]]) [[stack]]) (e.g. [Hopkins 99, prop. 11.6](#Hopkins99)). Explicitly in components:

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

Given a [[commutative Hopf algebroid]] $\Gamma$ over $A$, def. \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents},
then a **left [[comodule]]** over $\Gamma$ is

1. an $A$-[[module]] $N$;

1. an $A$-[[module]] [[homomorphism]] (co-action)

   $\Psi_N \;\colon\; N \longrightarrow \Gamma \otimes_A N$;

such that

1. (co-[[unitality]])

   $(\epsilon \otimes_A id_N) \circ \Psi_N = id_N$;

1. (co-action property)

   $(\Psi \otimes_A id_N) \circ \Psi_N = (id_\Gamma \otimes_A \Psi_N)\circ \Psi_N$.

A [[homomorphism]] between comodules $N_1 \to N_2$ is a homomorphism of underlying $A$-modules making [[commuting diagrams]] with the co-action morphism. Write

$$
  \Gamma CoMod
$$

for the resulting [[category]] of (left) comodules over $\Gamma$. Analogously there are right comodules.

=--

+-- {: .num_example #ComoduleStructureOnGroundRing}
###### Example

For $(\Gamma,A)$ a [[commutative Hopf algebroid]], then $A$ becomes a left $\Gamma$-comodule (def. \ref{CommutativeHopfAlgebroidComodule}) with coaction given by the right unit

$$
  A \overset{\eta_R}{\longrightarrow} \Gamma \simeq \Gamma \otimes_A A
  \,.
$$

=--

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

The required co-unitality property is the dual condition in def. \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents}

$$
  \epsilon \circ \eta_R  = id_A
$$

of the fact in def. \ref{CommutativeHopfAlgebroid} that identity morphisms respect sources:

$$
  id
    \;\colon\;
  A
    \overset{\eta_R}{\longrightarrow}
  \Gamma
    \simeq
  \Gamma \otimes_A A
    \overset{\epsilon \otimes_A id}{\longrightarrow}
  A \otimes_A A
    \simeq
  A
$$

The required co-action property is the dual condition

$$
  \Psi \circ \eta_R = (id \otimes_A \eta_R) \circ \eta_R
$$

of the fact in def. \ref{CommutativeHopfAlgebroid} that composition of morphisms in a groupoid respects sources

$$
  \array{
    A
      &\overset{\eta_R}{\longrightarrow}&
    \Gamma
    \\
    {}^{\mathllap{\eta_R}}\downarrow
      &&
    \downarrow^{\mathrlap{\Psi}}
    \\
    \Gamma \simeq \Gamma \otimes_A A
      &\underset{id \otimes_A \eta_R}{\longrightarrow}&
    \Gamma \otimes_A \Gamma
  }
  \,.
$$

=--

## Properties

### Co-free Comodules

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

Given a [[commutative Hopf algebroid]] $\Gamma$ over $A$, there is a [[free-forgetful adjunction]]

$$
  \Gamma CoMod
   \underoverset
     {\underset{forget}{\longrightarrow}}
     {\overset{co-free}{\longleftarrow}}
     {\bot}
  A Mod
$$

between the [[category]] of $\Gamma$-[[comodules]], def. \ref{CommutativeHopfAlgebroidComodule} and the [[category of modules]] over $A$, where the [[cofree functor]] is [[right adjoint]].

The co-free $\Gamma$-[[comodule]] on an $A$-module $N$ is $\Gamma \otimes_A N$ equipped with the [[coaction]] induced by the [[comultiplication]] $\Psi$ in $\Gamma$.

=--

The proof is [[formal dual|formally dual]] to the characterization of [[free modules]], but we spell it out for completenss:

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

A homomorphism into a co-free $\Gamma$-comodule is a morphism of $A$-modules of the form

$$
  f \;\colon\; N \longrightarrow \Gamma \otimes_A C
$$

making the following [[commuting diagram|diagram commute]]

$$
  \array{
    N &\overset{f}{\longrightarrow}& \Gamma \otimes_A C
    \\
    {}^{\mathllap{\Psi_N}}\downarrow
      &&
    \downarrow^{\mathrlap{\Psi \otimes_A id}}
    \\
    \Gamma \otimes_A N
      &\underset{id \otimes_A f}{\longrightarrow}&
    \Gamma \otimes_A \Gamma \otimes_A C
  }
  \,.
$$

Consider the composite

$$
  \tilde f
  \;\colon\;
  N
    \overset{f}{\longrightarrow}
  \Gamma \otimes_A C
    \overset{\epsilon \otimes_A id}{\longrightarrow}
  A \otimes_A C
    \simeq
  C
  \,,
$$

i.e. the "corestriction" of $f$ along the counit of $\Gamma$. By definition this makes the following square commute

$$
  \array{
    \Gamma \otimes_A N
      &\overset{id \otimes_A f}{\longrightarrow}&
    \Gamma \otimes_A \Gamma \otimes_A C
    \\
    {}^{\mathllap{=}}\downarrow
      &&
    \downarrow^{\mathrlap{id \otimes_A \epsilon \otimes_A id}}
    \\
    \Gamma \otimes_A N
      &\underset{id \otimes_A \tilde f}{\longrightarrow}&
    \Gamma \otimes_A C
  }
  \,.
$$

Pasting this square onto the bottom of the previous one yields

$$
  \array{
    N &\overset{f}{\longrightarrow}& \Gamma \otimes_A C
    \\
    {}^{\mathllap{\Psi_N}}\downarrow
      &&
    \downarrow^{\mathrlap{\Psi \otimes_A id}}
    \\
    \Gamma \otimes_A N
      &\underset{id \otimes_A f}{\longrightarrow}&
    \Gamma \otimes_A \Gamma \otimes_A C
    \\
    {}^{\mathllap{=}}\downarrow
      &&
    \downarrow^{\mathrlap{id \otimes_A \epsilon \otimes_A id}}
    \\
    \Gamma \otimes_A N
      &\underset{id \otimes_A \tilde f}{\longrightarrow}&
    \Gamma \otimes_A C
  }
  \,.
$$

Now due to co-unitality, the right right vertical composite is the identity on $\Gamma \otimes_A C$. But this means by the commutativity of the outer rectangle that $f$ is uniquely fixed in terms of $\tilde f$ by the relation

$$
  f = (id \otimes_A f) \circ \Psi
  \,.
$$

This establishes a [[natural bijection]]

$$
  \frac{
    N \overset{f}{\longrightarrow} \Gamma \otimes_A C
  }{
    N \overset{\tilde f}{\longrightarrow} C
  }
$$

and hence the adjunction in question.

=--

### Cotensor product of comodules

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

Consider a [[commutative Hopf algebroid]] $\Gamma$ over $A$, def. \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents}.
Any left comodule $N$ over $\Gamma$ (def. \ref{CommutativeHopfAlgebroidComodule}) becomes a right comodule via the coaction

$$
  N
    \overset{\Psi}{\longrightarrow}
  \Gamma \otimes_A N
    \overset{\simeq}{\longrightarrow}
  N \otimes_A \Gamma
    \overset{id \otimes_A c}{\longrightarrow}
  N \otimes_A \Gamma
  \,,
$$

where the isomorphism in the middle the is [[braiding]] in $A Mod$ and where $c$ is the conjugation map of $\Gamma$.

Dually, a right comodule $N$ becoomes a left comodule with the coaction

$$
  N
    \overset{\Psi}{\longrightarrow}
  N \otimes_A \Gamma
    \overset{\simeq}{\longrightarrow}
  \Gamma \otimes_A N
    \overset{c \otimes_A id}{\longrightarrow}
  \Gamma \otimes_A N
  \,.
$$

=--

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

Given a [[commutative Hopf algebroid]] $\Gamma$ over $A$, def. \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents},
and given $N_1$ a right $\Gamma$-comodule and $N_2$ a left comodule (def. \ref{CommutativeHopfAlgebroidComodule}), then their **[[cotensor product]]** $N_1 \Box_\Gamma N_2$ is the [[kernel]] of the difference of the two coaction morphisms:

$$
  N_1 \Box_\Gamma N_2
    \;\coloneqq\;
  ker
  \left(
    N_1 \otimes_A N_2
    \overset{\Psi_{N_1}\otimes_{A} id - id \otimes_A \Psi_{N_2} }{\longrightarrow}
    N_1 \otimes_A \Gamma \otimes_A N_2
  \right)
  \,.
$$

If both $N_1$ and $N_2$ are left comodules, then their cotensor product is the cotensor product of $N_2$ with $N_1$ regarded as a right comodule via prop. \ref{LeftComodulesToRightComodules}.

=--

e.g. ([Ravenel 86, def. A1.1.4](#Ravenel86)).

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

Given a [[commutative Hopf algebroid]] $\Gamma$ over $A$, def. \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents},
and given $N_1, N_2$ two left $\Gamma$-comodules (def. \ref{CommutativeHopfAlgebroidComodule}), then their [[cotensor product]] (def. \ref{CotensorProductOfComodules}) is commutative, in that there is an [[isomorphism]]

$$
  N_1 \Box N_2 \;\simeq\; N_2 \Box N_1
  \,.
$$

=--

(e.g. [Ravenel 86, prop. A1.1.5](#Ravenel86))

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

Given a [[commutative Hopf algebroid]] $\Gamma$ over $A$, def. \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents},
and given $N_1, N_2$ two left $\Gamma$-comodules (def. \ref{CommutativeHopfAlgebroidComodule}), such that $N_1$ is [[projective module|projective]] as an $A$-[[module]], then

1. The morphism

   $$
      Hom_A(N_1, A)
       \overset{f \mapsto (id \otimes_A f) \circ \Psi_{N_1}}{\longrightarrow}
      Hom_A(N_1, \Gamma \otimes_A A)
       \simeq
      Hom_A(N_1, \Gamma)
       \simeq
      Hom_A(N_1, A) \otimes_A \Gamma
   $$

   gives $Hom_A(N_1,A)$ the structure of a right $\Gamma$-comodule;

1. The [[cotensor product]] (def. \ref{CotensorProductOfComodules}) with respect to this right comodule structure is isomorphic to the hom of $\Gamma$-comodules:

   $$
     Hom_A(N_1, A) \Box_\Gamma N_2
     \simeq
     Hom_\Gamma(N_1, N_2)
     \,.
   $$

   Hence in particular

   $$
     A \Box_\Gamma N_2
      \;\simeq\;
     Hom_\Gamma(A,N_2)
   $$

=--

(e.g. [Ravenel 86, lemma A1.1.6](#Ravenel86))

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

In computing the second page of $E$-[[Adams spectral sequences]], the second statement in lemma \ref{ComoduleHomInTermsOfCotensorProduct} is the key translation that makes the comodule [[Ext]]-groups on the page be equivalent to a [[Cotor]]-groups. The latter lend themselves to computation, for instance via [[Lambda-algebra]] or via the [[May spectral sequence]].

=--

### Homological algebra of comodules
 {#HomologicalAlgebraOfHopfComodules}

We discuss aspects of the [[homological algebra]] in [[categories]] of comodules over commutative Hopf algebroids, def. \ref{CommutativeHopfAlgebroidComodule}.

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

If a [[commutative Hopf algebroid]] $\Gamma$ over $A$, def. \ref{CommutativeHopfAlgebroid}, \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents} is such that $\eta_L, \eta_R \colon A \longrightarrow \Gamma$ is a [[flat morphism]], then the [[category]] $\Gamma CoMod$ of [[comodules]] over $\Gamma$, def. \ref{CommutativeHopfAlgebroidComodule}, is an [[abelian category]].

=--

(e.g. [Ravenel 86, theorem A1.1.3](#Ravenel86))

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

It is clear that, without any condition the Hopf algebroid, $\Gamma CoMod$ is an [[additive category]].

We need to show that with the assumption that $\Gamma$ is flat over $A$, then this is also a [[pre-abelian category]] in that [[kernels]] and [[cokernels]] exist. Let $f \colon (N_1,\Psi_{N_1}) \longrightarrow (N_2,\Psi_{N_2})$ be a morphism of comodules, hence a [[commuting diagram]] in $A$[[Mod]] of the form

$$
  \array{
    N_1 &\stackrel{f}{\longrightarrow}& N_2
    \\
    \downarrow^{\mathrlap{\Psi_{N_1}}} && \downarrow^{\mathrlap{\Psi_{N_2}}}
    \\
    \Gamma \otimes_A N_1
    &\stackrel{id_\Gamma \otimes_A f}{\longrightarrow}&
    \Gamma \otimes_A N_2
  }
  \,.
$$

Consider the kernel $ker(f)$ of $f$ in $A$[[Mod]] and its image under $\Gamma \otimes_A (-)$

$$
  \array{
    ker(f) &\longrightarrow& N_1 &\stackrel{f}{\longrightarrow}& N_2
    \\
    \downarrow && \downarrow^{\mathrlap{\Psi_{N_1}}} && \downarrow^{\mathrlap{\Psi_{N_2}}}
    \\
    \Gamma \otimes_A ker(f)
      &\longrightarrow&
    \Gamma \otimes_A N_1
      &\stackrel{id_\Gamma \otimes_A f}{\longrightarrow}&
    \Gamma \otimes_A N_2
  }
  \,.
$$

By the assumption that $\Gamma$ is a [[flat module]] over $A$, also $\Gamma \otimes_A ker(f) \simeq ker(\Gamma \otimes_A f)$ is a [[kernel]]. By its [[universal property]] this induces uniquely a morphism as shown on the left, making the above [[commuting diagram|diagram commute]]. This means that the $A$-module $ker(f)$ uniquely inherits the structure of a $\Gamma$-comodule such as to make $ker(f) \to N_1$ a comodule homomorphism. By the same universal property it follows that $ker(f)$ with this comodule structure is in fact the kernel of $f$ in $\Gamma CoMod$.

The argument for the existence of [[cokernels]] proceeds [[formal dual|formally dually]]. Therefore it follows that the comparison morphism

$$
  coker(ker(f)) \longrightarrow ker(coker(f))
$$

formed in $\Gamma CoMod$ has underlying it the corresponding comparison morphism in $A Mod$. There this is an [[isomorphism]], hence it is an isomorphism also in $\Gamma CoMod$, and so the latter is not just a [[pre-abelian category]] but in fact an [[abelian category]] itself.

=--

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

If a [[commutative Hopf algebroid]] $\Gamma$ over $A$, def. \ref{CommutativeHopfAlgebroid}, \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents} is such that $\eta_L, \eta_R \colon A \longrightarrow \Gamma$ is a [[flat morphism]], then

1. every co-free $\Gamma$-[[comodule]], def. \ref{CoFreeComodules}, on an [[injective module]] over $A$ is an [[injective object]] in $\Gamma CoMod$;

1. $\Gamma CoMod$ has [[enough injectives]] (if the [[axiom of choice]] holds in the ambient [[set theory]]).

=--

(e.g. [Ravenel 86, lemma A1.2.2](#Ravenel86))

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

First of all, assuming the [[axiom of choice]], then the [[category of modules]] $A Mod$ has [[enough injectives]] (see [this proposition](injective+object#AbHasEnoughInjectives)).
Now by prop. \ref{CoFreeComodules} we have the [[adjunction]]

$$
  \Gamma CoMod
   \underoverset
     {\underset{forget}{\longrightarrow}}
     {\overset{co-free}{\longleftarrow}}
     {\bot}
  A Mod
 \,.
$$

Observe that the [[left adjoint]] is a [[faithful functor]] (being a [[forgetful functor]]) and that, by the proof of prop. \ref{CategoryOfHopfComodulesIsAbelianIfHopfAlgebroidIsFlat}, it is an [[exact functor]]. With this a standard lemma applies ([here](injective+object#TransferOfEnoughInjectivesAlongAdjunctions)) which says that

1. with $I \in A Mod$ an [[injective module]], then the co-free comodule $\Gamma \otimes_A I$ is an [[injective object]] in $\Gamma CoMod$;

1. for $N \in \Gamma CoMod$ any object, and for $i \colon U(N) \hookrightarrow I$ a monomorphism of $A$-modules into an injective $A$-module, then the [[adjunct]] $\tilde i \colon N \hookrightarrow \Gamma\otimes_A I$ is a monomorphism in $\Gamma CoMod$ (and into an injective comodule).

=--

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

Let $\Gamma$ be a [[commutative Hopf algebroid]] over $A$, def. \ref{CommutativeHopfAlgebroid}, \ref{CommutativeHopfAlgebroidDefinitionInExplicitComponents}, such that $\eta_L, \eta_R \colon A \longrightarrow \Gamma$ is a [[flat morphism]],
Let $N \in \Gamma CoMod$ be a Hopf [[comodule]], def. \ref{CommutativeHopfAlgebroidComodule}, such that the underlying $A$-module is a [[projective module]] (a [[projective object]] in $A$[[Mod]]).

Then  (assuming the [[axiom of choice]]) every co-free commodule, prop. \ref{CoFreeComodules}, is an $F$-[[acyclic object]] for $F$ the [[hom functor]] $Hom_{\Gamma CoMod}(N,-)$.

=--

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

We need to show that the [[derived functors in homological algebra|derived functors]] $R^{\bullet} Hom_{\Gamma}(N,-)$ vanish in positive degree on all co-free comodules, hence on $\Gamma \otimes_A K$, for $K \in A Mod$.

To that end, let $I^\bullet$ be an [[injective resolution]] of $K$ in $A Mod$. By prop. \ref{CategoryOfHopfComodulesIsAbelianIfHopfAlgebroidIsFlat} then $\Gamma \otimes_A I^\bullet$ is a sequence of [[injective objects]] in $\Gamma CoMod$ and by the assumption that $\Gamma$ is flat over $A$ it is an [[injective resolution]] of $\Gamma \otimes_A K$ in $\Gamma CoMod$. Therefore the derived functor in question is given by

$$
  \begin{aligned}
    R^{\bullet \geq 1} Hom_\Gamma(N, \Gamma \otimes_A K)
    & \simeq
    H_{\bullet \geq 1}( Hom_\Gamma( N, \Gamma \otimes_A I^\bullet ) )
    \\
    & \simeq H_{\bullet \geq 1}( Hom_A(N, I^\bullet) )
    \\
    & \simeq 0
  \end{aligned}
  \,.
$$

Here the second equivalence is the cofree/forgetful adjunction isomorphism of prop. \ref{CoFreeComodules}, while the last equality then follows from the assumption that the $A$-module underlying $N$ is a [[projective module]] (since [[hom functors]] out of [[projective objects]] are [[exact functors]] ([here](projective+object#EquivalenceOfDefinitionsInAbelian)) and since derived functors of exact functors vanish in positive degree ([here](derived+functor+in+homological+algebra#DerivedFunctorOfExactFunctor))).

=--

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

In the application to Hopf algebroids induced from [[commutative ring spectra]] $E$ ([below](#FromRingSpectra)), lemma \ref{CoFreeHopfComodulesAreHomNAcyclicForProjectiveN} is the key statement that identifies the entries of the second page of the $E$-[[Adams spectral sequence]] with [[Ext]]-groups of Hof comodules. See at _[Adams spectral sequence -- The second page](Adams+spectral+sequence#TheE2TermOfTheEAdamsSpectralSequence)_.

=--

### Ext-groups

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

If $(\Gamma,A)$ graded commutative Hopf algebra such that

1. the underlying algebra is free graded commutative;

1. $\eta \colon A \to \Gamma$ is a [[flat morphism]];

1. $\Gamma$ is generated by [[primitive elements]] $\{x_i\}_{i\in I}$

then the [[Ext]] of $\Gamma$-comodules from $A$ and itself is the (graded) polynomial algbra on these generators:

$$
  Ext_\Gamma(A,A) \simeq A[\{x_i\}_{i \in I}]
  \,.
$$

=--

([Ravenel 86, lemma 3.1.9](#Ravenel86), [Kochman 96, prop. 3.7.5](#Kochman96))

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

Consider the co-free left $\Gamma$-comodule ([prop.](commutative+Hopf+algebroid#CoFreeComodules))

$$
  \Gamma \otimes_A A[\{y_i\}_{i \in I}]
$$

and regard it as a [[chain complex]] of left comodules by defining a [[differential]] via

$$
  d \colon x_i \mapsto y_i
$$

$$
  d \colon y_i \mapsto 0
$$

and extending as a graded [[derivation]].

We claim that $d$ is a homomorphism of left comodules: Due to the assumption that all the $x_i$ are primitive we have on generators that

$$
  \begin{aligned}
    (id,d) ( \Psi(x_i)  )
    & =
    (id,d) ( x_i \otimes 1 + 1 \otimes x_i )
    \\
    & =
    \underset{= 0}{x_i  \otimes \underbrace{(d 1)} }
     +
    \underset{= y_i}{ 1 \otimes \underbrace{(d x_i)} }
    \\
    & =
    \Psi( d x_i )
  \end{aligned}
$$

and

$$
  \begin{aligned}
    (id,d)( \Psi(y_i) )
    & =
    (id,d) ( 1, y_i )
    \\
    & =
    (1, d y_i)
    \\
    & =
    0
    \\
    & =
    \Psi( 0 )
    \\
    & =
    \Psi(d y_i)
  \end{aligned}
  \,.
$$

Since $d$ is a graded derivation on a free graded commutative algbra, and $\Psi$ is an algebra homomorphism, this implies the statement for all other elements.

Now observe that the canonical [[chain map]]

$$
  (\Gamma \otimes_A A[\{y_i\}_{i \in I}]  ,\; d)
    \overset{\simeq_{qi}}{\longrightarrow}
  A
$$

(which projects out the generators $x_i$ and $y_i$ and is the identity on $A$), is a [[quasi-isomorphism]], by construction. Therefore it constitutes a co-free resolution of $A$ in left $\Gamma$-comodules.

Since the counit $\eta$ is assumed to be flat, and since $A$ is trivially a [[projective module]] over itself, prop. \ref{CoFreeHopfComodulesAreHomNAcyclicForProjectiveN} now implies that the above is an [[acyclic resolution]] with respect to the functor $Hom_{\Gamma}(A,-) \colon \Gamma CoMod \longrightarrow A Mod$. Therefore it computes the [[Ext]]-functor. Using that forming co-free comodules is [[right adjoint]] to forgetting $\Gamma$-comodule structure over $A$ ([prop.](commutative+Hopf+algebroid#CoFreeComodules)), this yields:

$$
  \begin{aligned}
    Ext^\bullet_\Gamma(A,A)
    & \simeq
    H_\bullet(Hom_\Gamma(A,  \Gamma \otimes_A A[\{y_i\}_{i \in I}] ), d)
    \\
    & \simeq
    H_\bullet(Hom_A(A,  A[\{y_i\}_{i \in I}] ), d= 0 )
    \\
    & \simeq
    Hom_A(A,  A[\{y_i\}_{i \in I}] )
    \\
    & \simeq
    A[\{y_i\}_{i \in I}]
  \end{aligned}
  \,.
$$

=--

## Examples

### From ring spectra
 {#FromRingSpectra}

The [[cosimplicial object|cosimplicial]] spectra of certain [[commutative ring spectra]] $E$ (their [[Amitsur complexes]]) yield commutative Hopf algebroids when truncated. The Hopf algebroids appearing this way govern the corresponding $E$-[[Adams spectral sequences]].

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

Call a [[commutative ring spectrum]] $E$ _flat_ if one, equivalently both, of the morphisms

$$
  \eta_L \coloneqq \pi_\bullet(e \wedge id)
  \;\colon\;
  E_\bullet \longrightarrow E_\bullet(E)
$$

$$
  \eta_r
  \;\coloneqq\;
  \pi_\bullet(id \wedge e) \colon E_\bullet \longrightarrow E_\bullet(E)
$$

is a [[flat morphism]].

=--

+-- {: .num_example }
###### Example

Examples of ring spectra that are flat according to def. \ref{FlatE} include
$E = $

* [[sphere spectrum|S]],

* [[HR]] for $R = \mathbb{F}_p$ a [[prime field]],

* [[MO]], [[MU]], [[MSp]],

* [[KO]], [[KU]].

=--

+-- {: .num_example}
###### Example

Examples of ring spectra that are _not_ flat in the sense of def. \ref{FlatE} include [[HA|H]][[integers|Z]], and $M S U$.

=--

The key consequence of the assumption that $E$ is flat in the sense of def. \ref{FlatE} is the following.

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

If $E$ is flat, def. \ref{FlatE}, then for all spectra $X$ there is a [[natural isomorphisms]]

$$
  E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(X)
    \stackrel{}{\longrightarrow}
  \pi_\bullet(E \wedge E \wedge X)
$$

and hence for all $n \in \mathbb{N}$ there are isomorphisms

$$
  \pi_\bullet(E^{\wedge^{(n+2)}}\wedge X  )
    \simeq
  \underset{n+1\,factors}{
  \underbrace{E_\bullet(E)
    \otimes_{\pi_\bullet(E)}
    \cdots
    \otimes_{\pi_\bullet(E)}
  E_\bullet(E)
  }}
    \otimes_{\pi_\bullet(E)}
  E_\bullet(X)
  \,.
$$

=--

(e.g. [Adams 96, part III, lemma 12.5](#Adams96), [Schwede 12, prop. 6.20](#Schwede12))

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

The desired natural homomorphism

$$
  E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(X)
  \longrightarrow
  \pi_\bullet(E \wedge E \wedge X)
$$

is given on $[\alpha] \in \pi_\bullet(E \wedge E)$ and $[\beta] \in \pi_\bullet(E \wedge X)$ by $([\alpha, \beta])\mapsto [(id \wedge \mu \wedge id) \circ (\alpha \wedge \beta)]$.

To see that this is an isomorphism, observe that by flatness of $E$, the assignment $X \mapsto E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(-)$ is a [[generalized homology]] functor, hence [[Brown representability theorem|represented]] by some spectrum. The above morphism, natural in $X$, thus constitutes a homomorphism  of generalized homology theories, and so for these to be equivalent it is sufficient to check that they induce isomorphisms on the point. This is manifestly the case.

Finally we get the claimed isomorphisms for all $n$ by [[induction]]:

$$
  \begin{aligned}
    \pi_\bullet(E^{\wedge^{n+2}} \wedge X)
    & \simeq
    \pi_\bullet(E \wedge E \wedge E^{\wedge^n} \wedge X))
    \\
    &\simeq
     E_\bullet(E)
         \otimes_{\pi_\bullet(E)}
     E_\bullet( E^{\wedge^n} \wedge X )
     \\
     & =
     E_\bullet(E)
         \otimes_{\pi_\bullet(E)}
     \pi_\bullet( E^{\wedge^{n+1}} \wedge X )
  \end{aligned}
  \,.
$$

=--

Now we may identify the commutative Hopf algebroids arising from flat commutative ring spectra.

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

If a [[commutative ring spectrum]] $E$ is flat according to def. \ref{FlatE}, then, via the isomorphism of proposition \ref{FlatnessOfEImpliesKeyConsequence}, the cosimplicial spectrum $E^{\wedge^\bullet} \wedge X$ (the $E$-standard resolution of $X$ from example \ref{StandardEResolution}) exhibits:

1. for $X = E$: [[Hopf algebroid]]-structure, def. \ref{CommutativeHopfAlgebroid}, remark \ref{CommutativeHopfAlgebroidSpelledOut}, on $E_\bullet(E)$ over $\pi_\bullet(E)$ -- called the **dual $E$-[[Steenrod algebra]]**;

1. for general $X$: [[comodule]]-structure on $E_\bullet(X)$ over the dual $E$-[[Steenrod algebra]].

=--

(e.g. [Baker-Lazarev 01, theorem 1.1](#BakerLazarev01))

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

Via prop. \ref{FlatnessOfEImpliesKeyConsequence}, the image under $\pi_\bullet(-)$ of the cosimplicial spectrum $E^{\wedge^\bullet}(E)$ is identified as on the right of the following diagram

$$
  \array{
    \pi_\bullet(E\wedge E \wedge E) &\simeq& E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(E)
    \\
    \uparrow^{\mathrlap{\pi_\bullet(id \wedge e \wedge id)}}
    &&
    \uparrow^{\mathrlap{\Psi}}
    \\
    \pi_\bullet(E \wedge E) &=& E_\bullet(E)
    \\
    {}^{\mathllap{\pi_\bullet(e \wedge id)}}\uparrow
     \downarrow^{\mathrlap{\pi_\bullet(\mu)}}
     \;\;\;\;\;\;
     \uparrow^{\mathrlap{\pi_\bullet(id \wedge e)}}
     &&
     {}^{\mathllap{\eta_L}}\uparrow
     \downarrow^{\mathrlap{\epsilon}}
     \uparrow^{\mathrlap{\eta_R}}
    \\
    \pi_\bullet(E) &=& \pi_\bullet(E)
  }
  \,.
$$

Analogously the [[coaction]] is induced as on the right of the following diagram

$$
  \array{
    \pi_\bullet(E\wedge E \wedge X) &\simeq& E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(X)
    \\
    \uparrow^{\mathrlap{\pi_\bullet(id \wedge e \wedge id)}}
    &&
    \uparrow^{\mathrlap{\Psi_X}}
    \\
    \pi_\bullet(E \wedge X) &=& E_\bullet(X)
  }
  \,.
$$

=--

+-- {: .num_example}
###### Example

Examples of [[commutative ring spectra]] $E$ for which the dual $E$-[[Steenrod algebra]] $E_\bullet(E)$ over $\pi_\bullet(E)$ of corollary \ref{HopfAlgebroidStructureOnDualEOperations} happens to be a [[commutative Hopf algebra]] over $\pi_\bullet(E)$ instead of a more general [[commutative Hopf algebroid]], according to remark \ref{HopfAlgebrasAsHopfAlgebroids}, includes the cases

$E = $

* [[HA|H]]$\mathbb{F}_p$,

* ...

=--

The key use of the Hopf coalgebroid structure of prop. \ref{HopfAlgebroidStructureOnDualEOperations} for the purpose of the $E$-[[Adams spectral sequence]] is that it is extra structure inherited from maps of spectra under smashing with $E$:

+-- {: .num_example #SmashingMapsWithEFactorsThroughSteenrodComoduleHomomorphisms}
###### Example

For $Y,N$ any two [[spectra]], the morphism (of $\mathbb{Z}$-[[graded abelian groups]]) given by [[smash product of spectra|smash product]] with $E$

$$
  \pi_\bullet(E \wedge -)
    \;\colon\;
  \pi_\bullet([Y,N])
    \longrightarrow
  Hom^\bullet_{Ab}(E_\bullet(Y), E_\bullet(N))
$$

factors through $E_\bullet(E)$-[[comodule]] [[homomorphisms]] over the dual $E$-[[Steenrod algebra]]:

$$
  \pi_\bullet(E \wedge -)
    \;\colon\;
  \pi_\bullet([Y,N])
    \longrightarrow
  Hom^\bullet_{E_\bullet(E)}(E_\bullet(Y), E_\bullet(N))
    \longrightarrow
  Hom^\bullet_{Ab}(E_\bullet(Y), E_\bullet(N))
  \,.
$$

=--

In order to make use of this we need to invoke a [[universal coefficient theorem]] in the following form.

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

If $E$ is among the examples [[sphere spectrum|S]], [[HR]] for $R = \mathbb{F}_p$, [[MO]], [[MU]], [[MSp]], [[KO]], [[KU]], then for all $E$-[[module spectra]] $N$ with [[action]] $\rho \colon E\wedge N \to N$
the morphism of $\mathbb{Z}$-[[graded abelian groups]]

$$
 \pi_\bullet[Y,N]
   \stackrel{\phi \mapsto \rho \circ (id\wedge \phi)}{\longrightarrow}
  Hom_{\pi_\bullet(E)}^\bullet(E_\bullet(Y), \pi_\bullet N)_\bullet
$$

(from the [[stable homotopy group]] of the [[mapping spectrum]] to the [[hom-object|hom groups]] of $\pi_\bullet(E)$-[[modules]])

is an [[isomorphism]].

=--

This is the [[universal coefficient theorem]] of ([Adams 74, chapter III, prop. 13.5](#Adams74)), see also ([Schwede 12, chapter II, prop. 6.20](#Schwede12)).

With this we finally get the following statement, which serves to identity maps of certain spectra with their induced maps on $E$-homology:

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

If the assumptions of prop. \ref{AdamsUCT} hold, then for $X,N$ any two [[spectra]], the morphism of $\mathbb{Z}$-[[graded abelian groups]] from example \ref{SmashingMapsWithEFactorsThroughSteenrodComoduleHomomorphisms} in the form

$$
 \pi_\bullet(E\wedge (-))
   \;\colon\;
 \pi_\bullet[Y, E\wedge N]
   \stackrel{}{\longrightarrow}
  Hom_{E_\bullet(E)}^\bullet(E_\bullet(Y), E_\bullet(Y)))
$$

is an [[isomorphism]].

=--

([Adams 74, part III, page 323](#Adams74))

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

By the general formula for expressing [[adjuncts]], the morphism fits into the following [[commuting diagram]]

$$
  \array{
    [Y, E \wedge N]
      &\stackrel{\pi_\bullet(E\wedge(-))}{\longrightarrow}&
    Hom_{E_\bullet(E)}(
      E_\bullet(Y),
      E_\bullet(E \wedge N)
    )
    \\
    {}^{\mathllap{{\phi \mapsto} \atop {\mu \circ (id \wedge \phi)}}}
      \downarrow^{\mathrlap{\simeq}}
    && \downarrow^{\mathrlap{\simeq}}
    \\
    Hom_{\pi_\bullet(E)}(E_\bullet(Y), E_\bullet(N))
      &\stackrel{\simeq}{\longleftarrow}&
    Hom_{E_\bullet(E)}(
      E_\bullet(Y),
      E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(E)
    )
  }
  \,,
$$

where

1. the right vertical map comes from the isomorphism of prop. \ref{FlatnessOfEImpliesKeyConsequence};

1. the bottom isomorphism is the cofree/forgetful [[adjunction]] isomorphism of prop. \ref{CoFreeComodules};

1. the the left vertical morphism is an isomorphism by prop. \ref{AdamsUCT}.

Therefore also the top morphism is an iso.

=--

## Related concepts

* [[Hopf E-infinity-algebra]]

* [[Deligne's theorem on tensor categories]]

## References

* {#Adams74} [[Frank Adams]],  _[[Stable homotopy and generalized homology]]_, Chicago Lectures in mathematics, 1974

* {#Ravenel86} [[Doug Ravenel]], chapter 2 and appendix A.1 of  _[[Complex cobordism and stable homotopy groups of spheres]]_, 1986 ([pdf](http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenelA1.pdf))

* {#Kochman96} [[Stanley Kochman]], _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#Hopkins99} [[Mike Hopkins]], section 5 of _[[Complex oriented cohomology theories and the language of stacks]]_, course notes 1999 ([pdf](https://www.math.rochester.edu/people/faculty/doug/otherpapers/coctalos.pdf))

* [[Paul Goerss]], _Quasi-coherent sheaves on the moduli stack of formal groups_, [pdf](http://www.math.northwestern.edu/~pgoerss/papers/modfg.pdf)

* [[Mark Hovey]], _Homotopy theory of comodules over a Hopf algebroid_, [pdf](http://math.wesleyan.edu/~mhovey/papers/comodule.pdf); _Morita theory of Hopf algebroids_ ([pdf](http://math.wesleyan.edu/~mhovey/talks/hopfalgebroids.pdf))

* Barry Walker, _Hopf algebroids and stacks_  ([pdf](http://www.math.umn.edu/~tlawson/old/stackstalk.pdf))

* {#BakerLazarev01} [[Andrew Baker]], [[Andrey Lazarev]], _On the Adams Spectral Sequence for R-modules_, Algebr. Geom. Topol. 1 (2001) 173-199 ([arXiv:math/0105079](http://arxiv.org/abs/math/0105079))

* {#BakerJeanneret02} [[Andrew Baker]], Alain Jeanneret, _Brave new Hopf algebroids and extensions of $MU$-algebras_, Homology Homotopy Appl. __4__:1 (2002), 163-173, [MR1937961](http://www.ams.org/mathscinet-getitem?mr=1937961), [euclid](http://projecteuclid.org/euclid.hha/1139840059)

See also

* {#Schwede12} [[Stefan Schwede]], chapter II, section 10.3 of  _[[Symmetric spectra]]_, 2012

[[!redirects commutative Hopf algebroids]]

[[!redirects graded commutative Hopf algebroid]]
[[!redirects graded commutative Hopf algebroids]]

[[!redirects commutative Hopf algebra]]
[[!redirects commutative Hopf algebras]]

[[!redirects graded commutative Hopf algebra]]
[[!redirects graded commutative Hopf algebras]]

[[!redirects super-commutative Hopf algebra]]
[[!redirects super-commutative Hopf algebras]]

[[!redirects supercommutative Hopf algebra]]
[[!redirects supercommutative Hopf algebras]]


commutative invertible magma

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

There should be a commutative version of a [[invertible magma]]. That leads to the concept of a _commutative invertible magma_.

## Definition

A __commutative invertible magma__ is a [[magma]] $(G,(-)\cdot(-):G\times G\to G)$ with a unary operation $(-)^{-1}:G \to G$ called the __inverse__ such that

* $a \cdot (b \cdot b^{-1}) = a$

for all $a,b \in G$.

## Examples

* Every [[commutative loop]] is a commutative invertible [[unital magma]].

* Every commutative invertible quasigroup is a commutative invertible magma.

* Every [[abelian group]] is a commutative invertible monoid.

* The empty magma is a commutative invertible magma.

## Related concepts

* [[invertible magma]] (non-commutative version)

* [[commutative magma]] (non-invertible version)

* [[commutative loop]] (unital version)

* [[commutative invertible quasigroup]] (divisible version)

* [[commutative invertible semigroup]] (associative version)


commutative invertible quasigroup

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

There should be a commutative version of a [[invertible quasigroup]] or an invertible version of a [[commutative quasigroup]]. That leads to the concept of a _commutative invertible quasigroup_.

## Definition

A __commutative invertible quasigroup__ is a [[commutative quasigroup]] $(G,\cdot,/)$ with a unary operation $(-)^{-1}:G \to G$ called the __inverse__ such that

* $a \cdot (b \cdot b^{-1}) = a$

for all $a,b \in G$.

### Without division

A __commutative invertible quasigroup__ is a [[commutative magma]] $(G,(-)\cdot(-):G\times G\to G)$ with a unary operation $(-)^{-1}:G \to G$ called the __inverse__ such that

* $a \cdot (b \cdot b^{-1}) = a$
* $(a \cdot b) \cdot b^{-1} = a$
* $(a \cdot b^{-1}) \cdot b = a$

for all $a,b \in G$.

## Examples

* Every [[commutative loop]] is a commutative invertible unital quasigroup.

* Every commutative invertible semigroup is a commutative [[associative quasigroup]].

* Every [[abelian group]] is a commutative invertible monoid.

* The empty quasigroup is a commutative invertible quasigroup.

## Related concepts

* [[invertible quasigroup]] (non-commutative version)

* [[commutative quasigroup]] (non-invertible version)

* [[commutative loop]] (unital version)

* [[commutative invertible semigroup]] (associative version)


commutative invertible semigroup

[[!redirects possibly empty abelian group]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

There should be an associative version of a [[commutative invertible magma]]. That leads to the concept of a _commutative invertible semigroup_.

## Definition

A __commutative invertible semigroup__ is a [[commutative magma|commutative]] [[semigroup]] $(G,(-)+(-):G\times G\to G)$ with a unary operation $-:G \to G$ called the __inverse__ such that $a + b + (-b) = a$ for all $a,b \in G$.

### With subtraction only

A __commutative invertible semigroup__ is a [[set]] $G$ with a binary operation $(-)-(-):G \times G \to G$ called __[[subtraction]]__ such that:

  * For all $a$ and $b$ in $G$, $a-a=b-b$
  * For all $a$ in $G$, $(a-a)-((a-a)-a)=a$
  * For all $a$ and $b$ in $G$, $a-((b-b)-b) = b-((a-a)-a)$
  * For all $a$, $b$, and $c$ in $G$, $a-(b-c)=(a-((c-c)-c)-b$

For any element $a$ in a commutative invertible semigroup $G$, the element $a-a$ is called an __identity element__, and the element $(a-a)-a$ is called the __inverse element__ of $a$. For all elements $a$ and $b$, __addition__ of $a$ and $b$ is defined as $a-((b-b)-b)$.

## Properties

* Every commutative invertible semigroup is a [[commutative quasigroup]].

## Examples

* Every [[abelian group]] is a commutative invertible semigroup.

* The empty magma is a commutative invertible semigroup that is not an abelian group.

* Every [[ring]] is a [[monoid object]] in commutative invertible semigroups.

* Similarly, every ([[non-associative algebra|non-associative]]) [[unital algebra|unital]] [[integers|$\mathbb{Z}$]]-[[algebra]] is an [[H-space]] object in commutative invertible semigroups.

## Related concepts

* [[invertible semigroup]] (non-commutative version)

* [[commutative invertible magma]] (non-associative version)

* [[commutative semigroup]] (non-invertible version)

* [[abelian group]] (unital version)

* [[pointed commutative invertible semigroup]]

* [[Tarski group]]


commutative localization

Given a unital ring (or only a monoid) $R$ and a **central** [[multiplicative subset]] $S\subset Z(R)\subset R$ (i.e. set containing $1\in R$ and with every two elements containing their product, and such that all its elements are central in $R$) the ring of fractions (monoid of fractions, respectively) $S^{-1} R$ is sometimes said to be the __commutative localization__ of $R$ at $S$; the same name is also given to the canonical map $R\to S^{-1} R$ of rings (monoids resp.). The ring of fractions is defined as the set of equivalence classes $(s,r)\in S\times R$ where $(s,r)\sim (t,r')$ iff $\exists u\in S$, $u s r' = u t r$ (if $R$ is an [[integral domain]] one can skip mentioning $u$ in this condition); the equivalence classes are called fractions and denoted $s^{-1}r$; by centrality of $S$ it is easy to guess the multiplication rule $s^{-1}r t^{-1} r' = (t s)^{-1} (r r')$ and for the addition one first takes the representatives with the same denominator and then adds the numerators. E.g. the formula $s^{-1}r + t^{-1}r' = (t s)^{-1} (t r + s r')$ will do, and we indeed get a ring $S^{-1} R$ with unit $1^{-1} 1$ together with the canonical  homomorphism of rings $R\to S^{-1} R$ given by $r\mapsto 1^{-1} r$.

[[Localization of commutative rings]] at [[multiplicative subsets]] is the standard example, but the centrality of $S$ is enough for the whole theory to pass through.

Commutative localization can be extended to left modules.

__Module of fractions__ $S^{-1} M$ is the left $S^{-1} R$-module $S^{-1} M$ equipped with the natural map of $R$-modules $M\to S^{-1}M$ and defined as follows:

The underlying set of $S^{-1} M$ consists of equivalence classes $s^{-1} m$ of pairs $(s,m)\in S\times M$ where $(s,m)\cong (t,n)$ iff there exist $u\in S$ such that $u t m = u s n$, the multiplication by scalar is defined by $(s^{-1} r)(t^{-1} m):= (t s)^{-1} (r m)$ and the addition is $s^{-1} m + t^{-1} n := (s t)^{-1}(t m + s n)$. The correspondence $Q_S : M\mapsto S^{-1} M$ extends to a functor ${}_R Mod\to {}_{S^{-1} R}Mod$. The forgetful functor $U: {}_{S^{-1} R}Mod \to {}_R Mod$ is [[fully faithful functor]] and there is a natural transformation of functors $Id\to U Q_S$ whose components are the $R$-module maps $M\to S^{-1}M$ given by $m\mapsto 1^{-1}m$.

It can be then shown that this elementary approach is equivalent to the definition via the [[extension of scalars]] formula $S^{-1}M = S^{-1} R\otimes_R M$.

The basic result is that the commutative localization $S^{-1} R$ is a [[flat module|flat]] left module over $R$, the property which holds for more general [[Ore localization]].

Commutative localization in which also $R$ is commutative is a basic procedure used in defining [[algebraic scheme]] as a [[locally ringed space]]. Another special case of this procedure is forming the [[quotient field]] of a commutative [[integral domain]].


commutative loop

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

### With multiplication, division, and identity

A __commutative [[loop (algebra)|loop]]__ is a [[commutative magma|commutative]] [[unital magma]] $(G,(-)\cdot(-):G \times G \to G,1:G)$ equipped with a binary operation $(-)/(-):G \times G \to G$ called __division__ such that $(x/y) \cdot y = x$ and $(x \cdot y)/y = x$.

### With division and identity

A __commutative loop__ is a [[pointed]] [[magma]] $(G,/,1)$ such that:

  * For all $a$ in $G$, $a/a=1$
  * For all $a$ in $G$, $1/(1/a)=a$
  * For all $a$ and $b$ in $G$, $a/(1/b) = b/(1/a)$

with __multiplication__ defined as $a \cdot b= a/(1/b)$.

### With multiplication, inverses, and identity

A __commutative loop__ is a [[commutative magma|commutative]] [[unital magma]] $(G, (-)\cdot(-):G\times G\to G,1:G)$ equipped with a __inverse__ $(-)^{-1}:G \to G$ such that $(x \cdot y^{-1}) \cdot y = x$ and $(x \cdot y) \cdot y^{-1} = x$.

## Examples

* Every [[abelian group]] is a commutative loop.

## Related concepts

* [[commutative quasigroup]]


commutative magma

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[magma]] $(S,\cdot)$ is called _commutative_ if its [[binary operation]] $(-)\cdot(-) \colon S \times S $ has the property that for all $x,y \in S$ then

$$
  x \cdot y = y \cdot x
  \,.
$$

## Examples

Examples include [[commutative monoids]], [[abelian groups]], [[commutative rings]], [[commutative algebras]] etc.

Another example of a commutative magma is a [[midpoint algebra]].

## Related concepts

* [[magma]] (noncommutative version)

* [[commutative invertible magma]] (invertible version)

* [[commutative unital magma]] (unital version)

* [[commutative semigroup]] (associative version)

[[!redirects commutative magmas]]

[[!redirects commutative operation]]
[[!redirects commutative operations]]

[[!redirects commutativity]]


commutative monad

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

## Idea

The idea of a commutative monad could be motivated as either

* The [[monad]] equivalent of a [[commutative algebraic theory]], generalized to either non-finitary theories or to categories other than [[Set]]. Intuitively, the operations involved have a commutativity property analogous to [[commutative monoid|the one that monoids can have]];
* A [[monad]] on a [[monoidal category]] whose underlying functor is [[monoidal functor|lax monoidal]], and whose unit and multiplication are [[monoidal natural transformation|monoidal]]. The interpretation is that the monad plays well with the monoidal product of the category, encoding for example products of formal expressions;
* A [[monad]] on a [[monoidal category]] whose [[strong monad|left-strength]] and [[strong monad|right-strength]] commute with each other.

It turns out that all these concepts coincide (see below, as well as the articles on [[commutative algebraic theories]] and [[monoidal monads]]).

## Definition

Let $(T,\mu,\eta)$ be a [[monad]] on a [[symmetric monoidal category]] $C$, equipped with a [[strong monad|left-strength]] $\sigma$ (and with the right-strength $\tau$ induced from $\sigma$ and the [[braiding]]).

We say that $T$ (or $\sigma$) is **commutative** if the following diagram commutes for all objects $X$ and $Y$ of $C$.
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
]
& T X \otimes T Y \ar{dl}[swap]{\sigma} \ar{dr}{\tau} \\
T(T X \otimes Y) \ar{d}[swap]{T\tau} && T(X\otimes T Y) \ar{d}{T\sigma} \\
TT(X\otimes Y) \ar{dr}[swap]{\mu} && TT(X\otimes Y) \ar{dl}{\mu} \\
& T(X\otimes Y)
\end{tikzcd}

## Properties

### Relation with monoidal monads
 {#RelationWithMonoidalMoands}

The structure of a commutative monad is closely related to that of a [[monoidal monad]], namely a [[monad#definition|monad in the bicategory]] of [[monoidal categories]], [[lax monoidal functors]] and [[monoidal natural transformations]].

For the details see [this Proposition](monoidal+monad#BijectSymmMonoidalMonadsWithCommStrongMonads) at *[[monoidal monad]]*.

### On closed and monoidal closed categories

One can define [[strong monad|left-strength and right-strength]] on a [[closed category]], in a way that's analogous to the definition in a [[monoidal category]], and on [[monoidal closed categories]] the two definitions coincide (see [[strong monad#on_closed_and_monoidal_closed_categories|strong monad - on closed and monoidal closed categories]]).
The same is true for the notion of commutativity: one can define commutative monads on closed categories, and on closed monoidal categories the definition coincides with the definition given in this article.

More in detail, let $C$ be a [[closed category]], and denote its [[internal homs]] by $[X,Y]$.
Let $t':[X,Y]\to [T X, T Y]$ be a left-strength for $T$ (as defined [[strong monad#strong_monads_are_enriched_monads|here]]), and let $s':T[X,Y]\to [X, T Y]$ be a right-strength (as defined [[strong monad#right-strength_and_pointwise_structure|here]]).
We say that $t'$ and $s'$ _commute_, or that $T$ is _commutative_, if the following diagram commutes.
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
]
& T{[X,Y]} \ar{dl}[swap]{s'} \ar{dr}{Tt'} \\
{[X,T Y]} \ar{d}[swap]{t'} && T{[T X, T Y]} \ar{d}{s'} \\
{[T X, T T Y]} \ar{dr}[swap]{{[T X,\mu]}} && {[T X, T T Y]} \ar{dl}{{[T X,\mu]}} \\
& {[T X, T Y]}
\end{tikzcd}
It can be shown by [[currying]] that on a [[monoidal closed category]], this condition is equivalent to the "monoidal" definition of commutativity.
A reference for this is [Kock '71, Section 2](#Kock71).

### Tensor product of algebras and multimorphisms

In many situations, [[algebra over a monad|algebras]] over a commutative monad are canonically equipped with a [[tensor product]] analogous to the [[tensor product of modules]] over a ring.
In particular, commutative monads come equipped with a notion of [[multimorphisms]] of algebras, analogous to [[bilinear]] and [[multilinear maps]].

On a [[closed category]], analogously, the category of algebras inherits an [[internal hom]], just like the one of [[modules]]. If the category is [[closed monoidal category|monoidal closed]], the [[tensor product]] and the [[internal hom]] are [[adjoint]] to each other, making the category of algebra itself monoidal closed. This, once again, generalizes the hom-tensor adjunction of [[modules]] and [[vector spaces]].

For the details, see [[tensor product of algebras over a commutative monad]], and the [[tensor product of algebras over a commutative monad#bimorphisms_and_universal_property|section on the universal property]] for multimorphisms, as well as [[internal hom of algebras over a commutative monad]].

## Examples

In many of the examples, one can see how the property of commutativity (as opposed to just the structure of a [[strong monad|strength]]) has really the meaning of the operations being commutative.

* The free _commutative_ monoid monad on [[Set]] is commutative, the free monoid monad is not.
* If $M$ is a [[monoid]], the [[action monad]] $X\mapsto M\times X$ on [[Set]] is commutative if and only if $M$ is commutative as a monoid.
* More generally, in an arbitrary [[symmetric monoidal category]], the [[action monad]] induced by tensoring with an [[internal monoid]] is commutative if and only the monoid object is commutative. (See for example [Brandenburg, Example 6.3.12](#Brandenburg2014).)

Since commutative monads are the same as monoidal monads, monads encoding structures which admit products are commutative:

* Most [[probability monads]] are commutative, with monoidal structure given by forming the [[product probability]].
* The [[power set]] monad is commutative, with monoidal structure given by forming the product of subsets.

For a concrete example, consider the free commutative monoid monad $T$, and given a set $X$, write the elements of $TX$ as formal sums, such as $x_1+x_2$.
The commutativity condition says that these two compositions have the same result,
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
row sep=0,%
]
TX \times TY \ar{r}{\sigma} & T(TX\times Y) \ar{r}{T\tau} & TT(X\times Y) \ar{r}{\mu} & T(X\times Y) \\
(x_1+x_2, y_1+y_2) \ar[mapsto]{r} & (x_1+x_2, y_1) + (x_1+x_2, y_2) \ar[mapsto]{r} & ((x_1, y_1) + (x_2, y_1)) + ((x_1, y_2) + (x_2, y_2)) \ar[mapsto]{r} & (x_1, y_1) + (x_2, y_1) + (x_1, y_2) + (x_2, y_2)
\end{tikzcd}
and
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
row sep=0,%
]
TX \times TY \ar{r}{\tau} & T(X\times TY) \ar{r}{T\sigma} & TT(X\times Y) \ar{r}{\mu} & T(X\times Y) \\
(x_1+x_2, y_1+y_2) \ar[mapsto]{r} & (x_1, y_1+y_2) + (x_2, y_1+y_2) \ar[mapsto]{r} & ((x_1, y_1) + (x_1, y_2)) + ((x_2, y_1) + (x_2, y_2)) \ar[mapsto]{r} & (x_1, y_1) + (x_1, y_2) + (x_2, y_1) + (x_2, y_2)
\end{tikzcd}
Note that these two expressions are not the same on the nose, they are the same because addition is commutative.

(See also the explanations at [[commutative algebraic theory]].)

## See also

* [[monoidal monad]], [[commutative algebraic theory]], [[strong monad]], [[tensorial strength]]
* [[tensor product of algebras over a commutative monad]]
* [[internal hom of algebras over a commutative monad]]

## References

* {#Kock70} [[Anders Kock]], *Monads on symmetric monoidal closed categories*, Arch. Math. **21** (1970) 1-10 [[doi:10.1007/BF01220868](https://doi.org/10.1007/BF01220868)]

* {#Kock71} [[Anders Kock]], *Closed categories generated by commutative monads*, Journal of the Australian Mathematical Society **12** 4 (1971) 405-424  [[doi:10.1017/S1446788700010272](https://doi.org/10.1017/S1446788700010272), [[KockMonoidalMonads.pdf:file]]]

* {#Kock72} [[Anders Kock]], *Strong functors and monoidal monads*, Arch. Math **23** (1972) 113–120 [[doi:10.1007/BF01304852](https://doi.org/10.1007/BF01304852), [pdf](http://home.imf.au.dk/kock/SFMM.pdf)]

* H. Lindner, _Commutative monads_ in _Deuxiéme colloque sur l'algébre des catégories_. Amiens-1975. Résumés des conférences, pages 283-288. Cahiers de topologie et géométrie différentielle catégoriques, tome 16, nr. 3, 1975.

* {#Keigher78} [[William Keigher]], _Symmetric monoidal closed categories generated by commutative adjoint monads_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 19 no. 3 (1978), p. 269-293  ([NUMDAM](http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1978__19_3_269_0), [pdf](http://archive.numdam.org/article/CTGDC_1978__19_3_269_0.pdf))

* {#Seal12} Gavin J. Seal, _Tensors, monads and actions_ ([arXiv:1205.0101](http://arxiv.org/abs/1205.0101))

* {#Brandenburg2014} [[Martin Brandenburg]], _Tensor categorical foundations of algebraic geometry_ ([arXiv:1410.1716](https://arxiv.org/abs/1410.1716))

[[!redirects commutative monads]]

[[!redirects commutative strength]]
[[!redirects commutative strengths]]

[[!redirects commutative strong monad]]
[[!redirects commutative strong monads]]


commutative monoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Monoid theory
+-- {: .hide}
[[!include monoid theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

### In Sets

A _commutative monoid_ is a [[monoid]] where the multiplication satisfies the commutative law:
$$x y = y x.$$

Alternatively, just as a [[monoid]] can be seen as a [[category]] with one object, a commutative monoid can be seen as a [[bicategory]] with one object and one morphism (or equivalently, a [[monoidal category]] with one object).

Commutative monoids with [[homomorphisms]] between them form a _[[category of commutative monoids]]_.

Every commutative monoid has the canonical structure of a [[module]] over the commutative [[rig]] $\mathbb{N}$. That is, [[CMon]] = $\mathbb{N}$-[[Mod]].

### In any symmetric monoidal category

More generally, the concept makes sense [[internalization|internal]] to any [[symmetric monoidal category]] (or even any [[braided monoidal category]]). See at _[[commutative monoid in a symmetric monoidal category]]_ for details.

### As a 0-truncated object

A **commutative monoid** is a [[n-truncated|0-truncated]] commutative [[H-monoid]]. It is also a [[n-truncated|0-truncated]] [[E2-algebra|$E_2$-space]].

## Examples

* An [[abelian group]] is a commutative monoid that is also a [[group]].

* The [[natural numbers]] (together with 0) form a commutative monoid under addition.

* Every bounded [[semilattice]] is an _idempotent_ commutative monoid, and every idempotent commutative monoid yields a semilattice, (see that entry).

Examples of [[commutative monoids in a symmetric monoidal category]]:

1. A commutative monoid in the [[symmetric monoidal category|symmetric monoidal]] [[category of vector spaces]] is a [[commutative algebra]];

1. A commutative monoid in the [[symmetric monoidal category|symmetric monoidal]] [[category of chain complexes]] of vector spaces is a [[differential graded-commutative algebra]];

1. A commutative monoid in the [[symmetric monoidal category|symmetric monoidal]] [[category of chain complexes of super vector spaces]] is a [[differential graded-commutative superalgebra]].

## Results on commutative monoids in Set

1. If a commutative monoid is finitely generated it is finitely presented.

2. Finitely generated commutative monoids have decidable word problems, the isomorphism problem for them is decidable, and indeed the first-order theory of finitely generated commutative monoids is decidable (see [KharlampovichSapir](#KharlampovichSapir)).

3. If a finitely generated commutative monoid is **cancellative** ($a + b = a' + b \Rightarrow a = a'$) then it embeds in a finitely generated abelian group.

4.  If a finitely generated commutative monoid is cancellative and **torsion-free** ($a + a + \cdots + a = 0 \Rightarrow a = 0$) then it embeds in a finitely generated free abelian group (or more concretely, $\mathbb{Z}^n$).   Conversely any submonoid of $\mathbb{Z}^n$ is cancellative and torsion-free (but not necessarily finitely generated).  A commutative monoid is called an **affine monoid** if it is isomorphic to a finitely generated submonoid of $\mathbb{Z}^n$, and there is an extensive theory of these, connected to [[toric variety|toric varieties]] (see [BrunsGubeladze](#BrunsGubeladze)).

5.  If a finitely generated commutative monoid is cancellative and **nonnegative** ($a + b = 0 \Rightarrow a,b = 0$), it embeds in a finitely generated [[free commutative monoid]], or more concretely, $\mathbb{N}^n$
(Thm. 3.11, [RosalesGarcía-Sánchez](#RosalesGarcía-Sánchez)).   Conversely, any finitely generated submonoid of $\mathbb{N}^n$ is cancellative and nonnegative (but not necessarily finitely generated).

## Related concepts

* [[commutative magma]]

* [[CMon]]

* [[group completion]]

* [[category of monoids]]

* [[super commutative monoid]]

* [[cocommutative comonoid]]

* [[braided monoidal groupoid]]

* [[symmetric monoidal groupoid]]

* [[k-tuply monoidal n-groupoid]]

* [[symmetric monoidal category]]

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

* [[commutative ∞-monoid]]

* [[E-infinity algebra]]

* [[periodic table]]

[[!redirects abelian monoids]]
[[!redirects commutative monoid]]
[[!redirects commutative monoids]]

[[!redirects abelian monoid]]
[[!redirects commutative]]

## References

The word problem for commutative monoids is discussed here:

* {#KharlampovichSapir} Olga G. Kharlampovich and Mark V. Sapir. Algorithmic problems in varieties, International Journal of Algebra and Computation 5.04n05 (1995), 379-602.   ([pdf](http://www.math.vanderbilt.edu/~msapir/ftp/pub/survey/survey.pdf))

For affine monoids and other finitely generated commutative monoids see:

* {#BrunsGubeladze} Winfried Bruns and Joseph Gubeladze, *Polytopes, Rings, and K-theory*, Springer, Berlin, 2009.   ([pdf of preliminary incomplete version](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.126.2385&rep=rep1&type=pdf))

* {#RosalesGarcía-Sánchez} José Carlos Rosales and Pedro A. García-Sánchez, *Finitely Generated Commutative Monoids*, Nova Publishers, 1999.

For a treatment of free commutative monoids in [[homotopy type theory]]/[[univalent foundations]]:

* Vikraman Choudhury, [[Marcelo Fiore]], _Free Commutative Monoids in Homotopy Type Theory_ ([arXiv:2110.05412](https://arxiv.org/abs/2110.05412))


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The notion of _commutative monoid_ (or _commutative monoid object_, _commutative algebra_, _commutative algebra object_) in a [[symmetric monoidal (infinity,1)-category]] is the [[(infinity,1)-category theory|(infinity,1)-categorical]] generalization of the notion of [[commutative monoid in a symmetric monoidal category]].
It is the commutative version of [[monoid in a monoidal (infinity,1)-category]].

Note that _commutative_ here really means $E_\infty$, in the sense of [[E-infinity operad]].

## Definition

A **commutative monoid** in a [[symmetric monoidal (infinity,1)-category]] $C$ is a lax symmetric monoidal $(\infty,1)$-functor

$$
  * \to C
  \,.
$$

In more detail, this means the following:

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

Given a [[symmetric monoidal (infinity,1)-category]] in its [[quasi-category|quasi-categorical]] incarnation as a coCartesian fibration of [[simplicial set]]s

$$
  p : C^\otimes \to N(FinSet_*)
$$

a **commutative monoid** in $C$ is a [[section]]

$$
  A : N(FinSet_*) \to C^\otimes
$$

such that $A$ carries collapsing morphisms in $FinSet_*$ to coCartesian morphisms in $C^\otimes$.
=--

## $(\infty,1)$-Category of commutative monoids

### Definition

For $C$ a [[symmetric monoidal (∞,1)-category]] write $CMon(C)$ for the $(\infty,1)$-category of commutative monoids in $C$.

### Properties

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

* $CMon(C)$ has all [[(∞,1)-coproduct]]s and these are computed as [[tensor product]]s in $C$.

* For $K$ a _[[sifted (infinity,1)-category]]_ , [[(∞,1)-colimits]] of shape $K$ exist in $CMon(C)$ and are computed in $C$ if $K$-colimits exist in $C$ are preserved by tensor product with any object.

* $CMon(C)$ has all [[(∞,1)-limit]]s and these are computed in $C$.

=--

This is ([Lurie DAG III, section 4](#LurieDAG3)) or ([Lurie HA, sections 3.2.2 and 3.2.3](#LurieHA)).

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

$(\infty,1)$-Colimits over simplicial diagrams exists in $CMon(C)$ and are computed in $C$ if they exist in $C$ and a preserved by tensor products.

=--

Because the [[simplex category]] is a [[sifted (infinity,1)-category]] (as discussed there).

## Examples

* A commutative monoid in [[∞Grpd]] is an [[E-∞ space]].

* A commutative monoid in the [[stable (infinity,1)-category of spectra]] is a [[commutative ring spectrum]] or [[E-infinity ring]].

## Related concepts

* [[monoid in a monoidal (infinity,1)-category]]
* [[infinity-algebra over an (infinity,1)-operad]]

* [[commutative monoid in a symmetric monoidal category]]
* [[module over a monoid]]

* [[∞-group completion]]

## References

* {#LurieDAG3} [[Jacob Lurie]], _[[Commutative Algebra]]_.

* {#LurieHA} [[Jacob Lurie]], _[[Higher Algebra]]_.

An equivalent reformulation of commutative monoids in terms [[(∞,1)-algebraic theories]] is in

* [[James Cranch]], _Algebraic Theories and $(\infty,1)$-Categories_ ([arXiv](http://arxiv.org/abs/1011.3243))

[[!redirects commutative monoid in a symmetric monoidal (∞,1)-category]]

[[!redirects commutative monoid in an (∞,1)-category]]
[[!redirects commutative monoid in an (infinity,1)-category]]
[[!redirects commutative monoid in a monoidal (∞,1)-category]]
[[!redirects commutative monoid in a monoidal (infinity,1)-category]]
[[!redirects commutative monoid in a symmetric monoidal (∞,1)-category]]
[[!redirects commutative monoid in a symmetric monoidal (infinity,1)-category]]

[[!redirects commutative monoids in an (∞,1)-category]]
[[!redirects commutative monoids in an (infinity,1)-category]]
[[!redirects commutative monoids in a monoidal (∞,1)-category]]
[[!redirects commutative monoids in a monoidal (infinity,1)-category]]
[[!redirects commutative monoids in a symmetric monoidal (∞,1)-category]]
[[!redirects commutative monoids in a symmetric monoidal (infinity,1)-category]]

[[!redirects commutative algebra in an (∞,1)-category]]
[[!redirects commutative algebra in an (infinity,1)-category]]
[[!redirects commutative algebra in a monoidal (∞,1)-category]]
[[!redirects commutative algebra in a symmetric monoidal (∞,1)-category]]
[[!redirects commutative algebras in an (∞,1)-category]]
[[!redirects commutative algebras in an (infinity,1)-category]]
[[!redirects commutative algebras in a monoidal (∞,1)-category]]
[[!redirects commutative algebras in a symmetric monoidal (∞,1)-category]]

[[!redirects commutative ∞-monoid]]
[[!redirects commutative ∞-monoids]]
[[!redirects commutative infinity-monoid]]
[[!redirects commutative infinity-monoids]]


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

#Contents#
* table of contents
{:toc}

## Idea

Generalizing the classical notion of [[commutative monoid]], one can define a _commutative monoid_ (or _commutative monoid object_) [[internalization|internal to]] any [[symmetric monoidal category]] $(C,\otimes,I)$.
These are [[monoids in a monoidal category]] whose multiplicative operation is commutative.
Classical commutative monoids are of course just commutative monoids in [[Set]] with the [[cartesian product]].

## Definition

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

Given a [[monoidal category]] $(\mathcal{C}, \otimes, 1)$, then a **[[monoid in a monoidal category|monoid internal to]]** $(\mathcal{C}, \otimes, 1)$ is

1. an [[object]] $A \in \mathcal{C}$;

1. a morphism $e \;\colon\; 1 \longrightarrow A$ (called the _[[unit]]_)

1. a morphism $\mu \;\colon\; A \otimes A \longrightarrow A$ (called the _product_);

such that

1. ([[associativity]]) the following [[commuting diagram|diagram commutes]]

   $$
     \array{
       (A\otimes A) \otimes A
         &\underoverset{\simeq}{a_{A,A,A}}{\longrightarrow}&
       A \otimes (A \otimes A)
         &\overset{A \otimes \mu}{\longrightarrow}&
       A \otimes A
       \\
       {}^{\mathllap{\mu \otimes A}}\downarrow
         && &&
       \downarrow^{\mathrlap{\mu}}
       \\
       A \otimes A
         &\longrightarrow&
         &\overset{\mu}{\longrightarrow}&
       A
     }
     \,,
   $$

   where $a$ is the associator isomorphism of $\mathcal{C}$;

1. ([[unitality]]) the following [[commuting diagram|diagram commutes]]:

   $$
     \array{
       1 \otimes A
         &\overset{e \otimes id}{\longrightarrow}&
       A \otimes A
         &\overset{id \otimes e}{\longleftarrow}&
       A \otimes 1
       \\
       & {}_{\mathllap{\ell}}\searrow
       & \downarrow^{\mathrlap{\mu}} &
       & \swarrow_{\mathrlap{r}}
       \\
       && A
     }
     \,,
   $$

   where $\ell$ and $r$ are the left and right unitor isomorphisms of $\mathcal{C}$.

Moreover, if $(\mathcal{C}, \otimes , 1)$ has the structure of a [[symmetric monoidal category]]  $(\mathcal{C}, \otimes, 1, \tau)$ with symmetric [[braiding]] $\tau$, then a monoid $(A,\mu, e)$ as above is called a **[[commutative monoid in a symmetric monoidal category|commutative monoid in]]** $(\mathcal{C}, \otimes, 1, \tau)$ if in addition

* (commutativity) the following [[commuting diagram|diagram commutes]]

  $$
    \array{
      A \otimes A
        && \underoverset{\simeq}{\tau_{A,A}}{\longrightarrow} &&
      A \otimes A
      \\
      & {}_{\mathllap{\mu}}\searrow && \swarrow_{\mathrlap{\mu}}
      \\
      && A
    }
    \,.
  $$

Note that this condition makes sense even if the [[braiding]] $\tau$ is not symmetric, as in a [[braided monoidal category]]. Such a monoid is also called a **braided monoid** in $(\mathcal{C}, \otimes, 1, \tau)$.

A [[homomorphism]] of monoids $(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2)$ is a morphism

$$
  f \;\colon\; A_1 \longrightarrow A_2
$$

in $\mathcal{C}$, such that the following two [[commuting diagram|diagrams commute]]

$$
  \array{
    A_1 \otimes A_1
      &\overset{f \otimes f}{\longrightarrow}&
    A_2 \otimes A_2
    \\
    {}^{\mathllap{\mu_1}}\downarrow && \downarrow^{\mathrlap{\mu_2}}
    \\
    A_1 &\underset{f}{\longrightarrow}& A_2
  }
$$

and

$$
  \array{
    1_{\mathcal{c}} &\overset{e_1}{\longrightarrow}& A_1
    \\
    & {}_{\mathllap{e_2}}\searrow & \downarrow^{\mathrlap{f}}
    \\
    && A_2
  }
  \,.
$$

Write $Mon(\mathcal{C}, \otimes,1)$ for the [[category of monoids]] in $\mathcal{C}$, and $CMon(\mathcal{C}, \otimes, 1)$ for its subcategory of commutative monoids.

=--

## Examples
 {#Examples}

+-- {: .num_example}
###### Example
**([[commutative rings]])**

Write $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ for the category [[Ab]] of [[abelian groups]], equipped with the [[tensor product of abelian groups]] whose [[tensor unit]] is the additive group of [[integers]]. With the evident [[braiding]] this is a [[symmetric monoidal category]].

A commutative monoid in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ is equivalently a [[commutative ring]].

=--

+-- {: .num_example}
###### Example
**([[differential graded-commutative algebras]])**

The [[category of chain complexes]] $Ch(Vect)$ with its [[tensor product of chain complexes]] carries a [[symmetric monoidal category|symmetric]] monoidal [[braiding]] given on elements in definite degree $n \in \mathbb{Z}$ by

$$
  \tau \;\colon; v \otimes W \mapsto (-1)^{ n_v n_w } w \otimes v
  \,.
$$

The corresponding commutative monoid objects are the [[differential graded-commutative algebras]].

=--

+-- {: .num_example}
###### Example
**([[differential graded-commutative superalgebras]])**

The [[category of chain complexes of super vector spaces]] $Ch(SuperVect)$ with its [[tensor product of chain complexes]] carries two [[symmetric monoidal category|symmetric]] monoidal [[braidings]] given on elements in definite bidegree $(n,\sigma) \in \mathbb{Z} \times \mathbb{Z}/2$ by

1. $\tau_{Deligne} \;\colon\; v \otimes w \mapsto (-1)^{ (n_v n_w + \sigma_v \sigma_w) } w \otimes v$;

1. $\tau_{Bernst} \;\colon\; v \otimes w \mapsto (-1)^{ (n_v + \sigma_v) (n_w + \sigma_w) } w \otimes v$.

The corresponding commutative monoid objects are the [[differential graded-commutative superalgebras]].

[[!include sign rules in homological superalgebra -- table]]

Since the two braidings above are equivalent ([this Prop](chain+complex+in+super+vector+spaces#EquivalenceTwoSymmetricMonoidalStructuresOnChSuperVect)) the corresponding two categories of [[differential graded-commutative superalgebras]] are also canonically [[equivalence of categories]]:

$$
    ComMon\left(  Ch(SuperVect), \otimes, \tau_{Deligne} \right)
    \;\simeq\;
    ComMon\left(  Ch(SuperVect), \otimes, \tau_{Bernst} \right)
$$

$$
  sdgcAlg_{Deligne}
  \;\simeq\;
  sdgcAlg_{Bernst}
$$

=--

+-- {: .num_example}
###### Example
**([[commutative ring spectra]], [[E-infinity rings]])**

Write $(SymSpec(Top_{cg}),\wedge, \mathbb{S}_{sym})$ and $(OrthSpec(Top_{cg}),\wedge, \mathbb{S}_{orth})$ and $([Top^{\ast/}_{cg,fin}, Top^{\ast/}_{cg}], \wedge, \mathbb{S} )$ for the categories, respectively of [[symmetric spectra]], [[orthogonal spectra]] and [[pre-excisive functors]], equipped with their [[symmetric monoidal smash product of spectra]], whose [[tensor unit]] is the corresponding standard incarnation of the [[sphere spectrum]].

A commutative monoid in any one of these three categories is equivalently a [[commutative ring spectrum]] in the strong sense: via the respective [[model structure on spectra]] it represents an [[E-infinity ring]].

=--

+-- {: .num_example}
###### Example
**(in a cocartesian monoidal category)**

Every object $A$ in a [[cocartesian monoidal category]] $C$ becomes a commutative monoid in a unique way: the multiplication must be the [[fold map]] $\nabla \colon A + A \to A$, and the counit must be the unique map $! \colon 0 \to A$.  Similarly every morphism in $C$ becomes a morphism of commutative monoid objects, so the category of commutative monoid objects in $C$ is isomorphic to $C$.

=--

+-- {: .num_example}
###### Example
**(in $CommMon$)**

Since the category $CommMon$ of commutative monoids (in $Set$) is cocartesian, the category of commutative monoids in $(CommMon,+)$ is again $CommMon$.   Finite coproducts of commutative monoids are also finite products, so the category of commutative monoids in $(CommMon,\times)$ is also $CommMon$.

=--

## Related concepts

* [[monoid]], [[commutative monoid]]
* [[monoid in a monoidal category]]
* [[category of monoids]]
* [[module over a monoid]]
* [[semilattice object]]
* [[commutative monoid in a symmetric monoidal (infinity,1)-category]]

## References

### General

Discussion including proof that/when the category of [[module objects]] is itself [[closed symmetric monoidal category|closed symmetric monoidal]]:

* [[Mark Hovey]], [[Brooke Shipley]], [[Jeff Smith]], pp. 15 in: *Symmetric spectra*, J. Amer. Math. Soc. **13** (2000) 149-208 [[arXiv:math/9801077](http://arxiv.org/abs/math/9801077), [doi:10.1090/S0894-0347-99-00320-3](https://doi.org/10.1090/S0894-0347-99-00320-3)]

See also:

* [MO/180673](http://mathoverflow.net/questions/180673/category-of-modules-over-commutative-monoid-in-symmetric-monoidal-category).

* [Marty 2009, 1.2 and 1.3](#Marty09)

* [[Martin Brandenburg]], Section 4.1 _Tensor categorical foundations of algebraic geometry_, [arXiv:1410.1716](http://arxiv.org/abs/1410.1716).

Lecture notes:

* [[Urs Schreiber]]: *[Algebras and Modules](Introduction+to+Stable+homotopy+theory+--+1-2#AlgebrasAndModules)*, in *[[Introduction to Stable Homotopy Theory]]* 1.2: *[[Introduction to Stable homotopy theory -- 1-2|Structured Spectra]]*

### Opposite categories
 {#ReferencesOppositeCategories}

Discussion of the [[opposite categories]] of commutative monoid objects and regarded as categories of generalized [[affine schemes]]:

* [[Bertrand Toën]], [[Michel Vaquié]], *Au-dessous de $Spec \mathbb{Z}$*, Journal of K-Theory **3** 3 (2009) 437-500 [[doi:10.1017/is008004027jkt048](https://doi.org/10.1017/is008004027jkt048)]

* {#Marty09} [[Florian Marty]], *Des Ouverts Zariski et des Morphismes Lisses en Géométrie Relative*, Ph.D. Toulouse (2009) [[theses:2009TOU30071](https://www.theses.fr/2009TOU30071), [[Marty-DesOuverts.pdf:file]]]

* [[Florian Marty]], *Relative Zariski Open Objects*, Journal of K-Theory **10** 1 (2012) 9-39  [[arXiv:0712.3676](https://arxiv.org/abs/0712.3676), [doi:10.1017/is011012004jkt176 ](https://doi.org/10.1017/is011012004jkt176 )]

* [[Abhishek Banerjee]], *The relative Picard functor on schemes over a symmetric monoidal category*, Bulletin des Sciences Mathématiques **135** 4 (2011) 400-419 [[doi:10.1016/j.bulsci.2011.02.001](https://doi.org/10.1016/j.bulsci.2011.02.001)]

* [[Abhishek Banerjee]], *On integral schemes over symmetric monoidal categories* [[arXiv:1506.04890](https://arxiv.org/abs/1506.04890)]

* [[Abhishek Banerjee]], *Noetherian Schemes over abelian symmetric monoidal categories*, International Journal of Mathematics **28** 07 (2017) 1750051 [[doi:1410.3212](https://arxiv.org/abs/1410.3212), [doi:10.1142/S0129167X17500513](https://doi.org/10.1142/S0129167X17500513)]

[[!redirects commutative monoid object]]
[[!redirects commutative monoid objects]]

[[!redirects commutative monoids in a symmetric monoidal category]]
[[!redirects commutative monoids in symmetric monoidal categories]]

[[!redirects commutative monoid object in a symmetric monoidal category]]
[[!redirects commutative monoid objects in a symmetric monoidal category]]
[[!redirects commutative monoid objects in symmetric monoidal categories]]

[[!redirects commutative monoids in a monoidal category]]
[[!redirects commutative monoids in monoidal categories]]

[[!redirects commutative monoid object in a monoidal category]]
[[!redirects commutative monoid objects in a monoidal category]]
[[!redirects commutative monoid objects in monoidal categories]]

[[!redirects braided monoid]]


commutative monoid in a symmetric 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]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Let $C$ be [[symmetric monoidal category]] and $C$ the category of [[commutative monoids]] in $C$.
When $C$ is further a [[model category]], there are certain conditions under which there is an induced [[model structure]] on $CMon(C)$, where the [[weak equivalences]] and [[fibrations]] are defined as in $C$.

## Distinction between E-infinity monoids

Recall that $CMon(C)$ can be described as the category of [[algebras over an operad]] over the [[operad]] [[Comm]].
If the operad [[Comm]] were [[cofibrant object|cofibrant]], then for the existence of the induced model structure on $CMon(C)$ it would be sufficient to require the [[monoid axiom in a monoidal model category|monoid axiom]], and to use the [[model structure on algebras over an operad]] as discussed there.
However [[Comm]] is in general not cofibrant, and this is the distinction between [[Comm]] and the [[E-infinity operad]]: the latter is a [[cofibrant replacement]] of the former.
This is why the [[model structure]] on [[commutative monoids]] may not always exist even when the model structure on [[E-infinity monoids in a symmetric monoidal model category]] does.

For example, take $C$ to be the category of [[chain complexes]] over a [[field]] of [[positive characteristic]].
Then the category of [[commutative monoids]] in $C$ is the category of [[commutative dg-algebras]].
This does not have an induced [[model structure]], as explained in [MO/23885/2503](http://mathoverflow.net/a/23885/2503).

See [Rectification](#Rectification) for some results on when the [[model structures]] on [[E-infinity monoids]] and [[commutative monoids]] are [[Quillen equivalent]], though.

## Model structure

Let $C$ be a [[symmetric monoidal model category]] and let $CMon(C)$ denote the category of [[commutative monoids]] in the underlying category of $C$.
Define a [[weak equivalence]] (resp. [[fibration]]) of [[commutative monoids]] to be a [[weak equivalence]] (resp. [[fibration]]) of the underlying objects of $C$.
Below we will give sufficient conditions for this to define a [[model structure]] on $CMon(C)$.

+-- {: .un_theorem}
###### Theorem
Suppose that

  * $C$ is [[combinatorial model category|combinatorial]],

  * $C$ is _freely powered_, i.e.

    * $C$ satisfies the [[monoid axiom in a monoidal model category]],

    * $C$ is [[left proper model category|left proper]] and ...

    * every cofibration in $C$ is a _power cofibration_, i.e. ...

Then the category $CMon(C)$ is a [[combinatorial model category]] with [[weak equivalences]] and [[fibrations]] as defined above.
=--

See ([Lurie](#LurieHA), Proposition 4.5.4.6).

Next we state a more general version of this result, for which we will require some set-theoretic assumptions.
Suppose that $C$ is [[cofibrantly generated model category|cofibrantly generated]] by a set of [[cofibrations]] $I$ and a set of [[trivial cofibrations]] $J$.
Let $I \otimes C$-cell denote the closure of $I \otimes C = \{ i \otimes \id_X : i \in I, X \in Ob(C) \}$ under [[cobase change]] and [[transfinite composition]], and similarly for $J \otimes C$-cell.

+-- {: .un_theorem}
###### Theorem
Suppose that

  * $C$ is [[cofibrantly generated model category|cofibrantly generated]] by a set of cofibrations $I$ and a set of fibrations $J$,

  * the [[domains]] of the morphisms in $I$ (resp. $J$) are [[small objects]] relative to $(I\otimes C)$-cell (resp. $(J\otimes C)$-cell),

  * $C$ satisfies the [[monoid axiom in a monoidal model category]]

  * $C$ satisfies the _commutative monoid axiom_, i.e. ...

Then the category $CMon(C)$ is a [[cofibrantly generated model category]] with [[weak equivalences]] and [[fibrations]] as defined above.

If $C$ is further [[simplicial model category|simplicial]] (resp. [[combinatorial model category|combinatorial]], [[tractable model category|tractable]]), then so is $CMon(C)$.
=--

See ([White 14](#White14), Theorem 3.2).

## Rectification

Rectification of $E_\infty$-monoids is the question of whether the [[weak equivalence]] between the [[operads]] [[Comm]] and the [[E-infinity operad]] induces a [[Quillen equivalence]] on the [[model category of algebras over an operad|model categories of algebras]].
Since the [[model category of algebras over an operad]] over the [[E-infinity operad]] is a presentation of the [[(infinity,1)-category]] of [[commutative monoids in a symmetric monoidal (infinity,1)-category]], rectification for $C$ is equivalent to saying that the [[(infinity,1)-category]] presented by the model structure on [[commutative monoids]] is equivalent to the [[(infinity,1)-category]] of [[commutative monoids in a symmetric monoidal (infinity,1)-category]] in the [[symmetric monoidal (infinity,1)-category]] presented by $C$.

The following cases are particularly interesting.

* Rectification holds in the [[model category]] of [[symmetric spectra]].
* It does not hold in the [[model category]] of [[simplicial sets]] or in the [[model category]] of [[simplicial abelian groups]].

See ([Lurie](#LurieHA), Theorem 4.5.4.7) for sufficient conditions for rectification to hold.
See also ([White 14](#White14), Paragraph 4.2) for more discussion.

A general rectification criterion for [[symmetric monoidal model categories]] is formulated in [PS 14](#PS14), Proposition 10.1.2 and Theorem 9.3.6.
It says that given a [[tractable]] symmetric monoidal model category that satisfies a certain
compact generatedness assumption
with a morphism of [[admissible operads]] $A\to B$
(e.g., $A=E_\infty$, $B=Comm$),
the Quillen adjunction between $A$-monoids and $B$-monoids induced by the morphism
of operads $A\to B$ is a Quillen equivalence
if and only if for any cofibration $s$ and any $n\ge0$
the morphism $(A_n\to B_n)\wedge_{\Sigma_n}s^{\wedge n}$ is a weak equivalence, where $\wedge$ denotes the [[pushout product]] with respect to the monoidal structure.

## Related concepts

* [[model structure on E-infinity monoids in a symmetric monoidal model category]]

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

* [[commutative monoid in a symmetric monoidal (infinity,1)-category]]

* [[model structure on algebras over an operad]]

* [[model structure on algebras over a monad]]

* [[model structure on simplicial T-algebras]]

## References

* {#White14} [[David White]], _Model Structures on Commutative Monoids in General Model Categories_, [arXiv:1403.6759](http://arxiv.org/abs/1403.6759).

* {#LurieHA} [[Jacob Lurie]], _[[Higher Algebra]]_.

* {#PS14} [[Dmitri Pavlov]] and [[Jakob Scholbach]], _Admissibility and rectification of colored symmetric operads_, [arXiv:1410.5675](http://arxiv.org/abs/1410.5675).

[[!redirects commutative monoids in a symmetric monoidal model category]]

[[!redirects model structure on commutative monoids]]
[[!redirects model structures on commutative monoids]]
[[!redirects model category of commutative monoids]]
[[!redirects model categories of commutative monoids]]
[[!redirects model category of commutative monoids in a symmetric monoidal model category]]
[[!redirects model categories of commutative monoids in a symmetric monoidal model category]]

[[!redirects model structure on commutative monoids in a symmetric monoidal model category]]


commutative monoid operad

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _commutative operad_ is the [[operad]] whose [[algebras over an operad]] are  [[commutative monoids]] in the ambient [[enriched category|enriching category]], hence the [[commutative algebras]] if interpreted in [[vector spaces]] and the [[differential graded-commutative algebras]] if interpreted in [[chain complexes]].

Its cofibrant [[resolutions]] in the [[model structure on operads]] are [[E-infinity operads]] whose [[algebras over an operad|algebras]] are [[E-infinity algebras]].

## Properties

* [[Koszul duality]] relates the commutative operad with the [[Lie operad]].

## Related concepts

* [[associative operad]]

* **commutative operad**

* [[Lie operad]]

[[!redirects commutative monoid operads]]

[[!redirects operad for commutative algebras]]
[[!redirects operads for commutative algebras]]

[[!redirects Comm]]

[[!redirects commutative algebra operad]]
[[!redirects commutative algebra operads]]

[[!redirects commutative operad]]
[[!redirects commutative operads]]


commutative monoidal category

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

#Contents#
* table of contents
{:toc}

## Idea

Commutative monoidal categories are [[symmetric monoidal categories]] whose  [[tensor product]] is _strictly_ [[associative]] and [[unital]] (as for [[permutative categories]]), but also strictly [[commutative monoid|commutative]], in that the [[associators]], [[unitors]], and [[braidings]] are all the  [[identity natural transformation]].

Notice that the [[coherence theorem for symmetric monoidal categories]] only says that [[symmetric monoidal categories]] are symmetric monoidally equivalent to [[strict monoidal categories]] whose [[braidings]], however, may not be given by the identity.

## Examples

Commutative monoidal categories are the natural type of category that [[Petri nets]] freely generate ([Meseguer-Montanari 90](#MeseguerMontanari90), [Baez-Master 18, Section 2](#BaezMaster18)).

The symmetric monoidal category of [[line bundles]] on a topological space, or [[smooth line bundles]] on a manifold, or [[invertible sheaf|invertible sheaves]] on a variety or scheme, is symmetric monoidally equivalent to a commutative monoidal category under the usual tensor product of these structures.

Any abelian group object in $Cat$ is a commutative monoidal category.

## Definition

A commutative monoidal category is a [[commutative monoid object]]  in [[Cat]]  with its [[cartesian product]].  Equivalently, it is an [[internal category]] in the category of [[commutative monoids]].

Explicitly, the data of a commutative monoidal category are:

* A commutative monoid of objects $C_1$,

* a commutative monoid of morphisms $C_0$,

* source and target [[monoid homomorphisms]] $s,t \colon C_1 \to C_0$,

* an identity assigning monoid homomorphism $i \colon C_0 \to C_1$ and,

* a homomorphism $\circ \colon C_1 \times_C_0 C_1 \to C_1$ representing composition.

These homorphisms are required to satisfy the axioms of a [[category]]. In particular, because composition is a commutative monoid homomorphism, it satisfies the [[interchange law]]

$$ (g \circ f) + (h \circ k) = (g + h) \circ (f + k) $$

whenever all composites are defined.

## Characterization up to equivalence

One might [[conjecture]] that a symmetric monoidal category is symmetric monoidally equivalent to a commutative monoidal category iff all its self-braidings

$$   B_{x,x} \colon x \otimes x \to x \otimes x $$

are identity morphisms.  Note that since a commutative monoidal category has this property, and this property is invariant under symmetric-monoidal equivalence, the "only if" part of the characterization is certainly true.

This conjecture was proved in [Kim 16](#Kim16) under the assumption that the objects of the monoidal category can be [[totally ordered]] (which is true, for instance, assuming the [[axiom of choice]], or assuming the weaker [[ultrafilter principle]] which states that every [[filter]] extends to an [[ultrafilter]]).

## References

* {#MeseguerMontanari90} [[José Meseguer]], [[Ugo Montanari]], _Petri Nets are Monoids_ ([pdf](https://core.ac.uk/download/pdf/82342688.pdf))

* {#BaezMaster18} [[John Baez]], [[Jade Master]], Section 2 of: _Open Petri Nets_ ([arXiv:1808.05415](https://arxiv.org/abs/1808.05415))

* [[James Dolan]], _[[jamesdolan:Algebraic Geometry|Doctrines of Algebraic Geometry]]_

* {#Kim16}[[Youngsoo Kim]], _A Note on Strict Commutativity of a Monoidal Product_, Pure and Applied Mathematics Journal Volume 5, Issue 5, October 2016, pp. 155-159, [url](https://doi.org/10.11648/j.pamj.20160505.13)

[[!redirects commutative monoidal categories]]
[[!redirects Commutative monoidal category]]
[[!redirects Commutative monoidal categories]]


commutative quasigroup

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A __commutative [[quasigroup]]__ is a [[commutative magma]] $(G,(-)\cdot(-):G \times G \to G)$ equipped with a binary operation $(-)/(-):G \times G \to G$ called __division__ such that $(x/y) \cdot y = x$ and $(x \cdot y)/y = x$.

## Examples

* Every [[commutative loop]] is a commutative quasigroup.

* Every [[commutative invertible quasigroup]] is a commutative quasigroup.

* The empty quasigroup is a commutative quasigroup.

## Related concepts

* [[associative quasigroup]]


commutative ring spectrum > history

see at _[[ring spectrum]]_


commutative semigroup

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[semigroup]] $(S,\cdot)$ is called _commutative_ if its [[binary operation]] $(-)\cdot(-) \colon S \times S $ has the property that for all $x,y \in S$ then

$$
  x \cdot y = y \cdot x
  \,.
$$

## Examples

Examples include [[commutative monoids]], [[abelian groups]], [[commutative rings]], [[commutative algebras]] etc.

## Related concepts

* [[semigroup]] (noncommutative version)

* [[commutative invertible semigroup]] (invertible version)

* [[commutative monoid]] (unital version)

* [[commutative magma]] (non-associative version)

[[!redirects commutative semigroups]]


commutative square

# Commutative squares
* table of contents
{: toc}

## Definition and notation

Let $C$ be a [[category]]. A **square** of morphisms of $C$  consists of objects $X,Y,Z,W$ of $C$ and morphisms $f\colon X \to Z$, $g\colon X \to Y$, $f'\colon Y \to W$, and $g'\colon Z \to W$. This is often pictured as a square
$$\array{& X & \overset{f}\rightarrow & Z & \\
          g & \downarrow &&\downarrow & g'\\
          &Y & \underset{f'}\rightarrow& W & \\
}$$

The square is **[[commutative diagram|commutative]]** if $g' \circ f = f' \circ g$.

The class of commutative squares in $C$ is written $\square C$.

## Structure

This class has partial compositions $\circ_1$ and $\circ_2$ which are vertical and horizontal:
$$  \array{ \bullet & {\to} & \bullet & \\
           \downarrow &&\downarrow \\
          \bullet  & {\to}& \bullet \\
            \downarrow &  & \downarrow\\
            \bullet  & \to & \bullet
}  \quad \quad
 \array{\bullet & {\to} & \bullet & \to & \bullet  \\
           \downarrow &&\downarrow && \downarrow  \\
          \bullet  & {\to}& \bullet & \to & \bullet   }
$$
thus forming a (strict) [[double category]], also written $\square C$, whose objects are those of $C$, whose horizontal and vertical 1-cells are given by morphisms in $C$, and whose 2-cells exhibit commutative squares. It contains the _vertical category_ $\square_1 C$ and the _horizontal category_ $\square _2 C$.

One can also form _multiple compositions_ $[a_{ij}]$ of arrays $(a_{ij})$, $i=1, \ldots, m; j=1, \ldots , n$,  of commutative squares   provided that in the obvious sense _adjacent squares are composible_. One checks by induction that:

_any composition of commutative squares is commutative._

## Applications

Let $2$ denote the [[walking arrow]]: the category with two objects $0,1$ and one arrow $0 \to 1$. This has the structure of [[cocategory]]. Then the class of commutative squares in $C$  can also be described as $Cat(2 \times 2, C)$.

If $D$ is a category, then $Cat(D, \square_1 C)$ can be regarded as the class of natural transformations of functors $D \to C$. Then the category structure $\square _2 C$ induces a category structure on $Cat(D,\square _1 C)$ giving the [[functor category]] $CAT(D,C)$: the category of functors and natural transformations. (This account is due to [[Charles Ehresmann]].)

One deduces that if also $E$ is a category then there is a natural bijection
$$Cat(E \times D, C) \cong (E, CAT(D,C)),$$
which thus states that the category of (small if you like!) categories is [[cartesian closed category|cartesian closed]].

The commutative squares serve as the morphisms in the [[arrow category]] of $C$, which is the functor category $CAT(2,C)$.

[[!redirects commutative square]]
[[!redirects commutative squares]]
[[!redirects commuting square]]
[[!redirects commuting squares]]


commutative triangle

# Commutative triangles
* table of contents
{: toc}

## Definition

Let $C$ be a [[category]]. A **triangle** of morphisms of $C$ consists of objects $X,Y,Z$ of $C$ and morphisms $f\colon X \to Y$, $g\colon Y \to Z$, and $h\colon X \to Z$. This is often pictured as a triangle
$$ \array {
  X & \overset{f}\rightarrow & Y \\
    & \searrow^{h}           & \downarrow^{g} \\
    &                        & Z
} $$

The triangle is **[[commutative diagram|commutative]]** if $h = g \circ f$.

## Characterisation

A commutative triangle is determined entirely by $f$ and $g$; therefore, a commutative triangle is equivalent to a [[composable pair]] of morphisms.

Accordingly, one rarely hears of commutative triangles on their own; instead, the concept only comes up when one already has a triangle and asks whether it commutes.  (This is different from the situation with [[commutative squares]].)

[[!redirects commutative triangle]]
[[!redirects commutative triangles]]