monodromy

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#### Homotopy theory
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[[!include homotopy - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

_Monodromy_ is the name for the [[action]] of the [[homotopy group (of an ∞-stack)|homotopy groups]] of a [[space]] $X$ on [[fibers]] of [[covering spaces]] or [[locally constant ∞-stacks]] on $X$.

## In point-set topology
 {#InPointSetTopology}

We discuss monodromy of [[covering spaces]] in elementary [[point-set topology]].

### Definition

+-- {: .num_defn #CoveringSpaceMonodromy}
###### Definition
**(monodromy of a [[covering space]])

Let $X$ be a [[topological space]] and $E \overset{p}{\to} X$ a [[covering space]]. Write $\Pi_1(X)$ for the [[fundamental groupoid]] of $X$.

Define a [[functor]]

$$
  Fib_E
    \;\colon\;
  \Pi_1(X)
    \longrightarrow
  Set
$$

to the [[category]] [[Set]] of [[sets]] as follows:

1. to a point $x \in X$ assign the [[fiber]] $p^{-1}(\{x\}) \in Set$;

1. to the [[homotopy class]] of a [[path]] $\gamma$ connecting $x \coloneqq \gamma(0)$ with $y \coloneqq \gamma(1)$ in $X$ assign the function $p^{-1}(\{x\}) \longrightarrow p^{-1}(\{y\})$ which takes $\hat x \in p^{-1}(\{x\})$ to the endpoint of a path $\hat \gamma$ in $E$ which lifts $\gamma$ through $p$ with starting point $\hat \gamma(0) = \hat x$

   $$
     \array{
       p^{-1}(x) &\overset{}{\longrightarrow}& p^{-1}(y)
       \\
       (\hat x = \hat \gamma(0)) &\mapsto& \hat \gamma(1)
     }
     \,.
   $$

This construction is well defined for a given representative $\gamma$ due to the unique path-lifting property of covering spaces ([this lemma](covering+space#CoveringSpacePathLifting)) and it is independent of the choice of $\gamma$ in the given homotopy class of paths due to the homotopy-lifting property ([this lemma](covering+space#CoveringSpacesHomotopyLifting)).  Similarly, these two lifting properties give that this construction respects composition in $\Pi_1(X)$ and hence is indeed a [[functor]].

=--

Hence this defines a "[[permutation representation|permutation]] [[groupoid representation]]" of $\Pi_1(X)$.

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

Given a [[homomorphism]] between two [[covering spaces]] $E_i \overset{p_i}{\to} X$, hence a [[continuous function]] $f \colon E_1 \to E_2$ which respects [[fibers]] in that the [[diagram]]

$$
  \array{
    E_1 && \overset{f}{\longrightarrow} && E_2
    \\
    & {}_{\mathllap{p_1}}\searrow && \swarrow_{\mathrlap{p_2}}
    \\
    && X
  }
$$

[[commuting diagram|commutes]], then the component functions

$$
  f\vert_{\{x\}} \;\colon\; p_1^{-1}(\{x\}) \longrightarrow p_2^{-1}(\{x\})
$$

are compatible with the monodromy $Fib_{E}$ (def. \ref{CoveringSpaceMonodromy}) along any [[path]] $\gamma$ between points $x$ and $y$ from def. \ref{CoveringSpaceMonodromy} in that the following [[diagrams]] of [[sets]] [[commuting diagram|commute]]

$$
  \array{
     p_1^{-1}(x) &\overset{f\vert_{\{x\}}}{\longrightarrow}& p_2^{-1}(x)
     \\
     {}^{\mathllap{Fib_{E_1}([\gamma])}}\downarrow
       &&
     \downarrow^{\mathrlap{ Fib_{E_2}([\gamma]) }}
     \\
     p_1^{-1}(y) &\underset{f\vert_{\{y\}}}{\longrightarrow}& p_2^{-1}(\{y\})
  }
  \,.
$$

This means that $f$ induces a [[natural transformation]] between the monodromy functors of $E_1$ and $E_2$, respectively, and hence that constructing monodromy is itself a functor from the [[category]] of [[covering spaces]] of $X$ to that of [[permutation representations]] of the [[fundamental groupoid]] of $X$:

$$
  Fib
   \;\colon\;
  Cov(X)
    \longrightarrow
  Set^{\Pi_1(X)}
  \,.
$$

=--

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

For any $\hat x \in p_1^{-1}(x)$ let $\hat \gamma$ be the unique path in $E$ with $\hat \gamma(0) = \hat x$ and $p \circ \hat \gamma = \gamma$. By definition \ref{CoveringSpaceMonodromy} we have

$$
  Fib_{E_1}([\gamma(f)])(\hat x) = \hat \gamma(1)
$$

and hence

$$
  f(Fib_{E_1}([\gamma(f)])(\hat x)) = f(\hat \gamma(1))
$$

Now $f \circ \hat \gamma$ satisfies $f \circ \hat \gamma(0) = f(\hat x)$ and $p \circ f \circ \hat \gamma = \gamma$ by the fact that $f$ preserves fibers. Hence by uniqueness of path lifting ([this lemma](covering+space#CoveringSpacePathLifting)), $f \circ \hat \gamma$ is the unique lift of $\gamma$ with starting point $f(\hat x)$. By def. \ref{CoveringSpaceMonodromy} this means that

$$
  Fib_{E_2}([\gamma])(f(\hat x))
  =
  f (\hat \gamma(1))
  \,.
$$

This is the equality to be shown.

=--

### Properties

+-- {: .num_remark}
###### Remark
**([[fundamental theorem of covering spaces]])**

The [[reconstruction of covering spaces from monodromy]] is an [[inverse functor]] to the monodromy functor. The resulting [[equivalence of categories]]

$$
  Cov(X)
    \underoverset
      {\underset{Fib}{\longrightarrow}}
      {\overset{Rec}{\longleftarrow}}
      {}
  Set^{\Pi_1(X)}
$$

between the [[category of covering spaces]] and the [[permutation representation|permutation]] [[groupoid representations]] of the [[fundamental groupoid]] is known as the _[[fundamental theorem of covering spaces]]_.

=--

+-- {: .num_example #CoveringSpaceFundamentalGroupoid}
###### Example
**([[fundamental groupoid]] of covering space)

Let $E \overset{p}{\longrightarrow} X$ be a covering space.

Then the [[fundamental groupoid]] $\Pi_1(E)$ of the total space $E$ is
[[equivalence of categories|equivalently]] the [[Grothendieck construction]]
of the [[monodromy]] functor $Fib_E \;\colon\; \Pi_1(X) \to Set$

$$
  \Pi_1(E)
   \;\simeq\;
  \int_{\Pi_1(X)} Fib_E
$$

whose

* [[objects]] are pairs $(x,\hat x)$ consisting of a point $x \in X$ and en element $\hat x \in Fib_E(x)$;

* [[morphisms]] $[\hat \gamma] \colon (x,\hat x) \to (x', \hat x')$ are morphisms $[\gamma] \colon x \to x'$ in $\Pi_1(X)$ such that $Fib_E([\gamma])(\hat x) = \hat x'$.

=--

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

By the uniqueness of the path-lifting ([this lemma](covering+space#CoveringSpacePathLifting)) and the very definition of the [[monodromy]] functor.

=--

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

Let $X$ be a [[path-connected topological space]] and let $E \overset{p}{\to} X$ be a [[covering space]]. Then the total space $E$ is

1. [[path-connected topological space|path-connected]] precisely if the [[monodromy]] $Fib_E$ is a [[transitive action]];

1. [[simply connected topological space|simply connected]] precisely if the [[monodromy]] $Fib_E$ is [[free action]].

=--

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

By example \ref{CoveringSpaceFundamentalGroupoid}.

=--

## In cohesive $\infty$-Toposes

Let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]] and $X \in \mathbf{H}$ any object. Then the [[locally constant ∞-stacks]] on $X$ are represented by morphisms $X \to LConst Core(\infty Grpd)$. By adjunction such morphisms are equivalent to [[(∞,1)-functors]] $\Pi(X) \to Core(\infty Grpd)$  This morphism exhibits the **monodromy** of the [[locally constant ∞-stack]].

Specifically, the restriction $\mathbf{B}\Omega_x \Pi(X) \hookrightarrow \Pi(X) \to \infty Grpd$ to the [[delooping]] $\mathbf{B}\Omega_x \Pi(X)$ of the [[loop space object]] $\Omega_x \Pi(X)$ at a chosen baspoint $x : {*} \to X$ is the **monodromy [[action]]** of loops based at $x \in X$ on the fiber of the locally constant $\infty$-stack over $x$.

## References

* [[eom]], _[Monodromy transformation](https://encyclopediaofmath.org/wiki/Monodromy_transformation)_

[[!redirects monodromies]]


monoid

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###Context###
#### Category theory
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[[!include category theory - contents]]
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#### Algebra
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[[!include higher algebra - contents]]
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#### Monoid theory
+-- {: .hide}
[[!include monoid theory - contents]]
=--
=--
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\tableofcontents

## Idea
 {#Idea}

In [[algebra]], by a *monoid* one means a collection ([[set]]) of elements equipped with a [[binary operation]] (a "multiplication operation") which is [[associativity|associative]] and has a [[unit element]].

Hence monoid structure on a set is a fairly rudimentary form of algebraic [[structure]] which [[underlying|underlies]] many familiar [[structures]] considered [[algebra]], such as that of *[[groups]]* (which are monoids with all [[inverse elements]]) and *[[rings]]* (which are [[abelian groups]] compatibly equipped with a *second* monoid structure).

Therefore, in the algebraic literature monoids are, conversely, often called *unital [[semi-groups]]*. The root "mono-" in "monoid" refers to the  single [[binary operation]] (cf. *[[duoid]]* and *[[dioid]]*).

The terminology of *[[magmas]]* is meant to invoke this rudimemtary but foundational nature of basic algebraic structures: Monoids are precisely the [[unital magma|unital]] [[associative magmas]].

The [[categorification]] of the notion of monoids is that of *[[monads]]* whose ubiquitous role in [[mathematics]] (together with their [associated](adjoint+functor#RelationBetweenAdjunctionsAndMonads) [[adjoint functors]]) is hard to overstate.

## Definitions

### Elementary definition

The classical definition of monoids in [[Sets]], as a [[unital magma|unital]] [[associative magma]]:

\begin{definition}\label{MonoidsInSets}
A **monoid** is a [[set]] $M$ equipped with the [[structure]] if

1. (**[[binary operation]]**) a [[map]] from the [[Cartesian product]] of the set with itself to itself

   $$
     (\text{-}) \cdot (\text{-})
       \,\colon\,
     M \times M \to M
   $$

1. (**[[neutral element]]**) an [[element]]

   $$1 \in M$$

such that the following [[equations]] are satisfied

1. (**[[associativity]]**)

   $$
     x, y, z \,\in\, M
     \;\;\;\;\;
       \vdash
     \;\;\;\;\;
     (x \cdot y) \cdot z
       \;=\;
     x \cdot (y \cdot z)
     \mathrlap{\,,}
   $$

* (**[[unitality]]**)

  $$
    x \in M
     \;\;\;\;\;
       \vdash
     \;\;\;\;\;
    1 \cdot x
      \;=\;
    x
      \;=\;
    x \cdot 1
    \mathrlap{\,.}
  $$

\end{definition}

### In a monoidal category
{#inamonoidalcategory}

See *[[monoid in a monoidal category]]*.

### In terms of string diagrams

The data of a monoid may be written in [[string diagrams]] as:

[[monoid-data-labeled.png:pic]]

Thanks to the distinctive shapes, one can usually omit the labels:

[[monoid-data-unlabeled.png:pic]]

The axioms $\mu \cdot (\eta \otimes M) = 1_M = \mu \cdot (M \otimes \eta)$ and $\mu \cdot (M \otimes \mu) = \mu \cdot (\mu \otimes M)$ then appear as:

[[monoid-axioms-unlabeled.png:pic]]

### As a one-object category
 {#AsAOneObjectCategory}

Equivalently, and more efficiently, we may say that a (classical) monoid is the [[hom-set]] of a [[category]] with a single [[object]], equipped with the structure of its unit element and composition.

More tersely, one may say that a monoid *is* a category with a single object, or more precisely (to get the proper morphisms and $2$-morphisms) a [[pointed object|pointed]] category with a single object.
But taking this too literally may create conflicts in notation.  To avoid this, for a given monoid $M$, we write $\mathbf{B}M$ for the corresponding category with single object $\bullet$ and with $M$ as its [[hom-set]]: the [[delooping]] of $M$, so that $M = Hom_{\mathbf{B}M}(\bullet, \bullet)$. This realizes every monoid as a monoid of [[endomorphism|endomorphisms]].

Similarly, a monoid in $(C,\otimes,I)$ may be defined as the [[hom-object]] of a $C$-[[enriched category]] with a single object, equipped with its composition and identity-assigning morphisms; and so on, as in the classical (i.e. $\mathbf{Set}$-enriched) case.

For more on this see also [[group]].

### As a strict monoidal category

An alternate way to view a monoid as a category is as a discrete [[strict monoidal category]] $\mathbf{C}$ where the elements of the monoid are the objects of $\mathbf{C}$, the binary operation of the monoid provides the tensor product bifunctor, and the identity of the monoid is the unit object. Preordered monoids then yield (non-discrete) strict monoidal categories with the morphisms witnessing the preorder in the usual way.

### $\mathcal{O}$-Monoids over an $(\infty,1)$-Operad

The notion of _associative monoids_ discussed above are controled by the
[[associative operad]]. More generally in [[higher algebra]], for $\mathcal{O}$ any [[operad]] or [[(infinity,1)-operad]], one can consider **$\mathcal{O}$-monoids**. ([Lurie, def. 2.4.2.1](#Lurie))

These are closely related to [[(infinity,1)-algebras over an (infinity,1)-operad]] with respect to $\mathcal{O}$ ([Lurie, prop. 2.4.2.5](#Lurie)).

## Properties

### Finite products (and sums)

Let $M$ be a monoid, and let $M^*$ be the [[free monoid]] on $M$ with canonical function $h:M \to M^*$ taking the elements of $M$ to the generators in $M^*$. The finite product operation on $M$ is a [[monoid homomorphism]]

$$\prod_{i = 0}^{\mathrm{len}(-) - 1}(-)(i):M^* \to M$$

from $M^*$ to $M$, where:

$$\prod_{i = 0}^{\mathrm{len}(\epsilon) - 1} \epsilon(i) = 1$$

$$\prod_{i = 0}^{\mathrm{len}(h(a)) - 1} (h(a))(i) = a$$

$$\left(\prod_{i = 0}^{\mathrm{len}(a) - 1} a(i)\right) \cdot \left(\prod_{i = 0}^{\mathrm{len}(b) - 1} b(i)\right) = \prod_{i = 0}^{\mathrm{len}(a b) - 1} (a b)(i)$$

If $M$ is written additively $(+, 0)$ instead of multiplicatively $(\cdot, 1)$, the operation is called finite sum, and is defined as

$$\sum_{i = 0}^{\mathrm{len}(-) - 1}(-)(i):M^* \to M$$

from $M^*$ to $M$, where:

$$\sum_{i = 0}^{\mathrm{len}(\epsilon) - 1} \epsilon(i) = 0$$

$$\sum_{i = 0}^{\mathrm{len}(h(a)) - 1} (h(a))(i) = a$$

$$\left(\sum_{i = 0}^{\mathrm{len}(a) - 1} a(i)\right) + \left(\sum_{i = 0}^{\mathrm{len}(b) - 1} b(i)\right) = \sum_{i = 0}^{\mathrm{len}(a b) - 1} (a b)(i)$$

## Examples

* A monoid in which every element has an inverse is a [[group]]. For that reason monoids are often known (especially outside category theory) as _semi-group_s. (But this term is often extended to monoids without identities, that is to sets equipped with any associative operation.)
* The set of [[endomorphism|endomorphisms]] of a given object in a category has a canonical monoid structure given by composition.

## Remarks on notation

It can be important to distinguish between a $k$-tuply monoidal structure and the corresponding $k$-tuply degenerate category, even though there is a map identifying them. The issue appears here for instance when discussing the universal $G$-bundle in its groupoid incarnation.  This is
$$
  G \to \mathbf{E}G \to \mathbf{B}G
$$
(where $\mathbf{E}G = G//G$ is the action groupoid of $G$ acting on itself). On the left we crucially have $G$ as a monoidal 0-category, on the right as a once-degenerate 1-category. Without this notation we cannot even _write down_ the universal $G$-bundle!

Or take the important difference between group [[representation]]s and group 2-algebras, the former being functors $\mathbf{B}G \to Vect$, the latter functors $G \to Vect$. Both these are very important.

Or take an abelian group $A$ and a codomain like $2Vect$. Then there are 3 different things we can sensibly consider, namely 2-functors
$$
  A \to 2Vect
$$
$$
  \mathbf{B}A \to 2Vect
$$
and
$$
  \mathbf{B}^2A \to 2Vect
  \,.
$$
All of these concepts are different, and useful. The first one is an object in the group 3-algebra of $A$. The second is a pseudo-representation of the group $A$. The third is a representations of the 2-group $\mathbf{B}A$. We have notation to distinguish this, and we should use it.

Finally, writing $\mathbf{B}G$ for the 1-object $n$-groupoid version of an $n$-monoid $G$ makes notation behave nicely with respect to nerves, because then realization bars $|\cdot|$ simply commute with the $B$s in the game: $|\mathbf{B}G| = B|G|$.

This behavior under nerves shows also that, generally, writing $\mathbf{B}G$ gives the right intuition for what an expression means. For instance, what's the "geometric" reason that a group representation is an arrow $\rho : \mathbf{B}G \to Vect$? It's because this is, literally, equivalently thought of as the corresponding classifying map of the vector bundle on $\mathbf{B}G$ which is $\rho$-associated to the universal $G$-bundle:

the $\rho$-associated vector bundle to the universal $G$-bundle is, in its groupoid incarnations,
$$
  \array{
    V
    \\
    \downarrow
    \\
    V//G
    \\
    \downarrow
    \\
    \mathbf{B}G
  }
  \,,
$$
where $V$ is the vector space that $\rho$ is representing on, and this is classified by the representation $\rho : \mathbf{B}G \to Vect$ in that this is the pullback of the universal $Vect$-bundle
$$
  \array{
    V//G
    &\to&
    Vect_*
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}G
    &\stackrel{\rho}{\to}&
    Vect
  }
  \,,
$$

In summary, it is important to make people understand that groups can be identified with one-object groupoids. But next it is important to make clear that not everything that can be identified should be, for instance concerning the crucial difference between the category in which $G$ lives and the 2-category in which $\mathbf{B}G$ lives.

## Related concepts

* [[monoid in a monoidal category]], [[category of monoids]]

* [[monoidal symmetric proset]]

* **monoid**, internal monoid/monoid object,

  * [[commutative monoid]], [[cancellative monoid]]

  * [[monoidal groupoid]], [[braided monoidal groupoid]], [[symmetric monoidal groupoid]]

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

  * [[algebra in an (∞,1)-category|monoid object in a (∞,1)-category]]

* [[topological monoid]]

* [[comonoid]], [[cocommutative comonoid]]

* [[duoid]]

* [[group]], [[group object]]

* [[ring]], [[ring object]]

* [[categorical algebra]]

* [[A3-type]]

[[!include oidification - table]]

## References

Beware that the term "monoid" was first used by

* [[Arthur Cayley]], *Second and Third Memoirs on Skew Surfaces*, Otherwise Scrolls, Phil. Trans. (1863 and 1869)

for certain [[surfaces]], quite unrelated to the modern meaning of the term.

Instead, what are now called *monoids* ([[unital magma|unital]] [[associative magmas]]) were called *groupoids* (now clashing with the modern use of *[[groupoid]]*) by

* [[Garrett Birkhoff]], *Hausdorff Groupoids*, Annals of Mathematics, Second Series **35** 2 (1934) 351-360 [[jstor:1968437](https://www.jstor.org/stable/1968437), [doi:10.2307/1968437](https://doi.org/10.2307/1968437)]

The modern terminology "monoid" for unital associative magmas is (according to [Hollings 2009, p. 529](#Hollings09)) due to

* [[Nicolas Bourbaki]], [[Éléments de Mathématique]] (1943)

For more on the history of the notion:

* {#Hollings09} Christopher Hollings, *The Early Development of the Algebraic Theory of Semigroups*, Archive for History of Exact Sciences **63** (2009) 497–536 [[doi:10.1007/s00407-009-0044-3](https://doi.org/10.1007/s00407-009-0044-3)]

* Math.SE, *[Who invented Monoid?](https://mathoverflow.net/q/338281/381)*

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

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

Properties of monoids expressed through properties of their [[toposes]] of [[presheaves]]:

* Jens Hemelaer, Morgan Rogers, _Monoid Properties as Invariants of Toposes of Monoid Actions_, [arXiv:2004.10513](https://arxiv.org/abs/2004.10513).

Formalization of [[monoid objects]] as [[mathematical structures]] in [[proof assistants]]:

in a context of plain [[Agda]]:

* [[Martín Escardó]], *[The Types of Magmas and Monoids](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html#magmasandmonoids)*, §4 in: *Introduction to Univalent Foundations of Mathematics with Agda* [[arXiv:1911.00580](https://arxiv.org/abs/1911.00580),  [webpage](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html)]

in a context of [[cubical type theory|cubical]] [[Agda]]:

* [[1lab]]: *[Algebra.Monoid](https://1lab.dev/Algebra.Monoid.html)*

[[!redirects monoid]]
[[!redirects monoids]]


monoid axiom in a monoidal model category

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+-- {: .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

The _monoid axiom_ is an extra condition on a [[monoidal model category]] that helps to make its [[model structure on monoids in a monoidal model category]] exist and be well behaved.

## Definition

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

We say a [[monoidal model category]] satisfies the **monoid axiom** if every morphism that is obtained as a [[transfinite composition]] of [[pushouts]] of [[tensor products]] of acyclic cofibrations with any object is a weak equivalence.

=--

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

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

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

=--

## Properties

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

Let $C$ be a

* [[cofibrantly generated model category]];

* [[symmetric monoidal category]];

* [[monoidal model category]].

Then if the monoid axiom holds for the set of generating acyclic cofibrations it holds for all acyclic cofibrations.

=--

([Schwede-Shipley 00, lemma 3.5](#SchwedeShipley)).

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

If a [[monoidal model category]] satisfies the monoid axiom and

* it is a [[cofibrantly generated model category]];

* all objects are [[small objects]],

then the [[transferred model structure]] along the [[free-forgetful adjunction]] $(F \dashv U) : Mon(C) \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} C$ exists on its [[category of monoids]] and hence provides a [[model structure on monoids in a monoidal model category|model structure on monoids]].

=--

([Schwede-Shipley 00, theorem 4.1](#SchwedeShipley))

## Examples

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

Monoidal model categories that satisfy the monoid axiom (as well as the other conditions sufficient for the [above theorem](#ExistenceOfModelstructureOnMonoids) on the existence of transferred model structures on categories of monoids) include

with respect to [[Cartesian product]]

* the standard [[model structure on simplicial sets]];

with respect to [[tensor product of chain complexes]]:

* the projective [[model structure on chain complexes]] for connective chanin complexes with fibrations the surjections in positive degree

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

* the [[model structure on symmetric spectra]]

* the [[model structure on orthogonal spectra]]

* the [[model structure for excisive functors]]

* the [[model structure on Gamma-spaces]]

=--

([Schwede-Shipley 00, section 5](#SchwedeShipley), [MMSS 00, theorem 12.1 (iii)](#MMSS00))

## References

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

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

[[!redirects model structure on monoids]]

[[!redirects monoid axiom]]


[[!redirects algebra in an (infinity,1)-category]]
[[!redirects algebra in an (infinity,1)-category]]
[[!redirects algebra in an (infinity,1)-category]]
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###Context###
#### Higher algebra
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[[!include higher algebra - contents]]
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#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Monoid theory
+-- {: .hide}
[[!include monoid theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _monoid_ (or _monoid object_, _algebra_, _algebra object_) in a [[monoidal (infinity,1)-category]] $C$ is the [[(infinity,1)-category theory|(infinity,1)-categorical]] generalization of [[monoid in a monoidal category]].

## Definition

For $C$ a [[cartesian monoidal (∞,1)-category]] with a monoidal structure determined by the [[(∞,1)-functor]]

$$
  p_\otimes : C^\otimes \to N(\Delta)^{op}
$$

a **monoid object** of $C$ is a [[lax monoidal (infinity,1)-functor|lax monoidal (∞,1)-functor]]

$$
  N(\Delta)^{op} \to C^\otimes
$$

This generalizes how, for [[monoidal categories]], [[monoid in a monoidal category|monoid objects]] are the same as lax monoidal functors

$$
  * \to C
  \,.
$$

A generalization to the case that the monoidal (∞,1)-category $C$ is not Cartesian is discussed in Section 4.1.3 of Higher Algebra.

## Examples

* A monoid in the [[stable (infinity,1)-category of spectra]] is an [[A-infinity ring|A-∞ ring]]. If it is a commutative monoid, it is an [[E-infinity ring|E-∞ ring]].

## Related concepts

* [[infinity-module]]

* [[monoid]]

* [[En-algebra]]

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

* [[category object in an (∞,1)-category]]

## References

definition 1.1.14 in

* [[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))

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monoid in a monoidal category

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#### Monoidal categories
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#### Monoid theory
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Generalizing the classical notion of [[monoid]], one can define a _monoid_ (or _monoid object_) in any [[monoidal category]] $(C,\otimes,I)$.
Classical monoids are of course just monoids in [[Set]] with the [[cartesian product]].

By the [[microcosm principle]], in order to define monoid objects in $C$, $C$ itself must be a "categorified monoid" in some way.  The natural requirement is that it be a [[monoidal category]].  In fact, it suffices if $C$ is a [[multicategory]].  Contrast this with a [[group object]], which can only be defined in a [[cartesian monoidal category]] (or a [[cartesian multicategory]]).

## Definition

Namely, a **monoid in $C$** is an object $M$ equipped with a multiplication $\mu: M \otimes M \to M$ and a unit $\eta: I \to M$ satisfying the **associative law**:

$$
\array{
  & (M \otimes M) \otimes M
    & \stackrel{\alpha}{\longrightarrow}
  & M \otimes (M \otimes M)
    & \stackrel{1 \otimes \mu}{\longrightarrow}
  & M \otimes M
  \\
  & {}_{\mu \otimes 1}\searrow
    &&
    && \swarrow_{\mu}
  &
  \\
  &&
  M \otimes M
    & \stackrel{\mu}{\longrightarrow}
  M
  &&
}
$$

and the **left and right unit laws**:

$$
\array{
  & I \otimes M
  & \stackrel{\eta \otimes 1}{\longrightarrow}
  & M \otimes M
  & \stackrel{1 \otimes \eta}{\longleftarrow}
  & M \otimes I
  \\
  &
  & {}_{\lambda}\searrow
  & {}_{\mu}\downarrow
  & \swarrow_{\rho}
  &
  \\
  &
  &
  & M
  &
  &
}
$$

Here $\alpha$ is the [[associator]] in $C$, while $\lambda$ and $\rho$ are the left and right [[unitor|unitors]].

## Morphism of monoids

The analogue of a monoid homomorphism, called a morphism of monoids, is a morphism, $\f: M \to M'$  between two monoid objects, satisfying the equations;

$f \circ \mu = \mu' \circ (f \otimes f)$

$f \circ \eta = \eta'$

corresponding to the commutative diagrams;

$$
\array{
  & M \otimes M
  & \stackrel{f \otimes f}{\longrightarrow}
  & M' \otimes M'
  \\
  & {}_{\mu}\downarrow
  &
  & \downarrow_{\mu'}
  \\
  & M
  & \stackrel{f}{\longrightarrow}
  & M'
}
$$

$$
\array{
  & I
  & \stackrel{\eta}{\longrightarrow}
  & M
  \\
  &
  & {}_{\eta'}\searrow
  & \downarrow_{f}
  \\
  &
  &
  & M'
}
$$

## As categories with one object

Just as the category of regular [[monoid|monoids]] is equivalent to the category of locally small (i.e. Set-enriched) categories with one object, the category of monoids in $C$ (with the obvious morphisms) is equivalent to the category of $C$-[[enriched category|enriched categories]] with one object.

## Properties

### Preservation by lax monoidal functors

Monoid structure is preserved by [[lax monoidal functors]]. Comonoid structure by [[oplax monoidal functors]]. See [[lax monoidal functor]] for more details.

### Category of monoids

For special properties of categories of monoids, see [[category of monoids]].

## Examples

\begin{example}
\label{AssociativeAlgebras}
A monoid in a [[monoidal category|monoidal]] [[category of modules]] [[Mod|$R Mod$]] (over any [[ground ring]] $R$ and equipped with the [[tensor product of modules]]) is an [[associative unital algebra]] over $R$.
\end{example}

\begin{example}\label{Rings}
As the special case of Exp. \ref{AssociativeAlgebras} for $R = \mathbb{Z}$ the [[integers]]:

A monoid object in the [[monoidal category]] [[Ab]] of [[abelian groups]] with the [[tensor product of abelian groups]],  is a *[[ring]]*.
\end{example}

\begin{example}
As the special case of Exp. \ref{AssociativeAlgebras} for $R = k$ a [[field]]:

 A monoid object in the category [[Vect]] of [[vector spaces]] (over any [[ground field]] $k$) with the [[tensor product of vector spaces]] is an *[[associative unital algebra]]* over $k$.
\end{example}

* A monoid object in [[Top]] (with cartesian product as the tensor product) is a [[topological monoid]].
* A monoid object in [[Ho(Top)]] is an [[H-monoid]].
* A monoid object in the category of monoids (with cartesian product as the tensor product) is a [[commutative monoid]].  This is a version of the [[Eckmann-Hilton argument]].
* A monoid object in the category of complete join-[[semilattice]]s (with its tensor product that represents maps preserving joins in each variable separately) is a unital [[quantale]].
* The category of [[pointed set]]s has a monoidal structure given by the [[smash product]]. A monoid object in this monoidal category is an [[absorption monoid]].
* Given any monoidal category $C$, a monoid in the monoidal category $C^{op}$ is called a [[comonoid]] in $C$.
* In a [[cocartesian monoidal category]], every object is a monoid object in a unique way.
* For any category $C$, the [[endofunctor]] category $C^C$ has a monoidal structure induced by composition of endofunctors, and a monoid object in $C^C$ is a [[monad]] on $C$.

These are examples of monoids internal to monoidal categories.  More generally, given any [[bicategory]] $B$ and a chosen object $a$, the [[hom-category]] $B(a,a)$ has the structure of a monoidal category.  So, the concept of monoid makes sense in any [[bicategory]] $B$: we define a **monoid in $B$** to be a monoid in $B(a,a)$ for some object $a \in B$.  This often called a [[monad]] in $B$.  The reason is that a monad in [[Cat]] is the same as monad on a category.

A monoid in a bicategory $B$ may also be described as the [[hom-object]] of a $B$-[[enriched category]] with a single object.

## Related concepts

* [[monoid]]

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

* [[idempotent monoid in a monoidal category]]

* [[module over a monoid]]

* [[category of monoids]]

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

[[!include reader-writer (co)monads -- table]]

## References

Original references (including the case of a [[commutative monoids in a symmetric monoidal category]], but see there for more):

* [[Jean Bénabou]], _Algèbre élémentaire dans les catégories_ (1964), C. R. Acad. Sci. Paris **258** (1964) pp.771-774, [gallica](https://gallica.bnf.fr/ark:/12148/bpt6k40102/f817)

* [[Saunders MacLane]], Sections III.6 & VII.3 in: *[[Categories for the Working Mathematician]]*, Springer  (1979, 2nd ed.) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* [[Hans-Joachim Baues]], [[Mamuka Jibladze]], [[Andy Tonks]], *Cohomology of monoids in monoidal categories*, in: *Operads: Proceedings of Renaissance Conferences*, Contemporary Mathematics **202**, AMS (1997) 137-166 [[doi:10.1090/conm/202](https://doi.org/10.1090/conm/202), preprint:[pdf](https://archive.mpim-bonn.mpg.de/id/eprint/1484/1/preprint_1995_121.pdf)]

* [[Francis Borceux]], [[George Janelidze]], [[Gregory Maxwell Kelly]], p. 7 in: _Internal object actions_, Commentationes Mathematicae Universitatis Carolinae (2005) Volume: 46, Issue: 2, page 235-255 ([dml:249553](https://eudml.org/doc/249553))

Discussion for [[commutative monoids in a symmetric monoidal category]] 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).

* [[Florian Marty]], Sections 1.2, 1.3 in: _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]]]

* [[Martin Brandenburg]], Section 4.1 of: _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]]*

In [[cartesian monoidal categories]]:

* [[John Michael Boardman]], *Algebraic objects in categories*, Chapter 7 of: _Stable Operations in Generalized Cohomology_ [[pdf](https://math.jhu.edu/~wsw/papers2/math/28a-boardman-stable.pdf), [[Boardman-StableOperations.pdf:file]]] in: [[Ioan Mackenzie James]] (ed.) _[[Handbook of Algebraic Topology]]_ Oxford 1995 ([doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7))

and here formalized as [[mathematical structures]] in [[proof assistants]]:

in a context of plain [[Agda]]:

* [[Martín Escardó]], *[The Types of Magmas and Monoids](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html#magmasandmonoids)*, §4 in: *Introduction to Univalent Foundations of Mathematics with Agda* [[arXiv:1911.00580](https://arxiv.org/abs/1911.00580),  [webpage](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html)]

in a context of [[cubical type theory|cubical]] [[Agda]]:

* [[1lab]]: *[Algebra.Monoid](https://1lab.dev/Algebra.Monoid.html)*

[[!redirects monoids in a monoidal category]]
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monoid object > history

< [[monoid object]]


monoid theory - contents

__monoid theory__ in [[algebra]]:

* [[monoid]], [[infinity-monoid]]

* [[monoid object]], [[monoid object in an (infinity,1)-category]]

  * [[semiring]], [[rig]], [[ring]], [[associative unital algebra]]

* [[Mon]], [[CMon]]

* [[monoid homomorphism]]

* [[trivial monoid]]

* [[submonoid]], [[quotient monoid]]

* [[divisor]], [[multiple]], [[quotient element]]

* [[inverse element]], [[unit]], [[irreducible element]]

* [[ideal in a monoid]]

* [[principal ideal in a monoid]]

* [[commutative monoid]]

  * [[tensor product of commutative monoids]]

* [[cancellative monoid]]

* [[GCD monoid]]

* [[unique factorization monoid]]

* [[Bézout monoid]]

* [[principal ideal monoid]]

* [[group]], [[abelian group]]

* [[absorption monoid]]

  * [[zero divisor]]

  * [[integral monoid]]

* [[free monoid]], [[free commutative monoid]]

* [[graphic monoid]]

* [[monoid action]]

* [[module over a monoid]]

* [[localization of a monoid]]

* [[group completion]]

* [[endomorphism monoid]]

* [[super commutative monoid]]

[Edit this sidebar](/nlab/edit/monoid+theory+-+contents)


monoidal (2,1)-category

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#### 2-Category theory
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#Contents#
* table of contents
{:toc}

## Definition

A _monoidal (2,1)-category_ is a [[(2,1)-category]] equipped equivalently with the following structure

* when regarded as a [[2-category]] the structure of a [[monoidal 2-category]];

* when regarded as an [[(∞,1)-category]] the structure of a [[monoidal (∞,1)-category]].

Similarly:

A _symmetric monoidal (2,1)-category_ is a [[(2,1)-category]] equipped equivalently with the following structure

* when regarded as a [[2-category]] the structure of a [[symmetric monoidal 2-category]];

* when regarded as an [[(∞,1)-category]] the structure of a [[symmetric monoidal (∞,1)-category]].

[[!redirects monoidal (2,1)-categories]]

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[[!redirects symmetric monoidal (2,1)-categories]]


monoidal (infinity,1)-category

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#### Higher algebra
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#Contents#
* automatic table of contents goes here
{:toc}

## Idea

The notion of _monoidal $(\infty,1)$-category_ is the analogue of the notion of [[monoidal category]] in the context of [[(infinity,1)-category|(∞,1)-categories]]. Monoidal $(\infty,1)$-categories form an [[(∞,2)-category]], [[Mon(∞,1)Cat]].

There are various ways to state the monoidal structure. One is in terms of fibrations over the [[simplex category]]. This is the approach taken in

*  [[Jacob Lurie]], _Noncommutative algebra_ ([arXiv](http://arxiv.org/abs/math/0702299))

Another is in terms of [[(∞,1)-operad]]s (see there). This approach has been taken in ([Francis](#Francis)).  Both are described below.

Just as many ordinary $(\infty,1)$-categories (particularly, all of those that are [[locally presentable (infinity,1)-category|locally presentable]]) can be presented by [[model categories]], many monoidal $(\infty,1)$-categories can be presented by [[monoidal model categories]].  See for instance [NikolausSagave15](#NikolausSagave15).

### Idea of the simplicial definition

As discussed at the end of the entry on
[[monoidal category]],
an ordinary [[monoidal category]] may be thought of as a lax functor

$$
  * \to \mathbf{B} Cat
$$

from the terminal category to the one-object 3-category whose single
[[hom-object]] is the [[2-category]] [[Cat]] of all categories and for which composition is the [[cartesian monoidal category|cartesian monoidal structure]] on [[Cat]].

More concretely, as also described there, such a lax functor is a
kind of [[descent]] object in
a [[weighted limit]] $lim^{\Delta} const_{\mathbf{B}Cat}$, namely
a diagram

$$
  \array{
    F \Delta^4 &\stackrel{=}{\to}& \mathbf{B} Cat
    \\
    \uparrow && \uparrow^{Id}
    \\
    F \Delta^3 &\stackrel{\alpha}{\to}& \mathbf{B} Cat
    \\
    \uparrow && \uparrow^{Id}
    \\
    F \Delta^2 &\stackrel{\otimes}{\to}& \mathbf{B} Cat
    \\
    \uparrow && \uparrow^{Id}
    \\
    F \Delta^1 &\stackrel{C}{\to}& \mathbf{B}Cat
    \\
    \uparrow && \uparrow^{Id}
    \\
    F\Delta^0 &\stackrel{\bullet}{\to}& \mathbf{B}Cat
  }
$$

where $F \Delta^n$ is the free 3-category of the $n$-[[simplex]]
(an [[oriental]]), where the horizontal morphisms are the chosen data
 -- the category $C$, product $\otimes$, associator $\alpha$ and,
in degree 4, the respect for the pentagon identity --
and the condition is that this commutes for all vertical morphisms
$F(\Delta^n \to \Delta^m)$.

So this is a 4-functor

$$
  F(\Delta^4) \to \mathbf{B}Cat
$$

subject to a certain constraint.

Using the general mechanism of [[generalized universal bundle]]s, this
classifies a [[Cat]]-bundle

$$
  \array{
    C^\otimes \to F(\Delta^4)
  }
  \,.
$$

With a bit more time than I have on the train one can figure out
that conversely suitable such fibrations are equivalent to monoidal
categories. Alternatively, one can read pages 5 and 6 of
_LurieNonCom_ cited below.

In any case, this motivates the following definition.

## Definition

### Plain monoidal $(\infty,1)$-category

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

A **monoidal ($\infty,1$)-category** $(C, \otimes)$
is

* a [[simplicial set]] $C^\otimes$;

* and a [[coCartesian fibration]] of simplicial
sets $ p_\otimes : C^\otimes \to N(\Delta)^{op}$

* such that for each $n \in \mathbb{N}$ the induced [[(infinity,1)-functor]]
$C^\otimes_{[n]} \to C^\otimes_{\{i,i+1\}}$ determines an
equivalence of [[(infinity,1)-category|(infinity,1)-categories]]

$$
  C^\otimes_{[n]}
  \to
  C^\otimes_{\{0,1\}}
  \times
  \cdots
  \times
  C^\otimes_{\{n-1,n\}}
  \simeq
  (C^\otimes_{[1]})^n
  \,.
$$

Here $\Delta$ is the [[simplex category]] and $N(\Delta)$ its [[nerve]].

=--

### $\mathcal{O}$-monoidal $(\infty,1)$-category

The following defines [[symmetric monoidal (∞,1)-categories]] and their variants, where the [[commutative operad]] is replaced by any other [[(∞,1)-operad]].

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

Let $\mathcal{O}^\otimes$ be an [[(∞,1)-operad]]. A **[[coCartesian fibration of (∞,1)-operads]]** is

1. a [[coCartesian fibration]] $p : \mathcal{C}^\otimes \to \mathcal{O}^\otimes$ of the underlying [[quasi-categories]];

1. such that the composite

   $$
     \mathcal{C}^\otimes \stackrel{p}{\to} \mathcal{O}^\otimes \to Fin_* = Comm
   $$

   exhibits $\mathcal{C}^\otimes$ as an [[(∞,1)-operad]].

In this case we say that the underlying [[(∞,1)-category]]

$$
  \mathcal{C}  = \mathcal{C}^\otimes \times_{\mathcal{O}^\otimes} \mathcal{O}
$$

is equipped by $p$ with the [[structure]] of an **$\mathcal{O}$-monoidal $(\infty,1)$-category**.

=--

This is ([Lurie, def. 2.1.2.13](#Lurie)).

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

For $\mathcal{O}$ = [[commutative operad|Comm]], an $\mathcal{O}$-monoidal $(\infty,1)$-category is a [[symmetric monoidal (∞,1)-category]].

=--

## Higher monoidal structure

While for an ordinary [[monoid]] there is just one notion of commutativity (either it is or it is not commutative), already a [[monoidal category]] distinguishes between being just [[braided monoidal category|braided monoidal]] or fully [[symmetric monoidal category|symmetric monoidal]].

This pattern continues, as expressed by the [[k-tuply monoidal n-category|periodic table of k-tuply monoidal categories]].

A [[higher category theory|higher category]] may be a [[k-tuply monoidal n-category]] or more generally [[k-tuply monoidal (n,r)-category]] for different values of $k$. The lowest value of $k= 1$ (since for $k = 0$ there is no monoidal structure at all) corresponds to monoidal product which is $\infty$-associative, i.e.  associative up to higher coherent homotopies, but need not have any degree of _commutativity_.

One says that an $n$-category is _symmetric monoiodal_ if it is "as monoidal as possible", i.e. $\infty$-tuply monoidal. In particular, in [[higher algebra|Noncommutative algebra]] and [[higher algebra|Commutative algebra]] we have

* the 1-fold monoidal (∞,1)-categories described here;

* [[symmetric monoidal (infinity,1)-category|∞-tuply monoidal (∞,1)-categories]].

### Operadic/algebraic definition of monoidal structure

For each $1 \leq n \leq \infty$ let $E_n$ denote the [[little n-disk operad|little n-disk]] [[operad]] whose [[topological space]] of $E_n^k$ of $k$-ary operations is the space of embeddings of $k$ $n$-dimensional disks (balls) in one $n$-dimensional disk without intersection, and whose composition operation is the obvious one obtained from gluing the big outer disks into given inner disks.

In [John Francis' PhD thesis](#Francis)  the theory of [[(∞,1)-categories]] equipped with an action of the $E_n$-[[operad]] is established, so that

* $(\infty,1)$-categories with an $E_1$-action are precisely [[monoidal (∞,1)-categories]]  -- 1-fold monoidal $(\infty,1)$-categories;

* $(\infty,1)$-categories with an $E_\infty$-action are precisely [[symmetric monoidal (∞,1)-categories]] -- $\infty$-tuply monoidal $(\infty,1)$-categories;

* $(\infty,1)$-categories with an $E_n$-action for $1 \lt n \lt \infty$ are the corresponding $n$-tuply monoidal $(\infty,1)$-categories in between.

>**Remark** The second statement is example 2.3.8 in [EnAction](#Francis). The first seems to be clear but is maybe not in the literature. Jacob Lurie is currently rewriting [[higher algebra|Higher Algebra]] such as to build in a discussion of $E_n$-operadic structures in the definition of $k$-tuply monoidal $(\infty,1)$-categories.

## Related concepts

* [[monoidal category]], [[monoidal (2,1)-category]], **monoidal $(\infty,1)$-category**

  * [[monoidal (∞,1)-functor]]

  * [[Picard ∞-group]]

* [[braided monoidal category]], [[braided monoidal (∞,1)-category]]

  * [[braided ∞-group]]

* [[symmetric monoidal category]], [[symmetric monoidal (2,1)-category]], [[symmetric monoidal (∞,1)-category]]

  * [[abelian ∞-group]]

* [[closed monoidal category]] ,  [[closed monoidal (∞,1)-category]]

* [[prime spectrum of a monoidal stable (∞,1)-category]]

## References

The simplicial definition for plain monoidal $(\infty,1)$-categories is definition 1.1.2 in

* [[Jacob Lurie]], _[[Noncommutative Algebra]]_ ([arXiv](http://arxiv.org/abs/math/0702299))

John Francis' work on [[little cubes operad]]-actions on  $(\infty,1)$-categories is in

* [[John Francis]], PhD thesis ([web](http://dspace.mit.edu/handle/1721.1/43792))
{#Francis}

The general notion of an $\mathcal{O}$-monoidal $(\infty,1)$-category is around definition 2.1.2.13 of

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

An introductory survey is in

* [[Moritz Groth]], _A short course on $\infty$-categories_ ([pdf](http://www.math.uni-bonn.de/~mgroth/InfinityCategories.pdf))

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

* {#NikolausSagave15} [[Thomas Nikolaus]], [[Steffen Sagave]], _Presentably symmetric monoidal infinity-categories are represented by symmetric monoidal model categories_. Algebr. Geom. Topol. 17 (2017), no. 5, 3189--3212. ([arXiv:1506.01475](http://arxiv.org/abs/1506.01475))

[[!redirects monoidal (infinity,1)-categories]]
[[!redirects monoidal (∞,1)-category]]
[[!redirects monoidal (∞,1)-categories]]


monoidal (infinity,1)-functor

## Idea

An [[(∞,1)-functor]] that is a [[homomorphism]] of [[monoidal (∞,1)-categories]].

## Examples

* [[smashing localization]]

* [[completion of a module]]

## Related concepts

* [[monoidal functor]]

* **monoidal $(\infty,1)$-functor**

* [[monoidal (∞,n)-functor]]

[[!redirects monoidal (∞,1)-functors)]]
[[!redirects monoidal (infinity,1)-functors]]

[[!redirects monoidal (∞,1)-functor]]


monoidal (infinity,n)-functor

[[!redirects monoidal (∞,n)-functor]]

## Idea

An [[(∞,n)-functor]] that is a [[homomorphism]] of [[monoidal (∞,n)-categories]].

## Related concepts

* [[monoidal functor]]

* [[monoidal (∞,1)-functor]]

* **monoidal $(\infty,n)$-functor**

[[!redirects monoidal (∞,n)-functors]]
[[!redirects monoidal (infinity,n)-functors]]


monoidal (∞,1)-functors

The analog in [[(∞,1)-category]] of [[monoidal functors]], now going between [[monoidal (∞,1)-categories]].


monoidal action of a monoidal category

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

# Contents
* table of contents
{:toc}

## Idea
A **monoidal actegory** is a [[pseudoaction]] in the [[2-category]] $\mathbf{MonCat}$ of [[monoidal categories]], [[strong monoidal functors]] and [[monoidal natural transformations]].
Roughly, it is an [[actegory]] whose _actee_ is monoidal and such that the action is compatible with this structure.

## Definition
Since pseudoactions are actions of [[pseudomonoids]], and since a pseudomonoid in $\mathbf{MonCat}$ is a [[braided monoidal category]], monoidal actegories are always actions of braided monoidal categories.

The following comes from Definition 5.1.1 in ([Capucci and Gavranović 2022](#CG22)):

\begin{definition}
Let $(\mathcal{M}, j, \otimes)$ be a braided monoidal category.
A left *monoidal $\mathcal{M}$-actegory* is a monoidal category $(\mathcal{C}, i,\boxtimes)$ together with a strong monoidal functor, called **action**, $\odot:\mathcal{M} \times \mathcal{C} \to \mathcal{C}$, and monoidal natural transformations $\eta_x : j \odot x \to x$, $\mu_{m,n,x} : (m \otimes n) \odot x \to m \odot (n \odot x)$ satisfying the coherence laws of unitor and multiplicator of an [[actegory]].
\end{definition}

All the coherence laws of a monoidal actegory are spelled out in Appendix A of ([Capucci and Gavranović 2022](#CG22)).

The fact $\odot$ is strong monoidal means it comes equipped with a structural morphism called **mixed interchanger**:
\[
\iota_{m,x,n,y} : (m \otimes n) \odot (x \boxtimes y) \to (m \odot x) \boxtimes (n \odot y).
\]
It would also come with $\upsilon : i \to j \odot i$, but the monoidality of $\eta$ makes it so that $\upsilon = \eta_i$, making the first redundant.

### Braiding
Consider pseudoactions in $\mathbf{BrMonCat}$ and $\mathbf{SymMonCat}$. Now carriers of the action are braided functors, and the actee are braided monoidal categories.

The following appear as 5.4.1 and 5.4.3 in ([Capucci and Gavranović 2022](#CG22)):

\begin{definition}
Let $(\mathcal{M}, j, \otimes, \sigma)$ be a symmetric monoidal category.
A left **braided monoidal $\mathcal{M}$-actegory** is a braided monoidal category $(\mathcal{C}, i, \boxtimes, \beta)$ equipped with a braided monoidal functor $\odot: \mathcal{M} \times \mathcal{C} \to \mathcal{C}$ and two monoidal natural transformations $\eta$ and $\mu$ as above.
Thus it satisfies the axioms of monoidal actegory and an additional compatibility axiom with the braided structure of $\mathcal{M}$ and $\mathcal{C}$:

\begin{tikzcd}
	{(m \otimes n) \odot (x \boxtimes y)} && {(m \odot x) \boxtimes (n \odot y)} \\
	{(n \otimes m) \odot (y \boxtimes x)} && {(n \odot y) \boxtimes (m \odot x)}
	\arrow["\iota", from=1-1, to=1-3]
	\arrow["\beta", from=1-3, to=2-3]
	\arrow["\iota"', from=2-1, to=2-3]
	\arrow["{\sigma \odot \beta}"', from=1-1, to=2-1]
\end{tikzcd}

If $\mathcal{C}$ is symmetric, then we call $(\mathcal{C}, \odot)$ a **symmetric monoidal actegory**.
\end{definition}

## Classifying objects
Actegories are famously equivalent to monoidal functors $\mathcal{M} \to [\mathcal{C},\mathcal{C}]$. We can thus say that $[\mathcal{C},\mathcal{C}]$ *classifies* actions on $\mathcal{C}$.

When the action is monoidal, braided or symmetric monoidal, the classifier is different:

* Monoidal actions are classified by $Z(\mathcal{C})$, i.e. the [[Drinfeld center]] of $\mathcal{C}$.
* Braided monoidal actions are classified by $\Sigma(\mathcal{C})$, i.e. the [[symmetric center]] of $\mathcal{C}$, which is the full subcategory of $\mathcal{C}$ spanned by those objects $x:\mathcal{C}$ such that $\beta_{x,-}\beta_{-,x} = 1$.
* Symmetric monoidal actions are classified by $\mathcal{C}$ itself. In fact a symmetric monoidal action $\odot$ of $\mathcal{M}$ on $\mathcal{C}$ gives a symmetric monoidal functor $- \odot i : \mathcal{M} \to \mathcal{C}$, and viceversa $F(-) \boxtimes -$ is a symmetric monoidal action whenever $\mathcal{M} \to \mathcal{C}$ is a [[braided monoidal functor]] between [[symmetric monoidal categories]].

This is all proven in Section 5.5 of [Capucci & Gavranović 2022](#CG22).

## See also
* [[actegories]], [[pseudoaction]]
* [[para construction]] ($\mathbf{Para}(\odot)$ is monoidal iff $\odot$ is)
* [[periodic table]]

## References

* {#CG22} [[Matteo Capucci]], [[Bruno Gavranović]], _Actegories for the working amthematician_ [[arXiv:2203.16351](https://arxiv.org/abs/2203.16351)]

[[!redirects monoidal actions of a monoidal category]]
[[!redirects monoidal actions of monoidal categories]]

[[!redirects monoidal actegory]]
[[!redirects braided actegory]]
[[!redirects symmetric actegory]]

[[!redirects braided monoidal actegory]]
[[!redirects symmetric monoidal actegory]]

[[!redirects monoidal action]]
[[!redirects braided monoidal action]]
[[!redirects symmetric monoidal action]]

[[!redirects monoidal actegories]]
[[!redirects braided monoidal actegories]]
[[!redirects symmetric monoidal actegories]]


monoidal adjunction

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

#Contents
* table of contents
{:toc}

## Idea

A **monoidal adjunction** is an [[adjunction]] between [[monoidal categories]] which respects the monoidal structure.

Since there are several types of [[monoidal functors]] (lax, colax, and strong) there are several types of "adjunctions between monoidal categories which respect the monoidal structure."  Namely, we could have:

* An adjunction in the [[2-category]] [[MonCat]] of [[monoidal categories]] and [[strong monoidal functors]].  In this case both the left and right adjoint are strong.

  $$
    \array{
      \underoverset
        {\underset{R \, \text{strong monoidal}}{\longrightarrow}}
        {\overset{L \, \text{strong monoidal}}{\longleftarrow}}
        {}
    }
  $$

  We call this a **strong monoidal adjunction**.

* {#MonoidalAdjnctionIdea} An adjunction in the 2-category [[MonCat]]${}_\ell$ of monoidal categories and [[lax monoidal functors]].  In this case the right adjoint is lax, while the left adjoint is necessarily strong (by [[doctrinal adjunction]]; see [here](/nlab/show/doctrinal+adjunction#strength)).

  $$
    \array{
      \underoverset
        {\underset{R \, \text{lax monoidal}}{\longrightarrow}}
        {\overset{L \, \text{strong monoidal}}{\longleftarrow}}
        {}
    }
  $$

  In fact, since the [[right adjoint]] of an [[oplax monoidal functor]] is necessarily a [[lax monoidal functor]] ([this prop.](oplax+monoidal+functor#OplaxAdjointToLax)), it is sufficient to demand that $L$ be strong monoidal.

  This version, which is one of the most frequently occurring, is often called simply a **monoidal adjunction**.

* The dual: an adjunction in the 2-category $MonCat_c$ of monoidal categories and colax monoidal functors, in which case the left adjoint is colax and the right adjoint is strong.  One might call this an **opmonoidal adjunction**.

* A mixed situation, in which the left adjoint is colax, the right adjoint is lax, and the lax and colax structure maps are [[mates]] under the adjunction.  This is a [[conjunction]] in the [[double category]] of monoidal categories and lax and colax monoidal functors, so we may call it a **monoidal conjunction** or a **lax/colax monoidal adjunction**.  By [[doctrinal adjunction]], given any adjunction between monoidal categories, if the right adjoint is lax monoidal, then the left adjoint automatically acquires a colax monoidal structure making the adjunction into a monoidal conjunction, and dually.

## Details
 {#Details}

As mentioned [above](#MonoidalAdjnctionIdea), the nature of monoidal adjunctions follows as a special case from generalities of [[doctrinal adjunctions]]. For the record, here is an explicit discussion:

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

Let

$$
  \mathcal{C}
  \underoverset
    {\underset{R}{\longrightarrow}}
    {\overset{L}{\longleftarrow}}
    {\bot}
  \mathcal{D}
$$

be a pair of [[adjoint functors]] between [[monoidal categories]], such that the [[left adjoint]] $L$ is a [[strong monoidal functor]] by [[natural isomorphisms]]

$$
  \mu_L(X,Y)
  \;\colon\;
  L(X) \otimes L(Y) \overset{\simeq}{\longrightarrow} L(X \otimes Y)
$$

and

$$
  e_L
    \;\colon\;
  1 \overset{\simeq}{\longrightarrow} L(1)
  \,.
$$

Then

1. the [[right adjoint]] $R$ becomes a [[lax monoidal functor]] via natural morphisms

   $$
     \mu_R(X,Y)
     \;\colon\;
     R (X) \otimes R(Y)
      \overset{\eta(R(X) \otimes R(Y))}{\longrightarrow}
     R L (R(X) \otimes R(Y))
      \underoverset{}{ R( {\mu_L^{-1}(R(X), R(Y))} ) }{\longrightarrow}
     R ( L  R(X) \otimes L R (Y) )
       \overset{R( \epsilon(X) \otimes \epsilon(Y) )}{\longrightarrow}
     R(X \otimes Y)
   $$

   and

   $$
     e_R
     \;\colon\;
     1
       \overset{\eta(1)}{\longrightarrow}
     R L(1)
       \overset{R(e_L^{-1})}{\longrightarrow}
     R(1)
     \,,
   $$

   where $\eta$ denotes the [[adjunction unit]] and $\epsilon$ denotes the [[adjunction counit]], as usual.

1. For any object $A \in \mathcal{D}$ carrying the structure of a [[monoid object]] $(A, \mu_A, e_A)$, then

   1. the [[unit of the adjunction]] $\eta(A) \;\colon\; A \longrightarrow R L(A)$ is a monoid [[homomorphism]] with respect to the canonically induced monoid structure on $R L(A)$ ([this prop.](monoidal+functor#MonoidsToMonoidsByLaxMonoidal)) given by

      $$
        \mu_{R L(A)}
          \;\colon\;
        R L(A) \otimes R L(A)
          \overset{\mu_R(L(A))}{\longrightarrow}
        R( L(A) \otimes L(A))
          \overset{R(\mu_L(A))}{\longrightarrow}
        R L(A \otimes A)
          \overset{R L(\mu_A)}{\longrightarrow}
        R L(A)
      $$

      and

      $$
        e_{R L(A)}
          \;\colon\;
        1
          \overset{e_R}{\longrightarrow}
        R(1)
          \overset{R(e_L)}{\longrightarrow}
        R L(1)
          \overset{R L(e_A)}{\longrightarrow}
        R L(A)
      $$

   1. similarly for the [[counit of the adjunction]].

=--

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

The first statement is discussed at _[[oplax monoidal functor]]_.

For the second statement, we need first need to check that the following [[commuting square|square commutes]]:

$$
  \array{
    A \otimes A
     &\overset{\eta(A) \otimes \eta(A)}{\longrightarrow}&
    R L (A) \otimes R L (A)
    \\
    {}^{\mathllap{\mu_A}}\downarrow
      &&
    \downarrow^{\mathrlap{\mu_{R L(A)}}}
    \\
    A &\underset{\eta(A)}{\longrightarrow}& R L(A)
  }
$$

Now by definition, the composite of the top and right morphism here
is the total diagonal composite in the following diagram:

$$
  \array{
  && A \otimes A
  &\overset{\eta(A \otimes A)}{\longrightarrow}&
  R L(A \otimes A)
  &\overset{R(\mu_L^{-1})}{\longrightarrow}&
  R( L(A) \otimes L(A) )
  \\
  &&
    {}^{\mathllap{\eta(A) \otimes \eta(A)}}\downarrow && {}^{\mathllap{ R L( \eta(A) \otimes \eta(A) ) }}\downarrow
  &&
    {}^{\mathllap{R( L(\eta(A))  \otimes L(\eta(A)) )}}\downarrow
  & \searrow^{\mathrlap{R(id \otimes id)}}
  \\
  \mu_R(L(A))
  &\colon&
  R L (A) \otimes R L ( A)
   &\overset{\eta(R L (A) \otimes R L (A))}{\longrightarrow}&
  R L (R L ( A) \otimes R L (A))
   &\underoverset{}{ R( {\mu_L^{-1}(R L(A))} ) }{\longrightarrow}&
  R ( L  R L (A) \otimes L R L (A) )
    &\overset{R( \epsilon(L(A)) \otimes \epsilon(L(A)) )}{\longrightarrow}&
  R(L(A) \otimes L(A))
  \\
  && && && && \downarrow^{\mathrlap{ R( \mu_L(A)) }}
  \\
  && && && && R L( A \otimes A )
  \\
  && && && && \downarrow^{\mathrlap{ R L (\mu_A) }}
  \\
  && && && && R L (A)
 }
$$

Here the top sqaures commute by naturality of $\eta$ and $\mu_L$,
and top right diagonal morphism is the [[identity morphism]], as shown, by the [[zig-zag identity]] for the adjunction $(L \dashv R)$. Therfore $R(\mu_L^{-1})$ cancels agains $R(\mu_L)$. so that the composite morphism in question becomes just $A \otimes A \overset{\eta(A \otimes A)}{\longrightarrow} R L(A \otimes A) \overset{ R L(\mu_A) }{\longrightarrow} R L(A)$. Again by the naturality of the [[adjunction unit]] $\eta$

$$
  \array{
    A \otimes A
      &\overset{\eta(A \otimes A)}{\longrightarrow}&
    R L (A \otimes A)
    \\
    {}^{\mathllap{ \mu_A }}\downarrow
      &&
    \downarrow^{\mathrlap{R L(\mu_A)}}
    \\
    A &\underset{\eta(A)}{\longrightarrow}& R L (A)
  }
$$

this equals $\eta(A) \circ \mu_A$, as required.

Finally we need to check that the following diagram commutes:

$$
  \array{
     && 1
     \\
     & {}^{\mathllap{e_A}}\swarrow && \searrow^{\mathrlap{ e_{R L (A)} }}
     \\
     A
     &&
       \underset{\eta(A)}{\longrightarrow}
     &&
    R L(A)
  }
$$

Now unwinding the above definitions of $e_{R L}(A)$ in terms of the definition of $e_R$ we find that

$$
  e_{R L (A)}
  \;\colon\;
  1
    \overset{\eta(1)}{\longrightarrow}
  R L(1)
    \overset{R(e_L^{-1})}{\longrightarrow}
  R(1)
  \overset{R(e_L)}{\longrightarrow} R L(1) \overset{R L(e_a)}{\longrightarrow} R L(A)
  \,.
$$

Here the two morphisms in the middle cancel, so that we are left just with

$$
  e_{R L(A)}
  \;\colon\;
  1
    \overset{\eta(1)}{\longrightarrow}
  R L(1)
   \overset{R L(e_A)}{\longrightarrow}
  R L (A)
  \,.
$$

That this equals $\eta(A)\circ e_A$, as required, follows by the naturality of $\eta$:

$$
  \array{
    1
      &\overset{\eta(1)}{\longrightarrow}&
    R L(1)
    \\
    {}^{\mathllap{e_A}}\downarrow && \downarrow^{\mathrlap{R L(e_A)}}
    \\
    A &\underset{\eta(A)}{\longrightarrow}& R L(A)
  }
  \,.
$$

The argument for the homomorphism property of the counit should be formally dual to the above.

=--

## Examples

+-- {: .num_example #Stabilization}
###### Example
**([[stabilization]] in [[stable homotopy theory]])**

The [[stabilization]] adjunction

  $$
    Ho(Spectra)
     \underoverset
       {\underset{\Omega^\infty}{\longrightarrow}}
       {\overset{\Sigma^\infty(-)}{\longleftarrow}}
       {\bot}
    Ho(Spaces^{\ast/})
       \underoverset
         {\underset{}{\longrightarrow}}
         {\overset{(-)_+}{\longleftarrow}}
         {}
    Ho(Spaces)
  $$

  between the [[classical homotopy categories]] $Ho(Spaces)$ and $Ho(Spaces^{\ast/})$ of ([[pointed topological space|pointed]]) [[topological spaces]] and the [[stable homotopy category]] $Ho(Spectra)$ is a monoidal adjunction, since the [[left adjoint]] $\Sigma^\infty(-)_+$ (forming the [[suspension spectrum]] of a space after freely [[pointed topological space|adjoining a basepoint]]) is [[strong monoidal functor|strong monoidal]] with respect to forming [[product topological spaces]] and forming [[smash  product of spectra]], respectively.

  In fact this is the [[derived functors]] of what is even a [[monoidal Quillen adjunction]] between the [[classical model structure on topological spaces]] and the stable [[model structure on orthogonal spectra]] ([this cor.](Introduction+to+Stable+homotopy+theory+--+1-2#StableMonoidalQuillenSuspensionSpectrumFunctor)), which implies (strong) monoidality of the derived functors on homotopy categories ([this prop.](Introduction+to+Stable+homotopy+theory+--+1-2#StrongMonoidalDerivedFunctorFromStrongMonoidalQuillenAdjunction)).

In detail, let

$$
  (L \dashv R)
   \;\colon\;
  \mathbb{S}_{orth}Mod_{stable}
     \underoverset
      {\underset{\Omega^\infty_{orth}}{\longrightarrow}}
      {\overset{\Sigma^\infty_{orth}}{\longleftarrow}}
      {\bot}
  Top^{\ast/}_{Quillen}
    \underoverset
       {\underset{}{\longrightarrow}}
       {\overset{(-)_+}{\longleftarrow}}
       {\bot}
  Top_{Quillen}
$$

be the Quillen adjunction on orthogonal spectra ([here](Introduction+to+Stable+homotopy+theory+--+1-2#SuspensionSpectrumStructuredStrictQuillenAdjunction)).
The left adjoint $L$ is a strong monoidal functor, and hence so is its [[derived functor]] $\Sigma^\infty(-)_+ \colon Ho(Top) \to Ho(Spectra)$ (by [this prop.](Introduction+to+Stable+homotopy+theory+--+1-2#StrongMonoidalDerivedFunctorFromStrongMonoidalQuillenAdjunction)).

{#DerivedLaxMonoidalStructure} We want to see that the structure of a lax monoidal functor which is induced on the derived right adjoint $\Omega^\infty(-) \colon Ho(Top) \to Ho(Spectra)$ via prop. \ref{MonoidalAdjunctionElementary} is the expected one, given on [[Omega-spectra]] $X$ and $Y$ by

$$
  \Omega^\infty(X) \wedge \Omega^\infty(X)
    =
   X_0 \wedge Y_0
    \overset{}{\to}
   (X \wedge Y)_0
    =
   \Omega^\infty( X \wedge Y )
  \,.
$$

To see this, observe that if $X$ and $Y$ are [[CW-spectrum|CW-]][[Omega-spectra]] and hence cofibrant and fibrant in $\mathbb{S}_{orth}Mod_{stable}$ then the derived lax monoidal structure is given by the total bottom composite in the following diagram
$$
  \array{
     R (X) \otimes R(Y)
      &\overset{\eta(R(X) \otimes R(Y))}{\longrightarrow}&
     R L (R(X) \otimes R(Y))
      &\underoverset{}{ R( {\mu_L^{-1}(R(X), R(Y))} ) }{\longrightarrow}&
     R ( L  R(X) \otimes L R (Y) )
       &\overset{R( \epsilon(X) \otimes \epsilon(Y) )}{\longrightarrow}&
     R(X \otimes Y)
     \\
     &{}_{\mathllap{ \text{derived } \atop {\text{adjunction unit}} }}\searrow&
     \downarrow^{\mathrlap{R j L (R(X) \otimes R(Y))}}
     &&
     \downarrow^{\mathrlap{R j ( L R (X) \otimes L R(Y) )}}
     &&
     \downarrow^{\mathrlap{ R j( X \otimes Y ) }}
     \\
      &&
     R P L (R(X) \otimes R(Y))
      &\underoverset{}{ R Q( {\mu_L^{-1}(R(X), R(Y))} ) }{\longrightarrow}&
     R P ( L  R(X) \otimes L R (Y) )
       &\overset{R Q( \epsilon(X) \otimes \epsilon(Y) )}{\longrightarrow}&
     R P (X \otimes Y)
  }
  \,,
$$

where we write for brevity $(L \dashv R) \coloneqq (\Sigma^\infty_{orth} \dashv \Omega^\infty_{orth})$, and  where $j \colon id \longrightarrow P$ denotes [[functorial factorization|functorial]] [[fibrant replacement]] (which exists since the [[small object argument]] applies in $\mathbb{S}_{orth}Mod_{stable}$). By functoriality of the replacement, all the squares commute, so that the derived lax monoidal structure on CW-Omega spectra is seen to be equivalently the underived one.

But that underived top horizontal composite is manifestly just the canonical isomorphism $X_0 \wedge Y_0 \simeq (X \wedge Y)_0$ (since $R$ simply picks the component space in degree-0  and $L$ preserves the component space in degree 0).

=--

+-- {: .num_example #ExponentialModality}
###### Example
**([[exponential modality]] in [[linear type theory]])**

In [[linear type theory]] (see there for more) the [[exponential modality]] $!$ may have [[categorical semantics]] as the [[comonad]] induced by a monoidal adjunction.

=--

## Related concepts

* [[projection formula]]

* [[enriched adjunction]]

[[!redirects monoidal adjunctions]]
[[!redirects opmonoidal adjunction]]
[[!redirects opmonoidal adjunctions]]
[[!redirects comonoidal adjunction]]
[[!redirects comonoidal adjunctions]]
[[!redirects monoidal conjunction]]
[[!redirects monoidal conjunctions]]

[[!redirects strong monoidal adjunction]]
[[!redirects strong monoidal adjunctions]]


monoidal bicategory

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

#Contents#
* table of contents
{:toc}

## Idea

A **monoidal bicategory** is a [[bicategory]] with a [[monoidal category|monoidal structure]], which is up-to-equivalence in a suitable bicategorical sense.  A concise definition is that a monoidal bicategory is a [[tricategory]] with one object.  Just as every tricategory is equivalent to a [[Gray-category]], every monoidal bicategory is equivalent to a **Gray-monoid**, i.e. a monoid in the monoidal category [[Gray]].

Just as monoidal categories also come in [[braided monoidal category|braided]] and [[symmetric monoidal category|symmetric]] versions, monoidal bicategories have *three* extra levels of commutativity (see the [[periodic table]] and the [[stabilization hypothesis]]):

* [[braided monoidal 2-categories]]
* [[sylleptic monoidal 2-categories]]
* [[symmetric monoidal 2-categories]]

There are also 2-categorical variants of other structures on monoidal 1-categories, such as:

* [[closed monoidal 2-category]]
* [[balanced monoidal 2-category]]
* [[compact closed 2-category]]

## Examples {#Examples}

+-- {: .un_example}
###### Example
For $R$ a commutative [[ring]], there is a [[symmetric monoidal bicategory]] $Alg(R)$ whose

* [[object]]s are $R$-[[algebra]]s;
* [[morphism]]s are algebra [[bimodule]]s;
* [[2-morphisms|k-morphism]] are bimodule homomorphisms.

The monoidal product is given by [[tensor product]] over $R$.
=--

By [[delooping]] this once, this gives an example of a [[tricategory]] with a single object.  The tricategory statement follows from theorem 21 of

* [[Richard Garner]], [[Nick Gurski]], _The low-dimensional structures that tricategories form_ ([arXiv:0711.1761](http://arxiv.org/abs/0711.1761))

This, and that the monoidal bicategory is even symmetric monoidal is given by the main theorem in

* [[Mike Shulman]], _Constructing symmetric monoidal bicategories_ ([arXiv:1004.0993](http://arxiv.org/abs/1004.0993))

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

For $V$ a cocomplete [[closed monoidal category|closed]] [[symmetric monoidal category]], there is a symmetric (indeed compact closed) monoidal bicategory $V Prof$ whose objects are small $V$-[[enriched categories]] and whose morphisms are $V$-enriched [[profunctors]].

=--

## Internalization

Monoidal bicategories provide a fruitful context for examples of the [[microcosm principle]]; various kinds of [[monoidal category]] can naturally be generalized to corresponding kinds of [[pseudomonoids]] internal to corresponding kinds of monoidal bicategories.  In some cases this is very immediate:

* Inside a monoidal bicategory one can define a [[pseudomonoid]], which when specialized to $Cat$ yields a [[monoidal category]].
* Inside a braided monoidal bicategory one can define a [[braided pseudomonoid]], which when specialized to $Cat$ yields a [[braided monoidal category]].
* Inside a sylleptic monoidal bicategory one can define a [[symmetric pseudomonoid]], which when specialized to $Cat$ yields a [[symmetric monoidal category]].
* Inside a balanced monoidal bicategory one can define a [[balanced pseudomonoid]], which  when specialized to $Cat$ yields a [[balanced monoidal category]].

Of course, these also specialize to the appropriate kinds of monoidal categories when specialized to monoidal 2-categories of [[enriched categories]], [[internal categories]], etc.

But for fancier kinds of monoidal structure, we need instead to consider monoidal bicategories like [[Prof]] instead.  An ordinary pseudomonoid in $Prof$ specializes to a [[promonoidal category]], but ordinary monoidal categories can be regarded as particular "representable" promonoidal ones.  In terms of the bicategory $Prof$, a [[Cauchy-complete]] monoidal category can be identified with a [[map pseudomonoid]], i.e. a pseudomonoid whose structure 1-morphisms are [[bicategory of maps|maps]] (left adjoints).  The non-Cauchy-complete case can be dealt with using a monoidal [[proarrow equipment]] instead.  Of course, this also generalizes to bicategories of enriched profunctors, internal profunctors, etc.

Having passed from $Cat$ to $Prof$, we can now define more kinds of pseudomonoids.  Note that $Prof$ has more structure than being merely (symmetric) monoidal: it is a [[compact closed bicategory]], i.e. all objects have duals.

* Inside any compact closed bicategory one can define a [[closed pseudomonoid]].  A representable closed pseudomonoid in $Prof$ specializes to a [[closed monoidal category]].
* Inside any compact closed bicategory one can define a [[star-autonomous pseudomonoid]] (a.k.a. a "Frobenius pseudomonoid", since it also categorifies a [[Frobenius algebra]]).  A representable star-autonomous pseudomonoid in $Prof$ specializes to a [[star-autonomous category]].
* Inside any compact closed bicategory one can define a [[compact closed pseudomonoid]].  A representable compact closed pseudomonoid in $Prof$ specializes to a [[compact closed category]].

In fact compact closedness of the whole monoidal bicategory is not necessary to assume; it suffices to assume only that the pseudomonoid itself has a dual.  (And in the star-autonomous case, this is actually a *consequence* of the definition.)

All the above definitions can be found in [Day-Street 97](#DS97), except for the star-autonomous case which is in [Day-Street 03](#DS03) and [Street 04](#Street04).

## Coherence theorems

* [[coherence theorem for monoidal bicategories]]
* [[coherence theorem for braided monoidal bicategories]]
* [[coherence theorem for symmetric monoidal bicategories]]

## Related concepts

* [[braided monoidal 2-category]]

* [[sylleptic monoidal 2-category]]

* [[symmetric monoidal 2-category]]

* [[Picard 3-group]]

* [[compact closed 2-category]]

* [[monoidal category]]

* [[monoidal (2,1)-category]]

* [[monoidal double category]]

* [[monoidal (∞,1)-category]]

* [[monoidal (∞,n)-category]]

* [[module 2-category]]

## References

Slightly different definitions of these various structures can be found in the following sequence of papers:

Definition of *braided Gray monoids*, i.e. [[Gray-category]]-monoids with a [[braiding]]:

* [[Mikhail Kapranov]], [[Vladimir Voevodsky]], *2-Categories and Zamolodchikov tetrahedra equations*, in: _Proc. Symp. Pure Math._ **56** Part 2 (1994), AMS, Providence, pp. 177--260 ([doi:10.1090/pspum/056.2](https://doi.org/10.1090/pspum/056.2))

* [[Mikhail Kapranov]], [[Vladimir Voevodsky]], *Braided monoidal 2-categories and Manin-Schechtman higher braid groups*, Journal of Pure and Applied Algebra Volume 92, Issue 3, 25 March 1994, Pages 241-267 (doi:10.1016/0022-4049(94)90097-3)

Correction of the KV definition by adding one axiom:

* [[John C. Baez]], [[Martin Neuchl]], *Higher-Dimensional Algebra I: Braided Monoidal 2-Categories*, Adv. Math. 121 (1996), 196-244 ([arXiv:q-alg/9511013](https://arxiv.org/abs/q-alg/9511013))

A definition of braided Gray-monoid equivalent to Baez-Neuchl, also defining [[sylleptic monoidal 2-category|sylleptic]] and [[symmetric monoidal 2-category|symmetric]] Gray-monoids:

* {#DS97}  [[Brian Day]], [[Ross Street]], *Monoidal Bicategories and Hopf Algebroids*, Advances in Mathematics Volume 129, Issue 1, 15 July 1997, Pages 99-157 ([doi:10.1006/aima.1997.1649](https://doi.org/10.1006/aima.1997.1649))

Further modification of the definition by adding an axiom relating to the tensor unit:

* [[Sjoerd E. Crans]], *Generalized Centers of Braided and Sylleptic Monoidal 2-Categories*, Advances in Mathematics, Volume 136, Issue 2, 25 June 1998, Pages 183-223 ([doi:10.1006/aima.1998.1720](https://doi.org/10.1006/aima.1998.1720))

* Paddy McCrudden, in "Balanced coalgebroids," defined braided and sylleptic monoidal *bicategories* (not Gray monoids).

A full definition of [[braided monoidal 2-category|braided]], [[sylleptic monoidal 2-category|sylleptic]], and [[symmetric monoidal 2-category|symmetric]] monoidal bicategories and assembing into a [[tricategory]]:

* [[Chris Schommer-Pries]], [Ph. D. thesis](https://arxiv.org/abs/1112.1000),

A [[rectification|strictification]] theorem relating all these definitions, along with a [[coherence theorem]] for the [[braided monoidal 2-category|braided]] case:

* [[Nick Gurski]], *Loop spaces, and coherence for monoidal and braided monoidal bicategories*, Advances in Mathematics Volume 226, Issue 5, 20 March 2011, Pages 4225-4265 ([arXiv:1102.0981](http://arxiv.org/abs/1102.0981), [doi:10.1016/j.aim.2010.12.007](https://doi.org/10.1016/j.aim.2010.12.007))

A [[coherence theorem|coherence]] and [[rectification|strictification]] theorem for the symmetric case (but the sylleptic case is perhaps still open):

* [[Nick Gurski]], [[Angélica M. Osorno]], *Infinite loop spaces, and coherence for symmetric monoidal bicategories*, Adv. Math. 246 (2013) 1-32 ([arXiv:1210.1174](https://arxiv.org/abs/1210.1174))

See also:

* {#DS03} [[Brian Day]] and [[Ross Street]], "Quantum categories, star autonomy, and quantum groupoids", 2003 ([arxiv:0301209](https://arxiv.org/abs/math/0301209))

* {#Street04} [[Ross Street]], *Frobenius monads and pseudomonoids* (2004) [[pdf](http://web.science.mq.edu.au/~street/Frob.pdf)]

  > (on [[Frobenius monads]])

[[!redirects Gray monoid]]
[[!redirects Gray-monoid]]
[[!redirects monoidal bicategories]]
[[!redirects Gray monoids]]
[[!redirects Gray-monoids]]

[[!redirects monoidal 2-category]]
[[!redirects monoidal 2-categories]]


monoidal categories - contents

**[[monoidal categories]]**

* [[enriched monoidal category]], [[tensor category]]

* [[string diagram]], [[tensor network]]

**With braiding**

* [[braided monoidal category]]

* [[balanced monoidal category]]

  * [[twist]]

* [[symmetric monoidal category]]

**With duals for objects**

* [[category with duals]] (list of them)

* [[dualizable object]] (what they have)

* [[rigid monoidal category]], a.k.a. [[autonomous category]]

* [[pivotal category]]

* [[spherical category]]

* [[ribbon category]], a.k.a. [[tortile category]]

* [[compact closed category]]

**With duals for morphisms**

* [[monoidal dagger-category]]

* [[symmetric monoidal dagger-category]]

* [[dagger compact category]]

**With traces**

* [[trace]]

* [[traced monoidal category]]

**Closed structure**

* [[closed monoidal category]]

* [[cartesian closed category]]

* [[closed category]]

* [[star-autonomous category]]

**Special sorts of products**

* [[cartesian monoidal category]]

* [[semicartesian monoidal category]]

* [[monoidal category with diagonals]]

* [[multicategory]]

**Semisimplicity**

* [[semisimple category]]

* [[fusion category]]

* [[modular tensor category]]

**Morphisms**

* [[monoidal functor]]

  ([[lax monoidal functor|lax]], [[oplax monoidal functor|oplax]], [[strong monoidal functor|strong]] [[bilax monoidal functor|bilax]], [[Frobenius monoidal functor|Frobenius]])

* [[braided monoidal functor]]

* [[symmetric monoidal functor]]

**Internal monoids**

* [[monoid in a monoidal category]]

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

* [[module over a monoid]]

**{#_examples}Examples**

* [[tensor product]]

* [[closed monoidal structure on presheaves]]

* [[Day convolution]]

**Theorems**

* [[coherence theorem for monoidal categories]]

* [[monoidal Dold-Kan correspondence]]

**In higher category theory**

* [[monoidal 2-category]]

  * [[braided monoidal 2-category]]

* [[monoidal bicategory]]

  * [[cartesian bicategory]]

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

  * [[little cubes operad]]

* [[monoidal (∞,1)-category]]

  * [[symmetric monoidal (∞,1)-category]]

* [[compact double category]]