monoidal monad

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###Context###
#### Higher algebra
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[[!include higher algebra - contents]]
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#### Monoidal categories
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[[!include monoidal categories - contents]]
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#### 2-Category theory
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[[!include 2-category theory - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

\begin{definition}
\label{MonoidalMonad}
\linebreak
A **monoidal monad** is a [[monad]] in the [[2-category]] of [[monoidal category|monoidal categories]], [[lax monoidal functor|lax monoidal functors]], and [[monoidal natural transformation|monoidal transformations]].
\end{definition}

This notion originates inside the statement of [Kock 1970, Thm. 3.2](#Kock70).

{#InComponents} In components, this means (cf. [Kock 1970, p. 8](#Kock70), review includes [Seal 2012, §1.2](#Seal12)):

Monoidal [[structure]] on a [[monad]]

$
  \mathcal{E}
  \,\colon\,
  \mathbf{C} \to \mathbf{C}
$

$
  c  \in \mathbf{C}
  \;\;\;\;
   \vdash
  \;\;\;\;
  ret^{\mathcal{E}}_c
  \,\colon\,
  c \to \mathcal{E}(c)
$

$
  c  \in \mathbf{C}
  \;\;\;\;
   \vdash
  \;\;\;\;
  join^{\mathcal{E}}_{c}
  \,\colon\,
  \mathcal{E} \circ \mathcal{E}(c)
  \to
  \mathcal{E}(c)
$

acting on a [[monoidal category]] $(\mathbf{C}, \otimes, \mathbb{1})$ is:

1. [[lax monoidal functor]]-[[structure]] on $\mathcal{E}$

   $
     \vdash
     \;\;\;\;
     \epsilon^{\mathcal{E}}
     \,\colon\,
     1\!\!1 \to \mathcal{E}(1\!\!1)
   $

   $
     c,c' \in \mathbf{C}
     \;\;\;\;
       \vdash
     \;\;\;\;
     \mu^{\mathcal{E}}_{c,c'}
     \,\colon\,
     \mathcal{E}(c) \otimes \mathcal{E}(c')
     \to
     \mathcal{E}(c \otimes c')
   $

1. such that the monad structure transformations $ret^{\mathcal{E}}$ and $join^{\mathcal{E}}$ are [[monoidal transformations]] in that together with the lax monoidal structure $\epsilon^{\mathcal{E}}$ and $\mu^{\mathcal{E}}$ they make the following [[commuting diagram|diagrams commute]]:

First of all, the [[lax monoidal functor|lax monoidal unit]] must coincide with the [[unit of a monad|monad unit]]

\[\label{LaxUnitIsMonadUnit}\]

\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
sep=large%
]
  &
  1\!\!1
  \ar[dl, equals]
  \ar[
    dr,
    "{
      \epsilon^{\mathcal{E}}
    }"
  ]
  \\
  1\!\!1
  \ar[
    rr,
    "{
      \mathrm{ret}^{\mathcal{E}}_{1\!\!1}
    }"{swap}
  ]
  &&
  \mathcal{E}(1\!\!1)
\end{tikzcd}

which already implies the unit diagram for the join operation:

\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
sep=large%
]
  &
  1\!\!1
  \ar[
    dl,
    "{
       \mathcal{E}(\epsilon^{\mathcal{E}})
       \circ
       \epsilon^{\mathcal{E}}
    }"{description}
  ]
  \ar[
    dr,
    "{
      \epsilon^{\mathcal{E}}
    }"
  ]
  \\
  \mathcal{E}\circ\mathcal{E}(1\!\!1)
  \ar[
    rr,
    "{
      \mathrm{join}^{\mathcal{E}}_{1\!\!1}
    }"{swap}
  ]
  &&
  \mathcal{E}(1\!\!1)
\end{tikzcd}

and then the two main conditions:

\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
sep=large%
]
  c \otimes c'
  \ar[dd, equals]
  \ar[
    rr,
    "{
      \big(\mathrm{ret}^{\mathcal{E}}_c\big)
      \otimes
      \big(\mathrm{ret}^{\mathcal{E}}_{c'}\big)
    }"
  ]
  &&
  \mathcal{E}(c) \otimes \mathcal{E}(c')
  \ar[
    dd,
    "{
      \mu^{\mathcal{E}}_{c,c'}
    }"
  ]
  \\
  \\
  c \otimes c'
  \ar[
    rr,
    "{
      \mathrm{ret}^{\mathcal{E}}_{c \otimes c'}
    }"
  ]
  &&
  \mathcal{E}(c \otimes c')
\end{tikzcd}

and

\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
sep=large%
]
  \big(\mathcal{E}\circ\mathcal{E}(c)\big)
  \otimes
  \big(\mathcal{E}\circ\mathcal{E}(c')\big)
  \ar[
    d,
    "{
      \mu^{\mathcal{E}}_{\mathcal{E}(c), \mathcal{E}(c')}
    }"
  ]
  \ar[
    rr,
    "{
      \big(
        \mathrm{join}^{\mathcal{E}}_c
      \big)
      \otimes
      \big(
        \mathrm{join}^{\mathcal{E}}_{c'}
      \big)
    }"
  ]
  &&
  \big(\mathcal{E}(c)\big)
   \otimes
  \big(
    \mathcal{E}(c')
  \big)
  \ar[
    dd,
    "{
      \mu^{\mathcal{E}}_{c,c'}
    }"
  ]
  \\
  \mathcal{E}
  \Big(
  \big(\mathcal{E}(c)\big)
  \otimes
  \big(\mathcal{E}(c')\big)
  \Big)
  \ar[
    d,
    "{
      \mathcal{E}\big(
        \mu^{\mathcal{E}}_{c,c'}
      \big)
    }"
  ]
  \\
  \mathcal{E} \circ \mathcal{E}(c \otimes c')
  \ar[
    rr,
    "{
      \mathrm{join}^{\mathcal{E}}_{c \otimes c'}
    }"{swap}
  ]
  &&
  \mathcal{E}(c \otimes c')
\end{tikzcd}

Moreover, if $\mathbf{C}$ is even a [[symmetric monoidal category]] with [[braiding]] $\sigma$, then a monoidal monad on $\mathbf{C}$ as above is a *symmetric monoidal monad* if the [[underlying]] [[monoidal functor]] is a *[[symmetric monoidal functor]]*.

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

The notion of monoidal monad is equivalent to the notion of _[[commutative monad]]_. To put it another way: to give a commutative [[strong monad|strength]] for a monad $T$ is to give a monoidal monad whose [[underlying]] monad is $T$. We explore this connection below.

=--

## Properties

### Relation to commutative strong monads
 {#RelationToCommutativeStrongMonads}

We discuss how monoidal monads [[functor|functorially]] give rise to [[strong monads]].

#### Strength

First to recall the notion of a [[strong monad]]:

Let $V$ be a [[monoidal category]]. We say a [[functor]] $T \colon V \to V$ is **strong** if there are given left and right [[tensorial strength|tensorial strengths]], namely [[natural transformations]] of the form:

$$
  \tau_{A, B}
  \;\colon\;
  A \otimes T(B)
  \to
  T(A \otimes B)
$$

$$\,$$

$$
  \sigma_{A, B}
  \;\colon\;
  T(A) \otimes B
  \to
  T(A \otimes B)
  \,,
$$

which are suitably compatible with one another: The full set of [[coherence]] conditions may be summarized by saying $T$ preserves the two-sided [[action of a monoidal category|monoidal action]] of $V$ on itself, in an appropriate [[2-category theory|2-categorical]] sense. More precisely: the two-sided action of $V$ on itself is a [[lax functor]] of [[2-categories]]

$$
  \tilde{V}
  \colon
  B V \times (B V)^{op}
  \to
  Cat
$$

where

* [[Cat]] denotes the 2-category of [[categories]], [[functors]] and [[natural transformations]],

* $B V$ denotes the [[delooping]] one-object 2-category of the monoidal category $V$,

* $(B V)^{op}$ denotes its [[opposite 2-category|1-cell dual]], hence the same 2-category except with [[1-morphism]] [[composition]] (here: [[tensor product]]) in reverse order),

* the two-sided strength means we have a [[structure]] of [[lax natural transformation]] $\tilde{V} \to \tilde{V}$.

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

In the setting where $V$ is _[[symmetric monoidal category|symmetric]]_ monoidal, we will assume that the left and right strengths $\tau$ and $\sigma$ are related by the symmetry in the obvious way, by a [[commutative square]]

$$
  \array{
    A \otimes T(B)
    &
    \stackrel
      {\tau_{A, B}}
      {\longrightarrow}
    &
    T(A \otimes B)
    \\
    \mathllap{^c}
    \big\downarrow
    & &
    \big\downarrow
    \mathrlap{^{T(c)}}
    \\
    T(B) \otimes A
    &
    \underset{
      \sigma_{B, A}}
      {\longrightarrow}
    & T(B \otimes A)
}$$

where the $c$'s are instances of the symmetry isomorphism.

=--

\begin{definition}
\label{StrongMonad}
There is a [[category]] of [[strong functors]] $V \to V$, whose [[morphisms]] are [[natural transformations]] $\lambda \colon S \to T$ which are compatible with the strengths in the obvious sense. Under [[composition]], this is a strict [[monoidal category]].

The [[monoid objects]] in this monoidal category are called **[[strong monads]]**.
\end{definition}

\begin{definition}
\label{CommutativeMonad}
A [[strong monad]] $(T \colon V \to V, m \colon T T \to T, u: 1 \to T)$ (def. \ref{StrongMonad}) is a **[[commutative monad]]** if there is an [[equality]] of [[natural transformations]] $\alpha = \beta$ where

* $\alpha$ is the composite

  $$T A \otimes T B \stackrel{\sigma_{A, T B}}{\to} T(A \otimes T B) \stackrel{T(\tau_{A, B})}{\to} T T(A \otimes B) \stackrel{m(A \otimes B)}{\to} T(A \otimes B).$$

* $\beta$ is the composite

  $$T A \otimes T B \stackrel{\tau_{T A, B}}{\to} T(T A \otimes B) \stackrel{T(\sigma_{A, B})}{\to} T T(A \otimes B) \stackrel{m(A \otimes B)}{\to} T(A \otimes B).$$

\end{definition}

#### From monoidal to commutative strong monads

\begin{definition}
\label{StrengthFromMonoidalness}
**(strength from monoidalness)**
\linebreak
For $(T \colon V \to V, u \colon id \to T, m \colon T T \to T)$  a monoidal monad (Def. \ref{MonoidalMonad}), with the monoidal monad-[[structure]] on the [[underlying]] [[functor]] denoted by

$$
  \alpha_{A, B}
  \,\colon\,
  T(A) \otimes T(B)
  \to
  T(A \otimes B),
  \qquad
  \iota = u(I)
  \,:\, I \to T(I)
  \,,
$$

Define [[tensorial strength|strengths]] on both the left and the right by:

$$
  \tau_{A, B}
  \;\coloneqq\;
  \big(
    A \otimes T(B)
      \overset{u_A \otimes id}{\to}
    T(A) \otimes T(B)
      \overset{\alpha_{A, B}}{\to}
    T(A \otimes B)
  \big)
  \,,
$$

$$\,$$

$$
  \sigma_{A, B}
  \;\coloneqq\;
  \big(
    T(A) \otimes B
      \overset{id \otimes u_B}{\to}
    T(A) \otimes T(B)
      \overset{\alpha_{A, B}}{\to}
    T(A \otimes B)
  \big)
  \,.
$$
\end{definition}

+-- {: .num_prop}
###### Proposition
The strong monad structures obtained from monoidal monads via Def. \ref{StrengthFromMonoidalness} are [[commutative monads]] (Def. \ref{CommutativeMonad}).
=--

+-- {: .proof}
###### Proof
In fact, the two composites

$$T A \otimes T B \stackrel{\sigma_{A, T B}}{\to} T(A \otimes T B) \stackrel{T(\tau_{A, B})}{\to} T T(A \otimes B) \stackrel{m(A \otimes B)}{\to} T(A \otimes B)$$

$$\,$$

$$T A \otimes T B \stackrel{\tau_{T A, B}}{\to} T(T A \otimes B) \stackrel{T(\sigma_{A, B})}{\to} T T(A \otimes B) \stackrel{m(A \otimes B)}{\to} T(A \otimes B)$$

are both equal to $\alpha_{A, B}$. We show this for the second composite; the proof is similar for the first. If $\alpha_T$ denotes the monoidal constraint for $T$ and $\alpha_{T T}$ the constraint for the composite $T T$, then by definition $\alpha_{T T}$ is the composite given by
$$T T X \otimes T T Y \stackrel{\alpha_T T}{\to} T(T X \otimes T Y) \stackrel{T\alpha_T}{\to} T T(X \otimes Y)$$
and so, using the properties of monoidal monads, we have a commutative diagram

$$\array{
 & & T T X \otimes T Y & \stackrel{\alpha_T}{\to} & T(T X \otimes Y) \\
 & ^\mathllap{u \otimes 1} \nearrow & \downarrow^\mathrlap{1 \otimes T u} & & \downarrow^\mathrlap{T(1 \otimes u)} \\
T X \otimes T Y & \stackrel{u \otimes T u}{\to} & T T X \otimes T T Y & \stackrel{\alpha_T T}{\to} & T(T X \otimes T Y) \\
 & ^\mathllap{1} \searrow & \downarrow^\mathrlap{m \otimes m} & \searrow^\mathrlap{\alpha_{T T}} & \downarrow^\mathrlap{T\alpha_T} \\
 & & T X \otimes T Y & & T T(X \otimes Y) \\
& & & ^\mathllap{\alpha_T} \searrow & \downarrow^\mathrlap{m} \\
 & & & & T(X \otimes Y)
}$$

which completes the proof.
=--

This construction is [[functorial]]:

\begin{proposition}
\label{cf}
Given monoidal monads $S$ and $T$ on a [[monoidal category]] $C$, a [[monad#the_bicategory_of_monads|morphism of monads]] $\alpha \colon S\Rightarrow T$ is a [[morphism]] of the induced [[strong monad]] structures (Def. \ref{StrengthFromMonoidalness}) if and only if it is a [[monoidal natural transformation]].
\end{proposition}

(e.g. [FPR (2019), Prop. C.5](#support))

This relation has a converse:

\begin{proposition}
\label{BijectSymmMonoidalMonadsWithCommStrongMonads}
\linebreak
For a [[monad]] $T$ on (the [[underlying]] [[category]] of) a [[symmetric monoidal category|symmetric]] [[closed monoidal category]], there is a [[bijection]] between the [[structure]] on $T$ of:

1. a [[commutative monad|commutative]] [[strong monad]]

1. a [[symmetric monoidal functor|symmetric]] [[monoidal monad]].

\end{proposition}

This is due to [Kock (1972)](#Kock72), Thm. 2.3), a detailed review is in  [GLLN08, §7.3, §A.4](#GLLN08).

### Tensor product of algebras and multimorphisms

See [[commutative monad#tensor_product_of_algebras_and_multimorphisms|here]].

### Monoidal structure on the Kleisli category

The [[Kleisli category]] of a monoidal monad $T$ on $C$ inherits the [[monoidal structure]] from $C$.
In particular, the [[tensor product]] is given

* On objects, by the tensor product $\otimes$ of $C$;
* On morphisms, given $k:X\to TA$ and $h:Y\to TB$, their product is the map $X\otimes Y \to T(A\otimes B)$ obtained by the composition
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
]
X\otimes Y \ar{r}{f\otimes g} & TA \otimes TB \ar{r}{\nabla} & T(A\otimes B),
\end{tikzcd}
where $\nabla$ is the monoidal multiplication of $T$.

* The associator and unitor are induced by those of $C$.

## Examples

See [[commutative monad#examples|examples of commutative monads]].

## See also

* [[strong monad]], [[tensorial strength]]
* [[monoidal functor]]
* [[commutative monad]], [[commutative algebraic theory]]

## References
 {#References}

The definition originally appears inside statement and proof of Thm. 3.2 in:

* {#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)]

establishing right away the relation to [[commutative monads|commutative]] [[strong monads]] (in the case that the underlying [[monoidal category]] is [[symmetric monoidal closed category|symmetric monoidal closed]]) which is further expanded on in:

* {#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)]

Further discussion:

* {#Lindner75} 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) [[numdam:CTGDC_1975__16_3_217_0](http://www.numdam.org/item/CTGDC_1975__16_3_217_0), [pdf](http://www.numdam.org/article/CTGDC_1975__16_3_217_0.pdf)]

* {#Keigher78} [[William Keigher]], *Symmetric monoidal closed categories generated by commutative adjoint monads*, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 19 no. 3 (1978) 269-293  [[numdam:CTGDC_1978__19_3_269_0](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)]

* Kosta Dosen, Zoran Petric, *Coherence for Monoidal Monads and Comonads*, Mathematical Structures in Computer Science , **20** 4 (2010) 545-561  [[arXiv:0907.2199](https://arxiv.org/abs/0907.2199), [doi:10.1017/S0960129510000034](https://doi.org/10.1017/S0960129510000034)]

* {#Seal12} [[Gavin J. Seal]], *Tensors, monads and actions*, Theory and Applications of Categories **28** 15 (2013) 403-434.  [[arXiv:1205.0101](http://arxiv.org/abs/1205.0101), [tac:28-15](http://www.tac.mta.ca/tac/volumes/28/15/28-15abs.html)]

  > (on the [[Eilenberg-Moore categories]] of monoidal monads)

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

Discussion in the context of [[monads in computer science]]:

* {#GLLN08} [[Jean Goubault-Larrecq]], [[Slawomir Lasota]], [[David Nowak]], *Logical Relations for Monadic Types*, Mathematical Structures in Computer Science, **18** 6 (2008) 1169-1217 [[arXiv:cs/0511006](https://arxiv.org/abs/cs/0511006), [doi:10.1017/S0960129508007172](https://doi.org/10.1017/S0960129508007172)]

A statement in the above text is from

* {#support} [[Tobias Fritz]], [[Paolo Perrone]] and Sharwin Rezagholi, Appendix C of: _Probability, valuations, hyperspace: Three monads on Top and the support as a morphism_, 2019 ([arXiv:1910.03752](https://arxiv.org/abs/1910.03752))

[[!redirects monoidal monads]]


monoidal natural transformation

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###Context###
#### Monoidal categories
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[[!include monoidal categories - contents]]
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#### Category theory
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[[!include category theory - contents]]
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=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _monoidal natural transformation_ is a [[natural transformation]] between [[monoidal functors]] that respects the monoidal structure.

## Definition

Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}})$ be two [[monoidal categories]]. Let $(F_i, \mu_i, \epsilon_i)$ for $i \in \{1,2\}$ be two [[lax monoidal functors]] $\mathcal{C}\longrightarrow \mathcal{D}$.

Then a **monoidal natural transformation** $f$ from $(F_1,\mu_1, \epsilon_1)$ to $(F_2,\mu_2, \epsilon_2)$ is a [[natural transformation]] $f(x) \;\colon\; F_1(x) \longrightarrow F_2(x)$ between the underlying functors that is compatible with the tensor product and the unit in that the following [[commuting diagram|diagrams commute]] for all objects $x,y \in \mathcal{C}$:

$$
  \array{
    F_1(x) \otimes_{\mathcal{D}} F_1(y)
      &\overset{f_x \otimes_{\mathcal{D}} f_y}{\longrightarrow}&
    F_2(x) \otimes_{\mathcal{D}} F_2(y)
    \\
    {}^{\mathllap{(\mu_1)_{x,y}}}\downarrow
     &&
    \downarrow^{\mathrlap{(\mu_2)_{x,y}}}
    \\
    F_1(x\otimes_{\mathcal{C}} y)
     &\underset{f_{x \otimes_{\mathcal{C}} y} }{\longrightarrow}&
    F_2(x \otimes_{\mathcal{C}} y)
  }
$$

and

$$
  \array{
    && 1_{\mathcal{D}}
    \\
    & {}^{\mathllap{\epsilon_1}}\swarrow && \searrow^{\mathrlap{\epsilon_2}}
    \\
    F_1(1_{\mathcal{C}})
     &&\underset{f_{1_{\mathcal{C}}}}{\longrightarrow}&&
    F_2(1_{\mathcal{C}})
  }
  \,.
$$

## Related concepts

* [[symmetric monoidal natural transformation]]
* [[icon]]

## References

* {#EK65} [[Samuel Eilenberg]], [[G. Max Kelly]], p. 474 in: *Closed Categories*, in:  [[Samuel Eilenberg|S. Eilenberg]], [[D. K. Harrison]], [[S. MacLane]], [[H. Röhrl]] (eds.): *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) 421-562 [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

Exposition includes

* [[John Baez]], _Some definitions everyone should know_, [pdf](http://math.ucr.edu/home/baez/qg-fall2004/definitions.pdf).

[[!redirects monoidal natural transformation]]
[[!redirects monoidal natural transformations]]
[[!redirects monoidal transformation]]
[[!redirects monoidal transformations]]


monoidal preorder

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###Context###
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
#### $(0,1)$-Category theory
+-- {: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

* a [[monoid]] with a [[preorder]] such that the binary operation preserves the preorder.

## See also

* [[enriched poset]]

* [[enriched set]]

## References

* [[John Baez]], [*Lecture 21 - Chapter 2: Monoidal Preorders*](https://forum.azimuthproject.org/discussion/2082/lecture-21-chapter-2-monoidal-preorders)

[[!redirects monoidal preorders]]
[[!redirects monoidal proset]]
[[!redirects monoidal prosets]]
[[!redirects monoidal poset]]
[[!redirects monoidal posets]]


monoidal Quillen adjunction

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

#Contents#
* table of contents
{:toc}

## Idea

The concept of _monoidal Quillen adjunction_ is a lift of the concept of _[[strong monoidal adjunctions]]_ ([[adjoint functors]] for which the [[left adjoint]] is a [[strong monoidal functor]] so that the [[right adjoint]] is, canonically, a [[lax monoidal functor]]) from the context of plain [[categories]] to that of [[model categories]].

## Definition

For $C$ and $D$ [[monoidal model categories]], a **lax monoidal Quillen adjunction**

$$
  (L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}}
  D
$$

is

* a [[Quillen adjunction]] $(L \dashv R)$ between the underlying [[model categories]];

* equipped with the structure of a [[lax monoidal functor]] on $R$ with respect to the underlying [[monoidal categories]]

* such that the induced structure of an [[oplax monoidal functor]] on $L$ satisfies:

  1. for all cofibrant objects $x,y \in D$ the oplax monoidal transformation

     $$
       \tilde\nabla_{x,y} : L(x \otimes y) \to L(x) \otimes L(y)
     $$

     is a weak equivalence in $C$

  1. for some (hence any) cofibrant [[resolution]] $q : \hat I_D \stackrel{\simeq}{\to} I_D$ of the monoidal unit object in $D$, the composite

     $$
       L(\hat I_D) \stackrel{L(q)}{\to} L(I_D) \stackrel{\tilde e}{\to} I_C
     $$

     with the oplax monoidal counit is a weak equivalence in $C$.

This is called a **strong monoidal Quillen adjunction** if $L$ is a [[strong monoidal functor]].  In this case the first condition above on $L$ is vacuous, and the second becomes vacuous if the unit object of $D$ is cofibrant.

If a monoidal Quillen adjunction is also a [[Quillen equivalence]] it is called a **monoidal Quillen equivalence**.

## Properties

### Recognition of monoidal Quillen adjunctions {#Recognition}

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

Let $(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$ be a [[Quillen adjunction]] between [[monoidal model categories]] and let $R$ be equipped with the strcuture of a [[lax monoidal functor]].

Then the following two conditions are sufficient for $(L \dashv R)$ to be a lax monoidal Quillen adjunction:

1. for some (hence any) cofibrant [[resolution]] $q : \hat I_D \stackrel{\simeq}{\to} I_D$ of the unit object in $D$, the composite morphism

   $$
     L(\hat I_D) \stackrel{L(q)}{\to} L(I_D) \stackrel{\tilde i}{\to} I_C
   $$

   is a weak equivalence, (wher $\tilde i$ is the [[adjunct]] of $i : I_D \to R(I_C)$);

1. the unit object $I_D$ _detects weak equivalences_ in that for every weak equivalence $f : X \to Y$ between fibrant objects the morphism $D^{\Delta^{op}}(Q I_D, f)$ of [[hom-object]]s in the category of [[simplicial object]]s in $D$ is an equivalence of Kan complexes, for $Q I_D$ a cofibrant resolution in the [[Reedy model structure]] $D^{\Delta^{op}}_{Reedy}$.

=--

This is proposition 3.16 in ([SchwedeShipley](#SchwedeShipley)).

### Lift to an adjunction on monoids {#LiftToAdjunctionOnMonoids}

We discuss how a monoidal Quillen adjunction $(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$ induces, under mild conditions, an adjunction $(L^{mon} \dashv R) : Mon(C) \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} Mon(D)$ on the corresponding [[categories of monoids]]. In the following section we discuss how this is itself a Quillen adjunction

The [[lax monoidal functor]] $R : C \to D$ induces (as described there) a functor $R : Mon(C) \to Mon(D)$ on monoids (which by slight abuse of notation we denote by the same symbol). Write $\nabla_{X,Y} : R X \otimes R Y \to R(X \otimes Y)$ for the lax monoidal structure on $R$. This induces canonically the structure of a [[oplax monoidal functor]] (as described there) on the left adjoint  $L : D \to C$. Write $\tilde\nabla : L(X \otimes Y) \to L X \otimes L Y$ for this oplax structure.

While $L$ will not extend to a functor on the [[category of monoids]] unless $R$ is a [[strong monoidal functor]] there is nevertheless an adjoint $L^{mon}$ to $R : Mon(C) \to Mon(D)$.

As described at [[category of monoids]], if $C$ has countable [[coproduct]]s preserved by the [[tensor product]], then we have a [[free functor]]/[[forgetful functor]] [[adjunction]]

$$
  (F \dashv U) : Mon(C) \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}}
   C
  \,,
$$

where $F(X)$ is the [[tensor algebra]] over the object $X$ in $(C, \otimes)$.

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

Let $(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$ be a pair of adjoint functors between monoidal categories where $R$ is a lax monoidal functor and $D$ has all small colimits.

Then the functor $R : Mon(C) \to Mon(D)$ has a left adjoint

$$
  L^{mon} : Mon(D) \to Mon(C)
$$

given by forming the [[coequalizer]]s

$$
  L^{mon} : B
    \mapsto
   \lim_{\to}
  (F_C L F_D B \stackrel{\to}{\to} F_C L B)
$$

in $Mon(C)$ of the following two morphisms

* the first one is the image under $F_C \circ L$ of the [[unit of an adjunction|adjunction counit]] $ F_D U_D B  \to B$;

* the second is the unique $C$-monoid morphism that restricts to the $C$-morphism

  $$
    L F_D B
    \simeq
   \coprod_{n \in \mathbb{N}} L( B^{\otimes n})
    \stackrel{\coprod \tilde \nabla}{\to}
   \coprod_{n \in \mathbb{N}}
   (L B)^{\otimes n}
   \simeq
   F_C L B
  $$

  which is componentwise given by the [[oplax monoidal functor|oplax monoidal structure]] on $L$ induced by the lax monoidal structure on $R$.

=--

This is considered on p. 305 of ([SchwedeShipley](#SchwedeShipley))

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

To see that $(L^{mon} \dashv R)$ first notice that a morphism of monoids

$$
  L^{mon} X \to Y
$$

is by the definition of coequalizer a morphism of monoids $f : F_C L X \to Y$ satisfying a condition. By the free property of $F_C L X$ this in turn is a morphism $f_1 : L X \to Y$ in $C$ which by $(L \dashv R)$ is a morphism $\tilde f_1 : X \to R Y$ in $C$. So we need to show that the condition satisfied by $f$ is precisely the condition that makes $\tilde f_1$ a morphism of monoids in that

$$
  \array{
    X \otimes X &\stackrel{\tilde f_1 \otimes \tilde f_1}{\to}& R  Y \otimes R  Y
    \\
    \downarrow && \downarrow
    \\
    && R ( Y \otimes  Y)
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{\tilde f_1}{\to}& R  Y
  }
$$

commutes. We insert the definition of the [[adjunct]] $\tilde f_1$ and the [[lax natural transformation|lax naturality]] square of $R$ to get

$$
  \array{
    X \otimes X
    &\to&
    R L X \otimes R L X
    &\stackrel{R f_1 \otimes R f_1}{\to}& R  Y \otimes R  Y
    \\
    \downarrow && \downarrow &=& \downarrow
    \\
    && R(L X \otimes L Y) &\stackrel{R f_1}{\to}& R ( Y \otimes  Y)
    \\
    \downarrow && && \downarrow
    \\
    X & &\stackrel{\tilde f_1}{\to}&& R  Y
  }
  \,.
$$

The [[adjunct]] of the left/bottom composite is

$$
  L(X\otimes X) \to L X \stackrel{f_1}{\to} Y
$$

while the adjunct of the top/right composite is that of the diagonal, which is

$$
  L(X \otimes X) \stackrel{\tilde \nabla}{\to} L X \otimes L X
  \stackrel{f_1 \otimes f_1}{\to} Y \otimes Y \to Y
  \,.
$$

This in turn is by the definition of $f$ in terms of its components equal to

$$
  L(X \otimes X) \stackrel{\tilde \nabla}{\to} L X \otimes L X
  \stackrel{f_2}{\to}Y
  \,.
$$

The coequalizer property says indeed precisely that these two adjuncts are equal.

=--

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

There is a [[natural isomorphism]]

$$
  L^{mon} \circ F_D
  \simeq
  F_C \circ L
  \,.
$$

=--

This is considered on p. 305 of ([SchwedeShipley](#SchwedeShipley)).

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

On a monoid $K$ the morphism

$$
  L^{mon} F K \to F L K
$$

is defined as a coequalizing morphism of monoids

$$
  F L F K \to F L K
  \,.
$$

This in turn is given by a morphism in $C$

$$
  L F K \to F L K
  \,.
$$

Take this to be given componentwise by the oplax counit $\tilde e$.

This does coequalize then:  for one route is

$$
  L( (K) \otimes (K)  ) \to L(K \otimes K) \stackrel{\tilde \nabla}{\to} L(K) \otimes L(K)
$$

and the other

$$
  K( (K) \otimes (K) ) \stackrel{\tilde \nabla}{\to} L K \otimes L K
  \stackrel{Id}{\to}
  L K \otimes L K
  \,.
$$

=--

### Lift to a Quillen equivalence on monoids {#LiftToQuillenAdjunctionOnMonoids}

We now describe how the adjunction  $(L^{mon} \dashv R)$ established above becomes a [[Quillen adjunction]] for the [[transferred model structure]]s on the categories of monoids, transferred along the [[stuff, structure, property|forgetful]]/[[free functor]] adjunction

$$
  (F_C \dashv U_C) : Mon(C) \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}}
  C
$$

and how it becomes a [[Quillen equivalence]] if $(L \dashv R)$ is a monoidal Quillen eqivalence.

See [[model structure on monoids]].

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

We assume for this section that the monoidal model category $C$

* is [[symmetric monoidal category|symmetric monoidal]];

* is a [[cofibrantly generated model category]]

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

=--

Then by ([SchwedeShipleyAlgebras](SchwedeShipleyAlgebras)) the [[transferred model structure|transferred]] [[model structure on monoids in a monoidal model category]] $Mon(C)$ exists.

Notice also that by cofibrant generation every cofibrant object in $Mon(C)$ is a [[retract]] of a $(F \dashv U)$-[[cell object]].

=--

+-- {: .un_theorem #LiftedQuillenAdjunction}
###### Theorem

Let $(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$ be a lax monoidal Quillen adjunction between [[monoidal model categories]] with cofibrant unit obects.

Then also the adjunction

$$
  (L^{mon} \dashv R)
  :
  Mon(C) \stackrel{\overset{L^{mon}}{\leftarrow}}{\underset{R}{\to}} Mon(D)
  \,,
$$

from [above](#AdjunctionOnMonoids) is a Quillen adjunction between the [[transferred model structure|transferred]] [[model structure on monoids|model structures on monoids]].

If the forgetful functors $U_C$ and $U_D$ [create](#CreatedModelStructure) model structures on monoids, then $(L^{mon} \dashv R)$ is a [[Quillen equivalence]] if $(L \dashv R)$ is.

=--

This is theorem 3.12 in ([SchwedeShipley](#SchwedeShipley)). Its proof uses the following technical lemmas.

Let $(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$ be a monoidal Quillen adjunction between monoidal model categories with cofibrant unit objects.

Suppose the adjunction

$$
  (L^{mon} \dashv R)
  :
  Mon(C) \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} Mon(D)
$$

described above exists (just as an adjunction, not yet assumed to be a Quillen adjunction).

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

The morphism

$$
  L^{mon} I_D \to I_C
$$

induced by the oplax counit $\tilde e : L I_D \to I_C$ of the oplax monoidal functor is an [[isomorphism]] of monoids.

=--

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

We have that $I_D$ and $I_C$ are the [[initial object]]s in $Mon(D)$ and $Mon(C)$, respectively.  Because $L^{mon}$ is [[left adjoint]], it preserves these initial objects, so that there is _some_ isomorphism as claimed. It is hence sufficient to show that the oplax counit induces a morphism of monoids at all, by  the universal property of the initial object it will be an isomorphism.

It is clear that

$$
  \coprod_n \mu^n {\tilde e}^{\otimes n} : F L I_D \to I_C
$$

is a morphism of monoids, because

$$
  \array{
    (L I)^{\otimes k} \otimes (L I)^{\otimes (n-k)}
    &\stackrel{\mu_I^k {\tilde e}^{\otimes k} \otimes \mu_I^{n-k}{\tilde e}^{\otimes (n-k)}}{\to}&
    I \otimes I
    \\
    \downarrow && \downarrow
    \\
    (L I)^{\otimes n}
    &\stackrel{\mu^n_I {\tilde e}^{n}}{\to}& I
  }
$$

commutes. So we have to show that this morphism
coequalizes the two morphisms in the definition  of $L^{mon} I_D$.
By the same argument as in the [above proof](#AdjunctionOnMonoids) this is equivalent to showing that

$$
  \array{
    I_C \otimes I_C &\stackrel{e \otimes e}{\to}&
    R I_C \otimes R I_C
    \\
    \downarrow && \downarrow
    \\
    && R (I_C \otimes I_C)
    \\
    \downarrow && \downarrow
    \\
    I_C &\stackrel{e}{\to}& R I_C
  }
$$

commutes. This follows from the unitality of the [[lax monoidal functor]] $R$.

=--

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

For every monoid $B \in Mon(D)$ which is an $(F \dashv U)$-[[cell object]], the $(L \dashv R)$-[[adjunct]]

$$
 \chi_B :  L B \to L^{mon} B
$$

to the morphism underlying the [[unit of an adjunction|unit]] $B \to R L^{mon} B$ is a weak equivalence.

=--

This is proposition 5.1 in ([SchwedeShipley](#SchwedeShipley)).

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

We first show this for $B = I_D$ the tensor unit in $D$, which in $Mon(D)$ is the [[initial object]]s:

* We claim hat the [[unit of an adjunction|adjunction unit]] $I_C \to R L^{mon} I_D \stackrel{\simeq}{\to} R(I_C)$ is the lax monoidal unit $e$ of $R$.

  To see this, use that by the [previous lemma](#VeryFirstTechnicalLemma) the $(L \dashv R)$-[[adjunct]] of $I \to R L^{mon} I \to R I$ is $L I \to L^{mon} I \stackrel{\coprod_n \mu^n {\tilde e}^{\otimes n}}{\to} I$. Here the first morphism factors through the single power of $L I$, hence this is indeed $\tilde e : L I_D \to I_C$.

  Therefore by the axioms on monoidal Quillen adjunctions the
$(L \dashv R)$-adjunct $\chi_I$ is a weak equivalence.

We now proceed from this by induction over the cells of the [[cell object]] $B$.

So assume now that we have already shown that on some cell object $B$ the morphism $\chi_B$ is a weak equivalence. We want to deduce then that that after forming a new monoid $P$ by cell attachment, i.e. by a [[pushout]]

$$
  \array{
    F K &\to& F K'
    \\
    \downarrow && \downarrow
    \\
    B &\to& P
  }
$$

for $K \to K'$ a cofibration in $D$, also $\chi_P : L P \to L^{mon} P $ is a weak equivalence.

Notice that since $L^{mon}$ is left adjoint also

$$
  \array{
    L^{mon} F K &\to& L^{mon} F K'
    \\
    \downarrow && \downarrow
    \\
    L^{mon} B &\to& L^{mon} P
  }
$$

is a pushout in $Mon(C)$, and by the natural isomorphism from [the above lemma](#LemmaOnNaturalIso) so is

$$
  \array{
    F L K &\to& F L K'
    \\
    \downarrow && \downarrow
    \\
    L^{mon} B &\to& L^{mon} P
  }
  \,.
$$

* We claim that $B$ is cofibrant and that we can without restriction assume $K$ and $K'$ to be cofibrant in $D$.

  The first statement follows from an inductive application of the construction of pushouts as discussed at [[category of monoids]] in the section free monoids. For the second statement notice that since $F$ is left adjoint and preserves pushouts in $D$, we have that $P$ is also the pushout of the diagram

  $$
    \left(
    \array{
      F B &\to& F( B \coprod_K K' )
      \\
      \downarrow && \downarrow
      \\
      B &\to& P
    }
    \right)
    =
    \left(
       \array{
          && F K &\to& F K'
          \\
          && \downarrow && \downarrow
          \\
          F B &\to& F B
          \\
          \downarrow &&&& \downarrow
          \\
          B && \to && P
       }
    \right)
    \,.
  $$

  Since cofibrations are preserved by the Quillen left adjoint $F$ and under pushout, it follows that also $B \coprod_K K'$ is cofibrant if $K \to K'$ is a cofibration. So $B \to B \coprod_K K'$ can be used in place of $K \to K'$.

Notice that this means that our pushout square is in fact a [[homotopy pushout]] square (as discussed there). In particular a weak equivalence of these pushout diagrams will induce a weak equivalence of the pushouts, so that is what we will establish.

We now proceed as in [[category of monoids]] in the section free monoids for getting the following statement about the object underlying $P$

This $P$ is a [[colimit]] of a sequence of cofibrations

$$
  P \simeq \lim_{\to} ( B := P_0 \hookrightarrow P_1 \hookrightarrow P_2 \hookrightarrow \cdots )
$$

such that each morphism $P_{n-1} \hookrightarow P_n$ is a pushout in $D$ of a particular cofibration $Q_n(K,K', B) \hookrightarrow (B \otimes K')^{\otimes n} \otimes B$

By the coresponding disccussion of these pushouts under $L^{mon}$ it follows that also $L^{mon} P$ is the colimit of a sequence of cofibrations betwen objects $R_n$ that are pushouts of these particular cofibrations.

And the morphism $\chi_P$ respects all that and sends

$$
  \chi_{P_n} : L P_n \to L^{mon} R_n
$$

at each stage of the cell attachments. So it is sufficient to show that the three components of these maps on the pushout squares are weak equivalences. Since we showed above that our pushout squares are actually [[homotopy pushout]] squares, this will imply that also $\chi_P$ is a weak equivalence.

This again works by proceeding as in [[category of monoids]] in the section free monoids.

=--

+-- {: .un_lemma #AnotherTechnicalLemma}
###### Lemma

If $U_C$ [creates](#CreatedModelStructure) the model structure on $Mon(C)$ and the unit in $C$ is cofibrant,  then a cofibrant $C$-monoid is also cofibrant as an object in $C$.

=--

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

This is once more a consequence of the lemma on pushouts at at [[category of monoids]] in the section free monoids.

=--

We have now collected all prerequisites and turn to the proof of the [theorem about lifted Quillen adjunctions](#LiftedQuillenAdjunction).

+-- {: .proof}
###### Proof of the theorem

That $(L^{mon} \dashv R)$ is a Quillen adjunction is clear, as the [[model structure on monoids]] has fibrations and acyclic fibrations those in the underlying category, and these are preserved by $R$.

So the essential statement is that it is a Quillen equivalence of $(L \dashv R)$ is.

First notice that since by assumption the [[model structure on monoids]] $Mon(D)$ is [created](#CreatedModelStructure) by $U_D$ it follows by definition that the cofibrant $B$ is a [[retract]] of a [[cell object]] in $Mon(D)$. Then the [above lemma](#OneTechnicalLemma) asserts that

$$
  \chi_B : L B \to L^{mon} B
$$

is a weak equivalence.

To prove the theorem, we have to show for every cofibrant $B \in Mon(D)$  and fibrant $Y \in Mon(C)$ that a morphism $B \to R Y$ is a weak equivalence in $Mon(D)$ (hence its underlying morphism in $D$) precisely if its adjunct $L^{mon} B \to Y$ is a weak equivalence in $Mon(C)$ (hence its underlying morphism in $C$).

By definition of [[adjunct]] we have that

$$
 (B \to R Y) = ( B \to R L^{mon} B  \to R Y)
 \,.
$$

By the [second lemma above](#AnotherTechnicalLemma) we have that $B$ is cofibrant also in $C$. Therefore, since $(L \dashv R)$ is a Quillen equivalence between $C$ and $D$, the right hand is a weak equivalence precisely if its $(L \dashv R)$-adjunct

$$
  L B \stackrel{\chi_B}{\to} L^{mon} B \to Y
$$

is a weak equivalence in $D$. But since $\chi_B$ is a weak equivalence, this is the case precisely if $L^{mon}B \to Y$ is a weak equivalence.

=--

## Examples

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

The [[stabilization]] adjunction

  $$
    \left(
    \Sigma^\infty(-)_+
    \dashv
    \Omega^\infty
    \right)
    \;\colon\;
    Ho(Spectra)
     \underoverset
       {\underset{\Omega^\infty}{\longrightarrow}}
       {\overset{\Sigma^\infty(-)}{\longleftarrow}}
       {\bot}
    Ho(Spaces)
  $$

  between the [[classical homotopy category]] $Ho(Spaces)$ 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. Hence this is a [[monoidal adjunction]].

  In fact this is the [[derived functors]] of what is even a [[monoidal Quillen adjunction]]

  $$
   (\Sigma^\infty_{orth}(-)_+ \dashv \Omega^\infty_{orth})
   \;\colon\;
    OrthSpec_{stable}
      \underoverset
        {\underset{\Omega_{orth}^\infty}{\longrightarrow}}
        {\overset{\Sigma_{orth}^\infty(-)_+}{\longleftarrow}}
        {}
    Top_{Quillen}
  $$

  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) modality of the derived functors on homotopy categories ([this prop.](Introduction+to+Stable+homotopy+theory+--+1-2#StrongMonoidalDerivedFunctorFromStrongMonoidalQuillenAdjunction)).

=--

$\,$

* Examples arise in the [[monoidal Dold-Kan correspondence]]. See there for details.

* The quivalence between [[module spectra]] and [[chain complexes]] is exhibited by monoidal Quillen equivalences. See [[module spectrum]] for details.

## Related concepts

* [[Quillen adjunction]]

* [[simplicial Quillen adjunction]]

* [[Quillen equivalence]]

* **monoidal Quillen adjunction**

## References

The notion of strong monoidal Quillen adjunction is def. 4.2.16 in

* [[Mark Hovey]], _Model Categories_ Mathematical Surveys and Monographs, Volume 63, AMS (1999) ([pdf](https://www.math.rochester.edu/people/faculty/doug/otherpapers/hovey-model-cats.pdf), [Google books](http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover))

The lax monoidal version is considered as definition 3.6 of

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

The statements involving pushouts along free monoid morphisms are discussed in lemma 6.2 of

* [[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))
{#SchwedeShipleyAlgebras}

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

[[!redirects monoidal Quillen equivalence]]
[[!redirects monoidal Quillen equivalences]]

[[!redirects monoidal Quillen functor]]
[[!redirects monoidal Quillen functors]]


monoidal structure map

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

For $F\colon C \to D$ a [[functor]] between [[categories]] that are equipped with the structure of [[monoidal categories]] $(C, \otimes)$, $(D,\otimes)$, a **lax monoidal structure map** is a [[natural transformation]]
$$
  \nabla_{x,y}\colon F(x) \otimes F(y) \to F(x \otimes y)
$$
that equips $F$ with the structure of a [[lax monoidal functor]].

Similarly, an **oplax monoidal structure map**, or **lax comonoidal structure map** is a [[natural transformation]]
$$
  \Delta_{x,y}\colon F(x \otimes y) \to F(x) \otimes F(y)
$$
that equips $F$ with the structure of an [[oplax monoidal functor]].

An (op)lax (co)monoidal structure map is sometimes called an __(op)lax (co)monoidal transformation__; however, this is *not* a laxification (a directed weakening) of any strong notion of [[monoidal natural transformation]] (which has nothing to laxify).

[[!redirects monoidal structure map]]
[[!redirects monoidal structure maps]]

[[!redirects lax monoidal structure map]]
[[!redirects lax monoidal structure maps]]
[[!redirects lax monoidal transformation]]
[[!redirects lax monoidal transformations]]

[[!redirects oplax monoidal structure map]]
[[!redirects oplax monoidal structure maps]]
[[!redirects lax comonoidal structure map]]
[[!redirects lax comonoidal structure maps]]
[[!redirects oplax monoidal transformation]]
[[!redirects oplax monoidal transformations]]
[[!redirects lax comonoidal transformation]]
[[!redirects lax comonoidal transformations]]


monoidal symmetric proset

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

# Contents
* table of contents
{:toc}

## Idea

In set theory, algebraic structures such as [[monoids]] are defined as a set with additional structure and equational axioms. However, in [[homotopy type theory]], there are two possible notions of what it means for a type to be a set, and what it means that a set has equational axioms. Traditionally in homotopy type theory, a set is defined using the [[identity type]], where every identity type between two elements $a:A$ and $b:A$ of a type $A$ is [[propositional truncation|propositionally truncated]]. However, equality in set theory is usually defined as an [[equivalence relation]], and one could directly translate the equivalence relation of equality into homotopy type theory, resulting in a [[symmetric proset]]. The equational axioms of algebraic structures could similarly be defined using the equivalence relation of the symmetric proset rather than the identity type. In the context of defining monoids, this results in the notion of a monoidal symmetric proset.

## Definition

In [[homotopy type theory]], a *monoidal symmetric proset* is a [[symmetric proset]] $(M, \equiv_M)$ with a binary function $(-)\cdot(-):M \times M \to M$, an element $1:M$, and witnesses of associativity, left unitality, right unitality, and extensionality
$$a:M, b:M, c:M \vdash \mathrm{assoc}(a, b, c):(a \cdot b) \cdot c \equiv_M a \cdot (b \cdot c)$$
$$a:M \vdash \mathrm{lunit}(a):1 \cdot a \equiv_M a$$
$$a:M \vdash \mathrm{runit}(a):a \cdot 1 \equiv_M a$$
$$a:M, b:M, c:M, d:M \vdash \mathrm{ext}(a, b, c, d):(a \equiv_M b) \times (c \equiv_M d) \to (a \cdot c) \equiv_M (b \cdot d)$$

A monoidal symmetric proset is **univalent** or a **[[monoid]]** if the canonical function
$$idtoequiv(a, b):(a =_M b) \to (a \equiv_M b)$$
is an equivalence of types for all elements $a:M$ and $b:M$.

## Properties

Every monoidal symmetric proset is (the delooping of) a [[(2, 1)-preorder]] with only 1 object.

## See also

* [[monoid]]
* [[symmetric proset]]
* [[(2, 1)-preorder]]
* [[monoidal groupoid]]
* [[monoidal category]]


monoidal topos

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

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

Every [[topos]] $\mathbf{H}$ is canonically a [[cartesian closed monoidal category]] $(\mathbf{H}, \times)$, with the [[tensor product]] being the [[cartesian product]] "$\times$". But if, in addition to this canonical product, $\mathbf{H}$ is equipped with _another_ (hence necessarily non-[[cartesian monoidal category|cartesian]]) [[monoidal category|monoidal]] [[tensor product]] [[structure]] "$\otimes$", then $(\mathbf{H}, \otimes)$ might be called a _monoidal topos_.

If the extra monoidal structure is also [[closed monoidal category|closed]], one also speaks, generally, of a *[[doubly closed monoidal category]]*-[[structure]] (with [[internal logic]] a *[[bunched logic]]*). Hence instead of "monoidal topos" it might make sense to speak of "doubly monoidal topos".

Of course, a topos may carry more than one non-cartesian monoidal category structure, and in some of the interesting examples, the various monoidal structures will be interlinked in some way. A typical example is the phenomenon of [[plethysm]].

## Examples

\begin{example}
\label{ViaDayConvolution}
For $(\mathcal{C}, \otimes)$ a small [[monoidal category]], the [[presheaf topos]] $[\mathcal{C}, Set]$ is naturally a [[doubly closed monoidal category|doubly closed monoidal]] topos with respect to the [[Day convolution]] tensor product.
\end{example}

\begin{example}\label{SliceToposOverAMonoidObject}
**(slice topos over a monoid object)**
\linebreak
For $\mathbf{H}$ a [[topos]] and $\mathbb{G} \in \mathbf{H}$ a [[monoid object]], the [[slice topos]]  $\mathbf{H}_{/\mathbb{G}}$ inherits a non-cartesian monoidal structure $\otimes_G$ induced from the product structure on $\mathbb{G}$:

$$
  \left[
    \array{
      X_1
      \\
      \downarrow^{\mathrlap{\chi_1}}
      \\
      \mathbb{G}
   }
   \;\;
  \right]
  \otimes_{\mathbb{G}}
  \left[
    \array{
      X_2
      \\
      \downarrow^{\mathrlap{\chi_2}}
      \\
      \mathbb{G}
   }
   \;\;
  \right]
  \;\;
  \coloneqq
  \;\;
  \left[
    \array{
      X_1 \times X_2
      \\
      \downarrow^{\mathrlap{(\chi_1, \chi_2)}}
      \\
      \mathbb{G} \times \mathbb{G}
      \\
      \downarrow^{\mathrlap{\cdot}}
      \\
      \mathbb{G}
    }
    \;\;\;\;\;
  \right]
  \,.
$$

Such monoidal toposes are always biclosed and appear as categories of [[local action functionals]] in the context of _[[motivic quantization]]_.
\end{example}

\begin{remark}
This Ex. \ref{SliceToposOverAMonoidObject} overlaps with the previous Ex. \ref{ViaDayConvolution} in the case where the given topos is [[Set|$Set$]] and the small category $\mathcal{C}$ is the [[discrete category]] on a [[small set]] $X$. In that case, the monoidal category structure is just a [[monoid]] [[structure]] on $X$ and the two definitions of a monoidal structure agree up to [[isomorphism]].
\end{remark}

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

The [[classifying topos for the theory of objects]] -- namely the [[presheaf topos]] $[FinSet, Set]$ on the [[opposite category]] of [[FinSet]], or equivalently the category of [[finitary monad|finitary]] [[endofunctors]] $End_f(Set)$ -- is naturally a monoidal topos under [[composition]] of endofunctors. A standard textbook reference is ([Johnstone D3.2](#Johnstone)). The [[enriched category theory]] over this monoidal category is discussed in ([Garner 13](#Garner13)).

=--

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

There are several variations on the theme of the previous example \ref{ClassifyingToposForTheoryOfObjects}, where one considers [[presheaves]] on the "$\underline{\;\;\;\;\;}$" monoidal category $M$ generated by a single object, where the "blank" may be filled in with some structure-connoting adjective. In the previous example, we had that $FinSet^{op}$ is the [[free construction|free]] [[cartesian monoidal category]] generated by a single object, but the blank might be filled in with "symmetric" ($M$ is the [[permutation category]] $\mathbb{P} = Core(FinSet)$), or "braided" (the [[braid category]] $\mathbb{B}$), or "semicartesian symmetric monoidal" ([[FinInj|$Fin_{inj}$]]: finite sets and [[injections]] between them), or "strict" (the discrete category $\mathbb{N}$), and so on. In each of these cases, the [[presheaf topos]] $E = [M^{op}, Set]$ acquires a [[Day convolution]] structure, but more interestingly, we have an [[equivalence of categories|equivalence]]

$$E \simeq \{E, E\}$$

where the right side is the category of cocontinuous $\underline{\;\;\;\;\;}$ monoidal endofunctors. By transferring the endofunctor composition across the equivalence, $E$ acquires another monoidal product structure. In the classical symmetric case, this goes under names like [[plethysm|plethystic monoidal product]], and in each such case the notion of monoid therein gives a notion of [[operad]]. See also [[generalized multicategory]]. For more on this see at _[operad -- A detailed conceptual treatment](http://ncatlab.org/nlab/show/operad#Conceptual)_.

=--

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

The category of [[dendroidal sets]] is a topos (the [[presheaf topos]] on the [[tree category]]) and naturally carries the non-cartesian [[Boardman-Vogt tensor product]].

=--

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

In [[Dolan 2011](#Dolan11)] is indicated a category of "toric [[quasicoherent sheaves]]" which is a topos and hence under tensor product of quansicoherent sheaves a (semi?)-monoidal topos.

=--

\begin{example}\label{TangentInfinityTopos}
  Given an [[(\infinity,1)-topos|$\infty$-topos]] $\mathbf{H}$, its [[tangent (infinity,1)-topos|tangent $\infty$-topos]] $T \mathbf{H}$ carries the [[external tensor product|external]] [[smash product of spectra]].

Similarly, given a [[stable (infinity,1)-category|stable $\infty$-category]] $\mathcal{A}$ (or more generally a [[Joyal locus]], see there for more), the [[(infinity,1)-category]] $\int_{\mathcal{X} \in Grpd_\infty} Func(\mathcal{X}, \mathcal{A})$ of [[infinity-Grpd|$Grpd_\infty$]]-[[parameterized stable homotopy theory|parameterized]] $\mathcal{A}$-objects (is an [[(infinity,1)-topos|$\infty$-topos]] and) should carry the corresponding [[external tensor product]].
\end{example}

## Related pages

* [[Day convolution]]

* [[doubly closed monoidal category]]

* [[bunched logic]]

## References

For references on [[Day convolution]] see there.

The monoidal [[classifying topos for the theory of objects]] is discussed for instance in

* {#Johnstone} [[Peter Johnstone]], section D3.2 in _[[Sketches of an Elephant]]_

* {#Garner13} [[Richard Garner]], _Lawvere theories, finitary monads and Cauchy-completion_ ([arXiv:1307.2963](http://arxiv.org/abs/1307.2963))

A monoidal classifying topos of "toric quasicoherent sheaves" is indicated in

* {#Dolan11} [[James Dolan]], _tannakian correspondence for toric varieties (sketch for a doctoral thesis)_, December 2011 ([web](http://notebook360x.blogspot.nl/2011/12/toposes-of-toric-quasicoherent-sheaves.html))

[[!redirects monoidal toposes]]
[[!redirects monoidal topoi]]

[[!redirects doubly monoidal topos]]
[[!redirects doubly monoidal toposes]]
[[!redirects doubly monoidal topoi]]

[[!redirects monoidal (infinity,1)-topos]]
[[!redirects monoidal (infinity,1)-toposes]]
[[!redirects monoidal (infinity,1)-topoi]]

[[!redirects doubly monoidal (infinity,1)-topos]]
[[!redirects doubly monoidal (infinity,1)-toposes]]
[[!redirects doubly monoidal (infinity,1)-topoi]]

[[!redirects monoidal (∞,1)-topos]]
[[!redirects monoidal (∞,1)-toposes]]
[[!redirects monoidal (∞,1)-topoi]]

[[!redirects doubly monoidal (∞,1)-topos]]
[[!redirects doubly monoidal (∞,1)-toposes]]
[[!redirects doubly monoidal (∞,1)-topoi]]


monoidally cocomplete category

A _monoidally cocomplete category_ is a (small-) [[cocomplete category]] $C$ bearing a [[monoidal category]] structure, such that the monoidal product $\otimes$ is [[cocontinuous functor|cocontinuous]] in each of its separate variables, i.e., $X \otimes -$ is cocontinuous for each $X$ and $- \otimes Y$ is cocontinuous for each $Y$. The term is due to [[Max Kelly]].

Similarly, one has the notions of symmetric monoidally cocomplete category, braided monoidally cocomplete category, cartesian monoidally cocomplete category, and so on.

Under nice conditions on the category $C$, the cocontinuity in separate variables of the monoidal product implies that $C$ is [[monoidal closed category|monoidal biclosed]]. For instance, if $C$ is [[locally presentable category|locally presentable]] or is [[total category|total]], then being symmetric monoidally cocomplete is equivalent to being symmetric monoidal closed.

Any cocomplete [[cartesian closed category]] is monoidally cocomplete.

[[Day convolution]] provides a construction of the free monoidally cocomplete category on a small monoidal category.

## Related pages

- [[monoidal category]]
- [[cocomplete category]]
- [[Day convolution]]

## References

* Geun Bin Im and G. M. Kelly, _A universal property of the convolution monoidal structure_, J. Pure Appl. Algebra 43 (1986), no. 1, 75-88.


monomial

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###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[sum|summands]] in a [[polynomial]] are called _monomials_.

## References

* Wikipedia, _[Monomial](https://en.wikipedia.org/wiki/Monomial)_

[[!redirects monomials]]


monomial matrix

+-- {: .rightHandSide}
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### Context
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

A **monomial matrix** with entries in a [[field]] $F$ is an $n \times n$ [[matrix]] with exactly one nonzero entry in each row and exactly one nonzero entry in each column. Under matrix multiplication, monomial matrices form a [[group]]; this group is isomorphic to a [[semidirect product group|wreath product]]

$$S_n \ltimes (F^\ast)^n$$

with respect to the canonical action of the [[permutation group]] $S_n$ on the $n^{th}$ power $(F^\ast)^n$ of the group of nonzero scalars.

The special case where the non-vanishing entries are 1 is _[[permutation matrices]]_

[[!redirects monomial matrices]]


monomorphism

> This entry is about the basic notion of "monic" in relation to [[morphisms]] in [[category theory]]. For the notion of "monic" in relation to [[polynomials]] in [[commutative algebra]], see at _[[monic polynomial]]_.

***

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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of  _monomorphism_ is the generalization of the notion of _injective map of sets_ from the [[category]] [[Set]] to arbitrary [[category|categories]].

The [[formal duality|formally dual]] concept is that of _[[epimorphism]]_, which similarly generalizes (or strengthens) the concept of [[surjective function]].

Common jargon includes "is a mono" or "is monic" for "is a monomorphism", and "is an epi" or "is epic" for "is an epimorphism", and "is an iso" for "is an isomorphism".

## Definition

A [[morphism]] $f \colon X \to Y$ in some [[category]] is called a _monomorphism_ (sometimes abbrieviated to _mono_, or described as being _monic_), if for every [[object]] $Z$ and every [[pair of parallel morphisms]] $g_1,g_2 \colon Z \to X$ then

$$
  \left(
    f \circ g_1
    \,=\,
    f \circ g_2
  \right)
  \;\Rightarrow \;
  \left(
    g_1 \,=\, g_2
  \right)
  \,.
$$

Stated more abstractly, this says that $f$ is a monomorphism if for every $Z$ the [[hom-functor]] $Hom(Z,-)$ takes it to an [[injective function]] between [[hom-sets]]

$$
  Hom(Z,X)
    \overset{\phantom{AA} f_\ast \phantom{AA}}{\hookrightarrow}
  Hom(Z,Y)
  \,.
$$

Since injective functions are precisely the monomorphisms in [[Set]] (example \ref{MonomorphismsInSet} below) this may be stated as saying that $f$ is a monomorphism if $Hom(Z,f)$ is a monomorphism for all objects $Z$.

Finally, $f$ being a monomorphism in a [[category]] $\mathcal{C}$ means equivalently that it is an [[epimorphism]] in the [[opposite category]] $\mathcal{C}^{op}$.

## Examples

### General

+-- {: .num_example}
###### Example
**(monomorphisms in preorders)

In a [[preorder]], all arrows are mono because they satisfy the required condition vacuously (any pair of parallel arrows is equal in a preorder).

=--

+-- {: .num_example #MonomorphismsInSet}
###### Example
**(monomorphisms in $Set$)

The monomorphisms in the category [[Set]] of [[sets]] and [[functions]] between them are precisely the [[injective functions]].

=--

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

Every [[isomorphism]] is both a monomorphism and an [[epimorphism]].

=--

But beware that the converse fails:

### Examples of monos that are epi but not iso
 {#ExamplesOfMonosThatAreEpiButNotIso}

The following lists some examples of [[morphisms]] that are both [[monomorphisms]] and [[epimorphisms]], but not necessarily [[isomorphisms]].

\begin{example}

In the [[category]] of [[Hausdorff topological spaces]], the inclusion $A \hookrightarrow X$ of a [[dense subspace]] is an [[epimorphism]].

\end{example}

See [this Prop.](Hausdorff+space#DenseSubspaceInclusionsAreEpimorphismsAmongHausdorffSpaces) for proof.

\begin{example}

In [[unital ring|unital]] [[Rings]], the canonical inclusion $\mathbb{Z} \overset{i}{\hookrightarrow} \mathbb{Q}$ of the [[integers]] into the [[rational numbers]] is an [[epimorphism]].

\end{example}

See [this Prop.](Ring#InclusionOfIntegersIntoRationalsIsEpimorphismOfRings) for proof.

## Properties

We list the following properties without their (easy) proofs. The proofs can be found spelled out (dually) at [[epimorphism]].

\begin{prop}\label{BasicCharacterizationOfMonomorphisms}
The following are equivalent:

* $f : x \to y$ is a monomorphism in $C$;

* $f$ is an [[epimorphism]] in the [[opposite category]] $C^{op}$;

* postcomposition with $f$ is a [[monomorphism]] in [[Set]]: that is, for all $c \in C$, $f \circ -: Hom(c,x) \to Hom(c,y)$ is an [[injection]];

* the [[commuting diagram]]
  $$
    \array{
      x &\stackrel{Id}{\to}& x
      \\
      {}^{Id}\downarrow && \downarrow^{f}
     \\
     x &\underset{f}{\to}& y
    }
  $$

  is a [[pullback]] diagram.

\end{prop}

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

If $f \colon x \to y$ and $g \colon y \to z$ are monomorphisms, so is their composite $g f$.  If $g f$ is an monomorphism, so is $f$.

=--

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

Every [[equalizer]]

$$
  t \to x \stackrel{\longrightarrow}{\longrightarrow} y
$$

is a monomorphism.

=--

The converse of the above proposition fails, and a mononomorphism that is the equalizer of some pair of morphisms is called a [[regular monomorphism]].

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

Monomorphisms are preserved by [[pullback]].

=--

(In an [[adhesive category]] they are also preserved by [[pushout]].)

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

Monomorphisms are preserved by any [[right adjoint]] [[functor]], or more generally any functors that preserves pullbacks.

=--

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

Monomorphisms are [[reflected limit|reflected]] by [[faithful functors]].

=--

We have seen some ways in which monomorphisms get along with limits.  Here is another:

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

Any morphism from a terminal  object is a monomorphism.  The product of monomorphisms is a monomorphism.
=--

Monomorphisms do not get along quite as well with colimits.  For example, the unique morphism from the initial object is not always an monomorphism, and the canonical morphisms from the summands into a coproduct, e.g. $i_1 : x_1 \to x_1 + x_2$, are not always monomorphisms (though these results do hold in $Set$). However, the unique morphism from the initial object is a monomorphism when the initial object is [[strict initial object|strict]], and the canonical morphisms into a coproduct are monomorphisms when the coproduct is a [[disjoint coproduct]].

## Variations

At [epimorphism](/nlab/show/epimorphism#Variations) there is a long list of variations on the concept of epimorphism.  Each of these, of course, has a dual notion for monomorphism, but the ones most commonly used are:

* [[split monomorphism]] = morphism which has a [[retraction]]
* [[normal monomorphism]] = a [[kernel]] of some morphism (in a category with [[zero morphisms]])
* [[regular monomorphism]] = an [[equalizer]] of some [[parallel pair]]
* [[strong monomorphism]] = a monomorphism [[orthogonality|right orthogonal]] to [[epimorphisms]]
* monomorphism

Frequently, regular and strong monos coincide.  For instance, this is the case in any [[quasitopos]], and also in [[Top]], where they are the subspace inclusions (the plain monomorphisms are the injective functions).

Sometimes, all monomorphisms are regular---this seems to happen a bit more frequently than for epimorphisms.  For instance, this is the case in any [[pretopos]] (including any [[topos]], such as [[Set]]), but also in any [[abelian category]], and also in the category [[Grp]].

In [[Ab]] and in any [[abelian category]], all monomorphisms are normal.  But this is not so in [[Grp]], where (despite the fact that all monomorphisms are regular), the normal monos are the inclusions of [[normal subgroups]] (hence the name).  In any [[Ab-enriched category]], all regular monos are normal, but not all monos need be regular.

In a [[Boolean topos]], such as [[Set]] (in [[classical mathematics]]), any monomorphism with [[inhabited set|inhabited]] domain is split.  Of course, no mono with empty domain and inhabited codomain can be split (in contrast to the dual case, where it can happen that all epimorphisms split -- the [[axiom of choice]]).

## Related concepts

* [[isomorphism]] classes of monomorphism define [[subobject|subobjects]].

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

* [[image]], [[coimage]]

* a category in which all morphisms are monomorphisms is called a _[[left cancellative category]]_

* [[jointly monic morphisms]]

## References

Textbook accounts:

* [[Saunders MacLane]], §I.5 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* [[Francis Borceux]], Section 1.7 in: *[[Handbook of Categorical Algebra]]* Vol. 1: *Basic Category Theory* [[doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858)]

[[!redirects monomorphisms]]
[[!redirects monic]]
[[!redirects monics]]
[[!redirects mono]]
[[!redirects monos]]
[[!redirects Monomorphism]]
[[!redirects Monomorphisms]]
[[!redirects Monic]]
[[!redirects Monics]]
[[!redirects Mono]]
[[!redirects Monos]]


monomorphism in a derivator

## Idea

A *monomorphism in a derivator* is the generalization to the context of a [[derivator]] of the notion of [[monomorphism]] in ordinary category theory.  Viewing a derivator as the "shadow" of an [[(∞,1)-category]], the notion of monomorphism therein coincides with the notion of [[monomorphism in an (∞,1)-category]].

## Definition

Let $\square$ denote the category
$$ \array{a & \to & b \\ \downarrow & & \downarrow \\ c & \to & d} $$
that is the "free-living [[commutative square]]", let $I$ be the [[interval category]] $(0\to 1)$, and let $p\colon \square \to I$ denote the functor collapsing $a,b,c$ to $0$ and sending $d$ to $1$.

Let $D$ be a [[prederivator]] and $f\colon X\to Y$ a morphism in $D(1)$.  By one of the axioms of a derivator, there exists an object $F\in D(I)$ representing $f$, which is unique up to non-unique isomorphism.  We say that $f$ is a **monomorphism** in $D$ if $p^*(F) \in D(\square)$ is a [[pullback in a derivator|pullback square]].

## Examples

* It is well-known and easy to verify that a morphism $f:A\to B$ in a 1-category is a [[monomorphism]] if and only if the square
  $$ \array{ A & \overset{id}{\to} & A \\ ^{id}\downarrow & & \downarrow^f \\ A &\underset{f}{\to} & B } $$
  is a [[pullback]].  Therefore, in representable prederivators this definition reduces to the usual notion of monomorphism.

* In the homotopy derivator of an $(\infty,1)$-category, one can check that this reduces to the usual notion of [[monomorphism in an (∞,1)-category]].

[[!redirects monomorphisms in a derivator]]
[[!redirects monomorphisms in derivators]]


monomorphism in an (infinity,1)-category

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###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _monomorphism in an $(\infty,1)$-category_ is the generalization of the notion of [[monomorphism]] from [[category theory]] to [[(∞,1)-category]] theory. It is the special case of the notion of [[n-monomorphisms]] for $n = 1$. In an [[(∞,1)-topos]] every morphism factors by an [[effective epimorphism in an (∞,1)-category|effective epimorphism]] (1-epimorphism) followed by a monomorphism through its [[n-image|1-image]].

The dual concept is that of an [[epimorphism in an (∞,1)-category]].

There is also the concept [[regular monomorphism in an (∞,1)-category]], but beware that this need not be a special case of the definition given here.

There are also a notions of (homotopy) monomorphism in [[model categories]] and [[derivators]].

## Definition

For $\mathcal{C}$ an [[(∞,1)-category]], a [[morphism]] $f \colon Y \to Z$ is a **monomorphism** if regarded as an object in the [[over quasi-category|(∞,1)-overcategory]] $\mathcal{C}_{/Z}$ it is a [[n-truncated object in an (∞,1)-category|(-1)-truncated object]].

Equivalently this means that the projection

$$
  \mathcal{C}_{/f} \longrightarrow \mathcal{C}_{/Z}
$$

is a [[full and faithful (∞,1)-functor]]. This is in [Higher Topos Theory](#ref) after Example 5.5.6.13.

Equivalently this means that for every object $X \in C$ the induced morphism

$$
  \mathcal{C}(X,f)
  \;\colon\;
  \mathcal{C}(X,Y)
    \longrightarrow
  \mathcal{C}(X,Z)
$$

of [[∞-groupoids]] is such that its image in the [[homotopy category of an (∞,1)-category|homotopy category]] exhibits $\mathcal{C}(X,Y)$ as a [[disjoint union|disjoint summand]] in a  [[coproduct]] decomposition of $\mathcal{C}(X,Z)$.

So if

$$
  \mathcal{C}(X,Y)
  \;=\;
  \underset{i \in \pi_0\mathcal{C}(X,Y)}{\coprod} \mathcal{C}(X,Y)_{i }
  \;\;\;\;\;\;
  \text{and}
  \;\;\;\;\;\;
  \mathcal{C}(X,Z)
  \;=\;
  \underset{j \in \pi_0(\mathcal{C}(X,Z)}{\coprod} \mathcal{C}(X,Z)_j
$$

is the decomposition into [[connected components]], then there is an [[injective function]]

$$
  j
    \,\colon\,
  \pi_0 \mathcal{C}(X,Y)
  \longrightarrow
  \pi_0 \mathcal{C}(X,Z)
$$

such that $\mathcal{C}(X,f)$ is given by component maps $\mathcal{C}(X,Y)_i \to \mathcal{C}(X,Z)_{j(i)}$ which are each an equivalence.

## Properties

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

For $Z$ an [[object]] of $\mathcal{C}$, write $Sub(Z)$

$$
  Sub(Z)
    \;\coloneqq\;
  \tau_{\leq -1}\big(  C_{/Z} \big)
  \,.
$$

for the category of [[subobjects in an (∞,1)-category|subobjects]] of $\mathcal{C}$.

=--

This is [[poset|partially ordered]] under inclusion.

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

If $\mathcal{C}$ is a [[presentable (∞,1)-category]], then $Sub(Z)$ is a [[small category]].

=--

This appears as [[Higher Topos Theory|HTT, prop. 6.2.1.3]].

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

Monomorphisms are stable under [[(∞,1)-pullback]]: if

$$
  \array{
    A &\to& B
    \\
    {}^{\mathllap{f'}}\big\downarrow && \big\downarrow^{\mathrlap{f}}
    \\
    C &\to& D
  }
$$

is a pullback diagram and $f$ is a monomorphism, then so is $f'$.

=--

This is a special case of the general statement that $k$-[[truncated]] morphisms are stable under pullback. ([[Higher Topos Theory|HTT, remark 5.5.6.12]]).

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

In an [[(∞,1)-topos]], monomorphisms are stable under [[(∞,1)-pushout]]: if

$$
  \array{
    A &\to& B
    \\
    {}^{\mathllap{f}}\big\downarrow
    &&
    \big\downarrow{}^{\mathrlap{f'}}
    \\
    C &\to& D
  }
$$

is a [[homotopy pushout]] diagram and $f$ is a monomorphism, then so is $f'$.

=--

([Rezk 19, p. 21](#Rezk19))

## Related pages

* The equivalence class of a monomorphism is a [[subobject in an (∞,1)-category]].

* The notion of monomorphism in an $(\infty,1)$-category can also be characterized in its underlying homotopy [[derivator]]; see [[monomorphism in a derivator]].

## References

The definition appears after example 5.5.6.13 in

* [[Jacob Lurie]], _[[Higher Topos Theory]]_{#ref}

with further discussion in section 6.2.

Lecture notes:

* {#Rezk19} [[Charles Rezk]], p. 10 of: _Lectures on Higher Topos Theory_, Leeds 2019 ([pdf](https://faculty.math.illinois.edu/~rezk/leeds-lectures-2019.pdf), [[RezkHigherToposTheory2019.pdf:file]])

For [[model categories]], see

* [[Fernando Muro]], _Homotopy units in A-infinity algebras_, [arXiv:1111.2723](http://arxiv.org/abs/1111.2723).

[[!redirects monomorphism in an (∞,1)-category]]
[[!redirects monomorphisms in an (∞,1)-category]]
[[!redirects monomorphisms in an (infinity,1)-category]]
[[!redirects homotopy monomorphism]]
[[!redirects homotopy monomorphisms]]


monopole

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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
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=--
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#Contents#
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## Cosmic monopoles

In a [[gauge theory]] with a degenerate [[vacuum]] (such as when a [[Higgs mechanism]] applies), the [[moduli space]] of [[vacua]] is the [[quotient]] $G/H$ (the [[coset]]) of the [[gauge group]] $G$ by the [[stabilizer subgroup]] $H \hookrightarrow G$ of any of these vacua ([[spontaneous symmetry breaking]]).

This means that gauge equivalence classes of vaccum configurations on a [[spacetime]] $X$ are given by [[homotopy]] classes of maps $X \to \Pi(G/H)$ (where the notation on the right denotes the underlying [[homotopy type]] of the [[coset]] space, $\Pi$ is the [[shape modality]]).

If [[spacetime]] is locally to be taken of the form $\mathbb{R} \times (\mathbb{R}^3 - D^3 \times \mathbb{R}^0)$, hence with a 3-dimensional ball-like piece taken out, them homotopy classes of maps $X \to \Pi(G/H)$ are classified by the second [[homotopy group]] $\pi_2(G/H)$. For a given nontrivial element here the correponding [[vacuum]] is said to contain a **cosmic monopole defect**. ("Cosmic" just because this effect is thought to be most relevant on scales of [[cosmology]].)

In other words this means that the vacuum strcture changes continuously as one moves around the string, but has a [[singularity]] on the locus of the string itself.

For more see at _[[QFT with defects]]_ the section _[Topological defects from spontaneously broken symmetry](http://ncatlab.org/nlab/show/QFT+with+defects#DefectsFromBrokenSymmetry)_.

## Examples

### Magnetic monopoles
 {#MagneticMonopoles}

#### Dirac monopole
 {#DiracMonopole}

In the original [[Dirac charge quantization]] argument an [[electromagnetic field]] is considerd _outside_ a given point in space (line in [[spacetime]]). Mathematically this just corresponds to considering electromagfnetism on the maifold $(\mathbb{R}^3-D^0) \times \mathbb{R}$, but physically this may be taken to model a situation where there is a magnetic monopole defect in the vavuum structure. The _[[Dirac monopole]]_.

This is the classical example of monopoles and often taken to be the default meaning. (In the context of the [[standard model of particle physics]], however, the default meaning is that of electroweak vacuum defect monopoles, discussed [below](#ElectroweakMonopoles).)

Technically, in the sense of the discussion at [[QFT with defects]] in the section _[Topological defects from spontaneous symmetry breaking](QFT%20with%20defects#DefectsFromBrokenSymmetry)_ what happens here is that the [[moduli space]] of the [[instanton sector]] of electromagnetism (= the [[classifying space]] of [[circle group]]-[[principal bundles]]) $B U(1)$ has the [[homotopy type]] of an [[Eilenberg-MacLane space]] $B U(1) \simeq K(\mathbb{Z},2)$. This is by definition a space such that [[homotopy classes]] of [[continuous functions]] from a [[sphere]] into it are classified by the [[integers]]: $\pi_2(B U(1)) \simeq \mathbb{Z}$.

But since there is a [[homotopy equivalence]]

$$
  (\mathbb{R}^3 - D^0)\times \mathbb{R} \simeq S^2
$$

of [[Minkowski spacetime]] with the [[worldline]] of a point removes, this means precisely that the possible [[instanton]] field configurations of the [[electromagnetic field]] on such a spacetime is given by an [[integer]]. This integer is called the _[[magnetic charge]]_ of the monopole defect. (Mathematically it is the [[first Chern class]] of the $U(1)$-[[principal bundle]] which underlies the electromagnetic field).

#### Kaluza-Klein monopole

The gravitational incarnation of the [[Dirac monopole]] under [[Kaluza-Klein compactification]] is the _[[Kaluza-Klein monopole]]_. See there for more.

#### Yang monopole and 't Hooft monopole

The generalization of the [[Dirac monopole]] from $U(1)$-[[Yang-Mills theory]] in 3+1 dimensional [[spacetime]] to $SU(2)$-[[Yang-Mills theory]] in 5+1 dimensional spacetime is the _[[Yang monopole]]_.

With the [[Higgs field]] included, there is the _[['t Hooft-Polyakov monopole]]_.

### Electroweak monopoles
 {#ElectroweakMonopoles}

For a $G$-[[gauge theory]] with [[spontaneous symmetry breaking]] of the [[vacuum]] from [[gauge group]] $G$ down to some [[subgroup]] $H \hookrightarrow G$, monopole [[topological defects]] in the vacuum structure are classified by the second [[homotopy group]] $\pi_2(G/H)$ of the [[coset space]] (as explained at _[[QFT with defects]]_).

For the gauge theory of the [[electroweak force]] with its [[Higgs mechanism]] that is of relevance in the [[standard model of particle physics]]. This would mean that monopoles are classified by

$$
  \pi_2( (SU(2) \times U(1))_{ew} / U(1)_{em} )
  \,.
$$

However, in ([Cho-Maison](#ChoMaison)) it is claimed that the true topology of the moduli space of vacua in electroweak symmetry breaking is topologically more interesting. The corresponding phenomenology of vacuum defect monopoles of this kind is discussed in ([Cho-Kim-Yoon13](#ChoKimYoon13)).

## Related concepts

* [[Yang-Mills monopole]]

* [[fiber bundles in physics]]

* [[domain wall]], [[cosmic string]]

* [[vortex]]

* [[caloron correspondence]]

* [[dyon]]

* [[Nahm transform]], [[Bogomolny equation]]

[[!include fields and quanta - table]]

## References

Survery:

* [[John Preskill]], *Magnetic Monopoles*, Ann. Rev. Nucl. Part. Sci. **34** (1984) 461-530 [[doi:10.1146/annurev.ns.34.120184.002333](https://doi.org/10.1146/annurev.ns.34.120184.002333)]

Textbook accounts:

* [[Mikio Nakahara]], Section 10.5.2 of: _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

* [[Yakov Shnir]], *Magnetic Monopoles*, Springer 2005 ([ISBN:978-3-540-29082-7](https://www.springer.com/gp/book/9783540252771))

The original argument that magentic monopoles may appear in [[grand unified theories]]:

* [[Gerard ’t Hooft]], *Magnetic Monopoles in Unified Gauge Theories*, Nucl. Phys. B **79** (1974) 276-284 [doi:10.1016/0550-3213(74)90486-6]

On the [[phenomenology]] of magnetic monopoles:

* [[Particle Data Group]], *Magnetic Monopoles*, Section 95 in: *The Review of Particle Physics* (2022) [[pdf](https://pdg.lbl.gov/2022/reviews/rpp2022-rev-mag-monopole-searches.pdf)]

On [[Yang-Mills monopoles]]:

* [[Paul Sutcliffe]], *BPS Monopoles*, Int. J. Mod. Phys. A12:4663-4706 (1997) ([arXiv:hep-th/9707009](https://arxiv.org/abs/hep-th/9707009))

Experimental search for magnetic monopoles via the [[Schwinger effect]]:

* [[Bobby Acharya]] et al., *First experimental search for production of magnetic monopoles via the Schwinger mechanism* ([arXiv:2106.11933](https://arxiv.org/abs/2106.11933))

A general account of [[vacuum]] [[QFT with defects|defects]] such as [[domain walls]] and monopoles:

* {#VilenkinShellard94} [[Alexander Vilenkin]], E. P. S. Shellard, *Cosmic strings and other topological defects*, Cambridge University Press (1994) [[ISBN:9780521654760](https://www.cambridge.org/ae/universitypress/subjects/physics/theoretical-physics-and-mathematical-physics/cosmic-strings-and-other-topological-defects?format=PB&isbn=9780521654760), [spire:1384873](https://inspirehep.net/literature/1384873)]

Detailed discussion of the [[phenomenology]] of [[electroweak force|electroweak]] monopoles is in

* {#ChoMaison} Y.M. Cho and D. Maison, Phys. Lett. B391, 360 (1997)

* {#ChoKimYoon13} Y. M. Cho, Kyoungtae Kim, J. H. Yoon, _Finite Energy Electroweak Dyon_ ([arXiv:1305.1699](http://arxiv.org/abs/1305.1699))

Discussion of monopole [[correlators]]:

* {#Iqbal14} [[Nabil Iqbal]], _Monopole correlations in holographically flavored liquids_, Phys. Rev. D 91, 106001 (2015) ([arXiv:1409.5467](https://arxiv.org/abs/1409.5467))

[[!redirects monopoles]]

[[!redirects cosmic monopole]]
[[!redirects cosmic monopoles]]

[[!redirects magnetic monopole]]
[[!redirects magnetic monopoles]]


monotone function

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# Monotone functions
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## Idea
 {#Idea}

A function between [[preordered sets]] is called *monotone* if it respects the (pre)[[ordering]].

When [[preordered sets]] are regarded as [[categories]] (namely: [[(0,1)-categories]]) then monotone functions are equivalently the *[[functors]]* between these.

## Definition

### In components

Let $S$ and $T$ be [[preordered sets]], that is [[sets]] equipped with a [[reflexive relation|reflexive]] and [[transitive relation|transitive]] binary [[relation]] $\leq$.  (By convention, the same symbol is used for both sets, even though technically it is not the same relation.)

Then a [[function]] $f$ from $S$ to $T$ is __monotone (increasing)__, __isotone__, __weakly increasing__, or __order-preserving__ if it preserves $\leq$:
$$ x \leq y \;\Rightarrow\; f(x) \leq f(y) $$
for all $x, y$ in $S$.

A __strictly increasing__ function is a weakly increasing function that is also [[injective function|injective]], at least if $S$ and $T$ are [[partial order|partially ordered]].  Between arbitrary preordered sets, however, it is probably better to accept as strictly increasing any weakly increasing function that is weakly injective in that $x \leq y$ whenever $f(x) = f(y)$; such a function must be injective if $S$ is a partial order (since $y \leq x$ will also follow) but not necessarily in general.

A function $f$ is __monotone decreasing__, __antitone__, __weakly decreasing__, or __order-reversing__ if it reverses $\leq$:
$$ x \leq y \;\Rightarrow\; f(y) \leq f(x) $$
for all $x, y$ in $S$.

A __strictly decreasing__ function is a weakly decreasing function that is also (weakly) injective.

### Category-theoretic

As a preordered set is the same thing as a [[category]] in which any two [[parallel morphisms]] are equal, so a monotone function is simply a [[functor]] between such categories.  An antitone function is a [[contravariant functor]].  That 'monotone' may be used for both matches that 'functor' may be used for both covariant and contravariant functors.

Strictly increasing (and strictly decreasing) functions are particularly important between [[linearly ordered sets]], where they are the most natural kind of morphism.  Between partially ordered sets in general (and between preordered sets using the stricter definition), the strictly increasing functions are simply the [[monomorphisms]] (if weakly increasing functions are taken as the morphisms).  If we use the weaker definition between preordered sets, then the strictly increasing functions correspond to [[pseudomonic functors]], which is an appropriate sort of [[higher category theory|higher]] monomorphism; this is one reason for preferring that definition.

The alternative sort of monotone function on a single proset $S$ is rather different; we mention it here largely because of the potential terminological confusion, but it might as well have its own article if we find a nice name for it.  As a functor, it is a functor for which every object is an [[algebra for a functor|algebra]]; the condition is part of the requirements of a [[Moore closure]] (a [[monad]] on $S$).

## Related concepts

* [[enriched monotone]]

* [[injective function]]

* [[surjective function]]

* [[equivariant function]]

* [[opposite poset]]

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Monsky-Washnitzer cohomology

#Contents#
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## Idea

_Monsky-Washnitzer cohomology_ is a [[cohomology theory]] for [[affine scheme|affine algebraic schemes]] over [[fields]] of [[positive characteristic]].

## References

The original articles:

* P. Monsky, G. Washnitzer, _Formal cohomology. I._
Ann. of Math. (2) 88 1968 181–217, [MR248141](http://www.ams.org/mathscinet-getitem?mr=248141), [doi](http://dx.doi.org/10.2307/1970571)

* P. Monsky, _Formal cohomology. II. The cohomology sequence of a pair_, Ann. of Math. (2) 88 1968 218–238, [MR244272](http://www.ams.org/mathscinet-getitem?mr=244272), [doi](http://dx.doi.org/10.2307/1970572)

See also:

[MathSciNet](http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&s7=%22Monsky-Washnitzer+cohomology%22&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All)

[Google Scholar](http://scholar.google.co.uk/scholar?q=%22Monsky-Washnitzer+cohomology%22&hl=en&lr=&btnG=Search)

[Google](http://www.google.com/search?hl=en&q=%22Monsky-Washnitzer+cohomology%22&btnG=Search)

[arXiv: Experimental full text search](http://search.arxiv.org:8081/?query=%22Monsky-Washnitzer+cohomology%22&in=)

[arXiv: Abstract search](http://front.math.ucdavis.edu/search?a=&t=&q=%22Monsky-Washnitzer+cohomology%22&c=&n=25&s=Abstracts)


Monster group

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## Idea

The **Monster group** $M$ is a [[finite group]] that is the largest of the [[sporadic finite simple group]]s. It has [[order of a group|order]]

$$
 \begin{aligned}
& 2^{46}\cdot 3^{20}\cdot 5^9\cdot 7^6\cdot 11^2\cdot 13^3\cdot 17\cdot 19\cdot 23\cdot 29\cdot 31\cdot 41\cdot 47\cdot 59\cdot 71
  \\
 & = 808017424794512875886459904961710757005754368000000000
 \end{aligned}
$$

and contains all but six (the '[[pariah groups]]') of the other 25 [[sporadic finite simple groups]] as [[subquotients]], called the _[[Happy Family]]_.

See also [[Moonshine]].

## History

The Monster group was predicted to exist by [[Bernd Fischer]] and [[Robert Griess]] in 1973, as a [[simple group]] containing the [[Fischer groups]] and some other sporadic simple groups as [[subquotients]]. Subsequent work by Fischer, Conway, Norton and Thompson estimated the order of $M$ and discovered other properties and subgroups, assuming that it existed. In a famous paper

* [[Robert Griess]], _The Friendly Giant_ , Inventiones (1982)

Griess proved the existence of the largest simple sporadic group. The author constructs "by hand" a non-associative but commutative algebra of dimension 196883, and showed that the [[automorphism group]] of this algebra is the conjectured friendly giant/monster simple group. The name "Friendly Giant" for the Monster did not take on.

After Griess found this algebra [[Igor Frenkel]], [[James Lepowsky]] and Meurman and/or Borcherds showed that the Griess algebra is just the degree 2 part of the infinite dimensional [[Moonshine vertex algebra]].

There is a school of thought, going back to at least [[Israel Gelfand]], that sporadic groups are really members of some other infinite families of algebraic objects, but due to numerical coincidences or the like, just happen to be groups (see [this nCafe post](http://golem.ph.utexas.edu/category/2006/09/mathematical_kinds.html)). One version of this, in the case of the Monster (and perhaps for other sporadic groups via [[Moonshine]] phenomena) is that what we know as the Monster is just a shadow of a [[2-group]], as the Monster can be constructed as an automorphism group of a [[conformal field theory]], a structure rich enough to have a automorphism 2-group(oid) (see [this nCafe discussion](http://golem.ph.utexas.edu/category/2008/10/john_mckay_visits_kent.html#c019440)).

## Presentation

### Via Coxeter groups

The Monster admits a reasonably succinct description in terms of [[Coxeter groups]]. Let $[n]$ denote the linear [[graph]] with vertices $0, 1, \ldots, n$ with an edge between adjacent numbers $i, i+1$ and no others. If $1$ is the terminal (1-element) graph, there is a map $0: 1 \to [n]$, mapping the vertex of $1$ to the vertex $0$. Regarding this as an object in the [[undercategory]] $1 \downarrow Graph$, let $Y_{443}$ be the [[coproduct]] of the three objects $0: 1 \to [4]$, $0: 1 \to [4]$, $0: 1 \to [3]$ in $1 \downarrow Graph$. This (pointed) graph has 12 elements and is shaped like a $Y$, with arms of length 4, 4, 3 emanating from a central vertex of valence $3$.

Regard $Y_{443}$ as a [[Coxeter diagram]]. The associated [[Coxeter group]] $C_{443}$ is given by a [[group presentation]] with 12 generators (represented by the vertices) of order $2$ (so 12 relators of the form $x^2 = 1$), with a relation $(x y)^3 = 1$ if $x, y$ are adjacent vertices (so 11 relators, one for each edge), and $x y = y x$ if $x, y$ are non-adjacent (55 more relators). This Coxeter group (12 generators, 78 relators) is infinite, but by modding out by another strange 'spider' relator

$$(a b_1 c_1 a b_2 c_2 a b_3 c_3)^{10} = 1$$

the resulting quotient $N$ turns out to be a [[finite group]]. Here $a$ is the central vertex of valence $3$, $b_1, c_1$ are on an arm of length $4$ with $b_1$ adjacent to $a$ and $c_1 \neq a$ adjacent to $b_1$; similarly for $b_2, c_2$ on the other arm of length $4$, and for $b_3, c_3$ on the arm of length $3$. See [here](http://www.maths.qmul.ac.uk/~jnb/web/Pres/Mnst.html) if this is not clear.

It turns out that $N$ has a [[center]] $C$ of order $2$, and the Monster $M$ is the quotient, i.e. the indicated term in the exact sequence

$$1 \to C \to N \to M \to 1.$$

This implicitly describes the Monster in terms of 12 generators and 80 relators.

Such "$Y$-group" presentations (Coxeter group based on a similar $Y$-diagram, modulo a spider relation) are linked to a number of finite simple group constructions, the most famous of which is perhaps $Y_{555}$ which is a presentation of the "Bimonster" (the [[wreath product]] of the Monster with $\mathbb{Z}/2$). See [Ivanov](#Iv) for a general description of these. The presentation of the Monster given above was established in [Ivanov2](#Iv2).

### Via automorphisms of a super vertex operator algebra
 {#ViaAutomorphisms}

There is a [[super vertex operator algebra]], the [[Monster vertex operator algebra]], whose [[automorphism group|group of]] of [[automorphisms of a VOA]] is the [[monster group]].

  ([Frenkel-Lepowski-Meurman 89](#FrenkelLepowskiMeurman89), [Griess-Lam 11](#GriessLam11))

## Related concepts

* [[Moonshine]],

* [[Monster vertex algebra]]

* [[Mathieu group]], [[Mathieu moonshine]]

## References

* [Adam P. Goucher](http://mathoverflow.net/users/39521/adam-p-goucher), _Presentation of the Monster Group_, ([MO comment 2013-09-15](http://mathoverflow.net/q/142216))

* {#Iv} Alexander Ivanov, _Y-groups via transitive extension_, Journal of Algebra, Volume 218, Issue 2 (August 15, 1999), 412–435. ([web](http://www.sciencedirect.com/science/article/pii/S0021869399978821))

*  {#Iv2} A. A. Ivanov, _Constructing the Monster via its Y-presentation_,
in Combinatorics, Paul Erdős is Eighty, Bolyai Society Mathematical Studies, Vol. 1 (1993), 253-270.

* {#FrenkelLepowskiMeurman89} [[Igor Frenkel]], [[James Lepowsky]], Arne Meurman, _Vertex operator algebras and the monster_, Pure and Applied Mathematics __134__, Academic Press, New York 1998. liv+508 pp. [MR0996026](http://www.ams.org/mathscinet-getitem?mr=996026)

* {#GriessLam11} [[Robert Griess]] Jr., Ching Hung Lam, _A new existence proof of the Monster by VOA theory_ ([arXiv:1103.1414](https://arxiv.org/abs/1103.1414))

* [[Andre Henriques]], _[$H^4$ of the monster](http://mathoverflow.net/questions/69222/h4-of-the-monster)_

Possible relation to [[bosonic M-theory]]:

* [[Alessio Marrani]], [[Michael Rios]], [[David Chester]], *Monstrous M-theory*, Symmetry **15** 2 (2023) 490; [[doi:10.3390/sym15020490](https://doi.org/10.3390/sym15020490), [arXiv:2008.06742](https://arxiv.org/abs/2008.06742)]

[[!redirects Monster]]

[[!redirects monster group]]

[[!redirects Monster simple group]]


monster model

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## Idea
A monster for a first-order theory $T$ is a sort of "equivariant Grothendieck universe" for the definable sets of that theory. The notion generalizes that of Weil's universal domain (which are just monster models of the theory [[theory of algebraically closed fields|ACF]] of algebraically closed fields.)

The idea, to quote Hrushovski (see more below) is to bring out the geometry of a situation with minimal intervention of particular algebraic structures, for example allowing us to avoid many base changes.

## Definition
The most important aspects of monster models are _saturation_ and _homogeneity_: all [[type (in model theory)|types]] over parameter sets of a certain size are realized, and all elementary embeddings of subsets can be extended to automorphisms. Most commonly the monster is assumed to be the size of an inaccessible cardinal $\kappa$, with $\kappa$-saturation (all types over parameter sets of size less than $\kappa$ are realized) and $\kappa$-homogeneity (all elementary embeddings of subsets of size less than $\kappa$ are extended by automorphisms).

## Facts about monsters

* Compactness and saturation yield, in the monster, that:

1. families of definable sets (definable over a small parameter set) with nonempty finite intersections have nonempty total intersection,

2. small unions of definable sets (definable over a small parameter set) are finitely supported, and

3. small intersections of definable sets (definable over a small parameter set) are finitely supported.

* All small ($ < \kappa = \mathbb{M} \models T$) models of $T$ elementarily embed into $\mathbb{M}$.

* Two tuples $a,b$ will have the same type $\operatorname{tp}(a/A) = \operatorname{tp}(b/A)$ over a small parameter set $A$ if and only if they are $\operatorname{Aut}(\mathbb{M}/A)$-conjugate.

* Sometimes it is useful to talk about large types, particularly those over all of $\mathbb{M}$ as a parameter set. These are called _global [[type (in model theory)|types]]_, and are usually not realized.

## Examples
Monster models in general tend to be hard to explicitly describe, except when theories are "well-behaved", e.g. stable (when type spatn the number of isomorphism classes of models in a certain cardinality is $1$). For example, both are true for $\mathsf{ACF}$.

By a back-and-forth argument (e.g. the one used by Cantor to prove the uniqueness of the [[DLO|countable dense linear order without endpoints]]), one can show that any two monster models are well-defined up to isomorphism, up to cardinality.

## Comparison with scheme approach to algebraic geometry
From Appendix B of Ehud Hrushovski's _Computing the Galois group of a linear differential equation_:

_"The distinctions between [treating formulas as primary objects, or taking seriously abstract algebraic structures to the point of ignoring formulas, or treating formulas as representable functors, or just working in a universal domain] relate to the revolution of schemes only at its surface. Grothendieck's real contribution, in this respect, was no the change of language, but the incorporation of infinitesimals, and of homological algebra, as intrinsic parts of the geometry. These go beyond Weil's original universal domain, not because it is a universal domain, but because it is a universal domain for the wrong class of structures (rings, and without nilpotents)..._

_The most serious objection to universal domains is that they apply only when the structures in question admit amalgamation; and preferably a certain finiteness property allowing a model companion. Do these properties hold for an appropriate presentation of schemes with infinitesimals, or for varieties with coherent sheaves over them? There has been little work on these questions; perhaps because of the view, widely held and published by model-theorists, that [[ACF]] remains the first-order theory underlying algebraic geometry; that Grothendieck somehow changed the logical way of looking at it, rather than the theory itself. The answer to the second question appears to be positive. For the first, Cherlin showed that commutative rings with infinitesimals do not have a model companion in the language of rings, essentially because the order of infinitesimals must be taken seriously. But once this is done, a positive answer may exist (Truncated valuation rings have a good model theory, and give a partial response.)"_

## Related concepts
* [[type (in model theory)]]
## References
- John Baldwin, _The Monster Model_ [(pdf)](http://homepages.math.uic.edu/~jbaldwin/pub/monster.pdf)

- John Baldwin, _How big should the monster model be? [(pdf)](http://homepages.math.uic.edu/~jbaldwin/pub/monster4.pdf)