monadic decomposition

# Monadic decompositions

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{: toc}

## Idea

Suppose $F_1: C_1 \rightleftarrows D : U_1$ is an [[adjunction]], with induced [[monad]] $T_1 = U_1 F_1$ on $C_1$.  Then we can form the [[Eilenberg-Moore category]] $C_2 \coloneqq C_1^{T_1}$, and the [[comparison functor]] $U_2 : D \to C_2$.  If $D$ has [[reflexive coequalizers]], then $U_2$ has a left adjoint $F_2$, with induced monad $T_2 = U_2 F_2$ on $C_2$, and we can iterate.

If $D$ is moreover [[cocomplete category|cocomplete]], we can pass to limits and obtain a tower of adjunctions indexed by all [[ordinal numbers]].  If this tower converges (which happens, for instance, if $D$ is [[well-copowered category|well-copowered]]), then it factors the original adjunction into an adjunction $F_\infty : C_\infty \rightleftarrows D : U_\infty$ that is a [[reflection]] (i.e. the induced monad $T_\infty = U_\infty F_\infty$ on $C_\infty$ is the identity) followed by an adjunction $C_1 \rightleftarrows C_\infty$ whose right adjoint $C_\infty \to C_1$ is a [[transfinite composite]] of [[monadic functors]], hence in particular [[faithful functor|faithful]] and [[conservative functor|conservative]].  This is called a/the **monadic decomposition** of the original adjunction, and produces a [[factorization system on a 2-category|factorization system]] on a suitable 2-category.

## References

* Applegate and Tierney, *Iterated cotriples*, [springerlink](https://link.springer.com/chapter/10.1007%2FBFb0060440)

* MacDonald and Stone, *The tower and regular decompositions*, [numdam](http://www.numdam.org/item/CTGDC_1982__23_2_197_0)

* Adamek and Herrlich and Tholen, *Monadic decompositions*, [sciencedirect](https://www.sciencedirect.com/science/article/pii/0022404989901291)

[[!redirects monadic decompositions]]
[[!redirects iterated monads]]
[[!redirects iterated comonads]]


monadic descent

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###Context###
#### Locality and descent
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#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
* table of contents
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## Idea

Monadic descent is a way to encode [[descent]] of [[fibered category|fibered categories]] (equivalently, by the [[Grothendieck construction]], of [[pseudofunctor]]s) that have the special property that they are [[bifibration|bifibrations]]. This allows the use of algebraic tools, notably [[monad|monads]] and related structures from [[universal algebra]].

A [[bifibration]] $E \to B$ comes naturally equipped not only with a notion of pullback, but also of pushforward. Combined these provide pull-push-[[monad|monads]] that may be used to encode the [[descent]] property of the fibration.

A morphism $f : b_1 \to b_2$ in the base $B$ induces an [[adjunction]] $F\dashv U$ where

$$
  F \;:\; E_{b_1} =: A\leftrightarrow B := E_{b_2} \;:\; U
$$

and we ask whether $U$ is a [[monadic functor]].

This is the original description of [[descent]] of presheaves with values in 1-categories due to [[Alexander Grothendieck]].

The archetypical and motivating example is that of the bifibration $Mod \to Ring$ of [[module]]s over [[Ring]]s.

## Definition
 {#Definition}

Let $\mathcal{C}$ be a [[category]] and $\mathcal{C}_{(-)}$ a [[bifibration]] over it. For $f \colon X \longrightarrow Y$ a [[morphism]] in $\mathcal{C}$ write

$$
  (f_! \dashv f^\ast \dashv f_\ast)
  \colon
  \mathcal{C}_X
  \stackrel{\overset{f_!}{\longrightarrow}}{\stackrel{\overset{f^\ast}{\longleftarrow}}{\underset{f_\ast}{\longrightarrow}}}
  \mathcal{C}_Y
$$

for the corresponding [[base change]] [[adjoint triple]], and write

$$
  (T_f \dashv J_f) \coloneqq (f^\ast f_! \dashv f^\ast f_\ast)
$$

for the induced [[adjoint pair]] of a [[monad]] $T$ and a [[comonad]] $J$.

There is a standard definition of a [[category]] $Desc_{\mathcal{C}}(f)$ of [[descent data]] for $\mathcal{C}_{(-)}$ along $f$, which comes with a canonical morphism

$$
  \mathcal{C}_{Y} \longrightarrow Desc_{\mathcal{C}}(f)
  \,.
$$

The morphism $f$ is called (with respect to the given bifibration $\mathcal{C}_{(-)}$)

* a _descent morphism_ if this comparison map is a [[full and faithful functor]];

* an _effective descent morphism_ if the comparison map is an [[equivalence of categories]].

Now the [[Bénabou–Roubaud theorem]] states that if $\mathcal{C}_{(-)}$ satisfies the [[Beck–Chevalley condition]], then descent data is equivalent to the structure of an [[algebra over a monad]] for $T_f$  (equivalently a coalgebra for $J_f$), hence is the [[Eilenberg–Moore category]] for these (co-)monads

$$
  Desc_{\mathcal{C}}(f)
  \simeq
  EM(T_f)
  \,.
$$

Therefore when $\mathcal{C}_{(-)}$ satisfies the BC condition, then a morphism $f$ _is effective descent_ precisely if $f^\ast \colon \mathcal{C}_{Y} \to \mathcal{C}_{X}$ is [[monadic]], and _is descent_ precisely if $f^\ast$ is of [[descent type]].

This is the monadic formulation of descent theory, "monadic descent".

(e.g. [Janelidze–Tholen 94, pp. 247-248 (3-4 of 37)](#JanelidzeTholen94)).

## Properties

The main theorem is Beck's [[monadicity theorem]].

Given a Grothendieck [[bifibration]] $p:E\to B$ and a morphism $f:b\to b'$ in the base category $B$, one can choose a [[direct image]] $f_!:E_b\to E_{b'}$ and an [[inverse image]] functor $f^*:E_{b'}\to E_b$, which form an [[adjunction]] $f_!\dashv f^*$. Under some conditions (see the [[Bénabou–Roubaud theorem]]), the morphism $f$ is an [[effective descent morphism]] (with respect to $p$ as a [[fibered category]]) iff the comparison functor for the [[monad]] induced by the adjunction $f_!\dashv f^*$ is monadic.

We should now see that some instances of categories of [[descent]] data are canonically equivalent to and can be reexpressed via [[Eilenberg–Moore category|Eilenberg–Moore categories]] of monads, or dually comonads.

## Examples

### Descent for the codomain fibration
 {#DescentForTheCodomainFibration}

Let $\mathcal{C}$ be a [[locally Cartesian closed category]] with [[coequalizers]] (e.g. a [[topos]]). Then effective descent morphisms for the [[codomain fibration]] are precisely the [[regular epimorphisms]]. ([Janelidze–Tholen 94, 2.4](#JanelidzeTholen94)).

Hence for $f \colon X \longrightarrow Y$ any morphism in $\mathcal{C}$ and

$$
  (f_! \dashv f^\ast \dashv f_\ast) \colon \mathcal{C}_{/X} \longrightarrow \mathcal{C}_{/Y}
$$

the induced [[base change]] [[adjoint triple]], then $\mathcal{C}_{/Y}$ is equivalent to the [[Eilenberg–Moore category]] of algebras over $f^\ast f_!$ (equivalently: of coalgebras of $f^\ast f_\ast$) precisely if $f$ is an [[effective epimorphism]].

(Use [conservative pullback along epimorphisms](conservative+functor#PullbackAlongEpimorphisms) in the [[monadicity theorem]].)

### Monadic descent of bundles {#ForCodomainFibs}

One of the most basic examples of [[bifibration]]s are [[codomain fibration]]s $cod : [I,C] \to C$. Accordingly, monadic descent applied to codomain fibrations archetypically exhibits the nature of monadic descent. We therefore spell out this example is some detail.

An object in a [[codomain fibration]] over $Y \in C$ is a morphism $P \to Y$, hence a [[bundle]] in $C$, in the most general sense of bundle. Therefore monadic descent with respect to codomain fibrations encodes [[descent]] of bundles.

Other examples of monadic descent often find a useful interpretation when relating them back to monadic descent for codomain fibrations. For instance (co)monadic descent for [[Sweedler coring]]s, discussed below, finds a natural geometric interpretion this way (as discussed in detail there).

We show in the following that for $cod : [I,C] \to C$ a [[codomain fibration]] and for $\pi : Y\to X$ a morphism in $C$, an algebra object in $[I,C]_Y$ over the [[monad]] $f^* f_*$ encodes and is encoded by a "geometric" [[descent]] datum: that it is

* a morphism $P \to Y$

* equipped with a transition function between its two pullbacks to double $Y \times_X Y$

* which satisfies on $Y \times_X Y \times_X Y$ the usual cocycle condition.

#### Motivation: failure of push-forward for principal bundles

Monadic methods can be applied to the study of descent of structures that cannot only be pulled back, such as [[principal bundle]]s, but that can also be pushed forward, such as [[vector bundle]]s (with suitable care taken) or more generally [[module]]s over functions rings (discussed at [[Sweedler coring]]).

Given a [[principal bundle]] $P \to X$ (a topological one, say, i.e. a morphism in [[Top]]) and a morphism of base spaces $f : X \to Z$, the would-be candidate for the push-forward of $P$ along $f$ is simply the composite map
$P \to X \to Z$, regarded as a total space $P \to Z$ living over $Z$.

While that always exists as such, it will in general not be a principal bundle anymore: the [[fiber]]s of $P \to Z$ over points $z \in Z$ consist of many copies of the fibers of $P \to X$ over points in $X$. Hence the shape of the fibers may change drastically as we push bundles forward.

So principal bundles do have a canonical notion of push-forward, but it leads outside the category of principal bundles and lands generally in some [[overcategory]].

On the other hand, as we will see in detail below, if we take a principal bundle $P \to X$ and

* first push it forward in this generalized sense to an object $P \to Z$ in the [[overcategory]] $Top/Z$

* and **then** pull back the result of that again along $X \to Z$ the result, while still not a principal bundle, is the total space $P$ of the bundle pulled back to the first term in the [[Čech nerve]] of $f : X \to Z$. This pullback is of central interest in the description of the geometric [[descent]] property of the bundle.

But the composite operation of pushforward of overcategories

$$
  push(f) : Top/X \to Top/Z
$$

followed by pullback

$$
  pull(f) : Top/Z \to Top/X
$$

is nothing but the [[monad]] associated to $f : X \to Z$ with respect to the [[codomain fibration|codomain bifibration]] $cod : [I,Top] \to Top$.

So by regarding principal bundles $P \to X$ more generally as just objects in the [[overcategory]] $Top/X$ we make the tools of monadic descent applicable to them.

#### The monad

Let $C$ be a [[category]] with [[pullback]]s. Then the [[codomain fibration]]

$$
  cod : [I,C] \to C
$$

is a [[bifibration]] (as described there, in detail). Its [[fiber]] over an object $X \in C$ is the [[overcategory]] $C/X$.

The direct image operation $push(f)$ associated to a morphism $\pi : Y \to X$ in $C$ is the functor

$$
  push(\pi) : C/Y \to  C/X
$$

obtained by postcomposition with $f$, which sends $(P \to Y) \in C/Y$ to the composite $P \to Y \stackrel{\pi}{\to} X$ in $C$, regarded as an object of $C/X$.

The inverse image operation $pull(f)$ associated to $f$ is the functor

$$
  C/Y \leftarrow C/X : pull(\pi)
$$

obtained by [[pullback]] in $C$ along $\pi$, which sends $(Q \to X) \in C/X$ the [[pullback]] $Q \times_X Y$, regarded as an object of $C/Y$ in terms of the canonical projection morphism $Q \times_X Y\to Y$ out of the pullback.

Write

$$
  T_\pi = pull(\pi) \circ push(\pi) : C/Y \to C/Y
$$

for the [[monad]] built from these two [[adjoint functor]]s.

#### The algebras over the monad: geometric descent data

We spell out in detail the data of an algebra over the above monad, and show that this encodes precisely the familiar geometric [[descent]] datum for a [[bundle]].

To that end, let $(P, \rho)$

$$
  P : {*} \to C/Y
  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
  \array{
      && C/Y
      \\
      & {}^{\mathllap{P}}
     \nearrow  &\Downarrow^{\rho}& \searrow^{\mathrlap{T}}
     \\
    {*} &&\stackrel{P}{\to}&& C/Y
  }
$$

be an [[monad|algebra over]] our monad. In components this is an object $P$ equipped with a morphism $\rho_P : T P \to P$.

The object $T P \in [I,C]_Y$ is given by

* first pushing $P \to Y$ forward along $\pi : Y \to X$ to the object $P \to Y \to X$.

* then pulling this back along $\pi$ to yield the left vertical morphism in

  $$
    \array{
      Y \times_X P &\to& P
      \\
      \downarrow && \downarrow
      \\
      && Y
      \\
      \downarrow && \downarrow^{\mathrlap{\pi}}
      \\
      Y &\stackrel{\pi}{\to}& X
    }
    \,.
  $$

  This [[pullback]] along a composite of morphisms may be computed as two consecutive pullbacks. The first one is

  $$
    \array{
      Y \times_X Y &\to& Y
      \\
      \downarrow && \downarrow^{\mathrlap{\pi}}
      \\
      Y &\stackrel{\pi}{\to}& X
    }
  $$

  which is the first term in the [[Čech nerve]] of $\pi$. So the total pullback is the pullback $P$ to $Y\times_X Y$:

  $$
    \array{
      (Y \times_X Y) \times_Y P &\to& P
      \\
      \downarrow && \downarrow
      \\
      Y \times_X Y &\to& Y
      \\
      \downarrow && \downarrow^{\mathrlap{\pi}}
      \\
      Y &\stackrel{\pi}{\to}& X
    }
    \,.
  $$

Therefore the action $\rho_T : T P \to P$ of our [[monad]] on $P$ is given in $C$ by a morphism

$$
  \array{
    (Y \times_X Y) \times_Y P &&\stackrel{\rho}{\to}&&
    P
    \\
    & \searrow && \swarrow
    \\
    && Y
  }
  \,.
$$

As an example, think of this in the context $C = Top$ with $\pi  \colon Y \to X$ coming from an open [[cover]] $\{U_i \to X\}$ of $X$ with $Y = \coprod_i U_i$, and with $P = Y \times G$ a trivial $G$-[[principal bundle]] for some [[group]] $G$. Then the space $Y \times_X Y = \coprod_{i j} U_i \cap U_j$ is the union of double intersection of covering patches, and $(Y \times_X Y) \times_Y P = (\coprod_{i j} U_i \cap U_j \times G)$ is to be thought of as the trivial $G$-principal bundle over $U_j$, restricted to the intersection. In this case our morphism $\rho$ acts as

$$
  \rho : \coprod_{i j} : (U_i \cap U_j \times G) \to \coprod_i U_i \times G
$$

and thus maps the trivial $G$-bundle over $U_j$ on the intersection with the trivial $G$-bundle over $U_i$. So it is a _transition function_. If this is a $G$-equivariant, it may be part of the  [[descent]] datum for the $G$-[[principal bundle]].

### Monadic descent _along_ principal bundles {#AlongPrincipalBundle}

#### Idea

In the [above section](#ForCodomainFibs) we considered monadic descent _of_ [[bundle]]s $P \to Y$ _along_ morphisms $f : Y \to X$.

Now we consider monadic descent _along_ morphisms $f : P \to X$ that happen to be $G$-[[principal bundle]]s, for some [[group object]] $G$. When considered with respect to the [[codomain fibration]] this describes the situation where we ask for a bundle $L \to P$ that sits over the total space of another (principal) bundle to descend down along that bundle map to $X$. So beware of the two different roles that bundles play here.

#### Setup

Let $C$ be a [[category]] with [[pullback]]s and let $G$ be an internal group in $C$.

Let $\nu: P\times G\to P$ be a right principal [[action]] and $p:P\times G\to P$ the projection. Let $\pi:P\to X$ be the [[coequalizer]] of $\nu$ and $p$. The [[principal bundle|principality condition]] says that $P\times G \to P\times_X P$ given by $(p,g)\mapsto (p,pg)$ is an [[isomorphism]].

$$
P\times G \overset{\nu}\underset{p}\rightrightarrows P \overset{\pi}\to X
$$

We do not assume $P$ to be trivial. We have also the two projections

$$
P\times_X P \overset{p_1}\underset{p_2}\rightrightarrows P \overset{\pi}\to X
$$

out of the [[pullback]], where $p_1,p_2$ make a [[kernel pair]] of $\pi$. Thus the principality condition is equivalent to saying that $\nu,p$ make also a [[kernel pair]] of its own [[coequalizer]]. The two diagrams above are truncations of augmented [[simplicial object]]s in $C$. We want to relate these objects to [[monad]]s.

#### The two different monads

We now describe the monadic descent along the morphism $\pi : P \to X$ from above for the [[codomain fibration]] $cod : [I,C] \to C$.

There are two monads acting on the [[overcategory]] $C/P$ whose underlying functors are

1. $T := \pi^* \pi_!$.

1. $\tilde T := p_! \nu^*$

The first monad, $T$ is the usual one for monadic descent along $\pi$ induced from a pair of [[adjoint functor]]s.

The second one, $\tilde T$, exists due to the principality of $P \to X$ and is defined as follows:

to construct the component $\mu_h$ of the transformation $\mu: p_! \nu^* p_!\nu^*\to p_!\nu^*$ where $h: L\to P$, by the universal property of the pullback there is an obvious map $\nu^* p_! \nu^* h$ to $p_! \nu^* h$

$$
  \array{
    \nu^* p_! \nu^* L
    \\
    & \searrow^{\mathrlap{\mu_h}}
    \\
    &&\nu^* L &\to& L
    \\
    &&
    \downarrow && \downarrow
    \\
    &&
    P &\stackrel{\stackrel{p}{\to}}{\underset{\to}{\nu}}&
    X
  }
  \,,
$$

which can be interpreted as a map $p_!\nu^* p_! \nu^* h\to p_* \nu^* h$ because the domains of the maps $p_!\nu^* p_! \nu^* h$ and $\nu^* p_! \nu^* h$ are the same by the definition and the commuting triangles can be checked easily.

The [[principal bundle|principality]] $P\times G \cong P\times_X P$ now induces the [[isomorphism]]

$$p_! \nu^* h \cong \pi^* \pi_! h$$

[[natural isomorphism|natural]] in  $h:L\to P$, read off from the double [[pullback]] diagram

$$
  \array{
   p_! \nu^* L &\stackrel{\simeq}{\to}& \pi^* \pi_! L &\to& L
   \\
   \downarrow && \downarrow && \downarrow^{\mathrlap{h}}
   \\
   P \times G &\stackrel{\simeq}{\to}& P \times_X P &\to& P
   \\
   && \downarrow && \downarrow^{\mathrlap{\pi}}
   \\
   && P &\to& X
  }
  \,.
$$

This rule extends to an isomorphism of monads

$$
  T \simeq \tilde T
  \,.
$$

As a corollary, the [[Eilenberg–Moore category|Eilenberg–Moore categories]] of the two monads are [[equivalence of categories|equivalent]]. Notice that the actions over the monad $p_! \nu^* $ are certain maps $p_!\nu^*h\to h$, hence $\nu^* h\to p^* h$ by adjointness. This matches one of the definitions for an [[equivariant sheaf]].

The map $ \pi : P\to X$ of the principal bundle is an **[[effective descent morphism]]** with respect to the [[codomain fibration]] if the comparison functor for any of the two above isomorphic monads above is an equivalence of categories.

### Monadic descent of modules

There is a [[bifibration]] $Mod \to Rings$ from the category of modules over any ring, mapping each module to the ring that it is a module over. This models, dually,  an algebraic version of [[vector bundle]]s over [[affine scheme]]s.

Comonadic descent for this bifibration (equivalently monadic descent for its formal dual, $Mod^{op} \to Rings^{op}$) is the same as descent for a [[Sweedler coring]]. See there for details and geometric interpretations.

### Gluing categories from localizations

Another example is in [[zoranskoda:gluing categories from localizations|gluing categories from localizations]].

## Higher category theoretical version

All the ingredients of monadic descent generalize from [[category theory]] to [[higher category theory]]. Accordingly, one may consider [[higher monadic descent]] that relates to [[∞-stack]]s as monadic descent relates to [[stack]]s. For more on this see

* [[higher monadic descent]].

## Related concepts

* [[surjective geometric morphism]]

* [[descent]]

  * [[cover]]

  * [[cohomological descent]]

  * [[descent morphism]]

  * **monadic descent**

    * [[Beck-Chevalley condition]]

    * [[monadicity theorem]]

    * [[Sweedler coring]]

    * [[higher monadic descent]]

    * [[descent in noncommutative algebraic geometry]]

* [[sheaf]], [[(2,1)-sheaf]], [[2-sheaf]] [[(∞,1)-sheaf]]

## References

The [[Bénabou–Roubaud theorem]] on monadic descent is due to

* {#BenabouRoubaud70} [[Jean Bénabou]], [[Jacques Roubaud]], _Monades et descente_, C. R. Acad. Sc. Paris, Ser. A **270**  (1970) 96-98 [[gallica:12148/bpt6k480298g/f100](http://gallica.bnf.fr/ark:/12148/bpt6k480298g/f100), [[BenabouRoubaud-MonadesEtDescente.pdf:file]],   English translation (by [[Peter Heinig]]): [MO:q/279152](https://mathoverflow.net/q/279152)]

Review and further developments:

* {#JanelidzeTholen94} [[George Janelidze]], [[Walter Tholen]], *Facets of descent I*, Applied Categorical Structures **2** 3 (1994) 245-281 [[doi:10.1007/BF00878100](https://doi.org/10.1007/BF00878100)]

* {#JanelidzeTholen97} [[George Janelidze]], [[Walter Tholen]], *Facets of descent II*, Applied Categorical Structures **5** 3 (1997) 229-248 [[doi:10.1023/A:1008697013769](https://doi.org/10.1023/A:1008697013769)]

* [[George Janelidze]], Manuela Sobral, [[Walter Tholen]], _Beyond Barr exactness: effective descent morphisms_, Ch. 8 of Categ. Foundations, (eds. Maria Cristina Pedicchio, Walter Tholen) Enc. Math. Appl. 2003

* Bachuki Mesablishvili, _Monads of effective descent type and comonadicity_, Theory and Applications of Categories __16__:1 (2006) 1-45, [link](http://tac.mta.ca/tac/volumes/16/1/16-01abs.html); _Pure morphisms of commutative rings are effective descent morphisms for modules—a new proof_, Theory Appl. Categ. 7(3), 38–42 (2000)

* [[Francis Borceux]], Stefan Caenepeel, [[George Janelidze]], _Monadic approach to Galois descent and cohomology_, [arXiv:0812.1674](http://arxiv.org/abs/0812.1674)
* S. Caenepeel, _Galois corings from the descent theory point of view,_ in: Fields Inst. Commun. __43__ (2004) 163–186
* [[Tomasz Brzeziński]], Adrian Vazquez Marquez, Joost Vercruysse, _The Eilenberg–Moore category and a Beck-type theorem for a Morita context_, J. Appl Categor Struct (2011) 19: 821 [doi](https://doi.org/10.1007/s10485-009-9217-0)

In triangulated setup there are several results including

* P. Balmer, _Descent in triangulated categories_, Mathematische Annalen (2012) 353:1, 109–125

Discussion in [[homotopy theory]] for [[(infinity,1)-monads]] is in

* {#Hess10} [[Kathryn Hess]], section 2 of _A general framework for homotopic descent and codescent_ ([arXiv:1001.1556](http://arxiv.org/abs/1001.1556))

[[!redirects comonadic descent]]


monadic functor

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###Context###
#### Category theory
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[[!include category theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
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#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

A [[functor]] $U \,\colon\, D\to C$ is _monadic_ iff it has a [[left adjoint]] $F \,\colon\, C\to D$ such that -- under the [relation between adjunctions and monads](monad#RelationBetweenAdjunctionsAndMonads) -- the [[adjoint functor|adjunction]] $F\dashv U$ is that induced by the [[monad]] which it induces -- in which case it is called a *[[monadic adjunction]]*.

In this situation $U$ is identified with the [[forgetful functor]] from the [[Eilenberg-Moore category]] $EM(U \circ F)$ of the monad $(U \circ F, \eta, U\epsilon_F)$ on $C$, and hence shares the properties of these [[forgetful functors]].

\begin{tikzcd}
  \mathcal{C}
  \ar[
    from=r,
    "{ U }"{description}
  ]
  \ar[
    r,
    shift left=16pt,
    "{ F }"
  ]
  \ar[
    r,
    shift left=9.5pt,
    phantom,
    "{ \scalebox{.62}{$\bot$} }"
  ]
  &
  \mathcal{D}
  \simeq
  \mathrm{EM}(\mathrm{ U \circ F })
\end{tikzcd}

The *[[monadicity theorem]]* characterizes monadic functors and makes these 'nice properties' precise.

Monadic functors are sometimes called *functors of effective descent (type)*. See the page on *[[monadic descent]]* for more on this aspect.

## Definition

Given a [[adjoint pair|pair]] of [[adjoint functors]] $F \colon C \to D :U$, $F \dashv U$, with [[unit of an adjunction|unit]] $\eta: Id_C \to U \circ F$ and counit $\epsilon: F \circ U \to Id_D$, one constructs a [[monad]] $\mathbf{T}=(T,\mu,\eta)$ setting $T = U \circ F: C \to C$, $\mu = U \epsilon F: T T = U F U F \to U F = T$.

Consider the [[Eilenberg-Moore category]] $C^{\mathbf{T}}$ of $T$-algebras ($T$-modules) in $C$. Clearly $U (\epsilon_M): T U M = U F U M \to U M$ is a $T$-action. In fact there is a canonical *[[comparison functor]]* $K^{\mathbf{T}} \colon D \to C^{\mathbf{T}}$ given on objects by $K(M) \coloneqq \big(U M, U (\epsilon_M) \big)$.  We then say that we have a (resp. strictly) [[monadic adjunction]] iff $K$ is an equivalence (resp. isomorphism) of categories.

\begin{definition}\label{MonadicFunctor}
**(monadic functor)**
\linebreak
A [[functor]] $U \colon D \to C$ is *monadic* (resp. *strictly monadic*) if it has a [[left adjoint]] $F \colon C\to D$ and the [[comparison functor]] $K^{\mathbf{T}} \colon D \to C^{\mathbf{T}}$ is an [[equivalence of categories]] (resp. an [[isomorphism]] of [[strict categories]]).
\end{definition}

In other words, up to [[equivalence of categories|equivalence]], monadic functors are precisely the [[forgetful functors]] defined on [[Eilenberg-Moore categories]] for [[monads]], and strictly monadic functors are the same as these forgetful functors up to [[isomorphism]] of [[strict categories]].

* If the [[comparison functor]] is only [[fully faithful]], a functor is sometimes called *[[premonadic functor|premonadic]]* or said to be *of [[descent type]]*.

\begin{definition}
A *category* $D$ is called *monadic* over a category $C$ if there is any functor $U \colon D \to C$ which is monadic (Def. \ref{MonadicFunctor}).
\end{definition}

## Properties

### Basic properties

Every monadic functor is

1. [[faithful functor|faithful]] (by the definition of [[Eilenberg-Moore category]])

1. [[conservative]] (by [Beck's monadicity theorem](monadicity+theorem#statement)).

Moreover:

\begin{proposition}\label{MonadicFunctorsCreateLimits}
**([[monadic functors]] [[created limit|create limits]])**
A [[monadic functor]]

1. [[created limit|creates]] all [[limits]] that exist in its [[codomain]];

1. creates all [[colimits]] that exist in its codomain and are preserved by the corresponding monad (or, equivalently, by the monadic functor itself).

\end{proposition}
(e.g. [MacLane 71, Exercise IV.2.2 (p. 138)](#MacLane71))

\begin{remark}
Beware that the [[class]] of monadic functors is *not* generally closed under [[composition]].

For a specific **[[counter-example]]**: the category of [[reflexive graphs]] is monadic over [[Set]] via the functor $RefGph \to Set$ sending a graph to its set of [[edges]], and the [[category of categories]] is monadic over [[reflexive graphs]] via the [[forgetful functor]] $Cat \to RefGph$, but $Cat$ is not monadic over $Set$ (via any functor whatsoever, since monadic categories over [[Set]] are [[regular categories]] which [[Cat]] is not).

This is an instance of a general phenomenon: Let $\mathcal{C}$ be a [[reflective subcategory]] of a [[presheaf category]] $\widehat{A}$ (e.g. any [[locally presentable category]] is of this form). Then the [[adjoint functor|adjunction]] between $\mathcal{C}$ and $\widehat{A}$ is monadic, and the [[adjoint functor|adjunction]] between $\widehat{A}$ and $\mathrm{Set}^{\mathrm{Ob} A}$ is also monadic. But the composite adjunction between $\mathcal{C}$ and $\mathrm{Set}^{\mathrm{Ob} A}$ is often not monadic. For instance, if it is monadic, then $\mathcal{C}$ must be a [[Barr-exact category]].
\end{remark}

\begin{proposition}
Consider a pair of [[adjoint functors|adjunctions]]:
\begin{tikzcd}
	A & B & C
	\arrow[""{name=0, anchor=center, inner sep=0}, "U", shift left=2, from=1-1, to=1-2]
	\arrow[""{name=1, anchor=center, inner sep=0}, "{U'}", shift left=2, from=1-2, to=1-3]
	\arrow[""{name=2, anchor=center, inner sep=0}, "{F'}", shift left=2, from=1-3, to=1-2]
	\arrow[""{name=3, anchor=center, inner sep=0}, "F", shift left=2, from=1-2, to=1-1]
	\arrow["\dashv"{anchor=center, rotate=90}, draw=none, from=2, to=1]
	\arrow["\dashv"{anchor=center, rotate=90}, draw=none, from=3, to=0]
\end{tikzcd}
If here $U' U$ is monadic, then $U$ is of [[descent type]] and the [[comparison functor]] has a [[adjoint functor|left adjoint]]. If $U'$ is furthermore [[conservative]] (and in particular if it is monadic), then $U$ is monadic.
\end{proposition}

This is Propositions 4 and 5 of [Bourn](#Bourn).

\begin{proposition}
A monadic functor is *strictly monadic* if and only if it is also an [[amnestic functor|amnestic]] [[isofibration]].
\end{proposition}
\begin{proof}
Clearly, a strictly monadic functor is an amnestic isofibration; and if a monadic functor $U$ is amnestic, then the [[comparison functor]] $K$ is also [[amnestic functor|amnestic]], and if $U$ is a monadic isofibration, so is $K$; therefore in this case $K$ must be an isomorphism of categories.
\end{proof}

### Monadicity theorem

Various versions of Beck's [[monadicity theorem]] (also: "tripleability theorem" in older literature) give sufficient, and sometimes necessary, conditions for a given functor to be monadic. There are also dual, [[comonadic functor|comonadic versions]].

### Monadic functors to Set

Monadic functors to the category [[Set]] have additional properties.  For example:

* Every monadic functor $U \,\colon\, D \to \mathrm{Set}$ is a [[solid functor]].

* A category is monadic over $\mathrm{Set}$ (i.e. it admits a monadic functor to $\mathrm{Set}$) if and only if is [[exact category|Barr exact]], [[cocomplete category|cocomplete]], and has a [[projective object|projective]] [[separator|generator]].

[[Vitale (1994)](#Vitale94)]

## Examples
 {#Examples}

\begin{example}
  \label{ReflectiveSubcategoryInclusion}
  Every [[reflective subcategory]]-inclusion is a [[monadic functor]]. See also [there](reflective+subcategory#ReflectiveSubcategoryInclusionIsMonadic).
\end{example}

A proof is spelled out for instance in [Borceux 1994, vol 2, cor. 4.2.4](reflective+subcategory#Borceux94b). A [[formal proof]] in [[cubical Agda]] is given in [1Lab](reflective+subcategory#1Lab). See also at _[idempotent monad -- Properties -- Algebras for an idempotent monad and localization](idempotent+monad#AlgebrasForAnIdempotentMonad)_.

## Related concepts

* [[monadic adjunction]], [[structure-semantics adjunction]]

* [[comonadic functor]], [[monadicity theorem]]

* [[monadic decomposition]]

## References

* {#MacLane71} [[Saunders Mac Lane]], Section VI in: _[[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]], Def. 4.4.1 in: *[[Handbook of Categorical Algebra]]*,   Vol. 2: *Categories and Structures*, Encyclopedia of Mathematics and its Applications **50** Cambridge University Press (1994) [[doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865)]

* [[Emily Riehl]], §5.3 in: _[[Category Theory in Context]]_, Dover Publications (2017) [[pdf](http://www.math.jhu.edu/~eriehl/context.pdf)]

* {#Bourn} [[Dominique Bourn]]. _Low dimensional geometry of the notion of choice_. Category Theory 1991, CMS Conf. Proc. Vol. 13. 1992.

* {#Vitale94} [[Enrico Vitale]], *On the characterization of monadic categories over $SET$*, Cahiers de topologie et géométrie différentielle catégoriques **35** 4 (1994) 351-358.  [[numdam:CTGDC_1994__35_4_351_0](http://www.numdam.org/item/?id=CTGDC_1994__35_4_351_0), [pdf](http://www.numdam.org/article/CTGDC_1994__35_4_351_0.pdf)]

[[!redirects monadic]]
[[!redirects monadic category]]
[[!redirects monadic functors]]

[[!redirects comonadic]]
[[!redirects co-monadic]]

[[!redirects comonadic functor]]
[[!redirects co-monadic functor]]
[[!redirects comonadic functors]]
[[!redirects co-monadic functors]]

[[!redirects effective descent type]]
[[!redirects functor of effective descent type]]
[[!redirects functors of effective descent type]]


monadicity theorem

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#### Algebra
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#Contents#
* table of contents
{:toc}

## Idea
  {#Idea}

In [[category theory]] and [[universal algebra]], a *monadicity theorem* serves to characterize/recognize whether a given [[functor]] is a [[monadic functor]].

## Statement

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

Given a [[functor]] $U \colon D \rightarrow C$, then a [[parallel pair]] $f,g : a \rightarrow b$ in $D$ is called **$U$-split** if the pair $U f, U g$ has a [[split coequalizer]] in $C$.  Specifically, this means that there is a diagram in $C$:

$$U a \;\underoverset{U f}{U g}{\rightrightarrows}\; U b \;\overset{h}{\rightarrow}\; c$$
such that $h \cdot U f = h \cdot U g$, and $h$ and $U f$ have respective [[sections]] $s$ and $t$ satisfying $U g \cdot t = s \cdot h$.  This implies that the arrow $h$ is necessarily a coequalizer of $U f$ and $U g$.

The functor $U$ is said to *create coequalizers of $U$-split pairs* if for any such $U$-split pair, there exists a coequalizer $e$ of $f,g$ in $D$ which is preserved by $U$, and moreover any [[fork]] in $D$ whose image in $C$ is a split coequalizer must itself be a coequalizer (not necessarily split).

=--

\begin{theorem}
\label{BeckMonadicityTheorem}
**(Beck's monadicity theorem,  tripleability theorem)**
\linebreak
A [[functor]] $U \colon D \rightarrow C$ is [[monadic]] (or tripleable) if and only if

1. $U$ has a [[left adjoint]], and

1. $U$ [[created limit|creates]] [[coequalizers]] of $U$-split pairs, def. \ref{CreationOfCoequalizersOfuSplitPairs}.

\end{theorem}

The proof is reproduced for instance in ([MacLane, p. 147-150](#MacLane), [Riehl 2017, 5/5/](#Riehl17)).

An equivalent, and sometimes easier, way to state these conditions is to say that

\begin{theorem}
\label{BeckMonadicityTheoremAlternative}
**(Beck's monadicity theorem -- alternative formulation)**
\linebreak
A functor $U \colon D \to C$ is monadic precisely if

1. $U$ has a [[left adjoint]],

1. $U$ reflects [[isomorphisms]] (i.e. it is [[conservative functor|conservative]]), and

1. if for a [[parallel pair]] $(f,g)$ in $D$ the [[image]] $\big(U(f),\, U(g)\big)$ has a [[split coequalizer]] in $C$ then $(f,g)$ has a [[coequalizer]] in $D$ which is [[preserved colimit|preserved]] by $U$.

\end{theorem}

(e.g. [Borceux, vol 2, Thm. 4.4.4](#Borceux))

These conditions are equivalent to those in Thm. \ref{BeckMonadicityTheorem},
because a conservative functor [[reflected limit|reflects]] any limits or colimits which exist in its domain and which it [[preserved limit|preserves]] (by [this Prop.](conservative+functor#ConservativeFunctorsReflectLimitsWhichTheyPreserve)), while monadic functors are always conservative.

## Variants

### The crude monadicity theorem
 {#CrudeMonadicityTheorem}

The __crude monadicity theorem__ gives a *sufficient*, but not necessary, condition for a functor to be monadic.  It states that a functor $U : D \rightarrow C$ is [[monadic]] if

1. $U$ has a left adjoint
2. $U$ reflects isomorphisms
3. $D$ has and $U$ preserves coequalizers of [[reflexive coequalizer|reflexive pairs]].

(Recall that a parallel pair $f,g : a \rightarrow b$ is __reflexive__ if $f$ and $g$ have a common [[section]].)  This sufficient, but not necessary, condition is sometimes easier to verify in practice.  In contrast to the crude monadicity theorem, the necessary and sufficient condition above is sometimes called the __precise monadicity theorem__.

A further advantage of crude monadicity is this: while in general the composite of monadic functors need not be monadic, if $U_1\colon D \to C$ satisfies the hypotheses of the crude monadicity theorem and $U_2\colon C \to B$ is any monadic functor then $U_2 \circ U_1$ is monadic.   Furthermore, if both $U_1$ and $U_2$ both obey the hypotheses of the crude monadicity theorem, so does $U_2 \circ U_1$.  See ([BarrWells](#BarrWells), Proposition 3.5.1) for these and further results.

### Duskin's monadicity theorem

Duskin's monadicity theorem gives a different sufficient, but not necessary, condition which refers only to quotients of [[congruences]].

\begin{theorem}\label{DuskinMonadicityTheorem}
**(Duskin monadicity theorem)**
\linebreak
A functor $U \colon D \to C$ is [[monadic functor|monadic]] if

1. $U$ has a [[left adjoint]],

1. $D$ and $C$ are [[finitely complete category|finitely
complete]],

1. $U$ [[created limit|creates]] [[coequalizers]] for [[congruences]] in $D$ whose [[images]] in $C$ have [[split coequalizers]].

\end{theorem}

We can weaken the hypothesis a bit further to obtain the theorem:

* A [[right adjoint]] between finitely complete categories is monadic if it creates [[quotients]] for [[congruences]].

As usual, we can also modify it by replacing reflection of limits by reflection of isomorphisms.

* A conservative right adjoint $U\colon D \to C$ between finitely complete categories is monadic if any congruence in $D$ which has a quotient in $C$ already has a quotient in $D$, and that quotient that is preserved by $U$.

If we view the objects of $D$ as underlying $C$-objects with structure, this says that any congruence in $D$ induces a $D$-structure on its quotient in $C$.  As with the crude monadicity theorem, this condition is sometimes easier to verify since quotients of congruences are often better-behaved than arbitrary coequalizers.  This is the case in many "algebraic" situations.

Duskin actually gave a slightly more precise version only assuming the categories $C$ and $D$ to have particular finite limits, rather than all of them.

### Monadicity over Set

In the case when the base category $C$ is [[Set]], one can further refine the requisite conditions.  Linton proved that a functor $U\colon D\to Set$ is monadic if and only if

1. $U$ has a left adjoint,
1. $D$ admits [[kernel pairs]] and [[coequalizers]],
1. A [[parallel pair]] $R \rightrightarrows S$ in $D$ is a kernel pair if and only if its image under $U$ is so, and
1. A morphism $A\to B$ in $D$ is a [[regular epimorphism]] if and only if its image under $U$ is so.

There are other versions of this theorem, including generalizations to monadicity over [[presheaf categories]], which can be viewed as analogues of [[Giraud's theorem]].

### Strict monadicity

The version of the monadicity theorem given in [[Categories Work]] uses a notion of "creation of limits" which fails to observe the [[principle of equivalence]], concluding that the comparison functor is an *isomorphism* of categories, rather than merely an equivalence.  But the versions mentioned above can be found in the exercises.

Note however that if $U: D \to C$ is an [[amnestic functor|amnestic]] [[isofibration]], then $U$ is monadic iff it is strictly monadic. For an application of this observation, see for example the discussion of algebraically free monads at [[free monad]].

## Examples and Applications

### Actions over sets

\begin{example}
\label{BaseChangeOfPresheavesAlongFullyFaithfulFunctor}
**([[base change]] of [[presheaves]] along [[essentially surjective functor]])**
\linebreak
Let
$$
  F
    \,\colon\,
  \mathcal{S}
    \overset{\phantom{---}}{\longrightarrow}
  \mathcal{S}'
$$
be a [[functor]] between [[small categories]] which is [[essentially surjective]]. Then for $\mathcal{C}$ any [[bicomplete category]] (e.g. [[Set]]), the corresponding [[precomposition]]-functor on [[category of presheaves|categories of]] $\mathcal{C}$-valued [[presheaves]]
\[
  \label{BaseChangeOfPresheavesAsMonadicFunctor}
  PSh
  \big(
    \mathcal{S}
    ;\,
    \mathcal{C}
  \big)
  \overset{\;\; F^\ast \;\;}{\longleftarrow}
  PSh
  \big(
    \mathcal{S}'
    ;\,
    \mathcal{C}
  \big)
\]

is [[monadic functor|monadic]]. To see this, we check the conditions in Thm. \ref{BeckMonadicityTheoremAlternative}:

1. By the assumption that $\mathcal{C}$ is [[bicomplete category|bicomplete]], the left and right [[Kan extensions]] of presheaves along $F$ exist and exhibit $F^\ast$ as the middle part of a ("[[base change]]") [[adjoint triple]] :

   $$
     F_! \,\dashv\, F^\ast \,\dashv\, F_\ast
     \;\;
     \colon
     \;\;
     PSh
     \big(
       \mathcal{S}
       ;\,
       \mathcal{C}
     \big)
     \overset{\phantom{----}}{\leftrightarrow}
     PSh
     \big(
       \mathcal{S}'
       ;\,
       \mathcal{C}
     \big)
     \,.
   $$

   This shows that $F^\ast$ has a left adjoint. In addition, it *is* a left adjoint and [[left adjoints preserve colimits|as such preserves all colimits]], hence in particular all [[coequalizers]].

1. The assumption that $F$ is [[essentially surjective functor|essentially surjective]] implies that $F^\ast$ is [[conservative functor|conservative]] (because [[isomorphisms]] of presheaves are [[natural transformation]] of the underlying functors which are [[natural isomorphisms]], which is the case iff their component [[morphisms]] for each [[object]] is an isomorphism.)

\end{example}

\begin{example}\label{GroupActions}
**([[group actions]]/[[G-sets]])**
\linebreak
Specializating Ex. \ref{BaseChangeOfPresheavesAlongFullyFaithfulFunctor} to the case (where $\mathcal{C} = $ [[Sets]], just for definiteness and) where $F \,\colon\,\mathcal{S} \longrightarrow \mathcal{S}'$ is the [[terminal category|point]]-inclusion
$$
  pt_{\mathbf{B}G}
  \,\colon\,
  \ast
  \overset{\phantom{---}}{\hookrightarrow}
  \mathbf{B}G
  \,.
$$
into the [[delooping groupoid]] of a ([[discrete group|discrete]]) [[group]], and observing that

1. $PSh\big(\ast \big) \;\simeq\;$ [[Sets]],

1. $PSh\big(\mathbf{B}G \big) \;\simeq\;$ [[G-set|$G$Sets]]

   is the category [[G-sets]], i.e. of $G$-[[group actions]] with [[equivariant functions]] between them,

gives that (eq:BaseChangeOfPresheavesAsMonadicFunctor) is the [[forgetful functor]]

$$
  (pt_{\mathbf{B}G})^\ast
  \,\colon\,
  G Sets \overset{\phantom{--}}{\longrightarrow} Sets
$$

which is hence [[monadic functor|monadic]]. This is of course the [[forgetful functor]] on the [[algebra over a monad|algebras]] of the [[monad]]

$$
  T \,\coloneqq\, G \times (-)
  \;\colon\;
  Sets \longrightarrow Sets
$$

which forms the [[Cartesian product]] with (the [[underlying]] [[set]]) of $G$, and whose monad product is

$$
  T \circ T
    \,=\,
  G \times G \times (-)
  \longrightarrow
  G \times (-)
    \,=\,
  T
$$

is given by the group operation.
\end{example}

\begin{example}\label{MonoidActions}
**([[monoid actions]] and monoid-oid actions)**
\linebreak
In generalization of Ex. \ref{GroupActions}, if $A$ is a [[monoid]] and $F \,\colon\, \mathcal{S} \longrightarrow \mathcal{S}'$ in Ex. \ref{BaseChangeOfPresheavesAlongFullyFaithfulFunctor} is the point inclusion
$$
  pt \,\colon\,\ast \longrightarrow \mathbf{B}A
$$
into the corresponding 1-object category (see [there](monoid#AsAOneObjectCategory)), then the monadic functor (eq:BaseChangeOfPresheavesAsMonadicFunctor)

$$
  (pt_{\mathbf{B}A})^\ast
  \;\colon\;
  A Sets \longrightarrow Sets
$$
is the [[forgetful functor]] on [[monoid actions]], which are the [[algebra over a monad|algebras]] of the monad $A \times (-)$.

In this sense, the general situation of Ex. \ref{BaseChangeOfPresheavesAlongFullyFaithfulFunctor} may be understood as monadicity of [[modules]] for a [monoid-oid](horizontal+categorification#Monoidoid) $\mathcal{S}'$ defined over a [monoid-oid](horizontal+categorification#Monoidoid) $\mathcal{S}$.
\end{example}

### Groups over sets
 {#GroupsOverSets}

We will use Duskin's monadicity theorem (Thm. \ref{DuskinMonadicityTheorem}) to prove that the [[forgetful functor]]
$$
  U
  \,\colon\,
  Grps
  \overset{\phantom{---}}{\longrightarrow}
  Sets
$$

from [[Groups]] to their underlying [[Sets]] is [[monadic functor|monadic]].
Of course, this is also easy to show by explicit computation, but it serves as a useful example of how to use a monadicity theorem.

We first need it to have a [[left adjoint]]: this is easy to show by a direct construction of [[free groups]], but we could also invoke the [[adjoint functor theorem]].

It is also easy to show that it is [[conservative functor|conservative]] (a [[bijection|bijective]] [[group homomorphism]] is a group isomorphism).

So it remains to consider congruences:

Since [[limits]] in $Grp$ are [[created limit|created]] in $Set$, a [[congruence]] in $Grp$ on a group $G$ is an [[equivalence relation]] on $G$ which is also a [[subgroup]] of $G\times G$.  This latter condition means that if $g_1\sim g_2$ and $h_1\sim h_2$, then also $g_1^{-1}\sim g_2^{-1}$ and $g_1 h_1 \sim g_2 h_2$.  Since $g\sim g$ for all $g$, it follows that $g\sim h$ if and only if $1\sim h g^{-1}$, so $\sim$ is determined by the subset $H\subseteq G$ of those $h\in G$ such that $1\sim h$.  This $H$ is clearly a [[subgroup]] of $G$, and moreover a [[normal subgroup]], since if $h\in H$ and $g\in G$ we have $1 = g^{-1} g \sim g^{-1} h g$, so $g^{-1} h g\in H$.  Conversely, it is easy to construct a congruence from any normal subgroup, so the two notions are equivalent.  It remains only to observe that the quotient of a group by a normal subgroup is, in fact, a quotient of its associated congruence in $Grp$, which is preserved by $U$.

Thus, by Duskin's monadicity theorem (Thm. \ref{DuskinMonadicityTheorem}), $U$ is monadic.

### Categories over computads

The monadicity theorem becomes more important when the base category $C$ is more complicated and harder to work with explicitly, and when the objects of $D$ are not obviously defined as "objects of $C$ with extra structure."  For instance, the category of [[strict 2-categories]] is monadic over the category of 2-[[globular sets]], essentially by definition, but it is much less trivial to show that it is *also* monadic over the category of [[2-computads]].  This latter fact can, however, be proven using the monadicity theorem.

### Monadic descent

The monadicity theorem also plays a central role in [[monadic descent]].

## In $(\infty,1)$-categories
 {#InInfinityCategories}

There is a version of the monadicity theorem for [[(∞,1)-monads]] in [section 3.4](http://arxiv.org/PS_cache/math/pdf/0702/0702299v5.pdf#page=107) of

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

There is also a 2-categorical approach using quasicategories in

* [[Emily Riehl]], [[Dominic Verity]] _The 2-category theory of quasi-categories_ ([arXiv](http://arxiv.org/abs/1306.5144)), _Homotopy coherent adjunctions and the formal theory of monads_ ([arXiv](http://arxiv.org/abs/1310.8279))

## Related concepts

* [[monadic functor]], [[comonadic functor]]

* [[functor of descent type]]

* [[monadic decomposition]]

* [[Bénabou-Roubaud's theorem]]

* [[monadic descent]]

## References

The original reference for the (crude and precise) monadicity theorems is an untitled manuscript of [[Jon Beck]] that was distributed around 1966 -- 1968. The following is a scan of a copy distributed at the *Conference Held at the Seattle Research Center of the Battelle Memorial Institute* in June -- July 1968, provided by John Kennison:

* {#Beck68} [[Jon Beck]], untitled manuscript, 1968 ([[Beck_MonadicityTheorem.pdf:file]])

see also:

* [[Michael Barr]], *Coalgebras in a category of algebras*, in: *Category Theory, Homology Theory and their Applications I*, Lecture Notes in Mathematics **86**, Springer (1969) 1-12 [[doi:10.1007/BFb0079381](https://doi.org/10.1007/BFb0079381)]

Textbook accounts:

* {#MacLane71} [[Saunders MacLane]], §VI.7 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5**  Springer (1971) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* {#Borceux} [[Francis Borceux]], Section 4 in volume 2 of  _[[Handbook of Categorical Algebra]]_, in 3 vols.

* {#BarrWells} Chapter 3 of [[Michael Barr]], [[Charles Wells]], _[[Toposes, Triples, and Theories]]_  Grundlehren der math. Wissenschaften 278, Springer-Verlag 1983.  Available as [TAC Reprint 12](http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html).

Further references:

* [[FGA explained]], [[SGA 1]],

* [[Fred Linton]], _Some aspects of equational categories_, Proceedings of the Conference on Categorical Algebra. Springer 1966.

* {#BenabouRoubaud70} [[Jean Bénabou]], [[Jacques Roubaud]], _Monades et descente_, C. R. Acad. Sc. Paris, Ser. A **270**  (1970) 96-98 [[gallica:12148/bpt6k480298g/f100](http://gallica.bnf.fr/ark:/12148/bpt6k480298g/f100), [[BenabouRoubaud-MonadesEtDescente.pdf:file]],   English translation (by [[Peter Heinig]]): [MO:q/279152](https://mathoverflow.net/q/279152)]

* [[Duško Pavlović]], Categorical interpolation: descent and the Beck-Chevalley condition without direct images,  Category theory Como 1990, pp. 306--325, Lecture Notes in Mathematics 1488, Springer 1991

* [[Pierre Deligne]], _[[Catégories Tannakiennes]]_ , Grothendieck Festschrift, vol. II, Birkhäuser Progress in Math. 87 (1990) pp. 111-195.

* Wikipedia: [monadicity theorem](http://en.wikipedia.org/wiki/Beck%27s_monadicity_theorem)

* John Bourke, _Two dimensional monadicity_, [arxiv/1212.5123](http://arxiv.org/abs/1212.5123)

* {#Riehl17} [[Emily Riehl]], §5.5 in: *[[Category Theory in Context]]*, Dover Publications (2017) [[pdf](http://www.math.jhu.edu/~eriehl/context.pdf), [book website](http://www.math.jhu.edu/~eriehl/context/)]

There is a version for [[Morita contexts]] instead of monads:

* [[Tomasz Brzezinski]], Adrian Vazquez Marquez, Joost Vercruysse, _The Eilenberg-Moore category and a Beck-type theorem for a Morita context_, Appl. Categ. Structures __19__ (2011), no. 5, 821–858 [MR2836546](http://www.ams.org/mathscinet-getitem?mr=2836546) [doi](http://dx.doi.org/10.1007/s10485-009-9217-0)

On Beck's theorem for [[pseudomonads]] ([[higher monadic descent]]):

* I. J. Le Creurer, [[Francisco Marmolejo]], [[Enrico M. Vitale]], _Beck's theorem for pseudo-monads_, J. Pure Appl. Algebra **173** 3 (2002) 293-313 [doi:10.1016/S0022-4049(02)00038-5]

Discussion for [[(infinity,1)-monads]]:

* [[Jacob Lurie]], section 4.7 of _[[Higher Algebra]]_

and realized in the context of [[quasi-categories]]:

* {#RiehlVerity13} [[Emily Riehl]], [[Dominic Verity]], section 7.2 of _Homotopy coherent adjunctions and the formal theory of monads_ ([arXiv:1310.8279](http://arxiv.org/abs/1310.8279))

[[!redirects monadicity theorems]]

[[!redirects Beck's Monadicity Theorem]]
[[!redirects Beck's Monadicity Theorem]]
[[!redirects Beck Monadicity Theorem]]
[[!redirects Barr-Beck theorem]]
[[!redirects monadicity theorem]]

[[!redirects Barr-Beck monadicity theorem]]

[[!redirects Beck's monadicity theorem]]
[[!redirects Beck's monadicity theorem]]
[[!redirects Beck monadicity theorem]]
[[!redirects tripleability theorem]]
[[!redirects Beck's tripleability theorem]]
[[!redirects Beck's tripleability theorem]]
[[!redirects Beck tripleability theorem]]
[[!redirects Beck's theorem]]
[[!redirects Beck's theorem]]
[[!redirects Beck theorem]]
[[!redirects crude monadicity theorem]]
[[!redirects crude tripleability theorem]]
[[!redirects monadicity]]


monads in Haskell

+-- {: .num_defn}
###### Definition
A monad in [[Haskell]] is defined to be a type class with two methods:

> $\mathbf{class}\;Monad\;m\;\mathbf{where}$

> $\gt\gt =::ma\to(a\to mb)\to mb$

> $return::a\to ma$

such that

> $x\gt\gt=f\equiv fx$

> $a\gt\gt=return\equiv a$

> $(a\gt\gt=f)\gt\gt=g\equiv a\gt\gt =(\lambda x\to fx\gt\gt =g)$

=--

+-- {: .num_remark}
###### Remark
$m$ being a [[monad]] $(m:a\to a, \mu:m^2\to m,\epsilon:id_a\to m)$ on some object $a$ in a [[2-category]] can be expressed in [[Haskell]] by

> $\mathbf{class}\;Monad\;m\;\mathbf{where}$

> $map::(a\to b)\to ma\to mb$

> $return::a\to ma$

> $join:: m(ma)\to ma$

We have the following translation of Haskell and [[category theory, language and library specification|category theoretical language]]:

> $join:=\mu$

> $return :=\epsilon$

Then from the category theoretical properties of $(m,\mu,\epsilon)$ we obtain (mixing the two languages)

> $map\; g\circ \epsilon\equiv \epsilon\circ g$

> $map \; \circ \mu\equiv \mu\circ map(map\; g)$

> $\mu\circ map\; \mu \equiv \mu\circ \mu$

> $\mu\circ \epsilon\equiv \mu\circ map\;\epsilon =id$

And with the following definitions

> $map\; f :=(\lambda\to a\gt\gt=(return\circ f))$

> $ join \; a=a\gt\gt id$

or alternatively

> $a\gt\gt =f:=join (map\;fa)$

one can verify that the two given definitions of a monad are equivalent.
=--

## References

category theory/monads, haskellwiki, [wiki](http://www.haskell.org/haskellwiki/Category_theory/Monads)


monads of probability, measures, and valuations

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In many [[categorical approaches to probability theory]], one considers a [[category]] of [[spaces]], such as [[measurable spaces]] or [[topological spaces]], and equips this category with a [[monad]] whose [[functor]] part assigns to each space $X$ a space $P X$ of [[measures]], [[probability measures]], or [[valuation (measure theory)|valuations]] on $X$, or a variation thereof.

For probability theory, this can be interpreted as adding to the [[points]] of a space $X$ new "random points", modelled as probability measures or valuations.
The old points, which we can think of as deterministic, are embedded in $P X$ via the unit of the monad $X\to P X$. Just as well, the [[Kleisli morphisms]] of $P$ can be seen as [[stochastic maps]].
(Monads can be seen as ways of extending our spaces and functions to account for new phenomena, see for example *[[extension system]]* and *[[monad in computer science]]*.)
Note that these probability measures are technically different from [[random elements]]: they rather correspond to the [[law of a random variable|laws]] of the random elements.

[[algebra over a monad|Algebras]] of probability and measure monads can be interpreted as generalized [[convex spaces]] or [[conical spaces]] of a certain kind. For [[probability theory]], in particular, the algebras of a probability monad can be seen as spaces equipped with a notion of [[expectation value]] of a [[random variable]].
The details vary depending on the monad and on the category under consideration.

Many choices of categories and of monads are possible, depending on which aspects of measure theory or probability one wants to study.
See [the table below](#detailed_list) for more details.

Note that "probability monad" and "measure monad" are not used here as technical terms with a definition. Here we just explain what most monads of this kind look like.
(There are ongoing works which may give a general, precise definition of probability monad.)
The term "probability monad" was coined by Giry (see [here](#giry80)), to refer to what we today call the [[Giry monad]].

## Functor, unit and multiplication

### The functor: random outcomes

The basic idea, that holds for all monads of this sort, is that of forming spaces of [[random elements]], or rather their [[law of a random variable|laws]].

Consider a category $C$, whose objects we can think of as "spaces of outcomes" with possibly extra structure, such as [[measurable spaces]], or [[topological spaces]], or even just [[sets]].
A _probability monad_ assigns to each space $X$ a space $P X$ which we can think of as a space of "laws of random outcomes on $X$" or other measure-like entities.
For example,

* The [[distribution monad]] assigns to each [[set]] the [[distribution monad#finite_distributions|space of finite distributions]] on it.
* The [[Giry monad]] assigns to each [[measurable space]] the [[Giry monad#definition|space of probability measures]] over it.
* The [[extended probabilistic powerdomain]] assigns to each [[topological space]] the [[extended probabilistic powerdomain#spaces_of_valuations|space of continuous valuations]] on it.
* The [[Kantorovich monad]] assigns to each [[metric space]] the [[Kantorovich monad#wasserstein_spaces|Wasserstein space]] over it.

(See also the table [below](#detailed_list).)

A possible interpretation of probability monads, which uses the interpretation of [[monads]] as encoding generalized [[theories]], is that $P X$ is a space of "formal generalized convex combinations" (in the sense of [[convex space]]) of points of $X$.
For instance, for the [[distribution monad]], given the set $\{heads,tails\}$ of possible outcomes of a coin flip, the set $P X$ contains for example the distribution that assigns probability $1/2$ to both outcomes. This can be thought of as the formal convex combination
$$
\frac{1}{2}\, heads + \frac{1}{2}\,tails.
$$

On the morphisms, the functor gives for example the [[pushforward of measures]].

### The unit: deterministic random variables

The [[unit]] of the [[monad]] gives, for each space of outcomes, a map $X\to P X$, which we can think of as picking out "those outcomes which are not really random", or "Dirac delta" (see [[Dirac measure]] and [[valuation (measure theory)#dirac_valuation|Dirac valuation]].

These can be seen as the laws of a [[deterministic random variable]].

### The multiplication: mixing probability measures.

The multiplication of the monad is a map $P P X\to P X$ for all objects $X$. This can be thought of as of "mixing" or "averaging" probability measures.

The following example is taken from [Perrone '19, Example 5.1.2](#notesperrone).
Suppose that you have two coins in your pocket. Suppose that one coin is fair, with "heads" on one face and "tails" on the other face; suppose the second coin has "heads" on both sides. Suppose now that you draw a coin randomly, and flip it.  We can sketch the probabilities in the following way:
  \begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
]
   &&& ? \ar{dll}[swap]{1/2} \ar{drr}{1/2} \\
   & \mathrm{coin} \,1 \ar{dl}[swap]{1/2} \ar{dr}{1/2} &&&& \mathrm{coin} \,2 \ar{dl}[swap]{1} \ar{dr}{0} \\
   \mathrm{heads} && \mathrm{tails} && \mathrm{heads} && \mathrm{tails}
  \end{tikzcd}
 Let $X$ be the set $\{heads,tails\}$. A coin gives a _law_ according to which we will obtain "heads" or "tails" so it determines an element of $P X$. Since the choice of coin is also random (we also have a _law on the coins_), the law on the coins determines an element of $P P X$.
 By averaging, the resulting overall probabilities are

  \begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
]
   & ? \ar{dl}[swap]{3/4} \ar{dr}{1/4} \\
   \mathrm{heads} && \mathrm{tails}
  \end{tikzcd}

 In other words, the "average" or "mixture" can be thought of as an assignment $E:P P X\to P X$, from laws of "random random variables" to laws of ordinary random variables.

## Algebras: expectation values

According to the interpretation of probability monads in terms of "formal generalized convex combinations", the [[algebra over a monad|algebras]] of a probability monad are then spaces equipped with a way of evaluating these expressions to a result, which we can see as a generalized "[[midpoint]]", "average", or "[[expectation value]]".
(Expectation values play a very important role in probability: probability monads can encode them via their algebras.)

In other words, algebras of probability monads can be thought of as generalizations of [[convex sets]], depending on the actual category and monad in question. For example,

* The [[algebra over a monad|algebras]] of the [[distribution monad]] are [[convex spaces]].
* The [[algebra over a monad|algebras]] of the [[Radon monad]] are [[compact space|compact]] convex subsets of [[locally convex topological vector spaces]].
* The [[algebra over a monad|algebras]] of the [[Kantorovich monad]] are closed convex subsets of [[Banach spaces]].

(See also the table [below](#detailed_list).)

If the measure are not required to be normalized, instead of (generalized) convex combinations one should think of linear combinations with non-negative coefficients, and so as algebra one gets a generalization of [[conical spaces]] instead.

## Kleisli morphisms: random maps

The basic interpretation of a Kleisli morphism, i.e. a map $X\to P Y$, is that of a _map with a random outcome_.
For example,

* For the [[distribution monad]], Kleisli morphisms are [[stochastic maps]] (or [[stochastic matrices]]).
* For the [[Giry monad]], Kleisli morphisms are [[Markov kernels]].
* ...and so on.

The [[Kleisli category|Kleisli composition]] recovers exactly the [[Chapman-Kolmogorov formula]] for the composition of stochastic maps. In the language of [[conditional probability]], this reads as
$$
p(z|x) = \sum_y p(z|y) \, p (y|x)
$$
for the discrete case, and as
$$
p(A|x) = \int_Y p(A|y) \, d p(y|x)
$$
for the continuous case.

(See [Giry's original article](#giry80) for the details, as well as the introductions given in [Perrone '18, Section 1.1](#thesisperrone) and [Perrone '19, Example 5.1.13](#notesperrone)).

[[Kleisli category|Kleisli categories]] of probability monads are often instances of [[Markov categories]].

## Monoidal structure: products and marginals

Probability monads are usually defined on [[monoidal categories]], in particular on [[cartesian monoidal categories]].
On product spaces $X\times Y$, the object $P(X\times Y)$ has the interpretation of containing [[joint distributions]]. Given a joint distribution, one can form the [[joint and marginal probability|marginals]] by pushing forward along the [[product]] projections $X\times Y\to X$ and $X\times Y\to Y$. This is in general a destructive operation, since [[stochastic dependence]] may be discarded.

Given [[marginal distributions]], one can form a canonical joint distribution by forming the [[product distribution]]. This is encoded by a [[monoidal monad|monoidal structure]] on the probability monad (or equivalently, a [[commutative monad|commutative strength]]), a map
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
row sep=0,%
]
PX\times PY \ar{r}{\nabla} & P(X\times Y)
\end{tikzcd}
which satisfies particular compatibility conditions.
The Giry monad is monoidal on the category [[Meas]] with the [[cartesian product]]. Similar statements are true for most other probability and measure monads.

When the monad is [[affine monad|affine]], one has that *a product probability is necessarily the product of its marginals*. This is the case for monads of probability measures, but not for monads of unnormalized measures.

More on this at [[joint and marginal probability]].

## Duality

(...work in progress...)

## Detailed list

  Monad ($P$)
  Category
  Elements/points of $P X$
  Extra structure of $P X$
  $P$-Algebras
  References

  [[distribution monad]] (a.k.a. finitary Giry monad, convex combination monad)
  [[Set]]
  [[convex combinations]] or finitely-[[support#in_measure_theory|supported]] [[probability measures]]
  (just a set)
  [[convex spaces]]
  [Fritz '09](#fritzconvex), [Jacobs '18](#jacobs)

  [[Giry monad]]
  [[Meas]]
  [[probability measures]]
  initial [[σ-algebra]] of evaluation maps
  Full characterization unknown. [[Giry monad#algebras_over_the_giry_monad|See also here]].
  [Lawvere '62](#lawvere62), [Giry '80](#giry80)

  [[Giry monad]]
  [[Polish space|Pol]]
  [[Borel measure|Borel probability measures]]
  [[initial topology]] of integration maps
  Full characterization unknown. [[Giry monad#algebras_over_the_giry_monad|See also here]].
  [Giry '80](#giry80)

  [[Probability monad on QBS]]
  [[quasi-Borel spaces]]
  Equivalence classes of [[random variables]]
  [[quasi-Borel structure]] inherited by the [[cartesian closed category|cartesian closed structure]]
  Full characterization unknown.
  [HKSY '17](#QBS)

  [[Radon monad]]
  [[compactum#category_of_compacta|Comp]]
  [[Radon measure|Radon probability measures]] (or [[continuous valuations]])
  [[weak topology]] w.r.t. continuous functions
  [[compact]] [[convex subsets]] of [[locally convex topological vector spaces]]
  [Swirszcz '74](#swirszcz), [Keimel '08](#radonkeimel)

  [[Radon monad#the_ordered_case|ordered Radon monad]]
  [[compact ordered spaces#categories_of_compact_ordered_spaces|CompOrd]]
  [[Radon measure|Radon probability measures]] (or [[continuous valuations]])
  [[weak topology]] w.r.t. continuous functions, [[stochastic order]]
  [[compact]] [[convex subsets]] of [[ordered vector space|ordered]] [[locally convex topological vector spaces]]
  [Swirszcz '74](#swirszcz), [Keimel '08](#radonkeimel)

  [[extended probabilistic powerdomain#the_measure_monad_on_top|measure monad on Top]]
  [[Top]]
  [[τ-additive measures]]
  [[extended probabilistic powerdomain#the_measure_monad_on_top|A-topology]], [[stochastic order]]
  Full characterization unknown. [[extended probabilistic powerdomain#algebras|See also here]].
  [F-P-R '19](#support)

  [[probabilistic powerdomain]]
  [[dcpo]], [[continuous domains]]
  [[continuous valuations]]
  [[stochastic order]]
  [[abstract probabilistic domains]] (continuous case)
  [J-P '89](#jonesplotkin)

  [[extended probabilistic powerdomain]]{#extprobpow}
  [[Top]], [[stably compact spaces]]
  [[continuous valuations]]
  [[extended probabilistic powerdomain#spaces_of_valuations|initial topology of evaluation maps]], [[stochastic order]]
  Full characterization unknown. [[extended probabilistic powerdomain#algebras|Dedicated section here]]
  [Heckmann '96](#heckmann96), [A-J-K '04](#jung), [GL-J '19](#algebras), [F-P-R '19](#support)

  [[valuation monad on locales]]
  [[Loc]]
  [[continuous valuations]]
  [[initial topology]] of evaluation maps
  ...
  [Vickers '11](#vmonad)

  [[Kantorovich monad]]
  [[Cauchy-complete|complete]] [[metric spaces]]
  [[Radon measure|Radon probability measures]] of finite first moment
  [[Kantorovich-Wasserstein metric]]
  closed [[convex subsets]] of [[Banach spaces]]
  [van Breugel '05](#vanbreugel), [F-P '19](#kantorovich19)

  [[Kantorovich monad#the_ordered_case|ordered Kantorovich monad]]
  [[Cauchy-complete|complete]] [[L-ordered]] [[metric spaces]]
  [[Radon measure|Radon probability measures]] of finite first moment
  [[Kantorovich-Wasserstein metric]], [[stochastic order]]
  closed [[convex subsets]] of [[ordered Banach spaces]]
  [F-P](#orderedkantorovich)

(...to be expanded...)

## See also

* [[monad]], [[algebra over a monad]], [[Kleisli category]]
* [[commutative monad]], [[affine monad]]
* [[monad (in computer science)]]
* [[Giry monad]], [[Markov kernel]], [[Stoch]]
* [[categorical probability]], [[Markov category]]
* [[joint and marginal distributions]]
* [[probability theory]], [[measure theory]]
* [[valuation (measure theory)]]

## References

* {#Lawvere62} [[W. Lawvere]], _The category of probabilistic mappings_, ms. 12 pages, 1962
([[lawvereprobability1962.pdf:file]])

* {#Giry80} Michèle Giry, _A categorical approach to probability theory_, Categorical aspects of topology and analysis (Ottawa, Ont., 1980), pp. 68–85, Lecture Notes in Math. **915** Springer 1982.

* {#swirszcz} T. Swirszcz, _Monadic functors and convexity_, Bulletin de l'Academie Polonais des Sciences 22, 1974 ([pdf](https://www.fuw.edu.pl/~kostecki/scans/swirszcz1974.pdf))

* {#radonkeimel} [[Klaus Keimel]], _The monad of probability measures over compact ordered spaces and its Eilenberg-Moore algebras_, Topology and its Applications, 2008 ([doi:10.1016/j.topol.2008.07.002](https://doi.org/10.1016/j.topol.2008.07.002))

* {#Heckmann96} Reinhold Heckmann, _Spaces of valuations_, Papers on General Topology and Ap-plications, 1996 ([doi:10.1111/j.1749-6632.1996.tb49168.x](https://doi.org/10.1111/j.1749-6632.1996.tb49168.x),[pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.45.5845&rep=rep1&type=pdf))

* {#jung} Mauricio Alvarez-Manilla, Achim Jung, [[Klaus Keimel]], _The probabilistic powerdomain for stably compact spaces_, Theoretical Computer Science 328, 2004. [Link here](https://www.sciencedirect.com/science/article/pii/S0304397504004074).

* {#jonesplotkin} C. Jones and [[Gordon. D. Plotkin]], _A probabilistic powerdomain of evaluations_, LICS 4, 1989. ([doi:10.1109/LICS.1989.39173](https://doi.org/10.1109/LICS.1989.39173))

* {#algebras} [[Jean Goubault-Larrecq]] and Xiaodong Jia, _Algebras of the extended probabilistic powerdomain monad_, ENTCS 345, 2019
([doi:10.1016/j.entcs.2019.07.015](https://doi.org/10.1016/j.entcs.2019.07.015))

* {#support} [[Tobias Fritz]], [[Paolo Perrone]] and Sharwin Rezagholi, _Probability, valuations, hyperspace: Three monads on Top and the support as a morphism_, Mathematical Constructions in Computer Science 31(8), 2021. [arXiv](https://arxiv.org/abs/1910.03752).

* {#vmonad} [[Steve Vickers]], _A monad of valuation locales_, 2011. [Link here](https://www.cs.bham.ac.uk/~sjv/Riesz.pdf).

* {#vanbreugel} Franck van Breugel, _The metric monad for probabilistic nondeterminism_, unpublished, 2005. ([pdf](http://www.cse.yorku.ca/~franck/research/drafts/monad.pdf))

* {#kantorovich19} [[Tobias Fritz]] and [[Paolo Perrone]], _A probability monad as the colimit of spaces of finite samples_, Theory and Applications of Categories 34, 2019. ([pdf](http://www.tac.mta.ca/tac/volumes/34/7/34-07.pdf))

* {#orderedkantorovich} [[Tobias Fritz]] and [[Paolo Perrone]], _Stochastic order on metric spaces and the ordered Kantorovich monad_, Advances in Mathematics 366, 2020. ([arXiv:1808.09898](https://arxiv.org/abs/1808.09898))

* {#jacobs} [[Bart Jacobs]], _From probability monads to commutative effectuses_, Journal of Logical and Algebraic Methods in Programming 94, 2018.
([doi:10.1016/j.jlamp.2016.11.006](https://doi.org/10.1016/j.jlamp.2016.11.006))

* {#fritzconvex} [[Tobias Fritz]], _Convex spaces I: definitions and examples_, 2009. ([arXiv:0903.5522](https://arxiv.org/abs/0903.5522))

* {#QBS} [[Chris Heunen]], Ohad Kammar, [[Sam Staton]] and Hongseok Yang, _A convenient category for higher-order probability theory_, Proceedings of LICS 2017. ([arXiv](https://arxiv.org/abs/1701.02547))

* [[Sean Moss]], [[Paolo Perrone]], _Probability monads with submonads of deterministic states_, LICS 2022. ([arXiv:2204.07003](https://arxiv.org/abs/2204.07003))

* [[Tobias Fritz]], Tomáš Gonda, [[Paolo Perrone]], Eigil Fjeldgren Rischel, _Representable Markov categories and comparison of statistical experiments in categorical probability_, Theoretical Computer Science 961, 2023. ([arXiv:2010.07416](https://arxiv.org/abs/2010.07416))

An account in terms of [[codensity monads]] in:

* Ruben Van Belle, _Probability monads as codensity monads_. Theory and Applications of Categories 38 (2022), 811–842, ([tac](http://tac.mta.ca/tac/volumes/38/21/38-21abs.html))

An introduction to some of the concepts can be also found in:

* {#notesperrone} [[Paolo Perrone]], _Notes on Category Theory with examples from basic mathematics_, Chapter 5. ([arXiv](http://arxiv.org/abs/1912.10642))

* {#thesisperrone} [[Paolo Perrone]], _Categorical probability and stochastic dominance in metric spaces_, PhD thesis, Chapter 1. ([pdf](http://www.paoloperrone.org/phdthesis.pdf))

* [[Tobias Fritz]] and [[Paolo Perrone]], _Monads, partial evaluations, and rewriting_, MFPS 36, 2020 - Section 6. ([arXiv](https://arxiv.org/abs/1810.06037))

and in the following video lectures:

* [[Arthur Parzygnat]], _Categorical probability_, video playlist. ([YouTube](https://www.youtube.com/playlist?list=PLSx1kJDjrLRSKKHj4zetTZ45pVnGCRN80))

* [[Paolo Perrone]], _What is a probability monad?_, tutorial. ([YouTube](https://youtu.be/3Da88Tgw_rM))

category: probability

[[!redirects probability monad]]
[[!redirects probability monads]]
[[!redirects measure monad]]
[[!redirects measure monads]]
[[!redirects valuation monad]]
[[!redirects valuation monads]]
[[!redirects monad of probability]]
[[!redirects monads of probability]]
[[!redirects monad of probability measures]]
[[!redirects monads of probability measures]]
[[!redirects monad of measures]]
[[!redirects monads of measures]]
[[!redirects monad of valuations]]
[[!redirects monads of valuations]]
[[!redirects probability and measure monad]]
[[!redirects probability and measure monads]]
[[!redirects measure and valuation monad]]
[[!redirects measure and valuation monads]]
[[!redirects monads of probability measures valuations]]
[[!redirects monads of probability measures and valuations]]
[[!redirects monads of probability, measures and valuations]]
[[!redirects monads of probability, measures, valuations]]


MonCat

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

#Contents#
* table of contents
{:toc}

## Idea

$MonCat_{s}$ or $MonCat_\ell$ is the [[2-category]] whose [[objects]] are [[monoidal categories]], [[1-morphisms]] are [[strong monoidal functors]] or [[lax monoidal functors]], respectively, and whose [[2-morphisms]] are [[monoidal natural transformations]].

## Related concepts

* [[monoidal adjunction]]

* [[change of enriching category]]

[[!include categories of categories - contents]]


MOND

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[standard model of cosmology]] including [[dark energy]] and [[dark matter]] is in very good agreement with observation on scales above those of galaxies, but in its standard version becomes problematic below this scale (there are variants that address this, such as [[fuzzy dark matter]]).

Curiously though, the particular behaviour of experimental data on these "small" cosmological scales turns out to have a remarkably simple and universal phenomenological fit by a simple modification of the [[force]] law of [[Newtonian mechanics]] ("Newton's third law", corrections from [[general relativity]] are typically very small for the effects in question).

_MOND_ is the abbreviation for this _modified Newtonian dynamics_ ([Milgrom 83 a](#Milgrom83a), [Milgrom 83 b](#Milgrom83b), [Milgrom 83 c](#Milgrom83c)). This refers to the proposal of modifying Newton's third law

$$
  F = m a
$$

relating the [[force]] experienced by a body of [[mass]] $m$ to its [[acceleration]] $a$ by an expression of the form

$$
  F = m \,\mu(a/a_0)\, a
$$

for some acceleration scale $a_0$ and some interpolating function $\mu \colon \mathbb{R} \to \mathbb{R}$.

The motivation is that choosing the constant $a_0$ and the function $\mu$ suitably, then such a modified formula fits the rotation-velocities of observed [[galaxies]] in dependence of the [[radius]] remarkably well, something which the [[standard model of cosmology]] with [[dark matter]] still has some problems with (but see [Kaplinghat-Turner 02](#KaplinghatTurner02), [BLSF 09](#BLSF09), [Chan 13](#Chan13)).

In 2020, David Merritt wrote a review of MOND and dark matter which stated that the observational evidence favors MOND over dark matter ([Merritt20](#Merritt20)).

## Relativistic extensions of MOND

The original MOND model by Milgrom is an ad-hoc such modification of the basic laws of physics and breaks many established properties of physics, such as the [[principle of equivalence]] in [[general relativity]] and various [[conservation laws]]. Thus, many theories have been developed to reconcile MOND with relativity, including:

* Tensor-Vector-Scalar gravity ([Bekenstein 04](#Bekenstein04), "TeVeS")

* Aether scalar tensor theory ([SkordisZlosnik21](#SkordisZlosnik21), [SkordisZlosnik22](#SkordisZlosnik22), [MMH23](#MMH23), "AeST)

* Bimetric MOND ([Milgrom22](#Milgrom22))

* Khronometric MOND ([Flanagan23](#Flanagan23))

In addition, there are also studies of relativistic MOND gravity in general ([TMZ23](#TMZ23), [THCZ23](#THCZ23)).

{#TheseTheories} These theories tend to postulate new [[relativistic field theory|relativistic]] [[field (physics)|fields]] adjoined to plain [[Einstein gravity]]. While the [[relativistic field theory|relativistic]] [[interactions]] of these hypothetical new fields is designed to mimic the originally postulated MOdfied Newtonian Dynamics in appropriate regime, to be [[phenomenology|phenomenologically]] viable they must otherwise be effectively unobservable and as such are rather like the [[dark matter]] [[field (physics)|fields]] whose introduction the original idea of MOND it was to avoid. Therefore, the difference between the oxymoronic "relativistic MOND" and mainstream [[dark matter]] is now more one of style and focus than of principle.

### Tensor-Vector-Scalar gravity

The first such proposal for such a theory apart from [[gravity]]+[[dark matter]] itself ([Kaplinghat-Turner 02](#KaplinghatTurner02), [BLSF 09](#BLSF09), [Chan 13](#Chan13)) is to add to Einstein gravity a unit vector field and a [[scalar field]] ([Bekenstein 04](#Bekenstein04), "TeVeS"), themselves thus otherwise unobserved "dark fields", as it were,  but, as opposed to say the [[axion]] dark matter candidate, not motivated beyond the desire to fit galaxy rotation curves.

From [Bekenstein 04, p. 9](#Bekenstein04) one sees explicitly that TeVeS is just Einstein-gravity coupled to peculiar "matter" fields:

<img src="https://ncatlab.org/nlab/files/TeVeSActionFunctional.jpg" width="600">

## Related concepts

* An alternative proposal is that of [[fuzzy dark matter]], where the theory of gravity is left as is, but one assumes that there is [[dark matter]] which is so extremely light that its [[de Broglie wavelength]] is of the scales of galaxies. Such [[fuzzy dark matter]] models make the same prediction as cold dark matter on cosmological scales, but change behaviour on scales of galaxies due to the [[quantum physics|quantum]] nature of these light particles becoming relevant here. This change of behaviour has been argued to account for the kind of effects that MOND is a fit for.

## References

### Original articles

The concept of MOND is due to

* {#Milgrom83a} [[Mordehai Milgrom]], _A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis_, Astrophysical Journal. 270: 365–370. (1983) doi:10.1086/161130.

* {#Milgrom83b} [[Mordehai Milgrom]], _A modification of the Newtonian dynamics - Implications for galaxies_, Astrophysical Journal. 270: 371–389. (1983) doi:10.1086/161131.

* {#Milgrom83c} [[Mordehai Milgrom]], _A modification of the Newtonian dynamics - Implications for galaxy systems_, Astrophysical Journal. 270: 384. (1983) doi:10.1086/161132

There are many relativistic extensions of MOND, which are listed in the following section:

### Relativistic MOND models
 {#ReferencesRelativisticModels}

On [[general relativity|relativistic]] MOND [[model (in theoretical physics)|models]]:

* {#Bekenstein04} [[Jacob Bekenstein]], *Relativistic gravitation theory for the modified Newtonian dynamics paradigm*, Physical Review D **70** 8 (2004) 083509, [[arXiv:astro-ph/0403694](https://arxiv.org/abs/astro-ph/0403694), [doi:10.1103/PhysRevD.70.083509](https://doi.org/10.1103/PhysRevD.70.083509)]

* {#SkordisZlosnik21} [[Constantinos Skordis]], [[Tom Zlosnik]], *A new relativistic theory for Modified Newtonian Dynamics*. Physical Review Letters. Volume 127, Issue 16. Published 15 October 2021 ([doi:10.1103/PhysRevLett.127.161302](https://doi.org/10.1103/PhysRevLett.127.161302), [arXiv:2007.00082](https://arxiv.org/abs/2007.00082))

* {#SkordisZlosnik22} [[Constantinos Skordis]], [[Tom Zlosnik]], *Aether scalar tensor theory: Linear stability on Minkowski space*. Physical Review D. Volume 106, Issue 10. Published 15 November 2022. ([doi:10.1103/PhysRevD.106.104041](https://doi.org/10.1103/PhysRevD.106.104041), [arXiv:2109.13287](https://arxiv.org/abs/2109.13287))

* {#MMH23} [[Tobias Mistele]], [[Stacy McGaugh]], [[Sabine Hossenfelder]], *Aether Scalar Tensor theory confronted with weak lensing data at small accelerations*, ([arXiv:2301.03499](https://arxiv.org/abs/2301.03499))

* {#Milgrom22} [[Mordehai Milgrom]], *Broader view of bimetric MOND*, Physical Review D, Volume 106, Issue 8. Published 10 October 2022. ([doi:10.1103/PhysRevD.106.084010](https://doi.org/10.1103/PhysRevD.106.084010), [arXiv:2208.10882](https://arxiv.org/abs/2208.10882))

* {#TMZ23} [[Daniel Thomas]], [[Ali Mozaffari]], [[Tom Zlosnik]], *Consistent cosmological structure formation on all scales in relativistic extensions of MOND* ([arXiv:2303.00038](https://arxiv.org/abs/2303.00038))

* {#Flanagan23} [[Eanna Flanagan]], *Khronometric theories of modified Newtonian dynamics* ([arXiv:2302.14846](https://arxiv.org/abs/2302.14846))

* {#THCZ23} Shuxun Tian, Shaoqi Hou, Shuo Cao, Zong-Hong Zhu, *Time evolution of the local gravitational parameters and gravitational wave polarizations in a relativistic MOND theory*, Physical Review D, Volume 107, Issue 4. Published 24 February 2023. ([doi:10.1103/PhysRevD.107.044062](https://doi.org/10.1103/PhysRevD.107.044062), [arXiv:2302.13304](https://arxiv.org/abs/2302.13304))

### Review

General review includes

* Benoît Famaey and Stacy S. McGaugh, _Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions_,2012, Living Reviews in Relativity, 15, 10  ([arXiv:1112.3960](https://arxiv.org/abs/1112.3960))

* Joe Silk, Gary A. Mamon, _The Current Status of Galaxy Formation_ ([arXiv:1207.3080](http://arxiv.org/abs/1207.3080))

* Wikipedia, _[Modified Newtonian dynamics](https://en.wikipedia.org/wiki/Modified_Newtonian_dynamics)_

* Wikipedia, _[Tensor–vector–scalar gravity](https://en.wikipedia.org/wiki/Tensor%E2%80%93vector%E2%80%93scalar_gravity)_

* {#Peebles22} [[Phillip James E. Peebles]], §5.7 "Why is MOND Successful but Unpoular" of: *Anomalies in Physical Cosmology* [[arXiv:2208.05018](https://arxiv.org/abs/2208.05018)]

* {#Merritt20} [[David Merritt]], *A Philosophical Approach to MOND: Assessing the Milgromian Research Program in Cosmology*. Cambridge University Press, 2020. ISBN:978-1108492690

### Experimental constraints

The stark failure of plain MOND to fit data on large cosmological scales is highlighted in

* {#Dodelson11} Scott Dodelson, _The Real Problem with MOND_,  Int. J. Mod. Phys. D, 20, 2749 (2011).  ([arXiv:1112.1320](https://arxiv.org/abs/1112.1320))

* Kris Pardo, [[David Spergel]], _What is the price of abandoning dark matter? Cosmological constraints on alternative gravity theories_ ([arXiv:2007.00555](https://arxiv.org/abs/2007.00555))

The instability of its relativistic completion by TeVeS was pointed out in

* Michael D. Seifert, _Stability of spherically symmetric solutions in modified theories of gravity_, Phys.Rev.D76:064002, 2007 ([arXiv:gr-qc/0703060](https://arxiv.org/abs/gr-qc/0703060))

{#ConstraintsFromMergingNeutronStars} The detection of [[gravitational waves]] coincident with [[electromagnetic radiation]] from merging [[neutron stars]] (event GW170817, [LIGO-Virgo 17](LIGO-Virgo#LIGOVirgo17)) constrains relativistic completions of MOND:

* Jose María Ezquiaga, Miguel Zumalacárregui, _Dark Energy after GW170817_ ([arXiv:1710.05901](https://arxiv.org/abs/1710.05901))

* Sibel Boran, Shantanu Desai, Emre Kahya, Richard Woodard, _GW170817 Falsifies Dark Matter Emulators_ ([arXiv:1710.06168](https://arxiv.org/abs/1710.06168))

The observation of a galaxy that does not exhibit the effect which MOND claims is universal (otherwise attributed to the presence of [[dark matter]]) is reported in

* {#vanDokkum18} Pieter van Dokkum et.al. _A galaxy lacking dark matter_, Nature volume 555, 2018 pages 629–632 ([arXiv:1803.10237](https://arxiv.org/abs/1803.10237), [doi:10.1038/nature25767](https://www.nature.com/articles/nature25767))

> Regardless of the formation history of NGC1052–DF2, its existence has implications for the dark matter paradigm. Our results demonstrate that dark matter is separable from galaxies, which is (under certain circumstances) expected if it is bound to baryons through nothing but gravity. The "[[bullet cluster]]" demonstrates that dark matter does not always trace the bulk of the baryonic mass, which in clusters is in the form of gas.  NGC1052–DF2 enables us to make the complementary point that dark matter does not always coincide with galaxies either:  it is a distinct "substance" that may or may not be present in a galaxy.   Furthermore,  and paradoxically, the existence of NGC1052–DF2 may falsify
alternatives to dark matter. In theories such as [[MOND]] and the recently proposed emergent gravity paradigm a "dark matter" signature should always be detected, as it is an unavoidable consequence of the presence of ordinary matter. In fact, it had been argued previously that the apparent absence of galaxies such as NGC1052–DF2 constituted a falsification of the standard cosmological model, and evidence for modified gravity.

The observation that wide binaries do not seem to have MOND effects:

* [[Charalambos Pittordis]], [[Will Sutherland]], *Wide Binaries from GAIA EDR3: preference for GR over MOND?*. The Open Journal of Astrophysics, Vol. 6, 2023, February 02, 2023. ([doi:10.21105/astro.2205.02846](https://doi.org/10.21105/astro.2205.02846), [arXiv:2205.02846](https://arxiv.org/abs/2205.02846))

The observation that SPARC galaxies prefer dark matter over MOND:

* [[Mariia Khelashvili]], [[Anton Rudakovskyi]], [[Sabine Hossenfelder]], *SPARC galaxies prefer Dark Matter over MOND* ([arXiv:2401.10202](https://arxiv.org/abs/2401.10202))

### MOND phenomenology from actual dark matter

Derivations of the MOND phenomenology from actual [[dark matter]] is discussed in the following articles:

* {#KaplinghatTurner02} Manoj Kaplinghat, Michael S. Turner, _How Cold Dark Matter Theory Explains Milgrom's Law_, Astrophys.J. 569 (2002) L19 ([arXiv:astro-ph/0107284](https://arxiv.org/abs/astro-ph/0107284))

* {#BLSF09} Jean-Philippe Bruneton, Stefano Liberati, Lorenzo Sindoni, Benoit Famaey, _Reconciling MOND and dark matter?_, Journal of Cosmology and Astroparticle Physics, Issue 03, pp. 021 (2009) ([arXiv:0811.3143](https://arxiv.org/abs/0811.3143))

* {#Chan13} Man Ho Chan, _Reconciliation of MOND and Dark Matter theory_, Phys. Rev. D, 88, 103501 (2013) ([arXiv:1310.6801](https://arxiv.org/abs/1310.6801))

and specifically for [[axion|axionic]] [[fuzzy dark matter]]:

* {#HOTW16} [[Lam Hui]], [[Jeremiah Ostriker]], [[Scott Tremaine]], [[Edward Witten]], _On the hypothesis that cosmological dark matter is composed of ultra-light bosons_,  Phys. Rev. D 95, 043541 (2017) ([arXiv:1610.08297](https://arxiv.org/abs/1610.08297))

and for fuzzy dark matter with [[superfluid]]-effects includes in

* {#BerezhianiKhoury15} [[Lasha Berezhiani]], [[Justin Khoury]], _Theory of Dark Matter Superfluidity_, Phys. Rev. D 92, 103510 (2015) ([arXiv:1507.01019](https://arxiv.org/abs/1507.01019))

* {#Khoury16} [[Justin Khoury]], _Another Path for the Emergence of Modified Galactic Dynamics from Dark Matter Superfluidity_, Phys. Rev. D 93, 103533 (2016) ([arXiv:1602.05961](https://arxiv.org/abs/1602.05961))

### Hybrid MOND dark matter models

Hybrid models which contain both [[MOND]] and [[dark matter]] are explored in

* [[Tobias Mistele]], *Cherenkov radiation from stars constrains hybrid MOND dark matter models*. Journal of Cosmology and Astroparticle Physics, Volume 2022, November 2022 ([doi:10.1088/1475-7516/2022/11/008](https://doi.org/10.1088/1475-7516/2022/11/008), [arXiv:2103.16954](https://arxiv.org/abs/2103.16954))

* [[Tobias Mistele]], *A novel Cherenkov radiation constraint for hybrid MOND dark matter models*, ([arXiv:2208.14308](https://arxiv.org/abs/2208.14308))

### Other references

For possible MOND effects in the Solar System:

* Cezary Migaszewski, *On the origin of extreme trans-Neptunian objects within Modified Newtonian Dynamics* ([arXiv:2303.13339](https://arxiv.org/abs/2303.13339))

MOND as an alternative to Planet Nine hypothesis:

* Katherine Jones-Smith, Harsh Mathur, *Modified Newtonian Dynamics as an Alternative to the Planet Nine Hypothesis* ([arXiv:2304.00576](https://arxiv.org/abs/2304.00576))


monic map in a dagger 2-poset

[[!redirects injective map in a dagger 2-poset]]

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## Definition ##

A morphism $f \in Hom_A(a,b)$ of a [[dagger 2-poset]] $A$ is a **monic map** if it is an [[injective morphism in a dagger 2-poset|injective]] [[map in a dagger 2-poset|map]].

The type of all monic maps in $Hom_A(a,b)$ is defined as

$$MonoMap(a, b) \coloneqq \sum_{f:Hom_A(a,b)} isInjective(f) \times isMap(f)$$

## See also ##

* [[injective morphism in a dagger 2-poset]]
* [[map in a dagger 2-poset]]
* [[category of monic maps]]


monic polynomial

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

***

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## Definition

Given a unital [[ring]] $k$, a _monic polynomial_ over $k$ is a [[polynomial]] with [[coefficients]] in $k$, whose highest order coefficient is $1$.

A [[root]] of a monic polynomial over $k$ is by definition an _[[algebraic integer]]_ over $k$.

Here _algebraic integer_ usually means _algebraic integer over $\mathbf{Z}$_. All algebraic integers form a [[field]] called the _[[integral closure]]_ of $\mathbf{Z}$ in $\mathbf{C}$.

On the other hand, for a [[number field]] $K$, an [[integer]] in $K$ is an algebraic integer over $\mathbf{Z}$ which is in $K$; all integers in $K$ form a __ring of integers__ $\mathcal{O}_K$ of the number field $K$.

## References

* Wikipedia, _[Algebraic number](http://en.wikipedia.org/wiki/Algebraic_integer)_

[[!redirects monic polynomials]]


Monica Guica

* [Spire page](https://inspirehep.net/authors/1019389)

## Selected writings

On [[Kerr black holes]] in the context of the [[holographic principle]] and the [[AdS-CFT correspondence]]:

* [[Monica Guica]], [[Thomas Hartman]], Wei Song, [[Andrew Strominger]], _The Kerr/CFT Correspondence_, Phys. Rev. D **80** (2009) 124008 [[arXiv:0809.4266](http://arxiv.org/abs/0809.4266), [doi:10.1103/PhysRevD.80.124008](https://doi.org/10.1103/PhysRevD.80.124008)]

On [[TT deformation|$T \bar T$ deformations]] and the [[AdS/CFT correspondence]]:

* [[Monica Guica]], _$T\bar T$ deformations and holography: review and open questions_, talk at *[[Strings 2022]]* [[indico:4940816](https://indico.cern.ch/event/1085701/contributions/4940816/), [slides](https://indico.cern.ch/event/1085701/contributions/4940816/attachments/2481655/4260285/slides.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=bfe514d2-b4c8-4f03-a2d1-33ee630b74f6)]

category: people


Monika Blanke

* [Institute page](https://www.ttp.kit.edu/memberpages/blanke)

* [Spire page](https://inspirehep.net/authors/1032646?ui-citation-summary=true)

## Selected writings

On apparent [[flavour anomalies]] in [[B-meson]] decays:

* [[Monika Blanke]], [[Andreas Crivellin]], Stefan de Boer, Teppei Kitahara, Marta Moscati, Ulrich Nierste, Ivan Nišandžić, _Impact of polarization observables and $B_c \to \tau \nu$ on new physics explanations of the $b \to c \tau \nu$ anomaly_ ([arXiv:1811.09603](https://arxiv.org/abs/1811.09603))

* [[Monika Blanke]], [[Andreas Crivellin]], Stefan de Boer, Teppei Kitahara, Marta Moscati, Ulrich Nierste, Ivan Nišandžić, _Addendum: "Impact of polarization observables and $B_c \to \tau \nu$ on new physics explanations of the $b \to c \tau \nu$ anomaly"_ ([arXiv:1905.08253](https://arxiv.org/abs/1905.08253))

* [[Monika Blanke]], _Flavour Physics from Present to Future Colliders_, talk at [LFC19: Strong dynamics for physics within and beyond the Standard Model at LHC and Future Colliders](https://indico.ectstar.eu/event/55/) (2019) ([arxiv:1910.10662](https://arxiv.org/abs/1910.10662))

and their potential explanation by [[leptoquarks]] in [[Randall-Sundrum model|Randall-Sundrum]]/[[GUT]]-models:

* {#BlankeCrivellin18} [[Monika Blanke]], [[Andreas Crivellin]], _$B$ Meson Anomalies in a Pati-Salam Model within the Randall-Sundrum Background_, Phys. Rev. Lett. 121, 011801 (2018) ([arXiv:1801.07256](https://arxiv.org/abs/1801.07256))

* [[Monika Blanke]], *Theory Perspective on Heavy Flavour Physics*, talk at [10th Edition of the Large Hadron Collider Physics Conference](https://indico.cern.ch/event/1109611/) (2022) [[arXiv:2207.07354](https://arxiv.org/abs/2207.07354), talk slides and recording: [indico.cern:4686945](https://indico.cern.ch/event/1109611/contributions/4686945)]

category: people


Monika Marquart

## Selected writings

On [[duality in string theory]]:

* {#Marquart02} [[Monika Marquart]], Section A.3 of: _Applications of Dualities in String Theory_, 2002 ([sundoc:02/02H150](https://sundoc.bibliothek.uni-halle.de/diss-online/02/02H150), [pdf](https://sundoc.bibliothek.uni-halle.de/diss-online/02/02H150/prom.pdf))

On [[F-theory]] and [[heterotic M-theory on ADE-orbifolds]]:

* [[Monika Marquart]], [[Daniel Waldram]], _F-theory duals of M-theory on $S^1/\mathbb{Z}_2 \times T^4 / \mathbb{Z}_N$_ ([arXiv:hep-th/0204228](https://arxiv.org/abs/hep-th/0204228))

category: people


Monique Hakim

Monique Hakim was a student of [[Alexander Grothendieck]].

* Oberwolfach entry [I](http://owpdb.mfo.de/detail?photo_id=1572), [II](http://owpdb.mfo.de/person_detail?id=1518)

* [Mathematics genealogy](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=130211)

## Selected writings

Her thesis

* *Topos annelés et schémas relatifs*, Ergebnisse der Mathematik und ihrer Grenzgebiete, **64** Springer (1972) [[doi:10.1007/978-3-662-59155-0](https://doi.org/10.1007/978-3-662-59155-0)]

systematically develops ([[algebraic geometry|algebraic]]) [[geometry]] via ([[locally ringed topos|locally]]) [[ringed toposes]].

See also at _[[Cole's theory of spectrum]]_. A generalization of this program from [[topos theory]] to [[(∞,1)-topos theory]] is in [[Jacob Lurie|Lurie]]'s _[[Structured Spaces]]_.

## Related entries

* [[ringed topos]]

category: people


monism

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## Idea

Monism is a philosophical position which states that there is only one valid approach to a particular field. This contrasts with [[pluralism]], which states that there are multiple equally valid approaches to a particular field.

## Monism in the foundations of mathematics

In the context of the [[foundations of mathematics]], monism states that there is only a one valid approach to the foundations of mathematics. Traditionally, the mathematical status quo regarded the [[material set theory]] of [[ZFC]] as the one valid approach to the foundations of mathematcs. Some category theorists take the [[categorical set theory]] of [[ETCS]] as the one valid approach to the foundations of mathematics, while some type theorists take [[Martin-Löf type theory]] or [[cubical type theory]] to be the one valid approach to the foundations of mathematics.

## See also

* [[philosophy]]

* [[pluralism]]

* [[epistemology of mathematics]]

## References

On the failure of monism in [[set theory]]:

* Neil Barton (2017). *Independence and Ignorance: How agnotology informs set-theoretic pluralism.* ([philsci-archive:14636](https://philsci-archive.pitt.edu/14636/)

## External links

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


monochromatic layer

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## Idea

For each [[prime]] $p \in \mathbb{N}$ and for each [[natural number]]  $n \in \mathbb{N}$ there is a [[Bousfield localization of spectra]]

$$
  L_n \coloneqq L_{E(n)}
  \,,
$$

where $E(n)$ is the $n$th [[Morava E-theory]] (for the given [[prime]] $p$). These arrange into the _chromatic tower_ which for each spectrum $X$ is of the form

$$
  X \to \cdots \to L_n X \to L_{n-1} X \to \cdots \to L_0 X
  \,.
$$

The [[homotopy fibers]] of each stage of the tower

$$
  M_n(X) \coloneqq fib(L_{E(n)}X \longrightarrow L_{E(n-1)}(X))
$$

is called the $n$th _monochromatic layer_ of $X$.

## Examples

### Chromatic layers of the sphere spectrum

Discussion of the first chromatic layer of the [[sphere spectrum]]
is due to ([Miller-Ravenel-Wilson 77](#MillerRavenelWilson77)). Review is in ([Knudsen 13](#Knudsen13)). This is the [[image of the J-homomorphism]].

## Related concepts

* [[chromatic convergence theorem]]

* [[smash product theorem]]

## References

General lecture notes include

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010, Lecture 34 _Monochromatic layers_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture34.pdf))
 {#LurieLect34}

Discussion of the chromatic layers of the [[sphere spectrum]] is in

* [[Haynes Miller]], [[Doug Ravenel]], Stephen Wilson, _Periodic phenomena in the Adams-Novikov spectral sequence_ ([JSTOR](http://www.jstor.org/stable/1971064))
 {#MillerRavenelWilson77}

Lecture notes on this  include

* Ben Knudsen, _First chromatic layer of the sphere spectrum = homotopy of the $K(1)$-local sphere_, talk at _[2013 Pre-Talbot Seminar](http://math.northwestern.edu/~htanaka/pretalbot2013/index.php)_ ([pdf](http://math.northwestern.edu/~htanaka/pretalbot2013/notes/2013-04-04-Rob-Legg-K1-local-sphere.pdf))
 {#Knudsen13}

* Johan Konter, _The $K(n)$-local sphere_, talk at _[Talbot 2013: Chromatic homotopy theory](http://math.mit.edu/conferences/talbot/index.php?year=2013&sub=talks)_ ([pdf](http://math.mit.edu/conferences/talbot/2013/8-Konter-LK(n)S.pdf))

* Drew Heard, _Resolutions of the $K(2)$-local sphere_, talk at _[Talbot 2013: Chromatic homotopy theory](http://math.mit.edu/conferences/talbot/index.php?year=2013&sub=talks)_ ([pdf](http://math.mit.edu/conferences/talbot/2013/19-Lawson-thefuture.pdf))

[[!redirects monochromatic layers]]

[[!redirects chromatic layer]]
[[!redirects chromatic layers]]