enriched monad

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## Idea

The notion of *enriched monads* is that of *[[monads]]* in the context of [[enriched category theory]]: For $V$ a [[base of enrichment]], a $V$-enriched monad is a [[monad]] [[internalization|internal]] to the [[2-category]] [[VCat]] of $V$-[[enriched categories]].

Generally (except in the base case $V = $ [[Set]]) the [[structure]] of a $V$-enriched monad on a $V$-[[enriched category]] $\mathbf{C}$ is *stronger* than that of the [[underlying]] monad on the underlying [[Set]]-category $C$, whence one also speaks of *[[strong monads]]* (*a priori* a different notion, which however coincides with that of enriched monads under mild conditions, such as when $V$ is [[closed monoidal category|closed]], see [there](strong+monad#LeftStrongMonadsAreEnrichedMonads)).

The concept of enriched monads is key for the application of [[monads in computer science]], since a [[monad]] coded verbatim in a [[functional programming language]] --- where [[function types]] $X \to Y$ are to be [[categorical semantics|interpreted]] not as external [[hom-sets]] but as [[internal homs]] in the ambient [[closed monoidal category]] $V$ of [[data types]] --- is really a $V$-enriched monad (hence typicall a [[strong monad]]).

## Definition

Let

* $(V, \otimes, I)$ be a [[symmetric monoidal category]] which serves as the [[base of enrichment]],

  with $I$ denoting its [[unit object]].

* $\mathbf{C}$ be a $V$-[[enriched category]]

  with [[underlying]] [[Set]]-category denoted $C$, and

  with [[hom-objects]] between any pais of [[objects]] $X, Y$ denoted $\mathbf{C}(X,Y) \,\in\, V$.

\begin{definition}
A *$V$-enriched monad* on $\mathbf{C}$ is, in [[Kleisli triple]]-presentation (eg. [McDermott & Uustalu 2022, Def. 5.8](#McDermottUustalu22)):

* for every object $X \,\in\, \mathbf{C}$, an object $T(X) \,\in\,\mathbf{C}$;

* for every object $X \,\in\,\mathbf{C}$, a morphism in $V$ of the form

  $$
   ret_X
     \;\colon\;
    I \to \mathbf{C}\big(X,T(X)\big)
  $$

  (the *[[monad unit]]*)

* for all [[pairs]] of objects $X,Y$ of $mathbf{C}$, a morphism  in $V$ of the form

  $$
    bind_{X,Y}
      \;\colon\;
    \mathbf{C}\big(X,T(Y)\big)
      \to
    \mathbf{C}\big(T(X),T(Y)\big)
  $$

  (the *[[Kleisli extension]]* or *bind*-operation)

such that the structural equations on a [[Kleisli triple]]

1. $bind(f) \circ ret_X = f$,

1. $bind(ret_X\big) = id$

1. $bind(g) \circ bind(f) = bind\big(bind(g) \circ f\big)$

hold for [[generalized elements]] $f$, $g$ of the [[hom-objects]] $\mathbf{C}\big(X,\,T(Y)\big)$, $\mathbf{C}\big(Y,\,T(Z)\big)$, which means that the following [[commuting diagram|diagrams commute]] in $V$ for all objects $X, Y, Z$ of $\mathbf{C}$:

\begin{tikzcd}[column sep=60pt]
  &
  \mathbf{C}\big(T(X),\, T(Y)\big)
  \ar[
    dr,
    "{
      \mathbf{C}\big(
        \mathrm{ret}_X
        ,\,
        T(Y)
     \big)
    }"{sloped}
  ]
  \\
  \mathbf{C}\big(
    X
    ,\,
    T(Y)
  \big)
  \ar[
    ur,
    "{
      \mathrm{bind}_{X,Y}
    }"{sloped}
  ]
  \ar[
    rr,
    equals
  ]
  &&
  \mathbf{C}\big(
    X
    ,\,
    T(Y)
  \big)
\end{tikzcd}

\begin{tikzcd}[column sep=60pt]
  &
  \mathbf{C}\big(
    X
    ,\,
    T(X)
  \big)
  \ar[
    dr,
    "{
      \mathrm{bind}_{X,X}
    }"{sloped}
  ]
  \\
  I
  \ar[
    ur,
    "{
      \mathrm{ret}_X
    }"{sloped}
  ]
  \ar[
    rr,
    "{
      \mathrm{id}_{T(X)}
    }"{swap}
  ]
  &&
  \mathbf{C}\big(
    T(X)
    ,\,
    T(Y)
  \big)
\end{tikzcd}

\begin{tikzcd}[sep=60pt]
  \mathbf{C}\big(
    Y
    ,\,
    T(Z)
  \big)
  \otimes
  \mathbf{C}\big(
    X
    ,\,
    T(Y)
  \big)
  \ar[
    rr,
    "{
      \mathrm{bind}_{Y,Z}
      \otimes
      \mathrm{Id}
    }"
  ]
  \ar[
    dd,
    "{
      \mathrm{bind}_{Y,Z}
      \,\otimes\,
      \mathrm{bind}_{X,Y}
    }"{description}
  ]
  &&
  \mathbf{C}\big(
    T(Y)
    ,\,
    T(Z)
  \big)
  \otimes
  \mathbf{C}\big(
    X
    ,\,
    T(Y)
  \big)
  \ar[
    d,
    "{
      \circ_{{}_{X, T(Y), T(Z)}}
    }"
  ]
  \\
  &&
  \mathbf{C}\big(
    X
    ,\,
    T(Z)
  \big)
  \ar[
    d,
    "{
      \mathrm{bind}_{X,Z}
    }"
  ]
  \\
  \mathbf{C}\big(
    T(Y)
    ,\,
    T(Z)
  \big)
  \otimes
  \mathbf{C}\big(
    T(X)
    ,\,
    T(Y)
  \big)
  \ar[
    rr,
    "{
      \circ_{{}_{T(X), T(Y), T(Z)}}
    }"{swap}
  ]
  &&
  \mathbf{C}\big(
    T(X)
    ,\,
    T(Z)
  \big)
\end{tikzcd}

\end{definition}

## Properties

### Relation to strong and monoidal monads
 {#RelationToStrongMonads}

\begin{proposition}
\label{EquivalenceBetweenStrengthAndEnrichment}
**(equivalence between $V$-strength and $V$-enrichment)**
For

* $V$ a [[closed monoidal category]]

* $\mathbf{C}$, $\mathbf{D}$ a [[pair]] of $V$-[[enriched categories]]

  which are also $V$-[[tensoring|tensored]] (=[[copower|copowered]])

then there is a [[bijection]] between the [[structures]] of

* $V$-[[strong functors]] and $V$-[[enriched functors]] $\mathbf{C} \to \mathbf{D}$

on the [[underlying]] [[functors]],

and for any pair $F$, $G$ of such a [[bijection]] between

* $V$-[[strong natural transformations]] and $V$-[[enriched natural transformation]] $F \to G$

It follows in particular that there is a [[bijection]] between

* $V$-[[strong monads]] and $V$-[[enriched monads]]

on such $\mathbf{C}$.
\end{proposition}
([Ratkovic 2012, §3.2](#Ratkovic12); [McDermott & Uustalu 2022, Prop. 5.4, 5.8](#McDermottUustalu22))

\begin{remark}
  For $V$ a [[closed monoidal category]] the further assumptions in Prop. \ref{EquivalenceBetweenStrengthAndEnrichment} apply to $\mathbf{C} = \mathbf{D} \coloneqq \mathbf{V}$ being the canonical self-enrichment of $V$.
\end{remark}

Moreover:

\begin{proposition}
\label{RelationBetweenCommutativeStrongMonadsAndSymmetricMonoidalMonads}
For $V$ a [[symmetric monoidal closed category]], there is a [[bijection]] between the [[structures]] of

* [[commutative monad|commutative]]$\;$[[strong monads]] and [[symmetric monoidal functor|symmetric]] [[monoidal monads]]

on [[underlying]] [[monads]] on $V$.

\end{proposition}
([Kock 1972, Thm. 3.2](monoidal+monad#Kock72), review in [GLLN08, §7.3, §A.4](monoidal+monad#GLLN08), [Ratkovic 2012, Prop. 3.3.9](#Ratkovic12))

Hence from combining Prop. \ref{EquivalenceBetweenStrengthAndEnrichment} with Prop. \ref{RelationBetweenCommutativeStrongMonadsAndSymmetricMonoidalMonads} we get:
\begin{prop}
For $V$ a [[symmetric monoidal category|symmetric]] [[closed monoidal category]] with $\mathbf{V}$ denoting its self-[[enriched category]], every [[symmetric monoidal functor|symmetric]] [[monoidal monad]] on $V$ gives the [[structure]] of a $V$-[[enriched monad]] on $\mathbf{V}$.
\end{prop}

## Examples

\begin{example}
In the case in of enrichment by [[truth values]], a monad is a [[closure operator]] on a [[poset]].
\end{example}

\begin{remark}
In the application of [[monads in computer science]], $\mathbf{C} = \mathbf{V}$ is typically the [[base of enrichment]] itself, canonically enriched over itself.
For classical computing this is typically a [[cartesian closed category]].

Since declaring a monad in a [[functional programming language]] means to define it in the [[internal language]] of $V$, such [[monads in computer science]] are actually enriched, see the discussion [there](monad+in+computer+science#RefinedIdea).

For example, if $V$ is the [[syntactic category]] of a [[programming language]], then all the definable monads in the language are $V$-enriched.
\end{remark}

## Related entries

* [[strong monad]]

* [[monoidal monad]]

* [[additive monad]]

* [[monad (in computer science)]]

* [[tensorial strength]]

* [[enriched adjunction]]

* [[enriched Lawvere theory]]

## References

* {#Ratkovic12} [[Kruna S. Ratkovic]], *Strength and Enrichment*, Section 3.2 of: *Morita theory in enriched context* (2012) [[arXiv:1302.2774](https://arxiv.org/abs/1302.2774), [hal:tel-00785301](https://theses.hal.science/tel-00785301/document)]

* {#McDermottUustalu22} [[Dylan McDermott]], [[Tarmo Uustalu]], Def. 5.6 in: *What Makes a Strong Monad?*, EPTCS **360** (2022) 113-133 [[arXiv:2207.00851](https://arxiv.org/abs/2207.00851), [doi:10.4204/EPTCS.360.6](https://doi.org/10.4204/EPTCS.360.6)]

See also:

* [[Max Kelly]], [[John Power]], *Adjunctions whose counits are coequalizers and presentations of finitary enriched monads*, Journal of Pure and Applied Algebra **89** (1993) [doi:10.1016/0022-4049(93)90092-8]

* [[John Power]], *Enriched Lawvere theories*, Theory and Applications of Categories **6** 7 (1999) 83-93 [[tac:6-07](http://www.tac.mta.ca/tac/volumes/6/n7/6-07abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/6/n7/n7.pdf)]

* [[Eduardo Dubuc]], *Kan Extensions in Enriched Category Theory*, Lecture Notes in Mathematics **145**, Springer (1970) [[doi:10.1007/BFb0060485](https://doi.org/10.1007/BFb0060485)]

[[!redirects enriched monads]]


enriched monoidal category

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## Idea

The notion of *enriched monoidal categories* is a compatible combination of the notions of *[[enriched categories]]* and *[[monoidal categories]]*. The main point is that the [[tensor product]]-[[functor]] on the [[underlying]] [[monoidal category]] is properly an [[enriched functor]] with respect to the [[underlying]] [[enriched category]].

Special cases include [[tensor categories]], which are ([[Vect]],[[tensor product of vector spaces|$\otimes$]])-enriched monoidal categories.

## Definition

### The monoidal 2-category of enriched categories

For $V$ a *[[braided monoidal category|symmetric monoidal]]* [[cosmos]],  the [[2-category]] [[VCat]] of $V$-[[enriched categories]] becomes a [[monoidal 2-category]] by declaring the [[tensor product]] $\mathbf{C} \otimes \mathbf{D}$ of a [[pair]] $\mathbf{C}$, $\mathbf{D}$ of $V$-[[enriched categories]] to have

* as [[set]] of [[objects]] the [[cartesian product]] of the given sets of objects:

  $$
    Obj(\mathbf{C} \otimes \mathbf{D})
      \;\coloneqq\;
    Obj(\mathbf{C}) \times Obj(\mathbf{D})
  $$

* as [[hom-objects]] the [[tensor product]] in $V$ between the given [[hom-objects]]:

  $$
    (\mathbf{C} \otimes \mathbf{D})\big((c,d), (c',d')\big)
    \;\coloneqq\;
    \mathbf{C}(c,c') \otimes \mathbf{D}(d,d')
    \,.
  $$

* [[composition]] obtained by the given composition operations, after using the [[braiding]] in $V$ to align factors:

  $$
    \array{
      (\mathbf{C} \otimes \mathbf{D})
        \big((c',d'), (c'', d'')\big)
      \otimes
      (\mathbf{C} \otimes \mathbf{D})
        \big((c,d), (c',d')\big)
      &\overset
         {\circ_{\mathbf{C} \otimes \mathbf{D}}}{
        \longrightarrow
        }
       &
      (\mathbf{C} \otimes \mathbf{D})
        \big((c,d), (c'',d'')\big)
      \\
      \mathllap{{}^\simeq}\Big\downarrow
      \phantom{---------}
      &&
      \Big\uparrow\mathrlap{{}^{\simeq}}
      \\
      \mathbf{C}(c',c'')
        \otimes
      \mathbf{D}(d',d'')
        \otimes
      \mathbf{C}(c,c')
        \otimes
      \mathbf{D}(d,d')
      \underoverset
        {braid}
        {\sim}
        {\to}
      \mathbf{C}(c',c'')
        \otimes
      \mathbf{C}(c,c')
        \otimes
      \mathbf{D}(d',d'')
        \otimes
      \mathbf{D}(d,d')
      &\overset{
        \circ_{\mathbf{C}} \otimes \circ_{\mathbf{D}}
      }{\longrightarrow}&
      \mathbf{C}(c,c'') \otimes \mathbf{D}(d,d'')
    }
  $$

### Enriched monoidal categories

A $V$-enriched monoidal category $\mathbf{C}$ is a [[pseudomonoid]] [[internalization|internal]] to the above monoidal 2-category $V Cat$.

This means mainly that

* $\mathbf{C}$ is a $V$-[[enriched category]]

* whose [[underlying]] category $C$ is equipped with [[monoidal category]]-[[structure]], hence with a [[tensor product]]-[[functor]]

  $$
    \otimes \,\colon\, C \times C \longrightarrow C
  $$

* which is compatibly lifted for each [[pair]] of [[pairs]] of [[objects]] of $C$ to a [[morphisms]] on [[hom-objects]] in $V$:

  $$
    \otimes_{\mathbf{C}}\big( (c,d),\, (c',c') \big)
    \,\colon\,
    \mathbf{C}(c,c') \otimes \mathbf{C}(d,d')
    \longrightarrow
    \mathbf{C}(c \otimes_C d ,\, c' \otimes_C d')
  $$

in a compatible way.

## Examples

(...)

## Related concepts

* [[cartesian closed enriched category]], [[locally cartesian closed enriched category]]

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

* [[enriched monoidal model category]], [[simplicial monoidal model category]]

## References

* [[Jacob Lurie]], Section 1.6 in: *Derived Algebraic Geometry II: Noncommutative Algebra* [[arXiv:math/0702299](https://arxiv.org/abs/math/0702299)]

  which is Def. 4.1.7.7 in *[[Higher Algebra]]* [[pdf](https://www.math.ias.edu/~lurie/papers/HA.pdf)]

  > (focus on [[sSet]]-enrichment for [[simplicial monoidal model categories]])

* {#BataninMarkl12} [[Michael Batanin]], [[Martin Markl]], Section 2 of: *Centers and homotopy centers in enriched monoidal categories*, Advances in Mathematics **230** 4–6 (2012) 1811-1858 [[doi:10.1016/j.aim.2012.04.011](https://doi.org/10.1016/j.aim.2012.04.011), [arXiv:1109.4084](https://arxiv.org/abs/1109.4084)]

* [[Michael Ching]], Def. 1.10 in: *Bar constructions for topological operads and the Goodwillie derivatives of the identity*, Geom. Topol. **9** (2005) 833-934 [[arXiv:math/0501429](https://arxiv.org/abs/math/0501429), [doi:10.2140/gt.2005.9.833](http://dx.doi.org/10.2140/gt.2005.9.833)]

* [[Rune Haugseng]], answer to *Definitions of enriched monoidal category*, [MO:a/315075](https://mathoverflow.net/a/315075/381) (2018)

* [[Scott Morrison]], [[David Penneys]], *Monoidal Categories Enriched in Braided Monoidal Categories*, International Mathematics Research Notices **2019** 11 June 2019 3527–3579 [[doi:10.1093/imrn/rnx217](https://doi.org/10.1093/imrn/rnx217), [arXiv:1701.00567](https://arxiv.org/abs/1701.00567)]

* [[Liang Kong]], Wei Yuan, Zhi-Hao Zhang, Hao Zheng, *Enriched monoidal categories I: centers* [[arXiv:2104.03121](https://arxiv.org/abs/2104.03121)]

[[!redirects enriched monoidal categories]]

[[!redirects monoidal enriched category]]
[[!redirects monoidal enriched categories]]


enriched monoidal model category

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## Idea

For $V$ a [[monoidal model category]], a *$V$-enriched monoidal model category* is a [[model category]] equipped with

1. the [[structure]] of a [[monoidal model category]],

1. the [[structure]] of a $V$-[[enriched model category]],

1. the compatibility structure that makes the [[underlying]] [[monoidal category|monoidal]] and [[enriched category|$V$-enriched]] structure a

   $V$-[[enriched monoidal category]].

## Examples

* For $V =$ [[classical model structure on simplicial sets|$Set_{Qu}$]] one typically speaks of *[[simplicial monoidal model categories]]*.

[[!redirects enriched monoidal model categories]]

[[!redirects monoidal enriched model category]]
[[!redirects monoidal enriched model categories]]


enriched monotone

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# Contents
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{: toc}

## Idea

The [[enriched category theory|enriched]] generalization of the notion of *[[monotone function]]*.

## Definition

Given two preorders $A, B$ [[enriched preorder|enriched in]] a [[monoidal poset]] $M$, an **$M$-enriched monotone** is a function $f: A \to B$ such that
$$o_A(x, y) \leq o_B(f(x), f(y))$$
for all $x \in A$ and $y \in A$.

## Examples

* Let $M$ be $\Omega$, the set of [[truth values]], so that $A$ and $B$ are [[preorders]]. The $\Omega$-enriched monotones between $A$ and $B$ are monotones between $A$ and $B$.

* Let $M$ be $(\mathbb{R}_{\geq 0}, \geq, +, 0)$, the [[non-negative number|non-negative]] [[Dedekind real numbers]] with the preorder relation being greater than $\geq$, the monoidal operation being addition $+$, and the monoidal unit being zero $0$, so that $A$ and $B$ are [[quasipseudometric spaces]]. The $\mathbb{R}_{\geq 0}$-enriched monotones between $A$ and $B$ are distance-decreasing maps between $A$ and $B$.

## See also

* [[enriched preorder]]

* [[enriched monotone preorder]]

* [[quasipseudometric space]]

[[!redirects enriched monotones]]


enriched multicategory

* [[Tom Leinster]], _Generalized enrichment for categories and multicategories_, [math.QA/9901139](http://arxiv.org/abs/math/9901139)

For an interesting application see [[relaxed multicategory]].

[[!redirects enriched multicategories]]


enriched natural transformation

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## Idea

The notion of _$V$-enriched natural transformation_ is the appropriate notion of [[homomorphism]] between [[enriched functors]], the analog in [[enriched category theory]] of the ordinary notion of [[natural transformation]] in ordinary [[category theory]].

$V$-Enriched natural transformations constitute the [[2-morphisms]] in the [[2-category]] [[VCat]].

## Definition
 {#Definition}

For $V$ a [[cosmos for enrichment]], let

* $\mathbf{C}$, $\mathbf{D}$ be a [[pair]] of $V$-[[enriched categories]]

  (we denote their [[hom-objects]] by $C(-, -)$ etc., instead of $\hom_C(-, -)$ or similar),

* $F, G \,\colon\, \mathbf{C} \longrightarrow \mathbf{D}$ a [[pair]] of $V$-[[enriched functors]] between them.

Then:

\begin{definition}
\label{EnrichedNaturalTransformation}
An **enriched natural transformation** between these enriched functors

$$
  \eta \colon F \longrightarrow G
$$

is a [[family]] of [[morphisms]] of $V$

$$
  \eta_c
  \;\colon\;
  I
    \longrightarrow
  D(F c, G c)
$$

(out of the [[tensor unit]] $I$ of $V$) [[indexed set|indexed]] by  $c\in Ob(C)$)

such that for any [[pair]] of objects $c, c' \,\in\, Obj(\mathbf{C})$ the following [[commuting diagram|diagram commutes]]:

\[\label{ExtranaturalitySquare}\]
\begin{tikzcd}
  &
  \mathbf{C}(c,c')
   \otimes
  I
  \ar[
    r,
    "{
       G_{c,c'}
        \otimes
       \eta_c
    }"
  ]
  &
  \mathbf{D}\big(G(c),\, G(c')\big)
  \otimes
  \mathbf{D}\big(F(c),\, G(c)\big)
  \ar[
    dr,
    "{ \circ^{\mathbf{D}} }"
  ]
  &[-40pt]
  \\
  \mathbf{C}(c, c')
  \ar[
    ur,
    "{ r^{-1} }"
  ]
  \ar[
    dr,
    "{ l^{-1} }"{swap}
  ]
  & & &
  \mathbf{D}\big( F(c) ,\,G(c') \big)
  \\
  &
  I
    \otimes
  \mathbf{C}(c, c')
  \ar[
    r,
    "{
       \eta_{c'}
         \otimes
       F_{c,c'}
    }"{swap}
  ]
  &
  \mathbf{D}\big(F(c'),\, G(c') \big)
  \otimes
  \mathbf{D}\big(F(c),\, F(c')\big)
  \ar[
    ur,
    "{
      \circ^{\mathbf{D}}
    }"{swap}
  ]
\end{tikzcd}

\end{definition}
Here

* $\otimes$ denotes the [[tensor product]] in $V$,

* $I$ denotes the [[tensor unit]] in $V$,

* $l$,$r$ denote the left and right [[unitors]] of $V$,

* $F_{c,c'}$, $G_{c,c'}$ denote the component [[maps]] of the [[enriched functors]] between [[hom-objects]],

* $\circ^{\mathbf{D}}$ denotes the [[composition]]-operation of $\mathbf{D}$.

The above diagram expresses the $V$-enriched version of [[commuting diagram|commutativity]] of the plain [[naturality square]]:

$$
  \array{
    c
    &&
    F(c) &\overset{ \eta_c }{\longrightarrow}& G(c)
    \\
    \mathllap{{}^{f}}
    \Big\downarrow
    &\mapsto\;\;\;&
    \mathllap{{}^{F(f)}}
    \Big\downarrow
    &&
    \Big\downarrow\mathrlap{{}^{G(f)}}
    \\
    c'
    &&
    F(c') &\underset{\eta_{c'}}{\longrightarrow}& G(c')
  }
$$

## Properties

### As 2-morphisms

In generalization of how for plain [[natural transformations]] there is a notion of [[horizontal composition]] ("[[whiskering]]") and [[vertical composition]], so for enriched natural transformations:

\begin{proposition}\label{RightWhiskering}
**(horizontal composition)**
\linebreak
Given

\begin{tikzcd}
  \mathbf{C}
  \ar[
    rr,
    bend left=30,
    "{ F }",
    "{\ }"{name=s, swap}
  ]
  \ar[
    rr,
    bend right=30,
    "{ G }"{swap},
    "{\ }"{name=t}
  ]
  \ar[
     from=s,
     to=t,
     Rightarrow,
     shorten=10pt,
     "{ \eta }"
  ]
  &&
  \mathbf{D}
  \ar[
    rr,
    "{ H }"
  ]
  &&
  \mathbf{E}
\end{tikzcd}

where

1. $\eta$ is an enriched natural transformation (Def. \ref{EnrichedNaturalTransformation})

1. $H$ is an [[enriched functor]]

then we obtain an enriched natural transformation of the form

\begin{tikzcd}
  \mathbf{C}
  \ar[
    rr,
    bend left=30,
    "{ H \circ F }",
    "{\ }"{name=s, swap}
  ]
  \ar[
    rr,
    bend right=30,
    "{ H \circ G }"{swap},
    "{\ }"{name=t}
  ]
  \ar[
     from=s,
     to=t,
     Rightarrow,
     shorten=10pt,
     "{ H(\eta) }"
  ]
  &&
  \mathbf{E}
\end{tikzcd}

with component maps given as the [[composition]]

$$
  I
  \overset{\eta_c}{\longrightarrow}
  \mathbf{D}\big(F(c),\,G(c)\big)
  \overset{H_{F(c), G(c)}}{\longrightarrow}
  \mathbf{E}\big(H \circ F(c),\,H \circ G(c)\big)
  \,.
$$
\end{proposition}

### Relation to strong natural transformations

For [[closed monoidal categories]] $V$ there is a close relation between $V$-enriched natural transformations and $V$-[[strong natural transformations]].

For the moment see at *[enriched monad -- relation to strong monads](enriched+monad#RelationToStrongMonads)* for more.

## Examples

\begin{example}
  With [[Set]] denoting the category of [[sets]] and [[functions]] equipped with its [[cartesian monoidal category|cartesian monoidal]]-[[structure]] (via [[Cartesian product]] of sets), [[Set]]-enriched natural transformations are just plain [[natural transformations]] between [[functors]] between [[locally small categories]].
\end{example}

\begin{example}
\label{StrictTwoNaturalTransformations}
 With *[[Cat]]* denoting the [[1-category]] of [[small category|small]] [[strict categories]] equipped with its [[cartesian monoidal category|cartesian monoidal]] [[structure]] (via forming [[product categories]]), $Cat$-enriched natural transformations are also known as *[[strict 2-natural transformations]]*.
\end{example}

## Related concepts

* [[strong natural transformation]]

* [[enriched functor]]

* [[enriched functor category]]

* [[enriched adjoint functor]]

## References

* {#EilenbergKelly65} [[Samuel Eilenberg]], [[G. Max Kelly]],  §I.10 of: *Closed Categories*, in: *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) 421-562 [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

* [[Max Kelly]], p. 9 of: _Basic Concepts of Enriched Category Theory_, Cambridge University Press, Lecture Notes in Mathematics **64** (1982), Reprints in Theory and Applications of Categories, **10** (2005) 1-136 [[tac:tr10](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf)]

* {#Borceux94} [[Francis Borceux]], def. 6.2.4 of: _[[Handbook of Categorical Algebra]] Vol 2_, Cambridge University Press (1994)

* [[Emily Riehl]], §3.5 in *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]

For more references see at _[[enriched category]]_.

[[!redirects enriched natural transformation]]
[[!redirects enriched natural transformations]]

[[!redirects enriched natural isomorphism]]
[[!redirects enriched natural isomorphisms]]


enriched preorder

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## Idea ##

Recall ([here](partial+order#AsACategoryWithExtraProperties)) that a [[preorder]] may be understood as a [[thin category|thin]] [[(0,1)-category]], hence as a thin [[enriched category|category enriched]] over the [[cartesian monoidal category|cartesian monoidal]] [[proset]] of [[truth values]].
In generalization, one may speak of enriching preorders over other [[monoidal posets]].

## Defintion ##

Let $(M, \leq, \wedge, \top)$ be a [[monoidal poset]]. A **$M$-enriched [[proset]]** or **proset enriched over/in $M$** is a [[set]] $P$ with a [[binary function]] $o:P \times P \to M$ such that

* for every $a \in P$, $b \in P$, and $c \in P$, $o(a, b) \wedge o(b, c) \leq o(a, c)$

* for every $a \in P$, $\top \leq o(a, a)$.

## Examples ##

* An ordinary [[poset]] is just an $\Omega$-enriched poset, with $\Omega$ the set of [[truth values]].

* A [[Lawvere metric space]] is a $\overline{\mathbb{R}_{\geq 0}}$-enriched [[proset]], where $\overline{\mathbb{R}_{\geq 0}}$ are the [[non-negative number|non-negative]] [[extended real number|extended]] [[Dedekind real numbers]].

* A [[quasipseudometric space]] is a $\mathbb{R}_{\geq 0}$-enriched proset, where $\mathbb{R}_{\geq 0}$ are the [[non-negative number|non-negative]] [[Dedekind real numbers]].

* A [[quasimetric space]] is a $\mathbb{R}_{\geq 0}$-enriched [[poset]], where $\mathbb{R}_{\geq 0}$ are the [[non-negative number|non-negative]] [[Dedekind real numbers]].

* A [[pseudometric space]] is a $\mathbb{R}_{\geq 0}$-enriched [[symmetric proset]], where $\mathbb{R}_{\geq 0}$ are the [[non-negative number|non-negative]] [[Dedekind real numbers]].

* A [[metric space]] is a $\mathbb{R}_{\geq 0}$-[[enriched set]], where $\mathbb{R}_{\geq 0}$ are the [[non-negative number|non-negative]] [[Dedekind real numbers]].

* A set with an [[irreflexive]] [[comparison]], such as an [[apartness relation]] or a [[linear order]], is an $\Omega^\op$-enriched poset, where the [[co-Heyting algebra]] $\Omega^\op$ is the [[opposite poset]] of the set of [[truth values]] $\Omega$.

## See also ##

* [[Lawvere metric space]]
* [[enriched category]] (for the (1, 1)-category version)
* [[enriched set]]
* [[enriched directed loop graph]]
* [[enriched monotone]]
* [[enriched monotone preorder]]

[[!redirects enriched preorders]]

[[!redirects enriched proset]]
[[!redirects enriched prosets]]

[[!redirects enriched poset]]
[[!redirects enriched posets]]


enriched Quillen adjunction

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#Contemts#
* table of contents
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## Definition

In [[enriched model category]] theory, an __enriched Quillen adjunction__ is an [[enriched adjunction]] whose underlying ordinary adjunction is a [[Quillen adjunction]] between ordinary [[model categories]].

Here "underlying" refers to the underlying ordinary category $C_0$ of any $V$-enriched category, defined by $C_0(x,y) = V(I,C(x,y))$.  (Recall that an enriched model category is an enriched category, together with a model structure on its underlying ordinary category, and some compatibility conditions.)

## Special cases

A special role is played by [[sSet]]-enriched Quillen adjunctions, for the standard [[model structure on simplicial sets]]. See [[simplicial Quillen adjunction]] for more on that.

[[!redirects enriched Quillen adjunctions]]


enriched Reedy category

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#Contents#
* table of contents
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## Idea

The notion of _enriched Reedy category_ is a combination of that of [[Reedy category]] and [[enriched category]].

The main motivation for studying [[Reedy category|Reedy categories]] is that they induce [[Reedy model structure]]s on [[functor category|functor categories]].

The motivation for studying enriched Reedy categories is that they induced [[Reedy model structure|enriched Reedy model structures]] on [[enriched functor category|enriched functor categories]].

## Definition

(...)

## Properties

Let $\mathcal{V}$ be a [[monoidal model category]]. Let $\mathcal{A}$ be a $\mathcal{V}$-enriched Reedy category and let $\mathcal{E}$ be a $\mathcal{V}$-[[enriched model category]]. Write $[\mathcal{A}, \mathcal{C}]$ for the [[enriched functor category]].

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

The enriched Reedy model structure on $[\mathcal{A}, \mathcal{C}]$ exists and is a $\mathcal{V}$-[[enriched model category]].

=--

([Angeltveit, theorem 4.7](#Angeltveit)).

## Related concepts

* [[Reedy model structure]], [[generalized Reedy model structure]]

## References

Enriched Reedy categories were introduced in

* Vigleik Angeltveit, _Enriched Reedy categories_ ([arXiv:0612137](http://arxiv.org/abs/math/0612137))
 {#Angeltveit}

The defintion is def. 4.1 there.

[[!redirects enriched Reedy categories]]


enriched set

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# Contents
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## Idea ##

Recall ([here](partial+order#AsACategoryWithExtraProperties)) that a [[set]] may be understood as a [[thin category|thin]] [[univalent dagger category|univalent]] (0,1)-dagger category, hence as a thin univalent [[enriched category|dagger-category enriched]] over the [[cartesian monoidal category|cartesian monoidal]] [[proset]] of [[truth values]].
In generalization, one may speak of enriching preorders over other [[monoidal posets]].

## Defintion ##

Let $(M, \leq, \otimes, 1)$ be a [[monoidal poset]]. A **$M$-enriched [[symmetric proset]]** or **symmetric proset enriched over/in $M$** is a [[set]] $P$ with a [[binary function]] $o:P \times P \to M$ such that

* for every $a \in P$, $b \in P$, and $c \in P$, $o(a, b) \otimes o(b, c) \leq o(a, c)$

* for every $a \in P$ and $b \in P$, $o(a, b) \leq o(b, a)$

* for every $a \in P$, $1 \leq o(a, a)$

An **$M$-enriched [[set]]** or **set enriched over/in $M$** is an $M$-enriched symmetric proset which additinally satisfies $1 \leq o(a, b) \Rightarrow a = b$.

For $M$-enriched sets, $o(a, b) = o(b, a)$, $o(a, a) = 1$, and $1 = o(a, b) \Rightarrow a = b$.

## Examples ##

* An ordinary [[set]] is just an $\Omega$-enriched set, with $\Omega$ the set of [[truth values]].

* A [[metric space]] is a $(\mathbb{R}_{\geq 0}, \geq, +, 0)$-enriched set, where $\mathbb{R}_{\geq 0}$ are the [[non-negative number|non-negative]] [[Dedekind real numbers]].

* Given any [[Archimedean integral domain]] $A$ and an Archimedean integral subdomain $B \subseteq A$, then $B$ is an $(A_{\geq 0}, \geq, +, 0)$-enriched set, where $A_{\geq 0}$ is the non-negative elements of $A$.

* A set with an [[apartness relation]] is an $\Omega^\op$-enriched symmetric proset, where the [[co-Heyting algebra]] $\Omega^\op$ is the [[opposite poset]] of the set of [[truth values]] $\Omega$.

## See also ##

* [[pseudometric space]]
* [[Archimedean integral domain]]
* [[enriched proset]]

[[!redirects enriched set]]
[[!redirects enriched sets]]
[[!redirects enriched symmetric proset]]
[[!redirects enriched symmetric prosets]]


enriched sheaf

Enriched sheaf theory has been introduced in

* F. Borceux, C. Quinteiro, _A theory of enriched sheaves_, Cahiers Topologie Géom. Différentielle Catég. 37 (1996), no. 2, 145--162 [MR1394507](https://mathscinet.ams.org/mathscinet-getitem?mr=1394507)

They consider a locally finitely presentable symmetrical monoidal closed category $\mathcal{V}$ and a small $\mathcal{V}$-enriched category $\mathcal{C}$. The category $[\mathcal{C}^\mathrm{op},\mathcal{V}]$ of $\mathcal{V}$-valued $\mathcal{V}$-enriched functors on the dual of $\mathcal{C}$ is considered as a category of enriched presheaves.
Axioms for $\mathcal{V}$-enriched Grothendieck topologies are introduced in terms of $\mathcal{V}$-subfunctors of representable functors (one could say $\mathcal{V}$-sieves). The main result of the article is a bijection between reflective $\mathcal{V}$-[[enriched localization]]s of $[\mathcal{C}^\mathrm{op},\mathcal{V}]$ preserving finite limits and $\mathcal{V}$-enriched Grothendieck topologies on $\mathcal{C}$ and also a bijection with universal $\mathcal{V}$-closure operations.

This is a generalization of a [[Gabriel-Popescu theorem]] and of a characterization of Grothendieck topoi as left exact reflective localizations of presheaf categories.

[[Gabriel filter]]s (Gabriel topologies) are the case of $\mathcal{V}$-enriched Grothendieck topologies when the enrichment is over the category of abelian groups.

See also [[zoranskoda:MR4328537]].

[[!redirects enriched sheaves]]


enriched slice category

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# Enriched slice categories
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## Idea

The notion of enriched slice categories is the generalization of the notion of [[slice categories]] from plain [[category theory]] of [[locally small categories]] to [[enriched category theory]] of $\mathcal{V}$-[[enriched categories]].

## Definition

\begin{definition}
Let

* $(V,\otimes,I)$ be a [[monoidal category]] with [[pullbacks]] (e.g. a [[Bénabou cosmos]]),

* $C$ a $V$-[[enriched category]],

* $x\in C$ an [[object]].

Then the *enriched slice category* (or *enriched over-category*) $C/x$ is the following $V$-enriched category:

* its [[objects]] are the [[morphisms]] $f \,\colon\, y\to x$ in (the [[underlying ordinary category]] of) $C$, i.e. morphisms $I \to C(y,x)$ in $V$.

* its [[hom-object]] from $f \,\colon\, y\to x$ to $g \,\colon\, z\to x$ is the [[pullback]] in $V$:

\begin{center}
\begin{tikzcd}
  (C/x)(f,g)
   \ar[r]
   \ar[d]
   \ar[dr,phantom,near start,"\lrcorner"]
   &
   C(y,z)
     \ar[d,"{g\circ -}"]
   \\
   I
     \ar[r,"\ulcorner f \urcorner"']
   &
   C(y,x)
\end{tikzcd}
\end{center}
\end{definition}

\begin{remark}
The [[formal duality|dual]] concept is the
*enriched co-slice category* or *enriched under-category*, the enriched generalization of the notion of *[[co-slice category]]*.
\end{remark}

## Properties

* $C/x$ can be described as the [[comma object]] of $Id \,\colon\,C\to C$ and $x \,\colon\, I \to C$ in the [[2-category]] $V Cat$, where $I$ denotes the [[unit enriched category]].

* All [[hom-objects]] of $C/x$ are [[augmentation|augmented]], i.e. equipped with a [[morphism]] to the [[unit object]].  This seems somewhat curious unless $V$ is [[cartesian monoidal category|cartesian monoidal]] (or at least [[semicartesian monoidal category|semicartesian]]) in which case $I$ is the [[terminal object]] $1$.

* Any morphism $f \,\colon\, x\to y$ in $C$ induces a $V$-[[enriched functor]] $f_! \,\colon\, C/x \to C/y$.  If $C$ has $V$-enriched pullbacks, then $f_!$ has a [[right adjoint|right]] $V$-[[enriched adjoint]] $f^*$.  If $C$ is a [[locally cartesian closed enriched category]], then $f^*$ has a further right $V$-adjoint $f_*$.

## Related concepts

* [enriched slice model structure](slice+model+structure#ModelStructureInheritsEnriched)

* [[slice of presheaves is presheaves on slice]]

## References

* MathOverflow, *Enriched slice categories* ([MO:q/259829](https://mathoverflow.net/q/259829))

[[!redirects enriched slice categories]]
[[!redirects enriched coslice category]]
[[!redirects enriched coslice categories]]
[[!redirects enriched co-slice category]]
[[!redirects enriched co-slice categories]]

[[!redirects enriched over category]]
[[!redirects enriched over categories]]
[[!redirects enriched over-category]]
[[!redirects enriched over-categories]]

[[!redirects enriched under category]]
[[!redirects enriched under categories]]
[[!redirects enriched under-category]]
[[!redirects enriched under-categories]]

[[!redirects slice enriched category]]
[[!redirects slice enriched categories]]


enriched type

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# Contents
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## Idea

In [[homotopy type theory]], [[types]] represent [[infinity-groupoids|$\infty$-groupoids]], so there should be something similar to [[enriched (infinity,1)-category|enriched]] $\infty$-groupoids in homotopy type theory. The following definition is an experimental definition of such an object:

## Definition

Let $C$ be the [[subtype]] of [[types]] in a [[type universe]] $\mathcal{U}$ which satisfy a certain [[predicate]] $P$: for all types $T:C$, there is a [[term]] $p:P(T)$.

Then a type $T$ is *$C$-enriched* if for all terms $a:T$ and $b:T$, there is a [[dependent type|dependent]] term $p(a,b):P(a = b)$, where $a = b$ is the [[identity type]] of $a$ and $b$ in $T$.

## Examples

* One could define [[Ab|$Ab$]]-enriched types, which are types whose identity types are all [[abelian groups]] (usually defined with a predicate $isAbelianGroup$).

* Similarly, [[sets]] are [[Prop|$Prop$]]-enriched types, and [[homotopy n-type|$n$-types]] are $(n - 1)$Type-enriched types.

* A (extended) lower Dedekind cut could be defined as a set $U:\mathcal{U}$ with an injection $i:U \hookrightarrow \mathbb{Q}$ which satisfy certain axioms, which are detailed in the article on [[Dedekind cuts]]. Then, one could define a (extended) [[Richman premetric space]] as a type whose identity types are (extended) lower Dedekind cuts, which in classical mathematics is the same as a (extended) [[metric space]].

## Categorical semantics

Under the [[relation between type theory and category theory]], enriched types should correspond to [[enriched (infinity,1)-category|enriched]] [[infinity-groupoids|$\infty$-groupoids]] in [[higher category theory]].

## See also

* [[enriched category]]
* [[enriched (infinity,1)-category]]

[[!redirects enriched types]]


enriched Yoneda lemma

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#Contents#
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## Idea

The _enriched Yoneda lemma_ is the generalization of the usual [[Yoneda lemma]] from [[category theory]] to [[enriched category theory]].

## Statement

We discuss here two forms of the Yoneda lemma.

Let $V$ be a ([[locally small category|locally small]]) [[closed monoidal category|closed]] [[symmetric monoidal category]], so that $V$ is enriched in itself via its [[internal hom]].

### Weak form

 A _weak form_ of the enriched Yoneda lemma says that given a $V$-[[enriched functor]] $F: C \to V$ and an object $c$ of $C$, the _[[set]]_ of $V$-[[enriched natural transformation]]s $\alpha: \hom_C(c, -) \to F$ is in [[natural isomorphism|natural bijection]] with the set of elements of $F(c)$, i.e., the set of morphisms $I \to F(c)$, obtained by composition:

$$I \stackrel{1_c}{\to} \hom_C(c, c) \stackrel{\alpha c}{\to} F(c)$$

### Strong form

Now suppose that $V$ is in addition (small-)complete (has all small [[limit]]s). Then, given a [[small category|small]] $V$-[[enriched category]] $C$ and $V$-[[enriched functor]]s $F, G: C \to V$, one may construct the _object_ of $V$-natural transformations as an enriched [[end]]:

$$V^C(F, G) = \int_c V(F(c), G(c))$$

(which may in turn be expressed as an ordinary limit in $V$). This is the [[hom-object]] in the [[enriched functor category]].

A _strong form_ of the enriched Yoneda lemma specifies a $V$-[[natural isomorphism]]

$$V^C(\hom_C(d, -), F) = \int_c V(\hom_C(d,c),F(c)) \cong F(d).$$

This implies the weak form by applying the functor $\hom(I, -): V \to Set$.

### Density form

If $V$ is also (small-)cocomplete, there is an equivalent dual formulation of the enriched Yondeda lemma as a $V$-[[natural isomorphism]]
$$\int^c \hom_C(c,d) \otimes F(c) \cong F(d)$$
induced by the morphisms
$$\hom_C(c,d) \otimes F(c) \to F(d)$$
adjoint to $F_{c,d}$.
This version is sometimes called the **density theorem** and sometimes simply called the enriched Yoneda lemma.

### Dual forms

For an [[enriched presheaf]] $H: C^{op} \to V$ there are dual forms obtained by replacing $C$ above with $C^{op}$.
These are
$$ \int_c V(\hom_C(c,d),H(c)) \cong H(d) $$
and
$$ \int^c \hom_C(d,c) \otimes H(c) \cong H(d). $$

## References

* {#Day70} [[Brian Day]], _On closed categories of functors_, Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics Vol. 137. Springer-Verlag, 1970, pp 1-38 ([pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/DayReport.pdf))

Textbook accounts include

* [[Max Kelly]], sect. 1.9 (weak form) and 2.4 (strong from) of _Basic Concepts of Enriched Category Theory_, Cambridge University Press, Lecture Notes in Mathematics 64 (1982) [(pdf)](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html)

* {#Borceux94} [[Francis Borceux]], theorem 6.3.5 of _[[Handbook of Categorical Algebra]]_, Volume 2, Cambridge University Press (1994)

* [[Niles Johnson]], [[Donald Yau]], _Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic K-Theory_ ([three volume book](https://nilesjohnson.net/En-monoidal.html), Section III.3.7.

* [[Fosco Loregian]], _Coend calculus_, Cambridge University Press 2021 ([arXiv:1501.02503](http://arxiv.org/abs/1501.02503), [doi:10.1017/9781108778657]( https://doi.org/10.1017/9781108778657), ISBN:9781108778657), discusses the Set-enriched versions of the Yoneda lemma in Section 2.2, where the density forms are called  the **ninja Yoneda lemma**.

Generalizations to the case that the enriching monoidal category is not [[closed monoidal category|closed]] or [[symmetric monoidal category|symmetric]] (using [[skew-symmetric category|skew-symmetric categories]] or [[tensored category|tensored categories]]) can be found in

* {#Street12} [[Ross Street]], _Skew-closed categories_ ([arXiv:1205.6522](https://arxiv.org/abs/1205.6522))

* {#Hinich16} [[Vladimir Hinich]], *Enriched Yoneda lemma*, Theory and Applications of Categories **31** 29 (2016) 833-838 [[tac:31-29](http://www.tac.mta.ca/tac/volumes/31/29/31-29abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/31/29/31-29.pdf)]


enrichment versus internalisation

* table of contents
{:toc}

## Idea

[[enriched category|Enriched categories]] and [[internal categories]] are both generalisations of categories to category-like structures that are defined in terms of another category. An enriched category is (usually) defined in terms of a [[monoidal category]], while an internal category is (usually) defined in terms of a category with [[finite limits]]. The two notions are not unrelated, but they are different. One difference is that in a $V$-enriched category, the objects still form a _set_ (or a proper class) while the arrows are replaced by objects of $V$, while in a category internal to $E$, both the set of objects _and_ the set of arrows are replaced by objects of $E$.

One way that internalisation and enrichment are related is that internal categories and enriched categories are both instances of [[monad|monads]] in bicategories (the bicategory of spans and the bicategory of matrices, respectively). Since monads in a bicategory $B$ are one-object categories enriched in the bicategory $B$, it is thus possible to capture $E$-internal categories as one-object categories enriched in the [[bicategory of spans]] $Span(E)$.

It is useful to contrast the notion of [[internal category]] with that of [[enriched category]], each of which may be considered as a generalization according to whether categories are [defined](category#definitions) (i) by a single collection of all morphisms or (ii) by a family of collections indexed by pairs of objects. In some cases, one of these generalizations is a special case of the other, but in general they are incomparable. These two presentations may be seen as different presentations of the theory of categories as a [generalized algebraic theory](generalized+algebraic+theory#relationship_to_enriched_categories).

[[internalization|Internalisation]] is a quite general phenomenon, of which [[internal categories]] are a particular case. However, the distinction between "internalization" and "enrichment" becomes less clear in generality.

For example, in addition to categories enriched over a monoidal category, one can define categories enriched over a [[bicategory]] or an [[virtual double category]]. It then turns out that a category enriched over the bicategory (or virtual double category) of [[span|spans]] in a lex category $C$ which has one object is precisely an internal category in $C$.

## Comparing internalisation and enrichment in the same category

While enriched categories and internal categories are defined with respect to different kinds of category, making it impossible to compare them in general, we can compare the two notions when our category $V$ or $E$ has enough structure. E.g. $K$ is an $\infty$-[[extensive category]], such as [[Set]] or [[simplicial set|simplicial sets]] (or more generally any [[Grothendieck topos]]), (small) $K$-enriched categories can be identified with $K$-internal categories whose object-of-objects is discrete (that is, a coproduct of copies of the terminal object).

We may also internalise a category in a [[internal category in a monoidal category|monoidal category]]. One might expect that the same relationship holds in this generalised setting, though it does not appear this has been worked out in the literature.

## Related concepts

* [[enriched category]] and [[internal category]]

* [[model structure on enriched categories]] and [[model structure on internal categories]]

## References

The equivalence between categories enriched in $K$ and categories internal to $K$ with a discrete object of objects is due to the appendix of:

* [[Andrée Bastiani]], [[Charles Ehresmann]], _Multiple functors. III. The cartesian closed category $Cat_n$_, Cahiers de topologie et géométrie différentielle 19.4 (1978): 387-443.

This relationship has also been studied in:

* [[Michał R. Przybyłek]], *Kategorie wewnętrzne a kategorie wzbogacone* (2008), master's thesis ([pdf](https://www.mimuw.edu.pl/~mrp/mgr_mrp.pdf))

* [[Thomas Cottrell]], [[Soichiro Fujii]], and [[John Power]], _Enriched and internal categories: an extensive relationship (2017).
* [[Bojana Femić]] and [[Enrico Ghiorzi]], _Internalization and enrichment via spans and matrices in a tricategory_, _Journal of Algebraic Combinatorics_ (2023): 1-54.

[[!redirects internalisation versus enrichment]]
[[!redirects internalization versus enrichment]]
[[!redirects internalisation vs enrichment]]
[[!redirects internalization vs enrichment]]
[[!redirects enrichment versus internalization]]
[[!redirects enrichment vs internalisation]]
[[!redirects enrichment vs internalization]]


Enrico Betti

* [Wikipedia entry](https://en.wikipedia.org/wiki/Enrico_Betti)

## Related $n$Lab entries

* [[Betti number]]

category: people


Enrico Bombieri

* [Website](https://www.ias.edu/math/people/faculty/eb)

# Selected writings

On the [[Mordell conjecture]]:

* {#Bombieri90} [[Enrico Bombieri]], _The Mordell conjecture revisited_, _Annali della Scuola Normale Superiore di Pisa - Classe di Scienze_,  Sér. 4, 17 no. 4 (1990), p. 615-640 ([Numdam](http://www.numdam.org/item?id=ASNSP_1990_4_17_4_615_0))

category: people


Enrico Fermi

* [Wikipedia entry](http://en.wikipedia.org/wiki/Enrico_Fermi)

## Related entries

* [[fermion]]

* [[neutrino]]

category: people

[[!redirects Fermi]]


Enrico Ghiorzi

* [Mathematics Genealogy page](https://genealogy.math.ndsu.nodak.edu/id.php?id=260270)

## Selected writings

On [[internal categories]]:

* [[Enrico Ghiorzi]], *Complete internal categories* ([arXiv:2004.08741](https://arxiv.org/abs/2004.08741))

category: people


Enrico Vitale

* [website](http://perso.uclouvain.be/enrico.vitale/)

## Selected writings

On the characterization of [[monadic functors]] over [[Set]]:

* {#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)]

On the [[category]] of ([[Hausdorff space|Hausdorff]]) [[compactly generated topological spaces]] being a [[regular category]]:

* {#CagliariMatovaniVitale95} F. Cagliari, S. Mantovani, [[Enrico Vitale]], *Regularity of the category of Kelley spaces*, Applied Categorical Structures volume 3, pages 357–361 (1995) ([doi:10.1007/BF00872904](https://link.springer.com/article/10.1007/BF00872904), [pdf](http://www.dm.unibo.it/~cagliari/articoli/Regularkelley.pdf))

On [[regular and exact completions]] (and on [[free coproduct completions]] etc.):

* [[Aurelio Carboni]], [[Enrico Vitale]], *Regular and exact completions*, Journal of Pure and Applied Algebra **125** 1–3  (1998) 79-116 (doi:10.1016/S0022-4049(96)00115-6)

On the [[exact completion]] of the [[classical homotopy category]] $Ho(Top)$ being a [[pretopos]]:

* [[Marino Gran]], [[Enrico Vitale]], *On the exact completion of the homotopy category*, Cahiers de Topologie et Géométrie Différentielle Catégoriques **39** 4 (1998)  287-297 [[numdam:CTGDC_1998__39_4_287_0](http://www.numdam.org/item/CTGDC_1998__39_4_287_0)]

For the characterisation of [[categories]] which are [[localizations]] of [[Maltsev varieties]]:

* [[Marino Gran]] and [[Enrico Vitale]], _Localizations of Maltsev varieties_, Theory and Applications of Categories **5** 12 (1999) 281-291 [[TAC](http://www.tac.mta.ca/tac/volumes/1999/n12/5-12abs.html)]

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]

On [[algebraic theories]]:

* [[Jiří Adámek]], [[Jiří Rosický]], [[Enrico Vitale]], *Algebraic theories*, Cambridge University Press (2011) [[doi:10.1017/CBO9780511760754](https://doi.org/10.1017/CBO9780511760754), [pdf](https://perso.uclouvain.be/enrico.vitale/gab_CUP2.pdf)]

category: people

[[!redirects Enrico M. Vitale]]
[[!redirects E. M. Vitale]]


entanglement

+-- {: .rightHandSide}
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###Context###
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

While in [[classical mechanics]] a ([[pure state|pure]]) [[state]] is an [[generalized element|element]] of an [[object]] in a [[cartesian monoidal category]], in contrast in [[quantum mechanics]] a pure state is an element of an object in a non-cartesian [[monoidal category]] (say of [[Hilbert spaces]]). As a result, in quantum mechanics a state of a compound [[physical system]] may not come from a pair of states of the two [[subsystems]], but instead be a nontrivial [[sum]] -- a _[[superposition]]_ -- of such. These non-classical combinations of states of subsystems are called _entangled states_.

## Definition

In [[quantum mechanics]] a [[state]] of a [[physical system]] is represented by a [[vector]] in some ([[Hilbert space|Hilbert]]-)[[vector space]] $H$. If the system is the composite of two [[subsystems]] with state spaces $H_1$ and $H_2$, respectively, then the state space of the total system is the [[tensor product]] $H = H_1 \otimes H_2$. The [[universal property]] of the [[tensor product]] gives a [[bilinear map]]

$$
  p : H_1 \times H_2 \to H_1 \otimes H_2
$$

which sends a pair of states $(\psi_1, \psi_2)$ to their tensor product $\psi_1 \otimes \psi_2$. States in the [[image]] of $p$ are called **product states** or **separable states**. An **entangled state** is a state which is _not_ a product state.

## Examples

Consider two [[quantum systems]], $A$ and $B$, with state vectors $|\Psi^{(A)}\rangle$ and $|\Psi^{(B)}\rangle$ respectively.  The combined state of the system may be described by a single state vector $|\Psi^{(AB)}\rangle=|\Psi^{(A)}\rangle \otimes |\Psi^{(B)}\rangle$.

As an example, suppose that in the basis $\{|0\rangle ,|1\rangle\}$, $|\Psi^{(A)}\rangle = \frac{1}{\sqrt{2}}\left(|0\rangle +|1\rangle\right)$.  This can be interpreted as system $A$ being in state $|0\rangle$ with probability 1/2 and state $|1\rangle$ with probability 1/2.  Suppose further that $|\Psi^{(B)}\rangle = |0\rangle$.  Then we have

$|\Psi^{(AB)}\rangle=|\Psi^{(A)}\rangle \otimes |\Psi^{(B)}\rangle=\frac{1}{\sqrt{2}}\left(|0\rangle +|1\rangle\right)\otimes|0\rangle=\frac{1}{\sqrt{2}}\left(|00\rangle +|10\rangle\right)$.

Such a state is said to be a **product state** because it is "factorable" or equivalently separable, i.e. it can be formed from some combination of individual states in the basis.

Compare the above example to the state

$|\Psi^{(AB)}\rangle=\frac{1}{\sqrt{2}}\left(|00\rangle +|11\rangle\right)$.

This state is not a product state since it cannot be formed from any combination of individual states in the given basis.  Such a state is known as an **entangled state** because it is said to be _non-factorable_ or _non-separable_.  The entangled states discussed above are, in fact, [[pure states]] rather than [[mixed states]] because they cannot be broken down further.  However, there is also a notion of entanglement for mixed states.

## Properties

### LOCC and SLOCC

The following refers to ([Coecke-Kissinger](#CoeckeKissinger)).

Often if [[multi-party state]]s can be inter-converted via local operations, they are considered to be the same. This can be made formal by the following definition.

+-- {: .num_defn}
###### Definition
Two states $|\Psi\rangle,|\Phi\rangle \in \bigotimes H_i$ are said to be equivalent up to local operations with classical communication (LOCC) if they can be inter-converted by a protocol involving any number of steps where (i) one party applies a local unitary operation $U : H_i \rightarrow H_i$ or (ii) one party sends some classical information to another.
=--

Such a protocol is reversible, so since protocols compose, this generates an equivalence relation. While this removes a good deal of redundancy from the study of entanglement, it is often useful to use an even more course-grained relation.

+-- {: .num_defn}
###### Definition
Two states $|\Psi\rangle,|\Phi\rangle \in \bigotimes H_i$ are said to be equivalent up to stochastic LOCC (SLOCC) if they can be inter-converted _with some non-zero probability_ a protocol involving any number of steps where (i) one party applies a an _arbitrary_ local operation $L : H_i \rightarrow H_i$ or (ii) one party sends some classical information to another.
=--

An example of a local stochastic operation is as follows. Suppose Alice and Bob share a state $|\Psi\rangle \in H_A \otimes H_B$ and Alice wishes to perform some operation $L$. Alice prepares an ancilla qubit $|0\rangle \in \mathbb{C}^2$ and performs a unitary operation

\[ U : \mathbb{C}^2 \otimes H_1 \rightarrow \mathbb{C}^2 \otimes H_1 \]

on her qubit as well as her part of the state $|\Psi\rangle$. She then measures the ancilla qubit. If she gets an outcome of $|0\rangle$, she has performed some operation $L : H_A \rightarrow H_A$ and if she gets outcome $|1\rangle$ she has performed $L' : H_A \rightarrow H_A$. The probability of Alice successfully performing $L$ is then the probability of getting the outcome of $|0\rangle$ when she performed her measurement.

+-- {: .num_theorem}
###### Theorem
Two states are SLOCC-equivalent iff they can be inter-converted by applying arbitrary invertible local operations (ILOs).
=--

Its easy to show using the Schur decomposition that there are only two SLOCC-equivalence classes in $\mathbb{C}^2 \otimes \mathbb{C}^2$, namely the product state class and the [[Bell state]] class. Perhaps more surprising is the following result to to Dur, Vidal, and Cirac. [2]

+-- {: .num_theorem}
###### Theorem
Any genuine tripartite state |$\Psi$> $\in \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ is SLOCC-equivalent to either |W> or |GHZ>;.
=--

By genuine, they mean a state that is not a product of smaller states. The two states are defined as:
\[ |W\rangle = |100\rangle + |010\rangle + |001\rangle
\qquad\qquad
|GHZ\rangle = |000\rangle + |111\rangle
\]

Each of these states yields the structure of a commutative [[Frobenius algebra]]. $|GHZ\rangle$ yields a special CFA and $|W\rangle$ yields an "anti-special" CFA. This structure serves to uniquely identity these states (up to SLOCC) in $\mathbb{C}^2$. [1]

## Related concepts

* [[entanglement entropy]]

* [[holographic entanglement entropy]]

* [[short-range entanglement]]

* [[tensor network]]

* [[quantum teleportation]]

* [[quantum error correction]]

[[!include states and observables -- content]]

## References

Early discussion of [[composite quantum systems]] and their [[quantum entanglement]]:

* [[Erwin Schrödinger]], *Discussion of Probability Relations between Separated Systems*, Mathematical Proceedings of the Cambridge Philosophical Society, **31** 4 (1935) 555-563 [[doi:10.1017/S0305004100013554](https://doi.org/10.1017/S0305004100013554)]

Introduction and review:

* [[Teiko Heinosaari]], [[Mário Ziman]], Section 6 of: *The Mathematical Language of Quantum Theory -- From Uncertainty to Entanglement*, Cambridge University Press  (2011) [[doi:10.1017/CBO9781139031103]( https://doi.org/10.1017/CBO9781139031103)]

* Dagmar Bruss, _Characterizing entanglement_, J. Math. Phys. 43, 4237 (2002) [arXiv:quant-ph/0110078](http://arxiv.org/abs/quant-ph/0110078) [doi](http://dx.doi.org/10.1063/1.1494474)

* [[Ingemar Bengtsson]], [[Karol Życzkowski]], Chapter 15 of: *Geometry of Quantum States --- An Introduction to Quantum Entanglement*, Cambridge University Press (2006) [[doi:10.1017/CBO9780511535048](https://doi.org/10.1017/CBO9780511535048)]

Extensive survey of the field:

* Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki: *Quantum entanglement*, Rev. Mod. Phys. **81** (2009) 865 [[arXiv:quant-ph/0702225](https://arxiv.org/abs/quant-ph/0702225), [doi:10.1103/RevModPhys.81.865](https://doi.org/10.1103/RevModPhys.81.865)]

As a notion in [[quantum information theory]]:

* [[John Watrous]], Chapter 6 of: *The Theory of Quantum Information*, Cambridge University Press (2018) [[doi:10.1017/9781316848142](https://doi.org/10.1017/9781316848142), [webpage](https://cs.uwaterloo.ca/~watrous/TQI/), [pdf](https://cs.uwaterloo.ca/~watrous/TQI/TQI.pdf)]

In relation to *[[correlation]]*:

* {#LuoLuo03} Shun-long Luo, You-feng Luo, *Correlation and Entanglement*, Acta Mathematicae Applicatae Sinica **19** (2003) 581–598 $[$[doi:10.1007/s10255-003-0133-z](https://doi.org/10.1007/s10255-003-0133-z)$]$

and in relation to [[quantum computation]]/[[quantum supremacy]]:

* [[John Preskill]]: *Quantum computing and the entanglement frontier*: pp. 63-80 in: *The Theory of the Quantum World -- Proceedings of the 25th Solvay Conference on Physics*, World Scientific (2013) [[arXiv:1203.5813](https://arxiv.org/abs/1203.5813), [doi:10.1142/8674](https://doi.org/10.1142/8674), slides: [pdf](https://simons.berkeley.edu/sites/default/files/docs/394/preskilljohn.pdf)]

and in the context of [[entanglement entropy]] of [[topological phases of matter]]:

* [[Bei Zeng]], [[Xie Chen]], [[Duan-Lu Zhou]], [[Xiao-Gang Wen]]:

  Sec. 1 of of: *[[Quantum Information Meets Quantum Matter]] -- From Quantum Entanglement to Topological Phases of Many-Body Systems*, Quantum Science and Technology (QST), Springer (2019) $[$[arXiv:1508.02595](https://arxiv.org/abs/1508.02595), [doi:10.1007/978-1-4939-9084-9](https://doi.org/10.1007/978-1-4939-9084-9)$]$

Exposition of entanglement as a phenomenon of non-[[Cartesian monoidal category|Cartesian]] [[monoidal categories]]:

* {#Baez04} [[John Baez]], *Quantum Quandaries: a Category-Theoretic Perspective*, in D. Rickles et al. (ed.) *The structural foundations of quantum gravity*, Clarendon Press (2006) 240-265 [[arXiv:quant-ph/0404040](https://arxiv.org/abs/quant-ph/0404040), [ISBN:9780199269693](https://global.oup.com/academic/product/the-structural-foundations-of-quantum-gravity-9780199269693)]

A discussion in [[quantum mechanics in terms of dagger-compact categories]] is in

* {#CoeckeKissinger} [[Bob Coecke]], [[Aleks Kissinger]], _The compositional structure of multipartite quantum entanglement_ ([arXiv:1002.2540](http://arxiv.org/abs/1002.2540))

See also

* W. Dür, G. Vidal, J. I. Cirac, _Three qubits can be entangled in two inequivalent ways_, Phys. Rev. A. __62__, 062314

Discussion in [[quantum optics]]:

* [[Tim Byrnes]], [[Ebubechukwu Ilo-Okeke]], Section 10 of: *Quantum Atom Optics: Theory and Applications to Quantum Technology*, Cambridge University Press 2021 ([arXiv:2007.14601](https://arxiv.org/abs/2007.14601), [CUP](https://www.cambridge.org/core/books/quantum-atom-optics/2D867888B5C666D3A936F1C942C99568))

A connection to algebraic geometry is proposed in

* Frédéric Holweck, Jean-Gabriel Luque, Jean-Yves Thibon, _Geometric descriptions of entangled states by auxiliary varieties_, J. Math. Phys. __53__, 102203 (2012); [doi](http://dx.doi.org/10.1063/1.4753989)

The following work included the consideration of identical particles into the study of quantum entanglement. In this case, the usage of partial trace may not be suitable and instead subsystems are described in terms of subalgebras. The work is in operator algebraic framework, based on usage of [[GNS construction]] and related to the consideration of von Neumann entropy.

* [[A. P. Balachandran]], [[T. R. Govindarajan]], Amilcar R. de Queiroz, A. F. Reyes-Lega, _Entanglement and particle identity: a unifying approach_, Phys. Rev. Lett. 110, 080503 (2013) [arxiv/1303.0688](http://arxiv.org/abs/1303.0688); _Algebraic approach to entanglement and entropy_, [arxiv/1301.1300](http://arxiv.org/abs/1301.1300); _Entanglement, particle identity and the GNS construction: a unifying approach_ [arxiv/1205.2882](http://arxiv.org/pdf/1205.2882) (earlier, longer version, overlapping with 1303.0688)

Use of [[homological algebra]] for quantifying entanglement:

* {#Mainiero2019} [[Tom Mainiero]], *Homological Tools for the Quantum Mechanic* (2019) [[arXiv:1901.02011](https://arxiv.org/abs/1901.02011), [10.10.11.6/handle/1/1339](http://10.10.11.6/handle/1/1339)]

category: physics

[[!redirects entangled state]]
[[!redirects entangled states]]

[[!redirects entangled]]
[[!redirects separable]]

[[!redirects quantum entanglement]]

[[!redirects entangled particles]]


entanglement entropy

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#### Measure and probability theory
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[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[entropy]] induced by [[entanglement]] in [[quantum physics]], essentially a synonym for *[[subsystem]] entropy*.

If $\rho$ is a [[quantum state]] ([[density matrix]]) of some [[quantum systems]] and $A$ is a [[subsystem]] with complementary subsystem $\overline{A}$, then its *entanglement entropy* is

$$
  S_A
  \;\coloneqq\;
  -
  Tr_{A}\big( \rho_A \ln \rho_A \big)
  \,,
  \;\;\;
  \text{where}
  \;\;\;
  \rho_A \;\coloneqq\; Tr_{\bar A}(\rho)
  \,.
$$

Special aspects:

A constant contribution to $S_A$ (i.e. independent of the choice of $A$, in a suitable sense) is called a **[[topological entanglement entropy]]** indicating **[[long-range entanglement]]** and "[[topological order]]", being proportional to the total [[quantum dimension]] of [[anyon]]-excitations. (See the references [below](#ReferencesTopologicalEntanglementEntropy).)

On the other extreme, for **[[short range entanglement]]** the entanglement entropy is thought to scale with the surface [[area]] of the subsystem $A$ (to the extent that this makes sense, say in say in [[lattice models]]), a behaviour reminiscent of [[Bekenstein-Hawking entropy]] of [[black holes]]. For more on this see at **[[holographic entanglement entropy]]**.

## Related concepts

* [[tensor network]]

* [[holographic entanglement entropy]]

* [[black hole entropy]] via [[AdS-CFT duality]]

[[!include states and observables -- content]]

## References

### General

* Matthew Headrick, _Lectures on entanglement entropy in field theory and holography_ ([arXiv:1907.08126](https://arxiv.org/abs/1907.08126))

In [[AQFT]]:

* [[Roberto Longo]], Feng Xu, _Von Neumann Entropy in QFT_ ([arxiv:1911.09390](https://arxiv.org/abs/1911.09390))

See also

* Wikipedia, _[Entropy of entanglement](http://en.wikipedia.org/wiki/Entropy_of_entanglement)_

[[!include topological entanglement entropy -- references]]

[[!redirects entanglement entropies]]

[[!redirects entropy of entanglement]]

[[!redirects topological entanglement entropy]]
[[!redirects topological entropy]]


entanglement island proposal for black hole paradox -- references

### Entanglement island proposal for BH information paradox
 {#EntanglementIslandProposalForBHInformationParadox}

#### Claims

Claim that the proper application of [[holographic entanglement entropy]] to the discussion of [[Bekenstein-Hawking entropy]] resolves the apparent [[black hole information paradox]]:

* [[Geoff Penington]], [[Stephen Shenker]], [[Douglas Stanford]], [[Zhenbin Yang]], _Replica wormholes and the black hole interior_,  J. High Energ. Phys. **2022** 205 (2022) \[doi:10.1007/JHEP05(2020)013, [arXiv:1911.11977](https://arxiv.org/abs/1911.11977)\]

* [[Ahmed Almheiri]], [[Thomas Hartman]], [[Juan Maldacena]], [[Edgar Shaghoulian]], [[Amirhossein Tajdini]], _Replica Wormholes and the Entropy of Hawking Radiation_, J. High Energ. Phys. **2020** 13 (2020)  \[doi:10.1007/JHEP05(2020)013, [arXiv:1911.12333](https://arxiv.org/abs/1911.12333)\]

Review aimed at readers with minimal background:

* [[Ahmed Almheiri]], [[Thomas Hartman]], [[Juan Maldacena]], [[Edgar Shaghoulian]], [[Amirhossein Tajdini]], _The entropy of Hawking radiation_, Rev. Mod. Phys. **93** 35002 (2021) \[doi:10.1103/RevModPhys.93.035002, [arXiv:2006.06872](https://arxiv.org/abs/2006.06872)\]

See also:

* [[Suvrat Raju]], _Lessons from the Information Paradox_ \[arXiv:2012.05770\]

#### Criticism

* Anna Karlsson, _Concerns about the replica wormhole derivation of the island conjecture_ ([arXiv:2101.05879](https://arxiv.org/abs/2101.05879))

* Harvendra Singh, *Islands and Icebergs contribute nothing to the Page curve (System with a symmetrical bath)* \[arXiv:2210.13970\]

Argument that the would-be proof of the island conjecture secretly works only for *massive* gravity:

* [[Hao Geng]], [[Andreas Karch]], *Massive Islands*,  J. High Energ. Phys. **2020** 121 (2020) \[doi:10.1007/JHEP09(2020)121, [arXiv:2006.02438](https://arxiv.org/abs/2006.02438)\]

* [[Hao Geng]], [[Andreas Karch]], [[Carlos Perez-Pardavila]], [[Suvrat Raju]], [[Lisa Randall]], [[Marcos Riojas]], [[Sanjit Shashi]], *Inconsistency of islands in theories with long-range gravity*, J. High Energ. Phys. **2022** 182 (2022) \[doi:10.1007/JHEP01(2022)182\]

* [[Hao Geng]], *Graviton Mass and Entanglement Islands in Low Spacetime Dimensions* \[arXiv:2312.13336\]

reviewed in:

* [[Hao Geng]], *Recent Progress in Quantum Gravity: Karch-Randall Braneworld, Entanglement Islands and Graviton Mass*, Washington (2022) \[hdl:1773/49427\]

and discussion of what to do about this problem:

* [[Hao Geng]], [[Andreas Karch]], [[Carlos Perez-Pardavila]], [[Suvrat Raju]], [[Lisa Randall]], [[Marcos Riojas]], [[Sanjit Shashi]], *Information Transfer with a Gravitating Bath*, SciPost Phys. **10** 103 (2021)  \[arXiv:2012.04671, [doi:10.21468/SciPostPhys.10.5.103](https://scipost.org/10.21468/SciPostPhys.10.5.103)\]

Claim that the whole argument relies on the false assumption that the [[split property]] holds for [[quantum gravity]]:

* [[Suvrat Raju]], *Failure of the split property in gravity and the information paradox*,  Classical and Quantum Gravity, **39** 064002 \[doi:10.1088/1361-6382/ac482b, [arXiv:2110.05470](https://arxiv.org/abs/2110.05470)\]

Review in:

* [[Nirmalya Kajuri]], *[Of Islands, Holograms and Saving Quantum Physics From a Black Hole Paradox](https://science.thewire.in/the-sciences/unitarity-entanglement-islands-holography-information/)*, Science -- The Wire (Nov 2022)

  > $[\ldots]$ a few sceptics have argued that while the calculations are correct, they don't help resolve the black hole information-loss paradox. The troubles stem from the reservoir attached to the anti-de Sitter universe. The physicists who authored the island papers assumed that gravity stopped at the boundary of the anti-de Sitter space and didn’t enter the reservoir. This is not an innocuous assumption. $[\ldots]$ The key takeaway is that the island way to recover information and save unitarity works perfectly well – if you slightly modify Einstein’s theory of gravity. These criticisms have been around for some two years now, and physicists are yet to resolve them in print. $[\ldots]$ physicists continue to publish papers by the hundreds about the entanglement islands but few attempt to answer whether the islands are compatible with the Einsteinian gravity of our universe. $[\ldots]$

See also:

* Saskia Demulder, Alessandra Gnecchi, Ioannis Lavdas, [[Dieter Lüst]], *Islands and Light Gravitons in type IIB String Theory* \[arXiv:2204.03669\]

* [[Emil Martinec]]: "Thus far, picturesque evocations of quantum wormholes remain largely just that – picturesque evocations." ([2nd Nov 2022](https://www.math.columbia.edu/~woit/wordpress/?p=13113#comment-241963))

Argument that discussion of the issue in the toy example of 2d [[JT gravity]] does not actually generalize to higher dimensions:

* {#LandgrenShekar24} Filip Landgren, Arvind Shekar, *Islands and entanglement entropy in $d$-dimensional curved backgrounds* \[arXiv:2401.01653\]


entire function

An __entire function__ is a [[complex number|complex]]-valued [[function]] of one complex variable that is defined and [[holomorphic function|holomorphic]] on the entire (hence the name) [[complex plane]] (as opposed to a [[partial function|partial]] holomorphic function, which might be defined only on some [[open subset]] of the plane).

The study of entire functions is a rather nontrivial branch of [[complex analysis]]; of course, with many applications.

category:analysis

[[!redirects entire function]]
[[!redirects entire functions]]

[[!redirects entire holomorphic function]]
[[!redirects entire holomorphic functions]]
[[!redirects total holomorphic function]]
[[!redirects total holomorphic functions]]
[[!redirects entire analytic function]]
[[!redirects entire analytic functions]]
[[!redirects total analytic function]]
[[!redirects total analytic functions]]


entire morphism in a dagger 2-poset

+-- {: .rightHandSide}
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=--

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

## Idea ##

A morphism similar to how entire relations are defined in [[Rel]]

## Definition ##

A morphism $f \in hom_A(a,b)$ of a [[dagger 2-poset]] $A$ is an **entire morphism** if $1_a \leq f^\dagger \circ f$.

## See also ##

* [[onto morphism in a dagger 2-poset]]
* [[functional morphism in a dagger 2-poset]]
* [[injective morphism in a dagger 2-poset]]
* [[map in a dagger 2-poset]]


entire relation

+-- {: .rightHandSide}
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### Context
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
=--
=--

A binary [[relation]] from a [[set]] $X$ to a set $Y$ is called **entire** if every element of $X$ is related to at least one element of $Y$.  This includes most examples of what the pre-Bourbaki literature calls a (total) [[multi-valued function]] (although that term usually implied some continuity or analyticity properties as well).  An entire relation is sometimes called **total**, although that has another meaning in the theory of [[partial order]]s; see [[total relation]].

A [[function]] is precisely a relation that is both entire and [[functional relation|functional]].

Like any relation, an entire relation $r$ can be viewed as a [[span]]
$$ \array {
  &                  & \Gamma_r \\
  & \swarrow_{\pi_r} &          & \searrow^{\phi_r} \\
X &                  &          &                   & Y
} $$
Such a span is a relation iff the pairing map from the **graph** $\Gamma_r$ to $X \times Y$ is an [[injection]], and such a relation is entire iff the **projection** map $\pi_r$ is a [[surjection]].

The [[axiom of choice]] says precisely that every entire relation contains a [[function]].  Failing that, the [[COSHEP]] axiom may be interpreted to say that, given $X$, there is a single surjection $\pi_X: \Gamma_X \to X$ such that every entire relation from $X$ contains a relation given by a span whose left leg is $\pi_X$.  In any case, entire relations may be preferable to functions in some contexts where the axiom of choice fails.

When [[internalization|internalising]] entire relations to a [[site]], one may want to replace the projection map $\pi_r: \Gamma_r \to X$ by a [[cover|covering family]].

## Related concepts

* [[axiom of fullness]]

[[!redirects entire relation]]
[[!redirects entire relations]]


entourage

# Entourages
* table of contents
{: toc}

## Idea

An _entourage_ (aka _vicinity_) is a [[binary relation]] of 'approximate equality' on a [[space]], generally a [[uniform space]].  Just as a [[topological space]] is given by its [[underlying set]] of points and an appropriate collection of [[open subsets]], so a [[uniform space]] is given by its underlying set of points and an appropriate collection of entourages.

Entourages are actually a uniformization of [[neighbourhoods]] rather than of open sets as such.  Given a point $x$, a neighbourhood $U$ of $x$ defines a generalized notion of distance from $x$; a point $y$ is [less than] this distance from $x$ iff $y \in U$.  Uniformizing this, an entourage $E$ defines a generalized notion of uniform distance; two points $x$ and $y$ are [less than] this distance from each other iff $(x,y) \in E$.  Or said another way, if $U$ is a neighbourhood of $x$, then any point $y$ that is sufficiently close to $x$ will be in $U$; if $E$ is an entourage, then any two points $x$ and $y$ that are sufficiently close together will be related by $E$.

## Definitions

The precise definition depends on the context.

*  In a [[metric space]], a relation $\approx$ is an __entourage__ if there exists a positive [[real number]] $\epsilon$ such that
   $$ d(x,y) \lt \epsilon \;\implies\; x \approx y ,$$
   where $x,y$ are points in the metric space and $d$ is the metric.

*  In a [[gauge space]], $\approx$ is an __entourage__ if there exists an $\epsilon$ and a gauging distance $d$ such that the preceding condition holds.

*  In a [[topological abelian group]], $\approx$ is an __entourage__ if there is a [[neighbourhood]] $N$ of the identity element such that
   $$ x/y \in N \;\implies\; x \approx y ,$$
   where $x,y$ are points in the metric space and $/$ is the division operation in the group.

*  In a nonabelian [[topological group]], there are two distinct notions of entourage, one using the same formula as above and the other using $y/x$ in place of $x/y$.

*  Of course, the most general kind of entourage is that occurring in the definition of a [[uniform space]], in the same way that [[open sets]] occur in the definition of a [[topological space]].

## Infinitesimal entourages

In [[nonstandard analysis]], every point $x$ in a [[topological space]] $X$ has an infinitesimal neighbourhood in the nonstandard extension $X^*$, called the [[halo]] (or monad) of $x$.  This is to be thought of as the set $\{ y \in X^* \;|\; y \simeq x \}$ of all of the [[hyperpoint]]s that are infinitely close ([[adequality|adequal]]) to $x$.  Similarly, any [[uniform space]] $X$ has an infinitesimal entourage in its nonstandard extension $X^*$, a binary relation $\{ x \in X^*,\; y \in X^* \;|\; y \simeq x \}$ that relates two hyperpoints iff they are infinitely close to each other.  (So the infinitesimal entourage is simply the [[adequality]] relation.)

[[!redirects entourage]]
[[!redirects entourages]]
[[!redirects vicinity]]
[[!redirects vicinities]]


entropic force

#Contents#
* table of contents
{:toc}

## Idea

An effective [[force]] appearing in a complex system in the [[thermodynamic limit]] as a result of the system's [[entropy]] increasing over time.

## References

* Wikipedia, _[Entropic force](http://en.wikipedia.org/wiki/Entropic_force)_

[[!redirects entropic forces]]


entropy

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

# Entropy
* table of contents
{: toc}

## Idea

Entropy is a measure of disorder, given by the amount of [[information]] necessary to precisely specify the [[state]] of a system.

Entropy is important in [[information theory]] and [[statistical physics]].

## Mathematical definitions

We can give a precise [[mathematics|mathematical]] definition of the entropy in [[probability theory]].

### Preliminary definitions

We will want a couple of preliminary definitions.  Fix a [[probability space]] $(X,\mu)$; that is, $X$ is a [[set]], and $\mu$ is a [[probability measure]] on $X$.

#### Surprisal

If $A$ is a [[measurable subset]] of $X$, then the __surprisal__ or __self-[[information]]__ of $A$ (with respect to $\mu$) is

\[
  \sigma_\mu(A) \coloneqq -\log \mu(A)
  \,.
\]

Notice that, despite the minus sign in this formula, $\sigma$ is a nonnegative function (since $\log p \leq 0$ for $p \leq 1$); more precisely, $\sigma$ takes values in $[0,\infty]$.  The term 'surprisal' is intended to suggest how surprised one ought to be upon learning that the event modelled by $A$ is true: from no surprise for an event with probability $1$ to infinite surprise for an event with probability $0$.

The __expected surprisal__ of $A$ is then

\[
  \label{EqA}
  h_\mu(A) \coloneqq \mu(A) \sigma_\mu(A) = -\mu(A) \log \mu(A) = -\log(\mu(A)^{\mu(A)})
\]

(with $h_\mu(A) = 0$ when $\mu(A) = 0$).  Like $\sigma$, $h$ is a nonnegative function; it is also important that $h_\mu$ is [[convex function|concave]].  Both $h_\mu(\nothing)$ and $h_\mu(X)$ are $0$, but for different reasons; $h_\mu(A) = 0$ when $\mu(A) = 1$ because, upon observing an event with probability $1$, one gains no information; while $h_\mu(A) = 0$ when $\mu(A) = 0$ because one expects never to observe an event with probability $0$.  The maximum possible value of $h$ is $\mathrm{e}^{-1} \log \,\mathrm{e}$ (so $\mathrm{e}^{-1}$ if we use [[natural logarithms]]), which occurs when $\mu(A) = \mathrm{e}^{-1}$.

We have not specified the base of the [[logarithm]], which amounts to a constant factor (proportional to the logarithm of the base), which we think of as specifying the [[unit of measurement]] of entropy.  Common choices for the base are $2$ (whose unit is the [[bit]], originally a unit of memory in computer science), $256$ (for the byte, which is $8$ bits), $3$ (for the trit), $\mathrm{e}$ (for the nat or neper), $10$ (for the bel, originally a unit of relative power intensity in telegraphy, or ban, dit, or hartley), and $\root{10}{10}$ (for the decibel, $1/10$ of a bel).  In applications to [[statistical physics]], common bases are exactly (since 2019)

\[
  \label{EqB}
\mathrm{e}^{10^{29}/1\,380\,649} \approx 10^{3.145\,582\,127\,704\,085\,743 \times 10^{22}} \qquad \text{(for the joule per kelvin)},
\]

or

\[
  \label{EqC}
\mathrm{e}^{104\,600\,000\,000\,000/207\,861\,565\,453\,831} \approx 1.654\,037\,938\,063\,167\,336 \qquad \text{(for the calorie per mole-kelvin)},
\]

and so on; although $\mathrm{e}$ is common in theoretical work (and then the unit of measurement is said to be [[Boltzmann's constant]] rather than the nat or neper).

#### Almost partitions

Recall that a __[[partition]]__ of a set $X$ is a [[family of subsets|family]] $\mathcal{P}$ of subsets of $X$ (the _parts_ of the partition) such that $X$ is the [[union]] of the parts and any two distinct parts are [[disjoint sets|disjoint]] (or better, for [[constructive mathematics]], two parts are equal if their intersection is [[inhabited subset|inhabited]]).

When $X$ is a probability space, we may relax both conditions: for the union of $\mathcal{P}$, we require only that it be a [[full set]]; for the intersections of pairs of elements of $\mathcal{P}$, we require only that they be [[null sets]] (or better, for constructive mathematics, that $A = B$ when $\mu^*(A \cap B) \gt 0$, where $\mu^*$ is the [[outer measure]] corresponding to $\mu$).

For definiteness, call such a collection of subsets a __$\mu$-almost partition__; a $\mu$-almost partition is _measurable_ if each of its part is measurable (in which case we can use $\mu$ instead of $\mu^*$).

### Entropy of a $\sigma$-algebra on a probability space

This is a general mathematical definition of entropy.

Given a probability [[measure space]] $(X,\mu)$ and a $\sigma$-[[sigma-algebra|algebra]] $\mathcal{M}$ of [[measurable sets]] in $X$, the __entropy__ of $\mathcal{M}$ with respect to $\mu$ is
\[
  \label{general} H_\mu(\mathcal{M}) \coloneqq \sup \{ \sum_{A \in \mathcal{F}} h_\mu(A) \;|\; \mathcal{F} \subseteq \mathcal{M},\; {|\mathcal{F}|} \lt \aleph_0,\; X = \biguplus \mathcal{F} \}
  \,.
\]

In words, the entropy is the [[supremum]], over all ways of expressing $X$ as an internal [[disjoint union]] of [[finite set|finitely many]] elements of the $\sigma$-algebra $\mathcal{M}$, of the sum, over these measurable sets, of the expected surprisals of these sets.  This supremum can also be expressed as a [[convergence|limit]] as we take $\mathcal{F}$ to be finer and finer, since $h_\mu$ is concave and the partitions are [[directed set|directed]].

We have written this so that $\mathcal{F}$ is a finite partition of $X$; without loss of generality, we may require only that $\mathcal{F}$ be a $\mu$-almost partition.  In [[constructive mathematics]], it seems that we *must* use this weakened condition, at least the part that allows $\bigcup \mathcal{F}$ to merely be full.

This definition is very general, and it is instructive to look at special cases.

### Entropy of a probability space

Given a probability space $(X,\mu)$, the __entropy__ of this probability space is the entropy, with respect to $\mu$, of the $\sigma$-algebra of *all* measurable subsets of $X$.

### Entropy of a partition of a probability space

Every measurable almost-partition of a measure space (indeed, any family of measurable subsets) generates a $\sigma$-algebra.  The __entropy__ of a measurable almost-partition $\mathcal{P}$ of a probability measure space $(X,\mu)$ is the entropy, with respect to $\mu$, of the $\sigma$-algebra generated by $\mathcal{P}$.  The formula (eq:general) may then be written
\[ \label{partition} H_\mu(\mathcal{P}) = \sum_{A \in \mathcal{P}} h_\mu(A) = -\sum_{A \in \mathcal{P}} \log(\mu(A)^{\mu(A)}) ,\]
since an infinite sum (of nonnegative terms) may also be defined as a supremum.  (Actually, the supremum in the infinite sum does not quite match the supremum in (eq:general), so there is a bit of a theorem to prove here.)

In most of the following special cases, we will consider only partitions, although it would be possible also to consider more general $\sigma$-algebras.

### Entropy of (a partition of) a discrete probability space

Recall that a __discrete probability space__ is a [[set]] $X$ equipped with a function $\mu\colon X \to ]0,1]$ such that $\sum_{i \in X} \mu(i) = 1$; since $\mu(i) \gt 0$ is possible for only countably many $i$, $X$ must be [[countable set|countable]].  We make $X$ into a measure space (with every subset measurable) by defining $\mu(A) \coloneqq \sum_{i \in A} \mu(i)$.  Since every inhabited set has positive measure, every almost-partition of $X$ is a partition; since every set is measurable, any partition is measurable.

Given a discrete probability space $(X,\mu)$ and a partition $\mathcal{P}$ of $X$, the __entropy__ of $\mathcal{P}$ with respect to $\mu$ is defined to be the entropy of $\mathcal{P}$ with respect to the probability measure induced by $\mu$.  Simplifying (eq:partition), we find

\[
  \label{EqD}
  H_\mu(\mathcal{P}) = -\sum_{A \in \mathcal{P}} \log((\sum_{i \in A} \mu(i))^{\sum_{i \in A} \mu(i)})
  \,.
\]

More specially, the __entropy__ of the discrete probability space $(X,\mu)$ is the entropy of the partition of $X$ into [[singletons]]; we find

\[
  \label{EqE}
  H_\mu(X) = \sum_{i \in X} h_\mu(i) = -\sum_{i \in X} \log(\mu(i)^{\mu(i)})
  \,.
\]

This is actually a special case of the entropy of a probability space, since the $\sigma$-algebra generated by the singletons is the power set of $X$.

Yet more specially, the __entropy__ of a [[finite set]] $X$ is the entropy of $X$ equipped with the uniform discrete probability measure; we find
\[ \label{Boltzmann} H_{unif}(X) = -\sum_{i \in X} \log((\frac{1}{|X|})^{\frac{1}{|X|}}) = \log {|X|} ,\]
which is probably the best known mathematical formula for entropy, due to [[Max Planck]], who attributed it to [[Ludwig Boltzmann]].  (Its [physical interpretation](#physical) appears below.)

Of all probability measures on $X$, the uniform measure has the [[maximum entropy]].

### Entropy with respect to an absolutely continuous probability measure on the real line

Recall that a [[Borel measure]] $\mu$ on an [[interval]] $X$ in the [[real line]] is __[[absolutely continuous measure|absolutely continuous]]__ if $\mu(A) = 0$ whenever $A$ is a [[null set]] (with respect to [[Lebesgue measure]]), or better such that $\mu(A) \gt 0$ whenever the Lebesgue measure of $A$ is positive.  In this case, we can take the [[Radon–Nikodym derivative]] of $\mu$ with respect to Lebesgue measure, to get an [[integrable function]] $f$, called the __probability distribution function__; we recover $\mu$ by
\[ \label{pdf} \mu(A) = \int_A f(x) \mathrm{d}x ,\]
where the integral is taken with respect to Lebesgue measure.

If $\mathcal{P}$ is a partition (or a Lebesgue-almost-partition) of an interval $X$ into [[Borel sets]], then the __entropy__ of $\mathcal{P}$ with respect to an integrable function $f$ is the entropy of $\mathcal{P}$ with respect to the measure induced by $f$ using the integral formula (eq:pdf); we find

\[
  \label{EqF}
  H_f(\mathcal{P})
  =
   -\sum_{A \in \mathcal{P}}
  \log\big(
     (\int_A f(x) \mathrm{d}x)^{\int_A f(x) \mathrm{d}x}
  \big)
\,.
\]

On the other hand, the __entropy__ of the probability distribution space $(X,f)$ is the entropy of the entire $\sigma$-algebra of all Borel sets (which is *not* generated by a partition) with respect to $f$; we find

\[
  \label{EqG}
  H_f(X) = -\int_{x \in X} \log(f(x)^{f(x)}) \mathrm{d}x
\]

by a fairly complicated argument.

+-- {: .query}
I haven\'t actually managed to check this argument yet, although my memory tags it as a true fact.  —Toby
=--

### Entropy of a density matrix

In the analogy between [[classical physics]] and [[quantum physics]], we move from probability distributions on a [[phase space]] to [[density operators]] on a [[Hilbert space]].

So just as the entropy of a probability distribution $f$ is given by $- \int f \log f$, so the __entropy__ of a density operator $\rho$ is

\[
  \label{EqH}
  H_\rho \coloneqq -Tr (\rho \log \rho)
  \,.
\]

using the [[functional calculus]].

These are both special cases of the entropy of a [[state]] on a $C^*$-[[C-star-algebra|algebra]].

There is a way to fit this into the framework given by (eq:general), but I don\'t remember it (and never really understood it).

### Relative entropy
 {#RelativeEntropy}

For two finite probability distributions $p$ and $q$, their **relative entropy** is

\[
  \label{EqJ}
  S(p/q) \coloneqq \sum_{k = 1}^n p_k(log p_k - log q_k)
  \,.
\]

Or alternatively, for $\rho, \phi$ two [[density matrix|density matrices]], their relative entropy is

\[
  \label{EqK}
  S(\rho/\phi) \coloneqq tr \rho(log \rho - log \phi)
  \,.
\]

There is a generalization of these definitions to [[state]]s on general [[von Neumann algebra]]s, due to ([Araki](#Araki)).

For more on this see _[[relative entropy]]_.

## Physical entropy {#physical}

As hinted above, any probability distribution on a [[phase space]] in [[classical physics]] has an entropy, and any [[density matrix]] on a [[Hilbert space]] in [[quantum physics]] has an entropy.  However, these are __microscopic entropy__, which is not the usual entropy in [[thermodynamics]] and most other branches of [[physics]].  (In particular, microscopic entropy is conserved, rather than increasing with time.)

Instead, physicists use *coarse-grained* entropy, which corresponds mathematically to taking the entropy of a $\sigma$-algebra much smaller than the $\sigma$-algebra of all measurable sets.  Given a classical system with $N$ microscopic degrees of freedom, we identify $n$ macroscopic degrees of freedom that we can reasonably expect to measure, giving a map from $\mathbb{R}^N$ to $\mathbb{R}^n$ (or more generally, a map from an $N$-dimensional microscopic phase space to an $n$-dimensional macroscopic phase space). Then the $\sigma$-algebra of all measurable sets in $\mathbb{R}^n$ [[pullback|pulls back]] to a $\sigma$-algebra on $\mathbb{R}^N$, and the __macroscopic entropy__ of a statistical state is the [[conditional entropy]] of this $\sigma$-algebra.  (Typically, $N$ is on the order of [[Avogadro's number]], while $n$ is rarely more than half a dozen, and often as small as $2$.)

If we specify a state by a point in $\mathbb{R}^n$, a macroscopic pure state, and assume a uniform probability distribution on its [[fibre]] in $\mathbb{R}^N$, then this results in the [[maximum entropy]].  If this fibre were a finite set, then we would recover Boltzmann\'s formula (eq:Boltzmann).  This is never exactly true in classical statistical physics, but it is often nevertheless a very good approximation.  (Boltzmann\'s formula actually makes better physical sense in quantum statistical physics, even though Boltzmann himself did not live to see this.)

A more sophisticated approach (pioneered by [[Josiah Gibbs]]) is to consider all possible mixed microstates (that is all possible probability distributions on the space $\mathbb{R}^N$ of pure microstates) whose [[expectation values]] of total energy and other [[extensive quantities]] (among those that are functions of the macrostate) match the given pure macrostate (point in $\mathbb{R}^n$).  We pick the mixed microstate with the [[maximum entropy]].  If this is a [[thermal state]], then we say that the macrostate has a [[temperature]], but it has an entropy in any case.

### Gravitational entropy

* gravitational entropy

  * [[Bekenstein-Hawking entropy]]

  * [[generalized second law of thermodynamics]]

## Related concepts

[[!include Renyi entropies -- table ]]

* [[relative entropy]]

* [[entanglement entropy]], [[holographic entanglement entropy]]

* [[entropic force]]

* [[dissipative system]]

## References
{#References}

### General
 {#ReferencesGeneral}

The concept of entropy was introduced, by [[Rudolf Clausius]] in 1865, in the context of [[physics]], and then adapted to [[information theory]] by [[Claude Shannon]] in 1948, to [[quantum mechanics]] by [[John von Neumann]] in 1955, to [[ergodic theory]] by [[Andrey Kolmogorov]] and Sinai in 1958, and to [[topological dynamics]] by Adler, Konheim and McAndrew in 1965.

Survey with an eye towards [[black hole entropy]]:

* [[Ted Jacobson]], _Entropy from Carnot to Bekenstein_ ([arXiv:1810.07839](https://arxiv.org/abs/1810.07839))

### In quantum probability theory
 {#ReferencesInQuantumProbabilityTheory}

Discussion of entropy in [[quantum probability theory]], hence for [[quantum states]] understood as positive linear functionals on the [[star-algebra]] [[algebra of observables|of observables]] ([[operator algebra|operator algebraic]] [[state on a star-algebra|states on star-algebras]]) and with their [[density matrices]] defined via the [[GNS construction]]:

Introduction and survey:

* [[Ingemar Bengtsson]], [[Karol Życzkowski]], Section 12.2 of: *Geometry of Quantum States --- An Introduction to Quantum Entanglement*, Cambridge University Press (2006) [[doi:10.1017/CBO9780511535048](https://doi.org/10.1017/CBO9780511535048)]

* [[A. P. Balachandran]], T. R. Govindarajan, Amilcar R. de Queiroz, A. F. Reyes-Lega, _Algebraic approach to entanglement and entropy_, Phys. Rev. A 88, 022301 (2013) ([arXiv:1301.1300](http://arxiv.org/abs/1301.1300))

* [[A. P. Balachandran]], A. R. de Queiroz, S. Vaidya, *Entropy of Quantum States: Ambiguities*, Eur. Phys. J. Plus 128, 112 (2013) ([arXiv:1212.1239](https://arxiv.org/abs/1212.1239), [doi:10.1140/epjp/i2013-13112-3](https://doi.org/10.1140/epjp/i2013-13112-3))

* Paolo Facchi, Giovanni Gramegna, Arturo Konderak, *Entropy of quantum states* ([arXiv:2104.12611](https://arxiv.org/abs/2104.12611))

* Erling Størmer, _Entropy in operator algebras_, Astérisque, no. 232 (1995), p. 211-230. ([web](http://www.numdam.org/item/AST_1995__232__211_0/), [pdf](http://www.numdam.org/article/AST_1995__232__211_0.pdf))

In quantum mechanics, the basic notion is the von Neumann entropy defined in terms of [[density matrix]]. For type III [[von Neumann algebra]]s the density matrix is not well defined (physically, the problem is usually in ultraviolet divergences). Von Neumann entropy is generalized to arbitrary semifinite von Neumann algebra in

* [[Irving Segal|I. E. Segal]], _A note on the concept of entropy_, J. Math. Mech. 9 (1960) 623--629

[Relative entropy](#RelativeEntropy) of [[states]] on  [[von Neumann algebras]] was introduced in

* {#Araki} [[Huzihiro Araki]], _Relative entropy of states of von Neumann algebras_, Publ. RIMS Kyoto Univ.
11 (1976) 809--833 ([pdf](https://ems.press/content/serial-article-files/2833))

A note relating I. Segal's notion to relative entropy is

* [[Roberto Longo]], [[Edward Witten]], _A note on continuous entropy_, [arXiv:2202.03357](https://arxiv.org/abs/2202.03357)

A large collection of references on quantum entropy is in

* [[Christopher Fuchs]], _References for research in quantum distinguishability and state disturbance_ ([pdf](http://www.perimeterinstitute.ca/personal/cfuchs/BigRef.pdf))

### Category theoretic and cohomological interpretations

A discussion of entropy with an eye towards the [[presheaf topos]] over the [[site]] of finite [[measure spaces]] is in

* [[Mikhail Gromov]], _In a search for a structure, Part I: On entropy_ (2012) ([pdf](https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/structre-serch-entropy-july5-2012.pdf))

* [[William Lawvere]], _State categories, closed categories, and the existence_ [of] _semi-continuous entropy functions_, IMA Preprint Series #86, 1984. ([web](https://www.ima.umn.edu/preprints/State-Categories-Closed-Categories-and-Existence-Semi-Continuous-Entropy-Functions))

Category-theoretic characterizations of entropy are discussed in

* [[John Baez]], *[[johnbaez:Entropy as a functor|Entropy as a functor]]*

* Cheuk Ting Li, *A Characterization of Entropy as a Universal Monoidal Natural Transformation* [[arXiv:2308.05742](https://arxiv.org/abs/2308.05742)]

A (co)homological viewpoint is discussed in

* Pierre Baudot, [[Daniel Bennequin]], _The homological nature of entropy_, Entropy, 17 (2015), 3253-3318. ([doi](https://doi.org/10.3390/e17053253))

(for an update see also the abstract of a talk of Baudot [here](https://calendar.math.cas.cz/content/information-cohomology-and-topological-information-data-analysis))

### Axiomatic characterizations
 {#ReferencesAxiomaticCharacterization}

After the concept of entropy proved enormously useful in practice, many people have tried to find a more abstract foundation for the concept (and its variants) by characterizing it as the unique measure satisfying some list of plausible-sounding axioms.

A characterization of [relative entropy](#RelativeEntropy) on finite-[[dimension|dimensional]] [[C-star algebras]] is given in

* {#Petz92} D. Petz, _Characterization of the relative entropy of states of matrix algebras_, Acta Math. Hung. 59 (3-4) (1992) ([pdf](http://www.renyi.hu/~petz/pdf/52.pdf))

A simple characterization of von Neumann entropy of [[density matrices]] (mixed [[quantum states]]) is discussed in

* Bernhard Baumgartner, _Characterizing Entropy in Statistical Physics and in Quantum Information Theory_ ([arXiv:1206.5727](http://arxiv.org/abs/1206.5727))

Entropy-like quantities appear in the study of many PDEs, with entropy estimates. For an intro see

* L. C. Evans, _A survey of entropy methods for partial differential equations_, Bull. Amer. Math. Soc. 41 (2004), 409-438 ([web](https://www.ams.org/journals/bull/2004-41-04/S0273-0979-04-01032-8/), [pdf](http://math.berkeley.edu/~evans/ams.entropy.pdf)); and longer Berkeley graduate course text: _Entropy and partial differential equations_. ([pdf](http://math.berkeley.edu/~evans/entropy.and.PDE.pdf))

[[!redirects entropy]]
[[!redirects entropies]]

[[!redirects von Neumann entropy]]
[[!redirects von Neumann entropies]]

[[!redirects Shannon entropy]]
[[!redirects Shannon entropies]]