relative category

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###Context###
#### Category theory
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[[!include category theory - contents]]
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#### Homotopy theory
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[[!include homotopy - contents]]
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#Contents#
* table of contents
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## Idea

The notion of a  **relative category** is an extremely weak version of a [[category with weak equivalences]], providing the bare minimum needed to present an [[(∞,1)-category]]. The idea was first explored in a series of papers [Dwyer & Kan 1980](#DwyerKan1980Localizations) on [[simplicial localization]] (see there for more). A [[model category]] structure on relative categories was constructed by [Barwick & Kan 2012](#BarwickKan12), presenting the [[(∞,1)-category of (∞,1)-categories]].

More generally, the notion of relative category captures the idea of having a distinguished [[wide subcategory]] of morphisms that should be preserved by functors. Relative categories are occasionally used for this purpose, unrelated to any higher categorical purpose.

## Definition

### Basic

A **relative category** $C$ is a [[pair]] $(und C, weq C)$, where $und C$ is a [[category]] and $weq C$ is a [[wide subcategory]]. A [[morphism]] in $weq C$ is said to be a **[[weak equivalence]]** in $C$. A **relative functor** $f : C \to D$ is a [[functor]] $f \colon und C \to und D$ that preserves weak equivalences in the obvious sense.

The **[[homotopy category]]** of a relative category $C$ is the ordinary category $Ho C$ obtained from $und C$ by [[localization|freely inverting]] the weak equivalences in $C$.

### Refinements

* A **semi-saturated relative category** is a relative category $C$ such that every [[isomorphism]] in $und C$ is a weak equivalence in $C$.

* A [[category with weak equivalences]] is a semi-saturated relative category $C$ such that $weq C$ has the [[two-out-of-three property]].

* A [[homotopical category]] is a relative category $C$ such that $weq C$ has the [[two-out-of-six property]]; note that this automatically makes $C$ a category with weak equivalences.

* A **saturated homotopical category** is a relative category $C$ such that a morphism is a weak equivalence _if and only if_ it is invertible in $Ho C$; note that any such relative category must be a homotopical category in particular.

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

The semi-saturation condition is precisely the condition on
the inclusion $weq C \to und C$ ensuring it is a
[[monomorphism in an (infinity,1)-category|monomorphism]] in the
(2,1)-category of 1-categories.

=--

## Examples

Any ordinary [[category]] $C$ gives rise to three relative categories in a functorial way:

* A **minimal relative category** $min C$, in which the only weak equivalences are [[identity morphism|identities]].

* A **minimal homotopical category** $min^+ C$, in which the only weak equivalences are [[isomorphism|isomorphisms]].

* A **maximal homotopical category** $max C$, in which every morphism is a weak equivalence.

## Properties

### Categories of relative categories

Let $\mathbf{RelCat}$ be the [[large category]] of [[small category|small]] relative categories and relative functors. It is a [[locally presentable category|locally finitely presentable]] [[cartesian closed category]], and we refer to the [[exponential object]] $[C, D]_h$ as the **relative functor category**. $\mathbf{RelCat}$ is, in particular, a ([[strict 2-category|strict]]) [[2-category]].

Let $\mathbf{SsRelCat}$ be the full subcategory of semi-saturated relative categories. The inclusion $\mathbf{SsRelCat} \hookrightarrow \mathbf{RelCat}$ has a left [[adjoint]] that preserves finite [[product|products]], so $\mathbf{SsRelCat}$ is a [[reflective subcategory|reflective]] [[exponential ideal]] of $\mathbf{RelCat}$.

There are then the following strings of [[adjunctions]]:
$$min \dashv und \dashv max \dashv weq : \mathbf{RelCat} \to \mathbf{Cat}$$
$$Ho \dashv min^+ : \mathbf{Cat} \to \mathbf{RelCat}$$
$$Ho \dashv min^+ \dashv und \dashv max \dashv weq : \mathbf{SsRelCat} \to \mathbf{Cat}$$
Moreover, because $min^+$ embeds $\mathbf{Cat}$ as a reflective exponential ideal in $\mathbf{SsRelCat}$ and in $\mathbf{RelCat}$, the functor $Ho : \mathbf{SsRelCat} \to \mathbf{Cat}$ preserves finite products.

### As enriched categories
 {#AsEnrichedCategories}

The application of relative categories as expressing the idea of having a subcategory of distinguished morphisms can be neatly packaged into viewing relative categories as [[enriched categories]] over the category of sets with distinguished subsets. An equivalent formulation is that of an *[[M-category]]* (see also at *[[F-category]]* for a [[2-category]]-theoretic version of this idea).

Let $PairSet$ be the [[cartesian closed category]] whose objects are pairs of small sets $(X, A)$ such that $A \subseteq X$, and whose morphisms $(X, A) \to (Y, B)$ are functions $f : X \to Y$ with $f(A) \subseteq B$.

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

$RelCat$ is isomorphic to the category of small $PairSet$-enriched categories.

=--

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

With minimal rephrasing, a small $PairSet$-enriched category consists of the data

* A small set of objects $C$
* For objects $X,Y$, a small set $C(X,Y)$ with a subset $W(X,Y)$
* For objects $X,Y,Z$, a composition $C(Y,Z) \times C(X,Y) \to C(X,Z)$
that restricts to $W(Y,Z) \times W(X,Y) \to W(X,Z)$
* For objects $X$, a choice of identity element $id_X \in W(X,X)$

that is subject to identity and associativity relations. It's immediate that this is exactly the same data as a relative category $(C,W)$. And under this identification, enriched functors and relative functors are the same thing.

=--

### Model structures

A relative category can be equipped with a [[weak model structure]],
left or right [[semimodel structure]], or a [[model structure]].
(These are listed from less restrictive to more restrictive structures.)

These structures do not change the underlying [[(∞,1)-category]].
However, the do provide constructions to perform computations
in the underlying [[(∞,1)-category]].
Different structures yield different constructions,
but all resulting answers are weakly equivalent.

### Presentation of $(\infty,1)$-categories

The theory of relative categories presents a theory of [[(∞,1)-category|(∞,1)-categories]] in the following sense:

+-- {: .num_theorem}
###### Theorem
There exists an adjunction
$$K_\xi \dashv N_\xi : \mathbf{RelCat} \to \mathbf{ssSet}$$
such that every left [[Bousfield localization of model categories|Bousfield localization]] of the [[Reedy model structure]] on the category $\mathbf{ssSet}$ of [[bisimplicial sets]] induces a [[cofibrantly generated model category|cofibrantly-generated]] [[proper model category|left proper]] [[model category|model structure]] on $\mathbf{RelCat}$ making the adjunction a [[Quillen equivalence]]. In particular, there exists a model structure on $\mathbf{RelCat}$ that is Quillen equivalent to the [[model structure for complete Segal spaces]] on $\mathbf{ssSet}$.
=--

This is discussed in further detail at [[model structure on categories with weak equivalences]].

## Related concepts

* [[category with weak equivalences]]

* [[simplicial localization]]

* [[F-category]]

* [[M-category]]

* [[marked simplicial set]]

## References
 {#References}

The terminology "relative categories" is due to

* {#BarwickKan12} [[Clark Barwick]], [[Daniel Kan]], _Relative categories: Another model for the homotopy theory of homotopy theories_. Indagationes Mathematicae **23** (2012) 42-68 [[arXiv:1011.1691](https://arxiv.org/abs/1011.1691), [doi:10.1016/j.indag.2011.10.002](https://doi.org/10.1016/j.indag.2011.10.002)]

while the basic idea goes back, at least, to:

* {#DwyerKan1980Localizations} [[William Dwyer]], [[Daniel Kan]], _Simplicial localizations of categories_, J. Pure Appl. Algebra **17** (1980) 267-284. [[doi:10.1016/0022-4049(80)90049-3](https://doi.org/10.1016/0022-4049%2880%2990049-3)]

in the context of [[simplicial localization]] (cf. the [references](category+with+weak+equivalences#References) at "[[category with weak equivalences]]").

[[!redirects relative categories]]
[[!redirects relative functor]]
[[!redirects relative functors]]


relative cohomology

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###Context###
#### Algebraic topology
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#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
* table of contents
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## Idea

Where _[[cohomology]]_ classifies [[cocycles]] on an [[object]] $X$ with [[coefficients]] in some object $A$, _relative cohomology_ for a map morphism $Y \to X$ classifies cocycles on $X$ that satisfy some condition when pulled back to $Y$, such as being trivializable there. Or rather: cocycles _equipped_ with some [[structure]] on their pullback to $Y$, such as a trivialization.

## Definition

We first give a general abstract definition and then reduce to certain special cases.

### General definition

Recall the general abstract definition of _[[cohomology]]_, as discussed there:

for $\mathbf{H}$ an [[(∞,1)-topos]], and $X, A \in \mathbf{H}$ two [[objects]], a _cocycle on $X$ with coefficients in $A$_ is a morphism $X \to A$, the [[∞-groupoid]] of cocycles, [[coboundaries]] and higher coboundaries is the [[(∞,1)-categorical hom space]] $\mathbf{H}(X,A)$ and the _cohomology_ classes themselves are the connected components / homotopy classes in there

$$
  H(X,A) := \pi_0 \mathbf{H}(X,A)
  \,.
$$

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

Let $i : Y \to X$ and $f : B \to A$ be two [[morphisms]] in $\mathbf{H}$. Then the **relative cohomology** of $X$ with coefficients in $A$ relative to these morphisms is the connected components of the $\infty$-groupoid of relative cocycles

$$
  H_{Y}^B(X,A) := \pi_0 \mathbf{H}^I(Y \stackrel{i}{\to} X\;,\; B \stackrel{f}{\to} A)
  \,,
$$

where $\mathbf{H}^I$ the [[arrow (∞,1)-topos]] of $\mathbf{H}$, hence the [[(∞,1)-category of (∞,1)-functors]] $I \to \mathbf{H}$, where $I$ is the [[interval category]].

=--

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

Often relative cohomology is considered for the special case where $A$ is a [[pointed object]], $B = *$ is the [[terminal object]] and $B \simeq *  \to A$ is the point inclusion. In this case we may write just

$$
  H_Y(X,A) := H_Y^*(X,A)
  \,.
$$

=--

+-- {: .num_prop #AsAPullback}
###### Remark

The $\infty$-groupoid of relative cocycles is the [[(∞,1)-pullback]] in

$$
  \array{
    \mathbf{H}^I(Y \to X, B \to A)
    &\to&
    \mathbf{H}(Y, B)
    \\
    \downarrow && \downarrow^{\mathrlap{f_*}}
    \\
    \mathbf{H}(X, A) &\stackrel{i^*}{\to}& \mathbf{H}(Y,A)
  }
  \,.
$$

This makes manifest the interpretation of relative cocycles as $A$-cocycles on $X$ whose restriction to $Y$ is equipped with a coboundary to a $B$-cocycle on $Y$.

If $B$ is the point then this means: $A$ cocycles on $X$ which are equipped with a trivialization on $Y$.

=--

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

If we fix an $A$-cocycle $\mathbf{c}$ on $X$, then we may consider the relative cohomology just of this cocycle, hence the [[homotopy fiber]] of the $\infty$-groupoid of relative cocycles over $\mathbf{c}$

$$
  \array{
    \mathbf{H}^I_{\mathbf{c}}(i,f) &\to& \mathbf{H}^I(i,f)
    \\
    \downarrow && \downarrow
    \\
    * &\stackrel{\mathbf{c}}{\to}& \mathbf{H}(X,A)
  }
  \,.
$$

By the [[pasting law]] it follows that this fiber is equivalently given by the following [[(∞,1)-pullback]]

$$
  \array{
    \mathbf{H}^I_{\mathbf{c}}(i,f) &\to& \mathbf{H}(Y,B)
    \\
    \downarrow && \downarrow
    \\
    * &\stackrel{i^* \mathbf{c}}{\to}& \mathbf{H}(Y,A)
  }
  \,.
$$

Comparison shows that this identifies $\mathbf{H}^I_{\mathbf{c}}(i,f)$ as the cocycle $\infty$-groupoid of the $[i^* \mathbf{c}]$-[[twisted cohomology]] of $Y$ with coefficients in the [[homotopy fiber]] of $f$.

See for instance the example of [twisted bundles on D-branes](#TwistedBundlesOnDBranes) below.

=--

### In chain complexes

A special case of the general definition of [[cohomology]] is _[[abelian sheaf cohomology]]_, obtained by taking the coefficient objects $A$ and $B$ to be in the image of [[chain complexes]] under the [[Dold-Kan correspondence]].

If moreover we restrict attention to the case that $B = *$, then by remark \ref{AsAPullback} the relative cohomology is given by the [[homotopy fiber]] of a morphism of cochain complexes $C^\bullet(X,A) \to C^\bullet(Y,A)$, presenting the morphism $\mathbf{H}(X, A) \to \mathbf{H}(Y,A)$. Such homotopy fibers are given for instance by dual [[mapping cone]] complexes. Accordingly, in the abelian case the $A$-cohomology on $X$ relative to $Y$ is the [[cochain cohomology]] of this mapping cone complex. This is the definition one finds in much traditional literature.

## Examples

* for [[K-theory]] see _[[relative K-theory]]_ (...)

### Twisted bundles on D-branes
 {#TwistedBundlesOnDBranes}

We consider an example of relative cohomology for the general case where $B \to A$ is not the point inclusion, but exhibits an additional twist, according to \ref{RelationToTwistedCohomology}.

This example is motivated from the [[physics]] of [[D-branes]] in [[type II string theory]], as well as from [[twisted K-theory]].

Let the ambient [[(∞,1)-topos]] be $\mathbf{H} := $ [[Smooth∞Grpd]]. For any $n \in \mathbb{N}$ consider the sequence of [[Lie groups]]

$$
  U(1) \to U(n) \to PU(n)
  \,,
$$

exhibiting the [[unitary group]] as a [[central extension of groups]] of the [[projective unitary group]]. The [[fiber]] sequence on [[delooping]] [[smooth ∞-groupoids]] induced by this is

$$
  \mathbf{B}U(1) \to \mathbf{B}U(n) \to \mathbf{B} PU(n) \stackrel{f}{\to} \mathbf{B}^2 U(1)
  \,.
$$

For $Y$ a [[smooth manifold]], the $f$-[[twisted cohomology]] of $X$ is classifies $U(n)$-principal [[twisted bundles]] on $Y$, as discussed there. These appear in [[string theory]]/[[twisted K-theory]], where the twist is a [[B-field]]/[[circle n-bundle with connection|circle 2-bundle]]/[[bundle gerbe]] classified by $Y \to \mathbf{B}^2 U(1)$.

But in these applications, this 2-bundle is the restriction of an ambient 2-bundle classified by $\mathbf{c} : X \to \mathbf{B}^2 U(1)$ on a [[spacetime]] $X$ along an embedding $Y \hookrightarrow X$ of the [[D-brane]] [[worldvolume]]. Therefore the [[moduli stack]] of total field configurations consisting of the ambient [[B-field]] and the [[gauge field]] on the D-brane (see [[Freed-Witten anomaly]] for the details of this mechanism) is the $\infty$-groupoid of $(Y \hookrightarrow X)$-relative cocycles with coefficients in $(\mathbf{B}PU \to \mathbf{B}^2 U(1))$.

## Related concepts

* [[relative homology]]

* [[Gauss-Manin connection]]


relative commutant

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## Definition ##

Let $\mathcal{A} \subset \mathcal{B}$ be an inclusion of $*-$ [[associative algebra|algebra]]s. The _relative commutant_ $\mathcal{A}^c(\mathcal{B})$ is defined by

$$
\mathcal{A}^c(\mathcal{B}) := \{ B \in \mathcal{B} : B A = A B, \; A \in \mathcal{A} \}
$$

If the algebras are [[operator algebras]] defined on a [[Hilbert space]], then
$$
\mathcal{A}^c(\mathcal{B}) = \mathcal{A}' \bigcap \mathcal{B}
$$

[[!redirects relative commutants]]


relative comonad

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#### Categorical algebra
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#### 2-Category theory
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[[!include 2-category theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A _relative comonad_ is what is to a _[[relative adjunction]]_ as a [[comonad]] is to an [[adjunction]].

## Related pages

* [[relative monad]]

* [[extension system]]

## References

* [[Benedikt Ahrens]], Régis Spadotti, _Terminal semantics for codata types in intensional Martin-Löf type theory_, ([arXiv:1401.1053](https://arxiv.org/abs/1401.1053))

[[!redirects relative comonads]]


relative complement

# Relative complements
* table of contents
{: toc}

## Idea

The relative complement of $A$ in $B$ is like the [[complement]] of $A$, but relative to $B$.

## Definitions

### For subsets

Given two [[subset]]s $S$ and $T$ of a given ambient [[set]] $X$, the __relative complement__ of $S$ in $T$ is the set
$$ T \setminus S = \{ a \;|\; a \in T \;\wedge\; a \notin S \} .$$
That is, $T \setminus S$ consists of those elements of $T$ that are not elements of $S$.  This is also called the __difference__ of the sets; $T \setminus S$ may even be written $T - S$.  (Compare the [[symmetric difference]].)

Note that the relative complement of $S$ in the ambient set $X$ is simply the [[complement]] $\setminus S$ of $S$; sometimes this is called the _absolute_ complement.  Conversely, $T \setminus S$ is the [[intersection]] of $T$ and $\setminus S$.

In [[constructive mathematics]], the above definitions still holds for any two [[decidable subsets]] of a given ambient set $X$.

### In material set theory

In [[material set theory]], all sets are subsets of the [[proper class]] $V$ of all sets (or all small things, if we might have atoms).  In this case, there is no absolute complement (except as a proper class), but we can still form the relative complement:
$$ T \setminus S = \{ a \in T \;|\; a \notin S \} .$$

### In a lattice

Given a [[lattice]] $L$ and two elements $x$ and $y$ of $L$, a __relative complement__ of $x$ relative to $y$ is an element $y \setminus x$ such that:

*  $ x \wedge (y \setminus x) = \bot $ (where $\wedge$ is the [[meet]] in the lattice and $\bot$ is the [[bottom]] of the lattice) and
*  $ y \leq x \vee (y \setminus x) $ (where $\vee$ is the [[join]] in the lattice).

In a [[poset]] that is not a lattice, the same definition applies, with the existence of the relevant meet, join, and bottom bring required for the complement to exist.  In any case, complements are unique.  In a [[proset]], we may speak of *a* complement, or [[the]] complement up to equivalence.

The term _Boolean ring_ is sometimes used for something like a [[Boolean algebra]] in which there may be no [[top]] element (and therefore no complements) but still relative complements; compare the notions of _ring_ vs _algebra_ at [[sigma-algebra]].  That is, a Boolean ring is a [[bounded lattice|bounded-below]] [[distributive lattice]] in which all relative complements exist.

In [[classical mathematics]], this definition includes the previous ones; the [[power set]] of $X$ is a Boolean algebra, while the class of all sets is (in the sense above) a large Boolean ring.

### Relative pseudocomplements

Given a [[lattice]] $L$ and two elements $x$ and $y$ of $L$, a __relative pseudocomplement__ of $x$ relative to $y$ is an element $y \setminus x$ such that:

*  $ x \wedge (y \setminus x) = \bot $ and
*  given any $z$ such that $ x \wedge z = \bot $, $ y \wedge z \leq y \setminus x $.

The same issues apply about posets, uniqueness, and prosets.  A [[bounded lattice|bounded-below]] [[distributive lattice]] in which all relative pseudocomplements exist may be called a [[Heyting algebra|Heyting ring]], although I don't know if anybody does.  The set-theoretic examples are now valid even in [[constructive mathematics]] (although constructivists don't add ‘pseudo’ in that context).

If a relative complement exists, then every relative pseudocomplement is a relative complement (and vice versa), so the same notation may be used.  (Of course, some authors may intend for the notation to *imply* existence of a relative complement, so some care is still needed.)

[[!redirects relative complement]]
[[!redirects relative complements]]

[[!redirects relative pseudocomplement]]
[[!redirects relative pseudocomplements]]

[[!redirects set difference]]
[[!redirects set differences]]


relative dimension

#Contents#
* table of contents
{:toc}

## For morphisms of schemes

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

Let $f : X \to Y$ be a [[morphism]] locally of [[finite type]] between two [[schemes]] $X$ and $Y$.  The **relative dimension of $f$ at a [[point]]** $y \in Y$ is the [[dimension]] of the [[fiber]] $f^{-1}(y)$.  If all the nonempty fibers $f^{-1}(y)$ ($y \in Y$) are [[pure dimension||purely]] of the same dimension $n$, then one says that $f$ is of **relative dimension** $n$.

=--

## References

* [Stacks project, `02NI`](http://stacks.math.columbia.edu/tag/02NI)

Though the definition does not seem to appear in [[EGA]], there are some relevant propositions in (EGA, IV_2, 5.6):

* [[Alexandre Grothendieck]], [[Éléments de géométrie algébrique]], _IV. Étude locale des schémas et des morphismes de schémas, Seconde partie._ Publications Mathématiques de l'IHÉS, 24 (1965).  See (5.6.5 - 5.6.8).

[[!redirects relative dimensions]]


relative entropy

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### Context
#### Measure and probability theory
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[[!include measure theory - contents]]
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# Contents
* table of contents
{: toc}

## Idea

The notion of _relative entropy_ of [[state|states]] is a generalization of the notion of [[entropy]] to a situation where the entropy of one [[state]] is measured "relative to" another state.

It is also called

* _Kullback-Leibler divergence_,

* _information divergence_, or

* _information gain_.

## Definition

### For states on finite probability spaces
 {#OnFiniteDimensionalSpaces}

For two finite [[probability distributions]] $(p_i)$ and $(q_i)$, their **relative entropy** is

$$
  S(p/q) := \sum_{k = 1}^n p_k(log p_k - log q_k)
  \,.
$$

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

$$
  S(\rho/\phi) := tr \rho(log \rho - log \phi)
  \,.
$$

### For states on classical probability spaces

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

For $X$ a [[measurable space]] and $P$ and $Q$ two [[probability measure|probability measures]] on $X$, such that $Q$ is absolutely continuous with respect to $P$, their relative entropy is the [[integral]]

$$
  S(Q|P) = \int_X log \frac{d Q}{d P} d P
  \,,
$$

where $d Q / d P$ is the [[Radon-Nikodym derivative]] of $Q$ with respect to $P$.

=--

### For states on quantum probability spaces (von Neumann algebras)

Let $A$ be a [[von Neumann algebra]] and let $\phi$, $\psi : A \to \mathbb{C}$ be two [[state|states]] on it (faithful, positive [[linear functional|linear functionals]]).

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

The **relative entropy $S(\phi/\psi)$ of $\psi$ relative to $\phi$ is

$$
  S(\phi/\psi) := - (\Psi, (log \Delta_{\Phi,\Psi}) \Psi)
  \,,
$$

where $\Delta_{\Phi,\Psi}$ is the relative [[modular operator]] of any [[cyclic vector|cyclic]] and [[separating vector]] representatives $\Phi$ and $\Psi$ of $\phi$ and $\psi$.

=--

This is due to ([Araki](#Araki)).

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

* This definition is independent of the choice of these representatives.

* In the case that $A$ is finite dimensional and $\rho_\phi$ and $\rho_\psi$ are [[density matrices]] of $\phi$ and $\psi$, respectively, this reduces to the [above definition](#OnFiniteDimensionalSpaces).

=--

## Relation to machine learning

The [[machine learning]] process has been characterized as a minimization of relative entropy ([Ackley, Hinton and Sejnowski 1985](#AckleyHintonSejnowski)).

## Related concepts

* [[Fisher metric]]
* [[information geometry]]
* [[machine learning]]

## References

Relative entropy of [[state on a star-algebra|states]] on  [[von Neumann algebras]] was introduced in:

* {#Araki} [[Huzihiro Araki]], _Relative Entropy of States of von Neumann Algebras_, Publications of the Research Institute for Mathematical Sciences, **11** 3 (1976), 809-833 ([pdf](https://ems.press/content/serial-article-files/2833), [doi:10.2977/prims/1195191148]( https://doi.org/10.2977/prims/1195191148))

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

* {#Petz} D. Petz, _Characterization of the relative entropy of states of matrix algebras_,  Acta Mathematica Hungarica, **59** (1992), 3-4. ([doi:10.1007/bf00050907](https://doi.org/10.1007/bf00050907))

A survey of entropy in [[operator algebra|operator algebras]] is in

* Erling Størmer, _Entropy in operator algebras_ ([pdf](http://www.claymath.org/publications/currentvolumes/connes60/Stormer.pdf))

A characterization of [[machine learning]] as a process minimizing relative entropy is proposed in

* {#AckleyHintonSejnowski} David H. Ackley, Geoffrey E. Hilton, Terrence J. Sejnowski. _A learning algorithm for Boltzmann machines_, Cognitive Science, **9** (1985), 147–169. ([web](https://www.sciencedirect.com/science/article/abs/pii/S0364021385800124))

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

[[!redirects Kullback-Leibler divergence]]
[[!redirects Kullback-Leibler information]]
[[!redirects information divergence]]
[[!redirects information gain]]


relative flat modality

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#### Modalities, Closure and Reflection
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[[!include modalities - contents]]
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#Content#
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{:toc}

## Idea

In a context of [[differential cohesion]] a [[reduction modality]] exhibits an inclusion of its [[modal types]] -- the [[reduced objects]]. Essentially the corresponding inclusion of the [[anti-modal types]] is exhibited by an induced [[modal operator]], the _relative flat modality_ $\flat^{rel}$.

Where the plain [[flat modality]] $\flat$ sends any object $X$ to the type $\flat X$ of its [[global points]], the relative flat modality instead sends it to the type $\flat^{rel} X$ of all [[infinitesimal disks]] (i.e. the [[infinitesimal neighbourhoods]] of all global points) in $X$.

See also at _[[differential cohesion and idelic structure]]_.

## Definition

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

Given [[differential cohesion]],

$$
  \array{
     \Re &\dashv& \Im &\dashv& \&
     \\
     && \vee && \vee
     \\
     && ʃ &\dashv& \flat &\dashv& \sharp
  }
$$

define operations $ʃ^{rel}$ and $\flat^{rel}$ by

$$
  ʃ^{rel} X \coloneqq (ʃ X) \underset{\Re X}{\coprod} X
$$

$$
  \flat^{rel} X \coloneqq (\flat X) \underset{\Im X}{\times} X
  \,.
$$

Hence $ʃ^{rel} X$ makes a [[homotopy pushout]] square

$$
  \array{
    \Re X &\longrightarrow& X
    \\
    \downarrow && \downarrow
    \\
    ʃ X &\longrightarrow& ʃ^{rel} X
  }
$$

and $\flat^{rel}$  makes a [[homotopy pullback]] square

$$
  \array{
    \flat^{rel} X &\longrightarrow& X
    \\
    \downarrow && \downarrow
    \\
    \flat X &\longrightarrow& \Im X
  }
  \,.
$$

We call $ʃ^{rel}$ the _relative shape modality_ and $\flat^{rel}$ the _relative flat modality_.

=--

## Properties

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

The relative shape and flat modalities of def. \ref{InducedRelativeShapeAndFlat}

1. form an [[adjoint pair]] $(ʃ^{rel} \dashv \flat^{rel})$;

1. whose (co-)[[modal types]] are precisely the properly infinitesimal types, hence those for which $\flat \to \Im$ is an [[equivalence]];

1. $ʃ^{rel}$ preserves the [[terminal object]].

=--

It follows that when $\flat^{rel}$ has a further [[right adjoint]] $\sharp^{rel}$ with equivalent [[modal types]] containing the [[codiscrete object|codiscrete types]], then this defines a [[level of a topos|level]]

$$
  \array{
     \flat^{rel} &\dashv& \sharp^{rel}
     \\
     \vee && \vee
     \\
     \flat &\dashv& \sharp
     \\
     \vee && \vee
     \\
     \emptyset &\dashv& \ast
  }
$$

hence an intermediate subtopos
$\infty Grpd \hookrightarrow \mathbf{H}_{infinitesimal}\hookrightarrow \mathbf{H}_{th}$ which is [[infinitesimal cohesion|infinitesimally cohesive]].

This happens notably for the model of [[formal smooth ∞-groupoids]] and all its variants such as formal [[complex analytic ∞-groupoids]] etc. But in this case $(\flat^{rel} \dashv \sharp^{rel})$ does not provide  [[Aufhebung]] for $(\flat \dashv \sharp)$.

(...)

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

The [[counit of a comonad|counit]] of the relative flat modality is a [[formally étale morphism]].

=--

[[!redirects relative shape modality]]
[[!redirects relative sharp modality]]


relative homology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

### In singular homology

Let $X$ be a [[topological space]] and $A \hookrightarrow X$ a [[subspace]]. Write $C_\bullet(X)$ for the [[chain complex]] of [[singular homology]] on $X$ and $C_\bullet(A) \hookrightarrow C_\bullet(X)$ for the [[chain map]] induced by the subspace inclusion.

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

The [[cokernel]] of this inclusion, hence the [[quotient]] $C_\bullet(X)/C_\bullet(A)$ of $C_\bullet(X)$ by the [[image]] of $C_\bullet(A)$ under the inclusion, is the **chain complex of $A$-relative singular chains**.

* A [[boundary]] in this quotient is called an **$A$-relative singular boundary**,

* a [[cycle]] is called an **$A$-relative singular cycle**.

* The [[chain homology]] of the quotient is the **$A$-relative singular homology of $X$**

  $$
    H_n(X , A)\coloneqq H_n(C_\bullet(X)/C_\bullet(A))
    \,.
  $$

=--

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

This means that a singular $(n+1)$-chain $c \in C_{n+1}(X)$ is an $A$-relative cycle if its [[boundary]] $\partial c \in C_{n}(X)$ is, while not necessarily 0, contained in the $n$-chains of $A$: $\partial c \in C_n(A) \hookrightarrow C_n(X)$. So it vanishes only "up to contributions coming from $A$".

=--

## Properties

### Long exact sequences
 {#LongExactSequences}

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

Let $A \stackrel{i}{\hookrightarrow} X$. The corresponding
relative homology sits in a [[long exact sequence]] of the form

$$
  \cdots
  \to
  H_n(A)
   \stackrel{H_n(i)}{\to}
  H_n(X)
  \to
  H_n(X, A)
   \stackrel{\delta_{n-1}}{\to}
  H_{n-1}(A)
   \stackrel{H_{n-1}(i)}{\to}
  H_{n-1}(X)
  \to
  H_{n-1}(X, A)
  \to
  \cdots
  \,.
$$

The [[connecting homomorphism]] $\delta_{n} \colon H_{n+1}(X, A) \to H_n(A)$ sends an element $[c] \in H_{n+1}(X, A)$ represented by an $A$-relative cycle $c \in C_{n+1}(X)$, to the class represented by the [[boundary]] $\partial^X c \in C_n(A) \hookrightarrow C_n(X)$.

=--

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

This is the _[[homology long exact sequence]]_ induced by the given [[short exact sequence]] $0 \to C_\bullet(A) \stackrel{i}{\hookrightarrow} C_\bullet(X) \to coker(i)  \simeq C_\bullet(X)/C_\bullet(A)  \to 0$ of chain complexes.

=--

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

Let $B \hookrightarrow A \hookrightarrow X$ be a sequence of two inclusions. Then there is a [[long exact sequence]] of relative homology groups of the form

$$
  \cdots \to H_n(A , B) \to H_n(X , B) \to H_n(X , A ) \to H_{n-1}(A , B) \to \cdots
  \,.
$$

=--

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

Observe that we have a (degreewise) [[short exact sequence]] of chain complexes

$$
  0 \to C_\bullet(A)/C_\bullet(B) \to C_\bullet(X)/C_\bullet(B) \to C_\bullet(X)/C_\bullet(A) \to 0
  \,.
$$

The corresponding [[homology long exact sequence]] is the long exact sequence in question.

=--

### Excision
 {#Excision}

Let $Z \hookrightarrow A \hookrightarrow X$ be a sequence of [[topological subspace]] inclusions such that the [[closure]] $\bar Z$ of $Z$ is still contained in the [[interior]] $A^\circ$ of $A$: $\bar Z \hookrightarrow A^\circ$.

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

In the above situation, the inclusion $(X-Z, A-Z) \hookrightarrow (X,A)$ induces [[isomorphism]] in relative singular homology groups

$$
  H_n(X-Z, A-Z) \stackrel{\simeq}{\to} H_n(X,A)
$$

for all $n \in \mathbb{N}$.

=--

Let $A,B \hookrightarrow X$ be two [[topological subspaces]] such that their [[interior]] is a [[cover]] $A^\circ \coprod B^\circ \to X$ of $X$.

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

In the above situation, the inclusion $(B, A \cap B) \hookrightarrow (X,A)$ induces [[isomorphisms]] in relative singular homology groups

$$
  H_n(B, A \cap B) \stackrel{\simeq}{\to} H_n(X,A)
$$

for all $n \in \mathbb{N}$.

=--

A proof is spelled out in ([Hatcher, from p. 128 on](#Hatcher)).

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

These two propositions are equivalent to each other. To see this identify
$B = X - Z$.

=--

### Homotopy invariance
 {#HomotopyInvariance}

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

Relative homology is homotopy invariant in both arguments.

=--

(...)

### Relation to reduced homology of quotient topological spaces
 {#RelationToQuotientTopologicalSpaces}

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

A [[topological subspace]] inclusion $A \hookrightarrow X$ is called a **good pair** if

1. $A$ is [[closed subset|closed]] inside $X$;

1. $A$ has an [[neighbourhood]] in $X$ which is a [[deformation retract]] of $A$.

=--

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

For $X$ a [[CW complex]], the inclusion of any subcomplex $A \hookrightarrow X$ is a good pair (called a _[[CW-pair]]_ $(X,A)$).

=--

This is discussed at  _[CW complex -- Subcomplexes](CW+complex#Subcomplexes)_.

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

If $A \hookrightarrow X$ is a [[topological subspace]] inclusion which is _good_ in the sense of def. \ref{GoodPair}, then the $A$-relative singular homology of $X$ coincides with the [[reduced singular homology]] of the [[quotient space]] $X/A$:

$$
  H_n(X , A)
  \simeq
  \tilde H_n(X/A)
 \,.
$$

=--

For instance ([Hatcher, prop. 2.22](#Hatcher)).

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

By assumption we can find a [[neighbourhood]] $A \stackrel{j}{\to} U \hookrightarrow X$ such that $A \hookrightarrow U$ has a [[deformation retract]] and hence in particular is a [[homotopy equivalence]] and so induces also isomorphisms on all [[singular homology]] groups.

It follows in particular that for all $n \in \mathbb{N}$ the canonical morphism $H_n(X,A) \stackrel{H_n(id,j)}{\to} H_n(X,U)$ is an [[isomorphism]], by prop. \ref{HomotopyInvariance}.

Given such $U$ we have an evident [[commuting diagram]] of pairs of [[topological spaces]]

$$
  \array{
    (X,A) &\stackrel{(id,j)}{\to}& (X,U) &\leftarrow& (X-A, U - A)
    \\
    \downarrow && \downarrow && \downarrow^{\mathrlap{\simeq}}
    \\
    (X/A, A/A) &\stackrel{(id,j/A)}{\to}& (X/A, U/A) &\leftarrow& (X/A - A/A, U/A - A/A)
  }
  \,.
$$

Here the right vertical morphism is in fact a [[homeomorphism]].

Applying relative singular homology to this diagram yields for each $n \in \mathbb{N}$ the [[commuting diagram]] of abelian groups

$$
  \array{
    H_n(X,A) &\underoverset{\simeq}{H_n(id,j)}{\to}& H_n(X,U) &\stackrel{\simeq}{\leftarrow}& H_n(X-A, U - A)
    \\
    \downarrow && \downarrow && \downarrow^{\mathrlap{\simeq}}
    \\
    H_n(X/A, A/A) &\underoverset{\simeq}{H_n(id,j/A)}{\to}& H_n(X/A, U/A) &\stackrel{\simeq}{\leftarrow}& H_n(X/A - A/A, U/A - A/A)
  }
  \,.
$$

Here the left horizontal morphisms are the above isomorphims induced from the deformation retract. The right horizontal morphisms are isomorphisms by prop. \ref{ExcisionA} and the right vertical morphism is an isomorphism since it is induced by a homeomorphism. Hence the left vertical morphism is an isomorphism ([[2-out-of-3]] for isomorphisms).

=--

### Relation to reduced homology
 {#RelationToReducedHomology}

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

Let $X$ be a [[inhabited set|inhabited]] [[topological space]] and let $x \colon  * \hookrightarrow X$ any point. Then the relative singular homology $H_n(X , *)$ is isomorphic to the absolute [[reduced singular homology]] $\tilde H_n(X)$ of $X$

$$
  H_n(X , *) \simeq \tilde H_n(X)
  \,.
$$

=--

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

This is the special case of prop. \ref{HomologyOfQuotientSpace} for $A$ a point.

=--

## Examples

### Basic examples

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

The [[reduced singular homology]] of the $n$-[[sphere]] $S^{n}$  equals the $S^{n-1}$-relative homology of the $n$-[[disk]] with respect to the canonical [[boundary]] inclusion $S^{n-1} \hookrightarrow D^n$: for all $n \in \mathbb{N}$

$$
  \tilde H_\bullet(S^n) \simeq H_\bullet(D^n, S^{n-1})
 \,.
$$

=--

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

The $n$-[[sphere]] is [[homeomorphism|homeomorphic]] to the $n$-[[disk]] with its entire [[boundary]] identified with a point:

$$
  S^n \simeq D^n/S^{n-1}
  \,.
$$

Moreover the boundary inclusion is evidently a _good pair_ in the sense of def. \ref{GoodPair}. Therefore the example follows with prop. \ref{HomologyOfQuotientSpace}.

=--

### Detecting homology isomorphisms

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

If an inclusion $A \hookrightarrow X$ is such that all relative homology vanishes, $H_\bullet(X , A) \simeq 0$, then the inclusion induces isomorphisms on all singular homology groups.

=--

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

Under the given assumotion the long exact sequence in prop. \ref{RelativeHomologyLongExactSequence} secomposes into short exact pieces of the form

$$
  0 \to H_n(A) \to H_n(X) \to 0
  \,.
$$

Exactness says that the middle morphism here is an [[isomorphism]].

=--

### Relative homology of CW-complexes
 {#RelativeHomologyOfCWComplexes}

Let $X$ be a [[CW-complex]] and write

$$
  X_0 \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow \cdots \hookrightarrow X
$$

for its [[filtered topological space]]-structure with $X_{n+1}$ the topological space obtained from $X_n$ by gluing on $(n+1)$-cells.

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

The relative singular homology of the filtering degrees is

$$
  H_n(X_k , X_{k-1})
  \simeq
  \left\{
    \array{
       \mathbb{Z}[Cells(X)_n] & if\; k = n
       \\
       0 & otherwise
    }
  \right.
  \,,
$$

where $Cells(X)_n \in Set$ denotes the set of $n$-cells of $X$ and $\mathbb{Z}[Cells(X)_n]$ denotes the [[free abelian group]] on this set.

=--

For instance ([Hatcher, lemma 2.34](#Hatcher)).

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

The inclusion $X_{k-1} \hookrightarrow X_k$ is clearly a _good pair_ in the sense of def. \ref{GoodPair}. The quotient $X_k/X_{k-1}$ is by definition of CW-complexes a [[wedge sum]] of $k$-[[spheres]], one for each element in $kCell$. Therefore by prop. \ref{HomologyOfQuotientSpace} we have an isomorphism $H_n(X_k , X_{k-1}) \simeq \tilde H_n( X_k / X_{k-1})$ with the [[reduced homology]] of this wedge sum. The statement then follows by the respect of reduced homology for wedge sums as discussed at _[Reduced homology - Respect for wedge sums](reduced+homology#RelationToWedgeSums)_.

=--

## Related concepts

* [[relative cohomology]]

* [[cellular homology]]

* [[Gauss-Manin connection]]

## References

A standard textbook account for relative singular homology is section 2.1 of

* [[Allen Hatcher]], _[Algebraic Topology](http://www.math.cornell.edu/~hatcher/AT/ATpage.html)_

[[!redirects relative singular homology]]


relative K-theory

# Idea

_relative K-theory_ is the _[[relative cohomology]]_ version of _[[K-theory]]_.

For the moment see at _[[topological K-theory]] - [The relative K-groups](topological+K-theory#TheRelativeKGroup)_

## Related concepts

* [[reduced K-theory]]

[[!redirects relative K-group]]
[[!redirects relative K-groups]]


relative Langlands program

#Contents#
* table of contents
{:toc}

## Idea

The _relative Langlands program_ is a generalization of the [[Langlands program]] that is phrased in terms of [[spherical varieties]] instead of directly in terms of [[reductive groups]]. It is expected to recover the usual Langlands program in the group case. In more recent work, instead of focusing on a spherical variety $X$, the main focus is on a [[Hamiltonian action|Hamiltonian]] [[G-space|$G$-space]] $M$, which is often related to $X$ via its [[cotangent bundle]] $M=T^{*}X$. The tools of [[geometric quantization]] can then be used, with $M$ playing the role of the classical [[phase space]] to be quantized.

Recent ongoing work by Ben-Zvi, Sakellaridis, and Venkatesh attempt to relate it to [[topological quantum field theory]] (as inspired by both the [[Kapustin-Witten TQFT]] as well as the analogies of [[arithmetic topology]]). This is expected as well to illuminate topics in special values of [[L-functions]].

## Quantization in the relative Langlands program

The reference for this section is [#Sakellaridis2021](#Sakellaridis2021).

Let $F$ be the [[global field|global]] [[base field]] and let $G$ be a [[reductive group]]. Locally, the idea of "quantization" in the relative Langlands program refers to obtaining a [[unitary representation]] $\omega_{v}$ of $G(F_{v})$ out of the [[Hamiltonian action|Hamiltonian]] [[G-space|$G$-space]] $M$, analogous to the theory of [[geometric quantization]]. We also want a "base vector" $\Phi_{v}^{0}\in \omega_{v}$ for almost all $v$. Globally, we want to put together these local representations and have an automorphic realization $\otimes'\omega_{v}\to C^{\infty}\big(G(F)\backslash G(\mathbb{A}_{F})\big)$. An example of this is given by the [[Segal-Shale-Weil representation]] and the formation of [[theta series]].

Once one has obtained the theta series, one may investigate the following:

* The integral of the theta series against an automorphic form. This is related to periods of L-functions.

* The projection of the $L^{2}$-norm of the theta series to an automorphic representation. This is related to the relative trace formula.

## References

* [[David Ben-Zvi]], [[Yiannis Sakellaridis]], [[Akshay Venkatesh]], _Relative Langlands duality_ ([pdf](https://www.math.ias.edu/~akshay/research/BZSVpaperV1.pdf)).

* [[David Ben-Zvi]], _Relative Geometric Langlands Duality_ ([MSRI lectures](https://www.msri.org/workshops/918/schedules/28232))

* [[David Ben-Zvi]], _Relative Langlands_ ([pdf](https://www.msri.org/workshops/918/schedules/28233/documents/50487/assets/88599))

* [[Yiannis Sakellaridis]], _Spherical Varieties, Functoriality, and Quantization_ ([arXiv:2111.03004](https://arxiv.org/abs/2111.03004))

* {#Sakellaridis2021}[[Yiannis Sakellaridis]], _Hamiltonian Spaces and Periods of Automorphic Forms_ ([slides](https://math.jhu.edu/~sakellar/KAST.pdf))

* Raphael Beuzart-Plessis, _The Relative Langlands Program_ ([YouTube] (https://www.youtube.com/watch?v=J4xgPDJCymw)) ([Part 2] (https://www.youtube.com/watch?v=w4qn_WCb0bk)) ([Part 3] (https://www.youtube.com/watch?v=0rXaBG58YEs))


relative loop space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A pointed topological pair is a [[pointed topological space]] $X=(X,x_0)\in Top_*$ equipped with a [[topological subspace]] $A\subset X$ containing the [[base point]] $x_0$.

Let $I=[0,1]$ be the unit [[closed interval]] and for $N\in\mathbb{N}$ let $J^N \coloneqq \partial I^N-\{0\}\times I^{N-1}$.

\begin{definition}
  The *relative loop space* of a pointed topological pair $(X,A)$ is the [[compact-open topology|space of]] [[continuous maps]] of the form $(I^N,\partial I^N,J^N)\rightarrow (X,A,x_0)$, denoted $\Omega^N(X,A)$.
\end{definition}

For $A = \{x_0\}$ this reduces to the ordinary notion of ([[iterated loop space|iterated]]) *[[loop space]]*.

The relative loop spaces allow us to define the relative homotopy groups of topological pairs.

\begin{definition}
For $N\geq 1$ the $N$-th relative homotopy set of a topological pair $(X,A)$, denoted $\pi_N(X,A)$ is defined as the set of connected components of the associated relative loop space, ie $\pi_N(X,A)\coloneqq \pi_0\Omega^N(X,A)$.

If $N\geq 2$ then $\pi_N(X,A)$ is a group, which is abelian if $N\geq 3$.
\end{definition}

## Homotopy theory of relative loop spaces

If we are interested in the [[homotopy theory]] of relative loop spaces (as in *[[recognition of relative loop spaces]]*) the above definition is not appropriate since there is no [[model category]]-[[structure]] on the category of topological pairs, as explained in [this mathoverflow discussion](https://mathoverflow.net/questions/128976/can-one-make-the-category-of-pairs-of-topological-spaces-a-model-category).

The solution here is to work in the category $Top^\to_*$ of continuous pointed maps equipped with the [[projective model structure on functors]]. If we start with the [[classical model structure on topological spaces|Quillen model structure]] on $Top$, the [[cofibrant objects]] in $Top^\to_*$ are the inclusions of CW-pairs, and if we start with the [[mixed model structure]] we get the maps homotopy equivalent to those.

We can then define relative loop spaces as loop spaces of homotopy fibers.

\begin{definition}
  For $N\geq 1$ the relative $N$-loop space functor is the right derivable functor
$$
\Omega^N_{rel}:Top_*^\to \to Top_*, \qquad \Omega^N_{rel}(\iota:A\rightarrow X)\coloneqq (A\times_X X^I)^{\mathbb{S}^{N-1}}.
$$
\end{definition}

For inclusions of topological pairs the two definitions of relative loop spaces are [[natural transformation|naturally]] [[homeomorphism|homeomorphic]].

## Related concepts

* [[loop space]]

* [[CW-pair]]

[[!redirects relative loop spaces]]


relative monad

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

#Contents#
* table of contents
{:toc}

## Idea

A _relative_ monad $T \colon \mathbf J \to \mathbf C$ is much like a [[monad]] except that it is not an [[endofunctor]] on one [[category]], but more generally a functor between two different categories. To even formulate such a notion, (for instance the definition of the [[unit]]), the two categories have to be related somehow, typically via a specified comparison functor $J \colon \mathbf J \to \mathbf C$, in which case we say that $T$ is a monad *relative to* $J$.

Ordinary monads are then the special case of $J$-relative monads where $J$ is the [[identity functor]].

In generalization of the [relation between adjunctions and monads](monad#RelationBetweenAdjunctionsAndMonads), relative monads are related to [[relative adjunctions]].

## Definitions
 {#Definition}

Let

* $\mathbf J, \mathbf C$ be [[categories]]

* $J \,\colon\, \mathbf J \to \mathbf C$ a [[functor]].

### As skew Kleisi triples
 {#AsSkewKleisliTriples}

The following definition is a variation of the [[Kleisli triple]]-formulation of ordinary [[monads]]:

\begin{definition}
\label{def}
A **relative monad $T$ on $J$** is

* a [[functor]] $T \,\colon\, \mathbf J \to \mathbf C$

equipped with

1. a *$J$-relative [[unit of a monad|unit]]*

   $\eta_X \,\colon\, J X \to T X$

   [[natural transformation|natural]] in $X \colon \mathbf J$,

1. a *$J$-relative [[Kleisli extension]]*

   $(-)^* \colon \mathbf{C}(J X, T Y) \to \mathbf{C}(T X, T Y)$

   natural in both $X, Y \colon \mathbf J$

such that for $X, Y, Z \colon \mathbf J$ and $k \colon J X \to T Y$ the following [[equations]] hold:

1. (*left [[unitality]]*)

   $k = k^* \circ \eta_X \;,$

1. (*right [[unitality]]*)

   $\eta_X^* = 1_{T X} \;,$

1. (*[[associativity]]*)

   $(\ell^* \circ k)^* = \ell^* \circ k^*$

   for every $k \colon J X \to T Y$ and $\ell \colon J Y \to T Z$.

\end{definition}
[[ACU14, Def. 2.1](#ACU14)]

Notice that for any $f \colon X \to Y$ in $\mathbf J$, one has
\[
  T f
  \;=\;
  \big(
    \eta_{J Y} \circ J f
  \big)^*
  \,.
\]

\begin{example}
**(ordinary [[monads]] as [[Kleisli triples]]/[[extension systems]])**
\linebreak
In the special case that $\mathbf{J} = \mathbf{C}$ and $J = id$ the [[identity functor]], Def. \ref{def} reduces to the definition of ordinary [[monads]], but in the guise known as "[[Kleisli triples]]" or "[[extension systems]]" which is nominally different from (though equivalent to) the way monads are traditionally presented in [[category theory]]/[[categorical algebra]] (namely as [[monoid objects]] [[internalization|internal to]] a [[functor category|category of]] [[endofunctors]]), but is exactly the form commonly used for [[monads in computer science]].
\end{example}

### As monoids in a skew-monoidal category

The notion of a *[[skew-monoidal category]]* is like that of a [[monoidal category]] except that the [[unitors]] and [[associators]] are not necessarily [[invertible morphism|invertible]]. [[monoid|Monoids]] may be defined in a [[skew-monoidal category]] analogously as to in a [[monoidal category]].

In the general case that $\mathbf{J}$ is distinct from $\mathbf{C}$,  the [[functor category]] $Func(\mathbf J, \mathbf C)$ lacks a natural [[monoidal category]] [[structure]] (as opposed to the case of [[endofunctors]] $Func(\mathbf{C}, \mathbf{C})$) so that the usual definition of [[monads]] as [[monoids]] cannot apply --- but a suitable "skew" variant works:

The (skew-)[[composition]] of [[functors]] $F,G \colon \mathbf J \to \mathbf C$ may be defined by first [[left Kan extension|extending]] $F$ along $J$ and then composing with $G$. The resulting composition product on $[\mathbf J, \mathbf C]$ is [[coherence|coherent]] but only [[lax natural transformation|laxly]] so, hence the need to appeal to skew-monoidal categories:

\begin{theorem}
**([ACU14, Thm. 3.4](#ACU14))**
\linebreak
Suppose $J \colon \mathbf J \to \mathbf C$ is such that $\mathrm{Lan}_J \colon [\mathbf J, \mathbf C] \to [\mathbf C, \mathbf C]$ exists (e.g. if $\mathbf J$ is small and $\mathbf C$ cocomplete).
Then $[\mathbf J, \mathbf C]$ is skew-monoidal, with unit $J$ and product $F \circ^J G = (\mathrm{Lan}_J F) \circ G$, and a relative monad is a monoid in $([\mathbf J, \mathbf C], J, \circ^J)$.
\end{theorem}

When $J:\mathbf J \to \mathbf C$ is a free completion of $\mathbf{J}$ under colimits from some set $\mathcal{F}$ of indexing types, then this skew-monoidal structure on $[\mathbf J, \mathbf C]$ is properly monoidal, since it is equivalent to the $\mathcal{F}$-colimit preserving functors $\mathbf C\to\mathbf C$, and the monoidal structure is just functor composition.

### Relative to a Profunctor

The above definition makes sense even more generally when $J$ is a [[profunctor]] $\mathbf J^{op} \times \mathbf {C} \to Set$, i.e., we require

1.  a unit $\eta_X \colon J(X, T X)$, a [[natural transformation|natural]] in $X \colon \mathbf J$, that is an element of the [[end]], $\eta \in \int_{X: \mathbf J}J(X, T X)$

2.  a _[[extension system|Kleisli extension]]_ $(-)^* \colon J(X, T Y) \to \mathbf C(T X, T Y)$ natural in both $X,Y \colon \mathbf J$

with essentially the same equations. This generalizes the previous definition by defining the profunctor to be $\mathbf C(J-,=)$.

## Examples

### Generic examples

\begin{example}
\label{RelativeMonadsInducedFromActualMonads}
**(relative monads induced from actual monads)**
\linebreak
Given

* a [[functor]]

  $J \,\colon\, \mathbf{J} \to \mathbf{C}$,

* an actual$\;$ [[monad]]

  $T \,\colon\, \mathbf{C} \to \mathbf{C}$

  with

  * [[unit of a monad|unit]]$\;$  $\eta^T_{(-)}$

  * [[Kleisli extension]]$\;$ $bind^T$

the composite

* $T \circ J \,\colon\, \mathbf{J} \to \mathbf{C}$

becomes a $\mathbf{J}$-relative monad (Def. \ref{def}) with

* relative unit

  $\eta^{T J}_X \,\coloneqq\, \eta^T_{J(X)}$

* relative Kleisli extension

  $
    bind^{T J}\big( J(X) \overset{k}{\to} T \circ J(Y) \big)
      \;\coloneqq\;
    bind^T\big( J(X) \overset{k}{\to} T \circ J(Y) \big)
    \,.
  $

\end{example}
This example is stated in [ACU14, Prop. 2.3 (1)](#ACU14).
\begin{proof}
The required conditions on the relative Kleisli structure of $T \circ J$ immediately reduce to those of the actual Kleisli structure of $T$:

Given

$$
  k \,\colon\, J(X) \to T \circ J(X')
  ,\;\;\;
  \ell \,\colon\, J(X') \to T \circ J(X'')
$$

we have

left unitality:

$$
  \begin{array}{l}
    bind^{T J}(k) \circ \eta^{T J}_{X}
    \\
    \;\equiv\;
    bind^T(k) \circ \eta^T_{J(X)}
    \\
    \;=\;
    k
  \end{array}
$$

right unitality:

$$
  \begin{array}{l}
    bind^{T J}\big(
      \eta^{T J}_{X}
    \big)
    \\
    \;\equiv\;
    bind^T\big(
      \eta^T_{J (X)}
    \big)
    \\
    \;=\;
    id_{T \circ J(X)}
  \end{array}
$$

associativity:

$$
  \begin{array}{l}
    bind^{T J}\big(
      bind^{T J}(\ell)
      \circ
      k
    \big)
    \\
    \;\equiv\;
    bind^{T}\big(
      bind^{T}(\ell)
      \circ
      k
    \big)
    \\
    \;=\;
    bind^T(\ell) \circ bind^T(k)
    \\
    \;\equiv\;
    bind^{T J}(\ell) \circ bind^{T J}(k)
  \end{array}
$$

\end{proof}

A concrete instance of Exp. \ref{RelativeMonadsFromActualMonads} is spelled out in Exp. \ref{LinearSpan} below.

\begin{example}
A relative monad on the embedding $J \colon \mathbf{FinSet} \to \mathbf {Set}$ is the same thing as an [[abstract clone]]. These are equivalent to [[finitary monads]] and single-sorted [[algebraic theories]].
\end{example}

\begin{example}
Fixing a category $\mathbb{V}$ with [[finite products]], to give a [[Freyd category]] is to give a strong relative monad on the [[Yoneda embedding]] $\mathbb{V}\to [\mathbb{V}^{\mathrm{op}},\mathbf{Set}]$.
\end{example}

\begin{example}
The [[presheaf category]]-construction $(\mathbb{C}\mapsto [{\mathbb{C}}^{\mathrm{op}},\mathbf{Set}])$ may be regarded as a [[relative pseudomonad]] on the inclusion $\mathbf{Cat}\to \mathbf{CAT}$. (See also *[[Yoneda structures]]*.)
\end{example}

\begin{example}
A [[monad with arities|monad on $\mathbf C$ with arities in $\mathbf{J}\subseteq \mathbf C$]] is the same thing as a relative monad for the embedding $\mathbf{J}\to \mathbf C$. (Here $\mathbf{J}\subseteq \mathbf{C}$ is required to be a [[dense subcategory]], so that to give a functor $\mathbf{J}\subseteq \mathbf{C}$ is to give a functor $\mathbf{C}\to\mathbf{C}$ preserving $J$-absolute colimits.)
\end{example}

### Specific examples

\begin{example}\label{LinearSpan}
**([[linear span]])**
\linebreak
We spell out the simple but maybe instructive example of the construction which sends a [[set]] $B$ to the [[vector space]] which it [[linear span|spans]], i.e. to the $B$-[[indexed set|indexed]] [[direct sum]] of some [[ground field]] $\mathbb{K}$, regarded in $\mathbb{K}$-[[vector spaces]].

In detail,
for $\mathbb{K}$ any [[ground field]], consider:

1. $\mathbf{J} \coloneqq $ [[Set]];

1. $\mathbf{C} \coloneqq \int_{B \colon Set} Vect_B$ (or "[[VectBund]]", for short, see there for more) the [[category]] of [[indexed sets]] of [[vector spaces]] -- hence of [[vector bundles]] over [[sets]] (i.e. over [[discrete topological spaces]])

   $$
     \array{
       \mathscr{H}
       \\
       \big\downarrow
       \\
       B
     }
   $$

   with possibly base-changing vector bundle [[maps]] between them:

   $$
     \array{
       \mathscr{H}
       &\longrightarrow&
       \mathscr{H}'
       \\
       \big\downarrow
       &&
       \big\downarrow
       \\
       B
       &\underset{f}{\longrightarrow}&
       B'
     }
   $$

1. the relativization functor given by sending a set to the [[trivial bundle|trivial]] [[tensor unit]]-bundle over it:

   \[
     \label{FormingTrivialTensorUnitBundle}
     \array{
       J
       &\colon&
       Set
       &\longrightarrow&
       \int_{B \colon Set} Vect_B
       \\
       &&
       B
       &\mapsto&
       B \times \mathbb{K}
     }
   \]

Notice that for each [[map]] $f \;\colon\; B \to B'$ of base sets, there is a [[base change]] [[adjoint triple]] of functors

\[
  \label{BaseChangeForIndexedSetsOfVectorBundles}
  f_! \dashv f^\ast \dashv f_\ast
  \;\;\colon\;\;
  Vect_{B}
  \leftrightarrow
  Vect_{B'}
  \,.
\]

In particular, for $S' = \ast$ the [[terminal object|terminal]] [[singleton set]], the left base change along the unique $p_X \colon S \to \ast$ is the operation which forms the [[direct sum]] of the ([[fiber]]-)vector spaces in the bundle, and regards the resulting [[vector space]] as a bundle over the point:

$$
  (p_B)_!
  \;\colon\;
  Vect_B
    \longrightarrow
  Vect_\ast \,=\, Vect
  \,.
$$

In view of this, we claim that the functor

$$
  \array{
    Q
    &\colon&
    Set
    &\longrightarrow&
    \int_{B \colon Set} Vect_{B}
    \\
    &&
    S
    &\mapsto&
    (p_B)_!\big( B \times \mathbb{K} \big)
    \mathrlap{
      \;\simeq\;
      \underset{b \colon B}{\bigoplus} \mathbb{K}
    }
  }
$$

which may be understood as sending a [[set]] to its $\mathbb{K}$-[[linear span]],

carries the structure of a monad relative to the functor $J$ from (eq:FormingTrivialTensorUnitBundle) with

1. [[unit of a monad|unit]] given by

   $$
     \array{
       \eta_{B}
       &\colon&
       B \times \mathbb{K}
       &\longrightarrow&
       \underset{b \colon B}{\bigoplus} \mathbb{K}
       \\
       &&
       (b,k) &\mapsto& (k)_b
     }
   $$

1. [[extension system|Kleisli extension]] given by

   $$
     \Big(
       B \times \mathbb{K}
         \xrightarrow{f}
       \underset{
         \mathclap{b' \colon B'}
       }{\oplus} \mathbb{K}
     \Big)
     \;\;\;\;\mapsto\;\;\;\;
     \Big(
       \underset{b \colon B}{\oplus} \mathbb{K}
       \xrightarrow{
         \big(
           f(b,-)
         \big)_{b \colon B}
       }
       \underset{b' \colon B}{\oplus} \mathbb{K}
     \Big)
   $$

This specific example may be understood as a special case of the general situation of relative monads induced from an actual monad (Exmp. \ref{RelativeMonadsInducedFromActualMonads}): Here the actual monad in question is:

$$
  \array{
    \triangle
    &\colon&
    \int_{B \colon Set} Vect_B
    &\longrightarrow&
    \int_{B \colon Set} Vect_B
    \\
    &&
    \left[
      \array{
        \mathscr{V}
        \\
        \big\downarrow\mathrlap{{}^p}
        \\
        B
      }
      \;\;\;
    \right]
    &\mapsto&
    p_! \mathscr{V}
  }
$$

This monad is in fact the [[reflective localization]] of the [[reflective subcategory]]-embedding of plain [[VectorSpaces]] into bundled/parameterized vector spaces:

$$
  Vect
  \;\simeq\;
  Vect_\ast
  \xhookrightarrow{\phantom{--}}
  \int_{B \colon Set} Vect_B
  \,.
$$

\end{example}

## Related pages

* [[relative adjunction]]

* [[relative comonad]]

* [[extension system]], [[Kleisli triple]], [[monad in computer science]]

* [[relative pseudomonad]]

* [[polymonad]]

## References

The concept was introduced, in the context of [[monads in computer science]], in:

* {#ACU14} [[Thorsten Altenkirch]], [[James Chapman]], [[Tarmo Uustalu]], *Monads need not be endofunctors*, Logical Methods in Computer Science **11** 1:3 (2015) 1–40 [[arXiv:1412.7148](https://arxiv.org/abs/1412.7148), [pdf](http://www.cs.nott.ac.uk/~txa/publ/jrelmon.pdf), doi:10.2168/LMCS-11(1:3)2015]

Abstract discussion via [[virtual equipments]]:

* [[Nathanael Arkor]], [[Dylan McDermott]], _The formal theory of relative monads_, 2023. [[arXiv:2302.14014](https://arxiv.org/abs/2302.14014)]

* [[Nathanael Arkor]], [[Dylan McDermott]], _Relative monadicity_, 2023. [[arXiv:2305.10405](https://arxiv.org/abs/2305.10405)]

Exposition:

* [[Nathanael Arkor]], *Relative monads and their many guises*, talk at *[[Category Theory Octoberfest]]* [2022](https://richardblute.ca/octoberfest-2022/) [slides:[pdf](https://richardblute.files.wordpress.com/2022/10/arkor-ofest.pdf), video:[mp4](https://richardblute.files.wordpress.com/2022/10/arkor-ofest-1.mp4)]

On [[distributive laws]] for relative monads:

* [[Gabriele Lobbia]], *Distributive Laws for Relative Monads*, Applied Categorical Structures **31** 19 (2023)  [[doi:10.1007/s10485-023-09716-1](https://doi.org/10.1007/s10485-023-09716-1), [arXiv:2007.12982](https://arxiv.org/abs/2007.12982)]

[[!redirects relative monads]]


relative point of view

# The relative point of view
* tic
{: toc}

## Idea

Very generally, the __relative point of view__ on a subject given by a [[category]] $C$ replaces the consideration of properties of [[objects]] of $C$ with properties of [[morphisms]] of $C$.  This is considered a generalisation, as an object $x$ is identified with [[the]] morphism from $x$ to a [[terminal object]] of $C$.  Of course, $C$ must *have* a terminal object for this generalisation to be possible.

Often one will fix an object $y$ and concentrate on objects of $C$ __over__ $y$; these form the [[over-category]] $C/y$.  The original category $C$ may be recovered as $C/1$, where $1$ is a terminal object.

## Examples

[[Alexander Grothendieck]] championed the relative point of view in [[algebraic geometry]], replacing [[schemes]] with [[relative schemes]]; here $C$ is [[Sch]].

In _[[The Joy of Cats]]_, the authors study [[concrete categories]] (categories over [[Set]] with certain properties) from the relative point of view; here $C$ is [[Cat]].

More generally, applying the relative point of view to [[category theory]] leads to the notion of [[category over a category]], which is helpful for studying concrete categories (as above) as well as [[fibred categories]].


relative pseudomonad

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

* table of contents
{: toc}

## Idea

The notion of *relative pseudomonad* generalizes that of [[pseudomonad]], to be relative to a [[2-functor]], analogous to the notion of [[relative monad]] in the one-dimensional setting.  This enables the construction of [[Kleisli bicategories]] of pseudomonads on [[Prof]] (e.g. for defining [[generalized multicategories]]) by means of [[pseudo-distributive laws]] as if the [[presheaf category]] construction were a [[monad]] with [[Prof]] as its [[Kleisli bicategory]], even though it is not for size reasons.

An example is the [[free cocompletion]] construction on a small category.

## References

* [[Marcelo Fiore]], [[Nicola Gambino]], [[Martin Hyland]], Glynn Winskel, *Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures*, [arxiv](https://arxiv.org/abs/1612.03678)

An early observation that the [[Prof|bicategory of profunctors]] should be a [[Kleisli category]] for a [[relative monad]] is Appendice I.1(b) of:

* [[René Guitart]]. _Monades involutives complémentées_, Cahiers de topologie et géométrie différentielle catégoriques **16** 1 (1975) 17-101 [[numdam:CTGDC_1975__16_1_17_0](http://www.numdam.org/item/CTGDC_1975__16_1_17_0)]

[[!redirects relative pseudomonad]]


relative scheme

$Sch/S$ is the [[slice category]] of [[schemes]] over a fixed scheme $S$. The schemes over $S$ are morphisms of schemes $f: R\to S$, called also **relative schemes** over $S$ or __$S$-schemes__. A morphism $g: (f:R\to S)\to (f':R'\to S)$ is a morphism $g:R\to R'$ of schemes such that $f'\circ g = f$; morphisms of $S$-schemes are also called morphisms of schemes over $S$, and pictured by commutative triangles.

Every scheme can be considered a $Spec(\mathbb{Z})$-scheme. For any (commutative unital) ring $k$, a __$k$-scheme__ is a synonym for $Spec(k)$-scheme.

Grothendieck has emphasised the **[[relative point of view]]**: the emphasis of the basic theory of schemes should not be on the properties of schemes, but on the properties of morphisms.

Many definitions of local properties of schemes, can be automatically generalized to morphisms, by looking at properties of preimages of the affine covers of the base scheme. For example, a morphism $f:R\to S$ of schemes is [[quasicompact]] (or $R$ is quasicompact as an $S$-scheme) if the preimage $f^{-1}(U)$ of any affine $U\subset S$ is a [[quasicompact scheme]]. One can also talk about [[projective scheme|projective]], [[affine scheme|affine]], [[quasiprojective scheme|quasiprojective]], [[proper scheme|proper]], [[separated scheme|separated]] etc. morphisms.

Relative schemes over a general ringed topos are developed in a thesis under Grothendieck's guidance:

* Monique Hakim, _Topos annelés et schémas relatifs_, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64. Springer 1972. vi+160 pp.

category: algebraic geometry

[[!redirects S-Sch]]
[[!redirects Sch/S]]
[[!redirects S-scheme]]
[[!redirects S-schemes]]
[[!redirects relative scheme]]
[[!redirects relative schemes]]


relative scheme > history

< [[relative scheme]]


relative spectrum

## Basic construction via gluing

If $S$ is a scheme and $\mathcal{G}$ is a sheaf of commutative $\mathcal{O}_S$-algebras which are [[quasicoherent sheaf|quasicoherent]] as sheaves of $\mathcal{O}_S$-modules. For any affine $U\subset S$, the [[spectrum of a ring|spectrum]] of $\mathcal{G}(U)$ is a $U$-scheme.
By gluing these spectra over all affine $U$,
one obtains an $S$-scheme $Spec(\mathcal{G})$, the __relative spectrum__ of $\mathcal{G}$ over $S$ (also called the global spectrum of $\mathcal{G}$).

An $S$-scheme which can be obtained as a relative spectrum over $S$ is said to be a __relative affine scheme__.

## Application to vector bundles and generalizations

If $V$ is a vector space, then the symmetric algebra $Sym(V^*)$ can be interpreted as the algebra of regular functions on $V$ considered as an (affine) space. Vector bundles in algebraic geometry are usually viewed as locally free sheaves of $\mathcal{O}$-modules. However, one can construct their total spaces, using the [[symmetric algebra]] over the structure sheaf, obtaining a sheaf of algebras and then use the relative spectrum.

In more detail, let $S$ be an [[algebraic scheme]] and $\mathcal{F}$ a [[quasicoherent sheaf]] of $\mathcal{O}_S$-modules.
Then the correspondence $\mathcal{G}:V \mapsto Sym_{\mathcal{O}_S(V)}(\mathcal{F}(V))$ for affine $V$ extends to a sheaf of $\mathcal{O}_S$-algebras, hence one can apply the relative spectrum to obtain an $S$-scheme
$Spec(Sym_{\mathcal{O}_S}\mathcal{F})$ which should be viewed as the total space of the "bundle" $\mathcal{F}$ .

## Literature

* Stacks project 27.3 [Relative spectrum via glueing](https://stacks.math.columbia.edu/tag/01LL), 27.4 [Relative spectrum as a functor](https://stacks.math.columbia.edu/tag/01LQ)

category: algebraic geometry

[[!redirects relative affine scheme]]


relatively compact subspace

# Relatively compact subspaces
* table of contents
{: toc}

## Idea

Relatively compact subspaces are a purely [[topological structure|topological]] notion somewhat analogous to [[bounded subspace|bounded]] or [[totally bounded subspace|totally bounded]] subspaces of a [[metric space|metric]] or [[uniform space|uniform]] space.

## Definition

A [[subspace]] (or equivalently [[subset]]) $A$ of a [[topological space]] $X$ is **relatively compact** if its [[topological closure|closure]] in $X$ is [[compact subspace|compact]].  We can make the same definition if $X$ is a [[locale]].

A related notion is that $A$ be contained in *some* compact subspace (not necessarily its closure).  This is weaker in general but equivalent when $X$ is [[Hausdorff space|Hausdorff]].

## Properties

The relatively compact subsets of a [[topological space]] $X$ form an [[ideal]] in the [[power set]] of $X$.  That is, the [[empty subspace]] is relatively compact, the [[union]] of two relatively compact subspaces is relatively compact, and any subspace contained in relatively compact subspace is itself relatively compact.

If $X$ is [[Hausdorff space|Hausdorff]], then every [[compact subspace]] is relatively compact, and hence so is any subspace contained in a compact subspace.

Iff $X$ is [[compact space|compact]], then every subspace is relatively compact.

In a [[Cartesian space]], the relatively compact subspaces are the same as the [[bounded subspaces]].

[[!redirects relatively compact subspace]]
[[!redirects relatively compact subspaces]]

[[!redirects relatively compact subset]]
[[!redirects relatively compact subsets]]


relatively k-compact morphism in an (infinity,1)-category

# Relatively compact morphisms

* table of contents
{: toc}

## Definition

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

For $\kappa$ some [[cardinal]], say a morphism $f : x \to y$ in $C$ is **relatively $\kappa$-compact** if for all [[(∞,1)-pullbacks]] along $h : y' \to y$ to $\kappa$-[[compact object in an (∞,1)-category|compact object]]s, $y'$, the pulled back object $h^* x'$ is itself a $\kappa$-compact object.

=--

## Characterization of $(\infty,1)$-toposes

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

A [[presentable (∞,1)-category]] $C$  is an [[(∞,1)-topos]] precisely if

1. it has [[universal colimits]];

1. for sufficiently large regular [[cardinal]]s $\kappa$, $C$ has a [[(sub)object classifier in an (infinity,1)-topos|classifying object]] for relatively $\kappa$-compact morphisms.

=--

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

In [[∞Grpd]] the relatively $\kappa$-compact morphisms, $X \to Y$, def. \ref{RelativelyKappaCompact} are precisely those all whose [[homotopy fibers]]

$$
  X_{y} := X \times_{Y} \{y\}
$$

over all [[objects]] $y \in Y$ are $\kappa$-[[small infinity-groupoids]].

=--

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

We may write $Y$ as an [[(∞,1)-colimit]] over itself (see there)

$$
  Y \simeq {\lim_{\to}}_{y \in Y} \{y\}
$$

and then use the fact that [[∞Grpd]] -- being an [[(∞,1)-topos]] -- has
[[universal colimits]], to obtain the [[(∞,1)-pullback]] diagram

$$
  \array{
    {\lim_{\to}}_{y \in Y} X_y
      &\stackrel{\simeq}{\to}
     & X
    \\
    \downarrow && \downarrow
    \\
    {\lim_{\to}}_{y \in Y} \{y\}
    &\stackrel{\simeq}{\to}&
    Y
  }
$$

exhibiting $X$ as an $(\infty,1)$-colimit of $\kappa$-small objects
over $Y$. By stability of $\kappa$-compact objects under
$\kappa$-small colimits (see [here](http://ncatlab.org/nlab/show/compact+object+in+an+%28infinity%2C1%29-category#StabilityUnderColimits))
it follows that $X$ is $\kappa$-compact if $Y$ is.

=--

This is due to [[Charles Rezk]]. The statement appears as [[Higher Topos Theory|HTT, theorem 6.1.6.8]].

[[!redirects relatively k-compact morphism in an (infinity,1)-topos]]


relativistic binary

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

#Contents#
* table of contents
{:toc}

## Idea

A [[gravity|gravitationally]] [[bound state]] of two massive objects in [[general relativity]]. Inspirals of relativistic binaries are a common source of [[gravitational waves]].

## Examples

* [[binary pulsar]]

## References

### General

Survey includes

* Matthew J. Benacquista and Jonathan M. B. Downing, _[Relativistic Binaries](https://www.emis.de/journals/LRG/Articles/lrr-2013-4/articlese4.html)_, Living Reviews in Relativity

See also

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

### Details

Advanced computational techniques via gravitational [[Feynman diagrams]]:

* {#Damour17} [[Thibault Damour]], _High-energy gravitational scattering and the general relativistic two-body problem_, Phys. Rev. D 97, 044038 (2018) ([arXiv:1710.10599](https://arxiv.org/abs/1710.10599))

* {#BiniDamour18} Donato Bini, [[Thibault Damour]], _Gravitational spin-orbit coupling in binary systems at the second post-Minkowskian approximation_, Phys. Rev. D 98, 044036 (2018) ([arXiv:1805.10809](https://arxiv.org/abs/1805.10809))

Reviewed in

* {#Damour19} [[Thibault Damour]], _On the gravitational interaction of spinning bodies_, talk at _[Souriau2019](http://souriau2019.fr/?page_id=2)_

### Gravitational wave signals

On prediction of [[gravitational wave]] profiles emitted by [[black hole]] [[relativistic binaries]]:

* {#Damour10} [[Thibault Damour]], _Gravitational Waves from Coalescing Binary Black Holes: Theoretical and Experimental Challenges_, talk in Munich, 2010 ([web](http://www.theorie.physik.uni-muenchen.de/MATH/seminars/asc_kolloquium/archiv_sose10/damour/video_damour/index.html), [youtube](https://www.youtube.com/watch?v=hVy3YBalpWE))

* {#BuonannoDamour98} [[Alessandra Buonanno]], [[Thibault Damour]], _Effective one-body approach to general relativistic two-body dynamics_, Phys.Rev. D59 (1999) 084006 ([arXiv:gr-qc/9811091](http://arxiv.org/abs/gr-qc/9811091))

* {#BuonannoDamour00} [[Alessandra Buonanno]], [[Thibault Damour]], _Binary black holes coalescence: transition from adiabatic inspiral to plunge_, IX Marcel Grossmann Meeting in Rome, July 2000 ([arXiv:gr-qc/0011052](http://arxiv.org/abs/gr-qc/0011052))

* {#DamourNagar} [[Thibault Damour]], [[Alessandro Nagar]], _An improved analytical description of inspiralling and coalescing black-hole binaries_, Phys.Rev.D79:081503,2009 ([arXiv:0902.0136](http://arxiv.org/abs/0902.0136))

* {#Damour14} [[Thibault Damour]], [[Alessandro Nagar]], _A new analytic representation of the ringdown waveform of coalescing spinning black hole binaries_, 	10.1103/PhysRevD.90.024054 ([arXiv:1406.0401](http://arxiv.org/abs/1406.0401))

Review of the theoretical predictions and their experimental verification is given in

* {#Damour16} [[Thibault Damour]], _Gravitational Waves and Binary Systems_, talk at IHES, Feb 2016 ([video recording](https://www.youtube.com/watch?v=qYong_D4VUI))

[[!redirects relativistic binaries]]

[[!redirects binary star system]]
[[!redirects binary star systems]]

[[!redirects rotating binary star system]]
[[!redirects rotating binary star systems]]


relativistic particle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _relativistic particle_ in [[physics]] is a model for the dynamics of a single [[brane|particle]] that is propagating in a [[spacetime]] subject to [[force]]s such as [[gravity]] and (if it is charged) the [[electromagnetic field]].

The generalization to [[supergeometry]] is the [[superparticle]].

## Definition

The **relativistic particle** is described by the [[sigma-model]] whose

* [[target space]] is a [[spacetime]] $(X,g)$,

  where the [[pseudo-Riemannian metric]] $g$ -- or rather the [[Levi-Civita connection]] $\nabla$ that it induces on the [[tangent bundle]] $T X$ -- encodes the background field of [[gravity]] acting on the particle;

* [[worldvolume]] is the [[real line]] $\Sigma = \mathbb{R}$ or the [[circle]] $\Sigma = S^1$, the _[[worldline]]_;

* [[background gauge field]] is [[connection on a bundle|connection]] $\nabla$ on a [[circle group]]-[[principal bundle]] over $X$, encoding a field of [[electromagnetism]] acting by [[Lorentz force]] on the particle;

* [[configuration space]] is the [[quotient]]

  $$
    Conf := C^\infty(\Sigma, X)// Diff(\Sigma)
  $$

  of the space (naturally a [[diffeological space]]) of [[smooth function]]s $\Sigma \to X$ ("trajectories");

* the exponentiated [[action functional]] is for given parameters $m \in \mathbb{R}$ (the particle's [[mass]]) and $q \in \mathbb{R}$ (the particle's [[charge]])

  $$
    \exp(i S(-))
     :
   [\gamma]
       \mapsto
     \exp(
     i m \int dvol(\gamma^* g))
     \;\;
     hol(\nabla,\gamma)
     \,,
  $$

  where the first terms is the integral of the [[volume form]] of the pullback of the background metric, and where the second term is the [[holonomy]] of the [[circle n-bundle with connection|circle bundle with connection]] around $\gamma$. In the case that the underlying [[circle]]-[[principal bundle]] is trivial, so that the [[connection on a bundle|connection]] is given by a [[differential forms|1-form]] $A \in \Omega^1(X)$, the [[action functional]] is

  $$
    \begin{aligned}
     S :
     [\gamma]
      \mapsto
      &
      S_{kin}([\gamma]) + S_{gauge}([\gamma])
      \\
      & =
      \int m dvol(\gamma^* g)
      +
      \int q \gamma^* A
    \end{aligned}
      \,,
  $$

  where the first summand is the _kinetic action_ and the second the _gauge interaction_ term.

The above action functional is called the [[Nambu-Goto action]] in dimension 1. Alternatively (and mandatorily for vanishing [[mass]] parameter), the kinetic action is replaced by the corresponding [[Polyakov action]].

## Properties

### Covariant phase space

We determine the [[covariant phase space]] of the theory: the space of solutions to the [[equations of motion]] and the [[presymplectic structure]].

We assume for simplicity that the class of the background circle bundle is trivial, so that the connection is equivalently given by a 1-form $A \in \Omega^1(X)$. Write $F = d A$ for its [[curvature]] 2-form: the [[field strength]] of the [[electromagnetic field]].

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

The [[variational calculus|variation]] of the gauge interaction term is

$$
  \delta \int_\Sigma \gamma^* A
  =
  - \int_\Sigma F(\dot \gamma, \delta \gamma)
  \,.
$$

=--

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

Let $\mathbb{R}^d \stackrel{\simeq}{\to} U \hookrightarrow X$ be a local [[coordinate patch]] with coordinates $\{x^\mu\}$ and assume that $\gamma$ takes values in $U$ (or at least that its variation is supported there, which we can assume without restriction of generality).
Then the variation is given by
is

$$
  \begin{aligned}
    \delta \int_\Sigma \gamma^* A
    & =
     \delta \int_\Sigma A_\mu(\gamma) \dot \gamma^\mu d \tau
    \\
    & = \int_\Sigma
        \left(
           (\partial_{\nu} A_\mu)(\gamma) \dot \gamma^\mu
           -
           \frac{d}{d\tau} (A_\nu(\gamma))
        \right)
        \delta \gamma^\nu
        d \tau
    \\
    & = \int_\Sigma
        \left(
           (\partial_{\nu} A_\mu)(\gamma) \dot \gamma^\mu
           -
           (\partial_\mu A_\nu)(\gamma)) \dot \gamma^\mu
        \right)
        \delta \gamma^\nu
        d\tau
  \end{aligned}
  \,.
$$

=--

The variation of the kinetic terms is slightly subtle due to the square root in

$$
  dvol(\gamma^* g) = \sqrt{g(\dot \gamma, \dot \gamma)} d\tau
  \,.
$$

To deal with this, we first look at variations of trajectories in a small region where $g(\dot \gamma, \dot \gamma)$ is non-zero. For such we can always find a [[diffeomorphism]] $\Sigma \stackrel{\simeq }{\to} \Sigma$ such that this term is constantly $= 1$ in this region (recall that configurations are diffeomorphism classes of smooth [[curve]]s, so we may apply such a diffeomorphism at will to compute the variation).

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

With the above choice of diffeomorphism gauge, the equations of motion are

$$
  g(\nabla_{\dot \gamma} \dot \gamma / {\vert \dot \gamma}\vert ,-)  = \iota_{\dot \gamma} F
  \,,
$$

where $\nabla$ is the [[covariant derivative]] with respect to the [[Levi-Civita connection]] of the metric $g$.

=--

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

Computing as before in local coordinates and parameterization such that $g(\dot \gamma, \dot \gamma) = 1$, the variation of the kinetic terms is

$$
  \begin{aligned}
    \delta \int_\Sigma dvol(\gamma^* g)
     & =
     \delta \int_\Sigma \sqrt{g(\dot \gamma, \dot \gamma)} d\tau
     \\
     & =
     \int_\Sigma
      \left(
          \frac{1}{2}(\partial_\mu g_{\nu \lambda}) \dot \gamma^\nu \dot \gamma^\lambda
          -
          \frac{d}{d\tau}(g_{\mu \nu} \dot \gamma^\nu)
      \right)
      \delta \gamma^\mu
d \tau
    \\
    & =
     \int_\Sigma
      \left(
          \frac{1}{2}(\partial_\mu g_{\nu \lambda}) \dot \gamma^\nu \dot \gamma^\lambda
          -
          2 (g_{\mu\nu}\ddot \gamma^\nu +
     (\partial_\lambda g_{\mu\nu}) \dot \gamma^\nu \dot \gamma^\lambda)
      \right)
      \delta \gamma^\mu
      d \tau
   \\
    & =
    - \int_\Sigma
     g_{\mu \mu}(
      \ddot \gamma^\nu + \Gamma^\nu{}_{\alpha \beta} \dot \gamma^\alpha \dot \gamma^\beta
    )
   \delta \gamma^\mu
   \\
    & =
    - \int_\Sigma
     g_{\mu \mu}(
      \nabla_{\dot \gamma} \dot \gamma^\nu
    )
   \delta \gamma^\mu
   \\
   & = -\int_\Sigma g(\nabla_{\dot \gamma} \dot \gamma, \delta \gamma)
  \end{aligned}
  \,.
$$

(Here $\Gamma^\cdot{}_{\cdot \cdot}$ are the [[Christoffel symbols]].)

This gives the equations of motion as claimed.

=--

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

The [[norm]] of the tangent vector along a physical trajectory is preserved:

$$
  \begin{aligned}
    \frac{d}{d \tau} \sqrt{g(\dot \gamma, \dot \gamma)}
     & \propto
     2 \frac{d}{d \tau} g(\nabla_{\dot \gamma} \dot \gamma, \dot \gamma)
     \\
    & \propto
     F(\dot \gamma, \dot \gamma)
     \\
     & 0
  \end{aligned}
  \,.
$$

Therefore if the assumption $g(\dot \gamma, \dot \gamma) \neq 0$ is satisfied at one instant, is is so everywhere along the curve.

=--

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

For vanishing [[background gauge field]] strength, $F = 0$, the equations of motion

$$
  \nabla_{\dot \gamma} \dot \gamma = 0
$$

express the [[parallel transport]] of the tangent vector along a physical trajectory. This identifies these trajectories with the [[geodesics]] of $X$.

=--

## Related concepts

* [[sigma model]], [[brane]]

  * [[particle]],

    **relativistic particle**, [[non-relativistic particle]],

    [[spinning particle]], [[superparticle]]

  * [[string]], [[spinning string]], [[superstring]]

  * [[membrane]]

* [[relativistic field theory]]

## References

* Giulio Ruffini, _Four approaches to quantization of the relativistic particle_ ([arXiv:gr-qc/9806058](http://arxiv.org/abs/gr-qc/9806058))

Discussion in terms of [[BV-formalism]] includes

* Jean M. L. Fisch and [[Marc Henneaux]], _Antibracket—antifield formalism for constrained hamiltonian systems_ ([doi](http://dx.doi.org/10.1016/0370-2693%2889%2990292-X))

* [[Alberto Cattaneo]], [[Michele Schiavina]], _On time_, Lett. Math. Phys. **107** (2017) 375-408 [[doi:10.1007/s11005-016-0907-x](https://doi.org/10.1007/s11005-016-0907-x), [arXiv:1607.02412](http://arxiv.org/abs/1607.02412)]

See also

* Benjamin Koch, Enrique Muñoz, _Path integral of the relativistic point particle in Minkowski space_ ([arXiv:2012.15242](https://arxiv.org/abs/2012.15242))

[[!redirects relativistic particles]]

[[!redirects quantum relativistic particle]]
[[!redirects quantum relativistic particles]]


relativistic quantum field theory

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

#Contents#
* table of contents
{:toc}

## Idea

In the context of the [[physics]] of [[field (physics)|fields]] one speaks of _relativistic_ ([[quantum field theory|quantum]]) [[field theory]] to indicate that the underlying spaces considered are [[Lorentzian manifold|Lorentzian]] [[spacetimes]] that model configurations of the field of [[gravity]] according to the theory of "[[general relativity]]", whence the name, and that the [[equations of motion]] respect this structure, in that they respect [[tangent space]]-wise [[Lorentz group|Lorentz invariance]] (i.e. the [[Cartan geometry]] strucure, see at _[[first-order formulation of gravity]]_).

This is in contrast to other variants of [[field theory]] such as for instance

* [[non-relativistic field theory]] typically used in [[solid state physics]], where certainly effects of [[general relativity]] are irrelevant, but where even effects of [[special relativity]] may often be ignored;

* [[Euclidean field theory]] where there is no time direction at all (really: [[statistical mechanics]]), which however, in good situations, may be related to relativistic field theory by [[Wick rotation]]. In particular, this provides, under suitable conditions, a definition of _[[thermal quantum field theory]]_, see there for more.

Mathematically, the hallmark of relativistic field theory is the key role played by [[causal locality]], notably in the rigorous mathematical formulation of relativistic [[perturbative quantum field theory]] via [[causal perturbation theory]] (also called [[perturbative AQFT]]).

This marks also the contrast to "[[Euclidean field theory]]", where the underlying "[[spacetimes]]" are [[Riemannian manifolds]] (with [[Euclidean space|Euclidean]] instead of [[Lorentzian manifold|Lorentzian]] [[signature of a quadratic form|signature]], whence the name). While in good situations relativistic and Euclidean field theory may be related by [[Wick rotation]] ([[Osterwalder-Schrader theorem]]) generally they are very different.

## Related concepts

* [[relativistic particle]]

* [[causal locality]]

* [[AQFT on curved spacetimes]]

* [[locally covariant perturbative algebraic quantum field theory]]

[[!redirects relativistic quantum field theories]]

[[!redirects relativistic field theory]]
[[!redirects relativistic field theories]]


relativity

* [[special relativity]]

* [[general relativity]]


relativization in algebraic geometry

[[Grothendieck]] insisted that all of the theory of [[schemes]] should in foundational works be developed in a relative setup, that is working in the [[slice category]] $Sch/X$ of schemes over a fixed ground scheme $X$. Moreover the properties should not be studied for schemes only but for morphisms of schemes instead, where the properties are seen often at a more fundamental level.

For local properties of schemes, there is a standard way to extend them to morphisms: a morphism $f:X\to Y$ satisfies a property $P$ iff there is a [[Zariski topology|Zariski]] [[cover]] $\{U_\alpha\}_\alpha$ of $Y$ by open subschemes such that for all $\alpha$ the preimage $f^{-1}(U_\alpha)$ is a scheme satisfying property $P$.

category: algebraic geometry


relaxed multicategory

# Relaxed multicategories
* table of contents
{: toc}

## Idea

Vertex algebras are intuitively analogues of associative algebras, except that certain singularities come up (in physics terminology this comes from [[OPE]]'s).

Relaxed multicategories can be thought of, according to Borcherds, as multicategories in which the morphisms may have some sort of singularity. In the language of pseudotensor categories (a variant of coloured operads) this is studied by Beilinson and Drinfeld. In connection to quantum groups this has been studied by Soibelman.

## Sketch of a definition

Let $T$ be the free [[monoid]] [[monad]] on [[Set]] and $T^+$ the monad induced by the adjunction of the [[forgetful functor]] from $T$-multicategories to $T$-graphs. Leinster defines a particular $T^+$-multicategory (see [[generalized multicategory]]) $V$. A **relaxed multicategory** is any $V$-[[enriched multicategory|enriched]] $T$-multicategory.

## Literature

* [[Tom Leinster]], _Generalized enrichment for categories and multicategories_, [math.QA/9901139](http://arxiv.org/abs/math/9901139), chapter 4

* [[Yan Soibelman]], _Meromorphic tensor categories_, [q-alg/9709030](http://arxiv.org/abs/q-alg/9709030); _The meromorphic braided categroy arising in quantum affine algebras_, [math.QA/9901003](http://arxiv.org/abs/math/9901003)

* Craig T. Snydal, _Relaxed multi category structure of a global category of rings and modules _, [math.CT/9912075](http://arxiv.org/abs/math/9912075)
* Craig T. Snydal, _Equivalence of Borcherds G-vertex algebras and axiomatic vertex algebras _, [math.QA/9904104](http://arxiv.org/abs/math/9904104)

* [[A. A. Beilinson]], [[V. Drinfeld]], _Chiral Algebras_, AMS 2004 (a preprint in various forms since around 1995, cf. [here](http://www.math.uchicago.edu/~mitya/langlands.html)).

[[!redirects relaxed multicategory]]
[[!redirects relaxed multicategories]]


relevance logic

# Relevance logics

* table of contents
{:toc}

## Idea

**Relevance logics**, also known as **relevant logics**, are non-[[classical logics]] designed to avoid what some consider to be paradoxical about classical [[implication]].  In particular, certain hypothetical propositions in which antecedent and consequent are irrelevant to one another occur as [[tautologies]] in [[classical logic]], whereas one might consider that a claim that '$P$ implies $Q$' suggests that the truth of $P$ gives us reason to accept the truth of $Q$.

## General description

There are a number of logics that have been considered in the context of "relevance".  Many of them can, at least roughly, be regarded as [[substructural logics]] like [[linear logic]].  Unlike linear logic, most relevance logics permit the [[contraction rule]], but not the [[weakening rule]].  Thus in a sense they are the "dual" (or complement) of [[affine logic]] which permits weakening but not contraction.

Generally speaking, most relevance logics contain the following [[logical connectives]]:

* A [[logical conjunction]] and a [[logical disjunction]], usually written $\wedge$ and $\vee$.  These are *additive* connectives, like those written $\&$ and $\oplus$ in linear logic.
* An [[implication]], usually written $\to$.  This is a *multiplicative* or *intensional* connective, like the [[linear implication]] $\multimap$ of linear logic.
* A [[negation]], often written $\sim$ but sometimes $\neg$.  This is also a multiplicative/intensional connective.  Most relevance logics are "multiplicatively classical" in that the [[double negation law]] $\sim\sim A\to A$ holds.  It follows that $\wedge$ and $\vee$ satisfy all the [[de Morgan laws]] with respect to $\sim$.

Other connectives that sometimes appear in relevance logics include:

* A [[biconditional]] $A\leftrightarrow B$, usually defined as $(A\to B) \wedge (B\to A)$, combining the multiplicative implications with the additive conjunction.
* A multiplicative [[truth]], like linear logic's $\mathbf{1}$.  This can be defined as $A\to A$.
* An additive truth and an additive falsity, like linear logic's $\top$ and $\mathbf{0}$.  There does not appear to be any consensus in the relevance logic community on the notations for the nullary connectives, so on this page we will use the linear logic notations.
* A [[multiplicative conjunction]] like linear logic's $\otimes$, usually called *fusion* or *cotenability* and denoted by $\circ$ or $*$.  Assuming multiplicative classicality, this is definable from the relevant implication as $\sim (A\to\sim B)$.  It can also be characterized as an adjoint to $\to$, that is $A\circ B \to C$ if and only if $A\to (B\to C)$.
* More rarely, a [[multiplicative disjunction]] like linear logic's $\invamp$, sometimes called *fission* (but for which there does not seem to be a standard notation).  This is also classically definable from $\to$ as $\sim A \to B$.  Note that multiplicative DNE $\sim\sim A\to A$ is then equivalent to multiplicative [[excluded middle]] $A\invamp \sim A$.  The multiplicative falsity (linear logic $\bot$) seems to be rarely mentioned.
* Also rarely (and apparently controversially), an additive negation $\neg$, which might be "additively classical" in the sense that $A\vee \neg A$.
* A [[modal operator]] $\Box$ akin to [[necessity]].  The operation $\top \to A$ also sometimes behaves like necessity (note the mismatch of the additive $\top$ with the multiplicative $\to$).

This description may give the impression that relevance logics could be obtained from linear logic by adding contraction and perhaps removing some undesired connectives.  This is close to true, but many relevance logics have additional features that this would not capture.  One notable one is that the [[additive conjunction]] still [[distributive lattice|distributes]] over the [[additive disjunction]]:
$$ A \wedge (B\vee C) \leftrightarrow (A\wedge B) \vee (A\wedge C) $$
although this is not true in linear logic, and simply adding contraction does not make it true.  (It is true linearly --- and relevance-ly --- that the multiplicative conjunction distributes over additive disjunction, but adding contraction does not collapse the two conjunctions.)  It can of course be added as an [[axiom]], but it is often better to give a modified [[sequent calculus]] that makes it true automatically, by using [[bunches]].  One such sequent calculus can be found in ([Mints](#Mints)); see also [[bunched logic]] and [[display logic]].

Some relevance logics also do without contraction.  This has been found particularly helpful in making them [[paraconsistent logic|paraconsistent/nontrivial]] even in the presence of things like the [[comprehension rule]] of [[naive set theory]].

## Some relevance logics

Historically, relevance logics have usually been formulated first as [[Hilbert systems]], and only later have equivalent [[sequent calculus]] and [[natural deduction]] systems been found.

### The logic R

The logic R+ ("**positive R**") contains $\wedge,\vee,\to$ only, with contraction and the distributive law.  Fusion $\circ$ is sometimes added, since it can be characterized as an adjoint to $\to$, and likewise the nullary connectives.  This logic can be formalized by a [[sequent calculus]] with [[bunches]] on the left whose [[structural connectives]] represent $\wedge$ and $\circ$, and single formulas on the right; see ([Mints](#Mints)).

The logic R adds to R+ a multiplicative classical negation $\sim$.  It is not clear whether it has a good sequent calculus, although if we also include an additive classical negation $\neg$ (which can be shown to be a conservative extension) then it can be represented in [[display logic]].  It is also similar to "classical [[bunched implication]]" with contraction added, in which case it may be easier to omit the additive negation.   For a [[Hilbert system]], see [SEP](http://plato.stanford.edu/entries/logic-relevance/logicr.html).

## Categorical semantics

Just as linear logic (with the [[exchange rule]], but not contraction or weakening) has semantics in [[symmetric monoidal categories]], and ordinary logic has semantics in [[cartesian monoidal categories]] (both perhaps with extra structure, such as being [[closed monoidal category|closed]] or [[star-autonomous category|star-autonomous]]), affine logic has semantics in [[semicartesian monoidal categories]] while relevance logics (with contraction) have semantics in [[relevance monoidal categories]].  The monoidal product represents the [[multiplicative conjunction]]; of course, the other connectives then have to be added in an appropriately related way.

##Related concepts

* [[linear logic]]
* [[bunched logic]]
* Some relevance logics are also [[paraconsistent logic|paraconsistent]].

##References

* [Stanford Encyclopedia of Philosophy](http://plato.stanford.edu/entries/logic-relevance/)

* G. E. Mints. Cut-elimination theorem for relevant logics, Zap. Nauchn. Sem. LOMI, 1972,	Volume 32, Pages 90–97. ([math-net.ru](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=znsl&paperid=2569&option_lang=eng)).
An English translation appears in the Journal of Soviet Mathematics 6 (1976) pp.422-8. ([doi](http://doi.org/10.1007/BF01084083))
 {#Mints}

* Peter W. O'Hearn and David J. Pym. The Logic of Bunched Implications. _Bulletin of Symbolic Logic_ 5(2):215-244. ([pdf](http://www.lsv.ens-cachan.fr/~demri/OHearnPym99.pdf)) ([doi](https://dx.doi.org/10.2307%2F421090))

* Stephen Read, _Relevant Logic: A Philosophical Examination of Inference_, 2012, ([pdf](https://www.st-andrews.ac.uk/~slr/Relevant_Logic.pdf))

[[!redirects relevance logics]]
[[!redirects relevant logic]]
[[!redirects relevant logics]]