extension of distributions

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###Context###
#### Functional analysis
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[[!include functional analysis - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

Given a suitable [[subspace]] inclusion $X \hookrightarrow \hat X$ and a [[distribution]] $u$ on $X$, then an _extension_  of $u$ to $\hat X$ is a distribution $\hat u$ on $\hat X$ whose [[restriction of distributions]] to $X$ coincides with $u$. If $u$ comes from an ordinary [[smooth function]] or else if $u$ is properly understood as a [[generalized function]], then this corresponds to an ordinary [[extension]]

$$
  \array{
    X &\overset{u}{\longrightarrow}& \mathbb{R}
    \\
    \downarrow & \nearrow_{\mathrlap{\hat u}}
    \\
    \hat X
  }
$$

Regarding that we the distributions are not the maps from the underlying space we need to replace pre-composition $X\to\tilde{X}$ by the appropriate [[pullback of distributions]] always defined say if $X\to\tilde{X}$ is
a submersion; in the case of an [[open embedding]]  (see [this example](pullback+of+a+distribution#RestrictionOfDistributions)) this operation is the _restriction of distributions_, the operation dual to the operation of extension by zero of test functions.

## Definition

For an inclusion of two open sets on a manifold $U\subset V$ there is an operator of _extension by zero_ $E_{V U}: C^\infty_0(U)\to C^\infty_0(V)$ where $E_{V U}(f)(x) = f(x)$ if $x\in U$ and $E_{U V}(f)(x) = 0$ otherwise.
The __[[restriction of distributions]]__ $\rho_{U V} : \mathcal{D}'(V)\to \mathcal{D}'(U)$ is then defined by
$$
\langle \rho_{U V}(\phi), f\rangle = \langle \phi, E_{V U} f\rangle,
\,\,\,\,\,\,\,f\in C^\infty_0(U),\,\phi\in \mathcal{D}'(V).
$$

([this example](pullback+of+a+distribution#RestrictionOfDistributions)).

Now the diagram in the idea section makes sense in the following way:
for an open embedding $X\hookrightarrow\hat{X}$, $\hat{u}\in\mathcal{D}'(\hat{X})$ __extends__ $u\in \mathcal{D}'(X)$ if $\rho_{X\hat{X}}(\hat{u}) = u$.

+-- {: .num_defn #ExtensionOfDistributions}
###### Definition
**([[extension of distributions]])**

Let $X \overset{\iota}{\subset} \hat X$ be an inclusion of [[open subsets]] of some [[Cartesian space]]. This induces the operation of [[restriction of distributions]]

$$
  \mathcal{D}'(\hat X)
    \overset{\iota^\ast}{\longrightarrow}
  \mathcal{D}'(X)
  \,.
$$

Given a [[distribution]] $u \in \mathcal{D}'(X)$, then an _[[extension]]_ of $u$ to $\hat X$ is a distribution $\hat u \in \mathcal{D}'(\hat X)$ such that

$$
  \iota^\ast \hat u
  \;=\;
  u
  \,.
$$

=--

## Properties

### Point-extensions
 {#SolutionSpaceOfPointExtensions}

Of particular interest is the special case of [[extension of distributions]] to a single point $p \in \hat X$, hence where $X = \hat X \setminus \{p\}$ is the [[complement]] of that point.

Contrary to ordinary [[smooth functions]], distributions ([[generalized functions]]) in general have more than one extension to a point, where the freedom is in choosing a [[distribution with point support]] at that point, hence some [[derivative of a distribution|derivative]] of the [[delta distribution]] [[support of a distribution|supported]] at that point.

However, if one requires that the [[scaling degree of a distribution|scaling degree]] of the extended distribution at the given point is compatible with that of the original distribution, then there is only a [[finite set]] of possible extensions, parameterized by the [[coefficients]] of a finite number of derivatives of the delta-distribution (prop. \ref{SpaceOfPointExtensions} below).

In the construction of [[perturbative quantum field theories]] via [[causal perturbation theory]], where the ("[[operator-valued distributions|operator-valued]]") distributions in questions are [[time-ordered product]] coefficients in the [[scattering matrix]] and where the point being extended to corresponds to the point where an [[interaction]] happens, this finite set of choices is identified with the [[renormalization|("re"-)normalization]] freedom.

We discuss specifically the space of solutions of extending a distribution on the [[complement]] $\mathbb{R}^n \setminus \{0\}$ of the origin inside a [[Cartesian space]] to the full space $\mathbb{R}^n$.

+-- {: .num_prop #ExtensionUniqueNonPositiveDegreeOfDivergence}
###### Proposition
**(unique [[extension of distributions]] with negative [[degree of divergence of a distribution|degree of divergence]])**

For $n \in \mathbb{N}$, let $u \in \mathcal{D}'(\mathbb{R}^n \setminus \{0\})$ be a [[distribution]] on the [[complement]] of the origin, with [[negative number|negative]] [[degree of divergence of a distribution|degree of divergence]] at the origin

$$
  deg(u) \lt 0
  \,.
$$

Then $u$ has a _unique_ [[extension of distributions]] $\hat u \in \mathcal{D}'(\mathbb{R}^n)$ to the origin with the same degree of divergence

$$
  deg(\hat u) = deg(u)
  \,.
$$

=--

([Brunetti-Fredenhagen 00, theorem 5.2](#BrunettiFredenhagen00), [Dütsch  18, theorem 3.35 a)](#Duetsch18))

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

Regarding uniqueness:

Suppose $\hat u$ and ${\hat u}^\prime$ are two extensions of $u$ with $deg(\hat u) = deg({\hat u}^\prime)$. Both being extensions of a distribution defined on $\mathbb{R}^n \setminus \{0\}$, this difference has [[support of a distribution|support]] at the origin $\{0\} \subset \mathbb{R}^n$. By [this prop.](point-supported+distribution#PointSupportedDistributionsAreSumsOfDerivativesOfDeltaDistibutions) this implies that it is a linear combination of [[derivative of a distribution|derivatives]] of the [[delta distribution]] [[support of a distribution|supported]] at the origin:

$$
  {\hat u}^\prime - \hat u
  \;=\;
  \underset{
    {\alpha \in \mathbb{N}^n}
  }{\sum}
  c^\alpha \partial_\alpha \delta_0
$$

for constants $c^\alpha \in \mathbb{C}$. But by [this example](scaling+degree+of+a+distribution#DerivativesOfDeltaDistributionScalingDegree) the [[degree of divergence of a distribution|degree of divergence]] of these [[point-supported distributions]] is non-negative

$$
  deg( \partial_\alpha \delta_0)
  =
  {\vert \alpha\vert} \geq 0
  \,.
$$

This implies that $c^\alpha = 0$ for all $\alpha$, hence that the two extensions coincide.

Regarding existence:

Let

$$
  b \in C^\infty_{cp}(\mathbb{R}^n)
$$

be a [[bump function]] which is $\leq 1$ and [[constant function|constant]] on 1 over a [[neighbourhood]] of the origin. Write

$$
  \chi \coloneqq 1 - b \;\in\; C^\infty(\mathbb{R}^n)
$$

> graphics grabbed from [Dütsch 18, p. 108](#Duetsch18)

and for $\lambda \in (0,\infty)$ a [[positive real number]], write

$$
  \chi_\lambda(x)
  \coloneqq
  \chi(\lambda x)
  \,.
$$

Since the [[product of distributions|product]] $\chi_\lambda u$ has [[support of a distribution]] on a [[complement]] of  a [[neighbourhood]] of the origin, we may extend it by zero to a distribution on all of $\mathbb{R}^n$, which we will denote by the same symbols:

$$
  \chi_\lambda u \in \mathcal{D}'(\mathbb{R}^n)
  \,.
$$

By construction $\chi_\lambda u$ coincides with $u$ away from a neighbourhood of the origin, which moreover becomes arbitrarily small as $\lambda$ increases.  This means that if the following [[limit of a sequence|limit]] exists

$$
  \hat u
  \;\coloneqq\;
  \underset{\lambda \to \infty}{\lim}
   \chi_\lambda u
$$

then it is an extension of $u$.

To see that the limit exists, it is sufficient to observe that we have a [[Cauchy sequence]], hence that for all $b\in C^\infty_{cp}(\mathbb{R}^n)$ the difference

$$
  (\chi_{n+1} u - \chi_n u)(b)
  \;=\;
  u(b)( \chi_{n+1} + \chi_n )
$$

becomes arbitrarily small.

It remains to see that the unique extension $\hat u$ thus established has the same scaling degree as $u$. This is shown in ([Brunetti-Fredenhagen 00, p. 24](#BrunettiFredenhagen00)).

=--

+-- {: .num_prop #SpaceOfPointExtensions}
###### Proposition
**(space of [[point-extensions of distributions]])

For $n \in \mathbb{N}$, let $u \in \mathcal{D}'(\mathbb{R}^n \setminus \{0\})$ be a [[distribution]] of [[scaling degree of a distribution|degree of divergence]] $deg(u) \lt \infty$.

Then $u$ does admit at least one [[extension of distributions|extension]] (def. \ref{ExtensionOfDistributions}) to a distribution $\hat u \in \mathcal{D}'(\mathbb{R}^n)$, and every choice of extension has the same [[degree of divergence of a distribution|degree of divergence]] as $u$

$$
  deg(\hat u) = deg(u)
  \,.
$$

Moreover, any two such extensions $\hat u$ and ${\hat u}^\prime$ differ by a linear combination of [[partial derivatives|partial]] [[derivatives of distributions]] of order $\leq deg(u)$ of the [[delta distribution]] $\delta_0$ [[support of a distribution|supported]] at the origin:

$$
  {\hat u}^\prime
  -
  \hat u
  \;=\;
  \underset{
    { \alpha \in \mathbb{N}^n }
    \atop
    { {\vert \alpha\vert} \leq deg(u) }
  }{\sum}
  q^\alpha
  \partial_\alpha \delta_0
  \,,
$$

for a finite number of constants $q^\alpha \in \mathbb{C}$.

=--

This is essentially ([Hörmander 90, thm. 3.2.4](#Hoermander90)). We follow ([Brunetti-Fredenhagen 00, theorem 5.3](#BrunettiFredenhagen00)), which was inspired by ([Epstein-Glaser 73, section 5](#EpsteinGlaser73)). Review of this approach is in ([Dütsch 18, theorem 3.35 (b)](#Duetsch18)), see also remark \ref{WExtensions} below.

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

For $f \in C^\infty(\mathbb{R}^n)$ a [[smooth function]], and $\rho \in \mathbb{N}$, we say that _$f$ vanishes to order $\rho$_ at the origin if all [[partial derivatives]] with multi-index $\alpha \in \mathbb{N}^n$ of total order ${\vert \alpha\vert} \leq \rho$ vanish at the origin:

$$
  \partial_\alpha f (0) = 0
  \phantom{AAA}
  {\vert \alpha\vert} \leq \rho
  \,.
$$

By [[Hadamard's lemma]], such a function may be written in the form

$$
  \label{ForVanishingOrderRhoHadamardExpansion}
  f(x)
  \;=\;
  \underset{
    {\alpha \in \mathbb{N}^n}
    \atop
    { {\vert \alpha \vert} = \rho + 1 }
  }{\sum}
  x^\alpha
  r_\alpha(x)
$$

for [[smooth functions]] $r_\alpha \in C^\infty_{cp}(\mathbb{R}^n)$.

Write

$$
  \mathcal{D}_\rho(\mathbb{R}^n)
    \hookrightarrow
  \mathcal{D}(\mathbb{R}^n)
    \coloneqq
  C^\infty_{cp}(\mathbb{R}^n)
$$

for the subspace of that of all [[bump functions]] on those that vanish to order $\rho$ at the origin.

By definition this is equivalently the joint [[kernel]] of the [[partial derivative|partial]] [[derivatives of distributions]] of order ${\vert \alpha\vert}$ of the [[delta distribution]] $\delta_0$ [[support of a distribution|supported]] at the origin:

$$
  b \in \mathcal{D}_\rho(\mathbb{R}^n)
  \phantom{AA}
  \Leftrightarrow
  \phantom{AA}
  \underset{
    {\alpha \in \mathbb{N}^n}
      \atop
    { {\vert \alpha\vert} \leq \rho }
  }
  {\forall}
    \left\langle \partial_\alpha \delta_0, b \right\rangle
  = 0
  \,.
$$

Therefore every [[continuous linear map|continuous linear]] [[projection]]

$$
  p_\rho
   \;\colon\;
  \mathcal{D}(\mathbb{R}^n)
    \longrightarrow
  \mathcal{D}_\rho(\mathbb{R}^n)
$$

may be obtained from a choice of _dual basis_ to the $\{\partial_\alpha \delta_0\}$, hence a choice of smooth functions

$$
  \left\{
    w^\beta \in C^\infty_{cp}(\mathbb{R}^n)
  \right\}_{ { \beta \in \mathbb{N}^n } \atop { {\vert \beta\vert} \leq \rho } }
$$

such that

$$
  \left\langle
    \partial_\alpha \delta_0
    \,,\,
    w^\beta
  \right\rangle
  \;=\;
  \delta_\alpha^\beta
  \phantom{AAA}
  \Leftrightarrow
  \phantom{AAA}
  \partial_\alpha w^\beta(0)
    \;=\;
  \delta_\alpha^\beta
  \phantom{AAAA}
  \text{for}\, {\vert \alpha\vert} \leq \rho
  \,,
$$

by setting

$$
  \label{SpaceOfSmoothFunctionsOfGivenVaishingOrderProjector}
  p_\rho
  \;\coloneqq\;
  id
  \;-\;
  \left\langle
  \underset{
    {\alpha \in \mathbb{N}^n}
    \atop
    { {\vert \alpha\vert} \leq \rho }
  }{\sum}
  w^\alpha \partial_\alpha \delta_0
  \,,\,
  (-)
  \right\rangle
  \,,
$$

hence

$$
  p_\rho
  \;\colon\;
  b
    \mapsto
  b
    -
  \underset{
    {\alpha \in \mathbb{N}^n}
    \atop
    { {\vert \alpha\vert} \leq \rho }
  }{\sum}
  (-1)^{{\vert \alpha\vert}} w^\alpha \partial_\alpha b(0)
  \,.
$$

Together with [[Hadamard's lemma]] in the form (eq:ForVanishingOrderRhoHadamardExpansion) this means that every $b \in \mathcal{D}(\mathbb{R}^n)$ is decomposed as

$$
  \label{ForExtensionOfDistributionsTestFunctionDecomposition}
  \begin{aligned}
  b(x)
  & =
  p_\rho(b)(x)
  \;+\;
  (id - p_\rho)(b)(x)
  \\
  & =
  \underset{
    { \alpha \in \mathbb{N}^n }
    \atop
    { {\vert \alpha \vert} = \rho + 1 }
  }{\sum}
  x^\alpha r_\alpha(x)
  \;+\;
  \underset{
    { \alpha \in \mathbb{N}^n }
    \atop
    { {\vert \alpha \vert} \leq \rho }
  }{\sum}
  (-1)^{{\vert \alpha \vert}}
  w^\alpha \partial_\alpha b(0)
  \end{aligned}
$$

Now let

$$
  \rho
  \;\coloneqq\;
  deg(u)
  \,.
$$

Observe that (by [this prop.](scaling+degree+of+a+distribution#ScalingDegreeOfDistributionsBasicProperties)) the [[degree of divergence of a distribution|degree of divergence]] of the [[product of distributions]] $x^\alpha u$ with ${\vert \alpha\vert} = \rho + 1$ is [[negative number|negative]]

$$
  \begin{aligned}
    deg\left(
      x^\alpha u
    \right)
    & =
    \rho - {\vert \alpha \vert}
    \leq
    -1
 \end{aligned}
$$

Therefore prop. \ref{ExtensionUniqueNonPositiveDegreeOfDivergence} says that each $x^\alpha u$ for ${\vert \alpha\vert} = \rho + 1$ has a unique extension $\widehat{ x^\alpha u}$ to the origin. Accordingly the composition $u \circ p_\rho$ has a unique extension, by (eq:ForExtensionOfDistributionsTestFunctionDecomposition):

$$
  \begin{aligned}
  \left\langle
    \hat u
    \,,\,
    b
  \right\rangle
  & =
  \left\langle
    \hat u
    ,
    p_\rho(b)
  \right\rangle
  +
  \left\langle
    \hat u
    ,
    (id - p_\rho)(b)
  \right\rangle
  \\
  & =
  \underset{
    { \alpha \in \mathbb{N}^n }
    \atop
    { {\vert \alpha \vert} = \rho + 1 }
  }{\sum}
  \underset{
    \text{unique}
  }{
  \underbrace{
    \left\langle
     \widehat{x^\alpha u}
      \,,\,
      r_\alpha
    \right\rangle
  }
  }
  \;+\;
  \underset{
    { \alpha \in \mathbb{N}^n }
    \atop
    { {\vert \alpha\vert} \leq \rho }
  }{\sum}
  \underset{
    \text{choice}
  }{
  \underbrace{
  \langle
    \hat u
    \,,\,
    w^\alpha
  \rangle
  }
  }
  \left\langle
    \partial_\alpha \delta_0
    \,,\,
    b
  \right\rangle
  \end{aligned}
$$

That says that $\hat u$ is of the form

$$
  \hat u
  \;=\;
  \underset{
    \text{unique}
  }{
  \underbrace{
    \widehat{
      u \circ p_\rho
    }
  }
  }
  +
  \underset{
    { \alpha \in \mathbb{N}^n  }
    \atop
    { {\vert \alpha\vert} \leq \rho }
  }{\sum}
  c^\alpha
  \,
  \partial_\alpha \delta_0
$$

for a finite number of constants $c^\alpha \in \mathbb{C}$.

Notice that for any extension $\hat u$ the exact value of the $c^\alpha$ here depends on the arbitrary choice of dual basis $\{w^\alpha\}$ used for this construction. But the uniqueness of the first summand means that for any two choices of extensions $\hat u$ and ${\hat u}^\prime$, their difference is of the form

$$
  {\hat u}^\prime - \hat u
  \;=\;
  \underset{
    { \alpha \in \mathbb{N}^n  }
    \atop
    { {\vert \alpha\vert} \leq \rho }
  }{\sum}
  ( (c')^\alpha - c^\alpha )
  \,
  \partial_\alpha \delta_0
  \,,
$$

where the constants $q^\alpha \coloneqq ( (c')^\alpha - c^\alpha ) \in \mathbb{C}$ are independent of any choices.

It remains to see that all these $\hat u$ in fact have the same degree of divergence as $u$.

By [this example](scaling+degree+of+a+distribution#DerivativesOfDeltaDistributionScalingDegree) the degree of divergence of the point-supported distributions on the right is $deg(\partial_\alpha \delta_0) = {\vert \alpha\vert} \leq \rho$.

Therefore to conclude it is now sufficient to show that

$$
  deg\left(
    \widehat{
      u \circ p_\rho
    }
  \right)
  \;=\;
  \rho
  \,.
$$

This is shown in ([Brunetti-Fredenhagen 00, p. 25](#BrunettiFredenhagen00)).

=--

+-- {: .num_remark #WExtensions}
###### Remark
**("W-extensions")**

Since in [Brunetti-Fredenhagen 00, (38)](#BrunettiFredenhagen00) the projectors (eq:SpaceOfSmoothFunctionsOfGivenVaishingOrderProjector) are denoted "$W$", the construction of [[extensions of distributions]] via the proof of prop. \ref{SpaceOfPointExtensions} has come to be called "W-extensions" (e.g [Dütsch 18](#Duetsch18)).

=--

## Examples

### Powers of Feynman propagators

Extension of powers of [[Feynman propagators]] on [[globally hyperbolic spacetimes]] to the [[diagonal]] are worked out explicitly in ([Hollands 07, section 3.4](#Hollands07)).

## References

The argument for the characterization of the point extension of distributions goes back to

* {#EpsteinGlaser73} [[Henri Epstein]] and [[Vladimir Glaser]], section 5 of _[[The role of locality in perturbation theory]]_, Annales Poincaré Phys. Theor. A 19 (1973) 211 ([Numdam](http://www.numdam.org/item?id=AIHPA_1973__19_3_211_0 ))

thereby laying the foundation for [[causal perturbation theory]]. A textbook account in [[functional analysis]] is in

* {#Hoermander14} [[Lars Hörmander]], theorem 3.2.4 of _The Analysis of Linear Partial Differential Operators I_ (Springer, 1990, 2nd ed.)

A more concise formulation and proof is due to

* {#BrunettiFredenhagen00} [[Romeo Brunetti]], [[Klaus Fredenhagen]], section 5.2 of _Microlocal analysis and interacting quantum field theories: Renormalization on Physical Backgrounds_, Commun. Math. Phys. 208 : 623-661, 2000 ([math-ph/9903028](https://arxiv.org/abs/math-ph/9903028))

reviewed in

* {#Duetsch18} [[Michael Dütsch]], theorem 3.35 of _[[From classical field theory to perturbative quantum field theory]]_, 2018

Exposition of the application to [[renormalization]] of [[Feynman diagrams]] is in

* {#Brouder10} [[Christian Brouder]], _Multiplication of distributions_, 2010 ([[BrouderProductOfDistributions.pdf:file]])

The refinement to the point-extension problem for distributions in the solution space of a given system of [[differential equations]] is discussed in

* {#BahnsWrochna12} Dorothea Bahns, Michał Wrochna, _On-shell extension of distributions_, [arXiv:1210.5448](https://arxiv.org/abs/1210.5448)

Examples and applications to [[renormalization]] in [[perturbative quantum field theory]] are discussed for instance in

* {#Hollands07} [[Stefan Hollands]], sections 3.3 and 3.4 of _Renormalized Quantum Yang-Mills Fields in Curved Spacetime_, Rev. Math. Phys. 20:1033-1172, 2008 ([arXiv:0705.3340](https://arxiv.org/abs/0705.3340))

For more on extension of distributions in [[renormalization]] see the references at _[[causal perturbation theory]]_ and _[[locally covariant perturbative quantum field theory]]_.

[[!redirects extensions of distributions]]
[[!redirects extension of a distribution]]

[[!redirects point-extension of a distribution]]
[[!redirects point-extensions of distributions]]

[[!redirects point-extension of distributions]]
[[!redirects point-extensions of distributions]]


extension of scalars

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

#Contents#
* table of contents
{:toc}

## Idea

The _extension of scalars_ of a [[module]] along a [[homomorphism]] of [[rings]] is the [[Isbell duality|algebraic dual]] of what geometrically is the [[pullback]] of [[bundles]] along a map of their base spaces (with respect to the discussion at _[modules - as generalized vector bundles](http://ncatlab.org/nlab/show/module#RelationToVectorBundlesInIntroduction)_).

Explicitly, extension of scalars along a [[ring]] [[homomorphism]] $f : R \to S$ is the operation on $R$-[[modules]] given by forming the [[tensor product of modules]] with $S$ regarded as an $R$-module via $f$.

There are similar functors for  [[bimodules]] and in some other categories.

## Definition

Let $R$ and $S$ be [[commutative rings]] and let
$f \colon R\to S$
be a [[homomorphism]] of [[rings]].

We discuss _extension of scalars_ along $f$ first [[category theory|general abstractly]] and then explicitly in components.

### General abstract

Write $R$[[Mod]] and $S$[[Mod]] for the [[categories of modules]] over $R$ and $S$, respectively.

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

Given a [[ring]] [[homomorphism]] $f : R \to S$ the **[[restriction of scalars]]** functor

$$
  f^* : S Mod \to R Mod
$$

is the [[functor]] that takes an $S$-[[module]] $N$ to the $R$-module $f^*N$ whose underlying [[abelian group]] is that of $N$ and whose $R$-[[action]] is given by

$$
  r \cdot n \coloneqq f(r)\cdot n \;\;\;\; for \; r \in R, n \in N
  \,.
$$

=--

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

The [[restriction of scalars]] functor, def. \ref{RestrictionOfScalars}, is the [[right adjoint]] in a pair of [[adjoint functors]]

$$
  ( f_! \dashv f^* )
  \;\colon\;
  S Mod
   \stackrel{\overset{f_!}{\longleftarrow}}
   {\underset{f^*}{\longrightarrow}}
  R Mod
  \,.
$$

=--

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

The [[left adjoint]] $f_! \colon R Mod \to S Mod$ in prop. \ref{AdjointPair} is called **extension of scalars** along $f$.

=--

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

A further [[right adjoint]] $f_*$ would be called _[[coextension of scalars]]_ along $f$.

=--

See also [[induced representation]] $\dashv$ [[restricted representation]].

### In components

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

Given a [[ring]] [[homomorphism]] $f : R \to S$, the _extension of scalars_ [[functor]] $f_!$ of def. \ref{ExtensionByAdjoint} is the functor

$$
  f_! \coloneqq  S \otimes_R (-) \,:\, R Mod \to S Mod
$$

given by [[tensor product of modules]] with $S$ regarded as an $S$-$R$-[[bimodule]]: the left [[action]] being the canonical action of $S$ on itself, the right being the [[restriction of scalars]]-action along $f$.

Explicitly, for $N \in R Mod$

* the elements of $f_! N$ are [[equivalence classes]] of pairs $(s,n) \in S \times N$ under the [[equivalence relation]] $ (s \cdot f(r), n) = (s, r\cdot n) $ for all $s \in S$;

* the left $S$-[[action]] is given by $s' \cdot(s,n) = (s' \cdot s,n)$.

=--

## Properties

### Geometric interpretation

Under [[Isbell duality]] extension of scalars turns into a statement about [[geometry]].

By definition the category

$$
  Ring^{op} \underoverset{Spec}{\colon \simeq}{\to}
  Aff
$$

of (absolute) [[affine schemes]] is the [[opposite category]] of [[Ring]].

Hence for $f : R \to S$ a [[ring]] [[homomorphism]], we have equivalently a morphism

$$
  Spec(f) : Spec(S) \to Spec(R)
$$

of affine schemes.

An $R$-[[module]] $N$ corresponds to the collection of [[sections]] of a "generalized [[vector bundle]]" over $Spec(R)$: something that has a [[quasicoherent sheaf]] of sections.

The [[pullback]] of this "bundle" along $Spec(f)$ has sections forming the module $f_! N$.

Generally, for any [[fibered category]] like [[Mod]]$\to Aff$ we may regard the [[inverse image functor]] as the extension of scalars.

For that reason if there is some other fibered category $\mathcal{F}$ over the opposite of some algebraic category $\mathcal{A}$ whose objects are considered "objects of scalars" one is inclined to call the inverse image functor, the extension of scalars.

### Frobenius extensions

Extension of scalars coincides with *[[coextension of scalars]]* (to make an [[ambidextrous adjunction]] with [[restriction of scalars]]) in the case of [[Frobenius extensions]], see there for more.

## Examples

* [[complexification]] is extension of scalars along the inclusion $\mathbb{R} \hookrightarrow \mathbb{C}$ of the [[real numbers]] into the [[complex numbers]].

* [[localization of a module]]

## Related concepts

* **extension of scalars** $\dashv$ [[restriction of scalars]] $\dashv$ [[coextension of scalars]]


extension system

> This page is about an alternative presentation of [[monads]] as popular for [[monads in computer science]].  For a different notion of "extension system" (that is to a [[bicategory]] what a [[closed category]] is to a [[monoidal category]]) see instead at *[[closed category]]*.

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

# Extension systems
* table of contents
{: toc}

## Idea
 {#Idea}

The notion of "*extension system*" ([Marmolejo & Wood (2010)](#MarmolejoWood10)), originally called "*[[algebraic theory]] in extension form*" ([Manes (1976), p. 32](#Manes76)) and previously also referred to as "*[[Kleisli triple]]*" ([Moggi (1991), Def. 1.2](#Moggi91)) is an equivalent way of presenting the [[mathematical structure|structure]] of a *[[monad]]* (on a [[category]]) that does not explicitly refer to [[composition]] ("iteration") of its [[underlying]] [[endofunctor]]. This is simpler for certain purposes, and in any case more natural for others, notably in the use of [[monads in computer science]]. Abstractly, the notion may be understood as specialising the definition of $J$-[[relative monads]] to the case where $J$ is the [[identity functor]]

Specifically, noticing that the [[endofunctor]] [[underlying]] a [[monad]] may be understood as constructing its [free algebras](algebra+over+a+monad#FreeAlgebras), which (lacking [[relations]]) tend to be "large" (say as concerns the [[cardinality]] of their [[underlying sets]]), the iterated application of this endofunctor produces ever larger objects, and some authors have pointed to avoiding this phenomenon as motivation for considering extension systems:

> {#MarmolejoWoodQuote} [[Marmolejo-Wood 10](#MarmolejoWood10)]: there is an important overarching reason to consider monads in this way. Extension systems allow us to completely dispense with the iterates $[$...$]$ of the underlying arrow. No iteration is necessary. A moment's reflection on the various terms of terms and terms of terms of terms that occur in practical applications suggest that this alone justifies the alternate approach. $[$...$]$ we note that extension systems in [[higher category theory|higher dimensional category theory]] provide an even more important simplication of monads. For even in dimension 2, some of the tamest examples are built on [[pseudofunctors]] that are difficult to iterate.

Major alternative motivation for extension systems comes from their understanding as [[monads in computer science]] ([[Kleisli triples]]); see there for more.

## Definition

An **extension system** ([Marmolejo-Wood 2010](#MarmolejoWood10)) on a [[category]] $C$ consists of

* For every [[object]] $A\in C$, an object $T A\in C$ and a [[morphism]] $\eta_A \colon A\to T A$, and

* For every [[morphism]] $f\colon B\to T A$ in $C$, a morphism $f^T \colon T B \to T A$ (the *Kleisli extension* of $f$), satisfying the following [[axioms]]:

  * For every $A$ we have $(\eta_A)^T = 1_{T A}$,

  * For every $f \colon B\to T A$, we have $f^T \circ \eta_B = f$, and

  * For every $f \colon B\to T A$ and $g:C \to T B$, we have $f^T \circ g^T = (f^T \circ g)^T$.

Given these data, we make $T$ a [[functor]] by $T f = (\eta_A \circ f)^T$, we define multiplication maps $\mu_A:T T A \to T A$ as $(1_{T A})^T$, and we verify that the result is a [[monad]].  Conversely, given a monad $(T,\eta,\mu)$, we define $f^T = \mu_A \circ T f$ and check the above axioms.  Thus, extension systems are [[equivalence|equivalent]] to monads.

### The Kleisli category

This presentation of a monad is especially convenient for defining the [[Kleisli category]] $C_T$ (whence the aternative terminology "[[Kleisli triple]]"):

its objects are those of $C$, its morphisms $B\to A$ are the morphisms $B\to T A$ in $C$, and the composite of $f:B\to T A$ with $g:C \to T B$ is $f^T \circ g$.

### The category of algebras

It is also possible to define [[algebras over a monad]] using this presentation.  A $T$-algebra consists of

* An object $B$, and

* For every morphism $h:X\to B$, a morphism $h^B : T X \to B$, such that

* For every $h:X\to B$ we have $h^B \circ \eta_X = h$, and

* For every $h:X\to B$ and $y:Y\to T X$ we have $h^B \circ y^T = (h^B \circ y)^B$.

### Strong monads

When $C$ is a [[cartesian closed category]], to make $T$ a [[strong monad]] we simply have to enhance the extension operation $(-)^T$ to an internal morphism $(T A)^B \to (T A)^{T B}$, or equivalently $T B \times (T A)^B \to T A$.  This morphism is known as "bind" in use of [[monads in computer science]].

## Related concepts

* [[monad in computer science]]

* [[relative monad]]

* [[polymonad]]

## References

The notion appears explicitly in:

* {#Manes76} [[Ernest G. Manes]], Sec 3, Ex. 12 (p. 32) of: *Algebraic Theories*, Springer (1976) [[doi:10.1007/978-1-4612-9860-1](https://doi.org/10.1007/978-1-4612-9860-1)]

and is expanded on considerably in

* {#MarmolejoWood10} [[Francisco Marmolejo]], [[Richard J. Wood]], *Monads as extension systems -- no iteration is necessary* [[TAC]] **24** 4 (2010) 84-113 [[tac:24-04](http://www.tac.mta.ca/tac/volumes/24/4/24-04abs.html)]

An earlier appearance in a different guise ("devices"):

* [[R. F. C. Walters]], Chapter I of: *A categorical approach to universal algebra*, Ph.D. Thesis (1970) [[anu:1885/133321](https://openresearch-repository.anu.edu.au/handle/1885/133321)]

The use of extension systems (there called [[Kleisli triples]]) as [[monads in computer science]] capturing [[computation]] with side effects is due to:

* {#Moggi91} [[Eugenio Moggi]], *Notions of computation and monads*, Information and Computation, **93** 1 (1991) [doi:10.1016/0890-5401(91)90052-4, [pdf](http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf)]

This way of presenting a monad is also closely related to [[continuation-passing style]], as described in

* [[Andrzej Filinski]], _Representing Monads_, POPL 1994. ([pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.43.8213&rep=rep1&type=pdf))

Generalization to [[pseudomonads]]:

* [[Francisco Marmolejo]], [[Richard J. Wood]], *Kan extensions and lax idempotent pseudomonads*, [[TAC]] **26** 1 (2012) 1-29 [[26-01](http://www.tac.mta.ca/tac/volumes/26/1/26-01abs.html)]

* [[Francisco Marmolejo]], [[Richard J. Wood]], *No-iteration pseudomonads*, [[TAC]] **28** 14 (2013) 371-402 [[tac:28-14](http://www.tac.mta.ca/tac/volumes/28/14/28-14abs.html)]

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[[!redirects Kleisli extension]]
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extension theorems -- table

| [[extension]] [[theorems]] | [[continuous functions]] | [[smooth functions]] |
|---|---|---|
|  **plain [[functions]]**           | [[Tietze extension theorem]] | [[Whitney extension theorem]] |
| **[[equivariant functions]]** | [[equivariant Tietze extension theorem]] |


extension type

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## Idea

In [[dependent type theories]] with more than one layer of type, function extension types are types which behave like [[function types]] except the [[domain]] is the outer layer  (or equivalent) of type rather than the normal layer of type.

## Definition

### In dependent type theory with second layer for propositions

In [[dependent type theory]] with a second layer of propositions $\phi \; \mathrm{prop}$, the [[inference rules]] for extension types are given as follows:

Formation rules for extension types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash \phi \; \mathrm{prop} \quad \Gamma, \phi \vdash u:A}{\Gamma \vdash \{A \vert \overline{\phi \to u}\}}$$

Introduction rules for extension types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash \phi \; \mathrm{prop} \quad \Gamma, \phi \vdash u:A \quad \Gamma \vdash v:A \quad \overline{\Gamma, \phi \vdash u \equiv v:A}}{\Gamma \vdash \mathrm{inS}(v):\{A \vert \overline{\phi \to u}\}}$$

Elimination rules for extension types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash \phi \; \mathrm{prop} \quad \Gamma, \phi \vdash u:A \quad \Gamma \vdash v:\{A \vert \overline{\phi \to u}\}}{\Gamma \vdash \mathrm{outS}(v):A}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash \phi \; \mathrm{prop} \quad \Gamma, \phi \vdash u:A \quad \Gamma \vdash v:\{A \vert \overline{\phi \to u}\}}{\overline{\Gamma, \phi \vdash \mathrm{outS}(v) \equiv u:A}}$$

Computation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash \phi \; \mathrm{prop} \quad \Gamma, \phi \vdash u:A \quad \Gamma \vdash v:A}{\Gamma \vdash \mathrm{outS}(\mathrm{inS}(v)) \equiv v:A}$$

Uniqueness rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash \phi \; \mathrm{prop} \quad \Gamma, \phi \vdash u:A \quad \Gamma \vdash v:\{A \vert \overline{\phi \to u}\}}{\Gamma \vdash \mathrm{inS}(\mathrm{outS}(v)) \equiv v:\{A \vert \overline{\phi \to u}\}}$$

\begin{remark}
The introduction and elimination operators $\mathrm{inS}$ and $\mathrm{outS}$ implicitly carries around the underlying proposition $\phi$, and these matter in the equations.

For instance, in type theories with [[coercive subtyping]], one may expect to coerce _stronger_ extension types into _weaker_ ones using the composition of $\mathrm{inS}$ and $\mathrm{outS}$, where we use $\Gamma, \phi \vdash \phi'$ to say that the proposition $\phi$ implies $\phi'$ in the second layer:

$$\frac
  {\Gamma, \phi \vdash \phi' \quad \Gamma \vdash v : \{A \vert \phi \to u \} }
  {\Gamma \vdash \mathrm{inS}(\mathrm{outS}(v)) : \{A \vert \phi' \to u \} }$$

But this coercion does not simplify directly to $v$ because the underlying proposition carried by the operators are not the same.
\end{remark}

### In type theory with shapes

In [[type theory with shapes]], there are three different layers - there are the cube layer and the tope layer, which is [[logic over type theory]] where the cubes are the types and the topes are the propositions in the logic over type theory, and finally there is the type layer, which is a [[dependent type theory]]. Note that this type layer is distinct from the cube layer regarded as a type theory.

Shapes are formed in the usual set-builder notation in [[set theory]]: given a cube $I$ and a predicate tope $t:I \vdash \phi$, one could construct the shape $\{t:I \vert \phi\}$. A cofibration in two-level type theory is an inclusion of shapes, which means shapes $\{t:I \vert \phi\}$ and $\{t:I \vert \psi\}$ with a predicate $t:I \vert \phi \vdash \psi$.

Formation rules for dependent extension types
$$\frac{\{t:I \vert \phi\} \; \mathrm{shape} \quad \{t:I \vert \psi\} \; \mathrm{shape} \quad t:I \vert \phi \vdash \psi \quad \Xi \vert \Phi \vdash \Gamma \; \mathrm{ctx} \quad \Xi \vert \Phi \vert \Gamma \vdash A \; \mathrm{type} \quad \Xi, t:I \vert \Phi, \phi \vert \Gamma \vdash a:A}{\Xi \vert \Phi \vert \Gamma \vdash \langle \{t:I \vert \psi\} \to A \vert_a^\phi \rangle \; \mathrm{type}}$$

## Related concepts

* [[dependent extension type]]
* [[function type]]

## References

* {#RiehlShulman17} [[Emily Riehl]], [[Michael Shulman]], *A type theory for synthetic $\infty$-categories* $[$[arXiv:1705.07442](https://arxiv.org/abs/1705.07442)$]$

* [[Jonathan Sterling]], [[Robert Harper]], *Logical Relations as Types: Proof-Relevant Parametricity for Program Modules*, Journal of the ACM, Volume 68, Issue 6, December 2021, Article No.: 41, pp 1-47. ([doi:10.1145/3474834](https://doi.org/10.1145/3474834), [arXiv:2010.08599](https://arxiv.org/abs/2010.08599))

* [[Daniel Gratzer]], [[Jonathan Sterling]], [[Carlo Angiuli]], [[Thierry Coquand]], [[Lars Birkedal]], *Controlling unfolding in type theory* ([arXiv:2210.05420](https://arxiv.org/abs/2210.05420))

* [[Tesla Zhang]], *Three non-cubical applications of extension types* ([arXiv:2311.05658](https://arxiv.org/abs/2311.05658))

For extension types in [[cubical type theory]], see section 3.5 of:

* [[Carlo Angiuli]], *Computational Semantics of Cartesian Cubical Type Theory*, Ph.D. Thesis, <https://www.cs.cmu.edu/~cangiuli/thesis/thesis.pdf>

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extensional category > history

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< [[concrete category]]


extensional function

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## Definition

### In category theory

An **extensional function** is a [[functor]] between [[setoids]] ([[thin category|thin]] [[groupoids]]).

### In type theory

There are multiple inequivalent notions of extensional function in type theory.

#### As functions that preserve equivalence relations

An **extensional function** is a [[function]] $f:A \to B$ between [[setoids]] $A$ and $B$ such that for all terms $a:A$ and $b:A$, $p:(a \equiv_A b) \to (f(a) \equiv_B f(b))$.

#### As functions that preserve identity types

An **extensional function** is a [[function]] $f:A \to B$ between [[types]] $A$ and $B$ such that for all terms $a:A$ and $b:A$, $p:(a =_A b) \to (f(a) =_B f(b))$.

If there exists an [[interval type]], then every function is an extensional function.

## See also

* [[functor]]
* [[thin category]]
* [[setoid]]
* [[equivalence relation]]
* [[identity types]]
* [[interval type]]

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extensional relation

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## Idea

Given an extensional relation $\prec$ on a set, two elements are equal if they cannot be distinguished by the structure of the set of elements that they are related to, that those elements are related to, and so on forever in one direction.  We generally think of $x \prec y$ as saying that $x$ is a 'member' of $y$, so that $\prec$ is extensional when "$x$ is determined by its members" as in the [[axiom of extensionality]] for material [[set theory]].

## Definitions

We begin with the historically first definition, which is correct for [[well-founded relations]].  We then consider ways to extend this to more general relations, where the last version seems to be most correct.

### Weak extensionality

A relation $\prec$ on a set $S$ is __weakly extensional__ if, given any elements $x$ and $y$ of $S$, $x = y$ whenever (for all $t$) $t \prec x$ if and only if $t \prec y$.

Interpreting $\prec$ as membership $\in$, this corresponds to the [[axiom of extensionality]] as usually stated for [[pure sets]]. However, it is really only appropriate when $\prec$ is [[well-founded relation|well-founded]], just as the usual axiom of extensionality must be strengthened in the absence of the [[axiom of foundation]].

However, when $\prec$ is well-founded, weak extensionality is equivalent to all the stronger notions below; thus in that case it is usually called just **extensionality**.

### Finsler extensionality

Let $\prec^*$ be the reflexive-[[transitive closure]] of the relation $\prec$ on $S$ (so $\prec^*$ is a [[preorder]]).  Given an element $y$ of $S$, let $S \downarrow^* y$ be the [[downset]] of $y$ under $\prec^*$:
$$ S \downarrow^* y = \{ x: S \;|\; x = t_0 \prec \cdots \prec t_n = y \} $$
with $n \geq 0$.  Note that $y$ itself belongs to $S \downarrow^* y$ (through $n = 0$, if in no other way), so we may take it to be a [[pointed set]].  Then $\prec$ is __Finsler-extensional__ if it is weakly extensional and, given any elements $x$ and $y$ of $S$, $x = y$ whenever $S \downarrow^* x$ and $S \downarrow^* y$ are isomorphic as pointed sets equipped with a relation $\prec$.

Note that this definition includes weak extensionality, which won\'t follow from the other half unless $\prec$ is well-founded (see the examples below).  It is possible to get weak extensionality free by using the [[transitive closure]] $\prec^+$ instead; that is, define $S \downarrow^+ y$ with $n \gt 0$ only.  But then you need another step; define $(S \downarrow^+ y)^\top$ to be a pointed set with a new point $\top$ adjoined to $S \downarrow^+ y$; let $x \prec \top$ if and only if $x \prec y$ in $S$ itself.  (For a well-founded relation, $(S \downarrow^+ y)^\top \cong S \downarrow^* y$; in general, however, $y$ may already belong to $S \downarrow^+ y$, yet $y \ne \top$ in $(S \downarrow^+ y)^\top$.)  Then $\prec$ is Finsler-extensional on $S$ if and only if $x = y$ whenever $(S \downarrow^+ x)^\top \cong (S \downarrow^+ y)^\top$ as pointed sets equipped with $\prec$.

It is immediate that Finsler extensionality implies weak extensionality; the converse may be proved (using [[induction]]) for well-founded relations.

### Scott extensionality

Given an element $y$ of $S$, let a __path__ to $y$ consist of a sequence
$$ x = t_0 \prec \cdots \prec t_n = y $$
for $n \geq 0$.  Then let $T_y$ be the set of paths to $y$; given paths $\vec{t}$ and $\vec{u}$ in $N_y$, say that $\vec{t} \to \vec{u}$ if $\vec{u} = (t_1, \ldots, t_m)$ for some $m \leq n$.  Then $T_y$ with the relation $\to$ is the __tree__ to $y$.

$S$ is __Scott-extensional__ if the only [[automorphism]] of any such tree $T_y$ (as a set equipped with a relation $\to$) is the [[identity function]] on $T_y$.

Then Scott extensionality implies Finsler extensionality, and the converse holds for well-founded relations.

### Strong extensionality

The strongest version of extensionality is motivated by the study of [[terminal coalgebra]]s and [[coinduction]].

Let $S$ be equipped with a binary relation $\prec$.  A __[[bisimulation]]__ on $(S,\prec)$ is a binary relation $\sim$ such that whenever $x \sim y$, for any $a \prec x$ there is a $b \prec y$ with $a \sim b$, and conversely for every $b \prec y$ there is an $a \prec x$ with again $a \sim b$.  We then say that $\prec$ is __strongly extensional__ if every bisimulation is contained in the identity relation; i.e., $x = y$ whenever $x \sim y$ for any bisimulation $\sim$.  In general, this is probably the best situation in which to say that $\prec$ is simply __extensional__.

Finsler and Scott extensionality may be understood as special cases of this for particular bisimulations $\sim$.  (So can strong extensionality, since any set equipped with a relation has a weakest bisimulation $\approx$.)  This is the approach taken by Aczel to study all the above notions of extensionality together.

In particular, strong extensionality implies Scott extensionality, and the converse holds for well-founded relations.  Thus, all forms of extensionality are equivalent for well-founded relations.

### Weak extensionality for arbitrary binary relations

Given sets $A$ and $B$, let $\prec$ be a binary relation on $A$ and $B$, or equivalently, a [[predicate]] on the cartesian product $A \times B$. Then $\prec$ is (weakly) extensional if, given any elements $x$ and $y$ of $B$, $x =_B y$ whenever for all $t:A$, $t \prec x$ if and only if $t \prec y$.

$$\forall x:B.\forall y:B.(x =_B y) \iff \forall t:A.(t \prec x) \iff (t \prec y)$$

### In homotopy type theory

In [[homotopy type theory]], a relation is an [[h-proposition]]-valued [[type family]]. A binary relation $\prec$ on a carrier type $S$ is **weakly extensional** if for all terms $a:S$ and $b:S$ the canonical function

$$\mathrm{idtoextcond}(a, b):(a =_S b) \to \prod_{c:S} (c \prec a) \simeq (c \prec b)$$

is an [[equivalence of types]]. This implies that the type $S$ is an [[h-set]].

## Examples

*  The [[axiom of extensionality]] in material [[set theory]] states membership is an extensional relation on the class of [[pure sets]].  (Note that the [[axiom of foundation]] states that membership is a well-founded relation, so one usually doesn\'t worry about the different notions of extensionality for ill-founded relations.)  More generally, the membership relation on the [[transitive closure]] or reflexive-transitive closure of a pure set is an extensional relation on a set.

*  Conversely, one can *define* [[pure set]]s in *structural* [[set theory]] in part as sets equipped with an extensional (and optionally well-founded) relation.

* In any [[single-sorted definition of a category|single-sorted definition]] of a [[topos]], there is a [[terminal object]] $1$ (represented by the [[identity morphism]] of the terminal object), one could define the relation $\prec$ on the set of morphisms $C$ as

$$a \prec b \coloneqq s(a) = 1 \wedge t(b) = 1 \wedge t(a) = s(b)$$

Due to the [[universal property]] of the [[terminal object]], $\prec$ is weakly extensional. However, $\prec$ is not strongly extensional. The relation $\sim$ defined by equality on every pair of functions $a \sim b \coloneqq a = b$, and in addition, $\mathrm{id}_2 \sim \mathrm{swap}$ and $\mathrm{swap} \sim \mathrm{id}_2$ for the identity function and the swap function on the set with two elements $2$, is a bisimulation on $\prec$, but it is not true that $\mathrm{id}_2 = \mathrm{swap}$.

* In any [[single-sorted definition of a category|single-sorted definition]] of a [[topos]], there is also an [[initial object]] $0$ (represented by the [[identity morphism]] of the initial object). One could define the relation $\in$ on the set of morphisms $C$ as

$$a \in b \coloneqq s(a) = 1 \wedge s(b) = 0 \wedge t(a) = t(b)$$

Due to the [[universal property]] of the [[initial object]], $\in$ is weakly extensional. However, $\in$ is not strongly extensional, for the same reason that $\prec$ defined above is not strongly extensional.

*  A [[well-order]] is precisely a well-founded, [[transitive relation|transitive]], extensional relation.  Removing well-foundedness here gives a theory of ill-founded [[ordinal number]]s.

*  On the set $\mathbf{2} = \{0,1\}$, now let $0 \prec 0$ and $0 \prec 1$ (but no other relationships).  This relation is not weakly extensional, although it does satisfy the other half of Finsler extensionality, since $\mathbf{2} \downarrow^* 0 \ncong \mathbf{2} \downarrow^* 1$.  However, $(\mathbf{2} \downarrow^+ 0)^\top \cong (\mathbf{2} \downarrow^+ 1)^\top$; this shows the necessity of defining Finsler extensionality as we do.

*  On the set $\mathbf{2}$ again, let $0 \prec 1$ and $1 \prec 0$ (but $0 \nprec 0$ and $1 \nprec 1$).  This relation is weakly extensional but not Finsler-extensional, since $\mathbf{2} \downarrow^* 0 \cong \mathbf{2} \downarrow^* 1$.  Yet $0$ and $1$ can hardly be distinguished by $\prec$ when there is an [[automorphism]] of $(\mathbf{2},\prec)$ that swaps them; this and the other examples below motivate the stronger notions of extensionality.

*  On the set $\mathbf{3} = \{0,1,2\}$, let $2 \nprec 0$ and $i \nprec i$ but all other relationships hold.  Then this relation is Finsler-extensional but not Scott-extensional, since $T_2 \cong T_1$.

*  Finally, on the set $\mathbf{2}$ again, let $1 \nprec 0$ but all other relationships hold.  Then this relation is Scott-extensional but not strongly extensional, since $0 \approx 1$.

## Extensional quotients

Weak extensionality is a kind of [[antisymmetric relation|antisymmetry]] condition: Let $x \leq y$ mean that $t \prec y$ whenever $t \prec x$.  Then $\leq$ is clearly a [[preorder]], which is antisymmetric (so a [[partial order]]) if and only if $\prec$ is weakly extensional.

Now suppose that if $t \prec x$ if and only if $t \prec y$ for all $t$, then $x \prec z$ if and only if $y \prec z$ for all $z$.  Then if we define $x \equiv y$ to mean that $x \leq y$ and $y \leq x$, then $\equiv$ is an equivalence relation such that $\prec$ descends to a weakly extensional relation on the [[quotient set]] $S/{\equiv}$.

Strongly extensional quotients are even easier to construct, although they do create additional relationships.  It is easy to see that the union of any family of bisimulations is a bisimulation, and therefore there is a largest bisimulation $\approx$ for any binary relation $\prec$ on $S$.  Moreover, $\approx$ is an equivalence relation (though not every bisimulation need be so), and $\prec$ descends to a strongly extensional relation on the quotient $S/\approx$.

## Simulations

Given two sets $S$ and $T$, each equipped with a strongly extensional relation $\prec$, a [[function]] $f\colon S \to T$ is a __[[simulation]]__ of $S$ in $T$ if

*  $f(x) \prec f(y)$ whenever $x \prec y$ and
*  given $t \prec f(x)$, there exists $y \prec x$ with $t = f(y)$.

Then sets so equipped form a [[category]] with simulations as [[morphism]]s.

Note that there is at most one simulation from $S$ to $T$; in fact, strong extensionality for $T$ is equivalent to saying that any two simulations $S \rightrightarrows T$ are equal.  Also, any simulation from $S$ to $T$ must be an [[injection]]; in fact, strong extensionality for $S$ is equivalent to saying that any simulation $S \to T$ is injective.

Thus, we have a (large) [[poset]] of sets equipped with extensional relations, and we can consistently interpret the simulations as [[subset]] inclusion.  This leads to the model of sets equipped with extensional relations as [[transitive set]]s.

## References

*  Peter Aczel; [Non-Well-Founded Sets](https://web.archive.org/web/20161020002632/https://standish.stanford.edu/pdf/00000056.pdf), especially Chapter 4.

For extensional relations in [[homotopy type theory]], see

* [[Håkon Robbestad Gylterud]], [[Elisabeth Bonnevier]], *Non-wellfounded sets in HoTT* ([arXiv:2001.06696](https://arxiv.org/abs/2001.06696))

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extensional type theory

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## Idea

_Extensional type theory_ denotes the flavor of [[type theory]] in which [[identity types]] satisfy the *reflection rule*, saying that if two terms are [[typal equality|typally equal]] then they are also [[judgmental equality|judgmentally equal]].

In particular, this implies that all [[identity types]] are [[propositions]] / of [[h-level 1]], and thus equivalently that all types are required to be [[h-sets]].  Therefore, extensional type theory is a [[set-level type theory]], and hence a form of [[set-level foundations]].  However, there are other set-level type theories, such as those obtained by adding [[UIP]] as an axiom.

__Note:__ *For a while, the nLab incorrectly used "extensional type theory" to refer to what we now call [[set-level type theory]].  If you encounter uses of this sort, please correct them.*

Extensional type theory is poorly behaved metatheoretically, and very difficult to implement in a proof assistant.  However, it is sometimes more convenient to work with informally, and there are conservativity theorems relating it to other set-level type theories that are better-behaved.

Type theory which is not extensional is called _[[intensional type theory]]_.

+-- {: .num_remark}
###### Remark
The word "extensional" in type theory (even when applied to identity types) sometimes refers instead to the axiom of  [[function extensionality]].  In general this property is orthogonal to the one considered here: function extensionality can hold or fail in both extensional and intensional type theory.

In particular, [[homotopy type theory]] is [[intensional type theory|intensional]] in that [[identity types]] are crucially _not_ demanded to be [[propositions]], but [[function extensionality]] is often assumed (in terms of these intensional identity types, of course) --- in particular, it follows from the [[univalence axiom]].  Indeed, univalence itself is arguably an [[extensionality principle]] for the universe (Hofmann and Streicher originally introduced it under the name "universe extensionality"), but it is inconsistent with "extensional type theory" in the sense considered here.
=--

+-- {: .num_remark}
###### Remark
The origin of the names "extensional" and "intensional" is somewhat confusing.  In fact they refer to the behavior of the [[definitional equality]].  The idea is that the [[identity type]] is always an "extensional" notion of equality (although it can be more or less extensional, depending on whether further extensionality principles like [[function extensionality]] and [[univalence]] hold).  Thus, if the definitional equality *coincides* with the identity type, as it does under the reflection rule, the former is also extensional, and so we call the type theory "extensional" --- while if the two equalities do *not* coincide, then the definitional equality has room to be more intensional than the identity type, and so we call the type theory "intensional".
=--

## Definition

The Martin-Lof definition of [[identity types]] as an [[inductive type]] family makes them intensional.  To make the type theory extensional, we add a rule that any [[inhabitant]] of an identity type $p:Id_A(x,y)$ induces a [[definitional equality]] between $x$ and $y$.  In other words, we have an "equality reflection rule" of the form

$$ \frac{p:Id_A(x,y)}{x\equiv y} $$

At first, this may appear to be only a "[[skeleton|skeletality]]" assumption, since it does not assert explicitly that $p$ is [[reflexive relation|reflexivity]] rather than a nontrivial [[automorphism|loop]].  However, we can derive this with the induction rule for identity types.  Consider the [[dependent type]]

$$ (x:A),(y:A),(p:Id_A(x,y)) \;\vdash\; Id_{Id_A(x,y)}(p,refl(x)). $$

This is well-typed because the reflection rule applied to $p$ yields a judgmental equality $x\equiv y$, so that we have $refl(x):Id_A(x,y)$.  Moreover, [[substitution|substituting]] $x$ for $y$ and $refl(x)$ for $p$ yields the type $Id_{Id_A(x,x)}(refl(x),refl(x))$, which is inhabited by $refl(refl(x))$.

Thus, by induction on identity, we have a term in the above type, witnessing a [[typal equality]] between $p$ and $refl(x)$.  Finally, applying the equality reflection rule again, we get a judgmental equality $p\equiv refl(x)$.

+-- {: .num_remark}
###### Remark
On the other hand, if in addition to the equality reflection rule we postulate that all equality proofs are definitionally equal to reflexivity, then we can derive the induction rule for identity types.  Extensional type theory is often presented in this form.
=--

A different, also equivalent, way of presenting extensional type theory is with a definitional [[eta-conversion]] rule for the identity types; see _[here](identity+type#EtaConversion)_.

## Properties

### Decidability

Extensional [[Martin-Löf type theory]] does not have [[decidability|decidable]] type checking.  See _[[intensional type theory]]_ for more on this.

## Related concepts

* [[axiom K]], [[axiom UIP]]

* [[intensional type theory]], [[homotopy type theory]]

* [[observational type theory]], [[higher observational type theory]]

## Examples

* [[NuPRL]]

* [[Andromeda]]

## References

* {#Hofmann95} [[Martin Hofmann]], _Extensional concepts in intensional type theory_, Ph.D. thesis, University of Edinburgh, (1995) ([ECS-LFCS-95-327](http://www.lfcs.inf.ed.ac.uk/reports/95/ECS-LFCS-95-327/), [[HofmannExtensionalIntensionalTypeTheory.pdf:file]])

* [[Ieke Moerdijk]], E. Palmgren, _Wellfounded trees in categories_. Ann. Pure Appl. Logic, 104:189-218 (2000)

* [[Ieke Moerdijk]], E. Palmgren, _Type theories, toposes and constructive set theory: predicative aspects of AST_,  Ann. Pure Appl. Logic, 114:155-201, (2002)

* [[Jonathan Sterling]], [[Carlo Angiuli]], [[Daniel Gratzer]], _Cubical syntax for reflection-free extensional equality_. In Herman Geuvers, editor, _4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)_, volume 131 of _Leibniz International Proceedings in Informatics (LIPIcs)_, pages 31:1-31:25. ([arXiv:1904.08562](https://arxiv.org/abs:1904.08562), [doi:10.4230/LIPIcs.FCSD.2019.31](https://doi.org/10.4230/LIPIcs.FCSD.2019.31))

* [[Jonathan Sterling]], [[Carlo Angiuli]], [[Daniel Gratzer]], _A Cubical Language for Bishop Sets_, Logical Methods in Computer Science, 18 (1), 2022. ([arXiv:2003.01491](https://arxiv.org/abs/2003.01491)).

[[!redirects extensional type theories]]
[[!redirects equality reflection]]
[[!redirects equality reflection rule]]


extensionality

\tableofcontents

\section{Disambiguation}

There are many principles in mathematics called 'extensionality'; what these principles all have in common are that they all characterize the notion of [[equality]] in that [[collection]]. Examples include

* [[function extensionality]] (for functions)

* [[equivalence extensionality]] (for equivalences)

* [[sequence extensionality]] (for sequences)

* [[tuple extensionality]] (for tuples)

* [[product extensionality]] (for products/pairs)

* [[univalence]]/[[universe extensionality]] (for small types of universes)

* [[propositional extensionality]] (for propositions)

* [[axiom of extensionality]] (for pure sets)

* [[extensional type theory]] (for terms, uses [[definitional equality]] for the extensionality condition)

There are other notions in mathematics which could be considered to be an extensionality principle. Examples include

* [[extensional relations]] and [[bisimulation]]

* [[antisymmetry]] (for elements of a [[poset]])

* [[connected relation|connectedness]] (for elements of a [[strict total order]])

* [[tight apartness relation|tight apartness]] (for elements of an [[inequality space]])

* [[separation axiom]] (for elements of a [[Kolmogorov topological space]])

* [[Rezk completion]] (for [[objects]] of a [[category]] or [[(infinity,1)-category]])

[[!redirects extensionality principle]]

[[!redirects extensionality principles]]

category: disambiguation


extensive 2-category

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

#Contents#
* table of contents
{:toc}

## Idea

The generalization of the notion of _[[extensive category]]_ from [[category theory]] to [[2-category theory]].

## Definition

A [[2-coproduct]] $A+B$ in a [[2-category]] is said to be **disjoint** if we have [[comma squares]]
$$\array{
  A^{\mathbf{2}} & \to & A & \quad &
  B^{\mathbf{2}} & \to & B & \quad &
  0 & \to & A & \quad &
  0 & \to & B \\
  \downarrow & \Downarrow & \downarrow & \quad &
  \downarrow & \Downarrow & \downarrow & \quad &
  \downarrow & \Downarrow & \downarrow & \quad &
  \downarrow & \Downarrow & \downarrow & \quad\\
  A & \to & A+B & \quad &
  B & \to & A+B & \quad &
  B & \to & A+B & \quad &
  A & \to & A+B & \quad}
$$
The first two say that $A\to A+B$ and $B\to A+B$ are [[ff morphism|ff]] and the second two say that they are disjoint subobjects of $A+B$.

A coproduct $A+B$ is said to be **universal** if for any morphism $Z\to A+B$, the pullbacks
$$\array{X & \to & Z & \leftarrow & Y\\
  \downarrow & & \downarrow & & \downarrow\\
  A & \to & A+B & \leftarrow & B}$$
exist and exhibit $Z$ as a coproduct $X+Y$.

Finally, we say a 2-category is **extensive** if it has finite coproducts which are disjoint and universal.  If it also has finite limits we say it is **lextensive**, and if it is also [[coherent 2-category|coherent]] we call it **positive**.  (Note that disjoint coproducts in a coherent 2-category are always universal.)

An extensive 2-category does satisfy 2-categorical versions of the additional characterizations of an [[nLab:extensive category|extensive 1-category]], but unlike in the 1-categorical case, these alternate conditions do not seem to suffice to characterize extensivity.  In particular, though, a 1-category is extensive as a 1-category iff it is so as a homwise-discrete 2-category.

## Properties

### Preservation

If $K$ is extensive, so is $K^{co}$, obviously.  Less obvious is:

+--{: .num_lemma}
###### Lemma
If $K$ is extensive, so are $gpd(K)$, $pos(K)$, and $disc(K)$.  In other words, if $K$ is extensive, so is its $(n+1)$-category $trunc_n(K)$ of $n$-[[truncated object|truncated objects]] for $0\le n$.
=--
+--{: .proof}
###### Proof
Since the three given categories are closed in $K$ under limits and strict initial objects, it suffices to show they are closed under coproducts.  First suppose given two morphisms $f,g:Z\to A_1+A_2$.  Then $f$ decomposes $Z=X_1+X_2$, and $g$ decomposes $Z=Y_1+Y_2$.  Then the inclusions $X_i \to Z = Y_1+Y_2$ also decompose each $X_i=X_{i 1} + X_{i 2}$.  Now if there exists a 2-cell $f\to g$, it induces a map from each $X_{i j}$ to the comma object of $A_1$ and $A_2$.  Since coproducts are disjoint and initials are strict, this implies that $X_{12}=X_{21}=0$.  Therefore, we have a decomposition $Z=X_{11}+X_{22}$ so that $f=f_1+f_2$ and $g=g_1+g_2$, where $f_i:X_{i i} \to A_i$ and $g_i:X_{i i} \to A_i$.

Now, by universality of the coproduct $X_{11}+X_{22}$, it follows that 2-cells $f\to g$ are determined uniquely by pairs of 2-cells $f_1\to g_1$ and $f_2\to g_2$.  Therefore, if $A_1$ and $A_2$ are groupoidal, any 2-cells $f_1\to g_1$ and $f_2\to g_2$ are invertible, and thus so is any 2-cell $f\to g$; so $A_1+A_2$ is groupoidal.  And if $A_1$ and $A_2$ are posetal, any parallel 2-cells $f_1 \;\rightrightarrows\; g_1$ and $f_2 \;\rightrightarrows\; g_2$ are equal, and thus so are any $f \;\rightrightarrows\; g$; so $A_1+A_2$ is posetal.  And of course the discrete case follows by combining these.
=--

However, the (0,1)-category (= poset) $Sub(1)$ of (-1)-truncated objects (= subterminal objects) does _not_ inherit extensivity, and in fact posets are almost never extensive: the only disjoint coproduct is $0+0$.

We also have:

+--{: .num_lemma}
###### Lemma
If $K$ is extensive, so are the [[fibrational slice]]s $Opf(X)$ and $Fib(X)$ for any $X\in K$.
=--

## References

This is due to

* [[Mike Shulman]], _[[michaelshulman:extensive 2-category]]_

[[!redirects extensive 2-categories]]


extensive category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Regular and Exact categories
+-- {: .hide}
[[!include regular and exact categories - contents]]
=--
#### Category Theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

* table of contents
{: toc}

## Idea

A [[category]] is _extensive_ if it has [[coproducts]] that interact well with a certain class of [[pullbacks]].  Variations (some only terminological) include _lextensive_, _disjunctive_, and _positive_ categories.  All of these come in _finitary_ and _infinitary_ versions (and, more generally, $\kappa$-ary versions for any [[arity class]] $\kappa$).

## Definitions
 {#Definition}

### Extensive categories

\begin{definition}
A __finitely extensive category__ (or __finitary extensive category__) is a [[category]] $E$ with [[finite coproduct|finite]] [[coproduct|coproducts]] such that one, and hence all, of the following equivalent conditions holds:

1. For any [[pair]] of [[objects]] $a, b$ the coproduct functor on [[slice categories]] is an [[equivalence of categories]]:

   \[
     \label{CoproductFunctorOnSliceCategoriesIsEquivalenceOfCategories}
     \array{
       E_{/a}
         \,\times\,
       E_{/b}
       &\xrightarrow{\;\;\;\simeq\;\;\;}&
       E_{/(a \sqcup b)}
       \\
       \left(
       \array{
         X
         \\
         \downarrow
         \\
         a
       }
       \right)
       ,\;
       \left(
       \array{
         Y
         \\
         \downarrow
         \\
         b
       }
       \right)
       &\mapsto&
       \left(
       \array{
         X \sqcup Y
         \\
         \downarrow
         \\
         a \sqcup b
       }
       \right)
     }
   \]

   (e.g. [Carboni, Lack & Walters 1993, Def. 2.1](#CarboniLackWalters93)).

1. {#DisjointAndPullbackStableCoproducts} [[pullback|Pullbacks]] of finite-coproduct injections along arbitrary morphisms exist and finite coproducts are [[disjoint coproduct|disjoint]] and [[pullback-stable colimit|stable under pullback]].

   (e.g [CLW93, Prop. 2.6, Lem. 2.11](#CarboniLackWalters93))

1. Pullbacks of finite-coproduct injections (and thus all coproduct injections) along arbitrary morphisms exist, and in any commutative diagram

   \[
     \label{HomIntoCoproductIsPullbackIfDomainIsCoproduct}
     \array{
       x & \to & z & \leftarrow & y
       \\
       \downarrow  & &   \downarrow  &  &  \downarrow
       \\
       a & \to & a+b & \leftarrow & b
     }
   \]

   the two squares are pullbacks if and only if the top row is a coproduct diagram

   (e.g. [CLW93, Prop. 2.2](#CarboniLackWalters93)).

1. Finite coproducts are [[van Kampen colimits]]. By definition, this is just to say that one of the previous two conditions holds.

\end{definition}

\begin{definition}
An __infinitary extensive category__ is a category $E$ with all (small) [[coproducts]] such that the following analogous equivalent conditions hold:

1. [[pullback|Pullbacks]] of [[coprojection|coproduct injections]] along arbitrary morphisms exist

   and small coproducts are [[disjoint coproduct|disjoint]] and [[pullback-stable colimit|stable under pullback]].

2. For any small-[[indexed set]] $(a_i)$ of objects, the coproduct functor of [[slice categories]], generalizing (eq:CoproductFunctorOnSliceCategoriesIsEquivalenceOfCategories), is an [[equivalence of categories]]:

   $$
     \underset{i \in I}{\prod}
     (E_{/a_i})
       \overset{\sim}{\longrightarrow}
     E_{/(\coprod_i a_i)}
     \,.
   $$

3. Pullbacks of finite-coproduct injections (and thus all coproduct injections) along arbitrary morphisms exist, and for any family of commutative squares
 $$\array{ x_i & \to & z \\\downarrow &&\downarrow^f \\ a_i & \to & \coprod_i a_i } $$
 in which the bottom family of morphisms is the coproduct injections and the right-hand morphism is always the same, the top family are the injections of a coproduct diagram (hence $z = \coprod_i x_i$) if and only if all the squares are pullbacks.

4. All small coproducts are [[van Kampen colimits]].

\end{definition}

\begin{remark}
In between finitary and infinitary extensive , a __$\kappa$-ary extensive category__ (for $\kappa$ a [[cardinal number]] or an [[arity class]]) is one with disjoint and stable coproducts of fewer than $\kappa$ objects.  The unqualified term **extensive category** can refer to either the finitary or infinitary version, depending on the author; the more usual meaning is the finitary version.
\end{remark}

\begin{remark}
Extensive categories are also called __positive categories__, especially if they are also [[coherent category|coherent]].  Note that any disjoint coproduct in a coherent category is automatically pullback-stable.  A positive coherent category which is also [[exact category|exact]] is called a [[pretopos]].  Infinitary pretoposes encapsulate all the exactness conditions of Giraud's theorem characterizing [[Grothendieck topos|Grothendieck toposes]] (the remaining condition is the existence of a small [[generating set]]).
\end{remark}

\begin{definition}
If an extensive category also has [[finitely complete category|finite limits]], it is called __lextensive__ or __disjunctive__.
\end{definition}
(Note that the more usual default meaning of 'disjunctive', unlike the other terms, is the infinitary case.)

\begin{remark}
The alternative definitions of finitary disjunctive refer only to binary coproducts, but they obviously imply analogous statements for $n$-ary coproducts for all finite $n \ge 1$.  Less obviously, they also imply the analogous statement for $0$-ary coproducts (that is, [[initial object]]s).  In this case, the statement is that the initial object 0 is _strict_ (any map $a\to 0$ is an isomorphism).

Furthermore, if binary coproducts are [[disjoint coproduct|disjoint]], then (at least assuming [[classical logic]]) any infinitary coproducts that exist are also disjoint, since
$$
  \bigsqcup_{a\in A} X_a
  \;\;\cong\;\;
   X_{a_0} \sqcup \bigsqcup_{a\neq a_0} X_a
  \;\;\cong\;\;
  X_{a_0} \sqcup X_{a_1} \sqcup \bigsqcup_{a\neq a_0,a_1} X_a
$$
for any $a_0, a_1\in A$.  Therefore, if a finitary-extensive category has infinitary pullback-stable coproducts, it is necessarily infinitary-extensive.  In particular, a cocomplete [[locally cartesian closed category]] is finitary extensive if and only if it is infinitary extensive.
\end{remark}

\begin{remark}
As a further special case of the preceding, since an [[elementary topos]] is finitary extensive, any cocomplete elementary topos is infinitary extensive.  However, in this case, one of the arguments for finitary extensivity applies directly to the infinitary case and does *not* require classical logic; see [[toposes are extensive]].
\end{remark}

\begin{remark}
See [[familial regularity and exactness]] for a generalization of extensivity and its relationship to [[exact category|exactness]].
\end{remark}

\begin{remark}
Any extensive category with [[finite products]] is automatically a [[distributive category]].
\end{remark}

### Superextensive sites

Any extensive category admits a [[Grothendieck topology]] whose [[cover|covering families]] are (generated by) the families of inclusions into a [[coproduct]] (finite or small, as appropriate).  We call this the **extensive [[coverage]]** or **extensive topology**.  The [[codomain fibration]] of any extensive category is a [[stack]] for its extensive topology.

In general, we call a [[site]] __superextensive__ if its underlying category is extensive, its covering families are generated by (finite or small) families, and its coverage includes the extensive one.  See [[superextensive site]] for more details.

### Pre-lextensive categories

Extensivity is an "exactness" condition, analogous to being a [[exact category]] or a [[pretopos]] (a pretopos being precisely a category that is exact and finitary-extensive).  The corresponding "regularity" condition analogous to being a [[regular category]] or a [[coherent category]] is not well-known, but is not hard to write down.

Let us say (without making any assertion that this is good or permanent terminology) that a category is **pre-lextensive** if

1. it has a [[strict initial object]] $0$ (equivalently, its [[subobject]] [[preorders]] have pullback-stable bottom elements), and
1. whenever $A\rightarrowtail X$ and $B\rightarrowtail X$ are disjoint subobjects (i.e. $A\cap B=0$), they have a pullback-stable union (which is then automatically a disjoint and stable coproduct).

This is intended to complete the table of analogies:

|some                       |all                    |
|---------------------------|-----------------------|
|[[regular category]]       |[[exact category]]     |
|[[coherent category]]      |[[pretopos]]           |
|pre-lextensive category    |lextensive category    |

Regular/exact categories have quotients of (some) [[congruences]].  Exact categories have quotients of all congruences, while regular ones have quotients only of congruences that are [[kernel pairs]].  Another way to say that is that in a regular category, you can conclude that the quotient of some congruence exists if you can exhibit another object of which the quotient would be a [[subobject]] if it existed.  Similarly, pre-/lextensive categories have [[disjoint unions]].  Lextensive categories have all disjoint unions (= [[coproducts]]), while in a pre-lextensive category you can conclude that a pair of objects $X,Y$ have a disjoint union if you can exhibit another object in which $X$ and $Y$ can be embedded disjointly.  Finally, coherent categories/pretoposes have both quotients and
disjoint unions, or equivalently quotients and not-necessarily-disjoint unions, with the same sort of relationship between the two.

Evidently a pre-lextensive category is lextensive as soon as any two objects can be embedded disjointly in a third.  Pre-lextensive categories also suffice for the interpretation of [[internal logic|disjunctive logic]].

Being pre-lextensive is also sufficient to define the extensive topology and show that it is subcanonical, since it implies that whatever disjoint coproducts exist must be pullback-stable.  The codomain fibration of a pre-lextensive category is not necessarily a stack for its extensive topology, but this condition is weaker than extensivity.  It is true, however, that if $C$ is a pre-lextensive category whose codomain fibration is a stack for its extensive topology, and in which the disjoint coproduct $1+1$ exists, then $C$ is extensive, for arbitrary disjoint (binary) coproducts can then be obtained by descent along the covering family $(1\to 1+1, 1\to 1+1)$.

## Examples

\begin{example}\label{FreeCoproductCompletionsAreExtensive}
  Any [[free coproduct completion]] is extensive.

  As one immediately checks condition (eq:CoproductFunctorOnSliceCategoriesIsEquivalenceOfCategories), e.g. [Carboni, Lack & Walters 1993, Prop. 2.4](#CarboniLackWalters93).
\end{example}

\begin{example}
An [[elementary topos]] (or, more generally, any [[pretopos]]) is finitary lextensive; a [[Grothendieck topos]] (or, more generally, any [[cocomplete category|cocomplete]] elementary topos) is infinitary lextensive.
\end{example}

\begin{example}
A [[quasitopos]] with disjoint coproducts, or more generally a [[locally cartesian closed category]] with disjoint coproducts, is extensive. (Of course not all quasitoposes have disjoint coproducts, one example being a [[complete Heyting algebra]].)
\end{example}

\begin{example}
The category [[Top]] of [[topological spaces]] is infinitary lextensive.  The category [[Diff]] of [[smooth manifolds]] is infinitary extensive, though it does not have all [[pullbacks]] (only those involving a [[cospan]] of [[transversal maps]]).
\end{example}

\begin{example}
The category of [[schemes]] is infinitary lextensive. In more detail: the category of functors $CRing \to Set$ is infinitary lextensive (since finite limits and small coproducts are computed pointwise in $Set$), then sheaves with respect to the Zariski topology on $CRing^{op}$ form an infinitary lextensive category (since finite limits and small coproducts are reflected back from $[CRing, Set]$ by applying a left exact reflection to the inclusion of sheaves in presheaves). Finally, the category of schemes, as a full subcategory of the Zariski sheaves, are closed under finite limits and small coproducts. (Some discussion of these points can be found at the [nForum](http://nforum.mathforge.org/discussion/5149/extensive-category/), particularly in comment #18.)
\end{example}

\begin{example}
The category of affine schemes (opposite to the category of commutative [[ring]]s with identity) is lextensive, but (perhaps contrary to geometric intuition) _not_ infinitary lextensive. Some details may be found [here](http://ncatlab.org/toddtrimble/published/Dippy+disproof+of+infinitary+extensivity+of+affine+schemes).
\end{example}

\begin{example}
The category [[Cat]] is infinitary lextensive.
\end{example}

\begin{example}
The category [[Vect]] is not even finitely extensive.
\end{example}

## Properties
 {#Properties}

\begin{proposition}
\label{InExtensiveCategoryCOnnectedObjectsArePrimitiveUnderCoproduct}
**(in extensive categories connected objects are primitive under coproduct)**
\linebreak
An object $X$ in an extensive category $\mathcal{C}$ is a [[connected object]] (in that the [[hom-functor]] $\mathcal{C}(X,-) \,\colon\, \mathcal{C} \to Set$ [[preserved colimit|preserves]] [[coproducts]]) if and only if in any [[coproduct]] decomposition $X \simeq U + V$, exactly one of $U$, $V$ is not the [[initial object]].
\end{proposition}

\begin{proof}
\label{ProofThatInExtensiveCategoryCOnnectedObjectsArePrimitiveUnderCoproduct}
In one direction, assume that $X$ is connected and consider an [[isomorphism]] $X \xrightarrow{\sim} X_1 \sqcup X_2$ to a [[coproduct]]. By assumption of connectedness, this morphism factors through one of the summands, say through $X_1$, as shown in the bottom row of the following diagram:

\begin{tikzcd}
    X_2
    \ar[r]
    \ar[d]
    \ar[dr, phantom, "\mbox{\tiny (pb)}"]
    &
    \varnothing
    \ar[r]
    \ar[d]
    \ar[dr, phantom, "\mbox{\tiny (pb)}"]
    &
    X_2
    \ar[d, "i_2"{right}]
    \\
    X
    \ar[r]
    \ar[
      rr,
      rounded corners,
      to path={
        -- ([yshift=-7pt]\tikztostart.south)
        -- node[below]{\scalebox{.7}{$\sim$}}
           ([yshift=-6pt]\tikztotarget.south)
        -- ([yshift=-0pt]\tikztotarget.south)
      }
    ]
    &
    X_1
    \ar[r, "i_1"{below}]
    &
    X_1 \sqcup X_2
    \mathrlap{\,.}
\end{tikzcd}
Consider then the [[pullback]] of the total bottom morphism along the inclusion $i_2$ of the other summand. Since pullbacks of isomorphisms are isomorphisms, the resulting top left object must be (isomorphic to) $X_2$, as shown. On the other hand, by the [[pasting law]] this pullback factors into two pullback squares, as shown above. But the pullback on the right gives the [[initial object]], since [[disjoint coproducts|coproducts are disjoint]] in an extensive category (see [above](#DisjointAndPullbackStableCoproducts)). This exhibits a morphisms $X_2 \to \varnothing$. But since the initial objects in extensive categories are [[strict initial objects]], this must be an isomorphism, $X_2 \simeq \varnothing$. By the same argument, $X_1$ cannot be an initial object, since otherwise $X$ would be too, which it is not by assumption of connectedness (Rem. \ref{InitialObjectIsNotConnected}). Hence we have shown that exactly one of the two summands in $X \simeq X_1 \sqcup X_2$ is initial.

In the other direction, assume that $X$ has non non-trivial coproduct decomposition and consider any [[morphism]] $f \colon X \to Y + Z$ into a coproduct. By [[extensive category|extensivity]],
this implies (eq:HomIntoCoproductIsPullbackIfDomainIsCoproduct)
a coproduct decomposition $X = U + V$
with $U \coloneqq f^\ast(i_Y)$ and $V \coloneqq f^\ast(i_Z)$.
But, by assumption, either $U$ or $V$ is initial, meaning that $X$ is [[isomorphism|isomorphic]] to either $V$ or $U$, respectively, so that $f$ factors through either $Y$ or $X$, respectively. In other words, $f$ belongs to exactly one of the two subsets $\hom(X, Y) \hookrightarrow \hom(X, Y + Z)$ or $\hom(X, Z) \hookrightarrow \hom(X, Y + Z)$.
\end{proof}

Another useful fact is that any extensive category with [[finite products]] is [[distributive category|distributive]].

## Related concepts

* [[extensive 2-category]]

* [[disjunctive (∞,1)-category]]

* [[intensive or extensive quantity|extensive quantity]]

* [[objective number theory]]

* [[disjunctive logic]]

* [[Gaeta topos]]

## References

* {#CarboniLackWalters93} [[Aurelio Carboni]], [[Stephen Lack]], [[Bob Walters|R. F. C. Walters]], _Introduction to extensive and distributive categories_, JPAA **84** (1993) pp. 145-158 (doi:10.1016/0022-4049(93)90035-R)

* [[Aurelio Carboni]], [[George Janelidze]], _Decidable (= separable) objects and morphisms in lextensive categories_ , JPAA **110** (1996) pp.210-240.

* [[Stephen Lack]], [[Enrico Vitale|Enrico M. Vitale]], _When do completion processes give rise to extensive categories?_ , JPAA **159** (2001) pp. 203-230 (doi:10.1016/S0022-4049(00)00060-8)

See also:

* {#catlistdiscussion} Freyd, Lack, Lawvere et al: [CATLIST discussion on extensive categories](https://www.mta.ca/~cat-dist/catlist/1999/extensive) 1996

[[!redirects extensive category]]
[[!redirects extensive categories]]

[[!redirects extensive topology]]
[[!redirects extensive topologies]]
[[!redirects extensive coverage]]
[[!redirects extensive coverages]]

[[!redirects lextensive category]]
[[!redirects lextensive categories]]

[[!redirects pre-lextensive category]]
[[!redirects pre-lextensive categories]]
[[!redirects prelextensive category]]
[[!redirects prelextensive categories]]

[[!redirects disjunctive category]]
[[!redirects disjunctive categories]]

[[!redirects infinitary extensive category]]
[[!redirects infinitary extensive categories]]

[[!redirects positive category]]
[[!redirects positive categories]]
[[!redirects positive coherent category]]
[[!redirects positive coherent categories]]


exterior

The __exterior__ of a [[subset]] of a [[topological space]] is the [[interior]] of its [[complement]], or equivalently the complement of its [[topological closure|closure]].  In more elementary terms, $x \in Ext A$ iff there is some [[neighbourhood]] of $x$ that is [[disjoint subsets|disjoint]] from $A$.

The exterior operation applied to [[open sets]], i.e., to a [[topology]] $\mathcal{O}$ of a space, coincides with the [[negation]] operation $\neg: \mathcal{O} \to \mathcal{O}$ when we view $\mathcal{O}$ as a [[frame]] or [[Heyting algebra]]. See also [[regular open set]].

[[!redirects exterior]]
[[!redirects exteriors]]


exterior algebra

+-- {: .rightHandSide}
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### Context
#### Super-Algebra and Super-Geometry
+-- {: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The **exterior algebra** $\Lambda V$ of a vector space is the [[free object|free]] [[graded-commutative algebra]] over $V$, where the elements of $V$ are taken to be of degree $1$.  (That is, the [[forgetful functor]] takes a graded-commutative algebra to its vector space of degree-$1$ elements.)

This construction generalizes to [[group representations]], [[chain complexes]], [[vector bundles]], [[coherent sheaves]], and indeed objects in any [[symmetric monoidal category|symmetric monoidal]]  [[linear category|linear categories]] with enough [[colimit|colimits]], where the tensor product distributes over those colimits (as in a [[2-rig]]).

## Explicit definition

We begin with the construction for vector spaces and then sketch how to generalize it.

### For vector spaces

Suppose $V$ is a [[vector space]] over a [[field]] $K$.  Then the **exterior algebra** $\Lambda V$ is generated by the elements of $V$ using these operations:

*  addition and scalar multiplication
*  an associative binary operation $\wedge$ called the **exterior product** or **wedge product**,

subject to these identities:

*  the identities necessary for $\Lambda V$ to be an [[associative algebra]]
*  the identity $v \wedge v = 0$ for all $v \in V$.

It then follows that $\Lambda V$ is a [[graded algebra]] where $\Lambda^p V$ is spanned by $p$-fold wedge products, that is, elements of the form

$$v_1 \wedge \cdots \wedge v_p $$

where $v_1, \dots, v_p \in V$.  It also follows that $\Lambda V$ is [[graded commutative algebra|graded commutative]]: that is, if $\omega \in \Lambda^p V$ and $\nu \in \Lambda^q V$, then

*  $\omega \wedge \nu = (-1)^{p q}\, \nu \wedge \omega$.

If $K$ is a field not of [[characteristic]] $2$, we may replace the relations

\[ \label{alternating} v \wedge v = 0 \]

by the relations

\[ \label{antisymmetric} v \wedge w = - w \wedge v \]

for all $v, w \in V$ ([Grassmann 1844, §37, §55](#Grassmann1844)).  If we can divide by $2$, then the relations (eq:antisymmetric) imply (eq:alternating), while the converse holds in any characteristic.

The exterior algebra of a vector space is also called the **Grassmann algebra** or **alternating algebra**.  It is also denoted $\bigwedge V$, $\bigwedge^\bullet V$, or $Alt V$.

### In general

More generally, suppose $C$ is any [[symmetric monoidal category]] and $V \in C$ is any object.  Then we can form the [[tensor power|tensor powers]] $V^{\otimes n}$.  If $C$ has countable coproducts we can form the [[coproduct]]

$$  T V = \bigoplus_{n \ge 0} V^{\otimes n} $$

(which we write here as a [[direct sum]]), and if the tensor product distributes over these coproducts, $T V$ becomes a [[monoid]] object in $C$, with multiplication given by the obvious maps

$$   V^{\otimes p} \otimes V^{\otimes q} \to V^{\otimes (p+q)} $$

This monoid object is called the [[tensor algebra]] of $V$.

The [[symmetric group]] $S_n$ acts on $V^{\otimes n}$, and if $C$ is a [[symmetric monoidal]] $\mathbb{K}$-[[linear category]] over a [[field]] $\mathbb{K}$ of [[characteristic]] zero, then we can form the antisymmetrization map

$$  p_A : V^{\otimes n} \to V^{\otimes n} $$

given by

$$ p_A = \frac{1}{n!} \sum_{\sigma \in S_n} sgn(\sigma) \, \sigma $$

The [[cokernel]] of $1 - p_A$ is called the $n$th **antisymmetric tensor power** or **alternating power** $\Lambda^n V$.  The coproduct

$$  \Lambda V = \bigoplus_{n \ge 0} \Lambda^n V $$

becomes a monoid object called the **exterior algebra** of $V$.

If $C$ is a linear category over a field of positive characteristic (or more generally, over a [[commutative ring]] in which not every positive integer is invertible, that is which is not itself an [[associative algebra|algebra]] over the [[rational numbers]]), then we need a different construction of $\Lambda^n V$; we define ... (please complete this!).

## Examples

### Over a super vector space

For $V$ a [[super vector space]], the exterior algebra $\Lambda V$ is often called the **Grassmann algebra** over $V$.  This $\Lambda V$ or $\wedge^\bullet V$ is the [[free object|free]] [[supercommutative algebra|graded commutative]] [[superalgebra]] on $V$.

Explicitly, this is the quotient of the [[tensor algebra]] $T V$ by the [[ideal]] generated by elements of the form

$$v \otimes w + (-1)^{deg v \cdot deg w } w \otimes v \, .$$

The product in this algebra is denoted with a wedge, and called the **wedge product**.  It obeys the relation

$$
  v \wedge w = - (-1)^{deg v \cdot deg w} w \wedge v
  \,.
$$

#### With an inner product

If $V$ is equipped with a [[bilinear form]] then there is also the [[Clifford algebra]] on $V$. This reduces to the Grassmann algebra for vanishing bilinear form.

But sometimes it is useful to consider the Grassmann algebra even in the presence of a non-degenerate bilinear form, in which case the inner product still serves to induce identifications between elements of the Grassmann algebra in different degree.

Let $V$ be $\mathbf{R}^3$ equipped with its standard inner product.  Then an element of $\Lambda^0 V$ is a scalar (a [[real number]]), an element of $\Lambda^1 V$ may be identified with a vector in the elementary sense, an element of $\Lambda^2 V$ may be identified with a [[bivector]] or [[pseudovector]], and an element of $\Lambda^3 V$ may be identified a [[pseudoscalar]].

More generally, let $V$ be $\mathbf{R}^n$, or indeed any real [[inner product space]].  Then an element of $\Lambda^p V$ is a $p$-vector as studied in [[geometric algebra]].  Using the inner product, we can identify $p$-vectors with $(n-p)$-pseudovectors.

On a [[manifold]] (or [[generalized smooth space]]) $X$, let $T^*X $ be the [[cotangent bundle]] of $X$.   Then we may define $\Lambda T^*X$ using the abstract nonsense describe earlier, taking $C$ to be the category of [[vector bundle|vector bundles]] over $X$.  Then a [[differential form]] on $X$ is a [[section]] of the vector bundle $\Lambda T^*X $.  If $X$ is an oriented (semi)-[[Riemannian manifold]], then we can identify $p$-forms with $(n-p)$-forms using the [[Hodge star]] operator.

### Semi-free dg-algebras

A [[semifree dga|semi-free dg-algebra]] is a [[dg-algebra]] whose underlying graded commutative algebra is free, i.e. is an exterior algebra. Examples include in particular [[Chevalley-Eilenberg algebra]]s of [[Lie algebra]]s, of $L_\infty$-[[L-∞ algebra|algebras]] and [[Lie ∞-algebroids]].

#### Differential forms / deRham complex

For $X$ a [[manifold]] consider the category of [[module]]s over its [[ring]] of smooth functions $C^\infty(X)$. One such module is $\Omega^1(X) = \Gamma(T^* X)$, the space of smooth sections of the [[cotangent bundle]] of $X$.

The [[deRham complex]] of $X$ is the exterior algebra

$$
  \Omega^\bullet(X) = \bigwedge_{C^\infty(X)} \Gamma(T^* X)
  \,.
$$

This is really a special case of the previous class of examples, as $\Omega^\bullet(X)$ equipped with the [[deRham complex|deRham differential]] is the [[Chevalley-Eilenberg algebra]] of the [[tangent Lie algebroid]].

## Related concepts

* [[decomposable differential forms]]

* [[body]]/[[soul]]

* [[Fock space]], [[exponential modality]]

* [[differential form]], [[de Rham complex]]

* [[Clifford algebra]], [[spin geometry]]

## References

The concept originates in

* {#Grassmann1844} [[Hermann Grassmann]], _[[Ausdehnungslehre]]_, 1844

where the graded-commutativity of the exterior product appears in  §37, §55.

Textbook account:

* [[Igor R. Shafarevich]], [[Alexey O. Remizov]]: §10 in: *Linear Algebra and Geometry* (2012) [[doi:10.1007/978-3-642-30994-6](https://doi.org/10.1007/978-3-642-30994-6), [MAA-review](https://maa.org/press/maa-reviews/linear-algebra-and-geometry)]

For the case of [[modules]] over a [[commutative ring]], see

* [[Stacks Project]], [Tag 00DM](http://stacks.math.columbia.edu/tag/00DM).

* [[Bourbaki]], _Algèbre_, chap. III, § 7.

Discussion of Grassmann algebras [[internalization|internal]] to any [[symmetric monoidal category]] is on p. 165 of

* {#Deligne90} [[Pierre Deligne]], _[[Catégories Tannakiennes]],  [[Grothendieck Festschrift]], vol. II, Birkhäuser Progress in Math. 87 (1990) pp.111-195.

See also at _[[signs in supergeometry]]_.

[[!redirects exterior algebra]]
[[!redirects exterior algebras]]
[[!redirects Grassmann algebra]]
[[!redirects Grassmann algebras]]
[[!redirects alternating power]]
[[!redirects alternating powers]]
[[!redirects exterior power]]
[[!redirects exterior powers]]
[[!redirects exterior product]]
[[!redirects exterior products]]
[[!redirects wedge product]]
[[!redirects wedge products]]


exterior bundle

+-- {: .rightHandSide}
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### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

For $X$ a [[smooth manifold]] its _exterior bundle_ is the [[vector bundle]] $\wedge^\bullet T^\ast X$ which is the [[direct sum]] ([[Whitney sum]]) of the bundle of [[differential n-forms]] for all $n \in \mathbb{N}$.

## Properties

If $(X,g)$ is a [[Riemannian manifold]], then the exterior bundle supports a canonical [[Dirac operator]], namely the [[Kähler-Dirac operator]] $d + d^\dagger$. The corresponding [[Fredholm operator]] $(d+ d^\ast)(1 + (d + d^\ast)^2)^{-1/2}$ constitutes a canonical class in the [[K-homology]] of $X$.

## Related concepts

* [[exterior algebra]]

* [[supersymmetric quantum mechanics]]

[[!redirects exterior bundles]]


exterior covariant derivative

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###Context###
#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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#### Lie theory
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[[!include infinity-Lie theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The combination of an [[exterior derivative]] with a [[covariant derivative]].

## Definition

### Definition on vector bundles
Let $p: E \to M$ be a [[vector bundle]] with a linear [[connection on a bundle|connection]] given by the [[covariant derivative]] $\nabla: \Gamma(T M) \times \Gamma(E) \to \Gamma(E)$. We let $\Omega(M, E)$ be the space of all [[differential form#twisted|differential forms with values in]] $E$. To define the exterior covariant derivative, we take the explicit formula of the [[exterior derivative]], and replace the usual [[derivative]] with the convariant derivative.

+-- {: .num_defn #DerivVect}
###### Definition
The **exterior covariant derivative** $d_\nabla: \Omega^p (M, E) \to \Omega^{p + 1}(M, E)$ is defined by the following formula: given a $p$-form $\Phi \in \Omega^p (M, E)$, its exterior covariant derivative is given by
$$
\begin{aligned}
(\mathrm{d}_\nabla \Phi)(X_0, \dots, X_p) =& \displaystyle\sum_{i = 0}^p (-1)^i \nabla_{X_i}\Phi (X_0, \dots, \hat{X_i}, \dots, X_p) \\
&+ \displaystyle\sum_{i \lt j} (-1)^{i + j} \Phi([X_i, X_j], X_0, \dots, \hat{X_i}, \dots, \hat{X_j}, \dots, X_p),
\end{aligned}
$$
where each $X_i$ is a [[vector field]] on $M$, and $\hat{X}$ means omission of $X$.
=--

In the case of the trivial bundle with the trivial connection, this gives the usual exterior derivative of a vector-valued differential form.

### Definition on principal $G$-bundles

Let $p: P \to M$ be a [[principal bundle|principal]] $G$-[[bundle]]. Let $\omega \in \Omega^1(P,\mathfrak{g})$ be a [[principal connection|connection]] on $P$. Let $H: T P \to T P$ be the horizontal projection given by $\omega$.

We let $V$ be a vector space, and $\rho: G \to \GL(V)$ be a [[representation]] of $G$ on $V$.

+-- {: .num_defn}
###### Definition
A [[differential form]] $\psi \in \Omega^k(P,V)$ is called:

* **horizontal**, if $\psi_q(v_1,\dots,v_k)=0$ whenever one of the vectors $v_i$ is [[vertical vector field|vertical]].

* **equivariant**, if $r_g^{*}\psi = \rho(g^{-1},\psi)$ for all $g\in G$, where $r_g: P \to P$ denotes the right action of $G$ on $P$.
=--

We denote by $\Omega^k_{\rho}(P,V)$ the space of horizontal and equivariant forms. Note that $\Omega_{\rho}(P,V)$ is in general not closed under the ordinary [[exterior derivative]]. There is a canonical isomorphism
$$
\Omega^k_{\rho}(P,V) \cong \Omega^k(M,P \times_{\rho}V),
$$
where $P \times_{\rho}V$ is the vector bundle [[associated bundle|associated]] to $P$ via the representation $\rho$.

+-- {: .num_defn}
###### Definition
 The **exterior covariant derivative** for forms on $P$
$$
\mathrm{d}_{\omega}: \Omega^k(P,V) \to \Omega^{k+1}(P,V)
$$
is defined by
$$
(\mathrm{d}_{\omega}\psi)_q(v_1,\dots,v_k) := (\mathrm{d}\varphi)_q(H v_1,\dots,H v_k).
$$
=--

+-- {: .num_prop}
###### Proposition
Every form in the image of $\mathrm{d}_{\omega}$ is horizontal. If a form $\psi$ is equivariant, $\mathrm{d}_{\omega} \psi$ is also equivariant.
=--

+-- {: .num_prop}
###### Proposition
The [[restriction]] of $\mathrm{d}_{\omega}$ to $\Omega^k_{\rho}(P,V)$ can be described in terms of the connection 1-form $\omega\in \Omega^1(P,\mathfrak{g})$ and the derivative $\mathrm{d}\rho: \mathfrak{g} \times V \to V$ of the representation $\rho$:
$$
\mathrm{d}_{\omega}(\psi) = \mathrm{d}\psi + \omega \wedge_{\mathrm{d}\rho} \psi\text{.}
$$
Here we have used the following general notation: if $U,V,W$ are vector spaces, $\varphi \in \Omega^p(M,V)$, $\psi\in\Omega^q(M,W)$ and $f: V \times W \to U$ is a linear map, we have $\varphi \wedge_{f} \psi \in \Omega^{p+q}(M,U)$.
=--

+-- {: .num_prop}
###### Proposition
Unlike the usual [[exterior derivative]], the exterior covariant derivative need not be [[nilpotent]] in general. Instead, we have
$$
\mathrm{d}_{\omega}(\mathrm{d}_{\omega}(\psi)) = \Omega \wedge_{\mathrm{d}\rho} \psi.
$$
In particular, $\mathrm{d}_{\omega} \circ \mathrm{d}_{\omega} = 0$ if $\omega$ is flat.
=--

+-- {: .num_defn}
###### Definition
The **exterior covariant derivative for forms on $M$**
$$
\mathrm{D}_{\omega}: \Omega^k(M,P \times_{\rho} V) \to \Omega^{k+1}(M,P \times_{\rho}V)
$$
is the map induced from $\mathrm{d}_\omega$ under the isomorphism
$\Omega^k_{\rho}(P,V) \cong \Omega^k(M,P \times_{\rho}V)$.
=--
Note that a connection on the principal bundle $p: P \to M$ [[induced connection|induces]] a connection on the [[associated bundle|associated vector bundle]] $P \times_\rho V$. Then the exterior covariant derivative in the sense of Definition \ref{DerivVect} coincides with this exterior covariant derivative.

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

* The [[connection on a bundle|connection]] $\omega$ itself is not in $\Omega_{\mathrm{Ad}}^1(P,\mathfrak{g})$: it is not horizontal.

* The [[curvature#curvature_of_a_connection|curvature]] of $\omega$ is $\Omega := \mathrm{d}_\omega(\omega)\in \Omega^2_{\mathrm{Ad}}(P,\mathfrak{g})$. Since $\mathrm{d}(\mathrm{Ad})=[-,-]$, we have
$$
\Omega = \mathrm{d}\omega + [\omega \wedge \omega]\text{.}
$$
The [[Bianchi identity]] is $\mathrm{d}_{\omega}\Omega=0$.

Under the isomorphism $\Omega^k_{\rho}(P,V) \cong \Omega^k(M,P \times_{\rho}V)$, the curvature $\Omega$ corresponds to a 2-form $F_{\omega} \in \Omega^2(M,\mathrm{Ad}(P))$, and the [[Bianchi identity]] corresponds to $\mathrm{D}_{\omega}F_{\omega} = 0$.
=--

## Applications

* In [[Yang-Mills theory]]...

## Related Concepts

* [[differential form]]

* [[connection on a bundle]]

* [[covariant derivative]]

* [[Yang-Mills theory]]

[[!redirects covariant exterior derivative]]
[[!redirects exterior covariant derivative]]
[[!redirects Exterior covariant derivative]]
[[!redirects Exterior Covariant Derivative]]