exterior differential system

# Contents #

* table of contents
{:toc}

# Idea #

There are several different ways to think about differential systems:

* the [[category theory|general abstract way]] which we shall put forward here:

  an exterior differential system is a sub-[[Lie-∞-algebroid]] $\mathfrak{a} \hookrightarrow T X$ of
  the [[tangent Lie algebroid]] $T X$ of a [[manifold]] that is
  the [[kernel]] of a morphism $p : T X \to \mathfrak{j}$
  of [[Lie-∞-algebroids]]:

  $$
    \mathfrak{a} := ker(p) \hookrightarrow T X \stackrel{p}{\to}
      \mathfrak{j}
  $$

* In the literature -- the the [references](#references)  below -- the term _exterior differential system_ is instead introduced and understood in the context of [[differential graded algebra|dg-algebra]] as a [[dg-ideal]] $J \subset \Omega^\bullet(X)$ inside the [[deRham dg-algebra]] of $X$ and all concepts there are developed from this perspective.

  From this the above perspective is obtained by noticing that from a dg-ideal $J$
  we are naturally led to form the quotient [[dg-algebra]] $\Omega^\bullet(X)/J$
  which is the [[cokernel]] of the inclusion $p^* : J \hookrightarrow \Omega^\bullet(X)$:

  $$
    J \stackrel{p^*}{\hookrightarrow} \Omega^\bullet(X) \to coker(p^*) =
    \Omega^\bullet(X)/J
    \,.
  $$

  The existing literature on exterior differential systems
  is actually a bit unclear about which
  [additional assumptions](#commonfurtherassumptions) on $J$ are supposed to be a crucial part of the definition. However, in most applications of interest --- see the [examples](#examples) below --- it turns out that $J$ is in fact a [[semifree dga]] (over $C^\infty(X)$).

  Here we take this as indication that

  * it makes good sense to understand exterior differential systems
    in the restricted sense where the [[dg-ideal]] $J$ is required
    to be a [[semifree  dga]];

  * the reason that the existing literature does present the
    desired extra assumptions on the [[dg-ideal]] $J$ in an
    incoherent fashion is due to a lack of global structural
    insight into the role of the definition of exterior differential
    systems.

  Because, recall that a [[Lie-∞-algebroid]] is -- effectively by definition -- the formal dual of a [[semifree dga|semifree]]
  $\mathbb{N}$-graded commutative [[dg-algebra]]. So precisely
  with that extra condition on $J$ all [[dg-algebra]]s in the above
  may be understood as [[Chevalley-Eilenberg algebra]]s of
  [[Lie-∞-algebroids]] and then the above [[cokernel]] sequence
  of [[dg-algebra]]s is precisely the formal dual of the
  [[kernel]] sequence of [[Lie-∞-algebroid]]s.

* Historically, one can trace back the basic idea of
  exterior differential systems to [[Eli Cartan]]'s work on
  [[partial differential equation]]s in terms of [[differential forms]]:

  for each system of partial differential equations

  $$
    \{
      F^\rho(\{x^\mu\}_{\mu}, \{f^j\}_j, \{\frac{\partial f^j}{\partial x^\mu}\} ) = 0
    \}_\rho
  $$

  there is a space $X$ and a [[dg-ideal]] $J \in \Omega^\bullet(X)$ such that solutions of the system of equations are given by **integral manifolds** of the exterior differential system determined by $J$.

The notion of an integral manifold of an exterior differential system
is crucial in the theory: in terms of $J$ it is a morphism
$\phi : \Sigma \to X$ of [[manifold]]s such that the
pullback of the ideal vanishes, $\phi^* J = 0$.

But this says precisely that $\phi$ extends to morphism of [[Lie-∞-algebroids]]

$$
  \phi : T \Sigma \to \mathfrak{a}
$$

with $CE(\mathfrak{a}) = \Omega^\bullet(X)/J$ as above. Therefore the relevance of the notion of _integral manifolds_ in the theory we take as another indication that
exterior differential systems are usefully thought of as being about [[Lie-∞-algebroids]].

# Definition #

+-- {: .num_defn}
###### Definition (exterior differential system)

An **exterior differential system** on a smooth [[manifold]] $X$
is a [[dg-ideal]] $J \subset \Omega^\bullet(X)$ of the
[[deRham dg-algebra]] $\Omega^\bullet(X)$ of $X$.

=--

Notice that $J$ being a [[dg-ideal]] means explicitly that

* $\forall \theta\in J \subset \Omega^\bullet(X), \omega \in \Omega^\bullet(X):
  \theta \wedge \omega \in J$

* the $\mathbb{N}$-grading $J = \oplus_{k \in \mathbb{N}} J_k $ on
  the [[dg-algebra]] $J$
  is induced from that of $\Omega^\bullet(X)$ in that $J_k = J \cap \Omega^\bullet(k)$

* $\forall \theta \in J \subset \Omega^\bullet(X) : d \theta \in J$

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

An **integral manifold** of an exterior differential system is
a submanifold $\phi : Y \hookrightarrow X$ such that the restriction of
all $\theta \in J$ to $Y$ vanishes: $\that|_Y = 0$.

=--

In other words, for an integral manifold the pullback of the
ideal $J$ along the inclusion map $\phi$ vanishes: $\phi^* J = 0$.

## Common further assumptions
  {#commonfurtherassumptions}

Often further assumptions are imposed on exterior differential systems.
Here are some:

* An exterior differential system is called **finitely generated**
if there is a finite set $\{\theta_k \in \Omega^\bullet(X)\}$
of [[differential form]]s such that $J$ is the [[dg-ideal]]
generated by these, so that

  $$
    J = \{ \sum_i f_i \theta_i  + \sum_j g_j d \theta_j| f_i, g_j \in C^\infty(X)\}
    \,.
  $$

* Often it is assumed that $J_0 = \mathbb{R}$.

  Dually in terms of [[Lie-∞-algebroids]]  this assumption means that $J$ is the [[Chevalley-Eilenberg algebra]] of a [[Lie-∞-algebroid]] that is just an [[L-∞-algebra]].

* +-- {: .num_defn}
  ###### Definition (strict independence condition)

  A **strict independence condition** on an exterior differential system $J \subset \Omega^\bullet(X)$ is an $n$-form $\omega \in \Omega^n(X)$ for some $n$ such that

  * $\omega$ is decomposable into a wedge product of $n$ 1-forms mod $J^n$

  * $\omega$ is pointwise not an element of $J$.

  For $(J, \omega)$ am exterior differential system with strict independence condition $\omega$, an **integral manifold** is now more restrictively an integral manifold $\phi : \Sigma \to X$ for $J$ but now such that $\phi^* \omega$ is a volume form on $\Sigma$ (i.e. pointwise non-vanishing).

  =--

# special cases #

Some special types of exterior differential systems carry their own names.

## Frobenius system ##

A **Frobenius system** is an exterior differential system
$J \subset \Omega^\bullet(X)$ that is locally generated as
a graded-commutative algebra from a set $\{\theta_j \in \Omega^1(U)\}_j $
of 1-forms.

Frobenius systems are in bijection with involutive subbundles of the
[[tangent bundle]] of $X$, i.e. subbundles $E \hookrightarrow T X$
such that for $v,w \in \Gamma(E) \subset \Gamma(T X)$ also the
Lie bracket of vector fields of $v$ and $w$ lands in $E$:
$[v,w] \in \Gamma(E) \subset \Gamma(T X)$:

* given a Frobenius system the sections of $\Gamma(E)$ are defined
  locally to be the joint [[kernel]] of the maps $\{\theta_i : \Gamma(T U) \to \mathbb{R}\}$.

* given ab involutive subbundle $E$ the corresponding Frobenius
  system is the collection of 1-forms that vanishes on $E$:

  $J = \{\theta \in \Omega^1(X) | \theta|_{E} = 0\}$.

Notice that the involutive subbundle may be thought of precisely
as a sub-[[Lie algebroid]]

$$
  \array{
     E &&\hookrightarrow&& T X
     \\
     & \searrow && \swarrow
     \\
     && X
  }
$$

of the [[tangent Lie algebroid]] (i.e. as a sub [[Lie-∞-algebroid]] that happens to be an ordinary [[Lie algebroid]]). And indeed, the [[Chevalley-Eilenberg algebra]] of $E$ is the quotient $\Omega^\bullet(X)/J$ of the [[deRham dg-algebra]] by the Frobenius system:

$$
  CE(E) = \Omega^\bullet(X)/J
  \,.
$$

### vertical tangent Lie algebroid ###

A special case of a [[Lie algebroid]] corresponding to a Frobenius system
is the [[vertical tangent Lie algebroid]] $T_{vert} Y$ of a map $\pi : Y \to X$.
This corresponds to the subbundle $ker(\pi_*) \subset T Y$ of vertical
vector fields on $Y$, with respecct to $\pi$. The corresponding
Frobenius system is that of **horizontal differential forms**

$$
  J = \Omega^\bullet_{hor}(Y) = \{\omega \in \Omega^1(Y)| \forall v \in ker(\pi_*): \omega(v) = 0\}
$$

and

$$
  CE(T_{vert} Y) = \Omega^\bullet(Y)/\Omega^\bullet_{hor}(Y)
$$

is the [[dg-algebra]] of **vertical differential forms** with respect to
$Y$.

This plays a central role in the theory of [[Ehresmann connection]]s and
[[schreiber:Cartan-Ehresmann ∞-connection|Cartan-Ehresmann ∞-connection]].

## systems of partial differential equations ##

A system

$$
  \{
    F^\rho : \mathbb{R}^n \times \mathbb{R}^s \times \mathbb{R}^{n \cdot s}
    \to
    \mathbb{R}
  \}_\rho
$$

of [[partial differential equation]]s in terms of variables $\{x^\mu\}_{\mu = 1}^n$ and functions $\{f^i\}_{i = 1}^s$ of the form

$$
  \{
    F^\rho((x^\mu), (f^i), \left(\frac{\partial f^i}{\partial x^\mu}\right))
    = 0
  \}
$$

is encoded by an exterior differential system on the
0-locus

$$
  X := \{(x,f,p) \in \mathbb{R}^n \times \mathbb{R}^s \times \mathbb{R}^{n s} |
    \forall \rho : F^\rho(x,f,p) = 0 \}
$$

of the $\{F^\rho\}_\rho$ (assuming that this is a manifold) with the
[[dg-ideal]] $J = \langle \theta_i \rangle_i$ generated by the 1-forms

$$
  \theta^i := d f^i - \sum_{\mu=1}^n p^i_\mu d x^\mu
  \,.
$$

Namely a solution to the system of partial differential equations is
precisely a [[section]] of the projection

$$
  X \to \mathbb{R}^n
$$

which defined an integral manifold of the exterior differential system.

# References #

The standard textbook is

* Bryant et al., _Exterior differential systems_

Course note are provided in

* Bryant, [[EDS-notes.pdf:file]]

[[!redirects exterior differential systems]]


exterior ring

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

# Contents
* table of contents
{: toc}

## Idea

A $\mathbb{Z}$-[[exterior algebra]].

## Definition

Given an [[abelian group]] $G$, the **exterior ring** $\Lambda(G)$ is the [[quotient ring]] of the [[tensor ring]] $T(G)$ by the [[ideal]] generated by the relations $g \cdot g$ for all $g \in G$.

### Universal property

Given an [[abelian group]] $G$, the **exterior ring** is a [[ring]] $\Lambda(G)$ with an abelian group [[homomorphism]] $g:G \to \Lambda(G)$ such that

* for every element $a:G$, $g(a) \cdot g(a) = 0$

* for every other ring $R$ with abelian group homomorphism $h:G \to R$ where for every element $a:G$, $h(a) \cdot h(a) = 0$, there is a unique ring homomorphism $i:\Lambda(G) \to R$ such that $i \circ g = h$.

## See also

* [[ring]]

* [[tensor ring]]

* [[super ring]]

* [[Clifford ring]]

* [[supercommutative ring]]

* [[differential ring]]

* [[exterior algebra]]

[[!redirects exterior rings]]


external direct sum of vector bundles

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

#Contents#
* table of contents
{:toc}

## Idea

The *external direct sum* or *external Whitney sum* of [[vector bundles]] is the operation which from a [[pair]] of [[vector bundles]] $\mathscr{V} \,\in\,$ [[Vect(X)]], $\mathscr{W} \,\in\,$ [[Vect(X)|$Vect(Y)$]] over possibly different base spaces $X,\ Y \,\in\,$ [[Top]] forms the [[direct sum of vector bundles]] ("[[Whitney sum]]") $(-)\bigoplus(-)$ of their [[pullback of a bundle|pullback]] to the [[product space]] $X \times Y$ of their base spaces:

$$
  \array{
    Vect(X) \times Vect(Y)
    &\oversey{\boxplus}{\longrightarrow}&
    Vect(X\times Y)
    \\
    \big(
      \mathscr{V}
      ,\,
      \mathscr{W}
    \big)
    &\mapsto&
    \big(
      (pr_X)^\ast \mathscr{V}
    \big)
    \oplus_{X \times Y}
    \big(
      (pr_Y)^\ast \mathscr{W}
    \big)
  }
$$

Since the [[Whitney sum]] $\oplus \,\colon\, Vect(X \times Y) \times Vect(X \times Y) \longrightarrow Vect(X \times Y)$ is the [[Cartesian product]] in [[Vect(X)|$Vect(X \times Y)$]], one may understand the external Whitney sum as the Cartesian product in the [[Grothendieck construction]] $\int_{X} Vect(X)$, see the the example [there](Grothendieck+construction#CartesianProductInGrothendieckConstruction).

## Related concepts

* [[external tensor product]]

* [[external tensor product of vector bundles]]

* [[direct sum of vector bundles]]

## References

Discussion in the context of [[topological K-theory]]:

* [[Max Karoubi]], §4.9 in: _K-Theory -- An introduction_, Grundlehren der mathematischen Wissenschaften **226** Springer (1978) [[pdf](https://webusers.imj-prg.fr/~max.karoubi/K.book/MK.book.pdf), [doi:10.1007%2F978-3-540-79890-3](https://link.springer.com/book/10.1007%2F978-3-540-79890-3)]

[[!redirects external direct sums of vector bundles]]

[[!redirects external Whitney sum of vector bundles]]
[[!redirects external Whitney sums of vector bundles]]


external tensor product

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###coContext###
#### Monoidal categories
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[[!include monoidal categories - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The concept of _external tensor product_ is a variant of that of [[tensor product]] in a [[monoidal category]] when the latter is generalized to [[indexed monoidal categories]] ([[dependent linear type theory]]).

## Definition

Consider an [[indexed monoidal category]] given by a [[Cartesian fibration]]

$$
  \array{
    Mod(-)
    \\
    \downarrow
    \\
    \mathbf{H}
  }
$$

over a [[cartesian monoidal category]] $\mathbf{H}$.

\begin{definition}\label{ExternalTensorProduct}
**(external tensor product)**
\linebreak
Given $X_1, X_2 \in \mathbf{H}$ the _external tensor product_ over these is the [[functor]]

$$
  \boxtimes \;\colon\;
  Mod(X_1)\times Mod(X_2)
  \longrightarrow
  Mod(X_1 \times X_2)
$$

given  on $A_1 \in Mod(X_1)$ with $A_2 \in Mod(X_2)$ by

$$
  A_1 \boxtimes A_2 \coloneqq (p_1^\ast A_1) \otimes_{X_1 \times X_2} (p_2^\ast A_2)
  \in Mod(X_1 \times X_2)
  \,,
$$

where $p_1, p_2$ denote the [[projection]] maps out of the [[Cartesian product]] $X_1 \times X_2 \in \mathbf{H}$.
\end{definition}

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

The external tensor product constitutes a [[tensor product]] on the total category $Mod$ of the given [[Grothendieck fibration]] $Mod(-)\to \mathbf{H}$; and with respect to this it is a [[monoidal fibration]].

=--

## Properties

### Relation to fiberwise tensor product

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

The fiberwise ("internal") tensor product over $X\in \mathbf{H}$ is recovered form the external one via a [[natural equivalence]]

$$
  A_1 \otimes_X A_2 \simeq \Delta_X^\ast (A_1 \boxtimes A_2)
$$

for $A_1, A_2 \in Mod(X)$, where $\Delta \colon X \longrightarrow X \times X$ is the [[diagonal]] in $\mathbf{H}$ on $X$.

=--

### Generation of $Mod(X_1 \times X_2)$ from external tensor products

Under suitable conditions on compact generation of $Mod(-)$ then one may deduce that $Mod(X_1 \times X_2)$ is generated under external product from
$Mod(X_1)$ and $Mod(X_2)$.

([Bondal-vdBerg 03](#BondalvdBerg03), [BFN 08, proof of prop. 3.24](#BFN08))

### In indexed monoidal categories
 {#CocontinuityAndAdjoints}

Suppose an [[indexed monoidal category]] which satisfies the [[motivic yoga]] ([[Wirthmüller context]]-form) in that:

1. the corresponding [[pseudofunctor]] takes values in [[adjoint functors]]

   $$
     \array{
       \mathllap{
         \mathbf{C} \,\colon\,
         \;
       }
       Base &\longrightarrow& Cat
       \\
       \mathcal{X} &\mapsto& \mathbf{C}_{\mathcal{X}}
       \\
       \Big\downarrow\mathrlap{{}^{f}}
       &&
       \mathllap{^{f_!}}\Big\downarrow
       \dashv
       \Big\uparrow\mathrlap{{}^{f^\ast}}
       \\
       \mathcal{Y}
       &\mapsto&
       \mathbf{C}_{\mathcal{Y}}
     }
   $$

1. between [[closed monoidal categories]]

1. the [[base change]] functors $f^\ast$ are

   1. besides being [[strong monoidal functors]]

   1. also [[strong closed functors]],

      hence the pushforward functors $f_!$ satisfy the [[projection formula]] ("[[Frobenius reciprocity]]")

   1. [[preserving colimits]]

1. pull-push through [[cartesian squares]] in $Base$ satisfies the [[Beck-Chevalley condition]].

Also assume that

* $Base$ is [[cocomplete]] and [[cartesian closed category|cartesian closed]],

* $\mathbf{C}_{\mathcal{X}}$ is cocomplete for each $\mathcal{X} \in Base$.

Then:

\begin{proposition}
  The external tensor product (Def. \ref{ExternalTensorProduct}) on the [[Grothendieck construction]] $\int \mathbf{C}$ [[preserves colimits]] in each variable.
\end{proposition}
\begin{proof}

Consider any [[object]]
$$
  \mathcal{W}_{\mathcal{Y}}
    \,\in\,
  \textstyle{\int} \mathbf{C}
$$
and any [[diagram]]
$$
  \begin{array}{ccc}
    \mathcal{V}_{\mathcal{X}}
    \,\colon\,
    I &\longrightarrow& \int \mathbf{C}
    \\
    i &\mapsto& \mathcal{V}(i)_{\mathcal{X}_i}
  \end{array}
$$
in the Grothendieck construction category $\int \mathbf{C}$.

Then with

* the description of colimits in Grothendieck constructions as (see [there](Grothendieck+construction#CoLimitsInAGrothendieckConstruction))

  $$
    \underset{\underset{i \in I}{\longrightarrow}}{lim}
    \big(
      \mathcal{V}(i)_{\mathcal{X}_i}
    \big)
    \;\simeq\;
    \left(
      \underset{\underset{i \in I}{\longrightarrow}}{lim}
      q^{\mathcal{X}_i}_!
      \mathcal{V}(i)
    \right)_{
      \underset{\underset{i \in I}{\longrightarrow}}{lim}
      \mathcal{X}_i
    }
  $$

  where

  $$
    q^{\mathcal{X}_i}
    \;\colon\;
    \mathcal{X}_i
      \longrightarrow
    \underset{\longrightarrow}{lim} \mathcal{X}
  $$

  denote the [[coprojections]] of the [[colimit]] of the [[underlying]] [[diagram]] in $Base$,

* the [[Beck-Chevalley condition]] for the following [[cartesian squares]] in $Base$

  $$
    \array{
      \mathcal{X}_i
      \times
      \mathcal{Y}
      &
      \overset{\; pr_{\mathcal{X}_i} \;}{\longrightarrow}
      &
      \mathcal{X}_i
      \\
      \mathllap{{q}^{\mathcal{X}_i} \times id_{\mathcal{Y}}}
      \Big\downarrow
      &&
      \Big\downarrow
      \mathrlap{{}^{ q^{\mathcal{X}_i} }}
      \\
      \underset{\underset{j \in I}{\longrightarrow}}{lim}
      \mathcal{X}_j
         \times
      \mathcal{Y}
      &
      \underset
        {pr_{\underset{\longrightarrow}{lim}\mathcal{X}}}
        {\longrightarrow}
      &
      \underset{\underset{j \in I}{\longrightarrow}}{lim}
      \mathcal{X}_j
      \mathrlap{\,,}
    }
  $$

* the fact that both $(-) \times \mathcal{Y}$ and $(-)\otimes (pr_{\mathcal{Y}})^\ast\mathscr{W}$ [[preserve colimits]] ([[left adjoints preserve colimits|being left adjoints]])

we obtain the following sequence of [[natural isomorphisms]]:

$$
  \begin{array}{ll}
  \Big(
  \underset{\underset{i \in I}{\longrightarrow}}{lim}
  \mathscr{V}(i)_{\mathcal{X}_i}
  \Big)
  \boxtimes
  \mathscr{W}_{\mathcal{Y}}
  \\
  \;\simeq\;
  \left(
  \big(
    \underset{\longrightarrow}{\lim}
    q^{\mathcal{X}}_!\mathscr{V}
  \big)_{\underset{\longrightarrow}{\lim}\mathcal{X}}
  \right)
  \boxtimes
  \mathscr{W}_{\mathcal{Y}}
  &
  \text{colimit in Groth. constr.}
  \\
  \;\simeq\;
  \Big(
    \big(
    (pr_{\underset{\longrightarrow}{lim}\mathcal{X}})^\ast
    (
      \underset{\longrightarrow}{\lim}
      q^{\mathcal{X}}_!\mathscr{V}
    )
    \big)
    \,\otimes\,
    \big(
    (pr_{\mathcal{Y}})^\ast
    \mathscr{W}
    \big)
  \Big)_{
    \big(\underset{\longrightarrow}{\lim}\mathcal{X}\big)
    \times
    \mathcal{Y}
  }
  &
  \text{def. of external tensor}
  \\
  \;\simeq\;
  \Big(
    \big(
      \underset{\longrightarrow}{\lim}
      (pr_{\underset{\longrightarrow}{lim}\mathcal{X}})^\ast
      q^{\mathcal{X}}_!
      \mathscr{V}
    \big)
    \,\otimes\,
    \big(
      (pr_{\mathcal{Y}})^\ast
      \mathscr{W}
    \big)
  \Big)_{
    \big(\underset{\longrightarrow}{\lim}\mathcal{X}\big)
    \times
    \mathcal{Y}
  }
  &
  \text{pullback preserves colimits}
  \\
  \;\simeq\;
  \Big(
    \big(
      \underset{\longrightarrow}{\lim}
      (q^{\mathcal{X}} \times id_{\mathcal{Y}} )_!
      (pr_{\mathcal{X}})^\ast
      \mathscr{V}
    \big)
    \,\otimes\,
    \big(
      (pr_{\mathcal{Y}})^\ast
      \mathscr{W}
    \big)
  \Big)_{
    \big(\underset{\longrightarrow}{\lim}\mathcal{X}\big)
    \times
    \mathcal{Y}
  }
  &
  \text{Beck-Chevalley}
  \\
  \;\simeq\;
  \bigg(
    \underset{\longrightarrow}{\lim}
    \Big(
    \big(
      (q^{\mathcal{X}} \times id_{\mathcal{Y}} )_!
      (pr_{\mathcal{X}})^\ast
      \mathscr{V}
    \big)
    \,\otimes\,
    \big(
      (pr_{\mathcal{Y}})^\ast
      \mathscr{W}
    \big)
    \Big)
  \bigg)_{
    \underset{\longrightarrow}{\lim}
    \big(
    \mathcal{X}
    \times
    \mathcal{Y}
    \big)
  }
  &
  \text{tensoring preserves colimits}
  \\
  \;\simeq\;
  \bigg(
    \underset{\longrightarrow}{\lim}
    (q^{\mathcal{X}} \times id_{\mathcal{Y}} )_!
    \Big(
    \big(
      (pr_{\mathcal{X}})^\ast
      \mathscr{V}
    \big)
    \,\otimes\,
    \big(
      (pr_{\mathcal{Y}})^\ast
      \mathscr{W}
    \big)
    \Big)
  \bigg)_{
    \underset{\longrightarrow}{\lim}
    \big(
    \mathcal{X}
    \times
    \mathcal{Y}
    \big)
  }
  &
  \text{projection formula}
  \\
  \;\simeq\;
  \underset{\underset{i \in I}{\longrightarrow}}{lim}
  \bigg(
  \Big(
    \big(
      (pr_{\mathcal{X}_i})^\ast \mathscr{V}(i)
    \big)
    \,\otimes\,
    \big(
      (pr_{\mathcal{Y}})^\ast \mathscr{W}
    \big)
  \Big)_{
    \mathcal{X}_i
    \times
    \mathcal{Y}
  }
  \bigg)
  &
  \text{colimit in Groth. constr.}
  \\
  \;\simeq\;
  \underset{\underset{i \in I}{\longrightarrow}}{lim}
  \Big(
    \mathscr{V}(i)_{\mathcal{X}_i}
    \,\boxtimes\,
    \mathscr{W}_{\mathcal{Y}}
  \Big)
  &
  \text{def of external tensor.}
  \end{array}
$$
\end{proof}

The following proposition still assumes the "motivic yoga" [above](#CocontinuityAndAdjoints), but in fact we need to assume the [[Beck-Chevalley condition]] only for the very special squares of this form:
$$
  \array{
    \mathcal{X} \times \mathcal{Y}
    &\overset{ f \times id }{\longrightarrow}&
    \mathcal{X}' \times \mathcal{Y}
    \\
    \mathllap{{}^{ pr_{\mathcal{X}} }}
    \Big\downarrow
      &&
    \Big\downarrow
    \\
    \mathcal{X}
      &\underset{\;\;\; f \;\;\;}{\longrightarrow}&
    \mathcal{X}'
  }
$$

\begin{proposition}
\label{PullAndPushAlongProductsOfMaps}
Given
$$
\array{
  f \,\colon\, \mathcal{X} &\longrightarrow& \mathcal{X}'
\\
g \,\colon\, \mathcal{Y} &\longrightarrow& \mathcal{Y}'
}
$$

we have

1. for $\mathscr{V} \,\in\, \mathbf{C}_{\mathcal{X}'}$ and $\mathscr{W} \,\in\, \mathbf{C}_{\mathcal{Y}'}$ [[natural isomorphism]] of this form:
   \[
     \label{PullbackOfExternalTensorAlongProductOfMaps}
     (f \times g)^\ast (\mathscr{V} \boxtimes \mathscr{W})
     \;\simeq\;
     \big(f^\ast \mathscr{V}\big)
     \boxtimes
     \big(g^\ast \mathscr{W}\big)
   \]

1. for $\mathscr{V} \,\in\, \mathbf{C}_{\mathcal{X}}$ and $\mathscr{W} \,\in\, \mathbf{C}_{\mathcal{Y}}$ [[natural isomorphism]] of this form:

   \[
     \label{PushforwardOfExternalTensorAlongProductOfMaps}
     (f \times g)_! (\mathscr{V} \boxtimes \mathscr{W})
     \;\simeq\;
     \big(f_! \mathscr{V}\big)
     \boxtimes
     \big(g_! \mathscr{W}\big)
     \mathrlap{\,.}
   \]

\end{proposition}
(The first statement is essentially immediate from the fact that pullback $(-)^*$ is assumed to be strong closed, but the second statement is not quite so immediate; it is discussed for the case of [[smash product]] of [[retractive spaces]] and [[parameterized spectra]] in [May & Sigurdsson (2006), Rem. 2.5.8, Prop. 13.7.2](#MaySigurdsson06), see also [Malkiewich (2019), Lem. 3.4.1](#Malkiewich19), [Malkiewich (2023), Lem. 2.5.1](#Malkiewich23), and is  mentioned in generality but without proof in [Shulman (2012), p. 624](#Shulman12).)
\begin{proof}
For the first statement (eq:PullbackOfExternalTensorAlongProductOfMaps) we have the following sequence of [[natural isomorphisms]]:
$$
  \begin{array}{ll}
      (f \times g)^\ast
      \big(
        \mathscr{V}
        \,\boxtimes\,
        \mathscr{W}
      \big)
      \\
      \;\simeq\;
      (f \times g)^\ast
      \Big(
        \big(
          \mathrm{pr}_{\mathbf{Y}}^\ast
          \mathscr{V}
        \big)
        \,\otimes_{\mathbf{Y} \times \mathbf{Y}'}\,
        \big(
          \mathrm{pr}_{\mathbf{Y}'}^\ast
          \mathscr{W}
        \big)
      \Big)
      &
      \text{by definition}
      \\
      \;\simeq\;
      \big(
        (f \times g)^\ast
        \mathrm{pr}_{\mathbf{Y}}^\ast
        \mathscr{V}
      \big)
      \otimes_{\mathbf{X} \times \mathbf{X}'}
      \big(
        (f \times g)^\ast
        \mathrm{pr}_{\mathbf{Y}'}^\ast
        \mathscr{W}
      \big)
      &
      \text{since pullback is strong closed}
      \\
      \;\simeq\;
      \big(
        \mathrm{pr}_{\mathbf{X}}^\ast
        f^\ast
        \mathscr{V}
      \big)
      \otimes_{\mathbf{X} \times \mathbf{X}'}
      \big(
        \mathrm{pr}_{\mathbf{X}'}^\ast
        g^\ast
        \mathscr{W}
      \big)
      &
      \text{by pseudo-functoriality}
      \\
      \;\simeq\;
      \big(f^\ast \mathscr{V}\big)
      \boxtimes
      \big(g^\ast \mathscr{W}\big)
      &
      \text{by definition.}
    \end{array}
$$

For the second statement (eq:PushforwardOfExternalTensorAlongProductOfMaps), first notice the special case where one of the maps is an [[identity morphism]] and the corresponding external tensor factor is the [[tensor unit]]; and notice here that external tensoring with the tensor unit is just pullback to a product (since pullback along the other leg preserves tensor units, being strong monoidal):
$$
  \mathscr{V} \boxtimes \mathbb{1}
  \;\simeq\;
  (pr_{\mathcal{X}})^\ast \mathscr{V}
  \,.
$$
This way we first find
$$
  \begin{array}{ll}
    (f \times id)_!
    \big(
      \mathscr{V} \boxtimes \mathbb{1}
    \big)
    \\
    \;\simeq\;
    (f \times id)_!
    \big(
      pr_{\mathcal{X}}^\ast \mathscr{V}
    \big)
    &
    \text{ by the above comment }
    \\
    \;\simeq\;
    pr_{\mathcal{X}'}^\ast\big(f_! \mathscr{V}\big)
    &
    \text{ using the BC-condition }
    \\
    \;\simeq\;
    (f_! \mathscr{V}) \boxtimes \mathbb{1}
    &
    \text{ by the above comment }
    \,,
  \end{array}
$$
from which the general case is obtained as follows:
$$
  \begin{array}{ll}
    (f \times g)_!
    \big(
      \mathscr{V}
      \boxtimes
      \mathscr{W}
    \big)
    \\
    \;\simeq\;
    (f \times g)_!
    \big(
      (\mathscr{V} \boxtimes \mathbb{1})
      \otimes
      (\mathbb{1} \boxtimes \mathscr{W})
    \big)
    &
    \text{by the above comment}
    \\
    \;\simeq\;
    (f \times g)_!
    \Big(
      \big(
      (id \times g)^\ast
      (\mathscr{V} \boxtimes \mathbb{1})
      \big)
      \otimes
      (\mathbb{1} \boxtimes \mathscr{W})
    \Big)
    &
    \text{by first claim and strong monoidalness}
    \\
    \;\simeq\;
    (f \times id)_!
    (id \times g)_!
    \Big(
      \big(
      (id \times g)^\ast
      (\mathscr{V} \boxtimes \mathbb{1})
      \big)
      \otimes
      \big(\mathbb{1} \boxtimes \mathscr{W}\big)
    \Big)
    &
    \text{by pseudo-functoriality}
    \\
    \;\simeq\;
    (f \times id)_!
    \Big(
      \big(\mathscr{V} \boxtimes \mathbb{1}\big)
      \otimes
      \big(
        (id \times g)_!
        \big(\mathbb{1} \boxtimes \mathscr{W}\big)
      \big)
    \Big)
    &
    \text{by the projection formula}
    \\
    \;\simeq\;
    (f \times id)_!
    \Big(
      \big(\mathscr{V} \boxtimes \mathbb{1}\big)
      \otimes
      \big(
        \mathbb{1} \boxtimes (g_!\mathscr{W})
      \big)
    \Big)
    &
    \text{by the special case above}
    \\
    \;\simeq\;
    (f \times id)_!
    \Big(
      \big(\mathscr{V} \boxtimes \mathbb{1}\big)
      \otimes
      (f \times id)^\ast
      \big(
        \mathbb{1} \boxtimes (g_!\mathscr{W})
      \big)
    \Big)
    &
    \text{by first claim and strong monoidalness}
    \\
    \;\simeq\;
    \Big(
      (f \times id)_!
      \big(\mathscr{V} \boxtimes \mathbb{1}\big)
    \Big)
      \otimes
      \big(
        \mathbb{1} \boxtimes (g_!\mathscr{W})
      \big)
    &
    \text{by the projection formula}
    \\
    \;\simeq\;
    \big(
      (f_!\mathscr{V}) \boxtimes \mathbb{1}
    \big)
    \otimes
    \big(
      \mathbb{1} \boxtimes (g_!\mathscr{W})
    \big)
    &
    \text{by the special case above}
    \\
    \;\simeq\;
    (f_! \mathscr{V})
    \boxtimes
    (g_! \mathscr{W})
    &
    \text{by the above comment}.
  \end{array}
$$
\end{proof}

\begin{corollary}
\label{PullPushAdjunctsOfExternalTensorProducts}
  The $\big((f \times g)_! \dashv (f \times g)^\ast\big)$-[[adjunct]] $\widetilde{f \boxtimes g}$ of an external tensor product of morphisms into the separate pullbacks
$$
  \mathscr{V}
    \boxtimes
  \mathscr{W}
  \overset{ \phi \boxtimes \gamma  }{\longrightarrow}
  (f^\ast \mathscr{V}') \boxtimes (g^\ast \mathscr{W}')
  \;\simeq\;
  (f \times g)^\ast
  (\mathscr{V}' \boxtimes \mathscr{W}')
$$
is the external tensor product of the separate [[adjuncts]]
$$
  \widetilde{ f \boxtimes g }
  \,\colon\,
  (f \times g)_!( \mathscr{V} \boxtimes \mathscr{W} )
  \,\simeq\,
  (f_! \mathscr{V}) \boxtimes (g_! \mathscr{W})
  \overset{
    \widetilde \phi \,\boxtimes\, \widetilde \gamma
  }{\longrightarrow}
  \mathscr{V}' \boxtimes \mathscr{W}'
  \,,
$$
where the natural isomorphisms shown are those form Prop. \ref{PullAndPushAlongProductsOfMaps}.
\end{corollary}
\begin{proof}
After restriction along the (non-full) [[subcategory]]-inclusion
$$
  \array{
    \mathbf{C}_{\mathcal{X}}
    \times
    \mathbf{C}_{\mathcal{Y}}
    &\longrightarrow&
    \mathbf{C}_{\mathcal{X} \times \mathcal{Y}}
    \\
    \big(\mathscr{V},\, \mathscr{W}\big)
    &\mapsto&
    \mathscr{V} \boxtimes \mathscr{W}
    \\
    \mathllap{{}^{ (\phi, \gamma) }}
    \Big\downarrow
    &&
    \Big\downarrow
    \mathrlap{{}^{ \phi \boxtimes \gamma }}
    \\
    \big(\mathscr{V}',\, \mathscr{W}'\big)
    &\mapsto&
    \mathscr{V}' \boxtimes \mathscr{W}'
  }
$$
we can clearly make the restriction of the functors $(f \times g)_!$ and $(f \times g)^\ast$ into adjoints by declaring the [[counit of an adjunction|adjunction counit]] $\epsilon$ to be the external tensor product of the $(f_! \dashv f^\ast)$- with the $(f_! \dashv g^\ast)$-[[adjunction counits]]. But by uniqueness of adjoints ([here](adjoint+functor#UniquenessOfAdjoints)) this must be isomorphic to the actual adjunction counit restricted to external tensor products:
$$
  \epsilon^{
    \big((f \times g)_! \dashv (f \times g)^\ast\big)
  }_{\mathcal{X} \boxtimes \mathcal{Y}}
  \;\;
  \simeq
  \;\;
  \epsilon^{ (f_! \dashv f^\ast) }_{\mathcal{X}}
  \,\boxtimes\,
  \epsilon^{ (g_! \dashv g^\ast) }_{\mathcal{Y}}
  \,.
$$
Now using the expression on the right together with Prop. \ref{PullAndPushAlongProductsOfMaps}
in the formula ([here](adjoint+functor#GeneralAdjunctsInTermsOfAdjunctionUnitCounit)) that expresses [[adjuncts]] as functor-images composed with the (co)unit gives that the adjunct is formed external tensor-factor wise, as claimed:
$$
  \begin{array}{ll}
    \widetilde{ \phi \boxtimes \gamma }
    \\
    \;\simeq\;
    \epsilon^{
      (f \times g)_! \dashv (f \times g)^\ast
    }_{ \mathscr{V}' \boxtimes \mathscr{W}' }
    \circ
    (f \times g)_!\big(\phi \boxtimes \gamma\big)
    &
    \text{ by the formula for adjuncts }
    \\
    \;\simeq\;
    \big(
      \epsilon^{ f_! \dashv f^\ast }_{\mathscr{V}'}
        \circ
      (f_! \phi)
    \big)
    \,\boxtimes\,
    \big(
      \epsilon^{ g_! \dashv g^\ast }_{\mathscr{W}'}
        \circ
      (g_! \gamma)
    \big)
    &
    \text{ by the previous discussion }
    \\
    \;\simeq\;
    \widetilde \phi
    \,\boxtimes\,
    \widetilde \gamma
    &
    \text{ by the formula for adjuncts. }
  \end{array}
$$
\end{proof}
\begin{proposition}
\label{ExternalPushoutProduct}
**(external [[pushout-product]])**
\linebreak
  In the situation above, the $\boxtimes$-[[pushout-product]] (with respect to the external tensor product) is given by
$$
  \big(
    \phi_f
  \big)
    \,\widehat{\boxtimes}\,
  \big(
    \gamma_g
  \big)
  \;\;\;
  \simeq
  \;\;\;
  \Big(
    \big(
      (pr_{X'})^\ast \phi
    \big)
    \,\widehat{\otimes}\,
    \big(
      (pr_{Y'})^\ast \gamma
    \big)
  \Big)_{ f \,\widehat{\times}\, g }
$$
\end{proposition}
where here we use the left-handed convention for component maps in morphisms in the Grothendieck construction:
$$
  \phi_f \,\colon\, \mathscr{V}_X \to \mathscr{V}'_{X'}
  \;\;\;\;\;\;
  \text{stands for}
  \;\;\;\;\;\;
  \begin{array}{rcl}
    X &\xrightarrow{f}& X'
    \\
    f_!\mathscr{V} &\xrightarrow{\phi}& \mathscr{V}'
  \end{array}
$$
\begin{proof}
By the general formula for colimits in Grothendieck constructions ([here](Grothendieck+construction#CoLimitsInAGrothendieckConstruction)) the underlying colimit in the base category is the evident one
\begin{tikzcd}
  X \times Y
  \ar[r, "{ \mathrm{id}\,\times\, g }"{description}]
  \ar[d, "{ f \,\times\, \mathrm{id} }"{description}]
  &
  X \times Y'
  \ar[d, "{ q_r }"{description}]
  \ar[ddr, bend left=15, "{ f \,\times\, \mathrm{id} }"{description}]
  &[-40pt]
  \\
  X' \times Y
  \ar[r, "{ q_l }"{description}]
  \ar[drr, bend right=15, "{ \mathrm{id} \,\times\, g}"{description}]
  &
  f\,\widehat{\times}\,g
  \ar[dr, dashed]
  \\[-20pt]
  && X' \times Y'
\end{tikzcd}
and the component map over the dashed morphism is the colimiting cocone of the diagram obtained from pushing the separate component maps along the [[coprojections]] $q_\cdot$ to form a span in $\mathbf{C}_{f \widehat{\times} g}$. This yields:
$$
  \begin{array}{ll}
  (f \widehat{\times} g)_!
  \Bigg(
    \bigg(
      (q_l)_!
      \Big(
        \big(
          (pr_{X'})^\ast \phi
        \big)
        \otimes
        \big(
          (pr_{Y})^\ast
          \mathrm{id}_{\mathscr{W}}
        \big)
      \Big)
    \bigg)
    \wedge
    \bigg(
      (q_r)_!
      \Big(
        \big(
          (pr_X)^\ast
          \mathrm{id}_{\mathscr{V}}
        \big)
        \otimes
        \big(
          (pr_{Y'})^\ast \gamma
        \big)
      \Big)
    \bigg)
  \Bigg)
  \\
  \;\simeq\;
  \Bigg(
    \bigg(
      (f \widehat{\times} g)_!
      (q_l)_!
      \Big(
        \big(
          (pr_{X'})^\ast \phi
        \big)
        \otimes
        \big(
          (pr_{Y})^\ast
          \mathrm{id}_{\mathscr{W}}
        \big)
      \Big)
    \bigg)
    \wedge
    \bigg(
      (f \widehat{\times} g)_!
      (q_r)_!
      \Big(
        \big(
          (pr_X)^\ast
          \mathrm{id}_{\mathscr{V}}
        \big)
        \otimes
        \big(
          (pr_{Y'})^\ast \gamma
        \big)
      \Big)
    \bigg)
  \Bigg)
  &
  \begin{array}{l}
  \text{since the left adjoint}\; (-)_!
  \\
  \text{preserves pushouts}
  \end{array}
  \\
  \;\simeq\;
  \Bigg(
    \bigg(
      (id \otimes g)_!
      \Big(
        \big(
          (pr_{X'})^\ast \phi
        \big)
        \otimes
        \big(
          (pr_{Y})^\ast
          \mathrm{id}_{\mathscr{W}}
        \big)
      \Big)
    \bigg)
    \wedge
    \bigg(
      (f \otimes id)_!
      \Big(
        \big(
          (pr_X)^\ast
          \mathrm{id}_{\mathscr{V}}
        \big)
        \otimes
        \big(
          (pr_{Y'})^\ast \gamma
        \big)
      \Big)
    \bigg)
  \Bigg)
  &
  \begin{array}{l}
  \text{by commutativity of}
  \\
  \text{the above diagram}
  \end{array}
  \\
  \;\simeq\;
  \Bigg(
    \bigg(
      \Big(
        \big(
          (pr_{X'})^\ast \phi
        \big)
        \otimes
        \big(
          (pr_{Y})^\ast
          \mathrm{id}_{f_!\mathscr{W}}
        \big)
      \Big)
    \bigg)
    \wedge
    \bigg(
      \Big(
        \big(
          (pr_X)^\ast
          \mathrm{id}_{f_!\mathscr{V}}
        \big)
        \otimes
        \big(
          (pr_{Y'})^\ast \gamma
        \big)
      \Big)
    \bigg)
  \Bigg)
   &
  \text{by the above result}
  \\
  \;\simeq\;
  \big(
    (pr_{X'})^\ast \phi
  \big)
  \,\displaystyle{\widehat{\otimes}}\,
  \big(
    (pr_{Y'})^\ast \gamma
  \big)
  &
  \text{by definition}
  \end{array}
$$
Here in the second-but-last step we used (eq:PushforwardOfExternalTensorAlongProductOfMaps).
\end{proof}

\linebreak

## Examples
 {#Examples}

\begin{example}
**([[external tensor product of vector bundles]])**
\linebreak

Given a [[pair]] of [[topological spaces]] $X,\,Y \,\in\, Top$ and a respective [[pair]] of [[topological vector bundles]] $\mathscr{V}_X \,\in\, $ [[Vect(X)|$Vect(X)$]] and $\mathscr{V}_X \,\in\, $ [[Vect(X)|$Vect(X)$]], their [[external tensor product of vector bundles|external tensor product]] is formed via [[pullback of bundles]] $(-)^\ast$ and ordinary [[tensor product of vector bundles]] $(-) \otimes (-)$ by the formula
$$
  \mathscr{V}_X
    \,\boxtimes\,
  \mathscr{W}_Y
  \;\;
  \equiv
  \;\;
  \Big(
  \big(
    (pr_X)^\ast \mathscr{V}
  \big)
  \otimes
  \big(
    (pr_Y)^\ast \mathscr{W}
  \big)
  \Big)_{X \times Y}
  \;\;\;
  \in
  \;
  Vect(X \times Y)
  \,,
$$
where we are denoting by
$$
  \array{
    && X \times Y
    \\
    &
    \mathllap{{}^{pr_X}}\swarrow
    &&
    \searrow\mathrlap{{}^{pr_Y}}
    \\
    X && && Y
  }
$$
the product [[projection]] maps in [[Top]].

This is a [[vector bundle]] over the [[product topological space]] $X \times Y$ whose [[fiber]] over a point $(x,y) \,\in\, X \times Y$ is the [[tensor product of vector spaces]] $\mathscr{V}_x \otimes \mathscr{W}_y$.
\end{example}

\begin{example}\label{CartesianProductInGrothendieckConstruction}
**([[Cartesian product]] in a [[Grothendieck construction]] is external cartesian product)**
\linebreak
  Given a [[contravariant functor|contravariant]] [[pseudofunctor]]
  $$
    \array{
       X
       \\
       \Big\downarrow\mathrlap{{}^{f}}
       \\
       Y
    }
    \;\;\;\;\;\mapsto\;\;\;
    \array{
       \mathcal{C}_X
       \\
       \Big\uparrow\mathrlap{{}^{f^\ast}}
       \\
       \mathcal{C}_Y
    }
  $$
  where the base category and all fiber categories $\mathcal{C}_{(-)}$ have [[Cartesian products]] and all [[base change]] morphisms $f^\ast$ [[preserved limit|preserve]] these products, then the [[Grothendieck construction]] $\int_X \mathcal{C}_X$ has cartesian products
(as a special case of a general result on limits in Grothendieck constructions, discussed [there](Grothendieck+construction#CoLimitsInAGrothendieckConstruction))
which is given on objects
$$
  \mathscr{V}_X \,\equiv\,
  \big(
    \mathscr{V} \,\in\, \mathcal{C}_X
  \big)
$$
by the formula
\[
  \label{CartesianProductInGrothendieckConstruction}
  \mathscr{V}_X
  \times
  \mathscr{W}_Y
  \;\;
  \simeq
  \;\;
  \Big(
    \big(pr_X^\ast \mathscr{V}\big)
    \,\times\,
    \big(pr_Y^\ast \mathscr{W}\big)
  \Big)_{X \times Y}
  \,.
\]
This is the general form of what might be called the "external cartesian product"; see also [this Proposition](free+coproduct+completion#FreeDistributiveCategoryMonad) at *[[free coproduct completion]]*.
\end{example}

## Related concepts

* [[external tensor product of vector bundles]], [[VectBund]]

* [direct product group representations](direct+product+group#Representations)

* [[doubly closed monoidal category]]

## References

The notion of the [[external tensor product of vector bundles]] originates in discussion of [[topological K-theory]]:

*  {#Atiyah67}  [[Michael Atiyah]], §2.6 in: *K-theory*, Harvard Lecture 1964 (notes by D. W. Anderson), Benjamin (1967) [[pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/atiyahk.pdf), [[AtiyahKTheory.pdf:file]]]

* {#Bott69} [[Raoul Bott]], p. 19 of: *Lectures on $K(X)$*, Benjamin (1969) [[pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/bottk.pdf), [[Bott-KTheory.pdf:file]]]

* {#Karoubi} [[Max Karoubi]], §4.9 in: _K-Theory -- An introduction_, Grundlehren der mathematischen Wissenschaften **226**, Springer (1978) [[pdf](https://webusers.imj-prg.fr/~max.karoubi/K.book/MK.book.pdf), [doi:10.1007/978-3-540-79890-3](https://doi.org/10.1007/978-3-540-79890-3)]

The notion of [[external tensor product of representations]] seems to be [[folklore]], it is mentioned in most textbooks but without any attribution:

* [[William Fulton]], [[Joe Harris]], Exc. 2.36 in: *Representation Theory: a First Course*, Springer (1991) [[doi:10.1007/978-1-4612-0979-9](https://link.springer.com/book/10.1007/978-1-4612-0979-9)]

* Caroline Gruson, Vera Serganova, p. 3 of: *From Finite Groups to Quivers via Algebras -- A Journey Through Representation Theory*, Springer (2018) [[doi:10.1007/978-3-319-98271-7](https://doi.org/10.1007/978-3-319-98271-7)]

The external tensor product of [[perverse sheaves]]:

* [[Volodymyr Lyubashenko]], *External tensor product of categories of perverse sheaves*, Ukr. Math. J. **53** 3 (2001) 311-322 [[arXiv:math/9911207](https://arxiv.org/abs/math/9911207)]

The external tensor product of of [[quasicoherent sheaves]]  in ([[derived algebraic geometry|derived]]) [[algebraic geometry]]:

* {#BondalvdBerg03} [[Alexei Bondal]], [[Michel Van den Bergh]], _Generators and representability of functors in commutative and noncommutative geometry_, Mosc. Math. J. **3** 1 (2003) 1-36 [[arXiv:math/0204218](https://arxiv.org/abs/math/0204218)]

* {#BFN08} [[David Ben-Zvi]], [[John Francis]], [[David Nadler]], _Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry_, J. Amer. Math. Soc. **23** 4 (2010) 909-966 [[arXiv:0805.0157](http://arxiv.org/abs/0805.0157), [ISSN:1088-6834](https://www.ams.org/journals/jams/2010-23-04/S0894-0347-10-00669-7)]

The external product on [[cobordism rings]]:

* {#Quillen71} [[Daniel Quillen]], §1.6 in: *Elementary Proofs of Some Results of Cobordism Theory Using Steenrod Operations*, Advances in Mathematics **7** (1971) 29–56 [<a href="http://dx.doi.org/10.1016/0001-8708(71)90041-7">doi:10.1016/0001-8708(71)90041-7</a>]

and on [[differential cobordism cohomology|differential cobordism rings]]:

* [[Ulrich Bunke]], [[Thomas Schick]], Ingo Schroeder, Moritz Wiethaup, §3.3.10 in: *Landweber exact formal group laws and smooth cohomology theories*, Algebr. Geom. Topol. **9** (2009) 1751-1790 [[arXiv:0711.1134](https://arxiv.org/abs/0711.1134), [doi:10.2140/agt.2009.9.1751](https://doi.org/10.2140/agt.2009.9.1751)]

and in the [[Hodge-filtered differential cohomology|Hodge-filtered version]]:

* [[Knut Bjarte Haus]], §4.1.8, §4.10 in: *Geometric Hodge filtered complex cobordism*, PhD thesis (2022) [[ntnuopen:3017489](https://ntnuopen.ntnu.no/ntnu-xmlui/handle/11250/3017489)]

* [[Knut B. Haus]], [[Gereon Quick]], §2.11 in: *Geometric Hodge filtered complex cobordism* [[arXiv:2210.13259](https://arxiv.org/abs/2210.13259)]

{#ReferencesExternalSmashProduct} The external [[smash product]] of [[retractive spaces]] and of [[parameterized spectra]]:

* {#MaySigurdsson06} [[Peter May]], [[Johann Sigurdsson]], §7.3 of: _[[Parametrized Homotopy Theory]]_, Mathematical Surveys and Monographs **132**, AMS (2006) [[arXiv:math/0411656](https://arxiv.org/abs/math/0411656)]

* {#Malkiewich19} [[Cary Malkiewich]], §3.1 of: *Parametrized spectra, a low-tech approach* [[arXiv:1906.04773](https://arxiv.org/abs/1906.04773), user guide: [pdf](https://people.math.binghamton.edu/malkiewich/users_guide_parametrized.pdf), [[Malkiewich-ParametrizedSpectraUserGuide.pdf:file]]]

* {#Malkiewich23} [[Cary Malkiewich]], §2.3 and §4.4 of: *A convenient category of parametrized spectra* [[arXiv:2305.15327](https://arxiv.org/abs/2305.15327)]

For general abstract literature dealing with the external tensor products see the references at _[[indexed monoidal category]]_ and at _[[dependent linear type theory]]_, such as

* {#Shulman12} [[Mike Shulman]], *Enriched indexed categories*, Theory Appl. Categ. **28** (2013) 616-695 [[arXiv:1212.3914](http://arxiv.org/abs/1212.3914), [tac:28-21](http://www.tac.mta.ca/tac/volumes/28/21/28-21abs.html)]

[[!redirects external tensor products]]

[[!redirects external product]]
[[!redirects external products]]

[[!redirects external tensor product of representations]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given two [[vector bundles]] $E_1 \to X_1$ and $E_2 \to X_2$, then their _external tensor product_ $E_1 \boxtimes E_2 \to X_1 \times X_2$ is the [[tensor product of vector bundles]] on the [[product]] [[space]] $X_1 \times X_2$ of the two [[pullback bundles]] to this space, along the canonical [[projection]] maps $pr_i \colon X_1 \times X_2 \to X_i $.

More abstracty, this is the [[external tensor product]] in the [[indexed monoidal category]] of [[vector bundles]] indexed over suitable [[spaces]].

## Definition

Let $X_1$ and $X_2$ be [[topological spaces]] and let $E_1 \to X_1$ and $E_2 \to X_2$ be [[topological vector bundles]].

The [[product topological space]] $X_1 \times X_2$ comes with two [[continuous function|continuous]] [[projection]] functions

$$
  X_1 \overset{pr_1}{\longleftarrow} X_1 \times X_2 \overset{pr_2}{\longrightarrow} X_2
  \,.
$$

This gives rise to the [[pullback bundles]] $pr_1^\ast E_1 \to X_1 \times X_2$ and $pr_2^\ast E_2 \to X_1 \times X_2$.

The _external tensor product_ $E_1 \boxtimes E_2$ is the [[tensor product of vector bundles]] of these pullback bundles:

$$
  E_1 \boxtimes E_2 \coloneqq (pr_1^\ast E_1) \otimes_{X_1 \times X_2} (pr_2^\ast E_2)
$$

which is again naturally a vector bundle over th product space

$$
  E_1 \boxtimes E_2 \longrightarrow X_1 \times X_2
  \,.
$$

## Properties

+-- {: .num_prop #ExternalProductTheoremIntopologicalKTheory}
###### Proposition
**([[external product theorem]] in [[topological K-theory]])**

For $X$ a [[compact Hausdorff space]] then the external tensor product of vector bundles over $X$ and over the [[2-sphere]] $S^2$ is an [[isomorphism]] of [[topological K-theory]] rings:

$$
  K(X) \otimes K(S^2) \overset{\simeq}{\longrightarrow} K(X \times S^2)
  \,.
$$

=--

## Related entries

* [[external direct sum of vector bundles]]

* [[direct sum of vector bundles]], [[inner product on vector bundles]]

* [[fundamental product theorem in topological K-theory]]

## References
 {#References}

The notion of external tensor product of vector bundles originates in discussion of [[topological K-theory]]:

*  {#Atiyah67}  [[Michael Atiyah]], §2.6 in: *K-theory*, Harvard Lecture 1964 (notes by D. W. Anderson), Benjamin (1967) [[pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/atiyahk.pdf), [[AtiyahKTheory.pdf:file]]]

* [[Raoul Bott]], p. 19 of: *Lectures on $K(X)$*, Benjamin (1969) [[pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/bottk.pdf), [[Bott-KTheory.pdf:file]]]

* {#Switzer75} [[Robert Switzer]], Rem. 13.51 in *Algebraic Topology -- Homotopy and Homology*, Grundlehren **212** Springer (1975) [[doi:10.1007/978-3-642-61923-6_12](https://doi.org/10.1007/978-3-642-61923-6_12)]

* {#Karoubi} [[Max Karoubi]], §4.9 in: _K-Theory -- An introduction_, Grundlehren der mathematischen Wissenschaften **226**, Springer (1978) [[pdf](https://webusers.imj-prg.fr/~max.karoubi/K.book/MK.book.pdf), [doi:10.1007/978-3-540-79890-3](https://doi.org/10.1007/978-3-540-79890-3)]

* {#Wirthmuller12} [[Klaus Wirthmüller]], p. 40 of: *Vector bundles and K-theory* (2012) [[[wirthmueller-vector-bundles-and-k-theory.pdf:file]]]

It is also briefly mentioned in a context of [[differential geometry]] in:

* [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], p. 84 of: *[[Connections, Curvature, and Cohomology]]* Volume 1: _De Rham Cohomology of Manifolds and Vector Bundles_, Academic Press (1973) [[ISBN:978-0-12-302701-6](https://www.elsevier.com/books/connections-curvature-and-cohomology-v1/greub/978-0-12-302701-6)]

Discussion in the context of the [[K-theory classification of topological phases of matter]]:

* [[Bruno Mera]], *The product of two independent Su-Schrieffer-Heeger chains yields a two-dimensional Chern insulator*, Phys. Rev. B **102** 155150 (2020) [[arXiv:2009.02268](https://arxiv.org/abs/2009.02268), [doi:10.1103/PhysRevB.102.155150](https://doi.org/10.1103/PhysRevB.102.155150)]

Discussion in the variant of categories of [[quasicoherent sheaves]]  in ([[derived algebraic geometry|derived]]) [[algebraic geometry]]:

* {#BondalvdBerg03} [[Alexei Bondal]], [[Michel Van den Bergh]], _Generators and representability of functors in commutative and noncommutative geometry_, Mosc. Math. J. **3** 1 (2003) 1-36 [[arXiv:math/0204218](https://arxiv.org/abs/math/0204218)]

* {#BFN08} [[David Ben-Zvi]], [[John Francis]], [[David Nadler]], _Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry_, J. Amer. Math. Soc. **23** 4 (2010) 909-966 [[arXiv:0805.0157](http://arxiv.org/abs/0805.0157)]

[[!redirects external tensor products of vector bundles]]


externalization

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

# Contents
* table of contents
{: toc}

## Idea

Many notions *[[internalization|internal]]* to a [[category]] $C$ (e.g.. [[internal groups]]) may be described alternatively in terms of [[functors]] ([[presheaves]], [[pseudofunctors]] etc.) with [[domain]] $C$, [[fibrations]] over $C$, and so on.  The process of replacing the internal structures on object or families of objects in $C$ by such "external"
structures involving the whole category $C$ is called the **externalization**. Sometimes external definitions give large versions (in the sense of set-theoretic size) of some internal notions.

For example, an [[internal groupoid]] (or even an [[internal category]]) $G$ in a [[finitely complete category]] $C$ gives rise to a [[Grothendieck fibration]]. A Grothendieck fibration equivalent to the externalization of an [[internal category]] is called [[small fibration]].

## References

* [[John Duskin]], _An outline of non-abelian cohomology in a topos: (I) The theory of bouquets and gerbes_, Cahiers de Topologie et Géométrie Différentielle Catégoriques __23__ no. 2 (1982), 165--191 [numdam](http://www.numdam.org/item?id=CTGDC_1982__23_2_165_0)


extranatural transformation

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

#Contents#
* table of contents
{: toc}

## Idea

**Extraordinary natural transformations**, or **extranatural transformations**, are what you get when you "bend the rules" for [[natural transformation]]s.  One intuitive approach to them is through [[string diagrams]]: every time you bend a string (that represents a component of a natural transformation) into a U-shape or upside-down U-shape, the U-shape represents a component of an extranatural transformation.  Thus, the rules for extranatural transformations mirror rules for ordinary [[natural transformations]], except they are bent into shapes with a covariant part and a contravariant part.  (Cf. interactions between particles and their corresponding [[antiparticle]]s.)

A transformation can also be ordinary-natural in some variables and extraordinary-natural in other variables.  Sometimes this sort of transformation is called a *generalized natural transformation*.  The late [[Max Kelly]] was fond of saying that really it's all the same basic concept, however, so why proliferate terminology needlessly?  So he would simply say a transformation was "natural" in all its arguments, both the "ordinary" and the "extraordinary" ones.

The calculus of natural and extranatural transformations is a very simple string diagram calculus; perhaps the most basic one.  It was first introduced by Eilenberg, Kelly, and Mac Lane in the mid 60's.

There is also a yet more general notion of [[dinatural transformation]].  However, there are few examples of dinatural transformations which are not extranatural.  Also, unlike extranatural transformations, dinatural transformations cannot be generalized to all [[enriched categories]] and do not admit a natural string diagram calculus.

## Examples

Consider the [[function set]] functor $\hom: Set^{op} \times Set \to Set$ (or more generally, the [[internal hom]] functor $\hom: V^{op} \times V \to V$ where $V$ is [[symmetric monoidal category|symmetric monoidal]] [[closed category|closed]]). The [[identity transformation]] $1: \hom \to \hom$ has components of the form

$$1_{x, y}: x^y \to x^y$$

and this of course is natural in each of the separate arguments $x, y$. String diagrammatically, this naturality would be represented by placing the domain over the codomain and linking the two instances of $x$ with a straight line and the two instances of $y$ with a straight line. Although it's trivial, let's at least record what naturality in say $y$ would mean: it means that for any morphism $g: y \to y'$ we have an equation of the form

\[ \label{naturality} 1_{x, y} x^g = x^g 1_{x, y'}: x^{y'} \to x^y \]

Now, the [[adjunction]] between [[tensor product]] and internal hom allow us to "bend" the transformation into another:

$$eval_{x, y}: x^y \otimes y \to x$$

in which the two instances of $y$ are linked by a U-shape. This gives a transformation which is natural in $x$ but not of course in $y$; rather, in $y$ we have an equation which is companion to (eq:naturality):

\[ \label{extranaturality} eval_{x, y} (x^g \otimes y) = eval_{x, y'} (x^{y'} \otimes g): x^{y'} \otimes y \to x \]

and we say in this case that $eval_{x, y}$ is **extranatural** in $y$. Notice how the extranatural variable $y$ in $eval_{x, y}$ appears once covariantly [in the tensor factor] and once contravariantly [in the exponent], but together on the same side of the arrow [here the domain]. (There is a nice string diagram picture for (eq:extranaturality) which the reader might like to draw at this point.)

Thus we already see that $eval_{x, y}$ is a "generalized natural" transformation, involving a mixture of naturality (in $x$) and extranaturality (in $y$).

The basic idea should now be clear, but let's give a few more examples. Starting with the identity transformation

$$1_{x, y}: x \otimes y \to x \otimes y$$

we can again bend it using the tensor-hom adjunction to form an arrow

$$coeval_{x, y}: x \to (x \otimes y)^y$$

where again $y$ appears once covariantly and once contravariantly, this time on the codomain side. The extranaturality in $y$ is the condition

$$(x \otimes g)^{y} coeval_{x, y} = (x \otimes y')^g coeval_{x, y'}$$

for every arrow $g: y \to y'$.

As these examples indicate, instances of ordinary naturality are typically transferred into instances of extranaturality by means of adjunctions. Indeed, basic instances of U-shapes or upside-down U-shapes in string diagrams come about through counits and units of adjunctions,

$$\varepsilon: F U \to 1 \qquad \eta: 1 \to U F$$

## Formalization {#Defn}

Let $F\colon A\times B\times B^{op} \to D$ and $G\colon A\times C\times C^{op}\to D$ be functors.  A family of morphisms
$$ \alpha_{a,b,c}\colon F(a,b,b) \to G(a,c,c) $$
for $a\in A$, $b\in B$, and $c\in C$ is said to be **natural**, or more precisely *ordinary-natural in $a$ and extranatural in $b$ and $c$*, if the following hold.

* For all $f\colon a\to a'$ in $A$ and all $b\in B$ and $c\in C$, the following square commutes (ordinary naturality in $a$):
  $$\array{&F(a,b,b) & \overset{F(f,1,1)}{\to} & F(a',b,b) &\\
  ^{\alpha_{a,b,c}} &\downarrow && \downarrow & ^{\alpha_{a',b,c}}\\
  &G(a,c,c)& \underset{G(f,1,1)}{\to} & G(a',c,c) &}$$
* For all $g\colon b\to b'$ in $B$ and all $a\in A$ and $c\in C$, the following square commutes (extranaturality in $b$):
  $$\array{&F(a,b,b') & \overset{F(1,1,g)}{\to} & F(a,b,b) &\\
  ^{F(1,g,1)} &\downarrow && \downarrow & ^{\alpha_{a,b,c}}\\
  &F(a,b',b')& \underset{\alpha_{a,b',c}}{\to} & G(a,c,c) &}$$
* For all $h\colon c\to c'$ in $C$ and all $a\in A$ and $b\in B$, the following square commutes (extranaturality in $c$):
  $$\array{&F(a,b,b) & \overset{\alpha_{a,b,c}}{\to} & G(a,c,c) &\\
  ^{\alpha_{a,b,c'}} &\downarrow && \downarrow & ^{G(1,h,1)}\\
  & G(a,c',c') & \underset{G(1,1,h)}{\to} & G(a,c',c) &}$$

It is convenient to draw "string diagrams" specifying in what variables a transformation is natural and extranatural.  For instance, the above transformation can be notated in this way:

<svg width="170" height="200" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:se="http://svg-edit.googlecode.com">

 <g>
  <title>Layer 1</title>
  <polyline se:connector="svg_19989_1 svg_19989_4" fill="none" stroke-width="2" stroke="#000" points="29.273,38 29.7109,101.5 30.1489,165" id="svg_19989_7"/>
  <text xml:space="preserve" text-anchor="middle" font-family="serif" font-size="24" stroke-width="0" stroke="#000000" fill="#000000" id="svg_19989_1" y="38" x="29">A</text>
  <text xml:space="preserve" text-anchor="middle" font-family="serif" font-size="24" stroke-width="0" stroke="#000000" fill="#000000" id="svg_19989_2" y="38" x="85">B</text>
  <text xml:space="preserve" text-anchor="middle" font-family="serif" font-size="24" stroke-width="0" stroke="#000000" fill="#000000" id="svg_19989_3" y="36" x="141">B</text>
  <text xml:space="preserve" text-anchor="middle" font-family="serif" font-size="24" stroke-width="0" stroke="#000000" fill="#000000" id="svg_19989_4" y="183" x="30">A</text>
  <text xml:space="preserve" text-anchor="middle" font-family="serif" font-size="24" stroke-width="0" stroke="#000000" fill="#000000" id="svg_19989_5" y="184" x="87">C</text>
  <text xml:space="preserve" text-anchor="middle" font-family="serif" font-size="24" stroke-width="0" stroke="#000000" fill="#000000" id="svg_19989_6" y="184" x="142">C</text>
  <path stroke-width="2" stroke="#000" fill="none" id="svg_19989_8" d="m85,42c3.5,17.5 6.5,44.5 28,45c21.5,0.5 28,-27 28,-44"/>
  <path transform="rotate(-180, 114, 130.5)" id="svg_19989_9" stroke-width="2" stroke="#000" fill="none" d="m86,108c3.5,17.5 6.5,44.5 28,45c21.5,0.5 28,-27 28,-44"/>
  <text xml:space="preserve" text-anchor="middle" font-family="serif" font-size="10" stroke-width="0" stroke="#000" fill="#000000" id="svg_19989_10" y="21" x="100">op</text>
  <text id="svg_19989_11" xml:space="preserve" text-anchor="middle" font-family="serif" font-size="10" stroke-width="0" stroke="#000" fill="#000000" y="166" x="103">op</text>
 </g>
</svg>

Similarly, a transformation from $f\colon (A,B,A^{op},C^{op}) \to X$ to $g\colon (C^{op},B,D^{op},D) \to X$ which is natural in $B$ and $C^{op}$ and extranatural in $A$ and $D$ would be notated in this way:

<svg width="250" height="200" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:se="http://svg-edit.googlecode.com">

 <g>
  <title>Layer 1</title>
  <text xml:space="preserve" text-anchor="middle" font-family="serif" font-size="24" stroke-width="0" stroke="#000000" fill="#000000" id="svg_43543_1" y="36" x="27">A</text>
  <text xml:space="preserve" text-anchor="middle" font-family="serif" font-size="24" stroke-width="0" stroke="#000000" fill="#000000" id="svg_43543_2" y="35" x="85">B</text>
  <text xml:space="preserve" text-anchor="middle" font-family="serif" font-size="24" stroke-width="0" stroke="#000000" fill="#000000" id="svg_43543_3" y="35" x="143">A</text>
  <text xml:space="preserve" text-anchor="middle" font-family="serif" font-size="24" stroke-width="0" stroke="#000000" fill="#000000" id="svg_43543_4" y="35" x="198">C</text>
  <text xml:space="preserve" text-anchor="middle" font-family="serif" font-size="24" stroke-width="0" stroke="#000000" fill="#000000" id="svg_43543_5" y="190" x="29">C</text>
  <text xml:space="preserve" text-anchor="middle" font-family="serif" font-size="24" stroke-width="0" stroke="#000000" fill="#000000" id="svg_43543_6" y="189" x="83">B</text>
  <text xml:space="preserve" text-anchor="middle" font-family="serif" font-size="24" stroke-width="0" stroke="#000000" fill="#000000" id="svg_43543_7" y="188" x="146">D</text>
  <text xml:space="preserve" text-anchor="middle" font-family="serif" font-size="24" stroke-width="0" stroke="#000000" fill="#000000" id="svg_43543_8" y="186" x="201">D</text>
  <path stroke-width="2" stroke="#000000" fill="#FF0000" id="svg_43543_9" d="m84,42l0,121"/>
  <path stroke-width="2" stroke="#000000" fill="#ffffff" id="svg_43543_10" d="m29,45c0.5,21 21.5,38 43,37"/>
  <path stroke-width="2" stroke="#000000" fill="#ffffff" id="svg_43543_12" d="m93,82c30,0 51,-14 50,-38"/>
  <path stroke-width="2" stroke="#000000" fill="#ffffff" id="svg_43543_13" d="m146,160c0.5,-23 10.5,-41.5 28,-42c17.5,-0.5 27.5,21.5 28,42"/>
  <path stroke-width="2" stroke="#000000" fill="#ffffff" id="svg_43543_14" d="m198,41c1.5,18 -22.5,36.5 -34,46c-11.5,9.5 -46.5,25.5 -71,36"/>
  <path stroke-width="2" stroke="#000000" fill="#ffffff" id="svg_43543_15" d="m76,130c-22,9.5 -46,22.5 -47,34"/>
  <text transform="matrix(1, 0, 0, 0.9, 0, 3.9)" xml:space="preserve" text-anchor="middle" font-family="serif" font-size="12" stroke-width="0" stroke="#000000" fill="#000000" id="svg_43543_16" y="16" x="156">op</text>
  <text id="svg_43543_17" transform="matrix(1, 0, 0, 0.9, 0, 3.9)" xml:space="preserve" text-anchor="middle" font-family="serif" font-size="12" stroke-width="0" stroke="#000000" fill="#000000" y="14.8889" x="213">op</text>
  <text id="svg_43543_18" transform="matrix(1, 0, 0, 0.9, 0, 3.9)" xml:space="preserve" text-anchor="middle" font-family="serif" font-size="12" stroke-width="0" stroke="#000000" fill="#000000" y="189.333" x="46">op</text>
  <text id="svg_43543_19" transform="matrix(1, 0, 0, 0.9, 0, 3.9)" xml:space="preserve" text-anchor="middle" font-family="serif" font-size="12" stroke-width="0" stroke="#000000" fill="#000000" y="184.888" x="162">op</text>
 </g>
</svg>

## Extranatural calculus

We set down a few basic lemmas which describe how extranatural transformations compose. These lemmas become very intuitive once one draws string diagrams to accompany them. (Cf. "yanking moves" in the string diagram calculus of adjunctions.)

+-- {: .un_lemma}
###### Lemma 1 ("stalactites")

Let $F \colon C^{op} \times C \to D$ and $G \colon \underbrace{C^{op} \times C \times \cdots \times C^{op} \times C}_{2n} \to D$ be functors. If $\alpha_{x, y_1, \ldots, y_{n-1}, z} \colon F(x, z) \to G(x, y_1, y_1, \ldots, y_{n-1}, y_{n-1}, z)$ is natural in $x,z$ and extranatural in $y_1, \ldots, y_{n-1}$, and $\beta_{y_1, \ldots, y_n} \colon G(y_1, y_1, \ldots, y_n, y_n) \to H$ (for some object $H$ of $D$) is extranatural in $y_1, \ldots, y_n$, then
$$\beta_{\underbrace{x, \ldots, x}_{n}} \alpha_{\underbrace{x, \ldots, x}_{n+1}}: F(x, x) \to H$$
is extranatural in $x$.
=--

+-- {: .un_lemma}
###### Lemma 2 ("stalagmites")
Let $G \colon \underbrace{C^{op} \times C \times \cdots \times C^{op} \times C}_{2n} \to D$ and $H \colon C^{op} \times C \to D$ be functors.  If $\alpha_{y_1, \ldots, y_n} \colon F \to G(y_1, y_1, \ldots, y_n, y_n)$ (for some object $F$ of $D$) is extranatural in $y_1, \ldots, y_n$, and $\beta_{x, y_1, \ldots, y_{n-1}, z} \colon G(x, y_1, y_1, \ldots, y_{n-1}, y_{n-1} ,z) \to H(x,z)$ is natural in $x, z$ and extranatural in $y_1, \ldots, y_{n-1}$, then
$$\beta_{\underbrace{x, \ldots, x}_{n+1}} \alpha_{\underbrace{x, \ldots, x}_{n}} \colon F \to H(x, x)$$
is extranatural in $x$.
=--

+-- {: .un_lemma}
###### Lemma 3 ("yanking")

Let $F, H$ be functors of the form $C \to D$, and let $G: \underbrace{C \times C^{op} \times C \times  \ldots \times C^{op} \times C}_{2n+1} \to D$ be a functor. If $\alpha_{x, y_1, \ldots, y_n} \colon F(x) \to G(x, y_1, y_1, \ldots, y_n, y_n)$ is natural in $x$ and extranatural in $y_1, \ldots, y_n$, and if $\beta_{y_1, \ldots, y_n, z} \colon G(y_1, y_1, \ldots, y_n, y_n ,z) \to H(z)$, is natural in $z$ and extranatural in $y_1, \ldots, y_n$, then
$$\beta_{\underbrace{x, \ldots, x}_{n+1}} \alpha_{\underbrace{x, \ldots x}_{n+1}}: F(x) \to H(x)$$
is natural in $x$.
=--

In fact, these lemmas essentially capture "all possible" ways in which extranatural transformations can be composed.  The general statement, which is obtained by combining these, is that if the graphs representing the two transformations can be composed without creating any cycles, then the transformations can be composed. In some detail, if there are no cycles in the composed graph, then it can be shown that each connected components has one of the forms of the compositions in Lemma 1-3.  It then follows from the lemmas (and from the fact that extranaturality can be determined by separation of variables) that each connected component can be replaced by a single edge, and the resulting graph defines the type of the composed transformation.

This can be found in the original paper about extranatural transformations:

* Eilenberg and Kelly, *A generalization of the functorial calculus*, J. Algebra 3 366--375 (1966)

With the operation of "loop-free composition," extranatural transformations with a given target form a [[paracategory]].  And as we vary the source and target categories, they assemble into an [[extraordinary 2-multicategory]].

## Profunctors

One abstract way to describe the structure of extranatural transformations is as an [[extraordinary 2-multicategory]].  Another abstract structure, which arguably arises more naturally in practice (but also includes more data than necessary), is a [[compact closed category|compact closed]] monoidal [[bicategory]], [[double category]], or [[proarrow equipment]].

More should go here, but for now see [[compact closed double category]].

## Related concepts

* [[bifunctor]]

* [[Kelly-Mac Lane graph]]

* [[proof net]]

* [[homotopy]]

* [[transfor]]

  * [[natural transformation]]

    * **extranatural transformation**, [[dinatural transformation]]

  * [[pseudonatural transformation]]

    * [[pseudo-extranatural transformation]]

  * [[lax natural transformation]]

* [[end]]

## References

* {#EilenbergKelly65} [[Samuel Eilenberg]], [[G. Max Kelly]],  §III.5 of: *Closed Categories*, in: *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) 421-562 [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

* [[Samuel Eilenberg]], [[G. M. Kelly]], *A generalization of the functorial calculus*, J. Algebra **3&& 3 (1966) 366-375 [[doi](http://dx.doi.org/10.1016/0021-8693%2866%2990006-8)]

* [[Max Kelly]], [[Saunders MacLane]], _Coherence in closed categories_, JPAA **1** (1971) 97-140 [[doi](http://dx.doi.org/10.1016/0022-4049%2871%2990013-2)]

A generalization to [[pseudonatural transformations]] can be found in

* J.C. Vidal, J.S. Tur, _A 2-categorial generalization of the concept of institution_, Studia Logica, 2010 [doi](https://10.1007/s11225-010-9268-0)
* Alexander S. Corner, _A universal characterisation of codescent objects_, [TAC](http://tac.mta.ca/tac/volumes/34/24/34-24abs.html) 34:24, 2019.

[[!redirects extranatural transformations]]
[[!redirects extraordinary natural transformation]]
[[!redirects extraordinary natural transformations]]
[[!redirects extranaturality]]
[[!redirects extraordinary naturality]]
[[!redirects extranatural]]


extremal black hole

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

## Idea

For a charged and/or rotating [[black hole]] ([[Kerr spacetime]], [[Reissner-Nordstrom spacetime]], [[Kerr-Newman spacetime]]) there is a [[minimum]] possible [[mass]] for given [[charge]] and [[angular momentum]]. When this minimum is attained, one hence speaks of an _extremal black hole_.

## Properties

### Near horizon geometry and JT gravity

The [[near horizon geometry]] of extremal black holes in $d$ dimensional spacetime  is [[anti de Sitter spacetime]] $AdS_{2}$ ([KLR 07](#KLR07), [AGJST 06](#AGJST06), [AY 07](#AY07), [ANT 07](#ANT07)).

The $AdS_2 \times S^{d-2}$ [[near horizon geometry]] of near-extremal black holes is well approximated by [[Jackiw-Teitelboim gravity]] ([NSST 18](#NSST18), [MTV 18](#MTV18)).

## Related concepts

* [[black holes in string theory]], [[black brane]]

* [[BPS state]]

* [[Bridgeland stability condition]]

## References

### General

* {#ANT07} Tatsuo Azeyanagi, Tatsuma Nishioka, Tadashi Takayanagi, _Near Extremal Black Hole Entropy as Entanglement Entropy via $AdS_2/CFT_1$_, Phys.Rev.D77:064005,2008 ([arXiv:0710.2956](https://arxiv.org/abs/0710.2956))

* [[Sean Carroll]], Matthew C. Johnson, [[Lisa Randall]], _Extremal limits and black hole entropy_, JHEP 0911:109,2009 ([arXiv:0901.0931](https://arxiv.org/abs/0901.0931))

* [[Ashoke Sen]], _Extremal black holes_, talk slides 2009 ([pdf](https://www.icts.res.in/sites/default/files/1250480845tifr.pdf))

See also

* Wikipedia, _[Extremal black hole](https://en.wikipedia.org/wiki/Extremal_black_hole)_

### BH entropy

On [[black hole entropy]] of (near-)[[extremal black holes]]:

* [[Luca Iliesiu]], *From statistical mechanics to microstate counting for extremal and near-extremal black holes*, talk at *[[Strings 2022]]* [[indico:4940850](https://indico.cern.ch/event/1085701/contributions/4940850), [slides](https://indico.cern.ch/event/1085701/contributions/4940850/attachments/2482365/4261646/Luca%20Iliesiu%20Tuesday-Reserach-Iliesiu-v2.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=96645a86-6d44-40ab-9abf-dd5a1920d881)]

### Near-horizon geometry and JT gravity

Discussion of the [[near horizon geometry]]

* {#KLR07} Hari K. Kunduri, James Lucietti, Harvey S. Reall, _Near-horizon symmetries of extremal black holes_, Class.Quant.Grav.24:4169-4190,2007 ([arXiv:0705.4214](https://arxiv.org/abs/0705.4214))

* {#AGJST06} Dumitru Astefanesei, Kevin Goldstein, Rudra P. Jena, [[Ashoke Sen]], Sandip P. Trivedi, _Rotating Attractors_, JHEP 0610:058,2006 ([arXiv:hep-th/0606244](https://arxiv.org/abs/hep-th/0606244))

* {#AY07} Dumitru Astefanesei, Hossein Yavartanoo, _Stationary black holes and attractor mechanism_, Nucl.Phys.B794:13-27,2008 ([arXiv:0706.1847](https://arxiv.org/abs/0706.1847))

via [[JT gravity]]:

* {#NSST18} Pranjal Nayak, Ashish Shukla, Ronak M Soni, Sandip P. Trivedi, V. Vishal, _On the Dynamics of Near-Extremal Black Holes_ ([arXiv:1802.09547](
https://arxiv.org/abs/1802.09547))

* {#MTV18} Upamanyu Moitra, Sandip P. Trivedi, V. Vishal, _Near-Extremal Near-Horizons_ ([arXiv:1808.08239](https://arxiv.org/abs/1808.08239))

### Actual observation
 {#ReferencesObservation}

Observational signature:

* Minyong Guo, Niels A. Obers, Haopeng Yan, _Observational signature of near-extremal Kerr-like black holes in a modified gravity theory at the Event Horizon Telescope_, Phys. Rev. D 98, 084063 (2018) ([arXiv:1806.05249](https://arxiv.org/abs/1806.05249))

Actual observation:

* Xinyu Dai, Shaun Steele, Eduardo Guerras, Christopher W. Morgan, Bin Chen, _Constraining Quasar Relativistic Reflection Regions and Spins with Microlensing_ ([arXiv:1901.06007](https://arxiv.org/abs/1901.06007))

  exposition in:

  [[Chandra Observatory]], _[Chandra Views Spinning Black Holes Across Cosmic Sea](https://scitechdaily.com/chandra-views-spinning-black-holes-across-cosmic-sea/)_, July 5 2019

[[!redirects extremal black holes]]


extremal epimorphism

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### Context
#### Category theory
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# Extremal morphisms
* table of contents
{: toc}

## Definition

An __extremal epimorphism__ (also sometimes called a _[[cover]]_) in a [[category]] $C$ is an [[epimorphism]] $e$ such that if $e = m \circ g$ where $m$ is a [[monomorphism]], then $m$ is an [[isomorphism]].

The dual notion is an __extremal monomorphism__: a monomorphism $m$ such that if $m = g \circ e$ where $e$ is an [[epimorphism]], then $e$ is an isomorphism.

## Remarks

* If $C$ has all [[equalizers]], then the assumption that $e$ is an epimorphism is redundant in the definition -- in this case any morphism admitting no non-trivial factorizations through monomorphisms is automatically epic.

* Any [[strong epimorphism]] is extremal.  The converse is true if $C$ has all [[pullbacks]].

* Any [[regular epimorphism]] is strong, and hence extremal.  The converse is true if $C$ is [[regular category|regular]].

* An [[image|image factorization]][^1] of a morphism $f$ is, by definition, a factorization $f = m \circ e$ where $m$ is a [[monomorphism]] and $e$ is an extremal epimorphism.

[^1]: Under a default meaning of [[image]] that makes reference to the class $M$ of all monomorphisms.

Of course, the dual properties are all true of extremal monomorphisms.  (See [[coequalizer]], [[monomorphism]], [[strong monomorphism]], [[pushout]], [[regular monomorphism]], [[coregular category]], [[coimage factorization]], [[epimorphism]].)

## Related concepts

* [[monomorphism]], [[epimorphism]]
* __extremal monomorphism__, __extremal epimorphism__
* [[strong monomorphism]], [[strong epimorphism]]
* [[strict monomorphism]], [[strict epimorphism]]
* [[regular monomorphism]], [[regular epimorphism]]
* [[normal monomorphism]], [[normal epimorphism]]
* [[effective monomorphism]], [[effective epimorphism]]
* [[split monomorphism]], [[split epimorphism]]

## References

* [[Francis Borceux]], Def. 4.3.2 in: *[[Handbook of Categorical Algebra]]* Vol. 1: *Basic Category Theory*, Encyclopedia of Mathematics and its Applications **50** Cambridge University Press (1994) ([doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858))

[[!redirects extremal morphism]]
[[!redirects extremal morphisms]]

[[!redirects extremal epimorphism]]
[[!redirects extremal epimorphisms]]
[[!redirects extremal epi]]
[[!redirects extremal epis]]
[[!redirects extremal epic]]
[[!redirects extremal epics]]

[[!redirects extremal monomorphism]]
[[!redirects extremal monomorphisms]]
[[!redirects extremal mono]]
[[!redirects extremal monos]]
[[!redirects extremal monic]]
[[!redirects extremal monics]]


extremal monomorphism

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#Contents#
* table of contents
{:toc}

## Definition

An **extremal monomorphism** in a [[category]] $C$ is a [[monomorphism]] $m$ such that if $m = g e$ where $e$ is an [[epimorphism]], then $e$ is an isomorphism.

## Remarks

The dual concept is [[extremal epimorphism]], and all properties of [[extremal epimorphism]]s apply in the dual sense to extremal monomorphisms.

In [[Top]] the extremal monomorphisms coincide with the [[regular monomorphism]]s, and these are exactly the subspace embeddings.


extremal quantum channel

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#### Quantum systems
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=--

#Contents#
* table of contents
{:toc}

## Idea

Since the space of [[quantum channels]] is a [[convex set|convex]] [[compact topological space]], every quantum channel may we written as a convex linear combination of the *extremal* points of this space. On [[finite-dimensional Hilbert spaces]] of [[dimension of a vector space|dimension]] 2 the extremal channels are exactly the [[unitary quantum channels]], but in higher dimensions there may be non-unitary extremal channels.

## Properties

### In terms of Kraus operators

> (old material which needs referencing)

A general quantum channel $T$, with Kraus operators $\{A\}_i$, is extremal if and only if the set

$$
\left \{A_{k}^{\dagger}A_{l} \right \}_{k,l\ldots N}
$$

is linearly independent.

As a [[unital quantum channel]] $T$ is extremal if and only if the set

$$
  \left \{A_{k}^{\dagger}A_{l} \oplus A_{l}A_{k}^{\dagger} \right \}_{k,l \ldots N}
$$

is linearly independent.

### Under tensor product

In the case of CPT or UCP maps the tensor product preserves extremality, but this is not always the case for UCPT maps. [[Miller & da Silva 2023](#MillerDaSilva23)]

A sketch of the proof for CPT maps is as follows. A finite sequence of vectors is linearly independent iff its Gram matrix is invertible. If $(A_k)_k$ and $(B_l)_l$ are Kraus operators for two CPT maps, then $(A_k \otimes B_l)_{k,l}$ are Kraus operators for the tensor product. Also, if both $(A_k)_k$ and $(B_l)_l$ are linearly independent, then $(A_k \otimes B_l)_{k,l}$ is also linearly independent. If we assume that $(A_k)_k$ and $(B_l)_l$ represent extreme CPT maps, then $(A_k^\dagger A_{k'})_{k,k'}$ and $(B_l^\dagger B_{l'})_{l,l'}$ are linearly independent. Let $G$ be the Gram matrix of $(A_k^\dagger A_{k'})_{k,k'}$ and $G'$ be the Gram matrix of $(B_l^\dagger B_{l'})_{l,l'}$, then $G$ and $G'$ are invertible. Let also $G''$ be the Gram matrix of $((A_k \otimes B_l)^\dagger (A_{k'} \otimes B_{l'}))_{(k,l),(k',l')}$. Extremality is preserved if $((A_k \otimes B_l)^\dagger (A_{k'} \otimes B_{l'}))_{(k,l),(k',l')}$ is linearly independent, which is equivalent to $G''$ being invertible. It follows that $G'' = G \otimes G'$ (Kronecker product). Using the properties of the Kronecker product and the invertibility of both $G$ and $G'$ we conclude that $G''$ is invertible.

A proof for the case of UCP maps may be obtained by the duality between CPT and UCP maps.

In the case of UCPT maps extremality is not always preserved. It is still preserved if one of the channels is over a space of dimension 2, since for dimension 2 the extreme UCPT maps are unitary channels. For higher dimensions one can construct counterexamples. A reason for this to happen is that if the UCPT maps are over a space of dimension $n$, then the Choi rank of an extreme UCPT map must be at most $\sqrt{2}n$. If we pick extreme UCPT maps over spaces of dimensions $n$ and $m$, one of them with rank greater than $\sqrt[4]{2}n$ and the other with rank greater than $\sqrt[4]{2}m$, then their tensor product have a rank greater than $\sqrt{2}nm$, so it can't be extreme.

## References

* {#Choi75} [[Man-Duen Choi]], Thm. 2.11 in: *Completely positive linear maps on complex matrices*, Linear Algebra and its Applications **10** 3 (1975) 285-290 [doi:10.1016/0024-3795(75)90075-0

* [[L. J. Landau]], [[Raymond F. Streater]],  *On Birkhoff's theorem for doubly stochastic completely positive maps of matrix algebras*, Linear Algebra and its Applications **193** (1993) 107-127 [doi:10.1016/0024-3795(93)90274-R]

* [[Christian B. Mendl]], [[Michael M. Wolf]], *Unital Quantum Channels -- Convex Structure and Revivals of Birkhoff's Theorem*, Commun. Math. Phys. **289**  (2009) 1057-1096 [[arXiv:0806.2820](http://arxiv.org/abs/0806.2820)]

* {#MillerDaSilva23} James Miller, S. T. da Silva, *On the Extremality of the Tensor Product of Quantum Channels* [[arXiv;2305.05795](https://arxiv.org/abs/2305.05795)]

[[!redirects extremal quantum channels]]


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#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Definition

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

A [[topological space]] is called _extremally disconnected_ if the [[closure]] of any [[open subset]] is still an open subset,
or, equivalently, the closures of disjoint open subsets are disjoint.

=--

## Properties

The extremally disconnected spaces can be characterized among the Hausdorff spaces by the fact that any continuous function mapping a dense subspace of an extremally disconnected space to a compact Hausdorff space, can be extended to the whole space. They can also be characterized among the completely regular spaces by the fact that the lattice of continuous functions mapping an extremally dis connected space to the unit interval is complete (cf. ([Giliman-Jerison, 1960](#GilimanJerison1960)), 3N and 6M). ([Strauss 1967](#Strauss67))

### As projective objects

\begin{theorem}\label{GleasonTheorem}
**([Gleason 1958](#Gleason58))**
\linebreak
Extremally disconnected topological spaces are precisely the [[projective objects]] in the [[category]] of [[compact topological space|compact]] [[Hausdorff topological spaces]], and in fact in
the category of regular Hausdorff [[topological spaces]] and [[proper maps]] between them.
\end{theorem}

See e.g. ([Bhatt-Scholze 13, below theorem 1.8](BhattScholze13)).
For the second part see ([Strauss 1967](#Strauss67)).

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

For $R$ a [[w-contractible ring]], the [[profinite set]] $\pi_0(Spec R)$
is an extremally disconnected [[profinite set]].

=--

Part of ([Bhatt-Scholze 13, theorem 1.8](#BhattScholze13)).

### In terms of logic

\begin{remark}
The extremal disconnectedness of a space is correlated with the property that the [[frame of opens|frame of its open subsets]] is a [[De Morgan Heyting algebra]] hence with the validity of the [[De Morgan law]] in [[logic]], since a result by Johnstone says that a [[topos]] $Sh(X)$ [[category of sheaves|of sheaves]] on a [[topological space]] $X$ is a [[De Morgan topos]] precisely when $X$ is extremally disconnected and this implies that all [[subobject lattices]] $sub(X)$ are De Morgan Heyting algebras, in particular $sub(1)$ corresponding to the frame of opens of $X$ (cf. [[De Morgan topos]] for details and references).
\end{remark}

### As a lifting property
 {#AsALiftingProperty}

Being extremally disconnected is equivalently a [[lifting property]] with respect to a surjective proper morphism of [[finite topological spaces]] (for more see the discussion at *[[separation axioms in terms of lifting properties]]*):

Namely that a topological space $X$ is extremally disconnected means equivalently that it solves the following lifting problems (from [[separation axioms in terms of lifting properties#IllustrationOfLiftingForExtremallyDisconnected|here]]):

\begin{tikzcd}
[
  column sep={between origins, 50pt},
  row sep={between origins, 40pt}
]
 \emptyset
  \ar[
    rr,
    "{ }"
  ]
  \ar[dd]
  && \boxed{
\boxed{{}^{\boxed{u}}\!\! \searrow_{\,c_1}} \,\,\, \boxed{{}_{c_2}\swarrow^{\boxed{v}}}
}
   \ar[dd]
  \\
  \\
  X
  \ar[
    rr,
    "{ \forall }"{below}
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \boxed{
    \overset{
      \boxed{
        \boxed{u}
        \;\;
        \,
        \;\;
        \boxed{v}}
      }{
        \underset{
           c
        }
        {
          \searrow \;\, \swarrow
        }
      }
  }
\end{tikzcd}

Here boxes indicate [[open subsets]] in a [[finite topological space]], and an arrow $x\to c$ means that $c$ is in the [[topological closure|closure]] of $x$. See at *[Background and notation](separation+axioms+in+terms+of+lifting+properties#BackgroundAndNotation)* for more.

One may deduce Gleason's theorem (Prop. \ref{GleasonTheorem}) in terms of these lifting properties:

Both being surjective and being proper are right lifting properties.
Hence, the existence of a [[weak factorization system]] generated by this morphism implies that each space admits a surjective proper map from an extremally disconnected space.

For more see at *[[separation axioms in terms of lifting properties]]* the section *[extremally disconnected spaces being projective](separation+axioms+in+terms+of+lifting+properties#ExtremallyDisconnectedSpacesBeingProjective)*.

## Examples

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

For $S$ any [[set]] regarded as a [[discrete topological space]],
its [[Stone-Cech compactification]] is extremally disconnected.

=--

(e.g. [Bhatt-Scholze 13, example 2.4.6](#BhattScholze13))

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

The [[profinite set]] underlying the [[p-adic integers]], regarded as a [[Stone space]], is _not_ extremally disconnected.

=--

(e.g. [Bhatt-Scholze 13, example 2.4.7](#BhattScholze13))

## Related concepts

* [[totally disconnected space]]

* [[De Morgan topos]]

## References

The result on projective spaces stems from

* {#Gleason58} [[Andrew M. Gleason]], _Projective topological spaces_, Ill. J. Math. **2** no.4A (1958) pp.482-489. ([euclid:ijm/1255454110](https://projecteuclid.org/euclid.ijm/1255454110))

* {#Strauss67} D. P. Strauss,  (1967). _Extremally Disconnected Spaces._ Proceedings of the American Mathematical Society, 18(2), 305. [doi:10.2307/2035286] (https://doi.org/10.2307/2035286)

Textbook account:

* [[Peter Johnstone]], _[[Stone Spaces]]_

* {#GilimanJerison1960} L.Giliman, M.Jerison. Rings of continuous functions. Van Nostrand, 1960.

Discussion in the context of the [[pro-etale site]]:

* {#BhattScholze13} [[Bhargav Bhatt]], [[Peter Scholze]], _The pro-étale topology for schemes_ ([arXiv:1309.1198](http://arxiv.org/abs/1309.1198))

[[!redirects extremally disconnected topological spaces]]

[[!redirects extremally disconnected profinite set]]
[[!redirects extremally disconnected profinite sets]]

[[!redirects extremally disconnected space]]
[[!redirects extremally disconnected spaces]]
[[!redirects extremally disconnected]]


extreme value theorem

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#### Analysis
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=--
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# The Extreme Value Theorem
* table of contents
{: toc}

## Idea

The classical _extreme value theorem_ states that a [[continuous function]] on the [[bounded subset|bounded]] [[closed interval]] $[0,1]$ with values in the [[real numbers]] does attain its [[maximum]] and its [[minimum]] (and hence in particular is a [[bounded function]]). This is a special case in [[analysis]] of the more general statement in [[topology]] that [[continuous images of compact spaces are compact]].

Although the Extreme Value Theorem (EVT) is often stated as a theorem about [[continuous maps]], it\'s really about [[semicontinuous maps]].

Recall that the [[lower real numbers]] are more general than the [[real numbers]] in just the way needed to guarantee that any [[inhabited subset|inhabited]] set of lower reals has a [[supremum]] (by including $\infty$, and in [[constructive mathematics]] by additionally relaxing the requirements of [[located real number|locatedness]]).  Consequently, the [[range]] of any lower-real-valued function with an [[inhabited set|inhabited]] [[domain]] must have a supremum.

The Extreme Value Theorem states that such a range must also have an [[infimum]] when certain conditions are met.  Specifically, we move to the realm of [[topology]], where the natural lower-real-valued functions are the [[semicontinuous map|lower semicontinuous]] ones.  So long as the [[domain]] of a lower semicontinuous map is [[compact space|compact]] (and inhabited), says the EVT, the range has an infimum, and furthermore this infimum is attained, becoming a [[minimum]].

## Statements

We first discuss the statement

* [for continuous functions](#ForContinuousFunctions)

and then the general case

* [for semicontinuous function](#ForSemicontinuousFunctions).

### For continuous functions
 {#ForContinuousFunctions}

+-- {: .num_prop}
###### Proposition
**(extreme value theorem)**

Let $C$ be a [[compact topological space]], and let

$$
  f \;\colon\; C \longrightarrow \mathbb{R}
$$

be a [[continuous function]] to the [[real numbers]] equipped with their [[Euclidean space|Euclidean]] [[metric topology]].

Then $f$ attains its [[maximum]] and its [[minimum]], i.e. there exist $x_{min}, x_{max} \in C$ such that for all $x \in C$ it is true that

$$
  f(x_{min})
    \leq
  f(x)
   \leq
  f(x_{max})
  \,.
$$

=--

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

Since [[continuous images of compact spaces are compact]] the image $f([a,b]) \subset \mathbb{R}$ is a [[compact topological space|compact]] [[subspace]].

Suppose this image did not contain its maximum. Then $\{(-\infty,x)\}_{x \in f([a,b])}$ were an [[open cover]] of the image, and hence, by its compactness, there would be a finite subcover, hence a finite set $(x_1 \lt x_2 \lt \cdots \lt x_n)$ of points $x_i \in f([a,b])$, such that the union of the $(-\infty,x_i)$ and hence the single set $(-\infty, x_n)$ alone would cover the image. This were in contradiction to the assumption that $x_n \in f([a,b])$ and hence we have a [[proof by contradiction]].

Similarly for the minimum.

=--

+-- {: .num_example }
###### Example
**(classical extreme value theorem)**

Let

$$
  f \;\colon\; [a,b] \longrightarrow \mathbb{R}
$$

be a [[continuous function]] from a [[bounded set|bounded]] [[closed interval]] ($a \lt b \in \mathbb{R}$)
regarded as a [[topological subspace]] of [[real numbers]] to the [[real numbers]], with the
latter regarded with their [[Euclidean space|Euclidean]] [[metric topology]].

Then $f$ attains its [[minimum]] $f(x_{min})$ and [[maximum]] $f(x_{max})$ and the [[image]] of $f$ is the [[closed interval]]

$$
  f([a,b]) = [f(x_{min}), f(x_{max})]
  \,.
$$

=--

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

Since [[continuous images of compact spaces are compact]] the image $f([a,b]) \subset \mathbb{R}$ is a [[compact topological space|compact]] [[subspace]].

By the [[Heine-Borel theorem]] the image, being compact, is a [[bounded set|bounded]] [[closed subset]], hence a [[finite set|finite]] union of bounded [[closed intervals]] and [[singleton]] subsets. By continuity of $f$, this union cannot be disjoint, for if $f([a,b])$ were a disjoint union $C_1 \sqcup C_2$ of closed inhabited subsets, then also the pre-image were a disjoint union of closed inhabited subsets $f^{-1}(C_1) \sqcup f^{-1}(C_2)$, contradicting the fact that the pre-image of the image is the connected interval $[a,b]$.

=--

### For semicontinuous functions
 {#ForSemicontinuousFunctions}

If $I$ is a [[compact space]], and if $f$ is an [[upper semicontinuous function]] from $I$ to the [[upper real numbers]] $[{-\infty,\infty}[$, then the [[range]] of $f$ is not only bounded above and not only has a finite [[supremum]], but it actually has a [[maximum]] value (unless $I$ is [[empty space|empty]]).

Similarly, and consequently (by replacing $f$ with $-f$), if $f$ is [[lower semicontinuous function|lower semicontinuous]] to the [[lower real numbers]] $]{-\infty,\infty}]$, then $\ran f$ is bounded below by a finite [[infimum]] which is its [[minimum]] value.  Consequently, if $f$ is [[continuous function|continuous]] to the [[real numbers]] $]{-\infty,\infty}[$, then $\ran f$ is [[bounded subset|bounded]] and has both a maximum and a minimum.)

In [[constructive mathematics]], this statement is correct in [[locale theory]] (in a sense that we should explain here!), but if we are speaking of pointwise functions defined on a real interval, then it fails in general.  However, we have approximate versions; in particular, it is constructive that $\ran f$ is bounded and has (if real-valued and pointwise-continuous) a real infimum and supremum.  However, the EVT interacts subtly with the [[fan theorem]]; the fan theorem is equivalent to the statement that whenever a pointwise-continuous real-valued function $f$ on $[0,1]$ satisfies $m \lt f \lt M$, then $m \lt \inf f$ and $\sup f \lt M$, which (once the existence of $\inf f$ and $\sup f$ is granted) may be viewed as a contrapositive form of the EVT.  (In fact, the fan theorem is separately equivalent to the statement that every continuous function on $[0,1]$ is [[uniformly continuous map|uniformly continuous]] and to the statement that every uniformly continuous function on $[0,1]$ has this contrapositive property.)  But without the fan theorem, even this contrapositive form cannot be proved (in marked contrast to similar theorems such as the [[intermediate value theorem]] and the [[mean value theorem]]), and in fact it is false in [[Russian constructivism]] (give a counterexample?).  (Does the full semicontinuous version follow from the fan theorem?)

## Related concepts

* [[continuous images of compact spaces are compact]]

## References

* Wikipedia, _[Extreme value theorem](https://en.wikipedia.org/wiki/Extreme_value_theorem)_

* Karin U. Katz, Mikhail G. Katz, Taras Kudryk, _Toward a clarity of the extreme value theorem_ ([arXiv:1404.5658](http://arxiv.org/abs/1404.5658))

[[!redirects extreme value theorem]]
[[!redirects extreme value theorems]]
[[!redirects Extreme Value Theorem]]
[[!redirects EVT]]


extremum

# Extrema
* table of contents
{: toc}

## Idea

In general, a __maximum__ is a [[top element]], a __minimum__ is a [[bottom element]], and an __extremum__ is either.  However, these terms are typically used in [[analysis]], where the [[order theory]] is secondary.  One also usually speaks of extrema of a [[function]], meaning a top or bottom element of the [[range]] of the function under its induced order as a [[subset]] of an ordered codomain.  In this context, one also considers *local* extrema of functions; a __local extremum__ of $f$ is an extremum of a [[restriction]] of $f$ to an [[open subspace]] of its original domain.  In any case, the extremum is __strict__ if the function takes the extreme value only once (in the relevant domain).

## Local extrema of differentiable functions

We list some sufficient and necessary conditions for a (nice) function $f$ on a [[smooth manifold]] to have a local extremum at a point $x$. As these are local conditions, we may assume $f$ is a function $U \to \mathbb{R}$ where $U$ is an [[open subset]] in a [[Cartesian space]] $\mathbb{R}^n$. These conditions fall under the rubric of "second derivative test".

Assume $f$ to be a twice-[[differentiable function]], and let $x$ in its [[domain]] be a [[critical point]]: a point where its [[derivative]] / [[Jacobian]] vanishes. Let $H_x(f)$ be the [[Hessian matrix]] of the function. Recall that $x$ is a **nondegenerate** critical point if the symmetric [[matrix]] $H_x$ is nondegenerate; equivalently, if $0$ is not an [[eigenvalue]] of $H_x$.

Let $x$ be a nondegenerate critical point. Then

* For $x$ to be a **strict local minimum** within some neighborhood, it is necessary and sufficient that $H_x(f)$ be a [[positive definite form]].

* For $x$ to be a **strict local maximum** within some neighborhood, it is necessary and sufficient that $H_x(f)$ be a [[negative definite form]].

(The only other possibility left for a nondegenerate critical point is that $H_x(f)$ be an [[indefinite form]], having a mix of positive and negative eigenvalues. In this case, $x$ is a **saddle point**. For more on this, see [[Morse theory]].)

If $x$ is a _degenerate_ critical point (so $0$ is an eigenvalue of $H_x$), we have:

* For $x$ to be a **local minimum**, it is necessary that $H_x(f)$ be a [[positive semidefinite form]], i.e., that all eigenvalues are nonnegative.

* For $x$ to be a **local maximum**, it is necessary that $H_x(f)$ be a [[negative semidefinite form]], i.e., that all eigenvalues are nonpositive.

These conditions are not sufficient. For a simple example, the origin in $\mathbb{R}^2$ is a critical point of $f(x, y) = x^3 - y^3$, where the Hessian is the zero matrix (hence positive semidefinite and negative semidefinite), but clearly the origin is neither a local maximum nor a local minimum.

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Eyal Markman

* [webpage](https://www.math.umass.edu/~markman/)

category: people


Eyup Yalcinkaya

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=248371)

## Selected writings

On [[equivariant principal bundles]] over the [[2-sphere]]:

* [[Eyup Yalcinkaya]], *Equivariant Principal Bundles over the 2-Sphere* ([arXiv:1902.06293](https://arxiv.org/abs/1902.06293))

category: people


Ezra Getzler

* [website](https://sites.northwestern.edu/getzler/)
* [list of publications](https://sites.northwestern.edu/getzler/Papers)

## Selected writings

On [[heat kernels]], [[Dirac operators]] and [[index theory]]:

* [[Nicole Berline]], [[Ezra Getzler]], [[Michèle Vergne]], *Heat Kernels and Dirac Operators*, Grundlehren __298__, Springer 2004 ([ISBN:9783540200628](https://www.springer.com/gp/book/9783540200628))

category: people

[[!redirects Getzler]]
[[!redirects E. Getzler]]


F'-theory

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

## Idea

The analogy of [[F-theory]] but for [[KK-compactification]] not on an [[elliptic fibration]] to 10d [[type IIB superstring theory]] with its $SL(2,\mathbb{Z})$-[[S-duality]] but to [[7d supergravity]] with its $SL(5,\mathbb{Z})$ [[U-duality]] group.

This is referred to as "F'-theory" in ([Kumar-Vafa 96](#KumarVafa96)). For the moment see at _[[U-duality]]_ for more.

## References

* {#KumarVafa96} Alok Kumar, [[Cumrun Vafa]], _U-Manifolds_, Phys.Lett. B396 (1997) 85-90 ([arXiv:hep-th/9611007](http://arxiv.org/abs/hep-th/9611007))