exact sequence of Hopf algebras

Drinfel'd looked at [[quantum group]]s as objects of the category [[opposite category|dual]] to the category of associative coassociative unital counital [[Hopf algebras]]. In (PW) a __short exact sequence of quantum groups__ is defined, as a sequence whose dual sequence of Hopf algebras is of the form

$$ B \stackrel{i}\hookrightarrow A\stackrel{p}\to A/I $$

where $B\hookrightarrow A$ is an inclusion of Hopf algebras, $I$ is a Hopf ideal in $A$ such that the quotient map $p$ is the cokernel of $i$ in the category of Hopf algebras. This corresponds to the exact sequence of affine group schemes when the Hopf algebras involved are commutative.

On the other hand, this definition does not work already for formal group schemes which correspond to cocommutative Hopf algebras. An additional problem is that according to a counterexample of (SchneiderESQGr) $B$ is not necessarily a normal Hopf algebra, nor $I$ a normal [[Hopf ideal]]  (contrary to blank assertion in (PW)). For these reasons, Schneider introduces a notion of __short strictly exact sequence of Hopf algebras__, as a sequence as above which satisfies the following 3 conditions

* (a) $B$ is a normal Hopf subalgebra of $A$

* (b) $A$ is right faithfully coflat over $A/I$

* (c) $p = coker(i)$ in the category of Hopf algebras.

There is a dual and equivalent by (SchneiderESQGr) set of conditions:

* (a') $I$ is a normal Hopf ideal in $A$

* (b') $A$ is right faithfully coflat over $A/I$

* (c') $i=ker(p)$ in the category of Hopf algebras.

It would be interesting to know which cohomology classifies all Hopf algebra by Hopf algebra extensions (there are also other kinds of "Hopf algebra extensions"). In full generality, this is an open question. Notice that in general questions on understanding various cocycles in bialgebra world is very partial, cf. [[bialgebra cocycle]].

It would also be interesting (though it may be known) how to characterize internally to the category of Hopf algebras what class of monomorphisms and epimorphisms correspond to the Hopf-algebraic conditions on $i$ and $p$ above.

Using [[Hopf monads]], in (BruguièresNatale) the theory of exact sequences/extensions of Hopf algebras (over an algebraically closed field of characteristics zero) is placed into the context of the exact seqences/extension theory of the corresponding categories of comodules. This is an interesting test case for general study of [[nonabelian cohomology]], more specifically a generalization of the classical [[Schreier's theory]].

## Literature ##

* (PW) [[B. Parshall]], J. Wang, _Quantum linear groups_, Memoirs AMS __439__

* (SchneiderESQGr) H-J. Schneider, _Some remarks on exact sequences of quantum groups_, Comm. Alg. __21__(9), 3337-3357 (1993)

* (BruguièresNatale) Alain Bruguières, Sonia Natale, _Exact sequences of tensor categories_, [arXiv:math.QA/1006.0569](http://arxiv.org/abs/1006.0569)

[[!redirects exact sequences of Hopf algebras]]


exact square

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# Exact squares
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## Definition

Let $A$, $B$, $C$, and $D$ be (usually [[small category|small]] [[strict category|strict]]) [[categories]], and consider a square of [[functors]]
$$
  \array{
    A
    &
      \overset{f}{\longrightarrow}
    &
    B
    \\
    \mathllap{^u}
    \Big\downarrow
    & \swArrow &
    \Big\downarrow
    \mathrlap{^v}
    \\
    C
    &\underset{g}{\longrightarrow}&
    D
  }
$$
that is filled by a [[natural transformation]] (which might be an [[identity natural transformation|identity]]).

This is said to be an **exact square** if the canonical [[Beck-Chevalley transformation]]
$$ u_! f^* \to g^* v_!, $$
relating functors with values in some other category $M$, is an [[isomorphism]].

Here $f^*\colon M^B\to M^A$ and $g^*\colon M^D\to M^C$ denote precomposition with $f$ and $g$ respectively, and $u_!\colon M^A\to M^C$ and $v_!\colon M^B\to M^D$ denote pointwise left [[Kan extensions]] along $u$ and $v$.  The Beck-Chevalley transformation is defined to be the composite
$$
  u_! f^*
   \longrightarrow
  u_! f^* v^* v_!
    \longrightarrow
  u_! u^* g^* v_!
    \longrightarrow
  g^* v_!
$$
of ([[counit of an adjunction|co]])[[unit of an adjunction|unit]] maps.

By the general calculus of [[mates]], this definition is equivalent to requiring that the dual transformation
$$ v^* g_* \to f_* u^* $$
is an isomorphism, where $f_*$ and $g_*$ denote pointwise *right* Kan extensions.

This definition refers to Kan extensions of diagrams with values in any category $M$ (whenever such Kan extensions exist).  If a square is exact only relative to diagrams defined in a *particular* category $M$, we may call it **$M$-exact**.

This is a "functional" definition, which generalizes correctly to other contexts (see below).  However, in the case of ordinary categories and Kan extensions, there is a precise combinatorial characterization of which squares are exact.  Thus, when using exactness to compute with Kan extensions in practice, we can use the combinatorial characterization to verify that a square is exact, then apply the definition above to compute one Kan extension in terms of another.

## Generalizations

The definition given above generalizes directly to squares inhabited by a 2-cell in any [[2-category]], and any suitable notion of "(pointwise) Kan extension" which may exist along morphisms in that 2-category.  This includes:

* Internal [[pointwise Kan extension]]s in any [[2-category]] with finite [[2-limits]].
* Enriched pointwise Kan extensions in a 2-category $V Cat$ of [[enriched categories]].  Note that this notion of "pointwiseness" is not the "internal" one in $V Cat$; see [here](Kan+extension#In2cat) for a counterexample.
* Pointwise Kan extensions can also be defined in any [[Yoneda structure]], and more or less equivalently in any [[2-category equipped with proarrows]].  This specializes both to the classical case, to internal pointwise extensions in a 2-category, and to the enriched case.
* In the world of [[(infinity,1)-categories]] we have a notion of [[homotopy exact square]], which can also be defined and characterized in the easier context of [[derivators]].  In fact, one can argue that derivators are basically designed as a context for the study of homotopy exact squares.

The combinatorial characterization of exact squares has a corresponding version for any of these generalizations, although in different cases it will be more or less explicit.

## Examples

### Comma squares

In ordinary category theory, any [[comma square]] is always exact.  In other words, if $A$ is the [[comma category]] $(v/g)$, with $f$ and $u$ the canonical projections, then the square is exact.  If additionally $C=*$ is the [[terminal category]], then $g$ simply picks out an object $d\in D$, and $A=v/d$ is the comma category of $v$ over $d$.  Exactness of this square then says that the value of $v_! F$ at $d\in D$ is the colimit of $F$ pulled back to $v/d$, i.e.
$$ (Lan_v F)(d) = colim_{v(b) \to d} F(b) $$
which is (for ordinary categories) one of the equivalent definitions of when a Kan extension is pointwise.

This example also makes clear why we define exactness only relative to pointwise Kan extensions.  If we allowed non-pointwise ones, then we would have hardly any exact squares.  Exactness is essentially a generalized version of pointwiseness, telling us how Kan extensions along one functor can be computed by transferring them to another functor, but for non-pointwise Kan extensions (which "exist by accident") we have basically no hope of doing this.

### Fully faithful functors

A functor $u\colon A\to B$ is a [[fully faithful functor]] if and only if the square
$$\array{A & \overset{id}{\to} & A\\
  ^{id}\downarrow && \downarrow^u\\
  A & \underset{u}{\to} & B}$$
is exact.  This just says that the unit $Id_A \to u^* u_!$ is an isomorphism, i.e. that left (and equivalently right) Kan extensions along $u$ are "honest" extensions.

### Final functors

A functor $u\colon A\to B$ is a [[final functor]] if and only if the square
$$\array{A & \overset{u}{\to} & B\\
  \downarrow && \downarrow\\
  *& \underset{}{\to} & *}$$
is exact.  Exactness of this square says that for $F\colon B\to M$, the canonical map $colim_A u^*F \to colim_B F$ is an isomorphism, which is one equivalent definition of when $u$ is final.

### Adjunctions

A functor $f: A \to B$ is left adjoint to $u: B \to A$ with unit $\eta: 1_A \implies uf$ if and only if the square
$$\array{A & \overset{id}{\to} & A \\
  ^f\downarrow & ^\eta\swArrow & \downarrow^{id}\\
  B& \underset{u}{\to} & A}$$
is exact. Dually, $f \dashv u$ with counit $\varepsilon: fu \implies 1_B$ if and only if the square
$$\array{B & \overset{u}{\to} & A\\
  ^{id}\downarrow & ^\varepsilon \swArrow & \downarrow^f\\
  B& \underset{id}{\to} & B}$$
is exact.

## Characterization

Exact squares can be characterized in several ways, which generalize in different directions.

### Using profunctors

Recall that for any functor $u\colon A\to C$, the left Kan extension $u_!$ can be computed as the colimit weighted by the corepresentable profunctor $C(u,1)$, while restriction $u^*$ along $u$ is the colimit weighted by the representable profunctor $C(1,u)$.  Moreover, given composable profunctors $H$ and $K$, an $H K$-weighted colimit is the same as a $K$-weighted colimit of an $H$-weighted colimit.  Therefore, $u_! f^*$ can be computed as a $C(u,1) \circ B(1,f)$-weighted colimit, while $g^* v_!$ can be computed as a $D(1,g) \circ D(v,1)$-weighted colimit.  Note that $D(1,g) \circ D(v,1) \cong D(v,g)$ by [[Yoneda reduction]].  Thus sufficient condition for a square as above to be exact is that the canonical induced transformation of profunctors
$$ C(u,1) \circ B(1,f) \to D(v,g) $$
is an isomorphism. If the categories are large, $C(u,1)\circ B(1,f)$ may not exist *a priori*, but there is still a 2-cell in the virtual equipment of functors and profunctors, and we ask that this cell $C(u,1), B(1,f)\to D(v,g)$ realizes the composition of $C(u,1)\circ B(1,f)$ as $D(v,g)$. Moreover, by considering colimits of the Yoneda embedding $B\to [B^{op},Set]$ it is easy to see that this condition is necessary as well.  Unraveling it, we see that it says that for all objects $b\in B$ and $c\in C$, the function
$$ \int^{a\in A} C(u(a),c) \times B(b,f(a)) \to D(v(b), g(c)) $$
is an isomorphism.

So far, this argument works to describe the exact squares in arbitrary [[enriched categories]], or more generally in any [[2-category equipped with proarrows]] where we have a sensible notion of profunctor that describes to the notion of pointwise Kan extension under consideration.  However, in the Set-based case, we can go further.  Given the construction of coequalizers in [[Set]], the above condition means that, calling $\psi \colon g u \to v f$ the given 2-cell, we have

1. For any morphism $\varphi\colon v(b) \to g(c)$ in $D$, there exists an $a\in A$ and morphisms $\alpha\colon u(a) \to c$ and $\beta\colon b\to f(a)$ such that $g(\alpha) \circ \psi_a \circ v(\beta) = \varphi$, and

2. For any $(a,\alpha,\beta)$ and $(a',\alpha',\beta')$ as above with $g(\alpha) \circ \psi_a \circ v(\beta) = g(\alpha') \circ \psi_{a'} \circ v(\beta')$, there is a [[zigzag]] of arrows connecting $a$ to $a'$ and rendering the evident induced diagram commutative.

We can state this equivalently as follows.  Given $b\in B$ and $c\in C$, define $(b/A/c)$ to be the category whose objects are triples $(a,\alpha,\beta)$ with $\alpha\colon u(a) \to c$ and $\beta\colon b\to f(a)$, and whose morphisms are morphisms $a\to a'$ making two triangles commute.  There is a functor
$$(b/A/c) \to D(v(b),g(c))$$
(considering the latter [[homset]] as a [[discrete category]]) which sends $(a,\alpha,\beta)$ to the composite $g(\alpha) \circ \psi_a \circ v(\beta)$, and the square is exact just when this functor is a bijection on [[connected component]]s.

#### Examples

+--{: .un_example}
###### Example
Note that in the case of a square
$$\array{A & \overset{u}{\to} & B\\
  \downarrow && \downarrow\\
  *& \underset{}{\to} & *}$$
this characterization reduces to saying that for any $b\in B$, the category $b/u$ is connected.  This is the standard combinatorial characterization of a [[final functor]].
=--

+--{: .un_example}
###### Example
This characterization implies directly that any [[cocomma square]] is exact, since the cocomma object of a cospan $C \overset{u}{\leftarrow} A \overset{f}{\to} B$ is precisely the [[cograph of a profunctor|cograph]] of the profunctor $C(u,1) \circ B(1,f)$.  This generalizes to any proarrow equipment (but unlike the case of comma squares, it does not generalize to [[homotopy exact squares]] unless we take a "homotopy cocomma object").
=--

### Using comma objects {#CommaObj}

Another approach is to argue as follows.  First note that any comma square of the form
$$\array{v/d & \overset{}{\to} & C\\
  \downarrow &\swArrow& \downarrow^v\\
  * & \underset{d}{\to} & D}$$
must be exact.  As observed above, this is one of the equivalent definitions of what it means to be a pointwise left Kan extension.  (Note that this ejects us from the world of enriched categories already.)  Dually,
$$\array{d/g & \overset{}{\to} & *\\
  \downarrow &\swArrow & \downarrow^d\\
  B & \underset{g}{\to} & D}$$
must also be exact, by definition of pointwise right Kan extensions.  Now suppose that a square
$$\array{A & \overset{f}{\to} & B\\
  ^u\downarrow & \swArrow & \downarrow^v\\
  C& \underset{g}{\to} & D}$$
is given.  The Beck-Chevalley transformation $u_! f^* \to g^* v_!$ between functors $C\to M$ will be an isomorphism as soon as it is an isomorphism componentwise, i.e. when evaluated at every object of $c$.  In other words, we want the transformation $c^* u_! f^* \to c^* g^* v_!$ to be an isomorphism.  Now consider the composite square
$$\array{u/c & \overset{q}{\to} & A & \overset{f}{\to} & B\\
  ^p\downarrow & \swArrow & \downarrow^u & \swArrow & \downarrow^v\\
  * & \underset{c}{\to}& C & \underset{g}{\to} & D.}$$
Since the left-hand square is of the form considered above, it is exact, so $p_! q^* \to c^* u_!$ is an isomorphism.  Thus $c^* u_! f^* \to c^* g^* v_!$ is an isomorphism if and only if the composite
$$ p_! q^* f^* \to c^* u_! f^* \to c^* g^* v_! $$
is an isomorphism, but this is just the Beck-Chevalley transformation for the composite square
$$\array{u/c & \overset{}{\to} & B\\
  \downarrow &\swArrow & \downarrow^v\\
  * & \underset{g(c)}{\to} & D.}$$
So the given square is exact just when all of these squares are exact.  But we can also play the same game dually, so the given square is exact just when for any $c\in C$ and $b\in B$, the square
$$\array{(b/A/c) & \overset{x}{\to} & *\\
  ^y\downarrow &\swArrow & \downarrow^{v(b)}\\
  * & \underset{g(c)}{\to} & D.}$$
is exact.  However, the comma square
$$\array{D(v(b),g(c)) & \overset{z}{\to} & *\\
  ^w\downarrow &\swArrow & \downarrow^{v(b)}\\
  * & \underset{g(c)}{\to} & D.}$$
is always exact, as observed previously, and by the universal property of a comma square, the previous one factors through this one via a functor $r\colon (b/A/c) \to D(v(b),g(c))$.  Hence we have $y = w r$ and $x = z r$, and the Beck-Chevalley transformation for the $(b/A/c)$ square factors as
$$y_! x^* \cong w_! r_! r^* z^* \to w_! z^* \overset{\cong}{\to} g(c)^* v(b)_!.$$
Hence it is an isomorphism if and only if $y_! x^* \to w_! z^*$ is an isomorphism, which is to say that colimits of constant diagrams on $(b/A/c)$ and on $D(v(b),g(c))$ agree.  But the colimit of a constant diagram is just a [[copower]] with the set of connected components of the domain category, so we recover the same characterization as before.

While this argument does not generalize to general enriched categories and proarrow equipments, it does generalize in a different direction.  The notion of [[derivator]] is essentially designed exactly so that this argument works, up until the last step: in a general derivator, colimits of constant diagrams may depend on more (or less) than the set of connected components of the domain.  For instance, in the derivator of [[∞-groupoids]], the colimit of a constant diagram is a copower with the [[nerve]] of the domain category, a finer invariant than its $\pi_0$.  It is a theorem of Cisinski that this is the finest possible: colimits of constant diagrams in a derivator never depend on anything more than the nerve of the domain.  Therefore, this yields a characterization of the [[homotopy exact squares]] for computing [[homotopy Kan extensions]] in derivators: the squares where the functor $(b/A/c) \to D(v(b),g(c))$ induces a weak equivalence of nerves.

##References

Some of the early theory is in

* [[René Guitart]], _Relations et carrés exacts,_ Ann. Sc. Math. Qué., juillet 1980, vol. IV, N° 2, p. 103-125.

[[!redirects exact squares]]


exact triangle

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#Contents#
* table of contents
{:toc}

## Definition

* [[long exact sequence]]

## Related concepts

* [[triangulated category]]

## References

Around remark 1.3.5 of

* [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_


exactness property

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#### Regular and Exact categories
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#### Limits and colimits
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[[!include infinity-limits - contents]]
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#### Category theory
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[[!include category theory - contents]]
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# Exactness properties
* table of contents
{: toc}

## Idea

An **exactness property** of a [[category]] asserts the existence of certain [[limits]] and [[colimits]], and moreover that the limits and colimits interact in a certain way.  Frequently, this includes stability of the colimits under [[pullback]], and also a condition expressing that some of the input data can be recovered from the colimit.

Many types of exactness can be expressed in terms of "colimits in the left-exact world".

Exactness properties of a functor refer to preservation of limits or colimits of certain kind, existence of adjoints and possibly their exactness properties.

## Examples

* As particular cases of [[familial regularity and exactness]] we have:
  * A [[regular category]] has [[quotient object|quotients]] of [[kernel pairs]] that are stable under pullback.
  * An [[exact category]] has [[quotient object|quotients]] of [[congruences]] that are stable under pullback.
  * A [[coherent category]] is regular and has finite [[unions]] of [[subobjects]] that are stable under pullback.
  * A [[pretopos]] is coherent and also extensive (see below).
  * A [[Grothendieck topos]] is an infinitary-pretopos that also has a [[generating set]].  Toposes play a universal role for lex colimits; see [Garner-Lack](#GarnerLack).

* An [[extensive category]] has [[coproducts]] that are disjoint and stable under pullback.

* An [[adhesive category]] has pushouts of monomorphisms that are stable under pullback and "van Kampen".

* An [[exhaustive category]] has colimits of sequences of monomorphisms that are pullback-stable and "exhaust" the colimit.

* More generally, having colimits of some class which are [[van Kampen colimit|van Kampen]] is an exactness property.

* Having [[filtered colimits]] which commute with [[finite limits]] is also an exactness property.

* Exact categories with pullback-stable [[reflexive coequalizers]] are an exactness notion.

* The various types of [[additive and abelian categories]] are [[Ab-enriched]] exactness properties. See also [[AT-category]].

* A [[site]] can be considered as a category with "exactness [[structure]]", or as a way of specifying certain exactness conditions which ought to hold after "completion".  See [[postulated colimit]] and [[exact completion]].

## References

* [[Richard Garner]] and [[Steve Lack]], *Lex colimits* ([arXiv:1107.0778](http://arxiv.org/abs/1107.0778))
 {#GarnerLack}

[[!redirects exactness properties]]
[[!redirects exactness properties of categories]]
[[!redirects lex colimit]]
[[!redirects lex colimits]]
[[!redirects exactness notion]]
[[!redirects exactness notions]]
[[!redirects notion of exactness]]
[[!redirects notions of exactness]]


example

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### Context
#### Mathematics
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[[!include mathematicscontents]]
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#Contents#
* table of contents
{:toc}

## Idea

In [[mathematics]], an *example* for a [[proposition]] with a [[universal quantifier]] $\big({\forall}_{x \in X} \; P(x)\big)$ or an [[existential quantifier]] $\big(\underset{x \in X}{\exists}\; P(x) \big)$ is a [[term]] $x$ of [[type]] $X$ for which $P(x)$ is [[true]] ([[inhabited type|inhabited]]).

## Examples

* [[examples of Kan extensions]]

* Every [[counterexample]] is an example of the negation of a proposition.

## Related concepts

[[!include mathematical statements --- contents]]

## See also

* [[archetypal example]]

[[!redirects examples]]


examples for Lagrangian BV


examples in topology

Basic examples

* [[sphere]], [[ball]], [[torus]], [[real line]], [[real n-dimensional space]], [[interval]], [[long line]]

Many classes come from standard constructions from other spaces or maps like

* [[quotient space]], [[subspace]] (as subset with induced topology), [[loop space]], [[free loop space]], [[suspension]], [[mapping cylinder]], [[mapping cone]], [[quotient space]], [[Fadell's configuration space]]

or from data which includes some parameters or algebraic data as input like

* [[Grassmanian]]s, [[flag varieties]]

and as representing object of some functor, or (more or less the same), as [[moduli space]]s of some structures

* [[classifying space]], [[Milnor construction]]

There are also classes of spaces which are often presented by standard recipes. For examples, manifolds may be presented by manifold atlases

Locally badly behaved spaces include

* [[Sierpinski space]], [[Warsaw circle]], [[fractal]], [[solenoid]]

For a compilation of examples important in algebraic topology see

* [[Neil Strickland]], _A bestiary of topological objects_ ([pdf](http://neil-strickland.staff.shef.ac.uk/courses/bestiary/bestiary.pdf))

Many examples are mentioned, or searched for, in the compendium of open problems

* J. van Mill, George M. Reed (editors), _Open problems in topology_, North Holland 1990, [pdf](http://futuretg.com/FTHumanEvolutionCourse/FTFreeLearningKits/01-MA-Mathematics%20and%20Preparation%20for%20University/010-MA10-UN03-09-Topology,%20Logic%20and%20Set%20Theory/Additional%20Resources/Jan%20van%20Mill,%20George%20M.%20Reed%20-%20Open%20Problems%20in%20Topology.pdf)


examples of adjoint functors

This entry lists examples for pairs of [[adjoint functor]]s.

For examples of the other [[universal construction]]s see

* [[limits and colimits by example]]

* [[examples of Kan extensions]].

#Free/forgetful functors#

The classical examples of pairs of [[adjoint functor]]s are $L \dashv R$ where the right adjoint $R : C' \to C$ [[stuff, structure, property|forgets structure]] in that it is a [[faithful functor]]. In these case the left adjoint $L : C \to C'$ usually is the [[free functor]] that "adds structure freely".

In fact, one usually turns this around and _defines_ the free $C'$-structure on an object $c$ of $C$ as the image of that object under the left adjoint (if it exists) to the functor $R : C' \to C$ that forgets this structure.

For instance

* forgetful right adjoint $R:$ [[Grp]] $\to$ [[Set]] forgets the group structure on a group and just remembers the underlying set -- the left adjoint  $L : Set \to Grp$ sends each set to the [free group](http://en.wikipedia.org/wiki/Free_group) over it.

#Nerves and realization#

For $C$ a category equipped with cosimplicial objects $\Delta_C : \Delta \to C$ and [[copower|tensored]] over $Set$;

$N_D : C \to Set^{\Delta^{op}}$

[[nerve]]

$$
  |-|_C : Set^{\Delta^{op}} \to C
$$

$$
  N_D(c) : \Delta^{op} \stackrel{\Delta_C^{op}}{\to} C^{op} \stackrel{C(-,C)}{\to} Set
$$

[[geometric realization|realization]]

$$
  |-|_C : Set^{\Delta^{op}} \to C
$$

$$
  |S_\bullet|_C = \int^{[n] \in \Delta} S_n \cdot \Delta_C[n]
$$

adjunction $|-|_C \dashv N_D$


examples of Frölicher spaces

{:myproof: .proof style="margin-left:2em;"}
{:mynumdef: .num_defn style="border:solid #cccccc;border-width:2px 1px;padding:0 1em;margin:0 1em;"}
{:goal: .un_remark style="border:solid #0000cc;background: #add8e6;border-width:2px 1px;padding:0 1em;margin:0 1em;"}

#Examples of [[Frölicher Spaces]]#
* automatic toc
{:toc}

## Manifolds as Frölicher Spaces ##

Any [[smooth manifold]] defines a Frölicher space with curves $C^\infty(\mathbb{R}, M)$ and functions $C^\infty(M, \mathbb{R})$.

## The Pinched Plane ##

Taking quotients in the category of Frölicher spaces is straightforward: the smooth functions are those that pull-back to smooth functions on the original space.

   As an example, consider the plane $\mathbb{R}^2$ quotiented out by the $x$-axis.

   +-- {: #quotientx style="text-align:center"}
   <svg width="188" height="197" xmlns="http://www.w3.org/2000/svg" se:nonce="83381" xmlns:se="http://svg-edit.googlecode.com" xmlns:xlink="http://www.w3.org/1999/xlink">
   <!-- Created with SVG-edit - http://svg-edit.googlecode.com/ -->
   <g>
   <title>Layer 1</title>
   <line fill="none" stroke-width="2" stroke="#000000" id="svg_83381_1" y2="1" x2="183" y1="1" x1="1"/>
   <line id="svg_83381_2" fill="none" stroke-width="2" stroke="#000000" y2="195" x2="187" y1="195" x1="5"/>
    <path id="svg_83381_3" d="m2,2c2,157 181,35 185,193" stroke-width="2" stroke="#000000" fill="none"/>
    <path id="svg_83381_4" d="m182,2c5,173 -179,18 -177,194" stroke-width="2" stroke="#000000" fill="none"/>
    <path id="svg_83381_5" d="m39,2c3,167 107,25 109,192" stroke-width="2" stroke="#9b9b9b" fill="none"/>
    <path id="svg_83381_6" d="m136,2c5,175 -86,17 -85,193" stroke-width="2" stroke="#9b9b9b" fill="none"/>
    <line fill="none" stroke-width="2" stroke="#9b9b9b" id="svg_83381_7" y2="193.99999" x2="96" y1="2" x1="94"/>
   </g>
   </svg>
   =--

   Let us write this as $X$.
This example is closely related to taking cones and suspensions in algebraic topology.
The smooth functions on $X$ are simple to describe: the set is equivalent to those smooth functions $\mathbb{R}^2 \to \mathbb{R}$ which are constant on the $x$-axis.

   Now let us consider the smooth curves.
Let $\alpha : \mathbb{R} \to X$ be a smooth curve.
We can partition $\mathbb{R}$ into two pieces: those points that are mapped to the squashed point in $X$ and those points that aren't.
Let us write $A$ for the set of points that are __not__ mapped to the squashed point.
Using bump functions it is easy to show that $A$ is open in $\mathbb{R}$.
As the quotient map $\mathbb{R}^2 \to X$ is bijective off the $x$-axis, the restriction of $\alpha$ to $A$ has a unique lift to $\mathbb{R}^2$.
Let us write $\alpha_x$ and $\alpha_y$ for the coordinate functions of this lift.
Again using bump functions, it is easy to show that $\alpha_x$ and $\alpha_y$ are smooth on $A$.
Furthermore, as the projection $(x,y) \mapsto y$ descends to a smooth function on $X$, $\alpha_y$ is actually the restriction to $A$ of a smooth function $\mathbb{R} \to \mathbb{R}$ which we shall also denote by $\alpha_y$.
Note that $A = \alpha_y^{-1}(0)$.

   The interesting part comes when looking at what happens to $\alpha_x$ at the boundary of $A$.
As $A$ is open, it is a disjoint union of open intervals.
The boundaries of each of these intervals forms part of the boundary of $A$ and it is simplest to start with these points.
For further simplicity, let us assume that $(0,1)$ is one of the components of $A$ and we are considering the boundary point $0$.
Thus we wish to consider $\lim_{t \to 0^+} \alpha_x(t)$.

   The general rule is simple to state: $\alpha_y$ must go to zero faster than $\alpha_x$ (and any of its derivatives) can go to infinity.

## Coequaliser Example ##

Let us give an example that shows that the category of Frölicher spaces is not [[locally cartesian closed category|locally cartesian closed]].
Consider a [[coequaliser]] diagram $\mathbb{R} \setminus \{0\} \to \mathbb{R} \amalg \mathbb{R}$ where the two maps are the inclusions into the two cofactors.

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   The coequaliser of this diagram is $\mathbb{R} \cup \{*\}$ where the $*$ is a doubled-point at $0$.  (So this is the well-known example of a non-Hausdorff [[manifold]].)
Thus any smooth function $\psi : \mathbb{R} \cup \{*\} \to \mathbb{R}$ has to satisfy $\psi(0) = \psi(*)$, which means that any smooth curve $\alpha : \mathbb{R} \to \mathbb{R} \cup \{*\}$ can choose whether to pass through $0$ or $*$ completely arbitrarily.

   We consider this as a coequaliser of spaces over $\mathbb{R}$ by taking the obvious map to $\mathbb{R}$ in each case.
The colimit is the same whether we work in the full category of Frölicher spaces or just those over $\mathbb{R}$.

   Now let $Y$ be any Frölicher space and consider it as a space over $\mathbb{R}$ via the [[constant function|constant]] zero map, $\omicron : Y \to \mathbb{R}$.
We take the [[fibred product]] over $\mathbb{R}$ of the coequaliser diagram.
Since $\mathbb{R} \setminus \{0\}$ has no points mapping to $0$, $(\mathbb{R} \setminus \{0\}) \times_{\mathbb{R}} Y = \emptyset$.
For the third space, we see that $(\mathbb{R} \amalg \mathbb{R}) \times_{\mathbb{R}} Y = Y \amalg Y$.
Thus the coequaliser diagram is now $\emptyset \to Y \amalg Y$.
The coequaliser is thus $Y \amalg Y$.
Note that smooth curves into $Y \amalg Y$ are of the form $(i, \beta)$ where $i$ is a constant in $\{0,*\}$ that distinguishes the cofactors and $\beta : \mathbb{R} \to Y$ is a smooth curve in $Y$.

   Let us consider the product over $\mathbb{R}$ of $\mathbb{R} \cup \{*\}$ with $Y$.
As a set, this is just $Y \amalg Y$ again.
However, as a Frölicher space it has different functions to those on $Y \amalg Y$.
The product over $\mathbb{R}$ is a subspace of $(\mathbb{R} \cup \{*\}) \times Y$ and thus a curve into it is smooth if and only if it is smooth into $(\mathbb{R} \cup \{*\}) \times Y$, whence it is smooth if and only if the projections to $\mathbb{R} \cup \{*\}$ and to $Y$ are smooth.
As we are considering curves in $(\mathbb{R} \cup \{*\}) \times_\mathbb{R} Y$, the projection to $\mathbb{R} \cup \{*\}$ must have image in $\{0,*\}$.
Thus _any_ curve is allowed by this and so the smooth curves into $(\mathbb{R} \cup \{*\}) \times_\mathbb{R} Y$ are of the form $(\alpha, \beta)$ where $\alpha$ is *any* function from $\mathbb{R}$ into $\{0,*\}$ and $\beta : \mathbb{R} \to Y$ is a smooth curve in $Y$.

   +-- {: .query}
   I notice that in some classically false versions of [[constructive mathematics]], the only functions from $\mathbb{R}$ to $\{0,*\}$ are the constant ones.  It would be nice if there were a nonclassical [[dream mathematics|dream universe]] in which the category of Frölicher spaces were locally cartesian closed!  Unfortunately, the counterexample can be saved by using a continuously parametrised coproduct $\coprod_{\mathbb{R}} \mathbb{R}$ instead of $\coprod_{\{0,*\}} \mathbb{R} = \mathbb{R} \amalg \mathbb{R}$.  ---Toby

   [[Andrew Stacey]] I think I'd be disappointed if locally cartesian became a property of what set theory was being used!  I suspect that the real reason this example works is the fact that $Y \to \mathbb{R}$ has such bad path-lifting properties and the coequaliser diagram that I chose is just a simple one that demonstrates it.

   I've been pondering how one might fix this.  I ought to write up Kriegl and Michor's extension of a manifold as, although they haven't done much with it, it contains some ideas that may be of interest.  What makes me think of it here is that that definition of a manifold (which isn't, by the way, the one in their weighty tome) has two parts: a space and its tangent space, and relationships between them.  This suggests that a smooth space, of whatever variety, should be more than just one space but some sort of diagram of spaces.

   This is also suggested by trying to generalise the Frolicher "idea" to non-set-based theories.  It's not immediately obvious how to make the saturation condition work, but the basic idea is to have objects as $(C,F,c)$ with $c : C \times F \to Hom$ and morphisms as $f : C_X \times F_Y \to Hom$.  The saturation condition, whatever it is, is what is needed to make this into a category.  So here it's obvious that objects are just special morphisms.  Feeding this back into the definition you get a glimmer of an idea that maybe a morphism (let me go back to genuine Frolicher spaces here for clarity) $f : X \to Y$ needs some sort of saturation condition as well as a compatibility condition.
   =--

   This is not the same as $Y \coprod Y$ and thus the functor $- \times_{\mathbb{R}} Y$ does not preserve colimits.
It cannot, therefore, be a left adjoint and so the category of Frölicher spaces is not locally cartesian closed.

   This example works because of the structure of $\mathbb{R} \cup \{*\}$.
If one were to work in an "input only" category, then the structure on $\mathbb{R} \cup \{*\}$ would be determined by those curves which lift to $\mathbb{R} \coprod \mathbb{R}$.
Such maps could not arbitrarily swap between $0$ and $*$ because up in $\mathbb{R} \coprod \mathbb{R}$ these two points are far apart.
Thus the subspace structure on $\{0,*\}$ in an "input only" category is _discrete_.
However, in the category of Frölicher spaces the outputs control the behaviour of quotients.
Functions out of $\mathbb{R} \cup \{*\}$ cannot detect the difference between $0$ and $*$.
Thus curves into $\mathbb{R} \cup \{*\}$ are allowed to swap between them with aplomb.
The subspace structure on $\{0,*\}$ is thus the _indiscrete_ structure.

   Working in the category of _Hausdorff_ Frölicher spaces (see [below](#hausdorff)) does not improve matters.
Then we need to replace each coequaliser but its Hausdorffification.
Now the distinction is clear since taking the product and then the coequaliser yields $Y \coprod Y$ as before but taking the coequaliser and then the product yields just $Y$.

   It is also worth pointing out that with the modification of the previous paragraph, this example only involves manifolds (assuming that $Y$ is chosen to be a manifold).  It therefore shows that a category extending that of smooth manifolds can __either__ be locally cartesian closed __or__ preserve limits and colimits from manifolds _but not both_.


examples of Kan extensions

This entry lists examples for pairs of [[Kan extension]]s.

For examples of the other [[universal construction]]s see

* [[limits and colimits by example]]

* [[examples of adjoint functors]].

#Yoneda extension#

* [[Yoneda extension]]

#Kan extension of presheaves and sheaves#

* [[restriction and extension of sheaves]]


excellent model category

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#Contents#
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## Idea

A [[monoidal model category]] $\mathbf{S}$ with extra properties guaranteeing a particularly good [[homotopy theory]] of $\mathbf{S}$-[[enriched categories]] is referred to in ([Lurie](#Lurie)) as an _excellent model category_.

(The term _excellent model category_ has also been used by [Lumsdaine and Shulman](#LumsdaineShulman) for a slightly different notion; see [[type-theoretic model category]].  This article is about Lurie’s sense.)

## Definition

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

Let $\mathbf{S}$ be a symmetric [[monoidal model category]]. It is called __excellent__ if

* it is a [[combinatorial model category]];

* the class of [[weak equivalences]] is stable under [[filtered colimits]];

* every monomorphism is a cofibration (hence in particular all objects are [[cofibrant]]);

* the class of cofibrations is closed under [[products]];

=--

This is ([Lurie, def. A.3.2.16](#Lurie)), except that he also includes the "invertibility hypothesis" ([see below](#InvertibilityHypothesis)).

## Examples

* The [[classical model structure on simplicial sets]].

* Any symmetric monoidal, locally presentable category with its [[trivial model structure]].

* The Cartesian model structure on [[marked simplicial sets]], with the cartesian product.

* The [[model structure for quasi-categories]] on simplicial sets.

## Properties

* One of Lurie's important original theorems was that if $\mathbf{S}$ is an excellent model category, $\mathbf{A}$ is a [[combinatorial model category|combinatorial]] $\mathbf{S}$-[[enriched model category]], and $C$ is a small $\mathbf{S}$-[[enriched category]], then the category $[C,\mathbf{A}]$ of enriched functors has both a [[projective model structure]] and an [[injective model structure]], which are both combinatorial.  However, it has since been shown by [MR13, Remark 3.8](#MR13) that excellence is unnecessary for this result.

## The invertibility hypothesis {#InvertibilityHypothesis}

[Lurie](#Lurie) originally included the an additional condition in the definition, called the **invertibility hypothesis**.  Fortunately, it was shown later by [Lawson](#Lawson) that the invertibility hypothesis is redundant: any model category satisfying the other axioms of an excellent model category always satisfies the invertibility hypothesis.

### Description of the invertibility hypothesis

The invertibility hypothesis states:

* for any homotopy equivalence $f$ in an $\mathbf{S}$-[[enriched category]] $C$, the localization functor $C \to C[f^{-1}]$ is a weak equivalence of $\mathbf{S}$-enriched categories.

The invertibility hypothesis requires some more explanation.  Firstly, by a "homotopy equivalence" we mean a morphism that becomes an isomorphism in the homotopy category $h C$, which is defined by applying to the hom-objects of $C$ the monoidal localization functor $\mathbf{S}\to h \mathbf{S}$ from $\mathbf{S}$ to its [[homotopy category]] (defined in the usual way for a model category by inverting its weak equivalences).  This makes $h C$ an $h \mathbf{S}$-enriched category, and we ask that $f$ be invertible in the underlying ordinary category of $h C$.

Secondly, by $C[f^{-1}]$ we mean a *homotopy invariant* notion of "localization".  One way to define this is to assume that $f$ is classified by a functor $[f]:\mathbf{2}_{\mathbf{S}} \to C$ that is a [[cofibration]] in the [[model structure on enriched categories]], where $\mathbf{2}_{\mathbf{S}}$ denotes the $\mathbf{S}$-enriched [[interval category]], and take the ordinary localization which can be defined as a pushout
$$\array{ \mathbf{2}_{\mathbf{S}} & \xrightarrow{[f]} & C \\ \downarrow && \downarrow \\ \mathbf{2}^\cong_{\mathbf{S}} & \to & C\langle f^{-1}\rangle.}$$
Here $\mathbf{2}^\cong_{\mathbf{S}}$ denotes the 2-object contractible groupoid regarded as an $\mathbf{S}$-category.  Another way to define it (without this assumption on $f$) is to factor the functor $\mathbf{2}_{\mathbf{S}}\to \mathbf{2}^\cong_{\mathbf{S}}$ through a cofibration $\mathbf{2}_{\mathbf{S}}\to E$ and weak equivalence $E\to \mathbf{2}^\cong_{\mathbf{S}}$ in the [[model structure on enriched categories]], and take the ordinary pushout
$$\array{ \mathbf{2}_{\mathbf{S}} & \xrightarrow{[f]} & C \\ \downarrow && \downarrow \\ E & \to & C[f^{-1}].}$$

Finally, by a "weak equivalence of $\mathbf{S}$-enriched categories" we mean a weak equivalence in the [[model structure on enriched categories]].

### Simplicial sets satisfy the invertibility hypothesis

We [Lurie](#Lurie) says (A.3.2.18) that this is "one of the main theorems" of [DK80](#DwyerKanLocalizations), but some further details may be helpful to understand how this comes about.  Note that this proof is not subsumed by Lawson's proof that all excellent model categories satisfy the invertibility hypothesis; instead Lawson *uses* this theorem to transfer the property from simplicial sets to all others excellent model categories.

The relevant theorem appears to be Proposition 10.4 of [DK80](#DwyerKanLocalizations) which states

> Let $V\to B\in \mathrm{s} O-Cat$ be a strong cofibration.  Then the induced map $B\to B[V^{-1}]$ is a weak equivalence if and only if every map of $\pi_0 B$ which is in the image of $\pi_0 V$ is invertible.

Here $\mathrm{s} O-Cat$ is the category of simplicially enriched categories with a fixed object set $O$.  Thus this proposition does not apply as written to the map $[f] : \mathbf{2}_{\mathbf{S}}\to C$ from the invertibility hypothesis, since it is not a bijection on objects.  Instead we have to replace $\mathbf{2}_{\mathbf{S}}$ by a simplicially enriched category $\mathbf{2}_{\mathbf{S},O}$ with set of objects $O = ob(C)$ in which the only nonidentity morphism "is" $f$.  Note that $[f]$ being a cofibration implies that it is injective on objects, so that $f$ has distinct domain and codomain; thus in $\mathbf{2}_{\mathbf{S},O}$ the unique nonidentity morphism is not an endomorphism and hence we don't have to worry about any nontrivial composites.

Now the construction of $B[V^{-1}]$ in [DK80](#DwyerKanLocalizations) is essentially the same as the pushout quoted above from [Lurie](#Lurie).  And $\pi_0 B$ denotes taking the homwise set of connected components of a simplicially enriched category, which can be factored as first taking the enriched homotopy category (which is a category enriched over $h(sSet)$) and then its underlying ordinary category; thus the assumption that "every map of $\pi_0 B$ which is in the image of $\pi_0 V$ is invertible" is equivalent to the assumption that $f$ is a homotopy equivalence.

Finally, a "strong cofibration" is defined in [DK80](#DwyerKanLocalizations) to be a cofibration with cofibrant domain, and $\mathbf{2}_{\mathbf{S},O}$ is easily checked to indeed be cofibrant (essentially because the unit object of $sSet$ is cofibrant).

### All excellent model categories satisfy the invertibility hypothesis

[Lawson](#Lawson) now shows that:

1. Cubical sets are monoidally Quillen equivalent to simplicial sets, hence also satisfy the invertibility hypothesis.
1. Any excellent model category admits a monoidal Quillen functor from cubical sets, hence also satisfies the invertibility hypothesis.

## References

* {#Lurie} [[Jacob Lurie]], _[[Higher Topos Theory]]_, Section A.3

* {#DwyerKanLocalizations} [[William Dwyer]], [[Daniel Kan]], _Simplicial localizations of categories_ , J. Pure Appl. Algebra 17 (1980), 267–284. ([pdf](http://www.nd.edu/~wgd/Dvi/SimplicialLocalizations.pdf))

* {#Lawson} [[Tyler Lawson]], _Localization of enriched categories and cubical sets_, [arXiv:1602.05313](https://arxiv.org/abs/1602.05313)

* {#MR13} [[M. Makkai]], [[J. Rosický]], _Cellular categories_, 2013, [arXiv:1304.7572](https://arxiv.org/abs/1304.7572)

* {#LumsdaineShulman} [[Peter LeFanu Lumsdaine]], [[Mike Shulman]], _Semantics of [[higher inductive types|Higher Inductive Types]], 2017, [arXiv:1705.07088](https://arxiv.org/abs/1705.07088)

[[!redirects excellent model categories]]


exception monad

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### Context
#### Computation
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#### Categorical algebra
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#### Type theory
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#Contents#
* table of contents
{:toc}

## Idea

In [[computer science]], an *exception* is a nominal failure of a program which however is *handled* in some manner, so that the failing computation still terminates in a controlled way.

In the simplest case this may mean that the program outputs an error message instead of continuing with the intended computation.

In terms of [[monads in computer science]], the *effect* of (possibly) throwing an exception is modeled by the [[monad]]:

1. which turns a given output [[data type]] into the [[coproduct type]]

   $$
     D \;\mapsto\; D \sqcup Msg
   $$

   with the intended type of handling/messages (typically $Msg = $ [[String (computer science)|string]]);

1. whose bind-operation on any $prog \colon D_1 \to D_2 \sqcup Msg$ is given by the [[codiagonal map]] $\nabla \;\colon\; Msg \sqcup Msg \;\to\; Msg$ as

   $$
     \phantom{\mathclap{\vert^{\vert^{\vert^{\vert^{\vert}}}}}}
     prog\big[-\big]
     \;\coloneqq\;
     D_2 \sqcup Msg
     \xrightarrow{\; prog \sqcup Id_{Msg} \;}
     D_3 \sqcup Msg \sqcup Msg
     \xrightarrow{\; id_{D_3} \sqcup \nabla \;}
     D_3 \sqcup Msg
     \,,
   $$

   meaning that whatever exception message has already been thrown gets carried further along.

If here $Msg = \ast$ is the [[unit type]], then the exception monad is also known as the *[[maybe monad]]*, modelling the effect that a program *may* fail, without however transmitting any further information about the failure.

Exception monads correspond to [[coreader comonads]] in the opposite category, because coproducts turn into [[products]] there.

## Related entries

* [[coreader comonad]]

* [[maybe monad]]

* [[writer monad]]

## References

Exceptions [[monads in computer science]]:

* {#Moggi91} [[Eugenio Moggi]], Exp. 1.1 in: *Notions of computation and monads*, Information and Computation, **93** 1 (1991) [doi:10.1016/0890-5401(91)90052-4, [pdf](http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf)]

* [[Philip Wadler]], §2.2 in: *Monads for functional programming*, in M. Broy (eds.) *Program Design Calculi* NATO ASI Series, **118** Springer (1992) [[doi;10.1007/978-3-662-02880-3_8](https://doi.org/10.1007/978-3-662-02880-3_8), [pdf](https://homepages.inf.ed.ac.uk/wadler/papers/marktoberdorf/baastad.pdf)]

* [[Jiří Adámek]], Stefan Milius, Nathan Bowler, Paul B. Levy, *Coproducts of Monads on Set*, 27th Annual IEEE Symposium on Logic in Computer Science (2012) [[arXiv:1409.3804](https://arxiv.org/abs/1409.3804), [doi:10.1109/LICS.2012.16](https://doi.org/10.1109/LICS.2012.16)]

and on exception handling [[module over a monad|modales]]:

* [[Gordon D. Plotkin ]], [[Matija Pretnar]], §1 in: *Handling Algebraic Effects*, Logical Methods in Computer Science, **9** 4 (2013) lmcs:705 [[arXiv:1312.1399](https://arxiv.org/abs/1312.1399), doi:10.2168/LMCS-9(4:23)2013]

Lecture notes:

* {#Uustalu21} [[Tarmo Uustalu]], p. 11 in: *Monads and Interaction Lecture 1* lecture notes for [MGS 2021](https://staffwww.dcs.shef.ac.uk/people/G.Struth/mgs21.html) (2021) [[pdf](https://cs.ioc.ee/~tarmo/mgs21/mgs1.pdf), [[Uustalu-Monads1.pdf:file]]]

See also:

* Wikipedia, *[Exception handling](https://en.wikipedia.org/wiki/Exception_handling)*

[[!redirects exception monads]]

[[!redirects exception]]
[[!redirects exceptions]]

[[!redirects exception handler]]
[[!redirects exception handlers]]

[[!redirects exception handling]]
[[!redirects exception handlings]]


exceptional collection

#Contents#
* table of contents
{:toc}

## Idea

In a [[derived category]] of [[chain complexes]], or more generally in a [[triangulated category]] or in fact in a [[stable (infinity,1)-category]], an _exceptional collection_ of object is a [[set]] of [[objects]] such that the [[Ext]]-groups ([[(infinity,1)-categorical hom-space|hom-spaces]]) between them are concentrated in degree 0. It is a _full exceptional collection_ if in addition it generates the category.

## References

In

* [[Alexei Bondal]], _Helices, representations of quivers and Koszul algebras_, in Helices and vector bundles, vol. 148 of London Math. Soc. Lecture Note Ser., pp. 75–95. Cambridge Univ. Press, Cambridge, 1990.

an exceptional collection in the [[derived category]] of [[coherent sheaves]]on a suitable [[Calabi-Yau variety]] [[cone]] is used to induce a [[quiver]] and make the derived category be equivalent to that of its [[quiver representations]].

See also

* Agnieszka Bodzenta, _DG categories and exceptional collections_ ([arXiv:1205.6148](http://arxiv.org/abs/1205.6148))

[[!redirects exceptional collections]]


exceptional field theory

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#Contents#
* table of contents
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## Idea

In the context of [[supergravity]] and [[string theory]], the term _exceptional field theory_ has come to be used for formulations of [[11-dimensional supergravity]] which make the ([[exceptional Lie group|exceptional]], whence the name) [[U-duality]] [[symmetry group]] structure manifest. This is in generalization of the "[[double field theory]]" formulation of 10d [[type II supergravity]] which makes (only) the [[T-duality]] symmetry manifest.

Accordingly, exceptional field theory is related to [[exceptional generalized geometry]] as [[double field theory]] is related to [[generalized complex geometry]].

A [[spacetime]] in exceptional field theory is locally modeled on the [[Cartesian product]]
$$ \mathbb{R}^{1,d-1} \times R $$
of a $d$-dimensional [[Minkowski space]] with the [[fundamental representation]] [[vector space]] $R$ of a form of the [[exceptional Lie group]] $E_{11-d}$ (the [[U-duality]] group). In the literature the former is called _external space_ and the latter _internal space_. Fields on this spacetime are subject to satisfy a certain [[differential equation]] derived from an [[invariant]] form of the representation and one considers a generalized [[isometry]] algebra on this space which fails the [[Jacobi identity]] by a term proportional to this contraint (e.g. [Baguet-Hohm-Samtleben 15, section 2](#BaguetHohmSamtleben15)).

## Related concepts

* [[U-duality]], [[exceptional Lie group]]

* [[exceptional generalized geometry]], [[exceptional tangent bundle]]

* [[non-geometry]], [[exotic branes]]

* [[double field theory]]

* [[generalized Scherk-Schwarz reduction]], [[gauged supergravity]], [[tensor hierarchy]], [[embedding tensor]]

* [[exceptional geometry]]

* [[universal exceptionalism]]

## References

Precursors include

* [[Olaf Hohm]], [[Henning Samtleben]], _U-duality covariant gravity_ ([arXiv:1307.0509](http://arxiv.org/abs/1307.0509))

The original articles are

* [[Olaf Hohm]], [[Henning Samtleben]], _Exceptional Form of $D=11$ Supergravity_, Phys. Rev. Lett. 111, 231601 (2013) ([arXiv:1308.1673](http://arxiv.org/abs/1308.1673))

* [[Olaf Hohm]], [[Henning Samtleben]], _Exceptional Field Theory I: $E_{6(6)}$ covariant Form of M-Theory and Type IIB_, Phys. Rev. D 89, 066016 (2014) ([arXiv:1312.0614](http://arxiv.org/abs/1312.0614))

* [[Olaf Hohm]], [[Henning Samtleben]], _Exceptional Field Theory II: $E_{7(7)}$_, Phys. Rev. D 89, 066017 (2014) ([arXiv:1312.4542](http://arxiv.org/abs/1312.4542))

* [[Olaf Hohm]], [[Henning Samtleben]], _Exceptional Field Theory III: $E_{8(8)}$_, 	Phys. Rev. D 90, 066002 (2014) ([arXiv:1406.3348](http://arxiv.org/abs/1406.3348))

Review:

* {#BermanBlair06} [[David Berman]], [[Chris D. A. Blair]], _The Geometry, Branes and Applications of Exceptional Field Theory_,  International Journal of Modern Physics A **35** 30 (2020) 2030014  [[arXiv:2006.09777](http://arxiv.org/abs/2006.09777), [doi:10.1142/S0217751X20300148](https://doi.org/10.1142/S0217751X20300148)]

Review for [[KK-compactification]] to [[5d supergravity]], hence for [[E6]]-[[U-duality]], includes

* {#BaguetHohmSamtleben15} Arnaud Baguet, [[Olaf Hohm]], [[Henning Samtleben]], _$E_{6(6)}$ Exceptional Field Theory: Review and Embedding of Type IIB_ ([arXiv:1506.01065](http://arxiv.org/abs/1506.01065))

Discussion of solitonic [[black brane]] (and [[exotic brane]]) solutions in terms of EFT includes

* {#BermanRudolph14} [[David Berman]], [[Felix Rudolph]], _Strings, Branes and the Self-dual Solutions of Exceptional Field Theory_ ([arXiv.1412.2768](http://arxiv.org/abs/1412.2768))

Discussion for [[E11]]:

* Alexander G. Tumanov, [[Peter West]], _$E_{11}$ and exceptional field theory_ ([arXiv:1507.08912](http://arxiv.org/abs/1507.08912))

* [[Guillaume Bossard]], [[Axel Kleinschmidt]], [[Ergin Sezgin]], _On supersymmetric E11 exceptional field theory_ ([arXiv:1907.02080](https://arxiv.org/abs/1907.02080))

* [[Guillaume Bossard]], [[Axel Kleinschmidt]], [[Ergin Sezgin]], _A master exceptional field theory_ ([arXiv:2103.13411](https://arxiv.org/abs/2103.13411))

Discussion for [[E9]]:

* Guillaume Bossard, Franz Ciceri, Gianluca Inverso, [[Axel Kleinschmidt]], [[Henning Samtleben]], _$E_9$ exceptional field theory II. The complete dynamics_ ([arXiv:2103.12118](https://arxiv.org/abs/2103.12118))

Generalization to exceptional [[super-spacetimes]]:

* Daniel Butter, [[Henning Samtleben]], [[Ergin Sezgin]], _$E_{7(7)}$-Exceptional Field Theory in Superspace_ ([arXiv:1811.00038](https://arxiv.org/abs/1811.00038))

Application to [[AdS4/CFT3]]:

* Oscar Varela, _Super-Chern-Simons spectra from Exceptional Field Theory_ ([arXiv:2010.09743](https://arxiv.org/abs/2010.09743))

See also:

On application to [[KK-reduction]] of [[D=10 supergravity]] and [[D=11 supergravity]] on [[squashed sphere|squashed]] [[7-spheres]]:

* {#DuboeufMalekSamtleben23} Bastien Duboeuf, [[Emanuel Malek]], [[Henning Samtleben]],
*Kaluza-Klein Spectrometry beyond Consistent Truncations: The Squashed $S^7$*,  J. High Energ. Phys. **2023** 62 (2023) [[arXiv:2212.01135](https://arxiv.org/abs/2212.01135), doi:10.1007/JHEP04(2023)062]

[[!redirects exceptional field theories]]
[[!redirects Exceptional field theories]]
[[!redirects Exceptional field theory]]


exceptional generalized geometry

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Exceptional structures
+-- {: .hide}
[[!include exceptional structures -- contents]]
=--
#### Duality in string theory
+-- {: .hide}
[[!include duality in string theory -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A variant of the idea of [[generalized complex geometry]] given by passing from generalization of [[complex geometry]] to generalization of [[exceptional geometry]]. Instead of by [[reduction of structure groups]] along inclusions like $O(d)\times O(d) \to O(d,d)$ it is controled by inclusions into [[split real forms]] of [[exceptional Lie groups]].

This serves to neatly encode [[U-duality]] groups in [[supergravity]] as well as higher [[supersymmetry]] of supergravity [[Kaluza-Klein mechanism|compactifications]]. See also at _[[exceptional field theory]]_ for more on this.

## Examples

### Higher supersymmetry
 {#HigherSupersymmetry}

Compactification of [[11-dimensional supergravity]] on a [[manifold]] of [[dimension]] 7 preserves $N = 1$ [[supersymmetry]] precisely if its [[generalized tangent bundle]] has [[G-structure]] for the inclusion

  $$
    SU(7) \hookrightarrow E_{7(7)}
  $$

of the [[special unitary group]] in dimension 7 into the [[split real form]] of [[E7]]. This is shown in ([Pacheco-Waldram 08](#PachecoWaldram08)).

One dimension down, compactification of 10-dimensional
[[type II supergravity]] on a 6-manifold $X$ preserves $N = 2$ supersymmetry precisely if the [[generalized tangent bundle]] $T X \otimes T^* X$ in the NS-NS sector admits [[G-structure]] for the inclusion

$$
  SU(3) \times SU(3) \hookrightarrow O(6,6)
  \,.
$$

This is reviewed in ([GLSW, section 2](#GLSW)).

## Related concepts

* [[non-geometric string theory vacua]]

* [[generalized complex geometry]], [[exceptional geometry]]

* [[generalized tangent bundle]], [[exceptional tangent bundle]]

* [[generalized Calabi-Yau manifold]], [[generalized G2-manifold]]

* [[Bn-geometry]], [[differential T-duality]]

* [[universal exceptionalism]]

## References

### General

Original articles:

* K. Koepsell, [[Hermann Nicolai]], [[Henning Samtleben]], _An exceptional geometry for d=11 supergravity?_, Class. Quant. Grav.17:3689-3702, 2000 ([arXiv:hep-th/0006034](http://arxiv.org/abs/hep-th/0006034))

* {#Hull07} [[Chris Hull]], _Generalised Geometry for M-Theory_, JHEP 0707:079 (2007) ([arXiv:hep-th/0701203](http://arxiv.org/abs/hep-th/0701203))

* {#PachecoWaldram08} Paulo Pires Pacheco, [[Daniel Waldram]], _M-theory, exceptional generalised geometry and superpotentials_, JHEP 0809:123, 2008 ([arXiv:0804.1362](http://arxiv.org/abs/0804.1362))

* {#GLSW} [[Mariana Graña]], [[Jan Louis]], Aaron Sim, [[Daniel Waldram]], _$E_{7(7)}$ formulation of $N=2$ backgrounds_ ([arXiv:0904.2333](http://arxiv.org/abs/0904.2333))

* G. Aldazabala, E. Andrésb, P. Cámarac, [[Mariana Graña]], _U-dual fluxes and generalized geometry_, JHEP 1011:083,2010 ([arXiv:1007.5509](http://arxiv.org/abs/1007.5509))

* [[Mariana Graña]], Francesco Orsi, _$N=1$ vacua in Exceptional Generalized Geometry_  ([arXiv:1105.4855](http://arxiv.org/abs/1105.4855))
 {#GranaOrsi}

* [[Mariana Graña]], Francesco Orsi, _N=2 vacua in Generalized Geometry_, ([arXiv:1207.3004](http://arxiv.org/abs/1207.3004))

* André Coimbra, Charles Strickland-Constable, [[Daniel Waldram]], _$E_{d(d)} \times \mathbb{R}^+$ Generalised Geometry, Connections and M theory_ ([arXiv:1112.3989](http://arxiv.org/abs/1112.3989))

* {#Baraglia11} [[David Baraglia]], _Leibniz algebroids, twistings and exceptional generalized geometry_, Journal of Geometry and Physics 62 (2012), pp. 903-934 ([arXiv:1101.0856](http://arxiv.org/abs/1101.0856))

* [[David Baraglia]], _Exceptional generalized geometry and $N = 2$ backgrounds_ ([pdf](http://mitchell.physics.tamu.edu/Conference/GeneralizedGeometries/Mon-1-Grana.pdf))

Review:

* Daniel Persson, _Arithmetic and Hyperbolic Structures in String Theory_ ([arXiv:1001.3154](http://arxiv.org/abs/1001.3154))

* Nassiba Tabti, _Kac-Moody algebraic structures in supergravity theories_ ([arXiv:0910.1444](http://arxiv.org/abs/0910.1444))

* [[Daniel Waldram]], *Fluxes, holography and uses of Exceptional Generalized Geometry*, talk at *[[Strings 2022]]*  [[indico:4940880](https://indico.cern.ch/event/1085701/contributions/4940880), [slides](https://indico.cern.ch/event/1085701/contributions/4940880/attachments/2482886/4262576/Daniel%20Waldram%20Strings2022-waldram.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=e222a3a9-ebf4-445c-ada0-f7b1d592d908) ]

  > on generalized [[G-structures]] and [[exceptional generalized geometry]] of [[flux compactifications]] and in [[AdS/CFT]]

Review of [[U-duality]] and [[exceptional generalized geometry]]  in [[KK-compactification]] of [[D=11 supergravity]]:

* [[Henning Samtleben]], *11D Supergravity and Hidden Symmetries*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2303.12682](https://arxiv.org/abs/2303.12682)]

Relation to [[Borcherds superalgebras]] is surveyed and discussed in

* Jakob Palmkvist, _Exceptional geometry and Borcherds superalgebras_ ([arXiv:1507.08828](http://arxiv.org/abs/1507.08828))

[[black branes]] in the exotic spacetime are discussed in

* Ilya Bakhmatov, [[David Berman]], [[Axel Kleinschmidt]], Edvard Musaev, Ray Otsuki, _[arXiv:1710.09740](https://arxiv.org/abs/1710.09740)_

{#ExceptionalSigmaModelReferences} The [[string]] and [[membrane]] [[sigma-models]] on exceptional spacetime (the "exceptional sigma models") are discussed in

* [[Yuho Sakatani]], [[Shozo Uehara]], _Branes in Extended Spacetime: Brane Worldvolume Theory Based on Duality Symmetry_, Phys. Rev. Lett. **117** 191601 (2016) [[arXiv:1607.04265](https://arxiv.org/abs/1607.04265)]

* [[Alex Arvanitakis]], [[Chris D. A. Blair]], _Type II strings are Exceptional_ ([arXiv:1712.07115](https://arxiv.org/abs/1712.07115))

* [[Alex Arvanitakis]], Chris Blair, _The Exceptional Sigma Model_ ([arXiv:1802.00442](https://arxiv.org/abs/1802.00442))

Generalized [[U-duality]]+[[diffeomorphism]] invariance in 11d:

* [[David Berman]], [[Martin Cederwall]], [[Axel Kleinschmidt]], [[Daniel C. Thompson]], _The gauge structure of generalised diffeomorphisms_, High Energ. Phys. 2013, 64 (2013) [[arXiv:1208.5884](http://arxiv.org/abs/1208.5884)]

For the [[worldvolume]] [[physical theory|theory]] of the [[M5-brane]] this is discussed in

* Machiko Hatsuda, Kiyoshi Kamimura, _M5 algebra and $SO(5,5)$ duality_ ([arXiv:1305.2258](http://arxiv.org/abs/1305.2258))

More on generalized exceptional [[sigma-models]]:

* David Osten, *On the universal exceptional structure of world-volume theories in string and M-theory* [[arXiv:2402.10269](https://arxiv.org/abs/2402.10269)]

### Super-exceptional generalized geometry
 {#SuperExceptionalGeometryReferences}

The combination/unification of [[exceptional generalized geometry]] with [[supergeometry]] used to be an open problem:

* [[Martin Cederwall]], p. 39 of _Fundamental issues in extended geometry_, 8th Mathematical Physics Meeting, Aug 2014 Belgrade, Serbia ([spire:1477275](http://inspirehep.net/record/1477275))

* [[Martin Cederwall]], Joakim Edlund, Anna Karlsson, p. 4, 7 of _Exceptional geometry and tensor fields_, J. High Energ. Phys. (2013) 2013: 28 ([arXiv:1302.6736](https://arxiv.org/abs/1302.6736))

Plausibility arguments that the bosonic body of the [[superspace]] underlying the [[M-theory Lie algebra]] serves as the unifying exceptional generalized geometry for [[M-theory]] for $n = 11$:

* [[Igor Bandos]], _Exceptional field theories, superparticles in an enlarged 11D superspace and higher spin theories_, Nucl. Phys. B925 (2017) 28-62 ([arXiv:1612.01321](https://arxiv.org/abs/1612.01321))

Arguments that super-exceptional M-geometry for $n = 11$ is in fact a further fermionic extension of that (to the "hidden supergroup" of D'Auria-Fre):

* [[Silvia Vaula]], _On the underlying $E_{11}$ symmetry of the $D= 11$ Free Differential Algebra_, JHEP 0703:010, 2007 ([arXiv:hep-th/0612130](http://arxiv.org/abs/hep-th/0612130))

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Higher T-duality of super M-branes]]_ ([arXiv:1803.05634](https://arxiv.org/abs/1803.05634))

* [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Higher T-duality of super M-branes|Higher T-duality in M-theory via local supersymmetry]]_, Physics Letters B **781** (2018)  [[arXiv:1805.00233](https://arxiv.org/abs/1805.00233)]

* Daniel Butter, [[Henning Samtleben]], [[Ergin Sezgin]], _$E_{7(7)}$ Exceptional Field Theory in Superspace_, JHEP 01 (2019) 087 [[arXiv:1811.00038](https://arxiv.org/abs/1811.00038), doi:10.1007/JHEP01(2019)087]

* {#FSS19c} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Super-exceptional embedding construction of the M5-brane|Super-exceptional geometry: origin of heterotic M-theory and super-exceptional embedding construction of M5]]_, Journal of High Energy Physics **2020** 107 (2020) [[arXiv:1908.00042](https://arxiv.org/abs/1908.00042), doi:10.1007/JHEP02(2020)107]

See also the references on the corresponding [[super-geometry]]-enhancement of [[type II geometry|type II]] [[generalized geometry]] ("[[doubled geometry]]"): _[doubled geometry -- References -- Doubled supergeometry](type+II+geometry#ReferencesDoubledSupergeometry)_.

### $E_6$, $E_7$, $E_8$

[[E6]],[[E7]], [[E8]]-geometry is discussed in

* Christian Hillmann, _Generalized E(7(7)) coset dynamics and D=11 supergravity_, JHEP 0903 (2009) 135 ([arXiv:0901.1581](http://arxiv.org/abs/0901.1581))

* {#GodazgarGodazgarNicolao13} Hadi Godazgar, Mahdi Godazgar, Hermann Nicolai, _Generalised geometry from the ground up_ ([arXiv:1307.8295](http://arxiv.org/abs/1307.8295))

* {#HohmSamtleben13} [[Olaf Hohm]], [[Henning Samtleben]], _Exceptional Form of $D=11$ Supergravity_, Phys. Rev. Lett. 111, 231601 (2013) ([arXiv:1308.1673](http://arxiv.org/abs/1308.1673))

(see also at [3d supergravity -- possible gaugings](3-dimensional+supergravity#PossibleGaugings)).

### $E_{10}$

The [[E10]]-geometry of [[11-dimensional supergravity]] compactified to the line is discussed in

* [[Thibault Damour]], [[Hermann Nicolai]], _Higher order M theory corrections and the Kac-Moody algebra E10_ ([arXiv:hep-th/0504153](http://arxiv.org/abs/hep-th/0504153))

* {#Nicolai08} [[Hermann Nicolai]], _Wonders of $E_{10}$ and $K(E_{10})$_ (2008) ([pdf](http://ipht.cea.fr/Pisp/pierre.vanhove/Paris08/talk_PDF/nicolai.pdf))

* [[Axel Kleinschmidt]], [[Hermann Nicolai]], _Standard model fermions and $K(E_{10})$_ ([arXiv:1504.01586](https://arxiv.org/abs/1504.01586))

### $E_{11}$

Literature discussing $E_{11}$ [[U-duality]] and in the context of [[exceptional generalized geometry]] of [[11-dimensional supergravity]].

Review includes

* [[Peter West]], section 17.5 of _[[Introduction to Strings and Branes]]_

* [[Fabio Riccioni]], _$E_{11}$ and M-theory_, talk at [Strings07](http://www.ift.uam.es/strings07/) ([pdf slides](http://member.ipmu.jp/yuji.tachikawa/stringsmirrors/2007/riccioni.pdf))

* [[Fabio Riccioni]], [[Peter West]], _The $E_{11}$ origin of all maximal supergravities_, JHEP 0707:063,2007 ([arXiv:0705.0752](http://arxiv.org/abs/0705.0752), [spire](http://inspirehep.net/record/749966/))

* [[Paul Cook]], _Connections between Kac-Moody algebras and M-theory_ PhD thesis ([arXiv:0711.3498](http://arxiv.org/abs/0711.3498))

* {#West16} [[Peter West]], _A brief review of E theory_ ([arXiv:1609.06863](http://arxiv.org/abs/1609.06863))

Original articles include the following:

The observation that $E_{11}$ seems to neatly organize the structures in [[11-dimensional supergravity]]/[[M-theory]] is due to

* {#West01} [[Peter West]], _$E_{11}$ and M theory_, Class. Quant. Grav., 18:4443–4460, 2001. ([arXiv:hep-th/0104081](http://arxiv.org/abs/hep-th/0104081))

A precursor to ([West 01](#West01)) is

* [[Bernard Julia]], _Dualities in the classical supergravity limits_ ([arXiv:hep-th/9805083](http://arxiv.org/abs/hep-th/9805083))

as explained in ([Henneaux-Julia-Levie 10](#HenneauxJuliaLevie10)).

The derivation of the [[equations of motion]] of [[11-dimensional supergravity]] and maximally supersymmetric [[5d supergravity]] from a [[vielbein]] with values in the [[semidirect product]] $E_{11}$ with its [[fundamental representation]] is due to

* {#West11} [[Peter West]], _Generalised geometry, eleven dimensions and $E_{11}$_, J. High Energ. Phys. (2012) 2012: 18 ([arXiv:1111.1642](http://arxiv.org/abs/1111.1642))

* Alexander G. Tumanov, [[Peter West]], _$E_{11}$ must be a symmetry of strings and branes_, Physics Letters B Volume 759, 10 August 2016, Pages 663–671 ([arXiv:1512.01644](https://arxiv.org/abs/1512.01644))

* Alexander G. Tumanov, [[Peter West]], _$E_{11}$ in $11d$_, Physics Letters B Volume 758, 10 July 2016, Pages 278–285 ([arXiv:1601.03974](https://arxiv.org/abs/1601.03974))

This way that elements of [[cosets]] of the [[semidirect product]] $E_{11}$ with its [[fundamental representation]] may encode [[equations of motion]] of [[11-dimensional supergravity]] follows previous considerations for [[Einstein equations]] in

* [[Abdus Salam]], J. Strathdee, _Nonlinear realizations. 1: The Role of Goldstone bosons, Phys. Rev. 184 (1969) 1750,

* [[Chris Isham]], [[Abdus Salam]], J. Strathdee, _Spontaneous, breakdown of conformal symmetry_, Phys. Lett. 31B (1970) 300.

* A. Borisov, V. Ogievetsky, _Theory of dynamical affine and conformal symmetries as the theory of the gravitational field_, Theor. Math. Phys. 21 (1973) 1179-1188 ([web](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tmf&paperid=3902&option_lang=eng))

* V. Ogievetsky, _Infinite-dimensional algebra of general covariance group as the closure of the finite dimensional algebras of conformal and linear groups_, Nuovo. Cimento, 8 (1973) 988.

Further developments of the proposed $E_{11}$ formulation of [[M-theory]] include

* [[Peter West]], _$E_{11}$, ten forms and supergravity_, JHEP0603:072,2006 ([arXiv:hep-th/0511153](http://arxiv.org/abs/hep-th/0511153))

* [[Fabio Riccioni]], [[Peter West]], _Dual fields and $E_{11}$_, Phys.Lett.B645:286-292,2007 ([arXiv:hep-th/0612001](http://arxiv.org/abs/hep-th/0612001))

* [[Fabio Riccioni]], [[Peter West]], _E(11)-extended spacetime and gauged supergravities_, JHEP 0802:039,2008 ([arXiv:0712.1795](http://arxiv.org/abs/0712.1795))

* [[Fabio Riccioni]], Duncan Steele, [[Peter West]], _The E(11) origin of all maximal supergravities - the hierarchy of field-strengths_, 	JHEP 0909:095 (2009) ([arXiv:0906.1177](http://arxiv.org/abs/0906.1177))

* [[Eric Bergshoeff]], I. De Baetselier, T. Nutma, _E(11) and the Embedding Tensor_ ([arXiv:0705.1304](http://arxiv.org/abs/0705.1304), [poster](http://mms.technologynetworks.net/posters/0364.pdf))

* Guillaume Bossard, [[Axel Kleinschmidt]], Jakob Palmkvist, [[Christopher Pope]], [[Ergin Sezgin]], _Beyond $E_{11}$_ ([arXiv:1703.01305](https://arxiv.org/abs/1703.01305))

Discussion of the [[semidirect product]] of $E_{11}$ with its $l_1$-[[representation]], and arguments that the [[charges]] of the [[M-theory super Lie algebra]] and in fact further brane charges may be identified inside $l_1$ originate in

* {#West04} [[Peter West]], _$E_{11}$, $SL(32)$ and Central Charges_, Phys.Lett.B575:333-
342,2003 ([arXiv:hep-th/0307098](http://arxiv.org/abs/hep-th/0307098))

and was further explored in

* {#KleinschmidtWest04} [[Axel Kleinschmidt]], [[Peter West]], _Representations of $G^{+++}$ and the role of space-time_, JHEP 0402 (2004) 033 ([arXiv:hep-th/0312247](http://arxiv.org/abs/hep-th/0312247))

* [[Paul Cook]], [[Peter West]], _Charge multiplets and masses for $E(11)$_, JHEP 11 (2008) 091 ([arXiv:0805.4451](http://arxiv.org/abs/0805.4451))

* [[Peter West]], _$E_{11}$ origin of Brane charges and U-duality multiplets_, JHEP 0408 (2004) 052 ([arXiv:hep-th/0406150](http://arxiv.org/abs/hep-th/0406150))

Relation to [[exceptional field theory]] is discussed in

* Alexander G. Tumanov, [[Peter West]], _$E_{11}$ and exceptional field theory_ ([arXiv:1507.08912](http://arxiv.org/abs/1507.08912))

Relation to [[Borcherds superalgebras]] is discussed in

* Pierre Henry-Labordere, [[Bernard Julia]], Louis Paulot, _Borcherds symmetries in M-theory_, JHEP 0204 (2002) 049 ([arXiv:hep-th/0203070](http://arxiv.org/abs/hep-th/0203070))

* {#HenneauxJuliaLevie10} [[Marc Henneaux]], [[Bernard Julia]], Jérôme Levie, _$E_{11}$, Borcherds algebras and maximal supergravity_ ([arxiv:1007.5241](http://arxiv.org/abs/1007.5241))

* {#Palmkvist12} Jakob Palmkvist, _Tensor hierarchies, Borcherds algebras and $E_{11}$_, JHEP 1202 (2012) 066 ([arXiv:1110.4892](http://arxiv.org/abs/1110.4892))

[[!redirects exceptional generalized geometry]]

[[!redirects exceptional generalized geometries]]

[[!redirects exceptional generalized complex geometry]]
[[!redirects exceptional generalized complex geometry]]

[[!redirects exceptional generalised geometry]]
[[!redirects exceptional generalised geometries]]

[[!redirects super-exceptional generalised geometry]]
[[!redirects super-exceptional generalised geometries]]
[[!redirects super-exceptional generalized geometry]]
[[!redirects super-exceptional generalized geometries]]


exceptional geometry

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Exceptional structures
+-- {: .hide}
[[!include exceptional structures -- contents]]
=--
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The classification of [[Riemannian manifolds]] with [[special holonomy]] contains two "exceptional" cases: [[G2-manifold|G2-holonomy]] in [[dimension]] 7, and [[Spin(7)-manifolds|Spin(7)-holonomy]] in dimension 8. Their study is the topic of _exceptional geometry_.

Sometimes more generally, exceptional geometry is understood to study [[spaces]] controled by [[exceptional Lie groups]] in some way.

\linebreak

## Properties

### In terms of twisted Cohomotopy

[[!include coset space structure on n-spheres -- table]]

## Related concepts

* [[exceptional Lie group]]

* [[exceptional generalized geometry]]

* [[exceptional field theory]]

* [[universal exceptionalism]]

## References

### General

General discussion is in

* [[Dominic Joyce]], _The exceptional holonomy groups and calibrated geometry_ ([pdf](http://gokovagt.org/proceedings/2005/ggt05-joyce.pdf))

* [[Simon Salamon]], _A tour of exceptional geometry_ ([pdf](http://calvino.polito.it/~salamon/T/tour.pdf))

* [[Simon Salamon]], _Self-duality and exceptional geometry_ ([pdf](http://calvino.polito.it/~salamon/G/baku.pdf))

Discussion of [[G2 manifolds]] is in

* Spiro Karigiannis, _$G_2$-manifolds -- Exceptional structures in geometry arising from exceptional algebra_ ([pdf](http://www.math.uwaterloo.ca/~karigian/talks/waterloo.pdf))

### In supergravity

Applications to [[KK-compactification]] of [[11d supergravity]] (see also at _[[M-theory on G2-manifolds]]_) is discussed in

* [[George Papadopoulos]], [[Paul Townsend]], _Compactifications of $D = 11$ supergravity on spaces of exceptional holonomy_ ([arXiv:hep-th/9506150](http://arxiv.org/abs/hep-th/9506150))

* K. Koepsell, [[Hermann Nicolai]], [[Henning Samtleben]], _An exceptional geometry for d=11 supergravity?_ ([arXiv:hep-th/0006034](http://arxiv.org/abs/hep-th/0006034))

* {#Hull} [[Chris Hull]], _Generalised Geometry for M-Theory_, JHEP 0707:079, 2007 ([arXiv:hep-th/0701203](http://arxiv.org/abs/hep-th/0701203))

For more along these lines see the references at _[[exceptional generalized geometry]]_.

### As M-brane target space

Discusssion of [[M-brane]] [[sigma-models]] on exceptional geometry [[target spaces]]:

* [[Yuho Sakatani]], [[Shozo Uehara]], _Branes in Extended Spacetime: Brane Worldvolume Theory Based on Duality Symmetry_, Phys. Rev. Lett. **117** 191601 (2016) [[arXiv:1607.04265](https://arxiv.org/abs/1607.04265), [talk slides](http://www2.yukawa.kyoto-u.ac.jp/~qft.web/2016/slides/sakatani.pdf)]

* [[Yuho Sakatani]], [[Shozo Uehara]], _Exceptional M-brane sigma models and $\eta$-symbols_ [[arXiv:1712.10316](https://arxiv.org/abs/1712.10316)]

[[!redirects exceptional geometries]]


exceptional inverse image

+-- {: .rightHandSide}
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### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[right adjoint]] to [[direct image with compact support]]

(...)

## References

* [[Dennis Gaitsgory]], [[Nick Rozenblyum]], 5.1.5 of _Ind-coherent sheaves on schemes_ ([pdf](http://www.math.harvard.edu/~gaitsgde/GL/IndCohSch.pdf), [web](http://www.math.harvard.edu/~gaitsgde/GL/))

## Related concepts

* [[inverse image]]

[[!redirects exceptional inverse images]]


exceptional isomorphism

+-- {: .rightHandSide}
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###Context###
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
#### Exceptional structures
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[[!include exceptional structures -- contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

Given a sequence of [[structures]] following some pattern, one loosely calls [[isomorphisms]] that pertain to some special elements in the sequence, without themselves following that pattern, _exceptional_ or _sporadic_.

## Examples

### Spin groups

The archetypical example is [[isomorphisms]] of [[spin groups]]. These appear in the infinite sequence $\big\{Spin(p) \vert p \in \mathbb{N}\big\}$, and for low values of $p$, but not generally, there are isomorphisms to other [[classical Lie groups]].

See at _[[spin group#ExceptionalIsomorphisms|spin group -- Exceptional isomorphisms]]_.

### Finite linear groups

There is an exceptional isomorphisms

$PSL_2(\mathbb{F}_5) \;\simeq\; I $

between the [[projective special linear group]] over the [[prime field]] $\mathbb{F}_5$ with the [[icosahedral group]].

See at [[icosahedral group#IsomorphismToSL25|Icosahedral group -- Exceptional isomorphism to PSL(F5)]].

## Related concepts

* [[exceptional Lie group]]

* [[exceptional Jordan algebra]]

* [[exceptional geometry]], [[exceptional generalized geometry]], [[exceptional field theory]]

* [[universal exceptionalism]]

## References

* {#Garrett13} [[Paul Garrett]], _Sporadic isogenies to orthogonal groups_ (July 2013) [[pdf](http://www.math.umn.edu/~garrett/m/v/sporadic_isogenies.pdf), [[Garrett-SporadicIsogenies.pdf:file]] ]

[[!redirects exceptional isomorphisms]]

[[!redirects sporadic isomorphism]]
[[!redirects sporadic isomorphisms]]


exceptional Lie algebra

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

## Idea

The _exceptional Lie algebras_ are the [[exceptional structures]] among the [[simple Lie algebras]]:

The classification of the [[simple Lie algebras]] consists of a few infinite series together with a [[finite number]] of examples, called _exceptional Lie algebras_. The [[simple Lie groups]] that these correspond to are the _[[exceptional Lie groups]]_, see there for more.

## Related concepts

* [[universal exceptionalism]]

## References

* [[John Baez]], _[Exceptional Lie algebras](http://math.ucr.edu/home/baez/octonions/node13.html)_, chapter 4 in _The Octonions_,  Bull. Amer. Math. Soc. 39 (2002), 145-205. ([web](http://math.ucr.edu/home/baez/octonions/octonions.html))

* J. R. Faulkner, J. C. Ferrar, _Exceptional Lie algebras and related algebraic and geometric structures, ([[FaulknerFerrar.pdf:file]])

* [[José Figueroa-O'Farrill]], _A geometric construction of the exceptional Lie algebras F4 and E8_ ([arXiv:0706.2829](http://arxiv.org/abs/0706.2829))

* Andrei Moroianu, [[Uwe Semmelmann]], _Invariant four-forms and symmetric pairs_ ([arXiv:1202.3407](http://arxiv.org/abs/1202.3407))

[[!redirects exceptional Lie algebras]]


exceptional Lie group

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#### Lie theory
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[[!include infinity-Lie theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _exceptional Lie groups_ are the [[exceptional structures]] among the [[simple Lie groups]].

The classification of [[simple Lie groups]] consists of four infinite series -- the [[classical Lie groups]], and five exceptional Lie groups, called

* [[G2]], [[F4]], [[E6]], [[E7]], and [[E8]].

On the level of [[Kac-Moody Lie algebras]]/[[Kac-Moody groups]] the E-series continues as

* [[E9]], [[E10]], [[E11]], ...

## Related concepts

* [[exceptional Lie algebra]]

* [[exceptional geometry]], [[exceptional generalized geometry]]

* [[universal exceptionalism]]

* [[Freudenthal magic square]]

## References

### Lie groups

* [[Ichiro Yokota]], _Exceptional Lie groups_ [[arXiv:0902.0431](http://arxiv.org/abs/0902.0431)]

### Lie algebras

Original articles include

* {#Freudenthal65} [[Hans Freudenthal]], _Lie groups in the foundations of geometry_, Advances in Mathematics, volume 1, (1965) pp. 145 - 190 ([dspace](http://dspace.library.uu.nl/handle/1874/17442))

The following are references on the [[Lie algebras]] underlying exceptional Lie groups.

Surveys include

* wikipedia, _[En](http://en.wikipedia.org/wiki/En_%28Lie_algebra%29)_

* J. R. Faulkner, J. C. Ferrar, _Exceptional Lie algebras and related algebraic and geometric structures, ([[FaulknerFerrar.pdf:file]])

* [[John Baez]], _[Exceptional Lie algebras](http://math.ucr.edu/home/baez/octonions/node13.html)_, chapter 4 in _The Octonions_,  Bull. Amer. Math. Soc. 39 (2002), 145-205. ([web](http://math.ucr.edu/home/baez/octonions/octonions.html))

Geometric constructions of [[exceptional Lie algebras]] are discussed in

* [[José Figueroa-O'Farrill]], _A geometric construction of the exceptional Lie algebras F4 and E8_ ([arXiv:0706.2829](http://arxiv.org/abs/0706.2829))

* Andrei Moroianu, [[Uwe Semmelmann]], _Invariant four-forms and symmetric pairs_ ([arXiv:1202.3407](http://arxiv.org/abs/1202.3407))

Cohomological properties are discussed in

* [[Skip Garibaldi]], _Cohomological invariants: Exceptional groups and spin groups_ Memoirs of the AMS 937 (2009)

[[!redirects exceptional Lie group]]
[[!redirects exceptional Lie groups]]


exceptional naturalism

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#### Philosophy
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#### Exceptional structures
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#Contents#
* table of contents
{:toc}

## Idea

In the [[philosophy of science]] and particularly the _[[philosophy of physics]]_, the [[philosophy|philosophical]] sentiment which expresses the following perspective on the description of [[physics]] by [[mathematics]] might deserve to be called _exceptional naturalism_ or similar:

> Since nature (reality) *is exceptional in that it has existence*, it is plausible that it is the [[exceptional structures]] among all [[mathematical structures|mathematical]] [[structures]] -- such as the exceptional examples in the classification of [[simple Lie groups]], the [[exceptional Lie groups]] -- that play a role in the mathematical description of nature, hence in [[physics]] and specifically in [[phenomenology]].

In that this perspective provides _a priori_ grounds for the way the universe is, it may be contrasted with [[empiricism]].

## Sources

Sources where a sentiment of exceptional naturalism has been expressed include the following:

[[David Olive]] (in [Olive & Landsberg 1989](#OliveLand89)) writes:

> The results...add the stunning new question as to why nature has an apparent predilection for exceptional structures such as the E groups.

and:

> I sense that the trail of exceptional structures is growing even warmer.

[Baez 95](#Baez95), in a context of the role of the [[monster group]] in [[physics]], writes:

> one may argue that the theory of our universe must be incredibly special, since out of all the theories we can write down, just this one describes the universe that actually exists. All sorts of simpler universes apparently don't exist. So maybe the theory of the universe needs to use special, "exceptional" mathematics for some reason, even though it's complicated

[Ramond 01](#Ramond01) writes, in a context of [[heterotic string theory]] and [[M-theory]]:

> Nature relishes unique mathematical structures.

and:

> The [[exceptional Lie algebra|Exceptional Algebras]] are most unique and beautiful among [[Lie algebra|Lie Algebras]], and no one should be surprised if Nature uses them.

[Witten 02a](#Witten02a), in the context of [[M-theory on G2-manifolds]] says, with regard to the [[exceptional Lie group]] [[G2]]:

> ...arise in [[KK-compactification|compactifying]] from [[11-dimensional supergravity|eleven]] to four dimensions on a compact seven-manifold $X$ of [[G2-manifold|G2-holonomy]]. This seems like an interesting starting point for making a model of the real world, which is certainly exceptional

[Witten 02b](#Witten02b), in the context of [[grand unified theory]] and [[heterotic string theory]] says:

> Describing nature by a group taken from an infinite family does raise an obvious question – why this group and not another? In addition to the three infinite families, there are five [[exceptional Lie groups]], namely [[G2]], [[F4]], [[E6]], [[E7]], and [[E8]]. Since nature is so exceptional, why not describe it using an exceptional Lie group?

[Ramond 03](#Ramond03) writes in a context of [[grand unification]]:

> In the Lie garden, one also finds five rare flowers, the [[exceptional Lie algebras|exceptional algebras]]: [[G2]], [[F4]], [[E6]], [[E7]] and [[E8]], their rank indicated by the subscripts. In view of Nature's fascination with unique structures, they merit further study.

[Boya 03](#Boya03) writes in view of the various occurrences of the [[octonions]] in [[M-theory]]:

> If the current [[M-theory]] is a _unique theory_, one should expect it to make use of singular, non-generic mathematical structures. Now it is known that many of the special objects in mathematics are related to [[octonions]], and therefore it is not surprising that this putative _theory-of-everything_ should display geometric and algebraic structures derived from this unique non-associative [[normed division algebra|division algebra]].

[Toppan 03](#Toppan03) writes:

> There is a growing interest in the logical possibility that [[exceptional  mathematical structures]] ([[exceptional Lie algebra|exceptional Lie and superLie algebras]], the [[exceptional Jordan algebra]], etc.) could be linked to an ultimate "exceptional" formulation for a [[theory of everything|Theory Of Everything]] (TOE). The maximal [[division algebra]] of the [[octonions]] can be held as the mathematical responsible for the existence of the [[exceptional structures]] mentioned above.

[Moore 14](#Moore14) writes, in a survey of the state of [[mathematical physics]] applied to fundamental high energy physics ("Physical Mathematics"):

> it must be said that much of Physical Mathematics has a predilection
for special, sporadic, and exceptional structures. $[$...$]$ I cannot forecast what stormy weather our field is destined to endure, but I can confidently forecast abundant [[moonshine]] in the years ahead. (section 11 "Exceptional structures")

A related comment in the context of [[F-theory]] [[GUT]] [[phenomenology]] requiring a point with [[ADE classification|E-type]] symmetry is in ([Vafa 15, slide 11](#Vafa15)):

>The [[landscape of string theory vacua|landscape]] concept typically goes against things being exceptional. Here we seem
to have evidence to the contrary.

[Penrose 2015](#Penrose15), thinking of _[[twistor space]]_, voices the idea that nature realizes very special mathematical spaces (in the first few minutes of the video recording [here](https://www.youtube.com/watch?v=kmYfYOW0vSg)).

[Harvey & Moore 2020](#HarveyMoore20) write:

> Physicists who wonder about the ultimate structure of our universe can be divided into two classes. The first class believes that our universe is chosen at random from a huge ensemble of possible universes. The second class believes that our universe and the fundamental laws that govern it are based on some beautiful and exceptional mathematical structure. String theory provides evidence for both points of view... The authors of the present paper are philosophically inclined towards the point of view of the second class of physicists and are thus interested in studying special points in the landscape of string compactifications that involve beautiful and exceptional mathematical structures. Certainly among these special points the ones associated to moonshine are amongst the most beautiful.

\linebreak

## Related entries

* [[exceptional geometry]]

* [[exceptional generalized geometry]]

* [[exceptional field theory]]

* [[multiverse]]

* [[absolute idealism]]

## References

* {#OliveLands89} [[David Olive]] and Peter Landsberg, _Introduction to String Theory: Its Structure and its Uses [and Discussion]_, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 329, No. 1605, Physics and Mathematics of Strings (Nov. 17, 1989), pp. 319-328, ([JSTOR](https://www.jstor.org/stable/38267))

* {#Baez95} [[John Baez]], _[This Week's Finds in Mathematical Physics (Week 66)](http://math.ucr.edu/home/baez/week66.html)_, October 10, 1995

* {#Boya03} Luis Boya, _Octonions and M-theory_ ([arXiv:hep-th/0301037](https://arxiv.org/abs/hep-th/0301037)).

* {#Ramond01} [[Pierre Ramond]], _Algebraic Dreams_ ([arXiv:hep-th/0112261](http://arxiv.org/abs/hep-th/0112261))

* {#Witten02a} [[Edward Witten]], _Deconstruction, $G_2$ Holonomy, and Doublet-Triplet Splitting_, ([arXiv:hep-ph/0201018](http://arxiv.org/abs/hep-ph/0201018))

* {#Witten02b} [[Edward Witten]], _Quest For Unification_, Heinrich Hertz lecture at [SUSY 2002](http://www.desy.de/susy02/) at DESY, Hamburg ([arXiv:hep-ph/0207124](http://arxiv.org/abs/hep-ph/0207124))

* {#Ramond03} [[Pierre Ramond]], _Exceptional Groups and Physics_ ([hep-th/0301050](http://arxiv.org/abs/hep-th/0301050))

* {#Toppan03} [[Francesco Toppan]], _Exceptional Structures in Mathematics and Physics and the Role of the Octonions_, ([arXiv:hep-th/0312023](https://arxiv.org/abs/hep-th/0312023))

* {#Moore14} [[Gregory Moore]], _[[Physical Mathematics and the Future]]_, talk at [Strings 2014](http://physics.princeton.edu/strings2014/)

* {#Vafa15} [[Cumrun Vafa]], _Reflections on F-theory_, 2015 ([pdf](http://f-theory15.mpp.mpg.de/talks/Vafa.pdf))

* {#Penrose15} [[Roger Penrose]], _Twistor theory_, talk at _[[New Spaces for Mathematics and Physics]]_, IHP Paris 2015 ([video recording](https://www.youtube.com/watch?v=kmYfYOW0vSg))

* [[Jeffrey A. Harvey]], [[Gregory W. Moore]], *Moonshine, Superconformal Symmetry, and Quantum Error Correction*,  J. High Energ. Phys. **2020**, 146 (2020). ([arXiv:2003.13700](https://arxiv.org/abs/2003.13700))

  > (in relation to the [[Monster vertex operator algebra]] and [[quantum error correction]])

[[!redirects universal exceptionalism]]


exceptional spinors and division algebras -- table

**[exceptional](spin+group#ExceptionalIsomorphisms) [[spin representation|spinors]] and [[real numbers|real]] [[normed division algebras]]**

| [[Lorentzian spacetime|Lorentzian]]
 [[spacetime]]
 [[dimension]] | $\phantom{AA}$[[spin group]] | [[normed division algebra]]  | $\,\,$ [[brane scan]] entry |
|--|--|--|--|
| $3 = 2+1$ | $Spin(2,1) \simeq SL(2,\mathbb{R})$ | $\phantom{A}$ $\mathbb{R}$ the [[real numbers]] | [[super 1-brane in 3d]]  |
| $4 = 3+1$ | $Spin(3,1) \simeq SL(2, \mathbb{C})$| $\phantom{A}$ $\mathbb{C}$ the [[complex numbers]] | [[super 2-brane in 4d]] |
| $6 = 5+1$ | $Spin(5,1) \simeq$ [[SL(2,H)]] | $\phantom{A}$ $\mathbb{H}$ the [[quaternions]] | [[little string]] |
| $10 = 9+1$ | [[Spin(9,1)]] ${\simeq}$ "[[SL(2,O)]]" | $\phantom{A}$ $\mathbb{O}$ the [[octonions]] | [[heterotic string|heterotic]]/[[type II string]] |

[[!redirects exceptional spinors table]]


exceptional structure

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#Contents#
* table of contents
{:toc}

## Idea

Various kinds of ([[mathematical structures|mathematical]]) [[structures]] have classifications by which the possible examples/cases of these structures are elements either

1. of some infinite sequence of such structures, following some systematic construction rule,

1. or else of a (typically [[finite set|finite]]) [[set]] of examples which follow no such rule.

In the second case, these "unsystematic" examples of the given kind of [[structure]] are then often called _exceptional_ or _sporadic_.

Exceptional structures are often related to one another. For example, the [[exceptional Lie group]] [[G2]] is the [[symmetry group]] of the [[octonions]], and the other [[exceptional Lie groups]] are related to the [[octonions]] in the [[Freudenthal magic square]], while the [[Albert algebra]] is the [[exceptional Jordan algebra]] of 3-by-3 hermitian matrices over the octonions. The [[Leech lattice]] is related to the [[Monster simple group]], and can also be connected with the octonions ([Wilson 09](#Wilson09)). Moreover, all these structures tend to appear as aspects of [[M-theory]].

[[!include exceptional structures -- contents]]

## References

See also:

* Wikipedia, _[Exceptional object](https://en.wikipedia.org/wiki/Exceptional_object#Monstrous_moonshine)_

In relation to the [[octonions]] and the [[Leech lattice]]:

* {#Wilson09} [[Robert A. Wilson]], _Octonions and the Leech lattice_, Journal of Algebra **322** 6 (2009) 2186-2190, [[pdf](http://www.maths.qmul.ac.uk/~raw/pubs_files/octoLeech1.pdf)), ([slides](http://www.maths.qmul.ac.uk/~raw/talks_files/Cambridge09.pdf)]

On [[exceptional Lie algebras]] and [[sporadic finite simple groups]] via [[del Pezzo surfaces]]:

* [[Yang-Hui He]], [[John McKay]], _Sporadic and Exceptional_ [[arXiv:1505.06742](https://arxiv.org/abs/1505.06742)]

On generalizations of exceptional structures, including [[E8]], [[octonions]] and the [[exceptional Jordan algebra]]:

* [[Piero Truini]], [[Michael Rios]], [[Alessio Marrani]], *The Magic Star of Exceptional Periodicity*, J. Phys.: Conf. Ser. **1194** (2019) 012106 [[arXiv:1711.07881](https://arxiv.org/abs/1711.07881), [doi:10.1088/1742-6596/1194/1/012106](https://iopscience.iop.org/article/10.1088/1742-6596/1194/1/012106)]

In relation to [[bosonic M-theory]], [[12d supergravity]], [[D=14 supersymmetry]] etc.:

* [[Michael Rios]], [[Alessio Marrani]], [[David Chester]], *Exceptional Super Yang-Mills in $D=27+3$ and Worldvolume M-Theory*, Phys. Lett. B **808** (2020) 135674 [[arXiv:1906.10709](https://arxiv.org/abs/1906.10709), [doi:10.1016/j.physletb.2020.135674](https://doi.org/10.1016/j.physletb.2020.135674)]

[[!redirects exceptional structures]]

[[!redirects exceptional mathematical structure]]
[[!redirects exceptional mathematical structures]]

[[!redirects sporadic structure]]
[[!redirects sporadic structures]]

[[!redirects sporadic mathematical structure]]
[[!redirects sporadic mathematical structures]]


exceptional structures -- contents

**[[exceptional structures]]**, [[exceptional isomorphisms]]

## Examples

* [[exceptional finite groups]]

  * [[monster group]]

  * [[Mathieu group]],

  * [[Conway group]]

* exceptional [[finite rotation groups]]:

  * [[tetrahedral group]]

  * [[octahedral group]]

  * [[icosahedral group]]

* [[exceptional Lie groups]]

  * [[G2]]

  * [[F4]]

  * [[E6]], [[E7]], [[E8]]

  and [[Kac-Moody groups]]:

  * [[E9]], [[E10]], [[E11]], ...

* [[Dwyer-Wilkerson H-space]]

* [[exceptional Lie algebras]]

* [[exceptional Jordan algebra]]

  * [[Albert algebra]]

* exceptional [[Jordan superalgebra]], $K_10$

* [[E8 lattice]], [[Leech lattice]]

* [[Cayley plane]]

## Interrelations

* [[supersymmetry and division algebras]]

* [[Freudenthal magic square]]

* [[moonshine]]

  * [[Mathieu moonshine]]

  * [[umbral moonshine]]

  * [[O'Nan moonshine]]

## Applications

* [[exceptional geometry]], [[exceptional generalized geometry]],

* [[exceptional field theory]]

## Philosophy

* [[universal exceptionalism]]


exceptional tangent bundle

#Contents#
* table of contents
{:toc}

## Idea

A generalization of the notion of _[[tangent bundle]]_ in _[[exceptional generalized geometry]]_.

For [[11-dimensional supergravity]] the bosonic [[body]] of  its [[typical fiber]] is

$$
  \mathbb{R}^{d-1,1} \oplus \wedge^2 (\mathbb{R}^{d-1,1})^\ast
 \oplus \wedge^5 (\mathbb{R}^{d-1,1})^\ast
$$

where the second summand corresponds to [[M2-brane]] charges and the third to [[M5-brane charges]].

This happens to coincide with the bosonic body of the [[M-theory super Lie algebra]], see there for more.

## Related concepts

* [[generalized tangent bundle]]

(...)

[[!redirects exceptional tangent bundles]]

[[!redirects exceptional generalized tangent bundle]]
[[!redirects exceptional generalized tangent bundle]]