Euclidean supermanifold

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[[!include supergeometry - contents]]
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#Contents#

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{:toc}

#Idea#

A _Euclidean supermanifold_ is a [[supermanifold]] that can be thought of as being equipped with a flat [[Riemannian metric]].

Alternatively, it is a [[supermanifold]] for which the transition functions of an [[manifold|atlas]] are restricted to be elements of the [[super Euclidean group]].

#Definition#

A **Euclidean supermanifold** of dimension $(p|q)$is a [[supermanifold]] that is quipped with an **$(X,G)$-structure** , where $X = \mathbb{R}^{p|q}$ and where $G$ is the [[super Euclidean group]] on $\mathbb{R}^{p|q}$.

Here an $(X,G)$-structure is defined as follows, essentially being a version of the discussion of _pseudogroups_ at [[manifold]].

**Definition** ([[Stephan Stolz|Stolz]], [[Peter Teichner|Teichner]]) A $(X,G)$-structure on a $(d|\delta)$-dimensional [[supermanifold]] $Y$ consists of

* a maximal [[manifold|atlas]] consisting of charts

  $$
    Y \sup_{opn} U_i
    \stackrel{\phi_i}{\to_\simeq}
    V_i \subset_{open} X
  $$

  (where on the left $Y_\red\supset_{open} (U_i)_{red}$) with $O_Y|_{(U_i)_{red} = O_{U_i}}$

* such that the transition function

  $$
    X \supset \phi_i(U_i \cap U_j)
    \stackrel{\phi_j \circ \phi_i^{-1}}{\to}
    \phi_j(U_i \cap U_j)
    \subset X
  $$

  is the restriction of a map

  $$
    X \simeq
    X \times pt
    \stackrel{id \times g}{\to}
    X \times G
    \stackrel{action}{\to}
    X
  $$

## family version ##

**definition** A family of $(X,G)$-(complex-, super-)manifolds is a map

$$
  \array{
    Y \\ \downarrow^p \\ S
  }
$$

together with a maximal atlas of charts

$$
  \array{
     Y \supset_{open} U_i &&\stackrel{\phi_i}{\to_\simeq}&&
      V_i \subset_{open} S \times X
     \\
     & \searrow && \swarrow
     \\
     && S
  }
$$

such that the transition maps

$$
  \array{
     S \times X \supset \phi_i(U_i \times U_j)
     &&\to&&
     \phi_j(U_i \cap U_j)
     \subset S \times X
     \\
     & \searrow && \swarrow
     \\
     && S
  }
$$

are the restriction of a map of the following form

$$
  S\times X \stackrel{Id \times g \times id}{\to}
  S \times G \times X
  \stackrel{Id \times action}{\to}
$$

for some

$$
  S \stackrel{g}{\to} G
$$

**example** a family $Y \to S$ of $(\mathbb{R}^d, Eucl(\mathbb{R}^d))$-manifolds, for $S$ an ordinary [[manifold]] is a [[submersion]] with flat [[Riemannian metric]] on the [[fiber]]s.

# ordinary Euclidean manifolds as Euclidean supermanifolds #

Specifically in 2-dimensions, an ordinary Spin-Eulidean manifold is one with $(\mathbb{R}^2, \mathbb{R}^2 \rtimes Spin(2))$-structure.

We want to regard this as a Euclidean supermanifold with
$(\mathbb{R}^{2|1}_{cs}, \mathbb{R}^{2|1}_{cs} \rtimes Spin(2))$-structure.

In general for two structures $(X,G)$ and $(X',G')$ we can transfer structures when we have a group homomorphisms

$G \to G'$ and with respect to that a

$G$-equivariant map $X' \to X$.

Then send every $(X,G)$-chart to the corresponding $(X',G')$-chart which as a subset of $X'$ is the inverse image of $X' \to X$.

This yields a functor

$$
  (X,G)-manifolds
  \to
  (X',G')-manifolds
  \,.
$$


Euclidean topology

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### Context
#### Topology
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[[!include topology - contents]]
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=--
=--

# Contents
* table of contents
{: toc}

## Definition

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

For $n \in \mathbb{N}$ a [[natural number]], write $\mathbb{R}^n$ for the [[Cartesian space]] of [[dimension]] $n$. The **Euclidean topology** is the [[topology]] on $\mathbb{R}^n$ characterized by the following equivalent statements

* it is the [[metric space|metric topology]] induced from the canonical structure of a [[metric space]] on $\mathbb{R}^n$ with [[distance]] function given by $d(x,y) = \sqrt{\sum_{i = 1}^n (x_i-y_i)^2}$;

* an [[open subset]] is precisely a subset such that contains an [[open ball]] around each of its points;

* it is the [[product topology]] induced from the standard topology on the [[real line]].
=--

## Properties

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

Two [[Cartesian space]]s $\mathbb{R}^k$ and $\mathbb{R}^l$ (with the Euclidean topology) are [[homeomorphic]] precisely if $k = l$.
=--

A proof of this statement was an early success of [[algebraic topology]].

## Related concepts

* [[Euclidean-topological infinity-groupoid]]

[[!redirects Euclidean topology]]
[[!redirects Euclidean topologies]]


Eudoxus real number

[[!redirects Eudoxus real numbers]]

# Contents #
* table of contents
{:toc}

## Idea

*Eudoxus real number* refers to a construction of the [[ordered field]] of [[real numbers]] discovered by [[Stephen Schanuel]]. The idea is that every real number $r$ can be represented by a [[relation]] on the [[integers]] that is "almost linear", e.g., the function that takes an integer $n$ to the [[floor]] of $r n$. Schanuel named this after [[Eudoxus]], who developed a theory of magnitude and proportion to account for the relations between the discrete and the continuous[^1].

[^1]: Quoting Schanuel: "[Eudoxus] noted that to measure
the ratio of a line segment to another with any desired accuracy, it isn't
necessary to cut either of them up -- instead multiply the 'numerator'
segment by a large integer -- thus pointing the way to a direct construction
of the reals as ring from the integers as mere additive group."

## Definition

A [[function]] $f: \mathbb{Z} \to \mathbb{Z}$ is *almost linear* if the quantity $f(m+n) - f(m) - f(n)$ taken over all $m, n \in \mathbb{Z}$ is [[bounded set|bounded]]. Almost linear functions are closed under the pointwise [[abelian group]] operations inherited from $\mathbb{Z}$ (i.e., $f + g$, $-f$), and are also closed under composition of functions.

Define an equivalence relation $\sim$ on the set of almost linear functions $AL(\mathbb{Z})$ by $f \sim g$ if $f - g$ is bounded. This equivalence relation respects the abelian group structure of $AL(\mathbb{Z})$, and if $f \sim f'$, then for any $g \in AL(\mathbb{Z})$ we have $g \circ f \sim g \circ f'$ and $f \circ g \sim f' \circ g$. We write $[f]$ for the equivalence class of $f$.

+-- {: .num_defn}
###### Definition
A **Eudoxus real number** is an equivalence class of $AL(\mathbb{Z})$ under $\sim$.
=--

The set of Eudoxus reals $\mathbb{E} \coloneqq AL(\mathbb{Z})/\sim$ becomes a [[commutative ring]] whose addition and multiplication are descended from $+$ and $\circ$ on $AL(\mathbb{Z})$. Already one distributive law $(f + f') \circ g = f \circ g + f' \circ g$ holds in $AL(\mathbb{Z})$, so as soon as one proves multiplication on $\mathbb{E}$ is commutative, one gets distributivity on the other side.

A Eudoxus real $[f]$ is deemed *positive* if it is represented by an $f$ such that for any $C$ there are infinitely many $m \in \mathbb{N}$ such that $f(m) \gt C$. It is easy to see this condition would then be satisfied by any representative function. As usual, define $[f] \lt [g]$ to mean $[g-f]$ is positive.

+-- {: .num_theorem}
###### Theorem
The commutative ring $\mathbb{E}$ equipped with $\lt$ is a [[real number|complete ordered field]].
=--

A careful proof and exposition has been given by [Arthan](#Arthan). The resulting construction is equivalent to the [[Cauchy reals]]; indeed for any almost linear function $f$, the sequence $f(n)/n$ is a [[Cauchy sequence]] of rationals that converges to the corresponding real number.

### As an initial algebra

An endofunction $f:\mathbb{Z}^\mathbb{Z}$ over the [[integers]] is *bounded* if

$$\exists C \in \mathbb{Z}. \forall m \in \mathbb{Z}. \vert f(m) \vert \lt C$$

and $f$ is *almost linear* if

$$\exists C \in \mathbb{Z}. \forall m \in \mathbb{Z}. \forall n \in \mathbb{Z}. \vert f(m + n) - f(m) - f(n) \vert \lt C$$

Let the set of almost linear endofunctions over the integers be defined as

$$AL(\mathbb{Z}) \coloneqq \{f\in \mathbb{Z}^\mathbb{Z} \vert \exists C \in \mathbb{Z}. \forall m \in \mathbb{Z}. \forall n \in \mathbb{Z}. \vert f(m + n) - f(m) - f(n) \vert \lt C \}$$

Let us define a *Eudoxus algebra* as a [[set]] $A$ with a function $\iota \in A^{AL(\mathbb{Z})}$ such that

$$\forall f \in AL(\mathbb{Z}). \forall g \in AL(\mathbb{Z}). (\exists C \in \mathbb{Z}. \forall m \in \mathbb{Z}. \vert f(m) - g(m) \vert \lt C) \implies (\iota(f) = \iota(g))$$

An *Eudoxus algebra homomorphism* between two Eudoxus algebras $A$ and $B$ is a function $f \in B^A$ such that

$$\forall a \in AL(\mathbb{Z}). f(\iota_A(a)) = \iota_B(a)$$

The *category of Eudoxus algebras* is the category $EudoxusAlg$ whose objects $Ob(EudoxusAlg)$ are Eudoxus algebras and whose morphisms $Mor(EudoxusAlg)$ are Eudoxus algebra homomorphisms. The set of **Eudoxus real numbers**, denoted $\mathbb{R}$, is defined as the initial object in the category of Eudoxus algebras.

### As a quotient inductive type

Let the proposition that an endofunction $f:\mathbb{Z} \to \mathbb{Z}$ over the [[integers]] is *bounded* be defined as follows:

$$isBounded(f) \coloneqq \left\Vert \sum_{C:\mathbb{Z}} \prod_{m:\mathbb{Z}} \vert f(m) \vert \lt C \ \right\Vert$$

and let the proposition that $f$ is *almost linear* be defined as follows:

$$isAlmostLinear(f) \coloneqq \left\Vert \sum_{C:\mathbb{Z}} \prod_{m:\mathbb{Z}} \prod_{n:\mathbb{Z}} \vert f(m + n) - f(m) - f(n) \vert \lt C \ \right\Vert$$

Let the type of almost linear endofunctions over the integers be defined as

$$AL(\mathbb{Z}) \coloneqq \sum_{f:\mathbb{Z} \to \mathbb{Z}} isAlmostLinear(f)$$

The type of **Eudoxus real numbers**, denoted as $\mathbb{R}$, is a [[higher inductive type]] generated by:

* a function $\iota: AL(\mathbb{Z}) \to \mathbb{R}$

* a dependent product of functions from propositions to identities:

$$ \eta : \prod_{f:AL(\mathbb{Z})} \prod_{g:AL(\mathbb{Z})} isBounded(f - g) \to (\iota(f) = \iota(g))$$

 * A set-truncator
$$\tau_0: \prod_{a:\mathbb{R}} \prod_{b:\mathbb{R}} isProp(a=b)$$

## Remarks

In an exchange at the categories list, Steve Schanuel says he got the idea from an old observation by (John) Tate on bilinear maps on $\mathbb{Z}$. The "attribution" to Eudoxus is best understood in the context of the footnote below. Particularly, it should not be thought that Eudoxus was 'constructing' the reals (whether of Dedekind type or Cauchy type) or constructing anything at all necessarily (rationals included), despite modern glosses on the spirit of Eudoxus's theory of magnitude that suggest more of a kinship to Dedekind cuts, and despite the fact that 'Eudoxus reals' as described here are equivalent to Cauchy reals, not Dedekind reals. Mainly, Schanuel is observing that by following a path suggested by Eudoxus, one can get to the reals directly, bypassing the need for the rationals as an intermediate construction.

There was a discussion on this at the [categories list](#thread).

## See also

* [[real number]]

* [[modulated Cauchy real number]]

* [[HoTT book real number]]

## References

Schanuel's discovery was unpublished by him, but was first disseminated by Ross Street:

* [[Ross Street]], _An efficient construction of the real numbers_, Gazette Australian Math. Soc. 12 (57–58), 1985. ([web](https://andrescaicedo.files.wordpress.com/2014/09/ross-street-an-efficient-construction-of-real-numbers.pdf))

It was subsequently noted several times in the literature that Street's sketch in that note on how to establish completeness isn't quite right. Street of course acknowledged this and gave an updated account here, mentioning for example corrected versions (e.g., by Arthan; see below):

* Ross Street, _Update on the efficient reals_, September 2003. ([web](http://maths.mq.edu.au/~street/reals.pdf))

* James Douglas, Rony Kirollos, Ben Odgers, Ross Street and Nguyen Hanh Vo, _The Efficient Real Numbers_, February 2004 ([web](http://science.mq.edu.au/~street/EffR.pdf))

Scattered remarks by Street, [[Michael Barr]], and Schanuel may be found at the categories list,

* Schanuel et al., Thread on reals à la Tate, Eudoxus, etc., categories list (year 2000).
 {#thread}

The quote attributed to Schanuel (see the footnote) was drawn from that thread.

A correct and fairly comprehensive account of the construction is by Arthan, who includes some useful historical material:

* Rob Arthan, _The Eudoxus Real Numbers_, [http://arxiv.org/abs/math/0405454](http://arxiv.org/abs/math/0405454) (May 24, 2004).
 {#Arthan}

A blog post on the Eudoxus real numbers:

* [[Emily Riehl]], *Constructing the real numbers as nearly multiplicative sequences*, [[n-Category Café]] ([web](https://golem.ph.utexas.edu/category/2023/09/constructing_the_real_numbers.html))


Eugene Cremmer

* [webpage](http://www.phys.ens.fr/~cremmer/)

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

## Selected writings

Introducing [[D=11 supergravity]]:

* {#CremmerJuliaScherk78} [[Eugene Cremmer]], [[Bernard Julia]], [[Joël Scherk]], _Supergravity in theory in 11 dimensions_, Phys. Lett. B **76** (1978) 409 [doi:10.1016/0370-2693(78)90894-8]

Formulation of [[D=11 supergravity]] in [[superspace]] (via [[supergeometry]]):

* {#CremmerFerrara80} [[Eugene Cremmer]], [[Sergio Ferrara]], *Formulation of Eleven-Dimensional Supergravity in Superspace*, Phys.Lett. B **91** (1980) 61 [doi:10.1016/0370-2693(80)90662-0]

## Related entries

* [[supergravity]], [[11-dimensional supergravity]]

* [[U-duality]], [[E7]]

category: people


Eugene Lerman

* [webpage](http://www.math.uiuc.edu/~lerman/)

## Selected writings

On [[orbifolds]] as [[stacks]] ([[topological stacks]]/[[differentiable stacks]]):

* {#Lerman08} [[Eugene Lerman]], *Orbifolds as stacks?*, Enseign. Math. **56** 3-4 (2010) 315-363 [[arXiv:0806.4160](http://arxiv.org/abs/0806.4160), [doi:10.4171/LEM/56-3-4](http://dx.doi.org/10.4171/LEM/56-3-4)]

category: people


Eugene Mele

* [webpage](https://www.physics.upenn.edu/people/standing-faculty/eugene-mele)

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

## Selected writings

Observation that [[graphene]] appears as a [[topological insulator]] when its [[spin-orbit coupling]] is resolved, due to a [[quantum spin Hall effect]]:

* {#KaneMele05a} [[Charles Kane]], [[Eugene Mele]], _Quantum Spin Hall Effect in Graphene_, Phys. Rev. Lett. 95, 226801 (2005) ([arXiv:cond-mat/0411737](https://arxiv.org/abs/cond-mat/0411737), [doi:10.1103/PhysRevLett.95.226801](https://doi.org/10.1103/PhysRevLett.95.226801))

Characterization of the resulting [[topological phase of matter]] via introduction fo the [[Kane-Mele invariant]]:

* {#KaneMele05b} [[Charles Kane]], [[Eugene Mele]],  *$Z_2$ Topological Order and the Quantum Spin Hall Effect*, Phys. Rev. Lett. **95** (2005) 146802 ([doi:10.1103/PhysRevLett.95.146802](https://doi.org/10.1103/PhysRevLett.95.146802))

On [[topological insulators]]:

* [[Liang Fu]], [[Charles Kane]], [[Eugene Mele]], *Topological Insulators in Three Dimensions*, Phys. Rev. Lett. **98** (2007) 106803 $[$[doi:10.1103/PhysRevLett.98.106803](https://doi.org/10.1103/PhysRevLett.98.106803)$]$

On [[topological semi-metals]]:

* N. P. Armitage, [[Eugene Mele]], [[Ashvin Vishwanath]], *Weyl and Dirac semimetals in three-dimensional solids*, Rev. Mod. Phys. **90** 015001 (2018) $[$[doi:10.1103/RevModPhys.90.015001](https://doi.org/10.1103/RevModPhys.90.015001)$]$

category: people

[[!redirects Gene Mele]]


Eugene Rabinovich

* [personal page](https://math.nd.edu/people/faculty/eugene-rabinovich/)

## Selected writings

On the [[CS-WZW correspondence]] via [[factorization algebras]]:

* {#GwilliamRabinovichWilliams2022} [[Owen Gwilliam]], [[Eugene Rabinovich]], [[Brian R. Williams]], *Factorization algebras and abelian CS/WZW-type correspondences*, Pure and Applied Mathematics Quarterly **18** 4 (2022) 1485–1553  [[arXiv:2001.07888](https://arxiv.org/abs/2001.07888), [doi:10.4310/PAMQ.2022.v18.n4.a7](https://dx.doi.org/10.4310/PAMQ.2022.v18.n4.a7)]

On [[Poisson holography]] via [[factorization algebras]]:

* [[Eugene Rabinovich]], *A Classical Bulk-Boundary Correspondence* [[arXiv:2202.12332](https://arxiv.org/abs/2202.12332)]

category: people


Eugene Speer

* [webpage](http://www.math.rutgers.edu/~speer/)

* [[zeta function regularization]] ("anatic regularization")

category: people


Eugene Wigner

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

## Selected writings

On [[group theory]] and its application in [[quantum physics]]:

* {#Wigner31} [[Eugene P. Wigner]]: *Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren*, Springer (1931) [[doi:10.1007/978-3-663-02555-9](https://doi.org/10.1007/978-3-663-02555-9), [pdf](https://digbib.ubka.uni-karlsruhe.de/volltexte/wasbleibt/33355459/33355459.pdf)]

* [[Eugene P. Wigner]]: *Group theory: And its application to the quantum mechanics of atomic spectra*, 5, Academic
Press (1959) [[doi:978-0-12-750550-3](https://www.elsevier.com/books/group-theory/wigner/978-0-12-750550-3)]

See also:

* _[[The Unreasonable Effectiveness of Mathematics in the Natural Sciences]]_

Co-introducing [[Jordan algebras]] as [[algebras of quantum observables]]:

* {#JordanvNeumannWigner34} [[Pascual Jordan]], [[John von Neumann]], [[Eugene Wigner]], *On an algebraic generalization of the quantum mechanical formalism*, Ann. Math. **35** (1934) 29-64 [[jstor:1968117](https://www.jstor.org/stable/1968117), [doi:10.1007/978-3-662-02781-3_21](https://doi.org/10.1007/978-3-662-02781-3_21)]

## Related entries

* [[time-reversal symmetry]]

* [[Wigner's theorem]]

* [[Wigner classification]]

* [[group contraction]]

* [[interpretation of quantum mechanics]]

category: people

[[!redirects Eugene P. Wigner]]


Eugenia Cheng

* [website](http:/eugeniacheng.com/)

## Selected writings

On [[iterated distributive laws]]:

* [[Eugenia Cheng]], *Iterated distributive laws*, Mathematical Proceedings of the Cambridge Philosophical Society **150** 3, (2011) 459-487  [[arxiv:0710.1120](https://arxiv.org/abs/0710.1120), [doi:10.1017/S0305004110000599](https://doi.org/10.1017/S0305004110000599)]

category: people


Eugenio Calabi

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

## Selected writings

On (in small part) the [[Calabi-Penrose fibration]]:

* {#Calabi67} [[Eugenio Calabi]], _Minimal immersions of surfaces in euclidean spheres_, J. Differential Geometry (1967), 111-125 ([euclid:jdg/1214427884](https://projecteuclid.org/euclid.jdg/1214427884))

* {#Calabi68} [[Eugenio Calabi]], _Quelques applications de l’analyse complexe aux surfaces d’aire minima_, Topics in Complex Manifolds (Ed. H. Rossi), Les Presses de l’Universit ́e de Montr ́eal (1968), 59-81 ([naid:10006413960](https://ci.nii.ac.jp/naid/10006413960))

## Related $n$Lab entries

* [[Calabi-Yau manifold]]

category: people


Eugenio Landi

* [personal page](https://eugeniolandi.github.io/)

## Selected writings

Discussion of [[geometric string structure|geometric]] string bordism in degreee 3 as a means to speak (via the [[Pontryagin-Thom theorem]]) about the [[third stable homotopy group of spheres]]:

* [[Domenico Fiorenza]], [[Eugenio Landi]], *Integrals detecting degree 3 string cobordism classes* [[arXiv:2209.12933](https://arxiv.org/abs/2209.12933)]

* [[Domenico Fiorenza]], *String bordism invariants in dimension 3 from $U(1)$-valued TQFTs*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#FiorenzaMar2023)]

category: people


Eugenio Moggi

* [website](http://www.disi.unige.it/person/MoggiE/)

## Selected writings

Introducing ([[strong monad|strong]]) [[monads in computer science]] (in the form of "[[extension systems]]", "[[Kleisli triples]]" as a form of [[computation]] with side-effects):

* {#Moggi89} [[Eugenio Moggi]], *Computational lambda-calculus and monads*, in: *Proceedings of the Fourth Annual Symposium on Logic in Computer Science* (1989) 14-23 [[doi:10.1109/LICS.1989.39155](https://doi.org/10.1109/LICS.1989.39155)]

* {#Moggi89b} [[Eugenio Moggi]], *An abstract View of Programming Languages*, LFCS report ECS-LFCS-90-113 (1989) [[web](http://www.lfcs.inf.ed.ac.uk/reports/90/ECS-LFCS-90-113), [[Moggi-AbstractView.pdf:file]]]

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

and on their [[monad transformers]]:

* [[Mauro Jaskelioff]], [[Eugenio Moggi]], *Monad Transformers as Monoid Transformers*, Theoretical Computer Science **411** 51–52 (2010) 4441-4466 [[doi:10.1016/j.tcs.2010.09.011](https://doi.org/10.1016/j.tcs.2010.09.011), [pdf](http://www.disi.unige.it/person/MoggiE/ftp/tcs10.pdf)]

Review:

* [[Nick Benton]], [[John Hughes]], [[Eugenio Moggi]], *Monads and Effects*, in: *Applied Semantics*, Lecture Notes in Computer Science **2395**, Springer (2002) 42-122 [[doi:10.1007/3-540-45699-6_2](https://doi.org/10.1007/3-540-45699-6_2)]

On [[categorical model of dependent types]]:

* [[Eugenio Moggi]], *A category-theoretic account of program modules*, Math. Structures Comput. Sci. **1** 1 (1991) 103-139 [[MR1108806 ](http://www.ams.org/mathscinet-getitem?mr=1108806), doi:10.1017/S0960129500000074]

category: people


Euler angle

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###Context###
#### Geometry
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[[!include higher geometry - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The 3 classical _Euler angles_ furnish a global parametrization and a smooth [[coordinate system]] on a big dense cell of the [[Lie group]] [[special orthogonal group|SO(3)]] of [[rotation]]s in 3 dimensions, or on its double cover, $SU(2)$ group. Its essential feature is that it supplies a multiplicative decomposition of arbitrary rotation into 3 elements belonging to distinguished 1-parametric subgroups of rotations $SO(2) = U(1)$ around 2 coordinate axes, where the classical choices are z-x-z and the angular parameters for the elements in distinguished $SO(2)$ are the Euler angles.

Generalized constructions apply to some other [[Lie groups]].

## Introducing Euler angles in intrinsic interpretation

We consider rotations of a rigid body in $\mathbf{R}^3$ with origin $O$. For the definitions we do not need to specify the unit vectors, but only the orientation on the axes.
We want to rotate a right handed triple of mutually orthogonal (semi)axes $OxOyOz$ into another, $Ox'Oy'Oz'$, by using rotations around coordinate axes. In this interpretation, the axes are around oriented axes which move together with the body.
We first move around $Ox$, then around $Oz$ of the rotated body, and finally around $Ox$ of the rotated body.

The basic notion is the line of nodes $n$ which is the intersection of the plane $\langle x,y\rangle$
(generated by vectors along $x$ and along $y$) and plane $\langle x',y'\rangle$.
Choose an orientation so that $OxOnOz$ be a right handed.
Regarding that both $x$ and $n$ are orthogonal to $z$ there is an angle $\phi$, $|\phi|\leq\pi$, such that if we rotate around $Oz$ by angle $\phi$, $Ox$ becomes
$On$, moreover this angle is unique except if $On = -Ox$, that is $|\phi| = \pi$.
Now rotate around $Ox$ of the rotated body, which is the same as $On$ of the original coordinate system, so that $Oz$ becomes $Oz'$. This is possible because $On$ is orthogonal to both of them by construction. The angle of rotation is $\theta$, which is the same as the original angle between $Oz$ and $Oz'$. Finally, we rotate around $Oz$ of the new system (which is $Oz'$ of the original problem) by the angle $\psi$ between from $Ox$ (which is the line of nodes, $On$ of the fixed system) to $Ox'$, again observing the orientation. Regarding that $OxOz$ went into $Ox'Oz'$, the third semiaxis is automatically fixed.

## Construction for $SU(2)$

(Following Vilenkin) Every $SU(2)$ matrix can be written in a form

$$
      u = \left(\array{ a & b \\ - \overline{b} & \overline{a} }\right)
$$
where $a,b\in\mathbf{C}$, $det(u) = 1$ and $|a|^2 + |b|^2 = 1$. The simplest parametrization is to take $|i|, arg(a)$ and $arg(b)$ as independent parameters.
If $a b \neq 0$ then there are unique parameters $\phi,\theta,\psi$ called Euler angles such that
$$
|a| = cos\frac{\theta}{2} ,
\,\,\,\,\,\,
arg(a) = \frac{\phi+\psi}{2},
\,\,\,\,\,\,
arg(b) = \frac{\phi-\psi+\pi}{2}
$$
and we can take $O\leq \phi\lt 2\pi$, $0\lt\theta\lt\pi$, $-2\pi\leq\psi\lt 2\pi$.
If $a=0$ or $b = 0$ then we can find triples $(\phi,\theta,\psi)$ satisfying the above system but the choice is not necessarily unique. In any case, the matrix $u$ is now

$$
u(\phi,\theta,\psi):= u = \left(\array{
cos\frac{\theta}{2} e^{i\frac{\phi+\psi}{2}}& i sin\frac\theta{2}e^{i\frac{\phi-\psi}{2}}
\\ i sin\frac\theta{2}e^{i\frac{\psi-\phi}{2}} & cos\frac\theta{2} e^{-i\frac{\phi+\psi}{2}}
}\right)
$$

There is a decomposition

$$
u(\phi,\theta,\psi) = \left(\array{
e^{\frac{i\phi}{2}}&0\\0&e^{-\frac{i\phi}{2}}
}\right)
\left(\array{
cos\frac\theta{2}&i sin\frac\theta{2}\\
i sin\frac\theta{2}&cos\frac\theta{2}
}\right)
\left(\array{
e^{\frac{i\psi}{2}}&0\\0&e^{-\frac{i\psi}{2}}
}\right)
$$

The standard homomorphism $SU(2)\to SO(3)$ sends this Euler angles to classical Euler angles for $SO(3)$; the map is surjective if we restrict to values
$\theta\in[0,2\pi]$ as the classical Euler angle bounds suggest.

## Related concepts

* [[Pauli matrix]]

## References

* Wikipedia, _[Euler angles](https://en.wikipedia.org/wiki/Euler_angles)_

* WolframMath, _[Euler angles](http://mathworld.wolfram.com/EulerAngles.html)_

* [[Lev Landau]], _Mechanics_

* _Understanding Euler angles_ (robotics, engineerings and aviation conventions) [web](http://www.chrobotics.com/library/understanding-euler-angles)

* N. Ja. Vilenkin, _Special functions and representation theory of groups $SO(n)$_

* L. C. Biedenharn, J. D. Louck, _The angular momentum in quantum physics_, Enc. Math Appl. __8__, Addison-Wesley 1981

* D. Brezov, C. Mladenova, I. Mladenov, _Vector decompositions of rotations_, J.Geom. Symmetry Phys. __28__ (2012) 67-103; _Vector parameters in classical hyperbolic geometry_, Proc. 15th Int. Conf. on Geometry, Integrability and Quantization, Ivaïlo M. Mladenov, A. Ludu, A. Yoshioka, eds. (Sofia: Avangard Prima, 2014) 79-105 [euclid](https://projecteuclid.org/euclid.pgiq/1436815703)

* S. Bertini, S. L. Cacciatori, B. L. Cerchiai, _On the Euler angles for $SU(N)$_, J. Math. Phys. __47__:4, id.043510 (2006) [doi](https://doi.org/10.1063/1.2190898) [arxiv:math-ph/0510075](https://arxiv.org/math/abs/0510075)

* S. L. Cacciatori, B. L. Cerchiai, _Euler angles for $G_2$_, J. Math. Phys. 46, 083512 (2005) [doi](https://doi.org/10.1063/1.1993549)

*  S. L. Cacciatori, F. Dalla Piazza, A. Scotti, _Compact Lie groups: Euler constructions and generalized Dyson conjecture_, Trans. Amer. Math. Soc. 369 (2017), 4709-4724 [doi](https://doi.org/10.1090/tran/6795)

[[!redirects Euler angles]]


Euler beta function

> This entry is about the concept in [[arithmetic]]. For the [[beta function]] related to [[renormalization group flow]] see there.

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

#Contents#
* table of contents
{:toc}

## Idea

The Euler beta function has been defined by [[Euler]] around 1730 by the so called Euler beta integral

$$
B(x,y) = \int_0^1 t^{x-1} (1-t)^{y-1} d t = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)},\,\,\,\,\,\,Re x\gt 0, \,\,\,\,Re y \gt 0,
$$

which can be expressed in terms of the gamma function as stated.

A multidimensional generalization is the [[Selberg integral]].

## Related concepts

* [[Gamma function]]

* [[factorial]]

## References

* G. E. Andrews, R. Askey, R. Roy, _Special functions_, Enc. of Math. and its Appl. __71__, Cambridge Univ. Press 1999
[[!redirects Euler's beta function]]
[[!redirects Euler beta integral]]


Euler bilinear form

Expression __Euler form__ may mean either a [[differential form]] ("Chern-Euler form") representing the [[Euler characteristic class]], but also a $\mathbf{Z}$-bilinear form on a Grothendieck group of a $k$-linear abelian or triangulated category, related to an alternative sum of dimensions of Ext-groups. This entry is about the latter.
The terminology is allegedly due [[Claus Michael Ringel]].

## Definition

If $C$ a $k$-linear [[abelian category]] such that $dim(Ext^i(M,N))\lt \infty$ for all $i$ and all objects $M,N$ in $C$, then the Euler form is a $\mathbf{Z}$-bilinear form $\langle -,-\rangle : K_0(C)\times K_0(C)\to\mathbf{Z}$ defined by
$$
\langle [M],[N]\rangle = \sum_{i\geq 0} (-1)^i dim_k(Ext^i(M,N)).
$$
Similarly, for suitable $k$-linear [[triangulated category]] $D$ one sets
$$
\langle [M],[N]\rangle = \sum_{i\geq 0} (-1)^i dim_k(Hom_D(M,\Sigma^i N))
$$

## Literature

* C. Geiß, _Derived tame algebras and Euler-forms_, Math Z. __239__ (2002) 829--862 [doi](https://doi.org/10.1007/s002090100349)
* Helmut Lenzing, _On the K-theory of weighted projective curves_, [arXiv:1702.03445](https://arxiv.org/abs/1702.03445)
* Helmut Lenzing, _A K-theoretic study of canonical algebras_, in: Representation theory of algebras (Cocoyoc, 1994), CMS Conf. Proc, 1996 [RG](https://www.researchgate.net/profile/Helmut-Lenzing/publication/265548116_A_K-theoretic_study_of_canonical_algebras/links/55ba4d0008aec0e5f43e995b/A-K-theoretic-study-of-canonical-algebras.pdf)
* [[Sefi Ladkani]], _Refined Coxeter polynomials_, [arXiv:2110.15329](https://arxiv.org/pdf/2110.15329)


Euler characteristic

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

#Contents#
* table of contents
{:toc}

## Idea

The **Euler characteristic** of an object is --  if it exists -- its [[category theory|categorical]] [[dimension]].

## Definitions

### Of a chain complex

There are two definitions of the Euler characteristic of a chain complex.  Let $R$ be a [[commutative ring]] and let $V$ be a [[chain complex]] of $R$-[[modules]].

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

If each $V_n$ is [[finitely generated object|finitely generated]] and [[projective object|projective]], then the **Euler characteristic** of $V$ is the alternating sum of their [[ranks]], if this is finite:
$$
  \chi(V)
  \,\coloneqq\,
  \sum_{n \in \mathbb{Z}} (-1)^n \, rk_R V_n
  \,.
$$
=--

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

If the [[homology group|homology]] of $V$ consists of finitely generated projective modules, then the **Euler characteristic** of $V$ is the alternating sum of its [[Betti number|Betti numbers]] (the [[ranks]] of its homology modules), if this is finite:
$$
  \chi(V)
  \,\coloneqq\,
  \sum_{n \in \mathbb{Z}} (-1)^n \, rk_R H_n(V)
  \,.
$$
=--

When both of these are defined, they are equal.  This is a consequence of the functoriality of the categorical definition (Definition \ref{EulerCharInSymMonCat}).

+-- {: .num_remark #InvariantUnderQIso}
###### Remark
Definition \ref{EulerCharOfChainComplex} shows that the Euler characteristic of chain complexes is invariant under the natural notion of [[equivalence in an (infinity,1)-category|equivalence]] of chain complexes: (zig-zags of) [[quasi-isomorphism|quasi-isomorphisms]].
=--

### Of a topological space (or $\infty$-groupoid)
 {#OfATopologicalSpace}

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

For $X$ a [[topological space]] and $R$ some [[ring]], its Euler characteristic over $R$ is the Euler characteristic, according to Def. \ref{EulerCharOfChainComplex}, of its [[homology]] [[chain complex]] (for instance [[singular homology]]), if this is finite: the alternating sum of its [[Betti numbers]]

\[
  \label{FormulaForTopologicalSpaceAsAltSumOfBettiNumbers}
  \begin{aligned}
    \chi(X)
    &
    \;\coloneqq\;
    \sum_{n \in \mathbb{N}}
    \,
    (-1)^n
    \,
    rk_{\mathbb{Z}} H_n(X, \mathbb{Z})
    \\
    &
    \;=\;
    \sum_{n \in \mathbb{N}}
    \,
    (-1)^n
    \,
    dim_{\mathbb{Q}} H_n(X, \mathbb{Q})
  \end{aligned}
  \,.
\]

=--

This is shown for $R = \mathbb{Z}$ the [[integers]] ([[integral cohomology]]), and equivalently for $R = \mathbb{Q}$ [[rational numbers]] ([[rational cohomology]]).
(Choosing a ring with torsion, however, might result in a different "Euler characteristic".)

This definition is usually known as the **Euler-Poincaré formula**.

\begin{remark}
This is the special case of the formula for the [[Lefschetz number]]: The Euler characteristic (eq:FormulaForTopologicalSpaceAsAltSumOfBettiNumbers) is the Lefschetz number of the [[identity map]], see
[there](Lefschetz+trace+formula#LefschetzNumber).
\end{remark}

Historically earlier was:

+-- {: .num_defn #EulerCharOfCWComplex}
###### Definition/Proposition

Let $X$ be a finite [[CW-complex]]. Write $cell(X)_k$ for the set of its $k$-cells. Then the Euler characteristic of $X$ is

$$
  \chi(X)
  \;=\;
  \sum_{k \in \mathbb{N}} (-1)^k \vert cell(X)_k \vert
  \,.
$$

=--

This is equivalently the Euler characteristic of the [[cellular homology|cellular]] chain complex of $X$ according to Definition \ref{EulerCharOfChainComplexPointset}.  Thus, since the homology of $X$ can be computed with cellular homology, this Euler characteristic agrees with the previous one.

In the special case that $X$ is a [[surface]], Def. \ref{EulerCharOfCWComplex} reduces to the historical definition by [[Leonhard Euler]], which implicitly was known already to Descartes around 1620:

+-- {: .num_defn #EulerCharOfSurface}
###### Definition/Proposition

Let $X$ be a convex [[polyhedron]]. Then its Euler characteristic is

$$
  \chi(X)
  \;\coloneqq\;
   \vert Vertices(X)\vert
   -
   \vert Edges(X)\vert
    +
   \vert Faces(X)\vert
  \;\in\;
  \mathbb{Z}
  \,,
$$

hence the number of [[vertices]] minus the number of [[edges]] plus the number of [[faces]] in the polyhedron.

=--

In particular if $X$ may be embedded into the [[2-sphere]], this means that

$$
  2 = \vert Vertices(X)\vert - \vert Edges(X)\vert +
   \vert Faces(X)\vert
  \,.
$$

By removing one point from the 2-sphere not contained in $X$, the result may be thought of as a [[planar graph]]. This has one [[face]] less than $X$ had (the one containing the point which was removed). Hence

+-- {: .num_cor #EulerFormulaForPlanarGraphs}
###### Corollary
**([[Euler formula for planar graphs]])**

For a planar [[graph]] $\Gamma$ we have

$$
  1 = \vert Vertices(\Gamma)\vert - \vert Edges(\Gamma)\vert +
   \vert Faces(\Gamma)\vert
  \,.
$$

=--

### Of an object in a symmetric monoidal category

All the definitions considered so far can be subsumed by the following general abstract one.

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

The Euler characteristic of a [[dualizable object]] in a [[symmetric monoidal category]] is its categorical [[dimension]]: the [[trace]] of its [[identity]] [[morphism]].
=--

Explicitly, this means the Euler characteristic of an object $X$ is the composite

$$ S \xrightarrow{\eta} X \otimes X^* \xrightarrow{\cong} X^* \otimes X \xrightarrow{\varepsilon} S $$

where $\eta$ and $\varepsilon$ are the [[unit of an adjunction|unit]] and counit of the dual pair $(X,X^*)$, and $S$ is the unit object of the symmetric monoidal category.  This subsumes the previous definitions as follows:

* In the category of chain complexes over a ring $R$, an object is dualizable if it is finitely generated overall, and projective in each degree.  Its dual is then given by $(X^*)_n = Hom_R(X_{-n},R)$.  The unit $\eta$ picks out $\sum x \otimes x^*$, where $\{x\}$ encompasses a basis for each $X_n$ and $\{x^*\}$ is the dual basis.

  The symmetry isomorphism $X \otimes Y \xrightarrow{\cong} Y \otimes X$ introduces a sign $x\otimes y \mapsto (-1)^{|y|} y\otimes x$, so that when we then evaluate, we get a contribution of $1$ for each $x$ of even degree and $-1$ for each $x$ of odd degree.  Thus we recover Def. \ref{EulerCharOfChainComplexPointset}.  Note that the unit object is $R$ itself in degree zero, so that we see $\chi(X)$ only as an element of $R$ (so, for instance, the Euler characteristic in this sense of a rank-$p$ free $(\mathbb{Z}/p)$-module is zero.

* In the [[derived category]] of chain complexes over $R$, an object is dualizable if it is quasi-isomorphic to one of the form above.  A similar argument shows that its Euler characteristic is then computed as in Def. \ref{EulerCharOfChainComplex}.

The Euler characteristic of a topological space or $\infty$-groupoid can also be defined *directly* with this approach, without a detour into homology.  Namely, if $X$ is any [[Euclidean Neighborhood Retract]], such as a finite [[CW complex]] or a [[smooth manifold]], then its [[suspension spectrum]] $\Sigma_+^\infty X$ is dualizable in the [[stable homotopy category]].  Its dual is the [[Thom spectrum]] of its stable [[normal bundle]], with the unit of the duality being the [[Thom collapse map]].  Its categorical Euler characteristic is then an endomorphism of the sphere spectrum, which can be identified with an integer (via $\pi_0^s(S) = \mathbb{Z}$). See around [DoldPuppe, corollary 4.6](#DoldPuppe)).

Since the categorical definition is purely in terms of the symmetric monoidal structure, Euler characteristics are preserved by any [[symmetric monoidal functor]] (as long as enough of its structure maps are isomorphisms).  Since chains and homology can be made into symmetric monoidal functors, it follows that all the ways of defining the Euler characteristic of a space agree.

See ([PontoShulman](#PontoShulman)) and the discussion at [[Thom spectrum]] for more on this.

### Homotopy cardinality ($\infty$-groupoid cardinality)
 {#HomotopyCardinality}

The [above](OfATopologicalSpace) Euler characteristic of a topological space is the alternating _sum_ over sizes of _homology_ groups. Similar in construction is the alternating _product_ of sizes of [[homotopy group|homotopy groups]]. This goes by the name _[[∞-groupoid cardinality]]_ or _[[homotopy cardinality]]_ . But [below](RelationBetweenDefinitions) we shall see that Euler characteristic of higher categories interpolates between this homotopical and the above homological notion.

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

For $X$ a [[topological space]] / [[homotopy type]] / [[∞-groupoid]], its **[[homotopy cardinality]]** or **[[∞-groupoid cardinality]]** is -- if it exists -- the [[rational number]] given by the alternating product of [[cardinality|cardinalities]] of [[homotopy group|homotopy groups]]

$$
  \chi_{homotop}(X)
  \;\coloneqq\;
  \sum_{[x] \in \pi_0(X)}
    \prod_{k =1}^\infty (|\pi_k(X,x)|)^{(-1)^k}
  \,.
$$

=--

### Of posets, groupoids and categories

The process of sending a [[category]] $C$ to its
[[geometric realization of categories]] ${\vert C \vert} \in $ [[Top]] $\simeq$ [[∞Grpd]] is a way to _present_ [[topological space|topological spaces]], and hence [[∞-groupoid|∞-groupoids]], by a category: we can think of $\vert C \vert$ as the [[Kan fibrant replacement]] of $C$: the [[universal property|universal solution]] to [[weak inverse|weakly inverting]] all [[morphism|morphisms]] of $C$.

Up to the relevant notion of [[equivalence in an (infinity,1)-category]] (which is [[weak homotopy equivalence]]), _every_ [[∞-groupoid]] arises as the [[nerve]]/[[geometric realization of categories|geometric realization]] of a category. In fact one can assume the category to be a [[poset]]. (This follows from the existence of the [[Thomason model structure]], as discussed in more detail there.)

Since the combinatorial data in a [[category]] and all the more in a [[poset]] $C$ is much smaller than in that of its [[Kan fibrant replacement]] $\vert C \vert$, it is of interest to ask if one can read off the Euler characteristic $\chi(\vert C \vert)$ already from $C$ itself. This is indeed the case:

#### Of finite categories
 {#LECharOfCategory}

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

Let $C$ be a finite [[category]]. A **weighting** on $C$ is a [[function]]

$$
 k^{(-)} \colon ob C \to \mathbb{Q}
$$

satsifying

$$
  \forall a \in ob C \;:\; \sum_{b \in ob C} \vert C(a,b) \vert k^b
  = 1
  \,,
$$

where $\vert C(a,b)\vert$ is the [[cardinality]] of the [[hom-set]] $C(a,b)$. A **coweighting** on $C$ is a weighting on the [[opposite category]] $C^{op}$.

If $C$ admits both a weighting $\{k^a\}$ and a coweighting $\{k_a\}$, then its **Euler characteristic** is

$$
  \chi(C) := \sum_{a \in ob C} k^a = \sum_{a \in ob C} k_a
  \;\;\;
  \in \mathbb{Q}
  \,.
$$

=--

The definition of Euler characteristic of [[poset|posets]] appears for instance in ([Rota](#Rota)). For [[groupoid|groupoids]] it has been amplified in [BaezDolan](#BaezDolan). The joint generalization to categories is due to ([Leinster](#Leinster)), where the above appears as def. 2.2.

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

Notice that this $\chi(C)$ is in general not an [[integer]], but a [[rational number]]. However in sufficiently well-behaved cases, discussed [below](RelationBetweenDefinitions), $\chi(C)$ coincides with the topological Euler characteristic $\chi(\vert C \vert)$ of its [[geometric realization of categories|geometric realization]]. Since that is integral, in these cases also $\chi(C)$ is.

=--

#### Of enriched and higher categories
 {#LECharOfEnrichedCat}

Let $V$ be an good enrichment category (for instance a [[cosmos]]) which itself comes equipped with a good notion of cardinality, in the form of a monoidal functor

$$
  |-| : V \to \mathbb{R}
  \,.
$$

Then the above formula for the Euler characteristic of a category verbatim generalizes to $V$-[[enriched categories]]. The ordinary case is recovered for $V = $ [[FinSet]] and $|-| : FinSet \to \mathbb{R}$ the ordinary [[cardinality]] operation.

Since [[strict infinity-categories]] can be understood as arising from iterative enrichment

$$
  Str2Cat \simeq Cat-Cat
$$

$$
  Str3Cat \simeq Str2Cat-Cat
$$

etc, this gives a notion of Euler characteristic of strict $\infty$-categories, hence in particular of [[strict infinity-groupoid|strict infinity-groupoids]].

One should be able to show that applied to strict $\infty$-groupoids this does reproduce [[homotopy cardinality]].

## Properties

### Compatibility with homotopy colimits
 {#UnderHomotopyColimits}

Euler characteristic behaves well with respect to the basic operations in [[homotopy theory]].

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

In a [[symmetric monoidal category|symmetric monoidal]] [[triangulated category]] with [[dualizable object|dualizable objects]] $X, Y, Z$, if

$$
  \array{
    Z &\to& Y
    \\
    \downarrow && \downarrow
    \\
    X &\to& W
  }
$$

is a [[homotopy pushout]], then the [tractial Euler characteristic](EulerCharInSymMonCat) of $W$ exists and is

$$
  \chi(W) = \chi(X) + \chi(Y) - \chi(Z)
  \,.
$$

=--

This is due to ([May, 1991](#May)).

### Relations between the various definitions
 {#RelationBetweenDefinitions}

The following propositions assert that and how the various definitions of Euler characteristics all suitably agree when thez jointly apply.

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

For $X$ a [[compact space|compact]] [[manifold]] let $P_T(X)$ be the [[poset]] of inclusions of [[simplices]] of a [[triangulation]] $T$ of $X$. Then the poset Euler characteristic of $P_T(X)$ coincides with the Euler characteristic of $X$ as a topological space

$$
  \chi(X) = \chi(P_T(X))
  \,.
$$

=--

This appears as ([Stanley, 3.8](#Stanley)).

The following proposition asserts that the definition \ref{EulerCharOfCat} of Euler characteristic of a category is indeed consistent, in that it does compute the Euler characteristic, def. \ref{EulerCharOfTopSpace} of the corresponding $\infty$-groupoid:

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

Let $C$ be a finite [[poset]] or, slightly more generally, a finite [[skeleton|skeletal]] [[category]] with no nontrivial [[endomorphism|endomorphisms]].

Write $\vert C \vert \in $ [[Top]] $\simeq$ [[∞Grpd]] for its [[geometric realization of categories|geometric realization]]. Then

$$
  \chi(C) = \chi(\vert C \vert)
  \,.
$$

=--

For posets this is due to Philipp Hall, appearing as ([Stanley, prop. 3.8.5]). For finite categories this is  ([Leinster, cor. 1.5, prop. 2.11](#Leinster)).

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

For $X$ a [[manifold]], regard its [[suspension spectrum]]

$$
  \Sigma^\infty X \in Ho(Spec)
$$

as an object in the [[stable homotopy category]]. Then its Euler characteristic as an object of a [[symmetric monoidal category]], def. \ref{EulerCharInSymMonCat} coincides with its topological Euler characteristic, def. \ref{EulerCharOfTopSpace}.

=--

This is due to ... (?)

+--{: .query}
[[Mike Shulman|Mike]]: Can the Euler characteristic of a category be recovered as the trace for a dualizable object in some symmetric monoidal category?
=--

### Relation between Euler characteristic and $\infty$-groupoid cardinality

For topological space / $\infty$-groupoids, there is both the notion of [homological Euler characteristic](#OfATopologicalSpace) as well as the notion of [homotopy cardinality](#HomotopyCardinality).

The latter looks a bit like the "exponential" of the former, so while similar to some extent they are very different notions, taken on face value. Still, the [Euler characteristic of a category](LECharOfCategory) or rather that [of a higher category](LECharOfEnrichedCat) does interpolate between the two notions:

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

For $C$ a finite [[groupoid]], its Euler characteristic as a category, def. \ref{EulerCharOfCat}, coincides with its homotopical Euler characteristic, def. \ref{EulerCharOfSpaceHomotopically} or [[groupoid cardinality]]

$$
  \chi(C) = \chi_{homotop}(C)
  \,.
$$

=--

This is noted in ([Leinster, example 2.7](#Leinster)).

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

For instance for $G$ a [[finite group]] let $B G \simeq K(G,1) \in $ [[Ho(Top)]] be its [[classifying space]]. This is the [[geometric realization of categories|geometric realization]] both of the one-object groupoids $*//G$ as well as of some finite [[poset]] $C$.

$$
  B G \simeq |*//G| \simeq |C|
  \,.
$$

By prop. \ref{PosteEulerCharGiveSpaceCharUnderRealization} we have that the categorical Euler characteristic of $C$ is the topological Euler characteristic of $B G$. But by prop. \ref{ReproducingGroupoidCardinality} we have that the categorical Euler characteristic of $*//G$ is the [[homotopy cardinality]] of $B G$.

=--

Typically for one and the same $\infty$-groupoid, Eucler characteristic and homotopy cardinality are never both well defined: if the series for one converges, that for the other does not.

However, by applying some standard apparent "tricks" on non-convergent series, often these can be made sense of after all, and then do agree with the other notion. For more on this see ([Baez05](#Baez05)).

### Basic relations for Euler characteristics of topological spaces

\begin{proposition}
\label{PoincareHopfTheorem}
**([[Poincaré–Hopf theorem]])**
\linebreak
(...)
\end{proposition}

\begin{proposition}
\label{EulerCharacteristicOfSpheres}
**(Euler characteristic of spheres)**
\linebreak
  The Euler characteristic of an even dimensional sphere is 2, that of an odd dimensional sphere vanishes:
$$
  \chi(S^{n})
  \;=\;
  \left\{
  \array{
    2 &\vert& n \, \text{is even}
    \\
    0 &\vert& \text{otherwise}
  }
  \right.
$$
\end{proposition}
(e.g.  [Libgober, p. 16](#Libgober))

\begin{proposition}
\label{EulerCharacteristicOfCoveringSpaces}
**(Euler characteristic of covering spaces)**
\linebreak
If $X \xrightarrow{q} Y$ is a [[covering space]] over a [[connected topological space]] $Y$ of [[degree of a continuous function|degree]] $deg(q) \in \mathbb{N}$ (and the [[ordinary homology]] of $X$ and $Y$ is finitely generated) then the [[Euler characteristic]] of $X$ is $deg(q)$ times that of $Y$:
$$
 \chi(X) \,=\, ord(G) \cdot \chi(Y)
  \,.
$$
\end{proposition}
(e.g. topospaces [here](https://topospaces.subwiki.org/wiki/Euler_characteristic_of_covering_space_is_product_of_degree_of_covering_and_Euler_characteristic_of_base))

\begin{proposition}
\label{EulerCharacteristicOfConnectedSums}
**([[Euler characteristic]] of [[connected sums]])**
\linebreak
Let $X$ and $Y$ be [[closed manifolds]] ([[topological manifold|topological]] or [[differentiable manifold|differentiable]])  of [[even number|even]] [[dimension]]. Then the [[Euler characteristic]] of their [[connected sum]] is the [[sum]] of their separate [[Euler characteristics]], minus 2:

$$
  \chi
  \big(
    X \sharp Y
  \big)
  \;=\;
  \chi
  \big(X\big)
  +
  \chi
  \big(Y\big)
  -2
$$
\end{proposition}
(e.g. [here](http://math.ucr.edu/~res/miscpapers/csums+echars.pdf))

### Gauss-Bonnet theorem

For $X$ an even-[[dimension]]al [[smooth manifold]], its Euler characteristic may also be given by [[integration]] of [[infinitesimal object|infinitesimal]] data: this is the statement of the higher dimensional [[Gauss-Bonnet theorem]].

## Related concepts

* [[magnitude of an enriched category]]

* [[Euler class]]

* [[orbifold Euler characteristic]]

* [[arithmetic genus]]

* [[C-field tadpole cancellation]]

## References

Review includes

* {#Libgober} Jonathan Libgober, _The Euler characteristic, Poincaré–Hopf theorem, and applications_ [[pdf](http://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Libgober.pdf), [[Libgober_EulerCharacteristic.pdf:file]]]

Textbook account:

* [[Edwin Spanier]],  p. 156 onwards in: _Algebraic topology_, McGraw-Hill  (1966)

Efremovich and Rudyak shown that the Euler characteristic is (up to the overall multiplicative factor) the only additive homotopy invariant of a finite CW complexes:

* V. A. Efremovich, Yu. B. Rudyak, _On the concept of the Euler characteristic_, Uspehi Mat. Nauk __31__:5(191) (1976), 239–240 [MR458412](http://www.ams.org/mathscinet-getitem?mr=458412), [Russian pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=3975&volume=31&year=1976&issue=5&fpage=239&what=fullt)

The description of Euler characteristics are categorical [[trace|traces]] in [[symmetric monoidal categories]] is discussed in section 4 of

* {#DoldPuppe} [[Albrecht Dold]], [[Dieter Puppe]], _Duality, trace and transfer_ , Proceedings of the Steklov Institute of Mathematics, (1984), issue 4

Behaviour of tracial Euler characteristic under [[homotopy colimit|homotopy colimits]] is discussed in

* {#May} [[Peter May]], _The additivity of traces in triangulated categories_ K-theory (1991) ([website](http://www.math.uiuc.edu/K-theory/0474/))

Textbooks on combinatorial aspects of Euler characteristic include

* {#Stanley} Richard P. Stanley, _Enumerative Combinatorics_ , Vol. I, Cambridge Studies in Advanced Mathematics 49, Cambridge University Press, corrected reprint (1997)

* {#Rota} [[Gian-Carlo Rota]], _On the foundations of combinatorial theory I: theory of Möbius functions_ , Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340–368.

The Euler characteristic of a smooth manifold as its [[dimension]] in the [[stable homotopy category]] is discussed in example 3.7 of

* {#PontoShulman} [[Kate Ponto]], [[Mike Shulman]], _Traces in symmetric monoidal categories_ ([pdf](http://www.sandiego.edu/~shulman/papers/traces_sym.pdf), [slides](http://www.sandiego.edu/~shulman/papers/ccrtraces.pdf)).

See [[Thom spectrum]] for more on this

The Euler characteristic of groupoids -- [[groupoid cardinality]] -- has been amplified in

* {#BaezDolan} [[John Baez]], [[James Dolan]], _From Finite Sets to Feynman Diagrams_ ([arXiv](http://arxiv.org/abs/math.QA/0004133))

An exposition with an eye towards the relation between Euler characteristic and $\infty$-groupoid cardinality is in

* [[John Baez]], _The mysteries of counting: Euler characteristic versus homotopy cardinality_ ([web](http://math.ucr.edu/home/baez/counting/)) Lecture notes (2005)
 {#Baez05}

A treatment of this relation using [[Morava K-theories]] is in

* Lior Yanovski, _Homotopy Cardinality via Extrapolation of Morava-Euler Characteristics_ ([arXiv:2303.02603](https://arxiv.org/abs/2303.02603))

The role of homotopy cardinality in [[quantization]] is touched on towards the end of

* [[Daniel Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]],  _[[Topological Quantum Field Theories from Compact Lie Groups]]_

The generalization of the definition of Euler characteristic from posets to categories is due to

* [[Tom Leinster]], _The Euler characteristic of a category_ ([arXiv:0610260](http://arxiv.org/abs/math/0610260))
  {#Leinster}

* [[Tom Leinster]], _The Euler characteristic of a category as a sum of a divergent series_ ([arXiv:0707.0835](http://arxiv.org/abs/0707.0835))

The compatibility of Euler characteristic of categories with [[homotopy colimits]] is discussed in

* [[Thomas M. Fiore]], [[Wolfgang Lück]], [[Roman Sauer]], *Euler Characteristics of Categories and Homotopy Colimits* [[arXiv:1007.3868](http://arxiv.org/abs/1007.3868)]

More on Euler characteristics of categories is in

* Kazunori Noguchi, _The Euler characteristic of infinite acyclic categories with filtrations_, [arxiv/1004.2547](http://arxiv.org/abs/1004.2547)

On "Negative sets" and Euler characteristic:

* [[André Joyal]], _Regle des signes en algebre combinatoire_ , Comptes Rendus Mathematiques de l'Academie des Sciences, La Societe Royale du Canada VII (1985), 285-290.

* [[Steve Schanuel]], _Negative sets have Euler characteristic and dimension_ , Lecture Notes in Mathematics 1488, Springer Verlag, Berlin, 1991, pp. 379-385.

* Daniel Loeb, _Sets with a negative number of elements_ , Adv. Math. 91 (1992), 64-74.

Resumming divergent Euler characteristics:

* William J. Floyd, Steven P. Plotnick, _Growth functions on Fuchsian groups and the Euler characteristic_ , Invent. Math. 88 (1987), 1-29.

* R. I. Grigorchuk, _Growth functions, rewriting systems and Euler characteristic_ , Mat. Zametki 58 (1995), 653-668, 798.

* James Propp, _Exponentiation and Euler measure_ , Algebra Universalis 49 (2003), 459-471. ([arXiv:math.CO/0204009](http://arxiv.org/abs/math/0204009)).

* James Propp, _Euler measure as generalized cardinality_ . ([arXiv](http://arxiv.org/abs/math/0203289)).

Euler characteristics of tame spaces:

* Lou van den Dries, _Tame Topology and O-Minimal Structures_ Cambridge U. Press, Cambridge, 1998. Chapter 4.2: Euler Characteristic.

Euler characteristics of groups:

* G. Harder, _A Gauss-Bonnet formula for discrete arithmetically defined groups_ , Ann. Sci. Ecole Norm. Sup. 4 (1971), 409-455.

* [[Jean-Pierre Serre]], _Cohomologie des groups discretes_ , Ann. Math. Studies 70 (1971), 77-169.

* [[Kenneth Brown]], _Euler characteristics, in Cohomology of Groups_ , Graduate Texts in Mathematics 182, Springer, 1982, pp. 230-272.

* O. Y. Viro, _Some integral calculus based on Euler characteristic_, in Topology and Geometry -- Rohlin Seminar, Springer Lec. Notes in Math. __1346__, 127--138 (1988) [doi](http://dx.doi.org/10.1007/BFb0082775)

Complex cardinalities:

* Andreas Blass, _Seven trees in one_ , Jour. Pure Appl. Alg. 103 (1995), 1-21. ([web](http://www.math.lsa.umich.edu/~ablass/cat.html))

* Robbie Gates, _On the generic solution to $P(X) = X$ in distributive categories_ , Jour. Pure Appl. Alg. 125 (1998), 191-212.

* [[Marcelo Fiore]], [[Tom Leinster]], _An objective representation of the Gaussian integers_ , Jour. Symb. Comp. 37 (2004), 707-716. ([arXiv](http://arxiv.org/abs/math/0211454))

* [[Marcelo Fiore]], [[Tom Leinster]], _Objects of categories as complex numbers_ , Adv. Math. 190 (2005), 264-277 ([arXiv:0212377](http://arxiv.org/abs/math/0212377))

* [[Marcelo Fiore]], _Isomorphisms of generic recursive polynomial types_ , to appear in 31st Symposium on Principles of Programming Languages (POPL04). ([ps](http://www.cl.cam.ac.uk/~mpf23/papers/Types/recisos.ps.gz))

* [[C. T. C. Wall]], _Arithmetic invariants of subdivision of complexes_, Canad. J. Math. 18(1966), 92-96, [doi](http://dx.doi.org/10.4153/CJM-1966-012-9), [pdf](http://cms.math.ca/cjm/v18/cjm1966v18.0092-0096.pdf)

$\mathbf{A}^1$-Euler characteristic

* Arcila-Maya, Niny; Bethea, Candace; Opie, Morgan; Wickelgren, Kirsten; Zakharevich, Inna, _Compactly supported $\mathbb{A}^1$-Euler characteristic and the Hochschild complex_, Topology and its Applications __316__  (2022) 108108 [doi](https://doi.org/10.1016/j.topol.2022.108108)

[[!redirects Euler characteristics]]


Euler class

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#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Euler class_ $\chi$ (or $e$) is a [[characteristic class]] of the [[special orthogonal group]], hence of [[oriented]] [[real vector bundles]].

The Euler class of the [[tangent bundle]] of a [[smooth manifold]] $X$, evaluated on its [[fundamental class]], is its [[Euler characteristic]] $\chi[X]$.

## Properties

### Cup square
 {#CupSquare}

For $E$ a [[vector bundle]] of [[even number|even]] [[rank of a vector bundle|rank]] $rank(E) = 2 k$, the  [[cup product]] of the [[Euler class]] with itself equals the $k$th [[Pontryagin class]]

\[
  \label{EulerSquareIsPontryagin}
  \chi(E) \smile \chi(E)
  \;=\;
  p_k(E)
  \,.
\]

(e.g. [Walschap 04, Section 6.3, p. 187](#Walschap04))

When the Euler class is represented by the [[Euler form]] of a [[connection]] $\nabla$ on $E$, which then is [[fiber]]-wise proportional to the [[Pfaffian]] of the [[curvature form]] $F_\nabla$ of $\nabla$, the relation (eq:EulerSquareIsPontryagin) corresponds to the fact that the product of a [[Pfaffian]] with itself is the [[determinant]]: $\big( Pf(F_\nabla) \big)^2 = det(F_\nabla)$.

### Whitney sum formula
 {#WhitneySumFormula}

+-- {: .num_prop #EulerClassOfWhitneySumIsCupProductOfEulerClasses}
###### Proposition
**([[Euler class]] takes [[Whitney sum]] to [[cup product]])**

The Euler class of the [[Whitney sum]] of two [[orthogonal group|oriented]] [[real vector bundles]] to the [[cup product]] of the separate Euler classes:

$$
  \chi( E \oplus F )
  \;=\;
  \chi(E)  \smile \chi(F)
  \,.
$$

=--

(e.g. [Walschap 04, Section 6.4](#Walschap04))

### Relation to top Chern class

\begin{prop}
  The [[top Chern class]] of a [[complex vector bundle]] $\mathcal{V}_X$ equals the [[Euler class]] $e$ of the underlying [[real vector bundle]] $\mathcal{V}^{\mathbb{R}}_X$:

$$
  \mathcal{V}_X
  \;
  \text{has complex rank}\;n
  \;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;
  c_n
  \big(
    \mathcal{V}_X
  \big)
  \;\;
    =
  \;\;
  e
  \big(
    \mathcal{V}^{\mathbb{R}}_X
  \big)
  \;\;\;\;
  \in
  H^{2n}
  \big(
   X;
   \,
   \mathbb{Z}
  \big)
  \,.
$$
\end{prop}

(e.g. [Bott-Tu 82 (20.10.6)](#BottTu82))

For more see at _[[top Chern class]]_.

### Poincaré–Hopf theorem

* [[Poincaré–Hopf theorem]]

[[!include Chern- and Pontrjagin forms -- section]]

### On unit sphere bundles

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

Let $X$ be a [[smooth manifold]] and $E \overset{\pi}{\longrightarrow} X$ an oriented [[real vector bundle]] of [[even number|even]] [[rank of a vector bundle|rank]], $rank(E) = 2k + 2$.

For any choice of [[connection on a bundle|connection]] $\nabla$ on $E$ ($SO(dim(X))$-connection), let $\chi(\nabla_E) \in \Omega^{2k}(X)$ denote the corresponding [[Euler form]].

Then the [[pullback of differential forms|pullback]] of the [[Euler form]] $\chi(\nabla_E)$ to the [[unit sphere bundle]] $S(E) \overset{S(\pi)}{\longrightarrow} X$ is [[exact differential form|exact]]

$$
  \big( S(\pi) \big)^\ast
  \chi(\nabla_E)
  \;=\;
  d \Omega
$$

such that the trivializing form has (minus) unit [[integration of differential forms|integral]] over any of the [[n-sphere|(2k+1)-sphere]]-[[fibers]] $S^{2k+1}_x \overset{\iota_x}{\hookrightarrow} S(E)$:

\[
  \label{FiberIntegrationOfTrivialization}
  \int_{S^{2k+1}} \iota_x^\ast \Omega
  \;=\;
  -1
  \,.
\]

=--

(e.g. [Walschap 04, Chapter 6.6, Thm. 6.1, p. 201-202](#Walschap04), [Poor 07, 3.68](#Poor07), [Nie 09](#Nie09))

### Fiber integration

+-- {: .num_prop  #FiberIntegrationOfCupPowersOfChiOver4Sphere}
###### Proposition

Let

$$
  \array{
    S^4
    &\longrightarrow&
    B Spin(4)
    \\
    &&
    \big\downarrow^{\mathrlap{\pi}}
    \\
    &&
    B Spin(5)
  }
$$

be the [[spherical fibration]] of [[classifying spaces]] induced from the canonical inclusion of [[Spin(4)]] into [[Spin(5)]] and using that the [[4-sphere]] is equivalently the [[coset space]] $S^4 \simeq Spin(5)/Spin(4)$ ([this Prop.](sphere#nSphereAsCosetSpace)).

Then the [[fiber integration]] of the odd [[cup product|cup powers]] $\chi^{2k+1}$ of the [[Euler class]] $\chi \in H^4\big( B Spin(4), \mathbb{Z}\big)$ (see [this Prop](Spin4#IntegralCohomologyOfClassifyingSpace))  are proportional to [[cup product|cup powers]] of the [[second Pontryagin class]]

$$
  \pi_\ast
  \left(
    \chi^{2k+1}
  \right)
  \;=\;
  2 \big( p_2 \big)^k
  \;\;\in\;\;
  H^4\big( B Spin(5), \mathbb{Z} \big)
  \,,
$$

for instance

$$
  \begin{aligned}
    \pi_\ast
    \big(
      \chi
    \big)
    & =
    2
    \\
    \pi_\ast
    \left(
      \chi^3
    \right)
    & =
    2 p_2
    \\
    \pi_\ast
    \left(
      \chi^5
    \right)
    & =
    2 (p_2)^2
  \end{aligned}
    \;\;\in\;\;
    H^4\big( B Spin(5), \mathbb{Z} \big)
  \,;
$$

while the [[fiber integration]] of the even [[cup product|cup powers]] $\chi^{2k}$ vanishes

$$
  \pi_\ast
  \left(
    \chi^{2k}
  \right)
  \;=\;
  0
  \;\;\in\;\;
  H^4\big( B Spin(5), \mathbb{Z} \big)
  \,.
$$

=--

([Bott-Cattaneo 98, Lemma 2.1](#BottCattaneo98))

## Related concepts

* [[Pontryagin class]], [[Stiefel-Whitney class]], [[Wu class]]

* [[I8|one-loop anomaly polynomial]]

## References

### General

* {#BottTu82} [[Raoul Bott]], [[Loring Tu]], Chapter 11 of  _[[Differential Forms in Algebraic Topology]]_, Graduate Texts in Mathematics 82, Springer 1982 ([doi:10.1007/BFb0063500](https://doi.org/10.1007/BFb0063500))

* [[Allen Hatcher]], _Euler and Pontryagin classes_, section 3.2 in _[Vector bundles and K-theory](http://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html)_ ([pdf](http://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf))

* Anant R. Shastri, section 2.1 of _Vector bundles and Characteristic Classes_ ([pdf](http://www.math.iitb.ac.in/~ars/seminar/bundle.pdf))

* Michael Hutchings, section 5 of _Cup product and intersections_ ([pdf](https://math.berkeley.edu/~hutching/teach/215b-2011/cup.pdf))

* {#Walschap04} Gerard Walschap, chapter 6.3 of: _Metric Structures in Differential Geometry_, Graduate Texts in Mathematics, Springer 2004 ([doi:10.1007/978-0-387-21826-7](https://link.springer.com/book/10.1007/978-0-387-21826-7))

Discussion of [[fiber integration]]:

* {#BottCattaneo98} [[Raoul Bott]], [[Alberto Cattaneo]], _Integral Invariants of 3-Manifolds_, J. Diff. Geom., 48 (1998) 91-133 ([arXiv:dg-ga/9710001](https://arxiv.org/abs/dg-ga/9710001), [euclid:jdg/1214460608](https://projecteuclid.org/euclid.jdg/1214460608))

Discussion for [[projective modules]]

* Satya Manda, _An overview of Euler class theory_ ([pdf](http://mandal.faculty.ku.edu/talks/amsTalk06.pdf))

See also

* Wikipedia [Euler class](http://en.wikipedia.org/wiki/Euler_class)

* Robert F. Brown, _On the Lefschetz number and the Euler class_, Transactions of the AMS __118__, (1965)  ([JSTOR](http://www.jstor.org/pss/1993952))

* Solomon Jekel, _A simplicial formula and bound for the Euler class_, Israel Journal of Mathematics __66__, n. 1-3, 247-259 (1989)

### Euler forms

Discussion of _[[Euler forms]]_ ([[differential form]]-representatives of Euler classes in [[de Rham cohomology]]) as [[Pfaffians]] of [[curvature forms]]:

* {#Chern44} [[Shiing-Shen Chern]], _A Simple Intrinsic Proof of the Gauss-Bonnet Formula for Closed Riemannian Manifolds_, Annals of Mathematics, Second Series, Vol. 45, No. 4 (1944), pp. 747-752 ([jstor:1969302](https://www.jstor.org/stable/1969302))

* {#BottTu82} [[Raoul Bott]], [[Loring Tu]], Chapter 11 of  _[[Differential Forms in Algebraic Topology]]_, Graduate Texts in Mathematics 82, Springer 1982 ([doi:10.1007/BFb0063500](https://doi.org/10.1007/BFb0063500))

* {#MathaiQuillen86} [[Varghese Mathai]], [[Daniel Quillen]], below (7.3) of _Superconnections, Thom classes, and equivariant differential forms_, Topology Volume 25, Issue 1, 1986 (10.1016/0040-9383(86)90007-8)

* {#Nakahara03} [[Mikio Nakahara]], Section 11.4.2 of: *[[Geometry, Topology and Physics]]*, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

* [Walschap 04, section 6.3](#Walschap04)

* {#Wu05} Siye Wu, Section 2.2 of _Mathai-Quillen Formalism_, pages 390-399 in _Encyclopedia of Mathematical Physics_ 2006 ([arXiv:hep-th/0505003](https://arxiv.org/abs/hep-th/0505003))

* {#Poor07} Walter A. Poor, 3.58 of _Differential Geometric Structures_, Dover Books on Mathematics, 2007

* {#Nie09} Zhaohu Nie, _Secondary Chern-Euler forms and the Law of Vector Fields_ ([arXiv:0909.4754](https://arxiv.org/abs/0909.4754)

* [[Hiro Lee Tanaka]], _Pfaffians and the Euler class_, 2014 ([pdf](http://www.hiroleetanaka.com/pdfs/2014-fall-230a-lecture-26-gauss-bonnet-chern.pdf))

* {#Nicolaescu18} [[Liviu Nicolaescu]], Section 8.3.2 of _Lectures on the Geometry of Manifolds_, 2018 ([pdf](https://www3.nd.edu/~lnicolae/Lectures.pdf), [MO comment](https://mathoverflow.net/a/117982/381))

[[!redirects Euler classes]]
[[!redirects Euler characteristic class]]
[[!redirects Euler form]]
[[!redirects Euler forms]]


Euler formula for planar graphs

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### Context
#### Graph theory
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#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Statement

For a [[connected graph|connected]] [[finite graph|finite]] [[planar graph]] $\Gamma$ the numbers of [[edges]], [[vertices]] and number of regions enclosed by edges ("[[faces]]") always satisfies the relation

$$
  1 = \vert Vertices(\Gamma)\vert - \vert Edges(\Gamma)\vert +
   \vert Faces(\Gamma)\vert
  \,.
$$

By passing to the [[one-point compactification]] of the [[plane]], which is the [[2-sphere]], we may think of the planar graph as a [[polyhedron]] embedded in the 2-sphere. Under this identification the above is a special case of the general formula for [[Euler characteristic]] of [[CW-complexes]]. See at _[Euler characteristic -- Of topological spaces](Euler+characteristic#OfATopologicalSpace)_.

## Applications

### In perturbative quantum field theory

That the [[loop order]] of a (planar) [[Feynman diagram]] is its contribution in powers of [[Planck's constant]] to the [[scattering amplitude]] is a consequence of Euler's formula. See at _[loop order -- Relation to powers in Planck's constant](loop+order#RelationToPowersInPlancksConstant)_

## References

* Wikiedia, _[Euler's formula](https://en.wikipedia.org/wiki/Planar_graph#Euler.27s_formula)_

[[!redirects Euler's formula for planar graphs]]

[[!redirects Euler characteristic of planar graphs]]
[[!redirects Euler characteristics of planar graphs]]

[[!redirects Euler characteristic of a planar graph]]
[[!redirects Euler characteristics of a planar graph]]


Euler integration

## Idea

A notion of [[integration]] using the [[Euler characteristic]] as a [[measure]].

## References

* Yuliy Baryshnikov, Robert Ghrist, _Euler integration over definable functions_ ([pdf](http://www.math.upenn.edu/~ghrist/preprints/definable.pdf))
* С. М. Гусейн-Заде, Интегрирование по отношению к эйлеровой характеристике и его приложения, УМН, 65:3 (393) (2010) 5–42, [link](http://mi.mathnet.ru/umn9357); Eng. transl.  Gusein-Zade, _Integration with respect to the Euler characteristic and its applications_, Russ. Math. Surv. __65__:3,2010, 399-432, [link](http://mi.mathnet.ru/eng/umn9357), [MR2011k:14011](http://www.ams.org/mathscinet-getitem?mr=2682719)

* [blog discussion](http://golem.ph.utexas.edu/category/2012/05/integrating_against_the_euler.html)


Euler number

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Several numbers are named after [[Euler]].

# Contents
* table of contents
{: toc}

## Euler's number

**Euler's number** is the [[irrational number]] typically denoted "[[e]]" and defined by the [[series]]

$$
  e \coloneqq \sum_{n \in \mathbb{N}} \frac{1}{n!}
  \,.
$$

This happens to also be the [[groupoid cardinality]] of the [[groupoid]] $core(FinSet)$, the [[core]] of [[FinSet]].

## Euler numbers

The **Euler numbers** (also called _secant numbers_) $E_n$, $n\geq 0$ are a sequence of [[integer|integers]] defined via the generating function

$$
\frac{1}{ch t}= \frac{2}{e^t+e^{-t}} = \sum_{n\geq 0} E_n \frac{t^n}{n!}
$$

All odd-numbered members of the sequence vanish: $E_{2k-1}=0$ for $k\in\mathbb{N}$. $E_0=1$, $E_2 = -1$, $E_4 = 5$, see [Euler number at wikipedia](http://en.wikipedia.org/wiki/Euler_number) for more and the [Abramowitz--Stegun handbook](http://en.wikipedia.org/wiki/Abramowitz_and_Stegunfor) for many Euler numbers.

+-- {: .num_remark}
###### Remark
Sometimes Euler numbers are defined as a different sequence that includes tangent numbers (and is non-alternating in sign):

$$\sec(x) + \tan(x) = \sum_{n \geq 0} E_n \frac{x^n}{n!}$$

where one has $1, 1, 1, 2, 5, 16, \ldots$ as the first few such Euler numbers. These $E_n$ satisfy the recurrence

$$2 E_{n+1} = \sum_i \binom{n}{i} E_i E_{n-i}$$

and count the number of alternating [[permutations]] $\pi$ on $\{1, 2, \ldots, n\}$, i.e. permutations such that $\pi(1) \gt \pi(2) \lt \pi(3) \gt \ldots$. See for example this $n$-Category Café [post](https://golem.ph.utexas.edu/category/2008/02/metric_spaces.html#c014963).
=--

Euler numbers generalize to __Euler polynomials__ $E_n(x)$ defined via the generating function

$$
\frac{2e^{tx}}{e^t+1}=\sum_{n\geq 0} E_n(x) \frac{t^n}{n!},
$$

so that $E_n = 2^n E_n(\frac{1}{2})$. The Euler polynomials satisfy the recursion $E_n(x+1)+E_n(x)=2x^n$ and may be conversely expressed via Euler numbers as

$$
E_n(x) = \sum_k \left(\array{n\\ k}\right) \frac{E_k}{2^k}\left(x-\frac{1}{2}\right)^{n-k}
$$

There is also a complementarity formula $E_n(1-x)=(-1)^n E_n(x)$.
See e.g. Louis Comtet, __Advanced combinatorics__, D. Reifel Publ. Co., Dordrecht-Holland, Boston 1974 [djvu](http://ftp.filekeeper.org/download/acm/books/Comtet%20Louis%20-%20Advanced%20Combinatorics.djvu)

## Eulerian numbers

There are also **Eulerian numbers** (forming a different, double sequence $A(n,k)$). Combinatorially, $A(n, k)$ counts the number of permutations $\pi$ of the set $\{1, 2, \ldots, n\}$ with $k$ _ascents_ (an ascent of $\pi$ is an element $j$ such that $\pi(j) \lt \pi(j+1)$). Alternatively, the double sequence can be defined recursively through the formula

$$x^n = \sum_k A(n, k) \binom{x+k}{n}.$$

For applications in algebraic geometry, see "Eulerian polynomials-[Hirzebruch]".

## Euler-Mascheroni constant

There is also a number, usually denoted as $\gamma$, called the _Euler-Mascheroni constant_. It is defined as a limit

$$\gamma \coloneqq \lim_{n \to \infty} \sum_{k=1}^n \frac1{k} - \log n$$

where $\log$ is the natural [[logarithm]]. It arises for example in discussions of the [[Gamma function]].

## Related concepts

* [[exponential map]]

* [[e]]

* [[pi]]

* [[golden ratio]]

[[!redirects Euler number]]
[[!redirects Euler numbers]]
[[!redirects Euler's number]]
[[!redirects Euler's numbers]]

[[!redirects Euler's constant]]

[[!redirects Eulerian number]]
[[!redirects Eulerian numbers]]

[[!redirects Euler polynomial]]
[[!redirects Euler polynomials]]


Euler product

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=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

The original _Euler product_ is an expression of the [[Riemann zeta function]] as an [[infinite product]] (instead of an [[infinite sum]] as originally defined) over all [[prime numbers]]. More generally one considers Euler product expressions for general [[zeta functions]] and [[L-functions]] (various types of L-functions are _defined_ by an Euler product, such as the [[Artin L-function]]).

When the [[Riemann zeta function]] is understood as an [[adelic integral]] via [[Iwasawa-Tate theory]], then the existence of Euler product expressions follows from the fact that the integration [[measure]] on the [[idele group]] $\mathbb{I}_{\mathbb{Q}}$ used in this context is a product of measures on all [[p-adic number]]-components:

if a function $f\colon \mathbb{I} \longrightarrow \mathbb{R}$ has product form $f = \underset{v}{\prod} f_v$ then

$$
  \int_{\mathbb{I}_{\mathbb{Q}}}
  \;
    f(x)
  \;
    {\vert x\vert}^s
  \;
  d^\times x
  \;=\;
  \underset{v}{\prod}
  \left(
  \int_{\mathbb{Q}_v^\times}
  \;f_v(x)\;
   {\vert x\vert}_v^s
  \;
  d_v^\times x
  \right)
  \,.
$$

(reviewed e.g. in [Garrett 11, 1.6](#Garrett11))

## References

* Wikipedia, _[Euler product](http://en.wikipedia.org/wiki/Euler_product)_

Discussion from the point of view of [[adelic integration]] in [[Iwasawa-Tate theory]] is for instance in

* {#Garrett11} [[Paul Garrett]], section 1.6 _Iwasawa-Tate on ζ-functions and L-functions_
([pdf](http://www-users.math.umn.edu/~garrett/m/mfms/notes_c/Iwasawa-Tate.pdf))

[[!redirects Euler products]]

[[!redirects Euler product form]]


Euler systems

\tableofcontents

## Idea

An _Euler system_ is a collection of classes in [[Galois cohomology]] satisfying certain compatibility conditions. The existence of a nontrivial Euler system can imply bounds on the [[Selmer group]] and thus the theory of Euler systems finds applications in the study of [[special values of L-functions]], for instance in Victor Kolyvagin's work on the [[Birch and Swinnerton-Dyer conjecture]].

## Definition

The reference for this section is Section 1.4 of [#LoefflerZerbes18](#LoefflerZerbes18).

\begin{definition}

Let $G_{\mathbb{Q}}$ be the [[absolute Galois group]] of $\mathbb{Q}$. Let $V$ be a [[Galois representation]] valued in $\mathbb{Q}_{p}$, and let $T\subset V$ be a $G_{K}$-stable $\mathbb{Z}_{p}$-lattice. Let $\Sigma$ be a finite set of primes containing $p$ and all ramified primes.

Let $P_{\ell}(V,t)$ be the local Euler factor at $\ell$, i.e.

$$P_{\ell}(V,t)=\det(1-t\cdot\rho(\Frob_{v}^{-1})).$$

An _Euler system_ for $(T,\Sigma)$ is a collection $\mathbf{c}=(c_{m})_{m}$, where $c_{m}\in H^{1}(\mathbb{Q}(\mu_{m}),T)$ satisfying the following compatibility conditions:

$$\mathrm{norm}_{\mathbb{Q}(\mu_{m})}^{\mathbb{Q}(\mu_{m\ell})}(c_{m\ell})=

\begin{cases}
c_{m}\;\if \ell\in\Sigma\;\or\;m\vert \ell\\
P_{\ell}(V^{*}(1),\sigma_{\ell}^{-1})\cdot c_{m}\;\otherwise
\end{cases}$$

where $\sigma_{\ell}$ is the image of $\mathrm{Frob}_{\ell}$ in $\mathrm{Gal}(\mathbb{Q}(\mu_{m})/\mathbb{Q})$.

\end{definition}

## Application to bounding the Selmer group

\begin{theorem}(Theorem 2 of [#LoefflerZerbes18](#LoefflerZerbes18))
Suppose $\mathbf{c}$ is an Euler system for $(T,\Sigma)$ with $c_{1}$ nonzero, and suppose that $V$ satisfies certain technical conditions. Then the strict Selmer group $\Sel_{\strict}(\mathbb{Q},V^{*}(1))$ is zero.
\end{theorem}

## Relation to motivic cohomology

One of the ways to construct examples of Euler systems is via [[motivic cohomology]] (see chapter 3 of [#LoefflerZerbes18](#LoefflerZerbes18)).

## References

{#LoefflerZerbes18} David Loeffler and Sarah Zerbes, _Euler Systems_, Arizona Winter School 2018 Notes ([pdf](https://swc-math.github.io/aws/2018/2018LoefflerZerbesNotes.pdf))


Euler's formula

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#Contents#
* table of contents
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Euler's formula is a fundamental relation between the [[exponential function]] and the [[trigonometric functions]].

## Statement

The [[exponential function]] $\exp: \mathbb{C} \to \mathbb{C}$ relates to [[sine]] and [[cosine]] $\cos, \sin \colon \mathbb{R} \to \mathbb{R}$ via

$$
  \exp(i x) = \cos(x) + i \sin(x)
$$

This can be interpreted as either a theorem (especially when $x$ is restricted to [[real numbers]] and $\cos, \sin$ have been defined independently of $\exp$), or as an implicit definition of $\cos, \sin$ (where more explicitly we have $\cos(x) \coloneqq \frac1{2}(\exp(i x) + \exp(-i x))$ and $\sin(x) \coloneqq \frac1{2 i}(\exp(i x) - \exp(-i x))$).

## Related concepts

* [[trigonometric identity]]

## References

Named after [[Leonhard Euler]].

* Wikipedia, _[Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula)_

[[!redirects Euler formula]]


Euler-Arnold equation

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* table of contents
{:toc}

## Idea

The _Euler-Arnold equation_ governs [[Hamiltonian dynamics on Lie groups]].

## References

The original influential is

* [[Vladimir Arnol'd]], _Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits._ Ann. Inst. Fourier (Grenoble) 16 (1966) fasc. 1, 319–361. ([MathSciNet](http://www.ams.org/mathscinet-getitem?mr=202082))

A survey is in

* [[Terence Tao]], _The Euler-Arnold equation_ ([web](http://terrytao.wordpress.com/2010/06/07/the-euler-arnold-equation/))

* Hernan Cendra, [[Jerrold Marsden]], Sergey Pekarsky, [[Tudor Ratiu]], _Variational principles for Lie-Poisson and Hamilton_pincaré equations_  ([pdf](http://www.google.de/url?sa=t&source=web&cd=1&ved=0CB4QFjAA&url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fdownload%3Fdoi%3D10.1.1.131.219%26rep%3Drep1%26type%3Dpdf&rct=j&q=Variational%20principles%20for%20Lie-Poisson%20and%20Hamilton-Poincar%C3%A9%20equations&ei=PMVbTsSrII-N-waYzbCnDA&usg=AFQjCNEgOCIlKEWeV5w6HX-00QEtj-gydQ&cad=rja))


Euler-Lagrange complex

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## Idea

Given a smooth [[bundle]] $E \to \Sigma$ over a [[smooth manifold]] $\Sigma$, then its _Euler-Lagrange complex_ is a [[resolution]] of the [[constant sheaf]] of [[locally constant functions]] on the [[jet bundle]] $J^\infty E$ by a [[chain complex]] of sheaves of certain [[differential forms]]. The Euler-Lagrange complex starts out as the complex of [[horizontal differential forms]] up to degree $n \coloneqq dim(\Sigma)$ the [[dimension]] of $\Sigma$, the following [[differential]] is

1. the [[Euler-Lagrange operator]] $\delta_{El}$

1. followed by the [[Helmholtz operator]] $\delta_{Helm}$

$$
 0 \to \mathbb{R}
 \to
  \Omega^{0,0}(J^\infty E)
  \stackrel{d_H}{\to}
  \Omega^{1,0}(J^\infty E)
  \stackrel{d_H}{\to}
  \cdots
  \stackrel{d_H}{\to}
  \Omega^{n,0}(J^\infty E)
  \stackrel{\delta_{EL}}{\to}
  \mathcal{F}^1(J^\infty E)
  \stackrel{\delta_{Helm}}{\to}
  \mathcal{F}^2(J^\infty E)
  \stackrel{\delta_V}{\to}
  \cdots
$$

Hence the elements in the Euler-Lagrange complex have the following interpretation

* in degree $dim(\Sigma)$: [[local Lagrangians]];

* in degree $dim(\Sigma)+1$: [[Euler-Lagrange equations of motion]];

* in degree $dim(\Sigma)-1$: trivial local Lagrangians;

* in degree $dim(\Sigma)+2$: [[obstructions]] for [[equations of motion]] to be variational, i.e. to be the Euler-Lagrange equations of a local Lagrangian.

## Properties

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

The [[cochain cohomology]] of the [[Euler-Lagrange complex]]

$$
 0 \to \mathbb{R}
 \to
  \Omega^{0,0}(J^\infty E)
  \stackrel{d_H}{\to}
  \Omega^{1,0}(J^\infty E)
  \stackrel{d_H}{\to}
  \cdots
  \stackrel{d_H}{\to}
  \Omega^{n,0}(J^\infty E)
  \stackrel{E}{\to}
  \mathcal{F}^1(J^\infty E)
  \stackrel{\delta_V}{\to}
  \mathcal{F}^2(J^\infty E)
  \stackrel{\delta_V}{\to}
  \cdots
$$

is [[isomorphism|isomorphic]] to the [[de Rham cohomology]] of the total space $E$ of the given fiber bundle.

=--

([Anderson 89, theorem 5.9](#Anderson89)).

## Related concepts

* [[variational sequence]]

## References

The Euler-Lagrange complex was recognized in

* {#Vinogradov78} [[Alexandre Vinogradov]], _A spectral sequence associated with a non-linear differential equation, and the algebro-geometric foundations of Lagrangian field theory with constraints_, Soviet Math. Dokl. 19 (1978), 144–148.

* {#Tulczyjew80} W. M. Tulczyjew, _The Euler-Lagrange resolution_, in Lecture Notes in Mathematics No. 836, Springer-Verlag, New York, 1980, pp. 22–48.

Review includes

* {#VinogradovKrasilshchik99} [[Alexandre Vinogradov]], I. S. Krasilshchik (eds.) _Symmetries and Conservation Laws for Differential Equations of Mathematical Physics_, vol. 182 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1999. ([pdf](ftp://softbank.iust.ac.ir/MathBooks/V/Vinogradov%20-%20Symmetries%20and%20Conservation%20Laws%20for%20Differential%20equations%20of%20mathematical%20physics.pdf))

* {#Anderson89} [[Ian Anderson]], _The variational bicomplex_, Utah State University 1989 ([[AndersonVariationalBicomplex.pdf:file]])

[[!redirects Euler complex]]
[[!redirects Euler complexes]]

[[!redirects Euler-Lagrange operator]]
[[!redirects Euler-Lagrange operators]]

[[!redirects Euler-Lagrange variational derivative]]
[[!redirects Euler-Lagrange variational derivatives]]

[[!redirects Euler-Lagrange derivative]]
[[!redirects Euler-Lagrange derivatives]]


Euler-Lagrange equation

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#### Algebraic Quantum Field Theory
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#### Equality and Equivalence
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=--
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# Contents
* table of contents
{: toc}

## Idea

In [[variational calculus]] the  **Euler-Lagrange [[equations]]** of a [[nonlinear functional]] arising from [[transgression of variational differential forms|transgression]] of a [[local Lagrangian density]] characterize the _[[extrema]]_ of that functional, hence its _[[critical locus]]_ (the "[[shell]]").

This originates from and is mainly used in [[physics]], specifically in [[Lagrangian field theory]], where the functional in question is the _[[action functional]]_ of a [[physical system]], and where its critical points encode the physically realized [[field histories]] by the _[[principle of extremal action]]_.

## Details

See at _[[A first idea of quantum field theory]]_ [this def.](A+first+idea+of+quantum+field+theory#EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime)

## Examples

* The Euler-Lagrange equations of the [[Einstein-Hilbert action]] are [[Einstein's equations]] of [[gravity]].

* [[Maxwell's equations]]

* [[Yang-Mills equations]]

## Related concepts

* [[Euler-Lagrange form]]

* [[equations of motion]]

  * [[Newton's laws of motion]]

  * [[Hamilton's equations]], [[de Donder-Weyl-Hamilton equation]]

  * [[Einstein's equations]]

* [[Schwinger-Dyson equation]], [[Ward identity]]

* [[variational calculus]]

* [[topos of laws of motion]]

## References

Named after [[Leonhard Euler]] and [[Joseph-Louis de Lagrange]].

Textbook account in the context of [[gauge theory]]:

* [[Marc Henneaux]], [[Claudio Teitelboim]], §1.1.1 in: _[[Quantization of Gauge Systems]]_, Princeton University Press (1992) [[ISBN:9780691037691](https://press.princeton.edu/books/paperback/9780691037691/quantization-of-gauge-systems), [jstor:j.ctv10crg0r](https://www.jstor.org/stable/j.ctv10crg0r)]

See also:

* Wikipedia, _[Euler-Lagrange equation](http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation)_.

* [[Robert Bryant]], [[Phillip Griffiths]], Daniel Grossman, _Exterior differential systems and Euler-Lagrange partial differential equations_, Chicago Lectures in Mathematics 2003, 205+xiv pp. [math.DG/0207039](http://arxiv.org/abs/math/0207039) [bookpage](http://www.press.uchicago.edu/ucp/books/book/chicago/E/bo3637779.html)

Discussion in the convenient context of [[smooth sets]]:

* _[Higher prequantum geometry II: The principle of extremal action -- comonadically](https://www.physicsforums.com/insights/higher-prequantum-geometry-ii-principle-extremal-action-comonadically/)_

* {#GiotopoulosSati23} [[Grigorios Giotopoulos]], [[Hisham Sati]], §5 in: *Field Theory via Higher Geometry I: [[schreiber:Smooth Sets of Fields]]* [[arXiv:2312.16301](https://arxiv.org/abs/2312.16301)]

[[!redirects Euler-Lagrange equations]]

[[!redirects Euler-Lagrange equation of motion]]
[[!redirects Euler-Lagrange equations of motion]]

[[!redirects equation of motion]]
[[!redirects equations of motion]]
[[!redirects equations of motion]]

[[!redirects law of motion]]
[[!redirects laws of motion]]


Euler-Lagrange form

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## Definition

Given a [[smooth function|smooth]] [[bundle]] $E \to \Sigma$ with $\Sigma$ a [[smooth manifold]] of [[dimension]] $n$, then a horizontal $n$-form $\mathbf{L}$ on the [[jet bundle]] $Jet(E)$ is a [[local Lagrangian]]. Its [[de Rham differential]] has a unique decomposition into a [[source form]] $\mathbf{E}$ and a horizontally exact form (with respect to the [[variational bicomplex]])

$$
  d \mathbf{L} = \mathbf{E} - d_H \theta
  \,.
$$

This source form $\mathbf{E}$ is the _Euler-Lagrange form_ of $\mathbf{L}$. It vanishes precisely at those points which are solutions to the [[Euler-Lagrange equations]] induced by $\mathbf{L}$.

The combination $\rho \coloneqq \mathbf{L} + \theta$ is the corresponding [[Lepage form]].

## References

* {#Zuckerman87} [[Gregg Zuckerman|G. J. Zuckerman]], _Action principles and global geometry_ , in Mathematical Aspects of String Theory, S. T. Yau (Ed.), World Scientific, Singapore, 1987, pp. 259€284. ([[ZuckermanVariation.pdf:file]])

[[!redirects Euler-Lagrange forms]]


Euler-MacLaurin formula

## Idea

_Euler-MacLaurin formula_ is a formula for approximation of integrals by sums. It involves [[Bernoulli numbers]].

## References

* wikipedia [Euler-MacLaurin formula](http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula)


Eva Silverstein

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

## Selected writings

On [[multi trace operators]] in [[AdS/CFT]]:

* [[Ofer Aharony]], [[Micha Berkooz]], [[Eva Silverstein]], _Multiple-Trace Operators and Non-Local String Theories_, JHEP 0108:006, 2001 ([arXiv:hep-th/0105309](https://arxiv.org/abs/hep-th/0105309))

* [[Ofer Aharony]], [[Micha Berkooz]], [[Eva Silverstein]], _Non-local string theories on $AdS_3$ times $S^3$ and stable non-supersymmetric backgrounds_, Phys. Rev. D65:106007, 2002 ([arXiv:hep-th/0112178](https://arxiv.org/abs/hep-th/0112178))

On [[M-theory on hyperbolic manifolds]]:

* G. Bruno De Luca, [[Eva Silverstein]], Gonzalo Torroba, *Hyperbolic compactification of M-theory and de Sitter quantum gravity* ([arXiv:2104.13380](https://arxiv.org/abs/2104.13380))

category: people


evaluation fibration of mapping spaces

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## Idea ##

In the context of a [[manifold structure of mapping spaces|mapping space]], an important family of [[smooth maps]] are the
[[evaluation maps]].  That is, for a [[sequentially compact]] [[Froelicher space|Frölicher space]] $S$ and a [[manifold]] $M$, we pick a [[point]] $p \in S$ and consider the map $ev_p \colon C^\infty(S,M) \to M$ given by $ev_p(\alpha) = \alpha(p)$.  This is [[smooth function|smooth]] (see [[smooth maps of mapping spaces]]).  At $q \in M$, the [[fibre]] is the space of smooth maps $S \to M$ which take $p$ to $q$.  This is again a smooth manifold (as shown in [[manifold structure of mapping spaces]]).  Providing $M$ has enough diffeomorphisms, this is the projection of a [[fibre bundle]].  That is to say, the sequence:
\[
C^\infty(S,p;M,q) \to C^\infty(S,M) \to M
\]
is a fibre bundle.

The remark about "enough diffeomorphisms" is the key to proving this.  To prove that this is a fibre bundle, we need to show that if $\beta \colon S \to M$ nearly takes $p$ to $q$ then we can deform $\beta$ to a map which takes $p$ to $q$ on the nose.  Knowing that $M$ has the structure of a manifold, we can interpret the statement "$\beta(p)$ is close to $q$" as meaning that we have fixed a [[coordinate chart|chart]] near $q$ and require that $\beta(p)$ be in the [[codomain]] of that chart.  We therefore have a good choice of deformation for $\beta(p)$ itself: deform it along the "straight line" from $q$ to $\beta(p)$ as defined by the chart.  The problem with this is that it only tells us what to do with $\beta(p)$, not with the rest of $\beta$.  So we need to drag the rest of $\beta$ along with $\beta(p)$.  This is where the diffeomorphisms come in: instead of moving $\beta(p)$ along a path, we deform the entire manifold using a diffeomorphism so that $\beta(p)$ is taken to $q$.  We do this using the methods of [[propagating flows]].


evaluation map

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#Contents#
* table of contents
{:toc}

## Definition
 {#Definition}

Let $\big(\mathcal{C}, \otimes, \mathbb{1}\big)$ be a [[closed monoidal category]]. For $S, D \in C$ two [[objects]], with [[internal hom]] $[S, \text{-}]$ [[right adjoint functor|right adjoint]] to the [[tensor product]] $S \otimes (\text{-})$, the *evaluation map*

$$
  ev \;\colon\; S \otimes [S, D]  \to D
$$

is the [[counit of an adjunction|counit]] of this [[adjoint functor|adjunction]], hence the [[adjunct]] of the [[identity]] morphism

$$
  Id \;\colon\; [S,D] \to [S,D]
  \,.
$$

Concretely for $\mathcal{C}$ = [[Sets]] or generally on [[generalized elements]], the evaluation map is given by evaluating a [[function]] at a value, whence the name:

$$
  \array{
    S \times [S,D] &\overset{ev}{\longrightarrow}& D
    \\
    \big(
      s
      ,\,
      f
    \big)
    &\mapsto&
    f(s)
    \mathrlap{\,.}
  }
$$

If $\mathcal{C}$ is moreover [[compact closed category|compact]] then for $D = \mathbb{1}$ the [[tensor unit]] we have that $[S, \mathbb{1}] \,=\, S^\ast$ is the [[dual object]] to $S$, whence the evaluation map becomes the pairing of objects with their duals
$$
  S \otimes S^\ast \overset{ev}{\longrightarrow} \mathbb{1}
  \,.
$$
Therefore, sometimes the pairing map on [[dual objects]] may be called "evaluation" even when the ambient category is not [[compact closed category|compact closed]].

## Properties

### Syntax and semantics
 {#SyntaxAndSemantics}

In a [[cartesian closed category]] the evaluation map

$$
  [X,Y]\times X \stackrel{eval}{\to} Y
$$

is the [[categorical semantics]] of what in the [[type theory]] [[internal language]] is the [[dependent type]] whose [[syntax]] is

$$
  f \colon X \to Y
  ,\;
  x \colon X
  \; \vdash \; f(x) \colon Y
$$

expressing  _[[function application]]_. On [[propositions]] ([[(-1)-truncation|(-1)-truncated]] [[types]]) this is the [[modus ponens]] [[deduction]] [[inference rule|rule]].

## See also

* [[coevaluation map]]

* [[state monad]], [[costate comonad]]

* [[concrete category]] (for the external version)

[[!redirects evaluation]]
[[!redirects evaluation maps]]
[[!redirects evaluation morphism]]
[[!redirects evaluation morphisms]]


evaluational category > history

[[!redirects evaluational category]]
[[!redirects evaluational categories]]

< [[concrete category]]


Evan Cavallo

* [webpage](http://www.cs.cmu.edu/~ecavallo/)

## Selected writings and talks

On [[generalized (Eilenberg-Steenrod) cohomology]] formulated in [[homotopy type theory]] (cf. *[[cohomology in homotopy type theory]]*):

* [[Evan Cavallo]], Section 3.2 of: *Synthetic Cohomology in Homotopy Type Theory* (2015) [[pdf](https://staff.math.su.se/evan.cavallo/works/thesis15.pdf), [[Cavallo-CohomologyInHoTT.pdf:file]]]

On the cube category with cartesian structure and one connection:

* [[Evan Cavallo]], *Cubes with one connection and relative elegance*, [[Homotopy Type Theory Electronic Seminar Talks]], 3 November 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Cavallo-2022-11-03-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=CrcExHaMDfk))

category: people


Evan Jenkins

* [personal page](http://ejenk.com/old-index.html)

## Selected writings

On [[braided 2-groups]]:

* [[Evan Jenkins]], _Extensions of groups by braided 2-groups_ ([arXiv:1106.0772](http://arxiv.org/abs/1106.0772))

category: people


Evan Patterson

* [website](https://www.epatters.org/)

### Selected writings

* [[Evan Patterson]], _Knowledge Representation in Bicategories of Relations_, ([arXiv:1706.00526](https://arxiv.org/abs/1706.00526))

* Rebekah Aduddell, James Fairbanks, Amit Kumar, Pablo S. Ocal, [[Evan Patterson]], Brandon T. Shapiro, _A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks_ ([arXiv:2301.01445](https://arxiv.org/abs/2301.01445))


even cohomology theory

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{:toc}

## Idea

A [[multiplicative cohomology theory|multiplicative]] [[cohomology theory]] $E$ is called _even_ if its [[cohomology ring]] is trivial in all odd degrees:

$$
  E^{2k+1}(X) = 0
  \,.
$$

## Properties

For an even cohomology theory $E$:

* There is an isomorphism of rings

$$ E^{\ast}(\mathbb{C}P^n) \cong E^{\ast}(S^0)[x]/\langle x^{n+1} \rangle,$$

where $|x|=2$.

* $E^{\ast}(\mathbb{C}P^{\infty} \times \ldots \times \mathbb{C}P^{\infty}) \cong E^{\ast}(S^0)[[x_1, \ldots, x_k]]$.

\begin{remark}\label{ComplexOrientation}
Any even cohomology theory is [[complex oriented cohomology theory|complex orientable]]; a choice of complex orientation gives an isomorphism

$$E^{\ast}(\mathbb{C}P^{\infty}) \cong E^{\ast}(S^0)[[x]].$$
\end{remark}
See [here](complex+oriented+cohomology+theory#ComplexOrientationOfEvenCohomologyTheories) at *[[complex oriented cohomology theory]]* (review also in [Mehrle 18, 1.1](#Mehrle18)).

## Related entries

* [[even periodic ring spectrum]]

* [[A Survey of Elliptic Cohomology - cohomology theories]]

## References

* {#Mehrle18} [[David Mehrle]], _Chromatic homotopy theory: Journey to the frontier_, Graduate workshop notes, Boulder 16-17 May 2018, ([pdf](http://pi.math.cornell.edu/~dmehrle/notes/conferences/cht/cht-notes.pdf), [[Mehrle_ChromaticHomotopyTheory.pdf:file]])

[[!redirects even cohomology theories]]


even number

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\linebreak

> An even number is that which is divisible into two equal parts.

> An odd number is that which is not divisible into two equal parts, or that which differs by a monad [=unit] from an even number.

> [[[Euclid]], Def. 6 & 7 of *[[Elements]]* Book VII (~ 400-300 BC), see [here](monad+disambiguation#ElementsVIIDef7)]

## Definition

An _even number_ is an [[integer]] $n \in \mathbb{Z}$ that is a multiple of 2, hence such that $n = 2 k$ for $k \in \mathbb{Z}$.

An _odd number_ is an integer that is not an even number.

## Properties

+-- {: .num_example #EvenAndOddIntegersAdjointModality}
###### Example
**([[adjoint modality]] of even and odd integers)**

Regard the [[integers]] as a [[preordered set]] $(\mathbb{Z}, \leq)$ in the canonical way, and thus as a [[thin category]].

Consider the [[full subcategory]] inclusions

$$
  \array{
    (\mathbb{Z}, \leq )
    & \overset{even}{\hookrightarrow}&
    (\mathbb{Z},\leq)
    \\
    n &\mapsto & 2 n
  }
  \phantom{AAAAA}
  \array{
    (\mathbb{Z}, \leq )
    & \overset{odd}{\hookrightarrow}&
    (\mathbb{Z},\leq)
    \\
    n &\mapsto & 2 n + 1
  }
$$

of the [[even number|even]] and the [[even number|odd]] [[integers]], as well as the functor

$$
  \array{
    (\mathbb{Z}, \leq )
    & \overset{\lfloor-/2\rfloor}{\longrightarrow}&
    (\mathbb{Z},\leq)
    \\
    n &\mapsto& \lfloor n/2 \rfloor
  }
$$

which sends any $n$ to the [[floor]] $\lfloor n/2 \rfloor$ of $n/2$, hence to the largest integer which is smaller or equal to the [[rational number]] $n/2$.

These functors form an [[adjoint triple]]

$$
  even \;\dashv\; \lfloor -/2 \rfloor \;\dashv\; odd
$$

and hence induce an [[adjoint modality]]

$$
  Even \;\dashv\; Odd
$$

on $(\mathbb{Z}, \leq)$ with

1. $Even \coloneqq 2 \lfloor -/2 \rfloor$ sending any integer to its "even floor value"

1. $Odd \coloneqq 2 \lfloor -/2 \rfloor + 1$ sending any integer to its "odd ceiling value".

=--

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

Observe that for all $n \in \mathbb{Z}$ we have

$$
  2 \lfloor n/2 \rfloor
    \overset{ \epsilon_n }{\leq}
  n
    \overset{ \eta_n }{\leq}
  2 \lfloor n/2 \rfloor  + 1
  \,,
$$

where the first inequality is an equality precisely if $n$ is even, while the second is an equality precisely if $n$ is odd. Hence this provides candidate [[unit of an adjunction|unit]] $\eta$ and [[counit of an adjunction|counit]].

Hence by [this characterization](adjoint+functor#UniversalArrow) of [[adjoint functors]]

1. the adjunction $\lfloor -/2 \rfloor \dashv odd$ is equivalent to the condition that

   for every $n \leq 2 k + 1$ we have $2 \lfloor n/2 \rfloor + 1  \leq 2 k + 1$;

1. the adjunction $even \dashv \lfloor -/2 \rfloor $ is equivalent to the condition that

   for every $2k \leq n$ we have $2k \leq 2 \lfloor n/2 \rfloor $,

which is readily seen to be the case.

=--

[[!redirects even numbers]]

[[!redirects odd number]]
[[!redirects odd numbers]]

[[!redirects odd natural number]]
[[!redirects odd natural numbers]]

[[!redirects odd integer]]
[[!redirects odd integers]]