generic point

Let $X$ be a [[topological space]].  A point $x \in X$ is called **generic** if the [[closure]] $\overline{\{x\}} = X$.

An [[integral scheme]] has a unique generic point, corresponding locally to the [[prime ideal]] $(0)$.

* wikipedia [generic point](https://en.wikipedia.org/wiki/Generic_point)

category: geometry, algebraic geometry

[[!redirects generic points]]


generic proof

# Generic proofs
* table of contents
{: toc}

## Definition

A **generic proof** in a [[category]] $C$ is a map ([[morphism]]) $\theta\colon \Theta\to\Lambda$ with all [[pullbacks]] such that, for every map $f\colon Y\to X$, there exists a (not necessarily unique) map $\nu\colon X\to \Lambda$ such that $f$ and the [[pullback|pulled back]] map $\nu^*\theta\;\colon\; \Theta \times_\Lambda X \to X$ each factor through each other, i.e. they are equivalent in the [[preorder reflection]] of the [[slice category]] $C/X$.

$$ \array {
   Y \\
   \uparrow \downarrow & \searrow^f \\
   \Theta \times_\Lambda X & \overset{\nu^*\theta}\longrightarrow & X \\
   \downarrow & & \downarrow\mathrlap{\nu} \\
   \Theta & \overset{\theta}\longrightarrow & \Lambda
} $$

The main interest of generic proofs is in the following:

+-- {: .un_theorem}
###### Theorem
If $C$ has [[finite limits]], then its [[ex/lex completion]] $C_{ex/lex}$ is a [[topos]] if and only if $C$ has [[weak dependent product]]s and a generic proof.
=--

In particular, if $C$ is a [[locally cartesian closed category]], then $C_{ex/lex}$ is a topos if and only if $C$ has a generic proof.

A counterpart of generic proofs in [[type theory]] is Russell's [[axiom of reducibility]].

## Axioms of choice

If $C$ is a [[topos]] satisfying the [[axiom of choice]], then its [[subobject classifier]] is a generic proof, since any $f\colon Y\to X$ factors through, and (assuming AC) is factored through by, its [[image]], a subobject of $X$.  In fact, for such a $C$, $C_{ex/lex}$ is equivalent to $C$.

The statement "$Set$ has a generic proof," or equivalently "$Set_{ex/lex}$ is a topos," is weaker than the full axiom of choice. Indeed, [Menni 2007](#Menni2007) shows that the exact completion of a Boolean Grothendieck topos with enough points is also a topos.

An even weaker statement is "$Set_{ex/lex}$ is [[well-powered category|well-powered]]."  Note that if $X$ is a set, then the subobject lattice of (the image of) $X$ in $Set_{ex/lex}$ is precisely the preorder reflection of $C/X$; thus this second statement says in particular that all such preorder reflections are essentially small.  This can be regarded as saying that although full AC may not hold, it is only violated in a "small way" at each set.  It is sufficient, for instance, to show that the category of [[anafunctors]] between two small categories is essentially small.

An instructive example is $C_{ex/lex}$ where $C = Set^{\to}$ (the [[arrow category]] of $Set$): this is an example of an exact completion of a topos that is well-powered but not itself a topos (because $C$ lacks a generic proof). It also shows, taking $D$ to be the [[coproduct completion]] of the [[walking arrow]], that we can have $D_{ex}$ (which is $Set^\to$ here) a topos but $(D_{ex})_{ex}$ not a topos. Details may be found in Menni's [2003 paper](#Menni2), section 5.3.

## References

* Matías Menni, "Exact completions and toposes", Ph.D. Thesis, U. Edinburgh 2000. ([web](http://www.lfcs.inf.ed.ac.uk/reports/00/ECS-LFCS-00-424/))

* Matías Menni, A characterization of the left exact categories whose exact completions are toposes, J. Pure Appl. Algebra 177 (2003) 287-301. ([pdf](http://www.lifia.info.unlp.edu.ar/papers/2003/Menni2003a.pdf))
  {#Menni2}

* Matías Menni, Cocomplete toposes whose exact completions are toposes, J. Pure Appl. Alg. 210 (2007), 511–520. ([pdf](http://www.lifia.info.unlp.edu.ar/papers/2007/Menni2007.pdf))
  {#Menni2007}

[[!redirects generic proof]]
[[!redirects generic proofs]]


generic representation theory

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

#Contents#
* table of contents
{:toc}

## Idea

This is an approach to [[linear representations]] of the [[general linear groups]] $GL_{n}(\mathbf{F}_{q})$ over [[prime fields]] which relates them to the [[Steenrod algebra]] in a very [[algebra|algebraic]] (and quite [[category theory|category theoretic]]) way.  It has been developed in a series of three papers by [[Nicholas Kuhn]].

From the first paper of the series:

\begin{imagefromfile}
        "file_name": "GenericRepresenations.png",
        "width": 600
\end{imagefromfile}

## References

* [[Nicholas Kuhn]], _Generic representations of the finite general linear groups and the Steenrod algebra: I, II, III_, Amer. J. Math. 116(1994), 327-360, [doi:10.2307/2374932](https://doi.org/10.2307/2374932), K-theory J. 8(1994), 395-428, [doi:10.1007/BF00961409](http://dx.doi.org/10.1007/BF00961409), K-theory J. 9(1995), 273-303, [doi:10.1007/BF00961668](http://dx.doi.org/10.1007/BF00961668).


Gennadi Sardanashvily

* [webpage](http://sardanashvily.blogspot.de/)

## related $n$Lab entries

* [[variational calculus]],

* [[jet bundle]], [[variational bicomplex]]

* [[fiber bundles in physics]],

  * [[Dirac charge quantization]], [[Aharonov-Bohm effect]]

category: people


Gennady Kasparov

* [webpage](http://as.vanderbilt.edu/math/bio/gennadi-kasparov)

category: people


genus

> this entry is about the notion of _genus_ in [[algebraic topology]]/[[cohomology]]. For classification of [[surfaces]] see instead the (related) entry _[[genus of a surface]]_, _[[genus of a curve]]_. There is also [[genus of a lattice]].

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Index theory
+-- {: .hide}
[[!include index theory - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Cobordism theory
+--{: .hide}
[[!include cobordism theory -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### Basic

For $R$ a ([[commutative ring|commutative]]) [[ring]], an $R$-valued **genus** is a ring [[homomorphism]] into $R$

$$
  \sigma : \Omega_\bullet  \to R
$$

from a [[cobordism ring]] for [[cobordisms]] with specified structure; typical choices being [[orientation]] or [[stable complex structure]]. Often the rationalization of such a morphism is meant, see below at _[Properties -- Rationalization](#Rationalization)_.

To emphasize that this is indeed a ring homomorphism and hence in particular respects the multiplicative structure, a genus is sometimes (especially in older literature) synonymously called a _multiplicative genus_.

### In stable homotopy theory and generalized cohomology theory
 {#InStableHomotopyTheory}

Since the [[cobordism ring]] is the ring of coefficients of the corresponding universal [[Thom spectrum]], e.g $M U$, $M SO$, so a genus may also be written as a ring homomorphism of the form

$$
  M SO_\ast \longrightarrow R
$$

or

$$
  M U_\ast \longrightarrow R
$$

respectively. Written this way it is immediate that genera arise naturally as the value on [[homotopy groups]] (the "[[decategorification]]" or "de-homotopification") of homomorphisms of [[E-∞ ring]] [[spectra]] from an actual universal [[Thom spectrum]] to some [[E-∞ ring]] $E$ with [[coefficient]] ring $R$

$$
  M SO \longrightarrow E
$$

or

$$
  M U \longrightarrow E
  \,.
$$

This in turn induces multiplicative morphisms of the [[cohomology theories]] [[Brown representability theorem|represented]] by these spectra (the domain being hence [[cobordism cohomology theory]]), and these multiplicative maps are the "families version" of the given genus/[[index]] ([Hopkins 94, section 3](#Hopkins94)).

Such homomorphisms in turn arise naturally from [[orientation in generalized cohomology|universal orientations in generalized E-cohomology]]. Namely such an orientation is a [[homotopy]] of the form

$$
  \array{
     \ast && \longleftarrow && B SO
     \\
       & {}_{\mathllap{0}}\searrow & \swArrow_\sigma  & \swarrow_{\mathrlap{J}}
     \\
     && B GL_1(E)
     \\
     && \downarrow
     \\
     && E Mod
  }
$$

(a trivialization of the $E$-[[(∞,1)-module bundle]] [[associated ∞-bundle|associated]] to the [[spherical fibration]] given by the [[J-homomorphism]]) and under forming [[homotopy colimits]] in $E$[[(∞,1)Mod]] this becomes an $E$-linear map

$$
  M SO \wedge E  \longleftarrow E
$$

hence a map

$$
  M SO  \longleftarrow E
  \,.
$$

At least in some important cases, genera seem to be naturally understood as encoding [[sigma-model]] [[quantum field theories]]. For $G$ some structure, the [[Thom spectrum]] $M G$ is the classifying space of [[manifolds]] with [[G-structure]], and hence may be thought of as classifying [[target spaces]] for [[sigma-models]]. The codomain spectrum $R$ itself may then be thought of as a classifying space for a certain class of QFTs, and hence the genus $\sigma : M G \to R$ can be thought of as assigning to any target space the corresponding [[sigma-model]].

This is for instance the case at least over the point for the [[A-hat genus]] $M Spin \to K O$, which may be thought of as sending manifolds with [[spin structure]] to the corresponding [[(1,1)-dimensional Euclidean field theories and K-theory|(1,1)-supersymmetric EFT]] ("[[spinning particle]]"); and for the [[Witten genus]] $M String \to tmf$, which can be thought of as sending a manifold with [[string structure]] to the corresponding [[(2,1)-dimensional Euclidean field theories and tmf|(2,1)-supersymmetric EFT]] ("[[heterotic string]]").

## Properties

### Rationalization
 {#Rationalization}

When the [[coefficient]] [[ring]] $R$ does not have additive [[torsion]], then any ring homomorphism

$$
  \phi \colon \Omega^{U/SO}_\bullet \longrightarrow R
$$

is determined already by its rationalization

$$
  \phi \colon \Omega^{U,SO}_\bullet\otimes \mathbb{Q} \longrightarrow R \otimes \mathbb{Q}
$$

which is traditionally denoted by the same symbol. The rational [[cobordism rings]] in turn are known to be [[polynomial rings]]

$$
  \Omega^U_\bullet\otimes \mathbb{Q} \simeq \mathbb{Q}[\mathbb{C}P^1,\mathbb{C}P^2, \cdots ]
$$

$$
  \Omega^{SO}_\bullet\otimes \mathbb{Q} \simeq \mathbb{Q}[\mathbb{C}P^2,\mathbb{C}P^4, \cdots ]
$$

whose generators are identified with the cobordism classes of the manifolds which are the complex [[projective spaces]], as indicated.

### Logarithm and Characteristic series
 {#LogarithmAndCharacteristicSeries}

#### Definition in components

Given a (rational) genus $\phi \colon \Omega^{U,SU}_\bullet\otimes \mathbb{Q} \to R \otimes \mathbb{Q}$ one defines (we follow ([Hopkins 94](#Hopkins94)))

1. its _logarithm_ to be the [[formal power series]] over $R \otimes \mathbb{Q}$ given by

   $$
     log_\phi(z) \coloneqq \sum_{n \in \mathbb{N}} \phi(\mathbb{C}P^n) \frac{z^{n+1}}{n+1};
   $$

1. its _characteristic series_ (or _Hirzebruch series_) to be the formal power series over $R \otimes \mathbb{Q}$

   $$
     K_\phi(z) \coloneqq \frac{z}{\exp_\phi(z)}
     \,,
   $$

   where $\exp_\phi$ is the inverse of the logarithm;

1. its _characteristic class_ as the [[universal characteristic class]] which via the [[splitting principle]] is fixed by its value on the universal line bundle as

   $$
     K_\phi(c_1) \in H^\bullet(B U(1),R \otimes \mathbb{Q})
   $$

   where $c_1 \in H^2(B U(1), \mathbb{Z})$ denotes the universal [[first Chern class]]; hence its value on a [[direct sum]]  $L_1 \oplus \cdots \oplus L_k$ of [[complex line bundles]] is

   $$
     \prod_{i} K_\phi(c_1(L_i))
     \,.
   $$

#### Definition via orientations in generalized cohomology
 {#HirzebruchSeriesViaOrientationsInGeneralizedCohomology}

Suppose that the given genus $\Omega_\bullet^{SO} \longrightarrow R$ indeed comes from an [[orientation in generalized cohomology]] (as discussed [above](#InStableHomotopyTheory)) hence from a [[homomorphism]] of [[E-∞ rings]]

$$
  \beta \;\colon\;  M SO \longrightarrow E
$$

for an [[E-∞ ring]] $E$ with [[homotopy groups]] $R\simeq \pi_\bullet(E)$.
(And suppose that $E$ defines a [[complex oriented cohomology theory]].)

This defines ([Ando-Hopkins-Rezk 10, prop. 2.11](#AndoHopkinsRezk10))
a universal [[orientation in generalized cohomology|orientation]] of real vector bundles and hence of [[complex vector bundles]] and hence of [[complex line bundles]] in $E$-cohomology

$$

  \beta \;\colon\; M U \longrightarrow M SO \stackrel{\beta}{\longrightarrow} E
  \,.
$$

Now rationally, i.e. for $E\otimes \mathbb{Q}$, there is a canonical such orientation, given by the composite

$$
  \alpha \;\colon\; M U \longrightarrow M SO \longrightarrow H \mathbb{Q} \simeq \mathbb{S}\otimes \mathbb{Q} \stackrel{}{\longrightarrow} E \otimes \mathbb{Q}
  \,.
$$

Thus, given any orientation $\beta$, its rationalization may be compared to $\alpha$. Since these rational orientations are equivalently trivializations of maps to $B GL_1(E \otimes \mathbb{R})$, their difference is a class $\beta/\alpha$ with coefficients in $GL_1(E\otimes R)$, hence over any space $X$ the difference is a class in $H^0(X, \pi_\bullet E\otimes \mathbb{Q})^\times$.

Specifically consider the [[delooping]] $X= B U(1)$ of the [[circle group]].
For this the [[cohomology ring]] is the [[power series]] ring in a single [[variable]] (the universal [[first Chern class]] $c_1(L)$).
Under the canonical inclusion $B U(1)\to B U$ both the above orientations $\beta$ and $\alpha$ pull back, so that we have a difference

$$
  K_\beta \coloneqq \beta/\alpha \in (E_\bullet \otimes \mathbb{Q})[ [ c_1(L) ] ]
  \,.
$$

This is the Hirzebruch series of $\beta$ ([Ando-Hopkins-Rezk 10, def. 3.10](#AndoHopkinsRezk10)).

If $F$ denotes the [[formal group law]] classified via $MU_\bullet \to M SO_\bullet \stackrel{\beta_\bullet}{\to} E_\bullet$ then

$$
  K_\beta(z) = \frac{z}{\exp_F(z)}
$$

### The Hirzebruch formula
 {#HirzebruchFormula}

The central theorem of ([Hirzebruch 66](#Hirzebruch66)) expresses the genus of an arbitrary ([[cobordism class]] of a) manifold $X$ of [[dimension]] $2n$ via the formula

$$
  \phi(X)
  =
  \langle \prod_{i = 1}^n K_\phi(x_i(T X)), [X] \rangle
$$

in terms of the Hirzebruch characteristic series $K_\phi$ discussed [above](#LogarithmAndCharacteristicSeries), and via the [[splitting principle]]:

This means that $\prod_{i = 1}^n K_\phi(x_i(T X))$ is the function of [[Chern classes]] $c_k$ (i.e. [[Pontryagin classes]] $P_{2k}$ and [[Euler classes]] $\chi$) obtained by rewriting the [[polynomial]] in the $x_i$ (the "[[Chern roots]]") as a polynomial in [[elementary symmetric polynomials]] $\sigma_k(x_1,\cdots, x_n)$ and then substituting for each of these by $c_k(T X)$.

(see also e.g. [ManifoldAtlas -- Genera -- 4.1 Construction](http://www.map.mpim-bonn.mpg.de/Formal_group_laws_and_genera#Construction)).

## Examples

### Todd genus

The [[Todd genus]] is the genus with logarithm

$$
  -log_{Todd}(1-x) = \sum_{n \in \mathbb{N}} \frac{x^n}{n}
$$

### Signature genus

The [[signature genus]];

### $\hat A$-genus
 {#AhatGenus}

The [[A-hat genus]] is the [[index of a Dirac operator]] coming from a [[spin bundle]] in KO-theory. It is given by the characteristic series

The [characteristic series](genus#LogarithmAndCharacteristicSeries) of the $\hat A$-genus is

$$
  \begin{aligned}
    K_{\hat A}(e)
     & =
    \frac{z}{e^{z/2} - e^{-z/2}}
    \\
     &=
    \exp\left(
       - \sum_{k \geq 2} \frac{B_k}{k} \frac{z^k}{k!}
     \right)
  \end{aligned}
  \,,
$$

where $B_k$ is the $k$th [[Bernoulli number]] ([Ando-Hopkins-Rezk 10, prop. 10.2](#AndoHopkinsRezk10)).

The $\hat A$-genus is an [[integer]] on manifolds with [[spin structure]].

### Elliptic genus

An [[elliptic genus]] is one whose logarithm is given by

$$
  log_{ell}(z) = \int_0^z (1-2 \delta t^2 + \epsilon t^4)^{-1/2} d t
$$

for constants $\delta, \epsilon$ with non-degenerate values $\delta^2 \neq \epsilon$ and $\epsilon = 0$.

For degenerate choices this reproduces the [[signature genus]] and the [[A-hat genus]] above, see at _[[elliptic genus]]_ for more. For non-degenerate values one may regard $\epsilon$ and $\delta$ as values of [[modular forms]] of the same name and hence regard all elliptic genera together as one single genus with [[coefficients]] in $MF_\bullet(\Gamma_0(2))$. This "universal" elliptic genus is the _[[Witten genus]]_.

### Witten genus

The  [[Witten genus]]

$$
  w \colon \Omega^{SO}_\bullet \longrightarrow \mathbb{Q}[ [q] ]
$$

is the genus with [[coefficients]] in
the [[power series]] ring $\mathbb{Q}[ [ q ] ]$
with characteristic series given by

$$
  \begin{aligned}
    K_w(z)
    & =
    \frac{z/2}{sinh(z/2)}
    \prod_{n \geq 1}
    \frac{(1-q^n)^2}{(1-q^n e^z)(1-q^n e^{-z})}
    \\
    & =
    \exp\left(
      \sum_{k \geq 2} G_k \frac{z^k}{k!}
    \right)
  \end{aligned}
  \,,
$$

where $G_k$ are the [[Eisenstein series]] ([Ando-Hopkins-Strickland 01](AndoHopkinsStrickland01), [Ando-Hopkins-Rezk 10, prop. 10.9](#AndoHopkinsRezk10)). (Notice that the constant term in $G_k$ is proportional to the $k$th [[Bernoulli number]], so that indeed the exponential expression matches that for the [A-hat genus above](#AhatGenus).)

On [[manifolds]] with [[spin structure]] the Witten genus takes values in $\mathbb{Z}[ [ q ] ]$

On manifolds with rational [[string structure]] it takes values in (the $q$-expansion of) [[modular forms]] for $SL_2(\mathbb{Z})$, meaning that setting $q = e^{2 \pi i \tau}$ then as a function $f$ of the parameter $\tau$ taking values in the [[upper half plane]] the Witten genus satisfies

$$
  f(-1/\tau) = (-\tau)^n f(\tau)
  \,.
$$

Finally on manifolds with actual [[string structure]] it takes values in [[topological modular forms]]. See at _[[Witten genus]]_ for more.

## Non-examples

### Euler characteristic

The [[Euler characteristic]] $X \mapsto \chi(X)$ is close to being  a genus, but is _not_ [[cobordism]] invariant

  (this is the [[index]] of the [[Dirac operator]] $D = d + d^\dagger$)

## Related concepts
 {#RelatedConcepts}

[[!include genera and partition functions - table]]

## References

The abstract concept of genus is due to [[Friedrich Hirzebruch]]. It had evolved out of the older concept of (arithmetic) [[genus of a surface]] via the concept of [[Todd genus]] introduced in

* [[John Arthur Todd]], _The arithmetical invariants of algebraic loci_, Proc. London Math. Soc. (2), Ser. 43, 1937,
190–225.

An review of the history is at the beginning of ([Hirzebruch-Kreck 09](#HirzebruchKreck09))

The theory of multiplicative sequences and characteristic series of genera is due to

* {#Hirzebruch56} [[Friedrich Hirzebruch]], *Neue topologische Methoden in der algebraischen Geometrie*, Ergebnisse der Mathematik und ihrer Grenzgebiete **9**, Springer (1956) [[doi:10.1007/978-3-662-41083-7](https://doi.org/10.1007/978-3-662-41083-7)]

* {#Hirzebruch66} [[Friedrich Hirzebruch]], *Topological methods in algebraic geometry*, Grundlehren **131**, Springer (1966) [[doi:10.1007/978-3-642-62018-8](https://doi.org/10.1007/978-3-642-62018-8), MR0202713 (34 #2573) Zbl 0843.14009]

Early review:

* {#HirzebruchGergerJung92} [[Friedrich Hirzebruch]], Thomas Berger, Rainer Jung, chapter 2 of: *Manifolds and Modular Forms*, Aspects of Mathematics **20**, Viehweg (1992), Springer (1994) [[doi:10.1007/978-3-663-10726-2](https://doi.org/10.1007/978-3-663-10726-2), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/hirzjung.pdf)]

Further review:

* [[Serge Ochanine]], _What is... an elliptic genus_?, Notices of the AMS, volume 56, number 6 (2009) ([pdf](http://www.ams.org/notices/200906/rtx090600720p.pdf))

* {#HirzebruchKreck09} [[Friedrich Hirzebruch]], [[Matthias Kreck]], _On the concept of genus in topology and complex analysis_, Notices of the AMS, volume 56, number 6 (2009) [pdf](http://www.ams.org/notices/200906/rtx090600713p.pdf)

* {#Hopkins94} [[Michael Hopkins]], section 2 of _Topological modular forms, the Witten Genus, and the theorem of the cube_, Proceedings of the International Congress of Mathematics, Zürich 1994 ([pdf](http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0554.0565.ocr.pdf))

* Manifold Atlas, _[Formal group laws and genera](http://www.map.mpim-bonn.mpg.de/Formal_group_laws_and_genera)_

* Wikipedia, _[Genus of a multiplicative series](http://en.wikipedia.org/wiki/Genus_of_a_multiplicative_sequence)_

* [](wwwmath.uni-muenster.de/reine/inst/lueck/lehre/.../summary6.)

Discussion in terms of [[orientations in generalized cohomology]] and specifically for the [[A-hat genus]] and the [[Witten genus]] is in

* {#AndoHopkinsStrickland01} [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], _Elliptic spectra, the Witten genus and the theorem of the cube_, Invent. Math. 146 (2001) 595–687 MR1869850

* {#AndoHopkinsRezk10} [[Matthew Ando]], [[Mike Hopkins]], [[Charles Rezk]], _Multiplicative orientations of KO-theory and the spectrum of topological modular forms_, 2010 ([pdf](http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf))

[[!redirects genera]]

[[!redirects Hirzebruch series]]


genus of a lattice

#Contents#
* table of contents
{:toc}

## Idea

A concept of [[genus]] for a [[lattice in a vector space]].

## References

* [[Jacob Lurie]], definition 7 in _[Tamagawa Numbers via Nonabelian Poincare Duality](http://www.math.harvard.edu/~lurie/282y.html)_, lecture 1 _Tamagawa numbers and Weil's conjectures_ ([pdf](http://www.math.harvard.edu/~lurie/282ynotes/LectureI-Weil.pdf))


genus of a number field

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

#Contents#
* table of contents
{:toc}

## Idea

Under the [[function field analogy]] it makes sense to regard a [[number field]] as the [[rational function|rational functions]] on an "[[arithmetic curve]] over [[F1]]". Accordingly, there is a sensible version of the concept of [[genus of a curve]] for number fields.

This _genus of a number field_ was originally introduced in ([Weil 39](#Weil39)) in a maybe somewhat ad hoc way. It is derived as being that definition which makes the [[Riemann-Roch theorem]] come out right in ([Neukirch 92, chapter III, def. (3.5)](#Neukirch92)).

## Examples

* The genus of the [[rational numbers]] themselves is 0.

## Properties

### Function field analogy

[[!include function field analogy -- table]]

## References

* {#Weil39} [[André Weil]], _Sur l'analogie entre les corps de nombres algébriques et les corps de fonctions algébriques_, Revue Scient. 77, 104-106, 1939

* {#Neukirch92} [[Jürgen Neukirch]], _Algebraische Zahlentheorie_ (1992), English translation _Algebraic Number Theory_,  Grundlehren der Mathematischen Wissenschaften 322, 1999 ([pdf](http://www.plouffe.fr/simon/math/Algebraic%20Number%20Theory.pdf))


genus of a surface

+-- {: .rightHandSide}
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### Context
#### Geometry
+-- {: .hide}
[[!include higher geometry - contents]]
=--
#### Complex geometry
+-- {: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The _genus_ of a [[surface]] (2-[[dimension|dimensional]] [[manifold]]) is an [[integer]] that helps classify the surface's [[homeomorphism]] class.  It gives the number of handles or 'cross caps' that are in a sum decomposition of the surface.

In [[complex analytic geometry]] a surface, being complex 1-dimensional, is called a [[complex curves]] and so one speaks of the [[genus of a curve]]. This holds also for [[algebraic curves]]. Even in [[arithmetic geometry]], there is _[[genus of a number field]]_.

## Properties

### Function field analogy

[[!include function field analogy -- table]]

## Related concepts

* [[genus of a curve]]

* [[arithmetic genus]]

* [[genus]]

* [[Riemann surface]]

* [[curve]], [[singular point of a curve]]

## References

* Wikipedia, _[Genus](http://en.wikipedia.org/wiki/Genus_%28mathematics%29)_

Various approaches to the classification of surfaces are described in introductory texts such as

* [[N. D. Gilbert]] and [[T. Porter]], Knots and Surfaces, Oxford U.P., 1994.

* _[How to see the genus](http://lamington.wordpress.com/2009/09/23/how-to-see-the-genus/)_

[[!redirects genus of a Riemann surface]]


geodesic

+-- {: .rightHandSide}
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### Context
#### Riemannian geometry
+-- {: .hide}
[[!include Riemannian geometry - contents]]
=--
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

On a [[Riemannian manifold]] $(X,g)$, a __geodesic__ (or geodesic line, geodesic path) is a path $x : I \to X$, for some (possibly infinite) [[interval]] $I$, which locally minimizes distance.

## Definition

One way to define a geodesic is as a [[critical point]] of the [[functional]] of length

$$
  (x : I \to M)
   \mapsto
   \int_I \sqrt{g(\stackrel{\cdot}{x}(t),\stackrel{\cdot}{x}(t))} dt
$$

on the appropriate space of curves $I \to X$, where $g$ is the [[metric]] tensor. This implies that it satisfies the corresponding [[Euler-Lagrange equation]]s, which in this case means that the [[covariant derivative]] (for the [[Levi-Civita connection]])

$$
\nabla_{\stackrel{\cdot}{x}} \stackrel{\cdot}{x} = 0
$$

vanishes.

In local coordinates, with [[Christoffel symbol]]s $\Gamma^i_{jk}$ the Euler-Lagrange equations for geodesics form a system

$$
\frac{d^2 x^i}{dt^2} + \sum_{jk} \Gamma^i_{jk}(x) \frac{d x^j}{dt} \frac{d x^k}{dt} = 0.
$$

So this means that a curve is a _geodesic_ if at every point its [[tangent vector]] is the [[parallel transport]] of the tangent vector at the start point along the curve.

## Minimizing geodesics

A geodesic may not *globally* minimize the distance between its end points.  For instance, on a 2-dimensional [[sphere]], geodesics are [[arcs]] of [[great circle]]s.  Any two distinct non-antipodal points are connected by exactly two such geodesics, one shorter than the other (you can go from Los Angeles to Boston directly across North America, or the long way around the world).

A geodesic which does globally minimize distance between its end points is called a **minimizing geodesic**.  The length of a minimizing geodesic between two points defines a distance function for any Riemannian manifold which makes it into a [[metric space]].

## Related concepts

* [[prime geodesic]]

* [[geodesic completeness]]

* [[totally geodesic submanifold]]

* [[variational calculus]]

## References

* en.wikipedia: [geodesic](http://en.wikipedia.org/wiki/Geodesic)
* Springer [[eom]]: Yu. A. Volkov, [Geodesic line](http://eom.springer.de/G/g044120.htm); Yu. A. Volkov, [Geodesic coordinates](http://eom.springer.de/G/g044060.htm); [Geodesic distance](http://eom.springer.de/G/g044080.htm); V.A. Zalgaller, [Geodesic geometry](http://eom.springer.de/G/g044100.htm); D.V. Anosov, [Geodesic flow](http://eom.springer.de/G/g044090.htm)
* Sh. Kobayashi, K. Nomidzu, _Foundations of differential geometry_, vol 1, 1963, vol 2, 1969, Wiley Interscience; reedition 1996 in series Wiley Classics; Russian ed.: Nauka, Moscow 1981.
* Arthur L. Besse, _Einstein manifolds_, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 10, Springer-Verlag 1987, xii + 510 pp. (for a review see [Bull. AMS](http://projecteuclid.org/euclid.bams/1183554925) and [MR88f:53087](ttp://www.ams.org/mathscinet-getitem?mr=88f:53087)); reprinted 2008, Springer Classics in Math.

[[!redirects geodesic line]]
[[!redirects geodesic path]]
[[!redirects geodesics]]

[[!redirects geodesic equation]]
[[!redirects geodesic equations]]


geodesic completeness

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

#Contents#
* table of contents
{:toc}

## Idea

A ([[pseudo-Riemannian manifold|pseudo-]])[[Riemannian manifold]] $(X,g)$ is called _geodesically complete_ (or just _complete_, for short) if each of its [[geodesics]] extends indefinitely, hence if the geodesic [[exponential map]] at every point $x \in X$ is defined on the full [[tangent space]] at that point, $\exp \colon T_x X \longrightarrow X$.

## Examples

The [[Euclidean spaces]] $\mathbb{R}^n$ and the ([[round sphere|round]] or [[squashed sphere|squashed]]) [[n-spheres]] $S^n$ are geodesically complete. But any [[open ball]] of [[finite number|finite]] [[radius]] inside $\mathbb{R}^n$ is not.

## Related concepts

* [[Clifford-Klein space form]]

  * [[spherical space form]]

* [[positive quaternion-Kähler manifold]]

## References

See also

* Wikipedia, _[Riemannian manifold -- Geodesic completeness](https://en.wikipedia.org/wiki/Riemannian_manifold#Geodesic_completeness)_

[[!redirects geodesically complete Riemannian manifold]]
[[!redirects geodesically complete Riemannian manifolds]]

[[!redirects geodesically complete]]

[[!redirects complete Riemannian manifold]]
[[!redirects complete Riemannian manifolds]]


geodesic convexity

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

# Contents
* table of contents
{: toc}

## Idea

The definition of _geodesic convexity_ is like that of _[[convexity]]_, but with straight lines in an [[affine space]] generalized to [[geodesics]] in a [[Riemannian manifold]] or [[metric space]].

## Definition

### In a Riemannian manifold

Let $(X,g)$ be a [[Riemannian manifold]] and $C \subset X$ a [[subset]]. We say that $C$ is

* **weakly geodesically convex** if for any two points from $C$ there exists exactly one minimizing [[geodesic]] in $C$ connecting them;

* **geodesically convex** if for any two points in $C$ there exists exactly one minimizing geodesic in $X$ connecting them, and that geodesic arc lies completely in $C$

* **strongly geodesically convex** if for any two points in $\overline{C}$ there exists exactly one minimizing geodesic in $X$ connecting them, and that geodesic arc lies completely in $C$, except possibly the endpoints; and furthermore there exists no nonminimizing geodesic inside $C$ connecting the two points.

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

The **convexity radius** at a point $p \in X$ is the [[supremum]] (which may be $+ \infty$) of $r \in \mathbb{R}$ such that for all $\eta \lt r$ the geodesic ball $B_p(r)$ is strongly geodesically convex.

The convexity radius of $(X,g)$ is the [[infimum]] over the points $p \in X$ of the convexity radii at these points.

=--

### In a metric space

For $X$ a [[metric space]], a *distance-preserving path* is a function $x \colon [a,b]\to X$ which is an [[isometry]].  This is a metric-space analogue of an "arc-length-parametrized minimizing geodesic" on a Riemannian manifold.  In particular, the existence of such a path implies that $d(x(a),x(b)) = b-a$.  We then say that $X$ is **geodesic** (or *geodesically convex*) if any two points can be connected by a distance-preserving path.

We say that $X$ is a **length space** if for any $x$ and $y$, the distance $d(x,y)$ is the [[infimum]] of the lengths of all continuous paths from $x$ to $y$.  The [[Hopf-Rinow theorem]] says that for a metric space in which every [[closed subset|closed]] [[bounded subset|bounded]] subset is [[compact space|compact]], being a length space is equivalent to being geodesic.

## Properties

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

At any point $p \in X$ of a Riemannian manifold, the convexity radius is positive.

=--

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

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

If $X$ is a [[compact space]] then the convexity radius of $(X,g)$ is positive.

=--

This is reproduced for instance as proposition 95 in ([Berger](#Berger))

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

Every [[paracompact manifold]] admits a complete [[Riemannian metric]] with bounded absolute sectional curvature and positive injectivity radius.

=--

This is shown in ([Greene](#Greene)).

## References

Original literature includes

* J. H. C. Whitehead, _Convex regions in the geometry of paths_ Quart. J. Math. 3, 33--42 (1932).
{#Whitehead}

* R. Greene, _Complete metrics of bounded curvature on noncompact manifolds_  Archiv der Mathematik Volume 31, Number 1 (1978)
{#Greene}

A review of geodesic convexity in Riemannian manifolds is in

* Isaac Chavel, _Riemannian geometry – A modern introduction_ Cambridge University Press (1993)
{#Chavel}

* Marcel Berger, _A panoramic view of Riemannian geometry_
{#Berger}

A categorical perspective on geodesic convexity for metric spaces can be found in

* _Taking categories seriously_, Reprints in Theory and Applications of Categories, No. 8, 2005, pp. 1–24. ([pdf](http://www.emis.de/journals/TAC/reprints/articles/8/tr8.pdf))

Much of the issues on geodesic convexity becomes more complicated when trying to generalize from Riemannian to [[Lorentzian manifolds]], as discussed at length at

* John K. Beem, Paul E. Ehrlich, Kevin L. Easley, _Global Lorentzian geometry_, Marcel Dekker, 1996, 635 pages
* John K. Beem, _Lorentzian geometry in the large_, Math. of gravitation I, Lorentzian geometry and Einstein equations, Banach Center Publications __41__, Inst. of Math. Polish Acad. of Sci. Warszawa 1997 [pdf](http://matwbn.icm.edu.pl/ksiazki/bcp/bcp41/bcp4111.pdf)

In particular, the conclusions of the Hopf-Rinow Theorem fail to hold for complete Lorentzian manifolds.

[[!redirects geodesically convex]]
[[!redirects strongly geodesically convex]]
[[!redirects geodesically convex subset]]
[[!redirects strongly geodesically convex subset]]
[[!redirects geodesically convex set]]
[[!redirects strongly geodesically convex set]]

[[!redirects convexity radius]]

[[!redirects length space]]
[[!redirects geodesic metric space]]
[[!redirects geodesic space]]
[[!redirects length spaces]]
[[!redirects geodesic metric spaces]]
[[!redirects geodesic spaces]]


geodesic flow

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $(X,g)$ a [[Riemannian manifold]] and $p \in X$ a point, the **geodesic flow** at $p$ is the map defined on an open neighbourhood of the origin in $(T_p X ) \times \mathbb{R}$ that sends $(v,r)$ to the endpoint of the [[geodesic]] that starts with tangent vector $v$ at $p$ and has length $r$.

(...)

## Definition

Let $(X,g)$ be a [[Riemannian manifold]]

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

(...) geodesic flow (...)

=--

The following are some auxiliary definitions that serve to analyse properties of geodesic flow (see [Properties](#Properties)).

For $p \in X$ a point and $r \in \mathbb{R}$ a positive real number, we write

$$
  B_p(r) = \{x \in X | d(p,x) \lt r\}  = \{ \exp( v) : T_p X \to X | |v| \lt r \} \subset X
$$

for set of points which are of [[distance]] less than $r$ away from $p$. As the propositions [below](#Properties) assert, for small enough $r$ this is [[diffeomorphic]] to an [[open ball]] and we speak of _metric balls_ or _geodesic balls_ .

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

For $p \in P$ a point, the **injectivity radius** $inj_p \in \mathbb{R}$ is the [[supremum]] over all values of $r \in \mathbb{R}$ such that the geodesic flow starting at $p$ with radius $r$ $\exp(-) : B_r(T_p X) \to X$ is a [[diffeomorphism]] onto its image.

The injectivity radius of $(X,g)$ is the infimum of the injectivity radii at each point.

=--

## Properties {#Properties}

### Properties of the injectivity radius

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

The injectivity radius is

* either equal to half the length of the smalled periodic geodesic,

* or equal to the smallest distance between two conjugate points.

=--

This appears for instance as scholium 91 in ([Berger](#Berger)).

### Lower bounds on the injectivity radius

There are several lower boundas on the [injectivity radius](#InjectivityRadius) of a Riemannian manifold.

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

The [[convexity radius]] is always less than or equal to half of the injectivity radius:

$$
  conv (X,g) \leq \frac{1}{2} inj(X,g)
  \,.
$$

=--

This appears for instance as proposition IX.6.1 in [Chavel](#Chavel), where it is attributed to M. Berger (1976). In ([Berger](#Berger)) it is proposition 95.

Let $R$ be the [[Riemann curvature]] tensor of $g$. For $p \in X$ the [[sectional curvature]] of a plane spanned by vectors $v,w \in T_p X$ is

$$
  K(v,w) :=
  \frac{R(v,w,v,w)}{g(v,v)g(w,w) - g(v,w)^2}
  \,.
$$

Say that $(X,g)$ is **complete** if, equivalently,

* with the [[distance]] function $X$ is a [[complete metric space]];

* $(X,g)$ is geodesically complete in that for all $v \in T_p X$ the flow $t \mapsto \exp_p(t v)$ exists for all $t \in \mathbb{R}$.

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

Let $(X,g)$ be complete and such that

1. the [[absolute value]] of the sectional curvature at all points is bounded from above;

1. the [[volume]] of the geodesic unit ball at all points is bounded from below.

Then the injectivity radius is positive.

=--

This is due to ([CheegerGromovTaylor](#CheegerGromovTaylor)). A survey is in ([Grant](#Grant)).

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

Every [[paracompact manifold]] admits a complete [[Riemannian metric]] with

* bounded absolute sectional curvature;

* positive convexity radius

* and hence with positive injectivity radius.

=--

This is shown in ([Greene](#Greene)).

## References

* Gabriel Paternein, _Geodesic flows_ Birkhäuser (1999)

The following is literature on injectivity radius estimates

A general exposition is in sectin 6 "Injectivity, Convexity radius and cut locuss" of

* Marcel Berger, _A panoramic view of Riemannian geometry_
{#Berger}

Also section IX of

* Isaac Chavel, _Riemannian geometry: a modern introduction_
{#Chavel}

A survey of the main estimates is in

* James Grant, _Injectivity radius estimates_ ([pdf](http://www.mat.univie.ac.at/~grant/papers/talk.pdf))
{#Grant}

The main theorem is due to

* [[Jeff Cheeger]], M. Gromov, and M. Taylor, _Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds_ , J. Differential Geom., 17 (1982), pp. 15–53.
{#CheegerGromovTaylor}

Older results on compact manifolds are in

* [[Jeff Cheeger]], _Finiteness theorems for Riemannian manifolds_ .

The existence of metrics with all the required propertiers for the injectivity estimates (completeness, bounded absolute sectional curvature) on paracompact manifolds is shown in

* R. Greene, _Complete metrics of bounded curvature on noncompact manifolds_  Archiv der Mathematik Volume 31, Number 1 (1978)
{#Greene}

More discussion of construction of Riemannian manifolds with bounds on curvature and volume is in

* John Lott, Zhongmin Chen, _Manifolds with quadratic curvature decay and slow volume growth_ ([pdf](http://math.berkeley.edu/~lott/science.pdf))

Analogous results for [[Lorentzian manifold]]s are discussed in

* Bing-Long Chen, Philippe G. LeFloch, _Injectivity Radius of Lorentzian Manifolds_ ([pdf](http://philippelefloch.files.wordpress.com/2009/12/2008-chenlefloch-cmp.pdf))

[[!redirects injectivity radius]]


geoemtric infinity-groupoid

## Idea

One might say **geometric [[∞-groupoid]]** or **geoemtric [[∞-stack]]** for an analog of the notion of [[geometric stack]] generalized from [[(2,1)-toposes]] to [[(∞,1)-toposes]].

## Related concepts

* [[étale ∞-groupoid]]

[[!redirects geometric infinity-groupoids]]

[[!redirects geometric ∞-groupoid]]
[[!redirects geometric ∞-groupoids]]


Geoff Cruttwell

Geoff Cruttwell is an assistant professor in the Department of Mathematics and Computer Science at Mount Allison University (and an adjunt Professor at the Dalhousie university mathematics department).

* [institutional homepage](https://www.mta.ca/gcruttwell/)

* [homepage](http://www.reluctantm.com/gcruttw/)

* [publication list](http://www.reluctantm.com/gcruttw/publications.html)

##Selected Publications

* {#Cruttwell}[[Geoff Cruttwell]], _Cartesian differential categories revisited,_ Mathematical Structures in Computer Science, Vol. 27 (1), pg. 70-91 (first published online in 2015)

category: people
[[!redirects G. Cruttwell]]
[[!redirects G.S.H. Cruttwell]]
[[!redirects Geoffrey Cruttwell]]


Geoff Cruttwell > history

[[!redirects Geoff Cruttwell>history]]
[[!redirects Geoff Cruttwell>history]]


Geoffrey Chew

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

## Writings

* {#QuarkOrBootstrap} _Quark or Bootstrap: Triumph or Frustration for Hadron Physics_, Physics Today, May 1970 ([web](https://pubarchive.lbl.gov/islandora/object/ir%3A144169/datastream/PDF/view), [[Chew70.pdf:file]])

## Related $n$Lab entries

* [[S-matrix]]

category: people


Geoffrey Mason

* [institute page](https://www.math.ucsc.edu/people/emeriti.php?uid=gem)

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=11734)

## Selected writings

On classification of strongly rational holomorphic [[vertex operator algebras]] of [[central charge]] $\leq 24$ (of relevance in [[heterotic string theory]] and [[monstrous moonshine]]):

* [[Chongying Dong]], [[Geoffrey Mason]], *Holomorphic Vertex Operator Algebras of Small Central Charges*, Pacific Journal of Mathematics **213** 2 (2004) 253–266 [[arXiv:math/0203005](https://arxiv.org/abs/math/0203005), [doi:10.2140/pjm.2004.213.253](http://dx.doi.org/10.2140/pjm.2004.213.253)]

category: people


Geoffrey Penington

* [Home page](https://physics.berkeley.edu/people/faculty/geoff-penington)

## Selected writings

On [[tensor networks]] in [[holographic entanglement entropy]]:

* Ning Bao, [[Geoffrey Penington]], Jonathan Sorce, Aron C. Wall, _Beyond Toy Models: Distilling Tensor Networks in Full AdS/CFT_ ([arXiv:1812.01171](https://arxiv.org/abs/1812.01171))

* Ning Bao, [[Geoffrey Penington]], Jonathan Sorce, Aron C. Wall, _Holographic Tensor Networks in Full AdS/CFT_ ([arXiv:1902.10157](https://arxiv.org/abs/1902.10157))

* Helia Kamal, [[Geoffrey Penington]], _The Ryu-Takayanagi Formula from Quantum Error Correction: An Algebraic Treatment of the Boundary CFT_ ([arXiv:1912.02240](https://arxiv.org/abs/1912.02240))

Claim that the proper application of [[holographic entanglement entropy]] to the discussion of [[Bekenstein-Hawking entropy]] resolves the apparent [[black hole information paradox]]:

* [[Geoff Penington]], _Entanglement Wedge Reconstruction and the Information Paradox_ ([arXiv:1905.08255](https://arxiv.org/abs/1905.08255))

* [[Geoff Penington]], [[Stephen Shenker]], [[Douglas Stanford]], [[Zhenbin Yang]], _Replica wormholes and the black hole interior_ ([arXiv:1911.11977](https://arxiv.org/abs/1911.11977))

On [[holographic entanglement entropy]] and [[quantum error correcting codes]]:

* [[Chris Akers]], [[Geoff Penington]], *Quantum minimal surfaces from quantum error correction* ([arXiv:2109.14618](https://arxiv.org/abs/2109.14618))

On [[quantum error correcting codes]] and [[black hole entropy]]:

* [[Chris Akers]], [[Netta Engelhardt]], [[Daniel Harlow]], [[Geoff Penington]], [[Shreya Vardhan]], *The black hole interior from non-isometric codes and complexity* [[arXiv:2207.06536](https://arxiv.org/abs/2207.06536)]

On [[von Neumann algebra factors]] [[algebra of observables|of observables]] for [[quantum field theory on curved spacetime|quantum field theory on]] [[de Sitter spacetime]]:

* [[Venkatesa Chandrasekaran]], [[Roberto Longo]], [[Geoff Penington]], [[Edward Witten]], *An Algebra of Observables for de Sitter Space*, Journal of High Energy Physics **2023**  82 (2023) [[arXiv:2206.10780](https://arxiv.org/abs/2206.10780), doi:10.1007/JHEP02(2023)082]

category: people

[[!redirects Geoff Penington]]


Geoffroy Horel

* [webpage](http://geoffroy.horel.org)

## Selected writings

On the [[topological Hochschild homology]] of the [[Lubin-Tate theory|Lubin-Tate ring spectrum]] via [[factorization homology]]:

* [[Geoffroy Horel]], *Higher Hochschild homology of the Lubin-Tate ring spectrum*, Algebr. Geom. Topol. **15** (2015) 3215-3252 [[arXiv:1311.2805](https://arxiv.org/abs/1311.2805), [doi:10.2140/agt.2015.15.3215](https://doi.org/10.2140/agt.2015.15.3215)]

A [[model category]]-[[structure]] on the category of [[simplicial categories]] (in the sense of [[simplicial objects in Cat]]) presenting [[(infinity,1)-categories|$(\infty,1)$-categories]]:

* {#Horel15} [[Geoffroy Horel]], *A model structure on internal categories*, Theory and Applications of Categories, **30** 20 (2015) 704-750 [[tac:30-20](http://www.tac.mta.ca/tac/volumes/30/20/30-20abs.html), [arXiv:1403.6873](http://arxiv.org/abs/1403.6873)]

## Related entries

* [[factorization homology]]

category: people


geogebra

__GeoGebra__ is a (Java-based) free software for graphing, algebra, geometry and statistics suited for educational use. Graphs in GeoGebra may have adjustable parameters, for example points of geometrical figures which can be changed with mouse, and the dependent parts of the graphs will change along (for example, in an image of a triangle with its orthocenter constructed we can change the vertices of the triangle and the orthocenter will change along in real time).

GeoGebra appears in desktop versions in several major operative systems (including Linux) and in the online web version which can be directly used online. The web version is hosted along with a large database of shared files which implement many tasks or educational demonstrations in GeoGebra. The desktop version has the option to export the graphs to various portable formats, e.g. SVG. Of course, the exported images are static (no graph parameters can be changed).

* [geogebra.org](http://www.geogebra.org/)
* Wikipedia: [geogebra](https://en.wikipedia.org/wiki/GeoGebra)

category: education, software


Geometric Algebra

> This page is about the book by Artin. For discussion of [[Clifford algebra]] see at _[[geometric algebra]]_.

***

* [[Emil Artin]]:

  \linebreak

  **Geometric Algebra**

  \linebreak

  Wiley 1957 (1988)

  \linebreak

  [ISBN:978-1-118-16454-9](https://www.wiley.com/en-ae/Geometric+Algebra-p-9781118164549)

  Wikipedia entry

  [ark:/13960/t4nk37034](https://archive.org/details/geometricalgebra033556mbp/page/n5/mode/2upa)

  [MAA review](https://maa.org/press/maa-reviews/geometric-algebra)

on [[classical Lie groups]] in a [[Klein geometry]] context.

## Contents

### Chapter I -- Preliminary notions

* [[set theory]]

* [[vector spaces]]

* [[dual linear space]]

* [[bilinear form]]

* [[linear equations]]

* [[group]], [[group theory]]

* [[field]], [[ordered field]]

### Chapter II -- Affine and Projective Geometry

* [[projective geometry]]

* [[dilatation operator]]

* [[Desargues' theorem]]

### Chapter III -- Symplectic and Orthogonal Geometry

* [[inner product]]

* [[symplectic form]]

### Chapter IV -- The General Linear Group

* [[general linear group]]

### Chapter V -- The Structure of Symplectic and Orthogonal Groups

* [[symplectic group]]

* [[orthogonal group]]

* [[Clifford algebra]]

category: reference


geometric algebra

> This page is about the [[associative unital algebra]] and [[N-graded module|$\mathbb{N}$-graded module]] that is isomorphic to a [[Clifford algebra]]. For the book by [[Emil Artin]] see instead at _[[Geometric Algebra]]_.

***

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Geometry
+-- {: .hide}
[[!include higher geometry - contents]]
=--
#### Super-Algebra and Super-Geometry
+-- {: .hide}
[[!include supergeometry - contents]]
=--
#### Spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

This definition is a modified version of the one given in [Aragón, Aragón, Rodríguez (1997)](#AragonEtAl97)

Given a [[commutative ring]] $R$, a **geometric algebra** is an [[N-graded module|$\mathbb{N}$-graded]] $R$-[[module]] $A$ with an [[associative unital algebra|associative unital]] [[bilinear function]] $(-)(-):A \times A \to A$ and a ring [[isomorphism]] $j:\langle A \rangle_0 \cong R$ such that

* for natural numbers $m:\mathbb{N}$ and $n:\mathbb{N}$, the product of every $m$-multivector and $n$-multivector is an $(m+n)$-multivector: for all $a \in \vert A \vert_m$ and $b \in \vert A \vert_n$, there exists $c \in \vert A \vert_{m+n}$ such that $a b = c$

* the product of every $1$-vector with itself is a $0$-vector: for all $a \in \langle A \rangle_1$ there exists $c \in \langle A \rangle_0$ such that $a^2 = c$.

The $0$-vectors are typically called **scalars** and $1$-vectors are just called **vectors**. An **$n$-blade** is an $n$-vector which could be written as a product of $n$ $1$-vectors.

## Relation to Clifford algebras

The [[categories]] of $R$-geometric algebras and $R$-[[Clifford algebras]] are [[equivalence of categories|equivalent]], similar to how the categories of [[abelian groups]] and $\mathbb{Z}$-[[modules]] are equivalent. As a result, geometric algebras and Clifford algebras are isomorphic algebras, and the two terms are interchangeable with each other. In fact, Clifford originally named Clifford algebras "geometric algebras", and it was later mathematicians who started using the term "Clifford algebra" for this object.

### In physics

The use of the term "geometric algebra" rather than "Clifford algebra" for this object continues in physics in a school of thought following [Hestenes 66](#Hestenes66) that emphasizes the usefulness of making this abstract algebra explicit in the exposition of geometry and physics, such as on 2- and 3-[[dimensional]] [[Cartesian spaces]] and on [[Minkowski spacetime]], and in physics, "geometric algebra" also refers to this school of thought.

In physics, geometric algebra may be seen as a third style of exposition and notation, in between 1) the traditional physics style of regarding the Clifford algebra in terms of [[matrix]] [[representations]], and 2) the traditional mathematics style of defining them as [[quotients]] of [[tensor algebras]] by ideals defined by [[quadratic forms]]. The idea is that while the former approach (1) suffers from its basis-dependency, the latter approach (2) tends to make the subject look more complicated to the novice and working physicist than it really is.

In textbooks on geometric algebra, the algebra is instead introduced without choosing matrix representations, but highlighting the obvious [[generators and relations]]-presentation over the definition via [[quotients]] of [[tensor algebras]]. (The general theory of Clifford algebras, including core results such as [[Bott periodicity]], is typically not mentioned in textbooks on geometric algebra). While the difference may seem inessential to the trained mathematician, it turns out that this perspectives helps open the subject to many working physicists. The enthusiasm about this perspective that [[David Hestenes]] reports to have experienced when he understood Clifford algebra in physics this way, back as a student, is what made geometric algebra become the school of thought that is today.

One point being made here is that the traditional physics textbook emphasis on the special geometry of the [[Cartesian space]] $\mathbb{R}^3$ with its exceptional [[vector product]] structure is outdated and contra-productive and can be replaced by a more elegant and more universal description in terms of [[bivector]] calculus (which canonically embeds into the [[Clifford algebra]]). In this respect geometric algebra may be thought of as a streamlined exposition of the geometry of [[rotation]] and [[spin]].

## Related concepts

* [[spin geometry]],

  [[spinor]], [[spin representation]]

* [[Dirac equation]]

* [[N-graded module]]

* [[non-associative geometric algebra]]

## References

On the definition of geometric algebras

* {#AragonEtAl97} G. Aragón, J.L. Aragón, M.A. Rodríguez (1997), Clifford Algebras and Geometric Algebra, _Advances in Applied Clifford Algebras_ Vol. 7 No. 2, pg 91–102, doi:10.1007/BF03041220, S2CID:120860757

The original account:

* {#Hestenes66}[[David Hestenes]],  _Space-time Algebra_. New York 1966, reprinted at Birkhäuser 2015 ([doi:10.1007/978-3-319-18413-5](https://www.springer.com/de/book/9783319184128))

Other accounts:

* {#Doran94} [[Chris Doran]], _Geometric Algebra and its Application to Mathematical Physics_, 1994 ([pdf](http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/DoranThesis.pdf))

* {#Sobczyk13} Garrett Sobczyk, _New Foundations in Mathematics: The Geometric Concept of Number_. New York (2013)

A textbook on [[mechanics]] written in this style is

* [[Chris Doran]], Anthony Lasenby, _Geometric algebra for physicists_, Cambridge University Press (2003) ([pdf](http://catdir.loc.gov/catdir/samples/cam033/2002035182.pdf))

See also

* Wikipedia, _[Geometric algebra](https://en.wikipedia.org/wiki/Geometric_algebra)_

[[!redirects geometric algebras]]


geometric analysis

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#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[analysis]]

* [[differential equations]]

* [[differential geometry]]

* [[differential topology]]

## References

* [[Sigurdur Helgason]], *Geometric Analysis on Symmetric Spaces*,  Mathematical Surveys and Monographs **39** (1994) [[doi:10.1090/surv/039](https://doi.org/10.1090/surv/039)]

* [[Jürgen Jost]], *Riemannian Geometry and Geometric Analysis*, Springer (2017) [[doi:10.1007/978-3-319-61860-9](https://doi.org/10.1007/978-3-319-61860-9)]

See also:

* Wikipedia, *[Geometric analysis](https://en.wikipedia.org/wiki/Geometric_analysis)*


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This entry provides hyperlinked keywords and further pointers to the literature for the articles

* [[Hisham Sati]], _Geometric and topological structures related to M-branes_ ,

  part I, Proc. Symp. Pure Math. 81 (2010), 181-236 ([arXiv:1001.5020](http://arXiv.org/abs/1001.5020)),

  part _II: Twisted $String$ and $String^c$ structures_, J. Australian Math. Soc. 90 (2011), 93-108 ([arXiv:1007.5419](http://arxiv.org/abs/1007.5419));

  part _III: Twisted higher structures_, Int. J. Geom. Meth. Mod. Phys. 8 (2011), 1097-1116 ([arXiv:1008.1755](http://arxiv.org/abs/1008.1755))

on phenomena of [[higher geometry]] and [[generalized cohomology]] encountered in [[string theory]] and specifically when going towards [[M-theory]].

Apart from original work this provides an exhaustive bibliography of the relevant existing literature, which we reproduce hyperlinked [below](#BibliographyI).

#Contents#
* table of contents
{:toc}

## Abstract

> We consider the topological and geometric structures associated with cohomological and homological objects in [[string theory|M-theory]]. For the latter, we have M2-branes and M5-branes, the analysis of which requires the underlying [[spacetime]] to admit a [[String structure]] and a [[Fivebrane structure]], respectively. For the former, we study how the fields in M-theory are associated with the above structures, with [[L-infinity algebra|homotopy algebras]], with [[twisted cohomology]], and with [[cohomology|generalized cohomology]]. We also explain how the corresponding charges should take values in [[topological modular form]]s. We survey background material and related results in the process.

## Higher cohomological charges
 {#HigherCohomologicalCharges}

Discussion of [[elliptic cohomology]] and [[Morava K-theory]] as the home for higher analogs of [[D-brane charges]] ([[M5-brane charge]], [[M9-brane]] charge...) and the corresponding [[orientation in generalized cohomology]] as higher [[quantum anomaly]] conditions (such as the [[Diaconescu-Moore-Witten anomaly]]):

* [[Hisham Sati]], [[Craig Westerland]], _Twisted Morava K-theory and E-theory_ ([arXiv:1109.3867](http://arxiv.org/abs/1109.3867))

* [[Igor Kriz]], [[Hisham Sati]], _M-theory, type IIA superstrings, and elliptic cohomology_, Adv. Theor.
Math. Phys. 8 (2004), no. 2, 345–394 ([arXiv:hep-th/0404013](http://arxiv.org/abs/hep-th/0404013))
 {#SatiKriz04}

* [[Lukas Buhné]], _[[Properties of Integral Morava K-Theory and the Asserted Application to the Diaconescu-Moore-Witten Anomaly]]_, Diploma thesis Hamburg (2011)
 {#Buhne11}

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{7twist} [[Hisham Sati]], _A higher twist in string theory_,
J. Geom. Phys. __59__ (2009) 369-373, [hep-th/0701232](http://arXiv.org/abs/hep-th/0701232).

{KSpin} H. Sati, _An approach to anomalies in M-theory via $KSpin$_,
J. Geom. Phys. __58__ (2008) 387-401,
[arXiv:hep-th/0705.3484](http://arxiv.org/abs/0705.3484).

{MO8} H. Sati, _On anomalies of $E_8$ gauge theory on String manifolds_,
[arXiv:0807.4940](http://arxiv.org/abs/0807.4940).

{SSS1} H. Sati, U. Schreiber and J. Stasheff, _$L_\infty$-connections and applications to String- and Chern-Simons $n$-transport_, in _Recent Developments in QFT_, eds. B. Fauser et al., Birkh{\"a}user, Basel (2008), [arXiv:0801.3480] (http://arXiv.org/abs/math.DG/0801.3480).

{SSS2} [[Hisham Sati]], [[Urs Schreiber]], and [[Jim Stasheff]],
_Fivebrane structures_, to appear in Rev. Math. Phys.
[arXiv:math.AT/0805.0564](http://arXiv.org/abs/0805.0564).

{SSS3} [[Hisham Sati]], [[Urs Schreiber]], and [[Jim Stasheff]], _Twisted differential String- and Fivebrane structures_, [arXiv:math.AT/0910.4001](http://arXiv.org/abs/0910.4001).

{SW} N. Seiberg and [[Edward Witten]], _Spin structures in string theory_,
Nucl. Phys. __B276__ (1986) 272-290.

{SWa} A. N. Schellekens and N.P. Warner, _Anomalies and modular invariance in string theory_, Phys. Lett. __B 177__ (1986) 317--323.

{SP} C. J. Schommer-Pries, _A finite-dimensional String 2-group_,
preprint 2009.

{Sch} J. H. Schwarz, _The power of M-theory_, Phys. Lett. __B367__ (1996) 97-103, [hep-th/9510086](http://arXiv.org/abs/hep-th/9510086).

{Lec} J. H. Schwarz, _Lectures on superstring and M-theory dualities_,
Nucl. Phys. Proc. Suppl. __55B__ (1997) 1-32,
[hep-th/9607201](http://arXiv.org/abs/hep-th/9607201).

{SVW} S. Sethi, C. Vafa, and E. Witten, _Constraints on low-dimensional string compactifications_,
Nucl. Phys. __B480__ (1996) 213-224,
[hep-th/9606122](http://arXiv.org/abs/hep-th/9606122).

{ST1} S. Stolz and P. Teichner, _What is an elliptic object?_, in Topology, geometry and quantum field theory, 247--343, Cambridge Univ. Press 2004.

{ST2} [[Stefan Stolz]] and [[Peter Teichner]], _Super symmetric field theories and integral modular functions_, [http://math.berkeley.edu/~teichner/Papers/SusyQFT.pdf](http://math.berkeley.edu/~teichner/Papers/SusyQFT.pdf)

{Stg} R. E. Stong, _Appendix: calculation of $\Omega^{Spin}_{11}(K(Z,4))$_, in Workshop on unified string theories (Santa Barbara, Calif., 1985), 430--437, World Sci. Publishing, Singapore, 1986.

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category: reference


geometric category

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#### Regular and Exact categories
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#### Category Theory
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# Contents
* table of contents
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## Definition

An **infinitary [[coherent category]]** or **geometric category** is a [[regular category]] in which the [[subobject]] [[poset]]s $Sub(X)$ have all _small_ [[union]]s which are [[pullback stability|stable under pullback]].

More generally, for $\kappa$ a [[regular cardinal]], a _$\kappa$-geometric category_, or _$\kappa$-coherent category_, is a [[regular category]] with unions for $\kappa$-small families of [[subobjects]], stable under pullback. (For $\kappa = \omega$ this reduces to the notion of _[[coherent category]]_, called a _pre-logos_ by Freyd--Scedrov.)

[Makkai-Reyes](#MakkaiReyes) call this a _$\kappa$-[[logical category]]_, while [Shulman](#Shulman) calls it a _$\kappa$-[[k-ary regular category|ary regular category]]_.  See also ([Butz-Johnstone, p. 12](#ButzJohnstone)).

## Properties

See [[familial regularity and exactness]] for a general description of the spectrum from regular categories through finitary and infinitary coherent categories.

### Well-poweredness

Frequently, geometric categories are additionally required to be [[well-powered category|well-powered]].  If a geometric category is well-powered, then its subobject posets are [[complete lattices]], hence also have all intersections.  Moreover, by the [[adjoint functor theorem]] for posets, it is a [[Heyting category]].

However, since [[geometric logic]] does not include [[implication]] or infinite [[conjunction]], this categorical structure should not necessarily be expected to exist in a category called "geometric" (and when they do exist, they are not preserved by [[geometric functors]]).  A requirement of well-poweredness is also inconsistent with the spectrum of [[familial regularity and exactness]].

Note, however, that if a geometric category has a small [[generating set]], then it is necessarily well-powered.  In particular, this applies to the [[syntactic category]] of any (small) [[geometric theory]], and also to any [[Grothendieck topos]].

## Related concepts

[[!include categories and logic - table]]

## References

Around lemma A1.4.18 in

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_

* [[Michael Makkai]], [[Gonzalo Reyes]], _First Order Categorical Logic_, Lecture Notes in Math. 611 (Springer-Verlag, 1977).
 {#MakkaiReyes}

* Casten Butz, [[Peter Johnstone]], _Classifying toposes for first order theories_, BRICS Report Series RS-97-20
 {#ButzJohnstone}

* [[Mike Shulman]], _Exact completions and small sheaves_, TAC 27(7), 2012 ([link](http://www.tac.mta.ca/tac/volumes/27/7/27-07abs.html)).
  {#Shulman}

[[!redirects geometric category]]
[[!redirects geometric categories]]
[[!redirects infinitary coherent category]]
[[!redirects infinitary coherent categories]]
[[!redirects infinitary-coherent category]]
[[!redirects infinitary-coherent categories]]

[[!redirects logical category]]
[[!redirects logical categories]]


geometric class field theory

#Contents#
* table of contents
{:toc}

## Idea

In the spirit of the [[function field analogy]], _geometric class field theory_ is analogous to [[class field theory]], but where the latter studies finite abelian extensions of [[global fields]], geometric class field theory studies finite [[covering spaces]] of suitable [[algebraic curves]] over any constant [[perfect field]] $k$, not necessarily a [[finite field]] and possibly of [[characteristic zero]]. In particular $k$ may be the [[complex numbers]] $\mathbb{C}$, in which case the theory is about covering of [[complex|curves]].

As such, geometric class field theory has become part of the [[geometric Langlands program]] and of higher dimensional class field theory.

## References

A brief survey of classical results is in

* [[Brian Conrad]], _Geometric global class field theory_ ([pdf](https://math.stanford.edu/%7Econrad/249BW09Page/handouts/geomcft.pdf))

A fairly comprehensive review of the theory is in the thesis

* {#Toth11} Peter Toth, _Geometric abelian class field theory_, 2011 ([web](http://dspace.library.uu.nl/handle/1874/206061))

Discussion in the context of the [[geometric Langlands correspondence]] includes

* {#BenZvi02} [[David Ben-Zvi]], _Geometric class field theory_, 2002 ([MSRI notes](http://www.msri.org/realvideo/ln/msri/2002/introstacks/ben-zvi/1/index.html))

See also

* [MO discussion](http://mathoverflow.net/q/73054/381)

Higher dimensional class field theory, using the [[Chow group with modulus]], is developed in

* [[Moritz Kerz]], [[Shuji Saito]], _Chow group of 0-cycles with modulus and higher dimensional class field theory_, [arXiv:1304.4400](http://arxiv.org/abs/1304.4400).

which is briefly summarized in

* _Chow group of 0-cycles with modulus and higher dimensional class field theory_, [pdf](http://preschema.com/documents/chowmodcft.pdf).


geometric computad

[[!redirects geometric computad]]

# Contents
* table of contents
{: toc}

## Idea

Geometric $n$-computads are [[free object|free]] [[geometric n-category| geometric n-categories]], whose notion of composition is modelled on [[manifold diagram|manifold diagrams]]. Geometric $n$-computads are a model for a [[semistrict n-category|semistrict]] flavor of [[n-categories]].

## Definition

\begin{terminology} We will say a [[manifold diagram|manifold $k$-diagram]] $(\mathbb{R}^k,f)$ is a *stratum $k$-type* if the diagram is of the form $(\mathrm{Cone}(S^{k-1}), \mathrm{cone}(\partial f))$. We call $(S^{k-1},\partial f)$ the *type boundary* of $f$.
\end{terminology}

Let $\mathcal{E} : \mathbf{Strat} \to \mathbf{Pos}$ denote the [entrance path poset functor](https://ncatlab.org/nlab/show/stratified+space#the_category_of_stratifications), that maps stratified spaces to their entrance path posets.

\begin{defn} A *geometric $n$-computad* is an $[n]$-graded set $C$ together with, for each $c \in C_i$, a stratum type $(\mathbb{R}^i, f_c)$ and a 'labeling' function $l_c : \mathcal{E}_0 f_c \to C_{\leq i}$ with $l_c\{0\} = c$, such that $k$-strata $s$ in $f_c$ have stratum type $f_{l_c(s)}$ and $l_{l_c(s)}$ factors through $l_c$ by the induced map $\mathcal{E}_0 f_{l_c(s)} \to \mathcal{E}_0 f_c$.
\end{defn}

In words, a computad $C$ is a bunch of '$i$-morphisms' (indexed by the sets $C_i$) for $i \leq n$, and each morphism has a stratum type whose boundary is consistently labeled in lower-dimensional morphisms.

## Remarks

* The remarkable observation about geometric computads is the following.  Classically, [building computads](https://ncatlab.org/nlab/show/computad#globular_computads) is an inductive *two-step* process: namely, usually one constructs computads inductively by, in the $n$th step, adding generating $n$-morphisms with boundary in existing $(n-1)$-morphisms, and *then* passing to the closure under composition and coherence conditions (which, passing back and forth between "new composites" and "new coherences" is often a pretty infinite process). In geometric computads, this is process becomes unnecessary: boundaries of $n$-morphisms are manifold $(n-1)$-diagrams (or, if you work dually, cell $(n-1)$-diagrams) which already can express all the composites and isotopies you could possibly need. This makes geometric computads *really* easy to work with.

* While it is straight-forward to define geometric computads (and related notions, such as their functors and certain transformations), a comprehensive theory of computads is still in development.

## Related pages

* [[manifold diagram]]

* [[geometric n-category]]

* [[homotopy.io]]

## References

A definition and further discussion can be found in

* {#Dorn23} [[Christoph Dorn]], _Nine short stories about geometric higher categories_, 2023 ([pdf](https://cxdorn.github.io/assets/pdfs/nine-stories.pdf))

The notion has been the topic of several talks and blog posts.

* [[Christoph Dorn]], [From trusses to geometric computads to combinatorial manifold diagrams](https://cxdorn.github.io/research/trusses/) (2022)

* {#functors} [[Christoph Dorn]], [The categorical Pontryagin-Thom construction](https://cxdorn.github.io/tcptc/) (2022) contains definitions of functors and certain transformations of geometric computads.

The finitely generated case can be efficiently manipulated using the proof-assistant [[homotopy.io]]

* [[Jamie Vicary]] et. al, [Introducing homotopy.io: A proof assistant for geometrical higher category theory](https://tqft.math.tecnico.ulisboa.pt/seminars?id=6607) (2022)

[[!redirects geometric n-computad]]
[[!redirects geometric computads]]


geometric dagger 2-poset

[[!redirects geometric dagger 2-posets]]

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

## Idea ##

A geometric dagger 2-poset is a dagger 2-poset whose [[category of maps]] is a [[geometric category]].

## Definition ##

A **geometric dagger 2-poset** is a [[dagger 2-poset]] $C$ such that

* There is an object $0 \in Ob(C)$ such that for each object $A \in Ob(C)$, there is a [[monic map in a dagger 2-poset|monic map]] $i_{0,A} \in Hom(0,A)$ such that for each object $B \in Ob(C)$ with a monic map $i_{B,A} \in Hom(B,A)$, there is a monic map $i_{0,B} \in Hom(0,B)$ such that $i_{B,A} \circ i_{0,B} = i_{0,A}$.

* For each object $A:Ob(C)$, [[set]] $S$, and family of objects $B:S \to Ob(C)$, with monic maps $i_{B(s),A} \in Hom(B(s),A)$ for each element $s \in S$, there is an object
$$\bigcup_{s \in S} B(s) \in Ob(C)$$
with monic maps
$$i_{\bigcup_{s \in S} B(s),A} \in Hom(\bigcup_{s \in S} B(s),A)$$
and for each element $s \in S$, there is a monic map
$$i_{B(s),\bigcup_{s \in S} B(s)} \in Hom(B(s),\bigcup_{s \in S} B(s))$$
such that
$$i_{\bigcup_{s \in S} B(s),A} \circ i_{B(s),\bigcup_{s \in S} B(s)} = i_{B(s),A}$$
and for every object $D \in Ob(C)$ with monic maps $i_{D,A} \in Hom(D,A)$ $i_{B,D} \in Hom(B,D)$, $i_{E,D} \in Hom(E,D)$ such that $i_{D,A} \circ i_{B,D} = i_{B,A}$ and $i_{D,A} \circ i_{E,D} = i_{E,A}$, there is a monic map
$$i_{\bigcup_{s \in S} B(s),D} \in Hom(\bigcup_{s \in S} B(s),D)$$
such that
$$i_{D,A} \circ i_{\bigcup_{s \in S} B(s),D} = i_{\bigcup_{s \in S} B(s),A}$$

* For each object $A \in Ob(C)$, $B \in Ob(C)$, $E \in Ob(C)$ with monic maps $i_{B,A} \in Hom(B,A)$, $i_{E,A} \in Hom(E,A)$, there is an object $B \cap E \in Ob(C)$ with monic maps $i_{B \cap E,A} \in Hom(B \cap E,A)$, $i_{B \cap E,B} \in Hom(B \cap E,B)$, $i_{B \cap E,E} \in Hom(B \cap E,E)$, such that $i_{B,A} \circ i_{B \cap E,B} = i_{B \cap E,A}$ and $i_{E,A} \circ i_{B \cap E,E} = i_{B \cap E,A}$, and for every object $D \in Ob(C)$ with monic maps $i_{D,A} \in Hom(D,A)$ $i_{D,B} \in Hom(D,B)$, $i_{D,E} \in Hom(D,E)$ such that $i_{B,A} \circ i_{D,B} = i_{D,A}$ and $i_{E,A} \circ i_{D,E} = i_{D,A}$, there is a monic map $i_{D,B \cap E} \in Hom(D,B \cap E)$ such that $i_{B \cap E,A} \circ i_{D,B \cap E} = i_{D,A}$.

* For each object $A \in Ob(C)$, $E \in Ob(C)$, with monic maps $i_{E,A} \in Hom(E,A)$ and set $S$, and family of objects $B:S \to Ob(C)$, elements $t \in S$ and monic map $i_{B(s),A} \in Hom(B(s),A)$, there is a [[unitary isomorphism]]
$$j_{B,C,E} \in E \cap (\bigcup_{s \in S} B(s)) \cong^\dagger \bigcup_{s \in S} E \cup B(s)$$

## Properties ##

* For each object $A \in Ob(C)$, the identity function $1_A \in Hom(A,A)$ is a monic map, and for each object $B:Ob(C)$ with a monic map $i_{B,A} \in Hom(B,A)$, $1_A \circ i_{B,A} = i_{B,A}$.

* The [[isomorphism classes]] of monic maps into every object $A$ is a [[frame]]. Since every monic map is a map, the [[category of maps]] is a [[geometric category]].

## Examples ##

The dagger 2-poset [[Rel]] of [[sets]] and [[relations]] is a geometric dagger 2-poset.

## See also ##

* [[dagger 2-poset]]
* [[coherent dagger 2-poset]]
* [[geometric dagger 2-poset]]
* [[Heyting dagger 2-poset]]
* [[Boolean dagger 2-poset]]
* [[elementarily topical dagger 2-poset]]


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* **geometric definition of higher categories**

* [[algebraic definition of higher categories]]

***

#Contents#
* table of contents
{:toc}

## Idea

In a **geometric definition** of [[(n,r)-categories]] [[composition]] of [[higher morphism]]s is not an _operation_ with a specified outcome but a _relation_ : the $(n,r)$-category is presented much like a [[directed space]] and [[k-morphism]]s are $k$-dimensional subspaces in there. When some of these $k$-morphisms are suitably adjacent, there is a guarantee that there exists a $k$-morphism that serves as their composite. But there may be several such. Instead of a rule for picking a specific one, subject to [[associativity]] constraints, there is a _contractible space of choices_ of possible composites.

From a geometric presentation of an $(n,r)$-category one can typically obtain an algebraic presentation by choosing composites. The contractibility of the space of choices becomes a [[coherence law]] satisfied by the collection of choices.

Conversely, one may typically think of the geometric presentation of an $(n,r)$-category as being the [[nerve]] of a corresponding algebraic presentation.

## Examples

* a geometric model for [[(∞,0)-categories]] = [[∞-groupoids]] are [[Kan complex]]es;

  An [[algebraic Kan complex]] is an algebraic model that is obtained from  a Kan complex by making choices for [[composition]], [[associator]] etc.

  * a geometric model for an [[n-groupoid]] is a $(n+1)$-[[coskeletal]] Kan complex.

* a geometric model for [[(∞,1)-categories]] are [[quasi-categories]]

  An [[algebraic quasicategory]] is an algebraic model that is obtained from  a quasi-category by making choices for [[composition]], [[associator]] etc.

  * a geometric model for a [[(1,1)-category]] = [[category]] is a 2-[[coskeletal]] [[quasi-category]]

* a geometric model for an [[(n,r)-category]] is an $(n,r)$-[[Theta space]]

* a geometric model for an [[(∞,n)-category]] is

  * an [[n-fold complete Segal space]]

* a geometric model for a general [[omega-category]] is supposed to be a [[weak complicial set]]

## Properties

When the [[(∞,1)-category]] of all [[(n,r)-categories]] is presented by a [[model category]], then typically geometric models are _cofibrant_ objects while algebraic models are typically _fibrant_ objects.

For instance all in the standard [[model structure on simplicial sets]], or the standard [[model structure for quasi-categories]] all objects are cofibrant.

## Related concepts

There are also geometric models for [[operad|operadic]] structures: _[[dendroidal sets]]_.

[[!redirects geometric definition of higher categories]]
[[!redirects geometric definitions of higher categories]]

[[!redirects geometric definition of higher category]]
[[!redirects geometric model of higher categories]]
[[!redirects geometric model of higher category]]