Global homotopy theory

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#### Representation theory
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This page collects material related to

* [[Stefan Schwede]],

  _Global homotopy theory_

  New Mathematical Monographs, 34

  Cambridge University Press, 2018

  xvi+828 pp

  [doi:10.1017/9781108349161](https://doi.org/10.1017/9781108349161)

  [arXiv:1802.09382](https://arxiv.org/abs/1802.09382)

on [[global equivariant stable homotopy theory]] in terms of [[orthogonal spectra]].

(The reader interested only in plain [[stable homotopy theory]] takes the collection $\mathcal{F}$ of groups on p. viii to be trivial throughout, to get just the theory of non-equivariant [[orthogonal spectra]]. See also the Schwede's lecture notes on _[[Symmetric spectra]]_.)

## Related texts

* [[Stefan Schwede]], _[[Lectures on Equivariant Stable Homotopy Theory]]_

* [[Charles Rezk]], _[[Global Homotopy Theory and Cohesion]]_ (on global homotopy theory being a [[cohesive (infinity,1)-topos]])

* [[Stefan Schwede]], _Orbispaces, orthogonal spaces, and the universal compact Lie group_ ([arXiv:1711.06019](https://arxiv.org/abs/1711.06019)) (on the relation to [[orbispaces]]/[[topological stacks]])

* [[Stefan Schwede]], Advanced Topics in Topology (V5D1): Global homotopy theory (Summer term 2020) [course website](http://www.math.uni-bonn.de/people/schwede/global.html) (includes list of lectures and bonus material, links to videos),
 [YouTube playlist](https://www.youtube.com/playlist?list=PLsU-oFZDWE_INhBDZSXLQVIa_m0CBb-Ys)

## Related $n$Lab entries

* [[orbifold cohomology]]

* [[ultracommutative ring spectrum]]

* [[global equivariant bordism homology theory]]

* [[orthogonal space]]

category: reference


Global Homotopy Theory and Cohesion

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#### Representation theory
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[[!include representation theory - contents]]
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#### Cohesive $\infty$-Toposes
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[[!include cohesive infinity-toposes - contents]]
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This page is to collect some links related to the text

* [[Charles Rezk]],

  _Global Homotopy Theory and Cohesion_

  2014

  ([pdf](https://rezk.web.illinois.edu/global-cohesion.pdf), [[Rezk14.pdf:file]])

on the [[global equivariant homotopy theory]] as a [[cohesive (∞,1)-topos]].

See at *[[cohesion of global- over G-equivariant homotopy theory]]*.

## Related $n$Lab entries

* [[Stefan Schwede]], _[[Global homotopy theory]]_

* [[singular cohesion]]

* [[Elmendorf's theorem]]

* [[orbispace]],

* [[orbifold]], [[orbifold cohomology]]

category: reference


global mixed Hodge theory

#Contents#
* table of contents
{:toc}

## Idea

The aim of global mixed Hodge theory is to make a [[global Hodge theory]] for pre-proper global analytic [[logarithmic geometry|logarithmic spaces]].

## Related subject

* [[global Hodge theory]]

* [mixed Hodge structure](http://ncatlab.org/nlab/show/Hodge+structure#mixed_hodge_structure)


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#### Model category theory
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#Contents#
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{:toc}

## Idea

The **global model structure on simplicial presheaves** $[C^{op}, sSet_{Quillen}]$ on a [[small category]] $C$ is the [[global model structure on functors]] on $C$ with values in $sSet_{Quillen}$, the standard [[model structure on simplicial sets]].

It [[presentable (∞,1)-category|presented]] the [[(∞,1)-category of (∞,1)-functors]] from $C^{op}$ to [[∞Grpd]], hence the [[(∞,1)-category of (∞,1)-presheaves]] on $C$.

The [[Bousfield localization|left Bousfield localizations]] of $[C^{op}, sSet]_{proj}$ are, up to [[Quillen equivalence]], precisely the [[combinatorial model category|combinatorial model categories]].

In particular, if $C$ carries the structure of a [[site]], then

* the left Bousfield localization of $[C^{op}, sSet_{Quillen}]$ at [[Cech cover]]s is the [[Cech model structure on simplicial presheaves]];

* the left Bousfield localization at [[hypercover]]s is the [[local model structure on simplicial presheaves]].

These localizations present the [[topological localization]] and [[hypercompletion]] of the [[(∞,1)-topos]] of $(\infty,1)$-presheaves on $C$ to the corresponding [[(∞,1)-topos]] of [[(∞,1)-category of (∞,1)-sheaves|(∞,1)-sheaves]]/[[∞-stack]]s on $C$.

## Definition

In every global model structure on simplicial presheaves on $C$ the weak equivalences are _objectwise_ those with respect to the standard [[model structure on simplicial sets]].

A morphism $f : A \to B $ in $[C^{op}, SSet]$ is, thus, a weak equivalence with respect to a global model structure precisely if for all $U \in Obj(C)$ the morphism

$$
  f(U) : A(U) \to B(U)
$$

is a weak equivalence of [[simplicial set]] (i.e. a morphism inducing isomorphisms of [[simplicial homotopy group]]s).

There are several choices for how to extend this notion of weak equivalences to an entire [[model category]] structure.
The two common extreme choices are

* the  **global projective** model structure has as **fibrations** the _objectwise_ fibrations of the standard [[model structure on simplicial sets]] (i.e. the [[Kan fibration]]s);

* the  **global injective** model structure has as **cofibrations** the _objectwise_ cofibrations with respect to the standard [[model structure on simplicial sets]] (i.e. the [[monomorphism]]s).

The other class of morphisms (cofibrations / fibrations) is in each case fixed by the corresponding [[weak factorization system|lifting property]].

## Properties

* Both projective and injective model structures define [[simplicial model category|proper simplicial model categories]].

## References

See also [[model structure on simplicial presheaves]].

The global projective model structure is originally due to

* A. K. Bousfield and D.M. Kan, _Homotopy limits completions and localizations_, Springer Lecture Notes in Math. 304 (2nd corrected printing), Springer-Verlag, Berlin--Heidelberg--New York (1987).

The fact that the global injective model structure yields a proper simplicial cofibrantly generated model category is originally due to

* [[Alex Heller]] _Homotopy Theories_, no. 383, Memoirs Amer. Math. Soc., Amer. Math. Soc., 1988.

The fact that the global projective structure yields a proper simplicial cellular model category is due to Hirschhorn-Bousfield-Kan-Quillen

* P. Hirschhorn, _Localizations of Model Categories_ ([web](http://www-math.mit.edu/~psh/))

A quick review of these facts is on the first few pages of

* Benjamin Blander, _Local projective model structure on simplicial presheaves_ ([pdf](http://www.math.uiuc.edu/K-theory/0462/combination2.pdf))

Details on the _projective_ global model structure is in

* [[Daniel Dugger]], _[[Universal Homotopy Theories]]_

[[!redirects global model structure on simplicial sheaves]]
[[!redirects global model structures on simplicial presheaves]]
[[!redirects global model structures on simplicial sheaves]]


global orbit category

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#### Homotopy theory
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#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _global orbit category_ is the unification of all $G$-[[orbit categories]] as the [[groups]] $G$ are allowed to vary.

The [[(∞,1)-presheaves]] over the global orbit category provide the [[base (∞,1)-topos]] over which the [[global equivariant homotopy theory]] (see there) is a [[cohesive (∞,1)-topos]] ([Rezk 14](#Rezk14)).

## Definition

The following defines the [[global equivariant indexing category]] $Glo$.

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

Write $Glo$  for the [[(∞,1)-category]] whose

* [[objects]] are [[compact Lie groups]];

* [[(∞,1)-categorical hom-spaces]] $Glo(G,H)$ are the [[geometric realizations]] of the [[Lie groupoid]] of smooth functors and smooth [[natural transformations]] $Top\infty Grpd(\mathbf{B}G, \mathbf{B}H)$.

=--

([Rezk 14, 2.1](#Rezk14))

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

Equivalent models for the global indexing category, def. \ref{GlobalIndexingCategory} include the category "$O_{gl}$" of ([May 90](#May90)).
Another variant is $\mathbf{O}_{gl}$ of ([Schwede 13](#Schwede13)).

=--

([Rezk 14, 2.4, 2.5](#Rezk14))

The following is the _[[global orbit category]]_.

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

Write

$$
  Orb \longrightarrow Glo
$$

for the non-full [[sub-(∞,1)-category]] of the global indexing category, def. \ref{GlobalIndexingCategory}, on the [[injection|injective]] group homomorphisms.

=--

([Rezk 14, 4.5](#Rezk14))

## Properties

### Relation to the local orbit category

The [[slice (∞,1)-category]] of the global orbit category $Orb$ (the version with [[faithful functors]] as morphisms) over $\mathbf{B}G$ is the local [[orbit category]] of $G$

$$
  Orb_{/\mathbf{B}G}  \simeq Orb_G
 \,.
$$

### Relation to orbispaces and $G$-spaces

The [[(∞,1)-category of (∞,1)-presheaves]] over the global orbit category is that of [[orbispaces]].

Accordingly, by the discussion [here](over-%28infinity%2C1%29-topos#SheavesOnBigSite), the [[slice (∞,1)-topos]] of orbispaces over $\mathbf{B}G$ is that of [[G-spaces]]

$$
  PSh_\infty(Orb)_{/\mathbf{B}G}
  \simeq
  PSh_\infty(Orb_{/\mathbf{B}G})
  \simeq
  PSh_\infty(Orb_G)
  \simeq
  G Space
$$

(where the last step is [[Elmendorf's theorem]]).

### Relation to equivariant homotopy theory

The [[(∞,1)-category of (∞,1)-presheaves]] on the [[global equivariant indexing category]] is the [[global equivariant homotopy theory]] and under the canonical projection is a [[cohesive (∞,1)-topos]] over [[∞Grpd]]. Its [[slice (∞,1)-topos]] over the terminal orbispace is cohesive over [[orbispaces]]

$$
  PSh_\infty(Glo)_{/\mathcal{N}} \to PSh_\infty(Orb)
 \,.
$$

## Related concepts

* [[orbifold cohomology]]

* [[cohesive relation between global- and G-equivariant homotopy theory]]

[[!include equivariant homotopy theory -- table]]

## References

* {#HenriquesGepner07} [[André Henriques]], [[David Gepner]], _Homotopy Theory of Orbispaces_ ([arXiv:math/0701916](http://arxiv.org/abs/math/0701916))

* {#Rezk14} [[Charles Rezk]], _[[Global Homotopy Theory and Cohesion]]_ (2014)

* {#Lurie} [[Jacob Lurie]], Section 3 of _Elliptic cohomology III: Tempered Cohomology_ ([pdf](http://www.math.harvard.edu/~lurie/papers/Elliptic-III-Tempered.pdf))

See also

* {#Schwede13} [[Stefan Schwede]], _Global homotopy theory_, 2013 ([pdf](http://www.math.uni-bonn.de/~schwede/global.pdf))

* {#May90} [[Peter May]], _Some remarks on equivariant bundles and classifying spaces_, Asterisque 191 (1990), 7, 239-253. International Conference on Homotopy Theory (Marseille-Luminy, 1988).


global quotient orbifold

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#Contents#
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## Idea

A _global quotient orbifold_ is an [[orbifold]] that is represented by a [[G-space]] $X$: To the extent that orbifolds are identified with [[topological groupoids]]/[[Lie groupoids]], a global quotient orbifold is represented by the [[action groupoid]] of $G$ [[action|acting]] on $X$. Understood as representing a [[topological stack]]/[[differentiable stack]], this is the [[quotient stack]] of $X$ by $G$. Understood in [[(∞,1)-topos theory]] this is the [[homotopy quotient]] $X \sslash G$.

The [[orbifold cohomology]] of global quotient orbifolds is a version of [[equivariant cohomology]] of $X$.

{#PresentableAndGoodOrbifolds} The orbifolds that are [[Morita equivalence|Morita-]][[equivalence in an (infinity,1)-category|equivalently]] global quotients of [[smooth manifolds]] by...

1. ... an [[group action|action]] of a [[Lie group]] are also called *[[presentable orbifolds]]*;

1. ...the action of a [[discrete group]] are called _[[good orbifolds]]_,

1. ... the action of a [[finite groups]] are called _[[very good orbifolds]]_ .

Non-good orbifolds are also called _bad orbifolds_ ([Thurston 92, Ch. 13, Def. 13.2.3](good+orbifold#Thurston92)). It is [[conjecture|conjectured]] ([Adem, Leida & Ruan 2007, Conj. 1.55](presentable+orbifold#AdemLeidaRuan07)) that *every* orbifold is [[presentable orbifold|presentable]].

## Related concepts

* [[action groupoid]], [[quotient stack]], [[homotopy quotient]]

* [[orbifold K-theory]], [[orbifold differential K-theory]]

* [[equivariant cohomology]]

## References

[[!include traditional orbifold cohomology -- references]]

[[!redirects global quotient orbifolds]]


global section

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### Context
#### Topos Theory
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#Contents#
* table of contents
{:toc}

## Idea

The _global sections_ of a [[bundle]] are simply its [[sections]]. When bundles are replaced by their [[sheaves]] of [[local sections]], then forming global sections corresponds to the [[direct image]] operation on sheaves with respect to the morphism to the [[terminal object|terminal]] [[site]]. This definition generalizes to objects in a general [[topos]] and [[(∞,1)-topos]].

## Definition

We start describing the more explicit notions of global sections of bundles and then work our way towards the more abstract notions in terms of [[topos]] theory.

### Of bundles

A __global [[section]]__ of a [[bundle]] $E \overset{p}\to B$ is simply a [[section]] of $p$, that is a map $s\colon B \to E$ such that $p \circ s = \id_B$.

$$
  \array{
    && E
    \\
    & {}^{\mathllap{s}}\nearrow & \downarrow^{\mathrlap{p}}
    \\
    B &\stackrel{id}{\to}& B
  }
  \,.
$$

The adjective 'global' here is used to distinguish from a **[[local section]]**: a *generalised* section over some subspace $i : U \hookrightarrow B$ which is a section of the map to $U$

$$
  i^* E := E|_U \to U
$$

from the [[pullback]]

$$
  \array{
    i^* E := E|_U &\to& E
    \\
    \downarrow && \downarrow
    \\
    U &\stackrel{i}{\to}& B
  }
  \,.
$$

Compare the notion of [[global point]], which is really the special case when $B$ is a [[terminal object]] (where the generalised section corresponds to a [[generalised element]]).  On the other hand, a global section of $E \overset{p}\to B$ in $\mathcal{C}$ *is* simply a global point in the [[slice category]] $\mathcal{C}/B$.

One often writes

$$
  \Gamma_U(E) := Hom_{\mathcal{C}/U}(U , E|_U)
$$

for the **set of global sections** over $U$ (or $\Gamma(U,E)$ or similar).

### Of sheaves on topological spaces

Every [[sheaf]] $A \in Sh(X) = Sh(Op(X))$ on (the [[site]] that is given by the [[category of open subsets]] of) a [[topological space]] $X$ is the sheaf of _[[local sections]]_  of its [[etale space]] [[bundle]] $E \to X$ in that

$$
  A : U \mapsto \Gamma_U(E)
$$

for every $U \in Op(X)$. For this reasons one often speaks of the value of a [[sheaf]] on some object as a set of sections, even if the corresponding bundle is never mentioned and doesn't really matter.

The set of global sections on $X$ is

$$
  \Gamma_X(A) = A(X) = Hom_{Sh(X)}(X, A)
  \,,
$$

where $X \in Sh(X)$ denotes the [[terminal object]] of the [[category of sheaves]] $Sh(X)$. Often this is written just using different notation

$$
  \Gamma_X(A) = Hom_{Sh(X)}(*,A)
$$

One notices that $\Gamma_X(-) : Sh(X) \to Set$ defined this way is the [[direct image]] functor on [[Grothendieck toposes]] that is induced from the canonical morphism $X \to *$ of [[topological space]]s (now "$*$" really denotes the [[point]] topological space!) and hence from the corresponding morphism of [[site]]s.

Again, this expression for global sections induces a relative version,
e.g. for sheaves on $S$-[[relative scheme|schemes]], the direct image functor goes into the base scheme $S$).

### Of objects in a general Grothendieck topos {#GeneralTopos}

The definition of global sections of sheaves on topological spaces in terms of the [[direct image]] of the canonical morphism to the terminal [[site]] generalizes to [[Grothendieck topos|sheaf toposes]] over arbitrary [[sites]].

For every [[Grothendieck topos]] $\mathcal{T}$, there is a [[geometric morphism]] (the *[[terminal geometric morphism]]*)

$$
  \Gamma \;\colon\; \mathcal{T}  \stackrel{\leftarrow}{\to} Set : LConst
$$

called the **global sections** functor. It is given by the [[hom-set]] out of the [[terminal object]]

$$
  \Gamma(-) = Hom_{\mathcal{T}}({*}, -)
$$

and hence assigns to each object $A\in \mathcal{T}$ its set of [[global element]]s $\Gamma(A) = Hom_{\mathcal{T}}(*,A)$.

The [[left adjoint]] $LConst : Set \to \mathcal{T}$ of the global section functor is the canonical [[Set]]-[[copower|tensoring]] functor

$$
  \otimes : Set \times \mathcal{T} \to \mathcal{T}
$$

applied to the [[terminal object]]

$$
  const = (-)\otimes {*} : Set \to \mathcal{T}
$$

which sends a set $S$ to the [[coproduct]] of $|S|$ copies of the terminal object

$$
  S \otimes {*} = \coprod_{s \in S} {*}
  \,.
$$

This is called the **constant object** of $\mathcal{T}$ on the set $S$. Notably when $\mathcal{T}$ is a [[Grothendieck topos|sheaf topos]] this is the **[[constant sheaf]]** $LConst_S$ on $S$.

$$
  \mathcal{T} \stackrel{\stackrel{LConst}{\leftarrow}}{\overset{\Gamma}{\to}} Set
  \,.
$$

If the topos $\mathcal{T}$ is a [[locally connected topos]] then the left adjoint functor $LConst$ is also a right adjoint, its left adjoint being the functor $\Pi_0 : \mathcal{T} \to Set$ that sends an object to its set of connected components.

### Of objects in an $(\infty,1)$-topos

The previous abstract definition generalizes straightforwardly to every context of [[higher category theory]] where the required notions of [[adjoint functor]] etc. are provided.

Notably in [[(∞,1)-category]] theory the global section functor on an [[∞-stack]] [[(∞,1)-topos]] $\mathbf{H}$ is the [[hom-functor]]

$$
  \Gamma(-)
  \;\coloneqq\;
  \mathbf{H}(*,-) \colon
  \;
  \mathbf{H} = Sh_{(\infty,1)}(C) \to Sh_{(\infty,1)}(*) = \infty Grpd
$$

of morphisms out of the terminal object.

This is indeed again the [[terminal geometric morphism]].

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

Let $\mathbf{H}$ be an [[∞-stack]] [[(∞,1)-topos]]. Then the [[∞-groupoid]] $Geom(\mathbf{H}, \infty Grpd)$ of [[geometric morphism|geometric]] [[(∞,1)-functor]]s is [[contractible]].

So $\infty Grpd$ is the [[terminal object]] in the [[(∞,1)-category]] of [[(∞,1)-topos]]es and geometric morphisms.

=--

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

This is [[Higher Topos Theory|HTT 6.3.4.1]]

=--

If the [[(∞,1)-topos]] is a [[locally contractible (∞,1)-topos]] then this is an [[essential geometric morphism]].

The composite [[(∞,1)-functor]] $\Gamma \circ LConst$ is the [[shape of an (∞,1)-topos|shape of]] $\mathbf{H}$.

## Related concepts

* [[local section]], [[flat section]]

* [[abelian sheaf cohomology]]

* [[ex-space]]

* [[global element]]

[[!redirects global sections]]

[[!redirects global sections functor]]
[[!redirects global section functor]]
[[!redirects global sections functors]]
[[!redirects global section functors]]

[[!redirects global section geometric morphism]]
[[!redirects global section geometric morphisms]]
[[!redirects global sections geometric morphism]]
[[!redirects global sections geometric morphisms]]


globalizer

#Content#

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#Idea#
A globalizer is a kind of higher order or [[hyperstructure]]d [[sheaf]] covering all the levels.

#References#
* Nils A. Baas, _Higher order architecture of collections of objects_, International Journal of General Systems vol. 44 (1), pp. 55-75, 2015. [arXiv:1409.0344](http://arxiv.org/abs/1409.0344)


globally equivariant spectrum

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## Idea

An [[object]] in the [[global equivariant stable homotopy theory]].

Hence a [[spectrum]] compatibly equipped with the structure of a $G$-[[equivariant spectrum]] as $G$ ranges over a suitable given class of [[groups]].

## Examples

* The global [[equivariant sphere spectrum]] for all the [[cyclic groups]] over the [[circle group]] is canonically a [[cyclotomic spectrum]] and as such is the [[tensor unit]] in the [[monoidal (infinity,1)-category]] of [[cyclotomic spectra]] (see there).

## References

* {#Schwede13} [[Stefan Schwede]], _[[Global homotopy theory]]_, 2013 ([pdf](http://www.math.uni-bonn.de/~schwede/global.pdf))

[[!redirects globally equivariant spectra]]

[[!redirects global equivariant spectrum]]
[[!redirects global equivariant spectra]]


globally hyperbolic Lorentzian manifold

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# Contents
* table of contents
{: toc}

## Idea

A [[Lorentzian manifold]] is called _globally hyperbolic_ if it admits a well-defined time evolution from initial data of [[field (physics)|physical fields]] on it.

## Definition

There are several equivalent definitions of global hyperbolicity. A simple one is:

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

A [[Lorentzian manifold]] (without [[boundary]]) is called **globally hyperbolic** if it contains a [[Cauchy surface]].

=--

In this form the characterization of global hyperbolicity appears for instance in the paragraph at the bottom of page 211 in ([HE](#LargeScale)).
The equivalence of this to more traditional definitions is ([HE, prop. 6.6.3](#LargeScale)) together with ([HE, prop. 6.6.8](#LargeScale)), due to ([Geroch1970](#Geroch)). The latter in fact implies the following stronger statement:

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

A [[Lorentzian manifold]] (without [[boundary]]) is **globally hyperbolic** if it admits a [[foliation]] by [[Cauchy surfaces]].

=--

See also ([Baer-Ginoux-Pfaeffle 07, theorem 1.3.10](#BaerGinouxPfaeffle07)).

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

So in particular for a globally hyperbolic spacetime $X$ there is a [[homeomorphism]]

$$
  \phi\colon \mathbb{R} \times \Sigma \to X
$$

from the [[product]] of the [[real line]] with a $(dim X - 1)$-[[dimension|dimensional]] [[smooth manifold]] $\Sigma$ and for each $t \in \mathbb{R}$ the image $\phi(t, \Sigma) \subset X$ is a [[Cauchy surface]] of $X$.

=--

A _time orientation_ of a globally hyperbolic Lorentzian spacetime is a choice of [[orientation]] of the factor $\mathbb{R}$ under the above homeomorphism.

## Related concepts

* [[AQFT on curved spacetimes]], [[locally covariant perturbative quantum field theory]]

* [[Cauchy problem]]

* [[Green hyperbolic partial differential equation]]

## References

Concise collection of definitions:

* {#MinguzziSánchez} E. Minguzzi, [[Miguel Sánchez]], §3.11 in: *The causal hierarchy of spacetimes*, in: *Recent Developments in Pseudo-Riemannian Geometry*, EMS ESI Lectures in Mathematics and Physics **4** (2008) 299-358 [[arXiv:gr-qc/0609119](https://arxiv.org/abs/gr-qc/0609119), [ISBN:978-3-03719-051-7](https://bookstore.ams.org/view?ProductCode=EMSESILEC/4)]

Survey:

* [[Miguel Sánchez]], *Globally hyperbolic spacetimes: slicings, boundaries and counterexamples*, Gen Relativ Gravit **54** 124 (2022) [[arXiv:2110.13672](https://arxiv.org/abs/2110.13672), [doi:10.1007/s10714-022-03002-6](https://doi.org/10.1007/s10714-022-03002-6)]

See also:

* {#LargeScale} Hawking, Ellis, section 6.6 of _The large-scale structure of Space-Time_ Cambridge (1973)

* {#BaerGinouxPfaeffle07} [[Christian Bär]], [[Nicolas Ginoux]], [[Frank Pfäffle]], _Wave Equations on Lorentzian Manifolds and Quantization_, ESI Lectures in Mathematics and Physics, European Mathematical Society Publishing House, ISBN 978-3-03719-037-1, March 2007, Softcover ([arXiv:0806.1036](https://arxiv.org/abs/0806.1036))

The fact that a single Cauchy surface implies a foliation by Cauchy surfaces is due to:

* {#Geroch} [[Robert Geroch]], §5, Thm. 11 in: *Domain of Dependence*, J. Math. Phys. **11** (1970) 437–449 [[doi:10.1063/1.1665157](https://doi.org/10.1063/1.1665157)]

The refinement of this statement to a smooth splitting:

* [[Antonio N. Bernal]], [[Miguel Sánchez]], _On smooth Cauchy hypersurfaces and Geroch's splitting theorem_, Commun. Math. Phys. **243** (2003) 461-470 [[arXiv:gr-qc/0306108v2](http://arxiv.org/abs/gr-qc/0306108), [doi:10.1007/s00220-003-0982-6](https://doi.org/10.1007/s00220-003-0982-6)]

* [[Antonio N. Bernal]], [[Miguel Sánchez]], *Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes*, Commun. Math. Phys. **257** (2005) 43-50 [[arXiv:gr-qc/0401112](https://arxiv.org/abs/gr-qc/0401112), [doi:10.1007/s00220-005-1346-1](https://doi.org/10.1007/s00220-005-1346-1)]

* [[Antonio N. Bernal]], [[Miguel Sánchez]], *Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions*, Lett. Math. Phys. **77** (2006) 183-197 [[arXiv:gr-qc/0512095](https://arxiv.org/abs/gr-qc/0512095), [doi:10.1007/s11005-006-0091-5](https://doi.org/10.1007/s11005-006-0091-5)]

[[!redirects globally hyperbolic Lorentzian manifolds]]

[[!redirects globally hyperbolic]]
[[!redirects globally hyperbolic Lorentzian manifold]]
[[!redirects globally hyperbolic pseudo-Riemannian manifold]]

[[!redirects globally hyperbolic spacetime]]
[[!redirects globally hyperbolic spacetimes]]

[[!redirects time orientation]]
[[!redirects time orientations]]


globe

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

#Contents#
* table of contents
{:toc}

## Idea

The cellular _$n$-globe_ is the globular analog of the cellular $n$-[[simplex]]. It is one of the basic [[geometric shapes for higher structures]].

## Definition

The **cellular $n$-globe** $G_n$ is the [[globular set]] represented by the object $[n]$ in the [[globe category]] $G$:

$$
  G_n := Hom_G(-,[n])
  \,.
$$

## Examples

The 0-globe is the singleton [[set]], the category with a single morphism.

The 1-globe is the [[interval category]].

The 3-globe looks like this

<img src="http://www.math.uni-hamburg.de/home/schreiber/pics/3globe.gif" alt=""/>

## Properties

### $n$-Category structure

There is a unique structure of a [[strict omega-category]], an [[n-category]] in fact, on the $n$-globe. This makes the collection of $n$-globes arrange themselves into a _co-globular $\omega$-category_, i.e. a functor

$$
  G \to \omega Cat
$$
$$
  [n] \mapsto G_n
  \,.
$$

### Relation to simplices

The _[[orientals]]_ translate between [[simplices]] and globes.

## References

See the references at _[[strict omega-category]]_ and at _[[oriental]]_.

[[!redirects n-globe]]
[[!redirects 1-globe]]
[[!redirects 2-globe]]
[[!redirects 3-globe]]
[[!redirects glob]]
[[!redirects globes]]
[[!redirects n-globes]]
[[!redirects 1-globes]]
[[!redirects 2-globes]]
[[!redirects 3-globes]]
[[!redirects globs]]

[[!redirects bigon]]
[[!redirects bigons]]


globe category

# The globe category
* table of contents
{: toc}

## Idea

The **globe category** $G$ encodes one of the main [[geometric shapes for higher structures]].  Its objects are the standard cellular $n$-[[globe|globes]], and presheaves on it are [[globular sets]].

It may also be called the [[globular category]], although that term has other interpretations.

## Definition

The **globe category** $G$ is
the category whose objects are the non-negative integers and
whose morphisms are generated from
$$
  \sigma_n : [n] \to [n+1]
$$
$$
  \tau_n : [n] \to [n+1]
$$
for all $n \in \mathbb{N}$ subject to the relations (dropping obvious subscripts)
$$
  \sigma\circ \sigma = \tau \circ \sigma
$$
$$
  \sigma\circ \tau = \tau \circ \tau
$$

### The reflexive globe category

If we add the generating morphisms
$$
  \iota_n : [n+1] \to [n]
$$
subject to the relations
$$
  \iota \circ \sigma = \mathrm{Id}
$$
$$
  \iota \circ \tau = \mathrm{Id}
  \,.
$$
we obtain the **reflexive globe category**.

## Remarks

* The globe category is used to define [[globular set|globular sets]].

* M. Roy has studied [[level of a topos|levels]] in the topos of [[globular set#ReflexiveGlobularSets|reflexive globular sets]] in the context of [[Bill Lawvere|F. W. Lawvere]]'s concept of [[Aufhebung]] (cf. [Kennett-Riehl-Roy-Zaks(2011)](#KRRZ11)).

## Related concepts

* [[globular set]]

* [[simplex category]]

* [[cube category]]

* [[cell category]]

## Reference

* {#KRRZ11} C. Kennett, [[Emily Riehl|E. Riehl]], M. Roy, M. Zaks, _Levels in the toposes of simplicial sets and cubical sets_ , JPAA **215** no.5 (2011) pp.949-961. ([preprint](http://arxiv.org/abs/1003.5944))

category: category


Globular

[[!redirects globular]]

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

> _The initial release of Globular works best in the Chrome browser. For discussion, go to the [nForum thread](http://nforum.ncatlab.org/discussion/6829/globular)._

> *Globular* may have been superceded by its successor tool _[[homotopy.io]]_, see there for more.

#Contents#
* table of contents
{:toc}

## Introduction ##

### Overview ###
 {#Overview}

_Globular_ is a web-based [[proof assistant]] for finitely-presented [[semistrict]] [[globular set|globular]] [[higher categories]] ("[[associative n-category|associative $n$-categories]]"). It allows one to formalize higher-categorical proofs in finitely-presented [[n-categories]], visualize them as [[string diagrams]], and share them with collaborators, or with the world. It is available at this address:

* [http://globular.science](http://globular.science)

**We recommend the Chrome browser.**

Globular currently operates up to the level of 4-categories. By the yoga of [[k-tuply monoidal n-category|k-tuply monoidal n-categories]], this also allows one to construct proofs in the following settings:

* Monoidal 3-category
* Monoidal 2-category, optionally with a braiding
* Monoidal category, optionally with a braiding or symmetry

The model of [[semistrict]] higher categories that _Globular_ uses puts all the weak structure in the interchangers, and higher associated structures. At the level of [[3-categories]], _Globular_ implements the axioms of a [[Gray category]]; since every tricategory is known to be equivalent to a semistrict 3-category, this means that any algebraic proof in a finitely-presented tricategory can be formalized. At the level of 4-categories, this is an implementation of a new definition of semistrict 4-category which is not yet published, and for which it is conjectured that every weak 4-category is equivalent to a semistrict 4-category.

_Globular_ is free to use, and [open-source](https://github.com/jamievicary/globular). There is a lot to be done, on both a theoretical and practical level, and anybody can get involved with development. If you are interested, please get in touch with [Jamie Vicary](http://www.cs.ox.ac.uk/people/jamie.vicary/). _Globular_ has been developed in the [Quantum Group](http://www.cs.ox.ac.uk/activities/quantum/) of the [Department of Computer Science](http://www.cs.ox.ac.uk/) at the [University of Oxford](http://www.ox.ac.uk/), by [Krzysztof Bar](https://www.cs.ox.ac.uk/people/krzysztof.bar/), Katherine Casey, [Aleks Kissinger](http://www.cs.ru.nl/A.Kissinger/), [Jamie Vicary](http://www.cs.ox.ac.uk/people/jamie.vicary/) and [Caspar Wylie]().

If you've never used Globular before, the [tutorial video](https://youtu.be/QXZJvXgs0xk) is a good place to start. To report a bug, please use the [issue tracker](https://github.com/jamievicary/globular/issues).

We are extremely grateful to John Baez, Manuel Bärenz, Bruce Bartlett, Eugenia Cheng, Chris Douglas, Eric Finster, Nick Gurski, André Henriques, Samuel Mimram and Dominic Verdon for useful discussions.

{#HowToCite} If you would like to cite _Globular_, you can use the BiBTeX entry

      @InProceedings{
      globular,
      author={Krzysztof Bar and Aleks Kissinger and Jamie Vicary},
      title={Globular: an online proof assistant for higher-dimensional rewriting},
      note={\href{http://ncatlab.org/nlab/show/Globular}{ncatlab.org/nlab/show/Globular}},
      pages={34:1--34:11},
      volume=52,
      year=2016,
      booktitle={Leibniz International Proceedings in Informatics}}

### Example proofs ###

To see what can be done with _Globular_, look at these example proofs. To navigate through them, use the _Project_ and _Slice_ controls in the top-right of the main diagram (see [Changing the view](#changingtheview).)

* **Frobenius implies associative** ([globular.science/1512.004](http://globular.science/1512.004)). In a monoidal category, if multiplication and comultiplication morphisms are unital, counital and Frobenius, then they are associative and coassociative.
* **Strengthening an equivalence** ([globular.science/1512.007](http://globular.science/1512.007)). In a 2-category, an equivalence gives rise to an adjoint equivalence, satisfying the snake equations.
* **Pentagon and triangle implies $\lambda_I = \rho_I$** ([globular.science/1512.002](http://globular.science/1512.002)). In a monoidal 2-category, if a pseudomonoid object satisfies pentagon and triangle equations, then it satisfies $\lambda_I = \rho_I$.
* **The antipode is an algebra homomorphism** ([globular.science/1512.011](http://globular.science/1512.011)). For a Hopf algebra structure in a braided monoidal category, the antipode is an algebra anti-homomorphism. (Thanks to [Dominic Verdon](https://www.cs.ox.ac.uk/people/dominic.verdon/) for formalizing this.)
* **The Perko knots are isotopic** ([globular.science/1512.012](http://globular.science/1512.012)). The [Perko knots](http://mathforum.org/mathimages/index.php/Perko_pair_knots) are a pair of 10-crossing knots stated by Little in 1899 to be distinct, but proven by Perko in 1974 to be isotopic. Here we give the isotopy proof. A nice feature is that the second and third Reidemeister moves do not have to be entered, since they are already implied by the 3-category axioms.
* **The pants bordism is commutative** ([globular.science/1601.005] (http://globular.science/1601.005)). For an oriented dualizable object in a monoidal 2-category, the associated pants bordism is commutative. (Based on unpublished notes by [Scott Carter](http://www.southalabama.edu/mathstat/personal_pages/carter/).)
* **K=P#P** ([globular.science/1605.003](http://globular.science/1605.003)). As knotted spheres in 4 dimensions, the Klein bottle is isotopic to the connect sum of two cross caps. (Based on the movie move proof in Section 2.3 of [_Knotted Surfaces and their Diagrams_](https://books.google.ca/books/about/Knotted_Surfaces_and_Their_Diagrams.html?id=zN7xBwAAQBAJ&redir_esc=y) by Scott Carter and Masahico Saito.)
* **The 1-twist spun trefoil is unknotted.** ([globular.science/1607.003](http://globular.science/1607.003)). For every classical (1,1)-knot, the knotted surface in $\mathbb{R}^4$ given by its 1-twist spin is unknotted. Here we give the explicit unknotting isotopy for the trefoil. (Based on the movie move proof in Carter and Saito. Formalization by Scott Carter.)
* **Swallowtail coherence** ([globular.science/1512.006](http://globular.science/1512.006)). In a 3-category, an adjunction of 1-morphisms gives rise to a coherent adjunction satisfying the swallowtail equations.
* **Butterfly coherence** ([globular.science/1605.002](http://globular.science/1605.002)). In a 4-category, an adjunction of 1-morphisms gives rise to a coherent adjunction satisfying the butterfly equations. (Developed by Krzysztof Bar and Jamie Vicary.)
* **Kirby calculus and 4-manifolds** ([globular.science/1808.001](http://globular.science/1808.001)). It's well-known that the braid groups are 3-endomorphism spaces in the free 3-category over a single object and a single (nontrivial) 2-morphism. In Kirby calculus (using the "dotted circle convention" by Akbulut), smooth 4-manifolds are specified by links. By adding a few more generators and relations (for handle slides and handle cancellations, a.k.a. Kirby moves), Kirby diagrams are the endomorphisms of the 2-identity in the previously mentioned 3-category. Some example manifolds are defined and some example diffeomorphisms are formalised. (By Manuel Bärenz)

## Basic functionality

### Overview

**Signatures and diagrams.** Using _Globular_ involves two main techniques: building the _signature_, and building a _diagram_ over the signature. The signature is the list of cells that are available, and the diagram is a particular composite of those cells. A diagram can be thought of as a composite cell in the free semistrict $\infty$-category generated by the signature.

The signature can be enlarged by building diagrams $S$ and $T$, and assigning them in turn as the source and target of a new cell, using the 'Source' and 'Target' buttons in the menu on the right of the interface. Any pair of diagrams can be chosen which satisfy the [[globular set|globularity conditions]] $s(S) = s(T)$ and $t(S) = t(T)$. The new cell can then be given a name and a colour, to make it easy to identify.

Diagrams are built by clicking on cells of the signature, and by clicking and dragging on the diagram itself. Signature cells of dimension $n+1$ act on $n$-diagrams by _rewriting_, replacing a subdiagram of shape $S$ and replacing it with a subdiagram of shape $T$.

The only things that Globular understands are $k$-cells, for some value of $k$. So if you want to build an $n$-category where an equation $f=g$ holds between $n$-cells, you have to do it by adding $(n+1)$-cells $a:f \to g$ and $b:g \to f$. If you then build some composite $C[f]$ involving $f$, you can apply the cell $a$ to obtain $C[g]$, and we interpret this as the _equation_ $C[f]=C[g]$. In a slogan, this is _equality via rewriting_. This is consistent with the basic premise of [[homotopy type theory]]: treat your proofs as first-order structures, which can themselves be reasoned about directly.

**Accounts.** If you create an account on the site, you can save workspaces to your private account, and resume work on them when you log back in to your account from any computer. You can also click 'Share', which shares a workspace privately with another user, and 'Publish', which creates a permanent public version of your workspace that can be linked to from a research paper, or from elsewhere on the web.

**Export and import.** Use the 'Export' feature to download a copy of your workspace to your local machine, and 'Import' to upload a previously-download workspace. This functionality only involves your browser, not the globular.science server, so you do not need to log in or even create an account to use these features. If you import a workspace, log in, and click 'Save', then a copy of the imported workspace will be saved to the server.

**Security and privacy.** The plain text of your password is not stored on the server. However, your private workspaces are saved in plain text, and as such should not be considered highly secure, as they could be accessed by someone with site administrator credentials. The administrators pledge not to access any user data, except from the registered email address, without the permission of that individual user. In other words: your private work is private! User data is backed up nightly to a secure server.

### Manipulating diagrams

A diagram is a composite of the cells in the current signature, and basic diagram manipulation can be performed by clicking on the signature, as follows, where we suppose our diagram $D$ is $k$-dimensional.

* **Creation.** If the current diagram is empty, clicking the icon of a cell in the signature constructs a diagram which consists solely of the selected generator.
* **Rewriting.** Selecting a $(k+1)$-cell $G$ from the signature displays a list of all the ways that its source $s(G)$ can be identified as a subdiagram of $D$. Choosing element of this list will then modify $D$, deleting the part of it which was chosen as matching the source $s(G)$, and replacing it with the target $t(G)$.
* **Attaching.** Selecting a $p$-cell $G$ for $p \leq k$ displays a list of all the ways that $G$ can be attached to $D$ along one of its boundaries. Choosing an element of this list performs the attachment.

The menu on the right-hand side of the screen gives further commands. Each of these commands has a shortcut key, which is also given.

* **Clear _c_.**  Deletes the current diagram.
* **Identity _i_.** Builds the identity $(k+1)$-diagram on the current $k$-diagram.
* **Source _s_.** Saves the current diagram as the source of a new generator.
* **Target _t_.** Saves the current diagram as the target of a new generator.
* **Restrict _r_.** Replaces the current diagram with the subdiagram currently being viewed (see [Changing the view](#changingtheview).)
* **Theorem _h_.** Creates a new theorem witnessing the current diagram (see [Invertibility](#invertibility).)
* **Graphic _g_.** Downloads a PNG representation of the current diagram.

Also, the user can click and click-and-drag on the diagram to produce various effects. When a command is ambiguous, a menu will be presented to allow the user to choose their preferred action.

* **Click.** This attempts to attach or rewrite, as appropriate for the location of the click, with an element of the signature. If an element of the signature is marked as invertible, then the higher structure this implies will also be considered for attachment.
* **Click-and-drag.** This attempts to attach or rewrite using an interchanger, or naturality for the interchanger, or invertibility of some cell. This is sensitive to both the click location and the direction of the drag, in a way which should be intuitive. This also allows cells to be cancelled from the top or bottom of a diagram.

Clicking and dragging is designed to work as if you were really 'touching' the strings. So if you want to braid one strand over another, click the strand to 'grab' it, and 'pull' it to one side. If you want to pull a vertex through a braiding, click the vertex to 'grab' it, and 'pull' it up or down through its adjacent braiding. Of course, _Globular_ will only carry out the command if the move you are attempting to make is actually valid in that location.

The most common move to perform is a basic interchanger, where two unrelated critical points with adjacent heights have their heights exchanged. Holding the Shift button while clicking-and-dragging tells _Globular_ to only look for basic interchanger moves. This speeds up the processing, and is particularly useful for applying long chains of interchangers.

In order to undo any change then click the back button in your browser. This feels very counter-intuitive but trust me it works!

### Changing the view ###
{#changingtheview}

_Globular_ can currently present string-diagram visualizations of 0-, 1- and 2-dimensional diagrams. Viewing a $k$-dimensional diagram for $k \gt 2$ is made possible by _projection_ and _slicing_, controlled by number scrollers which appear in the top-right of a diagram.

* **Project.** This determines the number of dimensions to be projected; it can take any value $p$ in the range $0 \leq p \leq k$. Projecting a dimension causes it to be effectively ignored by _Globular_; it is as if you are looking at a shadow of the true image. The resulting diagrams are exactly the _singularity graphics_ which are a common tool in [[Morse theory|higher Morse theory]].
* **Slice.** Having decided to project out $p$ dimensions, and supposing the image being displayed is $v$-dimensional (for $v \in \{0,1,2\}$), that leaves $k - p - v$ dimensions still to be handled. If this sum is positive, coordinates must be chosen using the slice controls to position the view within an appropriate $v$-dimensional subdiagram.

You can also zoom the main diagram using the scroll wheel of your mouse, and scroll around by clicking-and-dragging with the right mouse button (on a Mac trackpad, this is a two-fingered press-hard-and-drag. Yes, this really is a thing. If it doesn't seem to work, you're not pressing hard enough---the trackpad should 'click' under your fingers.)

### Technical implementation details

Globular is written in Javascript, and performs all its computations client-side, in the browser. Most of the computations it has to perform are linear in the size of the diagram, so this is quite computationally feasible, even for large proofs. The back-end is a Node.js server that allows the saving, sharing and publishing of workspaces. Graphics are implemented in SVG. Project data is compressed using the LZ4 algorithm.

## Singularities ##

### Overview ###

In a 2-category, a 1-cell and a 2-cell can be composed in one direction (horizontal composition), and two 2-cell can be composed in two directions (horizontal and vertical composition.) In general, in a traditional globular $n$-category, a $p$-cell and an $q$-cell can be composed in $\text{min}(p,q)$ different directions: along a common boundary 0-cell, or 1-cell, and so on, up to a common boundary $(\text{min}(p,q)-1)$-cell.

In _Globular_, composition works differently: a $p$-cell and a $q$-cell can be composed in just _one_ way: along a common boundary $(\text{min}(p,q)-1)$-cell. The reason for this design decision is that in terms of the graphical calculus, of all the different composites that an ordinary globular $n$-category allows, this is the only composition that results in a diagram in _generic position_. Nothing is lost by this restriction, since (conjecturally) the other composites can be recovered up to isomorphism by whiskering, in just the same way that horizontal composition of 2-cells in a 2-category can be expressed in terms of vertical composites and whiskering of 2-cell by identity 1-cell.

For every way that _Globular_ _doesn't_ let you compose cells---that is, for every $p,q$ and every $0 \leq n \lt \text{min}(p,q)$---there exists a $(p+q-n-1)$-cell, which _resolves_ the singularity that would be created if this composite were actually formed. These singularity resolutions manifest in _Globular_ as higher-dimensional cells, which allow parts of a diagram to move around each other; using them effectively is a key part of formalizing higher-dimensional proofs.
The main difficulty in the definition of semistrict $n$-categories is describing these structures.

In this section we list the singularities which _Globular_ recognizes. We organize them by categorical dimension, because to formalize a proof in an $n$-category, you may need to make use of every type of $k$-category singularity for $k \leq n$. Also, you can add any fixed natural number to the dimensions involved to get higher-dimensional versions of the same singularities; for example, interchangers arise as the composite of two $n$-cells along a common boundary $(n-2)$-cell.

### 2-category singularities ###

**Interchangers**

* _Composite:_ two $2$-cells composed along a common boundary 0-cell, forming a 3-cell resolution.
* _Interface:_ When unprojected, click the vertex to be interchanged and drag it up or down; if the interchanger is ambiguous because the source or target is the identity, then make your drag slightly to the left or right. When projected, grab the wire to be interchanged, and drag in the desired direction.

![Interchanger source](http://globular.science/nlab/interchanger-source.png)
$\mathraisebox{1.8cm}{\to}$
![Interchanger target](http://globular.science/nlab/interchanger-target.png)
$\qquad\qquad$
![Interchanger projection](http://globular.science/nlab/interchanger-projection.png)

### 3-category singularities ###

**Naturality of the interchanger in one variable**

* _Composite:_ a $2$-cell and a $3$-cell composed along a common boundary 0-cell, forming a 4-cell resolution.
* _Interface:_ ...

### 4-category singularities ###

**Naturality of naturality of the interchanger in one variable**

* _Composite:_ a 2-cell and a 4-cell composed along a common boundary 0-cell, forming a 5-cell resolution.
* _Interface:_ ...

**Naturality of the interchanger in two variables**

* _Composite:_ two 3-cells composed along a common boundary 0-cell, forming a 5-cell resolution.
* _Interface:_ ...

**Naturality of invertibility of the interchanger**

* _Composite:_ ...
* _Interface:_ ...

**Contractible 3rd Reidemeisterators**

* _Composite:_ ...
* _Interface:_ ...

## Other features

### Invertibility ###
{#invertibility}

A $k$-cell $f:A \to B$ in an $\infty$-category is _invertible_ when there is an inverse cell $f ^{-1} : B \to A$, and invertible $(k+1)$-cells $\mu: \mathrm{id}_A \to f ^{-1} \circ f$ and $\nu: \mathrm{id}_B \to f \circ f ^{-1}$. This is a co-inductive definition, giving rise to an infinite amount of higher-dimensional structure.

In _Globular_, selecting the 'Invertible' option for a cell in the signature allows this higher structure to be used in diagrams, by clicking or clicking-and-dragging at a point of a diagram or its boundary. If more than one piece of higher invertible structure could be inserted at the location of the click, a menu is presented and the user can choose which is intended.

### Popup labels

When you move your pointer over a diagram, a little label will pop up telling you what's underneath the pointer. Doing this near a vertex, you will see that Globular isn't really aware that the lines curve, so you will need to take care when clicking or clicking-and-dragging that Globular is correctly identifying the structure you are trying to interact with.

The labels are written using the following syntax, where 'X' is some arbitrary label denoting a $k$-cell, and where '.' is forward composition of $k$-cells. Note that every comma increases the dimension by 1.

* 'X inverse' describes the $k$-cell which is the inverse of $X$
* 'X, insert' describes the canonical $(k+1)$-cell of type $\text{id} _{s(X)} \to X . X^{-1}$
* 'X, cancel' describes the canonical $(k+1)$-cell of type $X . X^{-1} \to \text{id} _{s(X)}$
* 'Interchanger' describes the height-exchange cell
* 'Pull-through interchanger above', 'Pull-through interchanger underneath', 'Pull-through inverse interchanger above' and 'Pull-through inverse interchanger underneath' describe the 4 different ways that a cell can be pulled through an interchanger

Note that interchangers are of dimension at least 3, and pull-throughs are of dimension at least 4.

### Theorems

The theorem command allows a diagram to be stored, in a such way that it can be reused as a component in subsequent diagrams. The idea is that the diagram is a 'theorem' proving that its source can be rewritten into its target. The 'proof' is the diagram itself.

Concretely, let $D$ be the current diagram, which is $k$-dimensional, and let $s(D)$ and $t(D)$ be its source and target $(k-1)$-diagrams. Then the theorem command command creates two new generators:

 * A $k$-generator $T$ with source $s(D)$ and target $t(D)$
 * An invertible $(k+1)$-generator $P$ with source $T$ and with target $D$

The generator $T$ can be used in future proofs to 'stand in' for the diagram $D$. At any point, the generator $P$ can be used to replace $T$ with the content of diagram $D$. Since $P$ is invertible, it could also be used to identify $D$ as a subdiagram of a larger proof, and replace it with $T$, making the larger proof easier to understand.

### Coordinates

Composites produced by Globular are in generic position; for example, no two vertices ever appear at the same height. As a result, there is a simple coordinate system that can be used to refer to elements of a diagram. For a $k$-dimensional cell in an $n$-dimensional diagram, $(n-k+1)$ coordinates must be provided; that is, the number of coordinates is the codimension of the cell, plus 1. Moving the mouse cursor over a diagram in the screen shows the coordinate of the element underneath the cursor.

The coordinate system for an $n$-diagram is defined in the following way. An $n$-cell at height $y$ has coordinate $[y]$. A $k$-cell for $k \lt n$ has leading coordinate given by its height $z$ in the diagram, and subsequent coordinates given by its coordinates in the slice diagram at that height.

## Related entries

* [[homotopy.io]]

* [[rewriting]]

[[!include proof assistants and formalization projects -- list]]

## References
 {#References}

* [[Krzysztof Bar]], [[Aleks Kissinger]], [[Jamie Vicary]], *Globular: an online proof assistant for higher-dimensional rewriting*,  Logical Methods in Computer Science **14** 1 (2018) [[doi:10.23638/LMCS-14(1:8)2018](https://doi.org/10.23638/LMCS-14(1:8)2018), [arXiv:1612.01093](https://arxiv.org/abs/1612.01093)]

* [[Jamie Vicary]], _Globular tutorial video_, 2015-12-07 [15 minute video] (https://youtu.be/QXZJvXgs0xk)

* [[Jamie Vicary]], _Data Structures for Quasistrict Higher Categories_, referencing [arxiv:1610098] (https://arxiv.org/abs/1610.06908), 2016-12-05 talk at Simons Institute [48 minute video] (https://youtu.be/9w8t0Ze08l4)

category: software


globular category

*The* __globular category__ is probably the [[globe category]].

But *a* __globular category__ could also be an [[internal category]] or an [[enriched category]] in/over [[globular sets]].  For instance, a [[monoidal globular category]] is an internal category in globular sets equipped with additional structure.

See [[simplicial category]] for an analogous discussion.


globular operad

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of  _globular operads_ is a variant of that of [[operads]] on which certain [[algebraic definition of higher category|algebraic notions of higher category]] are based. The notion was introduced by [[Batanin]]; a globular operad is also called a _Batanin operad_.

A globular operad gives rise to a [[monad]] on the [[category]] of [[globular sets]]; one example is the free [[strict ∞-category]] monad $T$ on globular sets. The monads which so arise may be characterized precisely as [[cartesian monads]] on globular sets over $T$ (itself a cartesian monad).  This means that they are also examples of [[generalized multicategories]] relative to $T$.

## Definition ##

### Globular collections

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

A **globular collection** is a [[globular set]] $X$ equipped with a map

$$f: X \to T(1)$$

to $T(1)$, the underlying globular set of the free strict $\omega$-category on the terminal globular set. Hence the category $Coll$ is the [[slice category]]

$$Coll = Set^{Glob^{op}}/T(1)$$

=--

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

The category of collections carries a [[monoidal category|monoidal product]] $\circ$ defined as follows. Given collections $f: X \to T(1)$, $g: Y \to T(1)$, the underlying globular set of $X \circ Y$ is given by [[pullback]]

$$\array{
X \circ Y & \to & T(Y) \\
\downarrow & & \downarrow T(!) \\
X & \underset{f}{\to} & T(1)
}$$

and the requisite map $X \circ Y \to T(1)$ is given by the composite

$$X \circ Y \to T(Y) \overset{T(g)}{\to} T T(1) \overset{\mu(1)}{\to} T(1)$$

where $\mu: T T \to T$ denotes multiplication of the monad $T$. The monoidal unit is the collection $u(1): 1 \to T(1)$ where $u: Id \to T$ is the unit of $T$, and the associativity and unit constraints may be defined by means of universal properties, taking advantage of the fact that $T$ is cartesian.

=--

### Globular operads

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

A **globular operad** is a [[monoid]] in the [[monoidal category]] $Coll$, (with the monoidal structure given by  def. \ref{MonoidalStructureOnCollections}).

=--

### Monad of a globular operad

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

Each globular operad $f: P \to T(1)$, (as in def. \ref{GlobularOperad}), gives rise to a globular [[monad]] $M_P$ on $Set^{Glob^{op}}$. Abstractly, $M_P(X)$ is just the pullback

$$\array{
  M_P(X) & \to & T(X) \\
  \downarrow & & \downarrow T(!) \\
  P & \underset{f}{\to} & T(1)
}$$

and the multiplication and unit for $M_P$ may be worked out from the multiplication and unit for the globular operad $P$.

=--

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

A more concrete description of $M_P(X)$ may be worked out in terms of a concrete description of the free strict $\omega$-category $T(X)$. To describe this, first notice that every element $\tau$ of $T(1)$, which is essentially a pasting diagram built up out of globes of $1$, can be drawn as a globular set which we denote as $[\tau]$. The globes of $[\tau]$ are instances of globular cells as they appear in the pasting diagram $\tau$, and their sources and targets are then also instances of cells in $\tau$. (Batanin describes $T(1)$ in terms of trees, and the globular set $\tau$ is given formally in the tree language.)

Similarly, we can think of an element of $T(X)$ as a pasting diagram built out of globes in $X$, and such a pasting diagram can be thought of as having an underlying shape given by an pasting diagram $\tau$ in $T(1)$, together with a labeling of the pasting cells in $\tau$ by elements on $X$. The labeling is in fact just a morphism $[\tau] \to X$ of globular sets. Therefore we have an explicit formula for the set of $n$-cells of $T(X)$:

$$T(X)(n) = \sum_{\tau \in T(1)(n)} \hom([\tau], X)$$

and similarly, for a globular operad with underlying collection $f: P \to T(1)$,

$$M_P(X)(n) = \sum_{x \in P(n)} \hom([f(x)], X)$$

=--

### Categories of operators
 {#CategoryOfOperators}

The [[category of operators]] of a globular operad $A$ is (the [[syntactic category]] of) a homogeneous [[globular theory]] $i_A \colon \Theta_0 \to \Theta_A$ and every globular operad is characterized by its [[globular theory]]. See there for more details

## Examples

### The globular operad for $\omega$-categories

Write $\omega$ for the globular operad whose [[category of operators]], see [above](#CategoryOfOperators), is the [[Theta category]] $\Theta$.

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

The category $Str\omega Cat$ of [[strict ∞-categories]] is [[equivalence of categories|equivalent]] to that of algebras over the terminal globular operad. Hence it is the [[full subcategory]] of that of [[∞-graphs]] which satisfy the [[Segal condition]] with respect to the canonical inclusion $\Theta_0 \to \Theta$ that defines its [[globular theory]]: we have a [[pullback]]

$$
  \array{
    Str\omega Cat
     &\underoverset{\simeq}{N}{\to}& Mod_\Theta &\hookrightarrow&
PSh(\Theta)
    \\
    \downarrow^{\mathrlap{U}} && \downarrow^{} && \downarrow
   \\
   \omega Graph &\stackrel{\simeq}{\to}& Sh(\Theta_0) &\hookrightarrow& PSh(\Theta_0)
  }
  \,.
$$

=--

([Berger, theorem 1.12](#Berger))

### Weak $\omega$-categories

As a refinement of the above example:

In the Batanin (or Leinster) theory of $\infty$-categories, there is a universal contractible globular operad $f: K \to T(1)$, where each element $x \in K(n)$ is thought of as a _way_ of (weakly) pasting together the underlying shape $f(x)$. The contractibility implies that for every two different ways of pasting together the same shape, i.e., two elements $x, y \in K(n)$ such that $f(x) = f(y)$ and such that $x$ and $y$ have the same source and have the same target, there is an $(n+1)$-cell in $K(n+1)$ mediating between them, with source $x$ and target $y$, and which maps to the identity $(n+1)$-cell on $f(x)$.

A [[Batanin ∞-category]] is a globular set with a $K$-algebra structure.

## References

A review and characterization in terms of [[globular theories]] is in section 1 of

* [[Clemens Berger]], _A cellular nerve for higher categories_, Advances in Mathematics 169, 118-175 (2002) ([pdf](http://math1.unice.fr/~cberger/nerve.pdf))
 {#Berger}

Other work on globular operads :

* [[Camell Kachour]], Operads of higher transformations for globular sets and for higher magmas, Published in Categories and General Algebraic Structures with Applications (2015).

[[!redirects globular operads]]


globular set

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Graph theory
+-- {: .hide}
[[!include graph theory - contents]]
=--
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

# Globular sets
* table of contents
{: toc}

## Idea

Globular sets are to [[simplicial sets]] as [[globes]] are to simplices.

Much like [[simplicial sets]] underly common [[geometric definitions of higher categories]], so globular sets underly some [[algebraic definitions of higher categories]], see [below](#AsShapesForHigherCategories).

## Definition

### Basic definition

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

The **[[globe category]]** $\mathbb{G}$ is the [[category]] whose [[objects]] are the [[natural numbers]], denoted here $[n] \in \mathbb{N}$ (N.B. not to be confused with ordinals in any structural sense) and whose morphisms are [[generators and relations|generated]] from

$$
  \sigma_n : [n] \to [n+1]
$$
$$
  \tau_n : [n] \to [n+1]
$$

for all $n \in \mathbb{N}$ subject to the relations (dropping obvious subscripts)

$$
  \sigma\circ \sigma = \tau \circ \sigma
$$
$$
  \sigma\circ \tau = \tau \circ \tau
  \,.
$$

=--

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

A _globular set_, also called an *$\omega$-[[graph]]*, is a [[presheaf]] on $\mathbb{G}$. The [[category]] of globular sets is the [[category of presheaves]]

$$
  gSet \coloneqq PSh(\mathbb{G})
  \,.
$$

=--

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

This means that a globular set $X \in gSet$ is given by a collection of sets $\{X_n\}_{n \in \mathbb{N}}$ (the **set of $n$-[[globes]]**) equipped with [[functions]]

$$
  \{s_n,t_n \colon X_{n+1} \to X_n\}_{n \in \mathbb{N}}
$$

called the **$n$-source** and **$n$-target** maps (or similar), such that the **globular identities** hold: for all $n \in \mathbb{N}$

* $s_n \circ s_{n+1} = s_n \circ t_{n+1} $

* $t_n \circ s_{n+1} = t_n \circ t_{n+1} $.

=--

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

The globular identities ensure that two sequences of boundary maps

$$
  f_n \circ \cdots \circ f_{n+m-1} \circ f_{n+m} :
  S_{n+m+1} \to S_n
$$

with $n,m \in \mathbb{N}$ and for $f_k, \in \{s_k, t_k\}$ are
equal if and only if their last term $f_n$ coincides;
for all $n,m \in \mathbb{N}$ we have

$$
    s_n \cdots s_{n+1} \circ \cdots \circ
    s_{n+m} \circ i_{n+m} \circ \cdots \circ i_{n+1} \circ i_n
    =
    \mathrm{Id}
$$
$$
    t_n \cdots t_{n+1} \circ \cdots \circ
    t_{n+m} \circ i_{n+m} \circ \cdots \circ i_{n+1} \circ i_n
    =
    \mathrm{Id}
    \,.
$$

For $S$ a globular set we may therefore write unambiguously

$$
  s_n, t_n : S_{n+m+1} \to S_n
$$
$$
  i_n : S_n \to S_{n+m+1}
$$

with $i_n, s_n, t_m$ the sequence of $m$ consecutive identity-assigning, source or target maps, respectively.

=--

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

The presheaf definition can understood from the point of view of [[space and quantity]]: a _globular set_ is a space characterized by the fact that and how it may be _probed_ by mapping standard globes into it: the set $S_n$ assigned by a globular set to the standard $n$-globe $[n]$ is the set of $n$-globes in this space, hence the way of mapping a standard $n$-globe into this spaces.
=--

More generally:

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

A **globular object** $X$ [[internalization|in]] a [[category]] $\mathcal{C}$ is a [[functor]] $X : \mathbb{G}^{\mathrm{op}} \to \mathcal{C}$.

=--

### Reflexive globular sets {#ReflexiveGlobularSets}

If to the globe category we add additional generating morphisms

$$
  \iota_n : [n+1] \to [n]
$$

satisfying the relations

$$
  \iota \circ \sigma = \mathrm{Id}
$$
$$
  \iota \circ \tau = \mathrm{Id}
$$

we obtain the **reflexive globe category**, a presheaf on which is a
**reflexive globular set**.  In this case the morphism

$$
  i_n := S(\iota_n) : S_{n} \to S_{n+1}
$$

is called the $n$th **identity assigning map**; it satisfies the globular identities:

$$
  s \circ i = \mathrm{Id}
$$
$$
  t \circ i = \mathrm{Id}
$$

### $n$-globular sets

A presheaf on the full subcategory of the globe category containing only the integers $[0]$ through $[n]$ is called an **$n$-globular set** or an **$n$-graph**.  An $n$-globular set may be identified with an $\infty$-globular set which is empty above dimension $n$.

Note that a $1$-globular set is just a [[directed graph]], and a $0$-globular set is just a [[set]].

## Examples

* Any [[strict 2-category]] or [[bicategory]] has an underlying 2-globular set.  Likewise, any [[tricategory]] has an underlying 3-globular set.  Globular sets can be used as underlying data for [[n-categories]] as well; see for instance [[Batanin ∞-category]].

* A _[[strict omega-category]]_ is a globular set $C$ equipped in each degree with the structure of a [[category]] such that for every pair $k_1 \lt k_2 \in \mathbb{N}$ the induced structure on the 2-graph $C_{k_2} \stackrel{\to}{\to} C_{k_1} \stackrel{\to}{\to} C_0$ is that of a [[strict 2-category]].

* The **globular $n$-globe** $G_n$ is the globular set represented by $n$, i.e. $G_n(-) := Hom_G(-,n)$.

## Properties

### Grothendieck homotopy theory

The category of [[globes]] is not a [[weak test category]] according
to Scholium 8.4.14 in [Cisinski 06](#Cisinski06).

However, if we construct the free [[strict monoidal category]]
on the category of [[globes]], while ensuring that the terminal object becomes the monoidal unit,
then the resulting category of polyglobes is a [[test category]].

### As shapes for higher categories
 {#AsShapesForHigherCategories}

Globular sets may be used as a [[geometric shape for higher structures|geometric shape]] for [[algebraic definitions of higher categories]]:

* Equipped with strictly compatible [[composition]] [[structure]] on cells in any dimension, globular sets model *[[strict ∞-categories]]* (originally often: "[[omega-category|omega-categories]]", see also at *[[complicial sets]]*), historically one of the earliest notions of [[higher category theory]] but too restrictive to be useful for most purposes (in their further restriction to [[strict omega-groupoids]] they [are equivalent](strict+omega-groupoid#RelationToCrossedComplexes) just to [[crossed complexes]]).

* A more general ([[semi-strict infinity-category|semi-strict]]) notion of [[n-categories|$n$-categories]] modeled on globular sets are known as *[[associative n-category|associative $n$-categories]]*, see there for more, and see a corresponding [[proof assistant]]: *[[homotopy.io]]* (previously: *[[Globular]]*).

## Related concepts

* [[graph]]

* [[directed graph]]

  * **directed $n$-graph**

  * [[directed (n,r)-graph|directed $(n,r)$-graph]]

* [[ribbon graph]]

* [[Theta category]],

* [[globular theory]], [[globular operad]]

* Also related is the notion of [[computad]], which is similar to a globular set in some ways, but allows "formal composites" of $n$-cells to appear in the sources and targets of $(n+1)$-cells.

* [[simplicial object]]

  * [[simplicial set]]

  * [[simplicial object in an (∞,1)-category]]

* [[semi-simplicial object]]

  * [[semisimplicial set]]

* [[strict infinity-category]], [[strict omega-groupoid]]

* [[associative n-category]], [[Globular]]

## References

The definition is reviewed around def. 1.4.5, p. 49 of

* [[Tom Leinster]]: _Higher operads, higher categories_ ([arXiv:math/0305049](http://arxiv.org/abs/math/0305049))

See also

* [[Sjoerd Crans]], _On combinatorial models for higher dimensional homotopies_ ([web](https://web.archive.org/web/20130514164856/http://home.tiscali.nl/secrans/papers/comb.html))

* [[R. Street]], _The petit topos of globular sets_ , JPAA **154** (2000) pp.299-315.

* {#Cisinski06} [[Denis-Charles Cisinski]], *Les préfaisceaux comme types d'homotopie*, Astérisque **308** Soc. Math. France (2006), 392 pages [[numdam:AST_2006__308__R1_0](http://www.numdam.org/item/?id=AST_2006__308__R1_0) [pdf](http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf)]

The definition of globular set, without using that term, is in 2.2 and 2.3 of the following paper:

* [[Ronnie Brown]], [[Philip J. Higgins]], *The equivalence of $\infty$-groupoids and crossed  complexes*,  Cah. Top. G\'eom. Diff. 22 (1981) 371-386.

[[!redirects globular set]]
[[!redirects globular sets]]
[[!redirects reflexive globular set]]
[[!redirects reflexive globular sets]]
[[!redirects n-globular set]]
[[!redirects n-globular sets]]
[[!redirects reflexive n-globular set]]
[[!redirects reflexive n-globular sets]]
[[!redirects omega-graph]]
[[!redirects omega-graphs]]
[[!redirects ∞-graph]]
[[!redirects ∞-graphs]]
[[!redirects infinity-graph]]
[[!redirects infinity-graphs]]
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[[!redirects ∞-graphs]]
[[!redirects n-graph]]
[[!redirects n-graphs]]
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[[!redirects infinity-quivers]]
[[!redirects ∞-quiver]]
[[!redirects ∞-quivers]]
[[!redirects n-quiver]]
[[!redirects n-quivers]]
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[[!redirects directed omega-graphs]]
[[!redirects directed ∞-graph]]
[[!redirects directed ∞-graphs]]
[[!redirects directed infinity-graph]]
[[!redirects directed infinity-graphs]]
[[!redirects directed ∞-graph]]
[[!redirects directed ∞-graphs]]
[[!redirects directed n-graph]]
[[!redirects directed n-graphs]]
[[!redirects omega-directed graph]]
[[!redirects omega-directed graphs]]
[[!redirects ∞-directed graph]]
[[!redirects ∞-directed graphs]]
[[!redirects infinity-directed graph]]
[[!redirects infinity-directed graphs]]
[[!redirects ∞-directed graph]]
[[!redirects ∞-directed graphs]]
[[!redirects n-directed graph]]
[[!redirects n-directed graphs]]
[[!redirects reflexive omega-graph]]
[[!redirects reflexive omega-graphs]]
[[!redirects reflexive ∞-graph]]
[[!redirects reflexive ∞-graphs]]
[[!redirects reflexive infinity-graph]]
[[!redirects reflexive infinity-graphs]]
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[[!redirects reflexive n-quiver]]
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[[!redirects reflexive directed omega-graph]]
[[!redirects reflexive directed omega-graphs]]
[[!redirects reflexive directed ∞-graph]]
[[!redirects reflexive directed ∞-graphs]]
[[!redirects reflexive directed infinity-graph]]
[[!redirects reflexive directed infinity-graphs]]
[[!redirects reflexive directed ∞-graph]]
[[!redirects reflexive directed ∞-graphs]]
[[!redirects reflexive directed n-graph]]
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globular theory

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

#Contents#
* table of contents
{:toc}

## Idea

A _globular theory_ is much like an _[[algebraic theory]]_ / _[[Lawvere theory]]_ only that where the former has [[objects]] labeled by [[natural numbers]], a globular theory has objects labeled by [[pasting diagrams]] of [[globes]]. The [[models]] of "homogeneous" globular theories are precisely the algebras over [[globular operads]].

## Definition

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

Write

* $Str \omega Cat \in Cat$ for the category of [[strict ∞-categories]];

* $\Theta \hookrightarrow Str\omega Cat$ for the the **[[Theta category]]**, the [[full subcategory]] on the strict $\omega$-categories [[free construction|free]] on [[∞-graphs]]([[globular sets]]) (the [[pasting diagrams]]);

* $i \colon \Theta_0 \to \Theta$ for the [[wide subcategory|wide]] non-full subcategory on the morphism induced from morphisms of the underlying [[∞-graphs]]  (this means that these morphisms in $\Theta_0$ send $n$-globes to single $n$-globes, not to pastings of them).

* $\mathbb{G} \hookrightarrow \Theta_0$ for the [[full subcategory]] on the pasting diagrams given by a single globe -- the **[[globe category]]**.

=--

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

The **globular site** is the category $\Theta_0$ from def. \ref{CategoryOfGlobularSite} equipped with the structure of a [[site]] by taking the [[covering]] families to be the jointly [[epimorphism|epimorphic]] families.

=--

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

A **globular theory** (or rather its [[syntactic category]]) is a [[wide subcategory]] inclusion

$$
  i_A \colon \Theta_0 \to \Theta_A
$$

of the globular site, def. \ref{GlobularSite}, such that every [[representable functor]] $\Theta_A(-,T) \colon \Theta_A^{op} \to $ [[Set]] is a $\Theta_A$-model:

A **$\Theta_A$-[[model]]** is a [[presheaf]] $X \in PSh(\Theta_A)$ which
restricts to a [[sheaf]] on the globular site, $i_A^* X \in Sh(\Theta_0) \hookrightarrow PSh(\Theta_0)$.

Write

$$
  Mod_A \hookrightarrow PSh(\Theta_A)
$$

for the [[full subcategory]] of the [[category of presheaves]] on the $\Theta_A$-models. This is the category of $\Theta_A$-models.

=--

([Berger, def. 1.5](#Berger))

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

Given an globular theory $i_A \colon \Theta_0 \to \Theta_A$ a morphism in $\Theta_A$ is

* an **$A$-cover** if...;

* an **immersion** if...

=--

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

A globular theory $i_A \colon \Theta_0 \to \Theta_A$ is **homogeneous** if it contains a [[subcategory]] $\Theta^{cov}_A \to \Theta_A$ of $A$-covers, def. \ref{CoversAndImmersions} such that every morphism in $\Theta_A$ factors uniquely as an $A$-cover, def. \ref{CoversAndImmersions}, followed by an immersion, def. \ref{CoversAndImmersions}.

=--

([Berger, def. 1.15](#Berger))

## Properties

### Relation to $\omega$-graphs

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

The [[category of sheaves]] over the globular site
is [[equivalence of categories|equivalent]] to the category of [[∞-graphs]]

$$
  \omega Graph \simeq Sh(\Theta_0)
  \,.
$$

=--

([Berger, lemma 1.6](#Berger))

### Relation to globular operads

The ([[syntactic categories]] of) homogenous globular theories, def. \ref{HomogeneousTheory} are the [[categories of operators]] of [[globular operads]]:

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

A [[faithful functor|faithful]] [[monad]] $\underline{A}$ on [[∞-graphs]] encodes algebras over a [[globular operad]] $A$ precisely if

1. the induced globular theory $\Theta_A \hookrightarrow Alg_{\underline{A}}$ is homogeneous, def. \ref{HomogeneousTheory};

1. every $\underline{A}$-[[algebra over a monad|algebras]] factors uniquely into an $A$-cover followed by an $\underline{A}$-free $\underline{A}$-algebra morphism.

=--

([Berger, prop. 1.16](#Berger))

## Examples

### The theory of $\omega$-categories

The [[Theta category]] itself, equipped with the definition inclusion $i \colon \Theta_0 \to \Theta$, def. \ref{CategoryOfGlobularSite}, is the globular theory of [[∞-categories]].

In particular:

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

The category $Str\omega Cat$ of [[strict ∞-categories]] is [[equivalence of categories|equivalent]] to that of $\Theta$-models, def. \ref{GlobularTheory}. Hence it is the [[full subcategory]] of that of [[∞-graphs]] which satisfy the [[Segal condition]] with respect to the canonical inclusion $\Theta_0 \to Theta$: we have a [[pullback]]

$$
  \array{
    Str\omega Cat
     &\underoverset{\simeq}{N}{\to}& Mod_\Theta &\hookrightarrow&
PSh(\Theta)
    \\
    \downarrow^{\mathrlap{U}} && \downarrow^{} && \downarrow
   \\
   \omega Graph &\stackrel{\simeq}{\to}& Sh(\Theta_0) &\hookrightarrow& PSh(\Theta_0)
  }
  \,.
$$

=--

([Berger, theorem 1.12](#Berger))

## References

Section 1 of

* [[Clemens Berger]], _A cellular nerve for higher categories_, Advances in Mathematics 169, 118-175 (2002) ([pdf](http://math1.unice.fr/~cberger/nerve.pdf))
 {#Berger}

[[!redirects globular theories]]

[[!redirects globular site]]
[[!redirects cellular site]]


glueball

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
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## Idea

In [[quantum chromodynamics]], in fact already in pure [[special unitary group|SU(3)]] [[Yang-Mills theory]], a _glueball_ is a [[bound state]] of (just) [[gluons]] (a kind of [[confinement]]).

Glueballs are hard to detect in [[experiment]] but their existence is confirmed by [[lattice QCD]], also by the [[AdS-QCD correspondence]] (see there). According to [Greensite 11, section 8.5](#Greensite11):

> The [[caloron]] idea is probably the most promising current version of [[monopole]] [[confinement]] in pure non-abelian gauge theories, but it is  basically  (in certain [[gauge fixing|gauges]]) a superposition of [[monopoles]] with spherically  symmetric abelian fields, and this leads to the same questions raised in connection with monopole Coulomb gases.

## Properties

### In holographic QCD

In [[holographic QCD]], glueballs are [[AdS/CFT duality|dual]] to massless [[scalar fields]] in the $\sim$[[AdS spacetime|AdS]]-[[bulk spacetime]] (for instance, but not necessarily, the [[dilaton]]).

The following graphics shows predictions of glueball spectra from a simple [[AdS/QCD]]-model compared to [[lattice QCD]]-computations

\begin{imagefromfile}
  "file_name": "GlueballSpectrumFromAdSQCD.jpg",
  "width": 600,
  "caption": "From [Brower-Mathur-Tan 00](#BrowerMathurTan00)",
  "href": "https://arxiv.org/pdf/hep-th/0003115.pdf#page=22"
\end{imagefromfile}

>  Apparently the spin structure of [[type IIA supergravity]] does resemble the low mass glueball spin splitting. ([Brower-Mathur-Tan 00, p. 22](#BrowerMathurTan00))

## References

### General

The first article discussing glueballs is apparently

* [[Harald Fritzsch]], [[Peter Minkowski]], _Ψ-resonances, gluons and the Zweig rule_, Nuov. Cim. A, 1975,30: 393 ([doi:10.1007/BF02730295](https://doi.org/10.1007/BF02730295))

Review:

* Vincent Mathieu, _The Physics of Glueballs_, Zakopane, February 2009 ([pdf](https://itp.uni-frankfurt.de/~excitedqcd/Talks09/Mathieu_ZakopaneExQCD09.pdf), [[MathieuGlueballs09.pdf:file]])

* {#Greensite11} Jeff Greensite, _An Introduction to the Confinement Problem_, Lecture Notes in Physics, Volume 821, 2011 ([doi:10.1007/978-3-642-14382-3](https://link.springer.com/book/10.1007%2F978-3-642-14382-3))

See also

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

### In holographic QCD

Discussion of glueballs in [[holographic QCD]] ([[AdS/QCD]]):

* {#BrowerMathurTan00} Richard C. Brower, [[Samir Mathur]], Chung-I Tan, _Glueball Spectrum for QCD from AdS Supergravity Duality_, Nucl. Phys. B587:249-276, 2000 ([arXiv:hep-th/0003115](https://arxiv.org/abs/hep-th/0003115))

* Alex S. Miranda, C. A. Ballon Bayona, Henrique Boschi-Filho, Nelson R. F. Braga, _Glueballs at finite temperature from AdS/QCD_, Nucl. Phys. Proc. Suppl. 199:107-112, 2010 ([arXiv:0910.4319](https://arxiv.org/abs/0910.4319))

* Xue-Feng Li, Ailin Zhang, _Scalar glueball in a soft-wall model of AdS/QCD_,  Chinese Physics C, Volume 38, Number 1 ([arXiv:1309.7154](https://arxiv.org/abs/1309.7154))

* Cornélio Rodrigues Filho, _Glueballs in the Klebanov-Strassler Theory: Pseudoscalars vs Scalars_ ([arXiv:2011.12689](https://arxiv.org/abs/2011.12689))

[[!redirects glueballs]]

[[!redirects glue-ball]]
[[!redirects glue-balls]]


gluing function

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## Definition

+-- {: .num_defn #Charts}
###### Definition
**([[local chart]] and [[atlas]] and gluing function)

Given an $n$-dimensional [[topological manifold]] $X$ (def. \ref{TopologicalManifold}), then

1. an [[open subset]] $U \subset X$ and a [[homeomorphism]] $\phi \colon \mathbb{R}^n \overset{\phantom{A}\simeq\phantom{A}}{\to} U$ is also called a _[[local coordinate chart]]_ of $X$.

1. an [[open cover]] of $X$ by local charts $\left\{ \mathbb{R}^n \overset{\phi_i}{\to} U \subset X \right\}_{i \in I}$ is called an _[[atlas]]_ of the topological manifold.

1. denoting for each $i,j \in I$ the [[intersection]] of the $i$th chart with the $j$th chart in such an atlas by

   $$
     U_{i j} \coloneqq U_i \cap U_j
   $$

   then the induced homeomorphism

   $$
     \mathbb{R}^n
       \supset
        \phantom{AA}
     \phi_i^{-1}(U_{i j})
       \overset{\phantom{A}\phi_i\phantom{A}}{\longrightarrow}
     U_{i j}
       \overset{\phantom{A}\phi_j^{-1}\phantom{A}}{\longrightarrow}
     \phi_j^{-1}(U_{i j})
       \phantom{AA}
       \subset
     \mathbb{R}^n
   $$

   is called the _gluing function_ or _coordinate transformation_ from chart $i$ to chart $j$.

> graphics grabbed from [[The Geometry of Physics - An Introduction|Frankel]]

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## Related concepts

* [[transition function]]

[[!redirects gluing functions]]

[[!redirects coordinate transformation]]
[[!redirects coordinate transformations]]


gluino

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## Idea

The hypothetical [[superpartner]]-[[particle]] of the [[gluon]]. A [[gaugino]].

## Experiment

Experimental exclusion bounds of gluino [[rest masses]] due to the [[LHC]] [[experiment]] excludes gluinos/[[squarks]] of mass below about 3-4 [[TeV]] ([Particle Data Group Review 17, figure 113.2 and 113.9](#ParticleDataGroup17)):

[[LHCSuperpartnerMassExclusion2017.png:file]]

## Related entries

* [[split supersymmetry]]

* [[gaugino condensation]]

## References

Experimental exclusion as of 2017

* {#ParticleDataGroup17} [Particle Data Group Review 2017](http://pdg.lbl.gov/), 113. _Supersymmetry , part II (experiment)_ ([pdf](http://pdg.lbl.gov/2017/reviews/rpp2017-rev-susy-2-experiment.pdf))

Discussion of remaining possibilities from the point of view of the [[G2-MSSM]]:

* {#BuschmannGonzalesKane18} Malte Buschmann, Eric Gonzalez, [[Gordon Kane]], _Revisiting Gluinos at LHC_ ([arXiv:1803.04394](https://arxiv.org/abs/1803.04394))

See also

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

[[!redirects gluinos]]


gluon

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## Idea

The _gluon_ is the [[gauge boson]] of the $SU(3)$-[[Yang-Mills field]] in [[QCD]].

A [[vector boson]] in the [[standard model of particle physics]].

The [[bound states]] of gluons are [[glueballs]].

## Related concepts

* [[glueball]]

* [[quark-gluon plasma]]

* [[quark]]

* [[colour charge]], [[flavour (particle physics)]]

* [[gluino]]

## References

### General

(...)

[[!include classification of long-range forces -- references]]

[[!redirects gluons]]


glutos

* Vladimir Molotkov, _Glutoses: a generalization of topos theory_, [math.CT/9912060](http://arxiv.org/abs/math/9912060)

This work had no followers, $n$labers should maybe find out where to place it.

Examples include categories of:

*  [[manifolds]],
*  [[schemes]],
*  [[sheaves]] (which is why this work generalises [[topos theory]] in the sense of a [[Grothendieck topos]]).

[[!redirects glutos]]
[[!redirects glutoi]]
[[!redirects glutoses]]


Glynn Winskel

* [home page](http://www.cl.cam.ac.uk/~gw104/)

## Selected writings

On [[syntax]] and [[semantics]] of [[programming languages]]:

* [[Glynn Winskel]], *The Formal Semantics of Programming Languages*, MIT Press (1993) [[ISBN:9780262731034](https://mitpress.mit.edu/9780262731034/the-formal-semantics-of-programming-languages/), [pdf](https://www.cin.ufpe.br/~if721/intranet/TheFormalSemanticsofProgrammingLanguages.pdf)]

On [[transition systems]] and other [[models for concurrency]]:

* [[Mogens Nielsen]], [[Glynn Winskel]]: *Models for concurrency*, in: *Handbook of Logic in Computer Science* vol 4, Oxford Univ. Press (1994) 1-148 [[pdf](https://www.cl.cam.ac.uk/~gw104/winskel-nielsen-models-for-concurrency.pdf), [[WinskelNielsen-Concurrency.pdf:file]], [doi:10.5555/218623.218630](https://dl.acm.org/doi/10.5555/218623.218630), [ISBN:9780198537809](https://global.oup.com/academic/product/handbook-of-logic-in-computer-science-9780198537809?lang=en&cc=at)]

On [[transition systems]], [[bisimulations]] and [[open morphisms]]:

* {#JoyalNielsenWisnkel94} [[André Joyal]], [[Mogens Nielsen]], [[Glynn Winskel]]: *Bisimulation from open maps*, BRICS report series RS-94-7 (1994) [[doi:10.7146/brics.v1i7.21663](https://doi.org/10.7146/brics.v1i7.21663), [pdf](http://www.brics.dk/RS/94/7/BRICS-RS-94-7.pdf)], Information and Computation **127** 2 (1996) 164-185 [[doi:10.1006/inco.1996.0057](https://doi.org/10.1006/inco.1996.0057)]

On [[category theory|category theoretic]] generalizations of [[domain theory]]:

* [[Mikkel Nygaard]], [[Glynn Winskel]], *Domain theory for concurrency*, Theoretical Computer Science
**316** 1–3 (2004) 153-190 [[doi:10.1016/j.tcs.2004.01.029](https://doi.org/10.1016/j.tcs.2004.01.029), [pdf](https://www.cl.cam.ac.uk/~gw104/DomThy.pdf)]

## Related entries

* [[denotational semantics]]

* [[domain theory]]

category:people


gnarly problems in physics

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[[!include physicscontents]]
</div>

#Gnarly Problems in Physics#

This are puzzles that aren't very technical at first, that get deeper and more complicated the more you think about them.   [[John Baez]] has imagined writing a book on these someday.

#List of gnarly problems#

* [[light mill|Light Mills]]
* [dimension analysis](http://golem.ph.utexas.edu/category/2006/09/dimensional_analysis.html)


gnosticism

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## Related entries

* [[hermeticism]]

* [[mysticism]]

## References

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

* [[Georg Hegel]], _[[Lectures on the History of Philosophy]] -- [Gnostics](Lectures+on+the+History+of+Philosophy#Gnostics)_

[[!redirects gnosis]]


go

[[!redirects golang]]

Go, otherwise called golang, is a statically typed programming language promoted by Google.

Some pros and cons are listed in simple article <https://www.techworld.com/developers/java-vs-go-which-programming-language-should-i-learn-3672102>

* <a href="https://en.wikipedia.org/wiki/Go_%28programming_language%29"> wikipedia </a>


Goddard-Thorn no-ghost theorem

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## Idea

In [[string theory]] the _Goddard-Thorn theorem_ ([Goddard-Thorn 72](#GoddardThorn72), [Thorn 87](#Thorn87), [Thorn 12](#Thorn12)) establishes the consistency of the usual [[quantization]] of the [[bosonic string]] [[sigma-model]] for [[target space]] [[dimension]] 26, and of the [[NSR string|NSR]] [[superstring]] [[sigma-model]] for [[target space]] [[dimension]] 10.

In the original "old covariant quantization" approach to this problem the proof amounts to demonstrating that the [[inner product]] on the would-be [[Hilbert space]] of [[quantum states]] is indeed [[positive definite]]. Since would-be [[quantum states]] of [[negative number|negative]] [[norm]] have been known as "ghost states", the theorem has become widely known as the _no-ghost theorem_ of [[string theory]]. More modern derivations proceed via or compare to [[quantization]] via the [[BRST complex]] of the string [[sigma-model]].

In particular the theorem thus establishes the [[superstring]] excitations of vanishing [[mass]], and that (after [[GSO projection]]) these coincide with the quanta of [[perturbative quantum gravity|perturbative]] [[supergravity]] in 10 dimensions ([[type IIA supergravity]] or [[type IIB supergravity]]).

## Related entries

* [[string theory FAQ]]: _[Does string theory predict supersymmetry?](string+theory+FAQ#DoesSTPredictSupersymmetry)_

## References

The original articles are

* {#GoddardThorn72} [[Peter Goddard]], [[Charles Thorn]], _Compatibility of the dual Pomeron with unitarity and the absence of ghosts in the dual resonance model_, Phys. Lett., B 40, No. 2 (1972), 235-238 ([spire:74899](http://inspirehep.net/record/74899/))

* Richard C. Brower, _Spectrum-generating algebra and no-ghost theorem for the dual model_, Physical Review D 6.6 (1972): 1655 ([doi:10.1103/PhysRevD.6.1655](https://link.aps.org/doi/10.1103/PhysRevD.6.1655))

* {#Thorn87} [[Charles Thorn]], _A detailed study of the physical state conditions in covariantly quantized string theories_, Nuclear Physics B 286 (1987): 61-77 (doi:10.1016/0550-3213(87)90431-7)

* {#Thorn12} [[Charles Thorn]], _Improved Proof of the No-ghost Theorem for Fermion States of the Superstring_, Nuclear Physics B Volume 864, Issue 2, 11 November 2012, Pages 285-295 ([arXiv:1110.5510](https://arxiv.org/abs/1110.5510))

A textbook account is in

* [[Michael Green]], [[John  Schwarz]], [[Edward Witten]], section 2.2, 2.3 (bosonic) and section 4.3.2 (fermionic) of _Superstring theory_, Cambridge University Press 1987

The proof via DDF operators is also recalled in

* [[Urs Schreiber]], _DDF and Pohlmeyer invariants of (super)string_, JHEP0405:027, 2004 ([arXiv:hep-th/0403260](https://arxiv.org/abs/hep-th/0403260))

See also

* Wikipedia, _[Goddard–Thorn theorem](https://en.wikipedia.org/wiki/Goddard%E2%80%93Thorn_theorem)_

[[!redirects Goddard-Thorn theorem]]
[[!redirects Goddard-Thorn theorems]]

[[!redirects no-ghost theorem]]
[[!redirects no-ghost theorems]]

[[!redirects DDF operator]]
[[!redirects DDF operators]]


Goddard-Thorn theorem

## References

* Wikipedia, _[Goddard-Thorn theorem](http://en.wikipedia.org/wiki/Goddard–Thorn_theorem)_


Godement product

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## Idea

The __Godement product__ of two [[natural transformations]] between appropriate [[functors]] is their [[horizontal composition]] as 2-cells in the [[2-category]] [[Cat]] of [[categories]], functors and natural transformations:

\begin{tikzcd}[column sep=small]
\textbf{A}
 \arrow[to=1-5, bend left=50]{r}[name=F1]{F_1}
 \arrow[to=1-5, bend right=50]{r}[name=G1,label=below:$\scriptstyle G_1$]{}
 \arrow[phantom,Rightarrow,to path={(F1) -- node[] {$\big\Downarrow\normalsize\alpha$} (G1)}]{}
 &&&& \textbf{B}
 \arrow[to=1-9, bend left=50]{r}[name=F2]{F_2}
 \arrow[to=1-9, bend right=50]{r}[name=G2,label=below:$\scriptstyle G_2$]{}
 \arrow[phantom,Rightarrow,to path={(F2) -- node[] {$\big\Downarrow\normalsize\beta$} (G2)}]{}
 &&&& \textbf{C}
 \arrow[to=1-10, mapsto]
 & \textbf{A}
 \arrow[to=1-14, bend left=50]{r}[name=F3]{F_2 \circ F_1}
 \arrow[to=1-14, bend right=50]{r}[name=G3,label=below:$\scriptstyle G_2 \circ G_1$]{}
 \arrow[phantom,Rightarrow,to path={(F3) -- node[] {$\big\Downarrow\normalsize\beta\circ\alpha$} (G3)}]{}
 &&&& \textbf{C}
\end{tikzcd}

## Definition

For [[categories]] $A,B,C$, if $\alpha\colon F_1\to G_1\colon A\to B$ and $\beta\colon F_2\to G_2\colon B\to C$ are [[natural transformation]]s of [[functor]]s, the components $(\beta \circ \alpha)_M$ of the Godement product $\beta \circ \alpha\colon F_2\circ F_1\to G_2\circ G_1\colon A\to C$ (or $\alpha \ast \beta\colon F_1 ; F_2 \to G_1 ; G_2\colon A\to C$) are defined by any of the two equivalent formulas:
$$
(\beta\circ\alpha)_M = \beta_{G_1(M)}\circ F_2(\alpha_M)
$$
$$
(\beta\circ\alpha)_M = G_2(\alpha_M)\circ \beta_{F_1(M)}
$$
that can be rewritten using the [morphismwise notation](natural+transformations#morphismwiseDefn) into:
$$
(\beta\circ\alpha)_M = \beta(\alpha_M)
$$
that is:
$$
  \array{
    F_2(F_1(M))
    &
    \stackrel{F_2(\alpha_M)}{\to}
    &
    F_2(G_1(M))
    \\
    \beta_{F_1(M)}\downarrow
    &
    \searrow^{(\beta\circ\alpha)_M}
    &
    \downarrow \beta_{G_1(M)}
    \\ G_2(F_1(M))
    &
    \stackrel{G_2(\alpha_M)}{\to} & G_2(G_1(M))
  }
  \,.
$$

The [[interchange law]] in (general) $2$-categories (which in the case of $Cat$ boils down to assertion that the two formulas above are equivalent) is also sometimes called _Godement interchange law_.

The definition above is for the Godement product of $2$ natural transformations, but we can generalise from $2$ to any [[natural number]].  The Godement product of $0$ natural transformations is the __[[identity natural transformation]] on an [[identity functor]]__.

## Properties

The Godement product is strictly associative (so that [[Cat]] is a [[strict 2-category]]).

## References

Name after [[Roger Godement]].

[[!redirects Godement product]]
[[!redirects Godement products]]

[[!redirects horizontal composite of natural transformations]]
[[!redirects horizontal composites of natural transformations]]
[[!redirects horizontal composition of natural transformations]]
[[!redirects horizontal compositions of natural transformations]]