graph of a functor

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## Idea

In as far as the notion of [[functor]] generalizes that of [[function]] and that of [[profunctor]] generalizes that of [[relation]], the notion of _graph_ of a (pro)functor generalizes that of [[graph of a function]].

Just as the [[graph of a function]] $f : X \to Y$,
or more generally that of a [[relation]] $R \subset X \times Y$ for $X,Y \in Set = 0 Cat$ is nothing but the [[category of elements]] of the corresponding characteristic function $\chi_R : X \times Y \to (-1)Cat = \{0,1\}$, so the graph
of a functor $F\colon C \to D$, or more generally that of a [[profunctor]] $H : C^{op} \times D \to 0 Cat = Set$, is nothing but its [[category of elements]], a.k.a. [[Grothendieck construction]].

However, there are a number of different ways to construct such a category of elements, depending on the variance of morphisms in $C$ and $D$ that we include in it.  A one-variable functor $C\to Set$ has two categories of elements, one with a projection to $C$ (which is a [[discrete opfibration]]) and one with a projection to $C^{op}$ (which is a [[discrete fibration]]).  Similarly, a two-variable functor such as $H : C^{op}\times D\to Set$ has *four* categories of elements; thus a profunctor has four different "graphs".  In addition, there are two ways to make a functor into a profunctor, so a functor actually has *eight* graphs.

## Definition

### Graphs of a profunctor

Let $H:C^{op}\to D$ be a [[profunctor]]; it has the following four **graphs**.  In each case, the objects are triples $(c\in C, d\in D, x\in H(c,d)$, but we can take the morphisms $(c,d,x) \to (c',d',x')$ to be any of the following:

* Pairs $(f:c\to c', g:d\to d')$ such that $H(1,g)(x) = H(f,1)(x')$.  This graph comes with a projection to $C\times D$, which is a [[two-sided discrete fibration]] that is contravariant over $C$ and covariant over $D$.

* Pairs $(f:c'\to c, g:d\to d')$ such that $H(f,g)(x) = x'$.  This graph comes with a projection to $C^{op}\times D$, which is a [[discrete opfibration]].  It is also the [[comma category]] $(*\downarrow H)$:

  $$
  \array{
    Graph(H) &\to& {*}
    \\
    \downarrow & \swArrow & \downarrow
    \\
    C^{op} \times D &\stackrel{H}{\to}& Set
  }
  $$

* Pairs $(f:c\to c', g:d'\to d)$ such that $H(f,g)(x') = x$.  This graph comes with a projection to $C\times D^{op}$, which is a [[discrete fibration]]; it is the opposite of the previous category.

* Pairs $(f:c'\to c, g:d'\to d)$ such that $H(1,g)(x') = H(f,1)(x)$.  This graph comes with a projection to $C^{op}\times D^{op}$, which is a [[two-sided discrete fibration]] that is covariant over $C^{op}$ and contravariant over $D^{op}$; it is the opposite of the first graph.

If the profunctor is the [[hom profunctor]] of a category $C$, then the first graph is the [[arrow category]] of $C$ and the second graph is the [[twisted arrow category]] of $C$.

### Graphs of a functor

Let $F:C\to D$ be a functor, and let $F^\bullet : C^{op}\times D \to Set$ and $F_\bullet$ be the corresponding representable [[profunctors]]:

$$F^\bullet(c,d) = D(F c, d) \qquad F_\bullet(d,c) = D(d,Fc).$$

Then the graphs of $F^\bullet$ yield four graphs of $F$, all of whose objects are triples $(c\in C, d\in D, \phi:F c \to d)$, and whose morphisms $(c,d,\phi) \to (c',d',\phi')$ are:

* Pairs $(f:c\to c', g:d\to d')$ such that $\phi' \circ F f = g \circ \phi$.  This graph comes with a projection to $C\times D$, which is a [[two-sided discrete fibration]] that is contravariant over $C$ and covariant over $D$.  It is also the [[comma category]] $(F\downarrow Id_D)$:

  $$
  \array{
    Graph(F) &\to& C
    \\
    \downarrow & \swArrow & \downarrow^F
    \\
    D &\stackrel{Id_D}{\to}& D
  }
  $$

* Pairs $(f:c'\to c, g:d\to d')$ such that $\phi' = g \circ \phi \circ F f$.  This graph comes with a projection to $C^{op}\times D$, which is a [[discrete opfibration]]; it is the comma category $(* \downarrow F^\bullet)$.

* Pairs $(f:c\to c', g:d'\to d)$ such that $\phi = g \circ \phi' \circ F f$.  This graph comes with a projection to $C\times D^{op}$, which is a [[discrete fibration]]; it is the opposite of the preceeding graph.

* Pairs $(f:c'\to c, g:d'\to d)$ such that $\phi \circ F f = g \circ \phi'$.  This graph comes with a projection to $C^{op}\times D^{op}$, which is a [[two-sided discrete fibration]] that is covariant over $C^{op}$ and contravariant over $D^{op}$; it is the opposite of the first graph of $F^\bullet$.

Similarly, the graphs of $F_\bullet$ yield four graphs of $F$, all of whose objects are triples $(c\in C, d\in D, \psi : d \to F c)$, and whose morphisms $(c,d,\psi) \to (c',d',\psi')$ are:

* Pairs $(f:c\to c', g:d\to d')$ such that $F f \circ \psi = \psi\circ g$.  This graph comes with a projection to $C\times D$, which is a [[two-sided discrete fibration]] that is covariant over $C^{op}$ and contravariant over $D^{op}$.  It is also the [[comma category]] $(Id_D \downarrow F)$:

  $$
  \array{
    Graph(F) &\to& D
    \\
    \downarrow & \swArrow & \downarrow^{Id_D}
    \\
    C &\stackrel{F}{\to}& D
  }
  $$

* Pairs $(f:c'\to c, g:d\to d')$ such that $\psi' = F f \circ \psi \circ g$.  This graph comes with a projection to $C^{op}\times D$, which is a [[discrete opfibration]]; it is the comma category $(* \downarrow F^\bullet)$.

* Pairs $(f:c\to c', g:d'\to d)$ such that $\psi = F f \circ \psi' \circ g$.  This graph comes with a projection to $C\times D^{op}$, which is a [[discrete fibration]]; it is the opposite of the preceeding graph.

* Pairs $(f:c'\to c, g:d'\to d)$ such that $\psi \circ g = F f \circ \psi'$.  This graph comes with a projection to $C^{op}\times D^{op}$, which is a [[two-sided discrete fibration]] that is contravariant over $C$ and covariant over $D$; it is the opposite of the first graph of $F_\bullet$.

## For higher categories

### In general

For $n \leq \infty$, let $(n-1) Cat$ and $n Cat$ be a realization of the
notions of $n$-category of $(n-1)$-categories and of the $(n+1)$-category of $n$-categories,
respectively,
such that standard constructions of [[category theory]] work,
in particular a version of the [[Yoneda lemma]].
See [[higher category theory]].

Then with $C,D \in n Cat$ let $f : C \to D$ be a ($n$-)[[functor]].  By the general logic of [[profunctors]] this defines $n$-profunctors

$$
   F^\bullet : C^{op} \times D \stackrel{F^{op} \times Id}{\to} D^{op}
     \times D \stackrel{D(-,-)}{\to} (n-1) Cat
$$

$$
   F_\bullet : C \times D^{op} \stackrel{F \times Id}{\to} D^{op}
     \times D \stackrel{D(-,-)}{\to} (n-1) Cat
$$

We can then consider analogues of all eight kinds of graphs.  For instance, the second kind of graph of $F$ is the fibration $Graph(f) \to C^{op} \times D$ classified by $F^\bullet$.

### For $(\infty,1)$-categories

In the context of
$(\infty,1)$-[[Higher Topos Theory|category theory]], this graph may be taken to be the fibration classified by $\chi_f : C \times D^{op} \to (\infty,0)$ as described at  [[universal fibration of (∞,1)-categories]].

### For (0,0)-categories (sets)

To reproduce the ordinary notion of [[graph of a function]] let $(n,r) = (0,0)$.
then $(n,r)$-categories $X,Y$ are just sets and a functor $f : X \to Y$ is
just a function between sets. Moreover, the category of $(n-1,r) = (-1,0)$-categories
is the set $\{0,1\}$ of truth values, as described at [[(-1)-category]].
The [[profunctor]] corresponding to $f : X \to Y$ is therefore the
characteristic function

$$
  \chi_f : X \times Y \to \{0,1\}
$$

that maps

$$
  \chi_f(x,y) =
    \left\lbrace
      \array{
        1 & if f(x) = y
        \\
        0 & otherwise
      }
    \right.
    \,.
$$

(Notice that in this case $X^{op} = X$.)

The [[2-pullback]] of ${*} = {1} \to \{0,1\}$ along $\chi_f$ is
just the ordinary [[pullback]]

$$
\array{
  Graph(f) &\to& {*}
  \\
  \downarrow && \downarrow
  \\
  X \times Y
  &\stackrel{\chi_f}{\to}&
  \{0,1\}
}
$$

which identifies $Graph(f) \hookrightarrow X \times Y$ with the subset of pairs $(x,y)$
for which $f(x) = y$. This is the ordinary notion of [[graph of a function]].

## Related concepts

* **graph of a functor**, [[cograph of a functor]]
* [[cograph of a profunctor]]
* [[tabulator]]

[[!redirects graphs of a functor]]
[[!redirects graphs of functors]]
[[!redirects graph of a profunctor]]
[[!redirects graphs of a profunctor]]
[[!redirects graphs of profunctors]]
[[!redirects graph of a distributor]]
[[!redirects graphs of a distributor]]
[[!redirects graphs of distributors]]


graph of groups

[[!redirects graphs of groups]]

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## Idea

Given a [[groupoid infinity-action|group action]] on a [[graph]] $Y$ (with involution on the edges), one gets a quotient graph $X=Y/G$ together with the family of the stabilisers on the vertices and on the edges. Abstracting this gives a graph of groups.

## Definition

A graph of groups, $\Gamma(G,X)$, consists of

* a connected graph, $X$;

*  a function, $G$, which, for every vertex, $v\in V(X)$ assigns a group, $G_v$, and for each edge, $e\in E(X)$ assigns a group, $G_e$ such that $G_e= G_\overline{e}$

* For each edge, $e\in E(X)$, there exists a monomorphism, $\sigma::G_e\to G_{i(e)}$, where $i(e)$ is the initial vertex of the edge, $e$.

##Examples

In the usual form of Bass-Serre Theory a graph of groups is derived from the action of a group on a tree.

## Related entries.

* [[complex of groups]] - The theory of graphs of groups is the 1-dimensional precursor of that of  complexes of groups.

##References

* [[J.-P. Serre]], 1977, _Arbres, amalgames, SL2_, volume 46 of Astérisque, Société mathématique de France.

* [[J.-P. Serre]], 2003, _Trees_, Springer Monographs in Mathematics, Springer- Verlag Berlin

* [[H. Bass]], _Covering theory for graphs of groups_, Journal of Pure and Applied Algebra, 89, (1993), 3–47.

[[!redirects graphs of groups]]


graph product

## Idea

Given two [[groups]], two typical ways of combining them into a new group is by taking either their [[direct product group|direct product]] or their [[free product of groups|free product]]. Given a finite family of groups, you could repeatedly take binary direct products to get the direct product of the whole family, or repeatedly take binary free products to get the free product of the whole family. One could if they wish take a mix of direct products and free products. Given a [[simple graph|graph]] and a group assigned to each vertex, the graph product of this family of groups is a group which looks like a direct product if you look at two groups connected by an edge, and a free product otherwise.

## Definition

Let $\Gamma$ be a simple graph with vertex set $V$ and edge set $E \subseteq \binom{V}{2}$, and $\{G_v\}_{v \in V}$ a family of groups indexed by the vertex set. The graph product group $\Gamma G$ is given by first taking the free product of the groups, and then modding out by the [[normal subgroup]] generated by the relation which imposes commutativity for those elements which correspond to groups which are connected by an edge in the graph, i.e.

$$
  \Gamma G = \coprod G_v / R_\Gamma
$$

where $R_\Gamma$ is the normal subgroup generated by the relation $g_v g_u \sim g_u g_v$ for $\{u,v\} \in E$, $g_u \in G_u$, and $g_v \in G_v$.

## Examples

* If the graph has no edges, one recovers free products. If the graph is complete, one recovers direct products.

* Right-angled Artin groups

## References

The notion of graph products of groups was introduced by Elisabeth Green.

* [[Elisabeth Green]], _Graph products of groups_, PhD thesis, University of Leeds, (1990). [pdf](http://etheses.whiterose.ac.uk/236/)

António Veloso da Costa generalized the concept to monoids.

* [[António Veloso da Costa]], _Graph products of monoids_, Semigroup Forum, Volume 63, Issue 2, pp 247–277 (2001). [web](https://link.springer.com/article/10.1007/s002330010075)

Right-angled Artin groups

* [[Ruth Charney]], _An introduction to right-angled Artin groups_, Geom. Dedicata 125 (2007), 141–158. ([arxiv:0610668](https://arxiv.org/abs/math/0610668))


graph theory - contents

**[[graph theory]]**

[[graph]]

* [[vertex]], [[edge]]

* [[omega-graph]], [[hypergraph]]

* [[quiver]], [[n-quiver]]

[[category of simple graphs]]

### Properties

* [[graph distance]]

### Extra properties

* [[reflexive graph|reflexive]], [[directed graph|directed]]

* [[bipartite graph|bipartite]]

* [[planar graph|planar]]

### Extra structure

* [[reflexive graph|reflexive]] [[directed graph]] + [[unit law|unital]] [[associative]] [[composition]] = [[category]]

* [[ribbon graph]], [[combinatorial map]], [[topological map]], [[child's drawing]]

* [[vertex coloring]], [[clique]]


graph theory > history

< [[graph theory]]


graphene

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## Idea

Graphene is one of the [[solid]] [[phase of matter|phases]] of [[carbon]], appearing as a single-layer honeycomb lattice. This may be planar or tubular, etc.

{#PrimeExampleOfTopologicalPhase} Graphene is a prime example (and among the first to be discovered) of a [[topological phase of matter]]. Specifically:

1. at coarse resolution (currently accessible to [[experiment]]) it appears as a  ([[time-reversal symmetry|time-reversal]] and space inversion-symmetric) [[topological semi-metal]], due to a gap between its [[valence band]] and [[conduction band]] which closes only over two [[Dirac points]] in its [[Brillouin torus]];

1. at finer resolution -- namely when the [[spin-orbit coupling]] of the [[electrons]] is resolved (thought to be $\sim10^{-3}\,$[[meV]] $[$[MHSSKM06](#MHSSKM06)$]$ and hence too small for [[experiment|experimental]] observation, currently), which reveals a small energy gap opening at the two would-be Dirac points -- graphene appears as a ([[time-reversal symmetry|time-reversal symmetric]]) [[topological insulator]]  ([Kane & Mele 05a](#KaneMele05a)), whose non-trivial [[topological phase of matter|topological phase ]] is witnessed by the non-trivial [[Kane-Mele invariant]] in [[cyclic group of order two|$\mathbb{Z}/2$]] ([Kane & Mele 05b](#KaneMele05b)).

## Related concepts

## References

### General

The [[electronic band structure]] of graphene (reviewed in [WZLLJHD 12](#WZLLJHD12)) was predicted (long before the term was coined) already in

* [[Philip Russel Wallace]], *The Band Theory of Graphite*, Phys. Rev. **71** (1947) 622 ([doi:10.1103/PhysRev.71.622](https://doi.org/10.1103/PhysRev.71.622))

The synthesis/detection of graphene is due to

* [[Konstantin Novoselov]], [[Andre Geim]], S. V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, _Electric field effect in atomically thin carbon films_, Science __306__, no. 5696, pp. 666-669 (2004) ([doi:10.1126/science.1102896](http://dx.doi.org/10.1126/science.1102896), [arXiv:cond-mat/0410550](https://arxiv.org/abs/cond-mat/0410550))

(The procedure, won a Nobel Prize and the authors made it freely available without patenting.)

Further discussion of the [[electron band structure]] of graphene:

* A. H. Castro Neto, F. Guinea, N. M. R. Peres, [[Konstantin S. Novoselov]], and [[Andre K. Geim]], *The electronic properties of graphene*, Rev. Mod. Phys. **81** (2009) 109 $[$[doi:10.1103/RevModPhys.81.109](https://doi.org/10.1103/RevModPhys.81.109)$]$

Computation of the (tiny) [[spin-orbit coupling]] in graphene:

* {#MHSSKM06} Hongki Min, J. E. Hill, N. A. Sinitsyn, B. R. Sahu, Leonard Kleinman, A. H. MacDonald, *Intrinsic and Rashba spin-orbit interactions in graphene sheets*,
Phys. Rev. B **74** (2006) 165310 $[$[doi:10.1103/PhysRevB.74.165310](https://doi.org/10.1103/PhysRevB.74.165310)$]$

Observation that the [[spin-orbit coupling]] in graphene should open the gap at the [[Dirac points]] revealing a [[quantum spin Hall effect]] in graphene:

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

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

Review:

* {#WZLLJHD12} Nathan Weiss, Hailong Zhou, Lei Liao, Yuan Liu, Shan Jiang, Yu Huang, Xiangfeng Duan, *Graphene: an emerging electronic material*,  Adv Mater. **24** (43) (2012) 5782-825 ([doi:10.1002/adma.201201482](https://doi.org/10.1002/adma.201201482))

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

Review in the context of [[topological phases of matter]] and specifically topological [[semi-metals]]:

* [[Jérôme Cayssol]], [[Jean-Noël Fuchs]], Section 6 of: *Topological and geometrical aspects of band theory*, J. Phys. Mater. **4** (2021) 034007 ([arXiv:2012.11941](https://arxiv.org/abs/2012.11941), [doi:10.1088/2515-7639/abf0b5](https://doi.org/10.1088/2515-7639/abf0b5))

The 2+1 dim [[Dirac equation]] is used in modeling graphene:

* [[Konstantin Novoselov]], [[Andre Geim]], S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, A. A. Firsov, _Two-dimensional gas of massless Dirac fermions in graphene_,  Nature __438__, 197-200 (2005) [doi](http://dx.doi.org/10.1038/nature04233)

* María A. H. Vozmediano, _Renormalization group aspects of graphene_, [pdf](http://www.isis.stfc.ac.uk/groups/theory/downloads/the-strong-correlations-puzzle8120.pdf)

On [[non-perturbative effects]] in graphene:

* Juan Angel Casimiro Olivares, Ana Julia Mizher, Alfredo Raya, *Non-perturbative field theoretical aspects of graphene and related systems* [[arXi:2109.10420](https://arxiv.org/abs/2109.10420)]

Discussion via [[AdS-CFT in condensed matter physics]]:

* Jeong-Won Seo, Taewon Yuk, Young-Kwon Han, Sang-Jin Sin, *ABC-stacked multilayer graphene in holography* [[arXiv:2208.14642](https://arxiv.org/abs/2208.14642)]

On fractional charges in graphene:

* Chang-Yu Hou, Claudio Chamon, Christopher Mudry, *Electron fractionalization in two-dimensional graphenelike structures*, Phys. Rev. Lett. **98** (2007) 186809 [[arXiv:cond-mat/0609740](https://arxiv.org/abs/cond-mat/0609740), [doi:10.1103/PhysRevLett.98.186809](
https://doi.org/10.1103/PhysRevLett.98.186809)]

### Berry connection

On "fictitious" contributions to the [[Berry connection]] on the [[Brillouin torus]] of graphene:

* Mircea Trif, Pramey Upadhyaya, Yaroslav Tserkovnyak, *Theory of electromechanical coupling in dynamical graphene*, Phys. Rev. B **88** 245423 (2013) $[$[doi:10.1103/PhysRevB.88.245423](https://doi.org/10.1103/PhysRevB.88.245423), [arXiv:1210.7384](https://arxiv.org/abs/1210.7384)$]$

### Movement of Dirac points
 {#ReferencesMovementOfDiracPoints}

Moving the nodal points in graphene(-variants) by changing external parameters such as lattice anisotropy or [[strain]] (see also further discussion of external manipulation by strain [here](topological+insulator#ReferencesOnExternalManipulationViaStrain), and see the [references on momentum-space braiding](braid+group+statistics#ReferencesAnyonicBraidingInMomentumSpace) of band nodes):

* Cui-Lian Lia, *New position of Dirac points in the strained graphene reciprocal lattice*, AIP Advances **4** (2014) 087119 [[doi:10.1063/1.4893239](https://doi.org/10.1063/1.4893239)]

*  Marc Dvoraka, Zhigang Wu, *Dirac point movement and topological phase transition in patterned graphene*, Nanoscale **7** (2015) 3645-3650 [[doi:10.1039/C4NR06454B](https://doi.org/10.1039/C4NR06454B)]

* Zhenzhu Li, Zhongfan Liu, Zhirong Liu: *Movement of Dirac points and band gaps in graphyne under rotating strain*, Nano Research **10** (2017) 2005–2020 [[doi:10.1007/s12274-016-1388-z](https://doi.org/10.1007/s12274-016-1388-z)]

* Jian Kang, Oskar Vafek, *Non-Abelian Dirac node braiding and near-degeneracy of correlated phases at odd integer filling in magic angle twisted bilayer graphene*, Phys. Rev. B **102** (2020) 035161 [[arXiv:2002.10360](https://arxiv.org/abs/2002.10360), [doi:10.1103/PhysRevB.102.035161](https://doi.org/10.1103/PhysRevB.102.035161)]

Movement of Dirac points in a cousin of the [[Haldane model]]:

* Miguel Gonçalves, Pedro Ribeiro, Eduardo V. Castro, *Dirac points merging and wandering in a model Chern insulator*,  Europhysics Letters **124** 6 (2018) [[doi:10.1209/0295-5075/124/67003](https://iopscience.iop.org/article/10.1209/0295-5075/124/67003)]

and in [[photonic crystal]]-analogs:

* Yong-Heng Lu et al., *Observing movement of Dirac cones from single-photon dynamics*, Phys. Rev. B **103** 064304 (2021) [[doi:10.1103/PhysRevB.103.064304](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.103.064304)]

### Holographic description

On [[AdS-CMT duality]] for graphene-like systems via full [[supergravity]] (retaining the [[gravitino]]):

* [[Laura Andrianopoli]], [[Bianca L. Cerchiai]], [[Riccardo D'Auria]], [[Mario Trigiante]], *Unconventional Supersymmetry at the Boundary of $AdS_4$ Supergravity*,  J. High Energ. Phys. **2018** 7 (2018) [[arXiv:1801.08081](https://arxiv.org/abs/1801.08081), doi:10.1007/JHEP04(2018)007]

* [[Laura Andrianopoli]], [[Bianca L. Cerchiai]], [[Riccardo D'Auria]], A. Gallerati, R. Noris, [[Mario Trigiante]], [[Jorge Zanelli]], *$\mathcal{N}$-Extended $D=4$ Supergravity, Unconventional SUSY and Graphene*, JHEP 01 (2020) 084 [[arXiv:1910.03508](https://arxiv.org/abs/1910.03508), doi:10.1007/JHEP01(2020)084]

* Antonio Gallerati, *Supersymmetric theories and graphene*, in  *40th International Conference on High Energy physics* (ICHEP2020),  PoS **390** (2021) [[arXiv:2104.07420](https://arxiv.org/abs/2104.07420), [doi:10.22323/1.390.0662](https://doi.org/10.22323/1.390.0662)]

Exposition:

* [[Bianca L. Cerchiai]],  *Supergravity in a pencil*, [talk at](M-Theory+and+Mathematics#Cerchiai2020) *[[M-Theory and Mathematics]] [2020](M-Theory+and+Mathematics#2020)*, NYU Abu Dhabi [[[CerchiaiSlidesAtMTheoryAndMathematics2020.pdf:file]], video: [YT](https://youtu.be/xE7TmwyqqaU)]

* [[Bianca L. Cerchiai]], *Holography, Supergravity and Graphene*, talk at *106th online SIF Congress* (2020) [[pdf](https://agenda.infn.it/event/23656/contributions/120378/attachments/75347/96340/cerchiai_sif2020.pdf), [[Cerciai-SIF2020.pdf:file]]&r

category: physics


graphic category

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#### Category theory
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#### Monoid theory
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# Contents
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[[!redirects graphic monoid]]
[[!redirects graphics]]
[[!redirects unital left regular band]]
[[!redirects left regular band monoid]]

##Idea

A simple class of [[monoids]] and [[categories]] that permit an effective graphic display via their [[presheaf]] categories.

Graphic monoids arise in the study of [[arrangement of hyperplanes|hyperplane arrangements]] and are often called **left regular band monoids** in this context.

## Definition

A monoid $M$ is called _graphic_ if all $x,y\in M$ satisfy the so called _graphic identity_ $x y x=x y$ . A category $\mathcal{G}$ is called _graphic_ if the endomorphism monoid $End(x)$ is a graphic monoid for each object $x$.

### Remark

The terminology originates with Lawvere ([1989b](#Law89b)) where graphic monoids and categories $\mathcal{G}$ are demanded to be _finite_. This restriction is supported by the fact that finitely generated graphic monoids are necessarily finite ([see below](#fin_gen)).

_Graphic toposes_ are defined then as $FinSet^{\mathcal{G}^{op}}$ resulting in  presheaf categories of finite graph-like 'displays' that come with a well-behaved lattice of subtoposes permitting the kind of 'Hegelian analysis' that were a primary concern for [[Lawvere]] in ([Law89b](#Law89b), [Law91](#Law91), [Law02](#Law02)).

In particular, the graphic monoid $\Delta_1$ in the [example](#ex) and the eight-element [[Hegelian taco]] monoid of (Lawvere [1989b](#Law89b)) express diagrammatically configurations of [[level|essential subtoposes]] occuring in Lawvere's mathematical rendering of basic concepts of Hegel's dialectic logic.[^hegel]

[^hegel]: For more on Lawvere's approach to Hegelian dialectics see [[Aufhebung]], [[adjoint cylinder]], or [[Science of Logic]].

This bears some resemblance to the other context, namely hyperplane arrangements, in which such monoids mainly arise.

In the literature that approaches graphic monoids as special class of semigroups, they are called **unital left regular bands** or _left regular band monoids_ (cf. Brown ([2000](#brown00)), Aguiar-Mahajan ([2006](#AgMah06)), Margolis-Saliola-Steinberg ([2015](#MSS15))).

##Example{#ex}

The principal example of a graphic category is $\Delta_1$ the three element monoid $\{\delta_1,\delta_2,1\}$ with $\delta_i\delta_j=\delta_i$. The presheaf topos $Set^{\Delta_1^{op}}$ is the topos of [[reflexive graphs]].

To see this, recall that a _reflexive graph_ consists of a set of vertices $V$, a set of edges $E$, source and target maps $s, t : E \to V$, and a map $i : V \to E$ assigning to each vertex an 'identity edge' from that vertex to itself, so that $s(i(v)) = t(i(v)) = v$.

We can describe a reflexive graph without ever mentioning its vertices by using the identity edge $i(v)$ as a stand-in for $v$.  To do this, we
define operations $\delta_1, \delta_2 : E \to E$ by

$$  \delta_1(e) = i(s(e)), \qquad \delta_2(e) = i(t(e)) $$

and note that

$$ \delta_1(\delta_1(e)) = \delta_2(\delta_1(e)) = \delta_1(e) $$

$$ \delta_1(\delta_2(e)) = \delta_2(\delta_2(e)) = \delta_2(e) $$

Thus, the reflexive graph determines a functor $F : \Delta_1^{op} \to \Set$, where we think of the monoid $\Delta_1$ as a one-object category.  In other words, it gives a presheaf on $\Delta_1$.  Conversely, any presheaf on $\Delta_1$ determines a reflexive graph.

The more familiar two-sorted 'signature' for reflexive graphs can be obtained from the [[Cauchy completion]] $\overline\Delta_1$ by identifying the isomorphic descendents of $\delta_1,\delta_2$: the resulting $\tilde\Delta_1$ has two objects and is basically an augmentation with reflexivity data of the familiar diagram $V\rightrightarrows E$ that underlies the topos of ("irreflexive") directed graphs, or [[quiver|quivers]]. All three of $\Delta_1,\overline\Delta_1,\tilde\Delta_1$ are graphic categories and the resulting presheaf toposes are, of course, equivalent.

##Properties

* Lawvere calls the graphic identity _'the least common generalization of constant (x=c) and identity (x=1)_' ([1989b](#Law89b), p.53). By a _constant_ $c$ is meant here a $c\in M$ such that $c y=c$ for all $y\in M$. In the context of monoids arising from hyperplane arrangements constants are also called '_chambers_' (cf. [Aguiar-Mahajan 2006](#AgMah06)).

* In a graphic monoid every element is [[idempotent]] as can be directly seen from the graphic identity with $y=1$.

* It follows that a commutative graphic monoid is the same as a [[semilattice]], or equivalently, a commutative monoid where every element is idempotent.

* More generally, one can define a partial order on $M$ by $x\leq y$ iff $x y=y$ ('$x$ is a face of $y$'). In fact, the graphic identity is precisely the condition needed for the anti-symmetry of this relation on monoids whose elements are all idempotent - the reflexivity being a direct consequence of the idempotency.[^semigroups]

[^semigroups]: Semigroup theorists typically order a graphic monoid by $x\leq y$ if $y x=x$ because this is equivalent to $x M\subseteq y M$.

* The _free graphic monoid_ on a set $X$ is the set of totally ordered finite subsets of $X$, where the product of finite subsets $S$ and $T$ is the union $S \cup T$ ordered in such a way that the inclusion $S \hookrightarrow S\cup T$ is order-preserving, the inclusion $T \hookrightarrow S\cup T$ is order-preserving for all elements of $T$ not in $S$, and all the elements of $T$ not in $S$ are greater than all elements of $S$.  This is easy to understand from an example.  Consider the free graphic monoid on $X = \{a,b\}$.  This contains:

  1 (corresponding to the empty subset of $X$),

  $a$ (corresponding to the subset $\{a\}$),

  $b$ (corresponding to $\{b\}$),

  $a b$ (corresponding to $\{a,b\}$ ordered so that $a \lt b$),

  $b a$ (corresponding to $\{a,b\}$ ordered so that $b \lt a$),

  and nothing else, since for example $a b b = a b$ and $b a b = b a$ (since the union of ordered subsets $\{a,b\} \cup \{b\} = \{a, b\}$ inherits the same ordering that $\{a,b\}$ had).

* {#fin_gen}In particular, for a finite set of generators the free graphic monoid is finite and consists of all words without repetitions of which there are exactly $n!\sum_{i=0}^n \frac{1}{i!}$ over an alphabet with $n$ letters.[^schuetz]

[^schuetz]: The finiteness was basically already observed in Schützenberger ([1947](#Schütz47)). See also Lawvere ([1989b](#Law89b)).

* The variety (in the sense of universal algebra) of graphic monoids is generated by $\Delta_1$ (cf. Lawvere ([1991](#Law91))).

### Graphic monoids as shelves

There is a relation between graphic monoids and [[shelf|shelves]]:

+-- {: .num_prop #GraphicsAsShelves}
###### Theorem

A graphic monoid is the same as a _unital [[left shelf]]_, meaning a set equipped with a binary operation that obeys the left self-distributive law

$$  a (b c) = (a b)(a c) $$

and has an element $1$ serving as a left and right unit:

$$  1 a = a = a 1$$

=--

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

First start with a unital left shelf.  Note that the graphic identity
holds:

   $$ a  b = a (b 1) = (a b) (a 1) = (a  b)  a $$

   and also

   $$ a  b = a  (1  b) = (a  1)  (a  b) = a  (a  b) $$

   Using these and self-distributivity we can deduce the associative law as follows:

   $$\begin{aligned}
a  (b  c) &= (a  b)  (a  c) \\
&= ((a  b)  a)  ((a  b)  c) \\
&= (a  b)  ((a  b)  c) \\
&= (a  b)  c
\end{aligned}$$

   So, we have a graphic monoid.

   Conversely, suppose we start with a graphic monoid.  Then we can prove the left self-distributive law as follows:

   $$ a (b c) = (a b) c = (a b a) c = (a b)(a c)  $$

=--

Note in particular that, somewhat surprisingly, the **associativity** in a unital left shelf comes for free! For more background see the comments on:

   * [The origin of the word "quandle"](https://golem.ph.utexas.edu/category/2015/05/the_origin_of_the_word_quandle.html#c049149), $n$-Category Café.

## Related Pages

* [[graphic topos]]

* [[Hegelian taco]]

* [[shelf]]

* [[band]]

* [[rectangular band]]

* [[Aufhebung]]

* [[graph]]

* [[band]]

## References

* {#AgMah06}M. Aguiar, S. Mahajan, _Coxeter Groups and Hopf Algebras_ , AMS Providence 2006. ([draft](http://www.math.cornell.edu/~maguiar/monograph.pdf))

* {#brown00}K. S. Brown, _Semigroups, Semirings, and Markov Chains_ , J. Theor. Prob. **13** no.3 (2000) pp.871-938.

* R. Guitart, _Autocategories: II. Autographic algebras_ , Cah. Top. Géom. Diff. Cat. **LV** (2014) pp.151-160.

* N. Kimura, _The structure of idempotent semigroups I_ , Pacific
Journal of Mathematics **8** no.2 (1958) pp.257-275. ([pdf](http://msp.org/pjm/1958/8-2/pjm-v8-n2-p07-p.pdf))

* F. Klein-Barmen, _Über eine weitere Verallgemeinerung des Verbandsbegriffs_ , Math. Z. **46** (1940) pp.472-480.([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN00237921X))

* [[F. W. Lawvere]], _Qualitative distinctions between some toposes of generalized graphs_, Contemp. Math. **92** (1989) pp. 261-299.

* {#Law89b} [[F. W. Lawvere]], _Display of graphics and their applications, as exemplified by 2-categories and the Hegelian "taco"_  Proceedings of the first international conference on algebraic methodology and software technology University of Iowa, May 22-24 1989, Iowa City, pp. 51-74.

* {#Law91} [[F. W. Lawvere]], _More on graphic toposes_, Cah. Top. Géom. Diff. Cat. **XXXII** no. 1 (1991) pp.5-10. ([pdf](http://archive.numdam.org/article/CTGDC_1991__32_1_5_0.pdf))

* {#Law02} [[F. W. Lawvere]], _Linearization of graphic toposes via Coxeter groups_, JPAA **168** (2002) pp. 425-436. ([pdf](http://www.sciencedirect.com/science/article/pii/S0022404901001074/pdfft?md5=a4ca9bc67df6ae63ddf53c559bd71315&pid=1-s2.0-S0022404901001074-main.pdf))

* {#MSS15} S. Margolis, F. Saliola, B. Steinberg, _Cell Complexes, Poset Topology and the Representation Theory of Algebras Arising in
Algebraic Combinatorics and Discrete
Geometry_ , arXiv:1508.05446 (2015). ([abstract](http://arxiv.org/abs/1508.05446v1))

* M.-P. Schützenberger, _Sur certains treillis gauches_, C. R. Acad. Sci. Paris **225** (1947) pp.277–278. ([pdf](http://igm.univ-mlv.fr/~berstel/Mps/Travaux/A/1947TreillisGauchesCras.pdf)) {#Schütz47}


graphical quantum channel > history


graphical sets

##Definition ##
A **graphical set** is a functor from $\Phi^{op}$, the opposite of the [[graphical category]] to the [[category of sets]].


Grassmann algebra > history

< [[Grassmann algebra]]
[[!redirects Grassmann algebra - delete]]


Grassmannian

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

# Contents
* table of contents
{: toc}

## Idea

For $V$ a [[vector space]] and $r$ a [[cardinal number]] (generally taken to be a [[natural number]]), the **Grassmannian** $Gr(r,V)$ is the space of all  $r$-[[dimension|dimensional]] [[linear subspaces]] of $V$.

## Definition

For $n \in \mathbb{N}$, write $O(n)$ for the [[orthogonal group]] acting on $\mathbb{R}^n$. More generally, say that a [[Euclidean space|Euclidean]] [[vector space]] $V$ is an [[inner product space]] such that there exists a [[linear map|linear]] [[isometry]] $V \overset{\simeq}{\to} \mathbb{R}^n$ to $\mathbb{R}^n$ equipped with its canonical inner product. Then write $O(V)$ for the group of linear isometric [[automorphisms]] of $V$. For the following we regard these groups as [[topological groups]] in the canonical way.

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

For $n, k \in \mathbb{N}$ and $n \leq k$, then the $n$th **Grassmannian** of $\mathbb{R}^k$ is the [[coset]] [[topological space]].

$$
  Gr_n(\mathbb{R}^k) \coloneqq O(k)/(O(n) \times O(k-n))
  \,,
$$

where the [[action]] of the [[product group]] is via its canonical embedding $O(n)\times O(k-n) \hookrightarrow O(k)$.

Generally, for $W \subset V$ an inclusion of [[Euclidean space|Euclidean]] vector spaces, then

$$
  Gr_W(V) \coloneqq O(V)/(O(W)\times O(V-W))
  \,,
$$

where $V-W$ denotes the [[orthogonal complement]] of $W$ in $V$.

=--

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

The group $O(k)$ [[transitive action|acts transitively]] on the set of $n$-dimensional [[linear subspaces]], and given any such, then its [[stabilizer subgroup]] in $O(k)$ is isomorphic to $O(n)\times O(k-n)$. In this way the underlying set of $Gr_n(k)$ is in [[natural bijection]] to the set of $n$-dimensional linear subspaces in $\mathbb{R}^k$. The realization as a [[coset]] as above serves to equip this set naturally with a [[topology|topological space]].

=--

## Properties

### Relation to Stiefel manifolds and universal bundles

Similarly, the _real [[Stiefel manifold]]_ is the [[coset]]

$$
  V_n(k) \coloneqq O(k)/O(k-n)
  \,.
$$

The [[quotient]] [[projection]]

$$
  V_{n}(k)\longrightarrow Gr_n(k)
$$

is an $O(n)$-[[principal bundle]], with [[associated bundle]] $V_n(k)\times_{O(n)} \mathbb{R}^n$ a [[vector bundle]] of [[rank]] $n$. In the limit ([[colimit]]) that $k \to \infty$ is this gives a presentation of the $O(n)$-[[universal principal bundle]] and of the [[universal vector bundle]] of rank $n$, respectively.. The base space $Gr_n(\infty)\simeq_{whe} B O(n)$ is the [[classifying space]] for $O(n)$-[[principal bundles]] and rank $n$ vector bundles.

### CW-complex structure

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

The real Grassmannians $Gr_n(\mathbb{R}^k)$ and the complex Grassmannians $Gr_n(\mathbb{C}^k)$ admit the structure of [[CW-complexes]]. Moreover the canonical inclusions

$$
  Gr_n(\mathbb{R}^k)
  \hookrightarrow
  Gr_n(\mathbb{R}^{k+1})
$$

are subcomplex incusion (hence [[relative cell complex]] inclusions).

Accordingly there is an induced CW-complex structure on the [[classifying space]]

$$
  B O(n) \simeq \underset{\longrightarrow}{\lim}_k Gr_n(\mathbb{R}^k)
  \,.
$$

=--

A proof is spelled out in ([Hatcher, section 1.2 (pages 31-34)](#Hatcher)).

\begin{prop}\label{ComplexProjectiveSpaceIsOkaManifold}
**([[complex projective space]] is [[Oka manifold]])** \linebreak
Every [[complex projective space]] $\mathbb{C}P^n$, $n \in \mathbb{N}$, is an [[Oka manifold]]. More generally every [[Grassmannian]] over the [[complex numbers]] is an [[Oka manifold]].
\end{prop}
(review in [Forstnerič & Lárusson 11, p. 9](Oka+manifold#ForstnericLarusson11), [Forstnerič 2013, Ex. 2.7](Oka+manifold#Forstneric13))

## Examples

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

The [[Grassmannian]] $Gr_r(\mathbb{R}^{n+1})$ (def. \ref{RealAndComplexGrassmannian}) is

* for $r = 0$: the point;

* for $r = 1$: the [[real projective space]] $\mathbb{R}P^n$;

* for $r = n+1$: the point;

* for $r \gt n+1$: the empty space $\emptyset$.

=--

If $V$ is an [[inner product space]], then the [[orthogonal complement]] defines an [[isomorphism]] between $Gr(r,V)$ and $Gr(\dim V - r,V)$.

## Related concepts

* [[isotropic Grassmannian]]

* [[Lagrangian Grassmannian]], [[affine Grassmannian]]

* [[flag variety]], [[Schubert variety]]

* [[Stiefel manifold]]

* [[coset space]]

* [[projective space]]

* [[classifying space]], [[universal bundle]]

## References

Textbook accounts include

* {#Kochmann96} [[Stanley Kochmann]], section 1.3 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#Hatcher} [[Allen Hatcher]], section 1.2 of _Vector bundles & K-theory_ ([web](https://www.math.cornell.edu/~hatcher/VBKT/VBpage.html))

* SL Kleiman, _Geometry on Grassmannians and applications to splitting bundles and smoothing cycles_, Publications Mathématiques de l'IHÉS, 1969, numdam/[pdf](http://www.numdam.org/article/PMIHES_1969__36__281_0.pdf)

Lecture notes include

* [[Michael Hopkins]], _Grassmannian manifolds_ ([pdf](http://isites.harvard.edu/fs/docs/icb.topic1471353.files/grassmannian.pdf))

category: geometry, algebra

[[!redirects Grassmannians]]

[[!redirects Grassmannian manifold]]
[[!redirects Grassmannian manifolds]]


graviphoton

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### Context
#### Fields and quanta
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[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[Kaluza-Klein compactification]] on a [[circle principal bundle]] $(\widehat X, \widehat g)$, the _graviphoton field_ is the resulting [[gauge field]] ([[electromagnetic field]]) on the base of the fibration.

## Definition

If $v_5 \in \Gamma\big( T\widehat X\big)$ denotes the [[vertical vector field]] which corresponds to the [[isometry|isometric]] [[flow]] along the [[circle]] [[fibers]], the graviphoton field, regraded as a [[Cartan connection]] [[differential 1-form]] on the total space $\widehat X$ bundle is the contraction of the [[metric tensor]] $\widehat g$ with this vector field:

$$
  A \;\coloneqq\; \widehat g(v,-)
  \,.
$$

## Examples

### In type IIA string theory

In the [[duality between M-theory and type IIA string theory]], the graviphoton of the KK-compactification is the [[RR-field]] potential $C_1$ of [[type IIA string theory]].

### On D4-branes

The graviphoton as the [[RR-field]] potential $C_1$ of [[type IIA string theory]] then appears in the [[higher WZW term]] on the [[D4-brane]] ([CGNSW 96 (7.4)](#CGNSW96) [APPS97b (51)](#APPS97b))
as the _[[theta angle]]_ in [[D=5 super Yang-Mills theory]]:

$$
  \mathbf{L}_{D4}^{WZ}
  \;\propto\;
  C_1 \wedge \langle F \wedge F\rangle
  \,.
$$

## Related concepts

* [[dilaton]]

## References

### General

See also

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

### In string theory

The [[graviphoton]] of the [[duality between M-theory and type IIA string theory]], as the [[RR-field]]-potential $C_1$ in the [[higher WZW term]] of the [[D4-brane]]:

* {#CGNSW96} [[Martin Cederwall]], Alexander von Gussich, [[Bengt Nilsson]], Per Sundell, Anders Westerberg, around (4.6) of _The Dirichlet Super-p-Branes in Ten-Dimensional Type IIA and IIB Supergravity_, Nucl.Phys. B490 (1997) 179-201 ([arXiv:hep-th/9611159](http://arxiv.org/abs/hep-th/9611159))

* {#APPS97b} [[Mina Aganagic]], Jaemo Park, Costin Popescu, [[John Schwarz]], Section 6 of _Dual D-Brane Actions_, Nucl. Phys. B496 (1997) 215-230 ([arXiv:hep-th/9702133](https://arxiv.org/abs/hep-th/9702133))

* {#FSS19c} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Theorem 7.4 of _[[schreiber:Super-exceptional embedding construction of the M5-brane|Super-exceptional geometry: origin of heterotic M-theory and super-exceptional embedding construction of M5]]_

For more see at _[Green-Schwarz sigma model -- References -- For D-branes](Green-Schwarz+action+functional#ReferencesForDBranes)_.

Discussion as the [[theta angle]] of the [[Witten-Sakai-Sugimoto model]] for [[QCD]] on [[D4-branes]]:

* Si-wen Li, around (3.1) of _The theta-dependent Yang-Mills theory at finite temperature in a holographic description_ ([arXiv:1907.10277](https://arxiv.org/abs/1907.10277))

[[!redirects graviphotons]]


Gravitation

This pages collects material related to the textbook

* [[Charles Misner]], [[Kip Thorne]], [[John Wheeler]]:

  \linebreak

  **Gravitation**

  \linebreak

  W. H. Freeman, San Francisco (1973)

  ISBN:9780716703440, 9780691177793

  [inspire:95654](https://inspirehep.net/literature/95654)

  Wikipedia entry

on the [[theory (physics)|theory]] of [[Einstein gravity]] ([[general relativity]]).

## Related entries

* [[rubber-sheet analogy of gravity]]

## References

* Wikipedia _[Gravitation (book)](http://en.wikipedia.org/wiki/Gravitation_%28book%29)_

category: reference


gravitational anomaly

#Contents#
* table of contents
{:toc}

## Idea

The standard [[quantum anomaly]] induced by [[fermions]] coupled to a dynamical [[Riemannian metric]]/[[field (physics)|field]] of [[gravity]].

Explicit discussion as an anomalous [[action functional]] is in ([Freed 86, examples 1.5](#Freed86)).

## Related concepts

* [[conformal anomaly]]

## References

* [[Daniel Freed]], _Determinants, torsion, and strings_,  Comm. Math. Phys. 107:3 (1986) 483-513. ([Euclid](http://projecteuclid.org/euclid.cmp/1104116145))
  {#Freed86}

For more references see at _[[quantum anomaly]]_.

[[!redirects gravitational anomalies]]


gravitational constant

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### Context
#### Gravity
+-- {: .hide}
[[!include gravity contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

# Newton\'s gravitational constant
* table of contents
{: toc}

## Idea

[[Isaac Newton|Newton]]\'s [[gravity|gravitational]] constant provides a [[unit of measurement|unit]] conversion between units of [[spacetime]] and units of [[mass]]/[[energy]].

## Motivation

According to [[Albert Einstein|Einstein]]\'s theory of [[general relativity]], the [[Einstein tensor]] $G$ of the [[pseudoRiemannian metric]] $g$ on [[spacetime]] is proportional to the [[stress-energy tensor]] $T$ of the [[matter]] and [[radiation]] in spacetime.  (For simplicity, let any [[cosmological constant]] or other [[dark energy]] be included in $T$.)  The simplest way to state this proportion is as an equality:
$$ G = T $$
or (with [[tensor]] indices)
$$ G_{a,b} = T_{a,b} .$$

However, conventional [[units of measurement]] don\'t allow this equality, and so we write
$$ G_{a,b} = k T_{a,b} ,$$
where the constant $k$ is about $1.67732 \times 10^{-9} \m^3 \kg^{-1} \s^{-2} $ in [[SI unit]]s.  (Depending on how one handles the conversion between the space and time components of $G$ and $T$, there could be extra factors of the [[speed of light]] constant $2.99792458 \times 10^8 \m \s^{-1}$.)

This factor is essentially the gravitational constant; however, the __gravitational constant__ is traditionally taken to be smaller by a factor of $8 \pi$:
$$
G = 6.67408(31) \times 10^{-11} \m^3 \kg^{-1} \s^{-2},
$$
where the standard uncertainty (about 46ppm) is in parentheses. Yes, the same letter is used for this constant as for the Einstein tensor! Although this yields
$$ G_{a,b} = 8 \pi G T_{a,b} $$
in general relativity, it gives the simplest formula for the [[fictional force]] of gravity in the nonrelativistic limit:
$$ F = G \frac{m_1 m_2}{r^2} $$
for the [[force]] exerted by either [[mass]] $m_i$ on the other at a distance $r$.  (It was in this context that Newton used the constant.)

## Measurement

There is good stuff to say here about how we only know this constant to about 6 significant digits.

## Related concepts

[[!include fundamental scales -- contents]]

## References

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

[[!redirects gravitational constant]]
[[!redirects Gravitational constant]]
[[!redirects Newton's constant]]
[[!redirects Newton&39;s constant]]
[[!redirects Newton's constant]]
[[!redirects Newton\'s constant]]
[[!redirects Newton's constant]]


gravitational instanton

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#### Riemannian geometry
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The analog of [[Yang-Mills instantons]] in [[gravity]] (...)

## Related concepts

* [[Yang-Mills instanton]]

* [[Kaluza-Klein monopole]]

* [[Taub-NUT spacetime]]

* [[exotic sphere]]

## References

The original articles:

* {#EguchiFreund67} [[Tohru Eguchi]], [[Peter Freund]], _Quantum Gravity and World Topology_, Phys. Rev. Lett. 37, 1251 (1967) ([doi:10.1103/PhysRevLett.37.1251](https://doi.org/10.1103/PhysRevLett.37.1251))

* {#BelavinBurlankov76} [[Alexander Belavin]], D. Burlankov, _The renormalisable theory of gravitation and the Einstein equations_, Physics Letters A Volume 58, Issue 1, 26 July 1976, Pages 7-8 (doi:10.1016/0375-9601(76)90530-2)

* [[Tohru Eguchi]], [[Andrew Hanson]], _Gravitational instantons_, General Relativity and Gravitation volume 11, pages 315–320 (1979) ([doi:10.1007/BF00759271](https://doi.org/10.1007/BF00759271))

Review:

* [[Tohru Eguchi]], [[Peter Gilkey]], [[Andrew Hanson]], Section 10.2 of: _Gravitation, gauge theories and differential geometry_, Physics Reports Volume 66, Issue 6, December 1980, Pages 213-393 (doi:10.1016/0370-1573(80)90130-1)

See also

* Wikipedia, _[Gravitational instanton](https://en.wikipedia.org/wiki/Gravitational_instanton#References)_

On gravitational instanton number counted by [[first Pontrjagin class]]:

* [Eguchi-Freund 67, around (1)](#EguchiFreund67)

* [Belavin-Burlankov 76](#BelavinBurlankov76)

* Serdar Nergiz, Cihan Saclioglum, equation (27) in: _A Quasiperiodic Gibbons--Hawking Metric and Spacetime Foam_, Phys. Rev. D 53, 2240 (1996) ([arXiv:hep-th/9505141](https://arxiv.org/abs/hep-th/9505141))

Argument that [[exotic spheres]] may be regarded as gravitational instantons:

* {#Witten85} [[Edward Witten]], p. 12 of: _Global gravitational anomalies_, Comm. Math. Phys. Volume 100, Number 2 (1985), 197--229.  ([EUCLID](http://projecteuclid.org/euclid.cmp/1103943444))

* Randy A. Baadhio, _On the global gravitational instanton and soliton that are homotopy spheres_, Journal of Mathematical Physics 32, 2869 (1991) ([doi:10.1063/1.529078](https://doi.org/10.1063/1.529078))

[[!redirects gravitational instantons]]

[[!redirects gravitational instanton number]]
[[!redirects gravitational instanton numbers]]


gravitational wave

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=--
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+--{: .hide}
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## Idea

The [[physics|physical]] [[theory (physics)|theory]] of [[Einstein]]-[[gravity]] ("[[general relativity]]") predicts, via [[Einstein equations|Einstein's equations of motion]], that [[vacuum]] [[spacetime]] may show a kind of oscillations of the [[field]] of [[gravity]] roughly analogous to [[electromagnetic waves]] in [[electromagnetism]]. (Both kinds of waves are oscillation of the [[field (physics)|field]] itself and do not depend on some "medium" such as a water wave does.) In fact, since the [[force]] of [[gravity]] is reflected in the [[pseudo-Riemannian geometry]] of [[spacetime]], a gravitational wave is a kind of periodic distortion of spacetime geometry itself.

## Experimental observation

At the end of the 20th century, there had accumulated excellent but indirect evidence for gravitational waves from the observation of [[binary pulsars]] ([Hulse-Taylor 75](#HulseTaylor75)). Their observed rotational motion loses energy precisely to the extent that [[general relativity]] predicts is being radiated away by gravitational waves.

On 11 Febrary 2016 was the announcement of the first direct detection by the [[LIGO]] collaboration, using laser interferometry, of a gravitational wave signal coming from the in-spiral and merge of a pair of [[black holes]] ([LIGO 2016](#LIGO16)). The signal was a chirp at 35-250 Hz (converted to audio in [this looped recording](https://youtu.be/TWqhUANNFXw)), detected as coming from the sky over the southern hemisphere on 14 September 2015.

Further direct detections of gravitational wave events followed. The event GW170817 [LIGO-Virgo17](LIGOVirgo17) showed gravitational waves from merging [[neutron stars]] coincident with the corresponding [[electromagnetic radiation]].

## Related concepts

* [[soft graviton theorem]]

* [[electromagnetic wave]]

* [[pp-wave spacetime]], [[M-wave]]

* [[radiation]]

## References
 {#References}

### General

The first article that correctly derived gravitational waves from the [[Einstein equations]] is

* {#Einstein18} [[Albert Einstein]], _Über Gravitationswellen_, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften Berlin (1918), 154-167

In particular this correctly stated that gravitational waves require a [[quadrupole moment]] as a [[source]] (e.g. a rotating [[binary star system]]) and not just a [[dipole moment]] (e.g. an oscillating charge) as for [[electromagnetic waves]] (the [[graviton]] has [[spin]] 2, the [[photon]] has spin 1...), thereby correcting a mistake to this effect in the earlier article

* {#Einstein16} [[Albert Einstein]], _Näherungsweise Integration der Feldgleichungen der Gravitation_, Sitzung der physikalisch-mathematischen Klasse vom 22. Juni 1916 ([web](http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/echo/einstein/sitzungsberichte/BGG54UCY/index.meta&ww=0.7142&wh=0.7142&wx=0.2315))

The reality of gravitational wave solutions however kept being a cause of concern for many years (Einstein himself was concerned that the linearization approximation used in their derivation might have been too coarse), for a brief account of the early history see

* {#Steinicke05} [[Wolfgang Steinicke]], _Einstein and the Gravitational waves_, Astron. Nachr. / AN 326 (2005), No. 7 – Short Contributions AG 2005 Köln ([pdf](http://www.hs.uni-hamburg.de/DE/GNT/events/pdf/steinicke05.pdf))

* [[Sean Carroll]], _[Einstein vs. Physical Review](http://www.preposterousuniverse.com/blog/2005/09/16/einstein-vs-physical-review/)_, 2005

A modern walk through the derivation of gravitational waves from linearization of [[Einstein's equations]] may be found for instance on pages 5-24 of

* {#Guidry13} Mike Guidry, _[Astrophysics 616: General Relativity, Black Holes, and Cosmology](http://eagle.phys.utk.edu/guidry/astro616/)_ -- _Chapter 21 -- Gravitational waves_ ([pdf](http://eagle.phys.utk.edu/guidry/astro616/lectures/lecture_ch21.pdf))

See also

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

Review in view of modern gravitational wave detection (see [below](#ReferencesExperimentalObservation)):

* Rong-Gen Cai et al., *The Gravitational-Wave Physics*, National Science Review 4 (2017) 687-706 ([arXiv:1703.00187](https://arxiv.org/abs/1703.00187), [doi:10.1093/nsr/nwx029](https://doi.org/10.1093/nsr/nwx029))

* Ligong Bian et al., *The Gravitational-Wave Physics II: Progress*, Sci. China Phys. Mech. Astron. 2021 ([arXiv:2106.10235](https://arxiv.org/abs/2106.10235), [spire:1869296](https://inspirehep.net/literature/1869296))

* Frans Pretorius, *A Survey of Gravitational Waves* [[arXiv:2306.03797](https://arxiv.org/abs/2306.03797)]

Discussion in view of [[quantum gravity]]:

* Amaury Micheli, Patrick Peter, *Quantum cosmological gravitational waves?*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2211.00182](https://arxiv.org/abs/2211.00182)]

### Theoretical predictions
  {#eferencesTheoreticalPredictions}

Discussion of theoretical predictions for events that have a chance to yield detectable gravitational wave signals includes:

* {#Damour10} [[Thibault Damour]], _Gravitational Waves from Coalescing Binary Black Holes: Theoretical and Experimental Challenges_, talk in Munich, 2010 ([web](http://www.theorie.physik.uni-muenchen.de/MATH/seminars/asc_kolloquium/archiv_sose10/damour/video_damour/index.html), [youtube](https://www.youtube.com/watch?v=hVy3YBalpWE))

In particular, the computation of the signal from the coalescence of two inspiralling black hole [[relativistic binary|binaries]] is due to

* {#BuonannoDamour98} [[Alessandra Buonanno]], [[Thibault Damour]], _Effective one-body approach to general relativistic two-body dynamics_, Phys.Rev. D59 (1999) 084006 ([arXiv:gr-qc/9811091](http://arxiv.org/abs/gr-qc/9811091))

* {#BuonannoDamour00} [[Alessandra Buonanno]], [[Thibault Damour]], _Binary black holes coalescence: transition from adiabatic inspiral to plunge_, IX Marcel Grossmann Meeting in Rome, July 2000 ([arXiv:gr-qc/0011052](http://arxiv.org/abs/gr-qc/0011052))

* {#DamourNagar} [[Thibault Damour]], [[Alessandro Nagar]], _An improved analytical description of inspiralling and coalescing black-hole binaries_, Phys.Rev.D79:081503,2009 ([arXiv:0902.0136](http://arxiv.org/abs/0902.0136))

* {#Damour14} [[Thibault Damour]], [[Alessandro Nagar]], _A new analytic representation of the ringdown waveform of coalescing spinning black hole binaries_, 	10.1103/PhysRevD.90.024054 ([arXiv:1406.0401](http://arxiv.org/abs/1406.0401))

* [[Thibault Damour]], _Radiative contribution to classical gravitational scattering at the third order in $G$_ ([arXiv:2010.01641](https://arxiv.org/abs/2010.01641))

Review of the theoretical predictions and their experimental verification is given in

* {#Damour16} [[Thibault Damour]], _Gravitational Waves and Binary Systems_, talk at IHES, Feb 2016 ([video recording](https://www.youtube.com/watch?v=qYong_D4VUI))

* {#Damour19} [[Thibault Damour]], _On the gravitational interaction of spinning bodies_, talk at _[Souriau2019](http://souriau2019.fr/?page_id=2)_  ([video recording](https://www.youtube.com/watch?v=8HO3DLKircs))

Discussion using the [[string theory results applied elsewhere|string theoretic]] [[KLT relation]]/[[double copy]]-approach for computing higher order corrections to gravitational wave-signatures of [[relativistic binary]] mergers for use with [[LIGO]]:

* {#BCRSSZ19} [[Zvi Bern]], Clifford Cheung, Radu Roiban, Chia-Hsien Shen, Mikhail P. Solon, Mao Zeng, _Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order_, Phys. Rev. Lett. 122, 201603 (2019) ([arXiv:1901.04424](https://arxiv.org/abs/1901.04424))

Discussion in relation to the [[soft graviton theorem]]:

* Arnab Priya Saha, Biswajit Sahoo, [[Ashoke Sen]], _Proof of the Classical Soft Graviton Theorem in $D=4$_ ([arXiv:1912.06413](https://arxiv.org/abs/1912.06413))

### Experimental observation
 {#ReferencesExperimentalObservation}

Indirect detection of gravitational waves based on energy loss of a [[binary pulsar]] system is due to

* {#HulseTaylor75} Hulse, Taylor, _Discovery of a pulsar in a binary system_, The astrophysical journal, 195:L51-53, 1975 ([web](http://adsabs.harvard.edu/full/1975ApJ...195L..51H))

The first proposal of the [[LIGO]]-type experiment for the detection of gravitational waves is due to

* M. Gertsenshtein, I. Postivoit, _On the detection of low frequency gravitational waves_, Soviet Physics JETP, volume 16, number 2, 1963 ([pdf](http://www.jetp.ac.ru/cgi-bin/dn/e_016_02_0433.pdf))

Direct detection of gravitational waves by the [[LIGO]] experiment is reported in

* {#LIGO16} B. P. Abbott et al. ([[LIGO]] Scientific Collaboration and [[Virgo]] Collaboration), _Observation of Gravitational Waves from a Binary Black Hole Merger_, Phys. Rev. Lett. 116, 061102, 11 February 2016, doi:[10.1103/PhysRevLett.116.061102](http://dx.doi.org/10.1103/PhysRevLett.116.061102)

* {#LIGOVirgo17} B. P. Abbott et al. ([[LIGO]] Scientific Collaboration and [[Virgo]] Collaboration), _GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral_ Phys. Rev. Lett. 119, 161101, 2017 ([doi:10.1103/PhysRevLett.119.161101](https://doi.org/10.1103/PhysRevLett.119.161101))

Review:

* Salvatore Vitale, _The first five years of gravitational-wave astrophysics_ ([arXiv:2011.03563](https://arxiv.org/abs/2011.03563))

[[!redirects gravitational waves]]

[[!redirects gravitational wave spacetime]]


gravitino

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## Idea

In [[quantum field theory]] the term _gravitino_ refers to the [[supersymmetry|superpartner]] of the [[graviton]], a [[Rarita-Schwinger field]] of [[spin]] $3/2$ that appears in [[supergravity]].

In [[supergravity]] a field configuration is a [[connection on a bundle|connection]] locally given by a  [[Lie algebra-valued form]]

$$
  (E, \Omega, \Spi) : T X \to \mathfrak{siso}(d,1)
$$

on [[spacetime]] with values in the [[super Poincare Lie algebra]]. Its components $\Psi$ in the [[spin group]] [[representation]] $\Gamma \subset \mathfrak{siso}(d)$ is the **gravitino** field.

The name derives from the fact that the other two components are identified in [[gravity]] with the [[graviton]] field.

## Related concepts

* [[gravitino condensation]]

## References

### General

See also

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

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

### As a dark matter candidate

Discussion of the gravitino as a [[dark matter]] candidate:

* {#EllisOlive10} [[John Ellis]], [[Keith Olive]], _Supersymmetric Dark Matter Candidates_ ([arXiv:1001.3651](http://arxiv.org/abs/1001.3651))

A proposal for super-heavy [[gravitinos]] as [[dark matter]], by embedding [[D=4 N=8 supergravity]] into [[E10]]-[[U-duality]]-invariant [[M-theory]]:

* {#MeissnerNicolai18a} [[Krzysztof A. Meissner]], [[Hermann Nicolai]], _Standard Model Fermions and Infinite-Dimensional R-Symmetries_, Phys. Rev. Lett. 121, 091601 (2018) ([arXiv:1804.09606](https://arxiv.org/abs/1804.09606))

* {#MeissnerNicolai18b} [[Krzysztof A. Meissner]], [[Hermann Nicolai]], _Planck Mass Charged Gravitino Dark Matter_, Phys. Rev. D 100, 035001 (2019) ([arXiv:1809.01441](https://arxiv.org/abs/1809.01441))

following the proposal towards the end of

* {#GellMann83} [[Murray Gell-Mann]], introductory talk at _[Shelter Island II](https://en.wikipedia.org/wiki/Shelter_Island_Conference)_, 1983 ([[Gell-Mann_ShelterIslandII_1983.pdf:file]])

  in: _Shelter Island II: Proceedings of the 1983 Shelter Island Conference on Quantum Field Theory and the Fundamental Problems of Physics_. MIT Press. pp. 301–343. ISBN 0-262-10031-2.

Further duscussion:

* [[Krzysztof A. Meissner]], [[Hermann Nicolai]], _Supermassive gravitinos and giant primordial black holes_ ([arXiv:2007.11889](https://arxiv.org/abs/2007.11889))

* [[Krzysztof A. Meissner]], [[Hermann Nicolai]], *Evidence for a stable supermassive gravitino with charge $2/3$?* [[arXiv:2303.09131](https://arxiv.org/abs/2303.09131)]

[[!redirects gravitinos]]


gravitino condensation

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## Idea

[[gravitino]] [[condensation]], leading to [[supersymmetry breaking]] in [[supergravity]] theories.

## Related concepts

* [[gaugino condensation]]

## References

Survey:

* Nick Houston, _Gravitino condensation, supersymmetry breaking and inflation_ ([arXiv:1512.08210](https://arxiv.org/abs/1512.08210))

As a source for [[cosmic inflation]]:

* Jean Alexandre, Nick Houston, Nick E. Mavromatos, _Dynamical Supergravity Breaking via the Super-Higgs Effect Revisited_, Phys. Rev. D 88, 125017 (2013) ([arXiv:1310.4122](https://arxiv.org/abs/1310.4122))

* Jean Alexandre, Nick Houston, Nick E. Mavromatos, _Inflation via Gravitino Condensation in Dynamically Broken Supergravity_,  International Journal of Modern Physics D, Volume 24, Issue 04, April 2015 ([arXiv:1409.3183](https://arxiv.org/abs/1409.3183))

and in particular as a source for the [[Starobinsky model of cosmic inflation]]:

* Jean Alexandre, Nick Houston, Nick E. Mavromatos, _Starobinsky-type Inflation in Dynamical Supergravity Breaking Scenarios_, Phys. Rev. D 89, 027703 (2014) ([arXiv:1312.5197](https://arxiv.org/abs/1312.5197))

[[!redirects gravitino condensations]]

[[!redirects gravitino condensate]]
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graviton

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## Idea

The _graviton_ is the (hypothetical) quantum of the [[field (physics)|field]] of [[gravity]], i.e., the quanta of the theory of [quantum gravity](http://ncatlab.org/nlab/show/quantum+gravity).

## Details

In [[first-order formulation of gravity]] a field configuration is locally a [[Lie algebra-valued form]]

$$
  (E, \Omega) : T X \to \mathfrak{iso}(d)
$$

with values in the [[Poincare Lie algebra]].

This is a [[vielbein]] $E$ and a [[spin connection]] $\Omega$. This together is the **graviton field**.

A graviton has [spin](http://ncatlab.org/nlab/show/spin) $2$, and is massless. We can see that it has spin $2$ from the fact that the source of gravity is $T$, the [energy-momentum tensor](http://ncatlab.org/nlab/show/stress-energy+tensor), which is a second-rank tensor. It can be shown that a massless spin-$2$ particle has to be a graviton. The basic concept behind this is that massless particles have to couple to conserved currents - the stress-energy tensor $T$, the source of gravity.

In [[supergravity]] this is accompanied by the [[gravitino]].

## Related concepts

* [[soft graviton theorem]]

* [[boson]]

  * [[gauge boson]]

    * [[photon]], [[gluon]],

    * **graviton**,

      * [[dual graviton]]

      * [[giant graviton]]

* [[fermion]]

  * [[electron]], [[quark]],

  * [[gravitino]], [[gaugino]] , [[dilatino]]

## References

### General

See the references at _[quantum gravity -- and an effective perturbative field theory](quantum+gravity#ReferencesAsAnEffectiveFieldTheory)_

On potential [[experiments]] detecting gravitons:

* [[Freeman Dyson]], *Is the graviton detectable?*, International Journal of Modern Physics A **28** 25 (2013) 1330041 [[doi:10.1142/S0217751X1330041X](https://doi.org/10.1142/S0217751X1330041X)]

* Tony Rothman, Stephen Boughn, *Can Gravitons be Detected?*, Foundations of Physics **36** (2006) 1801–1825 [[doi:10.1007/s10701-006-9081-9](https://doi.org/10.1007/s10701-006-9081-9)]

* Tony Rothman, Stephen Boughn, *Aspects of graviton detection: graviton emission and absorption by atomic hydrogen*, Classical and Quantum Gravity **23** 20 (2006)  5839 [[doi:10.1088/0264-9381/23/20/006](https://iopscience.iop.org/article/10.1088/0264-9381/23/20/006)]

* Daniel Carney, Valerie Domcke, Nicholas L. Rodd, *Graviton detection and the quantization of gravity* [[arXiv:2308.12988](https://arxiv.org/abs/2308.12988)]

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

[[!redirects gravitons]]


gravity

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## Idea

A [[field (physics)|field]] configuration of the [[physical theory]] of _gravity_ on a [[spacetime]] $X$ is equivalently

* a [[vielbein]] field, hence a [[reduction of the structure group]] of the [[tangent bundle]] along $\mathbf{B} O(n-1,1) \to \mathbf{B}GL(n)$, defining a [[pseudo-Riemannian metric]];

* a [[connection on a bundle|connection]] that is locally a [[Lie algebra-valued 1-form]] with values in the [[Poincare Lie algebra]].

  $$
    (E, \Omega) : T X \to \mathfrak{iso}(d-1,1)
  $$

  such that this is a [[Cartan connection]].

(This parameterization of the gravitational field is called the [[first-order formulation of gravity]].) The component $E$ of the connection is the [[vielbein]] that encodes a [[pseudo-Riemannian metric]] $g = E \cdot E$ on $X$ and makes $X$ a [[pseudo-Riemannian manifold]]. Its quanta are the [[graviton]]s.

The [[non-propagating field]] $\Omega$ is the [[spin connection]].

The [[action functional]] on the space of such connection which defines the [[classical field theory]] of gravity is the [[Einstein-Hilbert action]].

More generally, [[supergravity]] is a [[gauge theory]] over a [[supermanifold]] $X$ for the [[super Euclidean group|super Poincare group]]. The field of supergravity is a Lie-algebra valued form with values in the [[super Poincare Lie algebra]].

$$
  (E,\Omega, \Psi) : T X \to \mathfrak{siso}(d-1,1)
$$

The additional [[fermion]]ic field $\Psi$ is the [[gravitino]] field.

So the [[configuration space]] of gravity on some $X$ is essentially the [[moduli space of Riemannian metrics]] on $X$.

## Details

> for the moment see *[[D'Auria-Fre formulation of supergravity]]* for further details

## Related concepts

* [[Einstein-Hilbert action]], [[Einstein equation]]

* [[rubber-sheet analogy of gravity]]

* [[general covariance]]

* [[Penrose-Hawking theorem]], [[cosmic censorship hypothesis]]

* [[weak gravity conjecture]]

* [[first order formulation of gravity]]

  * [[Plebanski formulation of gravity]], [[gravity as a BF-theory]], [[teleparallel gravity]]

  * [[Chern-Simons gravity]]

* [[Einstein-Maxwell theory]], [[Einstein-Yang-Mills theory]]

* [[higher order curvature corrections]]

* [[supergravity]]

* [[D=2 gravity]]

  * [[Liouville theory]]

  * [[Jackiw-Teitelboim gravity]]

* [[D=5 gravity]]

* [[spacetime]]

  * [[black hole]]

  * [[gravitational wave]]

* gravitational entropy

  * [[Bekenstein-Hawking entropy]]

  * [[generalized second law of thermodynamics]]

* [[energy condition]]

* [[positive energy theorem]]

* [[asymptotic safety]]

* [[cosmology]]

  * [[standard model of cosmology]]

* [[MOND]]

* [[computer experiment]]: [[gevolution]]

## References

### General

Historical texts:

* {#Infeld1962} [[Leopold Infeld]] (ed.), *Relativistic Theories of Gravitation*, Proceedings of a conference held in Warsaw and Jablonna 1962, Pergamon Press (1964) [[pdf](https://cds.cern.ch/record/2282975/files/warsaw-1962.pdf)]

Textbook accounts:

* [[Charles Misner]], [[Kip Thorne]], [[John Wheeler]], _[[Gravitation]]_ (1973)

* [[Robert Wald]], *General Relativity*, University of Chicago Press (1984) [[doi:10.7208/chicago/9780226870373.001.0001](https://doi.org/10.7208/chicago/9780226870373.001.0001), [pdf](http://fma.if.usp.br/~mlima/teaching/PGF5292_2021/Wald_GR.pdf)]

* [[Pietro Fré]], *Gravity, a Geometrical Course*, Volume 1: *Development of the Theory and Basic Physical Applications*, Spinger (2013) [[doi:10.1007/978-94-007-5361-7](https://doi.org/10.1007/978-94-007-5361-7)]

* [[Pietro Fré]], *Gravity, a Geometrical Course*, Volume 2: *Black Holes, Cosmology and Introduction to Supergravity*, Springer (2013) [[doi:10.1007/978-94-007-5443-0](https://doi.org/10.1007/978-94-007-5443-0)]

  > on [[black holes]], [[cosmology]] and the ([[D'Auria-Fré formulation of supergravity|D'Auria-Fré formulation of]]) [[supergravity]]

Lecture notes:

* [[Matthias Blau]], _Lecture notes on general relativity_ ([web](http://www.blau.itp.unibe.ch/GRLecturenotes.html))

* Emil T. Akhmedov, _Lectures on General Theory of Relativity_ ([arXiv:1601.04996](https://arxiv.org/abs/1601.04996))

* Pietro Menotti, _Lectures on gravitation_ ([arXiv:1703.05155](https://arxiv.org/abs/1703.05155))

See also

* {#Coley18} [[Alan Coley]], _Mathematical General Relativity_ ([arXiv:1807.08628](https://arxiv.org/abs/1807.08628))

On gravity in relation to [[thermodynamics]]:

* [[Thanu Padmanabhan]], *Gravity and/is Thermodynamics*,  Current Science, **109** (2015) 2236-2242 [[doi:10.18520/v109/i12/2236-2242](https://doi.org/10.18520/v109/i12/2236-2242)]

* [[Thanu Padmanabhan]], *Exploring the Nature of Gravity*, talk notes [[arXiv:1602.01474](https://arxiv.org/abs/1602.01474)]

Discussion of classical gravity via its [[perturbative quantum field theory]]:

* {#Scharf01} [[Günter Scharf]], _[[Quantum Gauge Theories -- A True Ghost Story]]_, Wiley 2001

* Gustav Uhre Jakobsen, _General Relativity from Quantum Field Theory_ ([arXiv:2010.08839](https://arxiv.org/abs/2010.08839))

This way the theory of gravity based on the standard [[Einstein-Hilbert action]] may be regarded as just an [[effective quantum field theory]], which makes some of its notorious problems be non-problems:

* [[John Donoghue]], _Introduction to the Effective Field Theory Description of Gravity_ ([arXiv:gr-qc/9512024](http://arxiv.org/abs/gr-qc/9512024))

Relation of the [[first-order formulation of gravity]] to [[BF-theory]]:

* [[Alberto S. Cattaneo]], [[Leon Menger]], [[Michele Schiavina]], *Gravity with torsion as deformed BF theory* [[arXiv:2310.01877](https://arxiv.org/abs/2310.01877)]

See also the references at _[[general relativity]]_.

### Covariant phase space
 {#CovariantPhaseSpaceReferences}

The (reduced) [[covariant phase space]] of gravity (presented for instance by its [[BV-BRST complex]], see there fore more details) is discussed for instance in

* [[Romeo Brunetti]], [[Klaus Fredenhagen]], _Towards a Background Independent Formulation of Perturbative Quantum Gravity_ ([arXiv:gr-qc/0603079](http://arxiv.org/abs/gr-qc/0603079))

which is surveyed in

* Katarzyna Rejzner, _The BV formalism applied to classical gravity_ ([pdf](http://rejzner.com/talks/Karlsruhe2011.pdf))

Careful discussion of [[observables]] in gravity is in

* {#Khavkine14} [[Igor Khavkine]],  _Local and gauge invariant observables in gravity_, talk at [Operator and Geometric Analysis on Quantum Theory](http://www.science.unitn.it/~moretti/convegno/convegno.html), preprint [arXiv:1503.03754](https://arxiv.org/abs/1503.03754)

Further discussion of the phase space of gravity in [[first-order formulation of gravity|first-order formulation]] via [[BV-BFV formalism]]:

* [[Michele Schiavina]], _BV-BFV approach to general relativity_ (2015) [[pdf](http://user.math.uzh.ch/cattaneo/schiavina.pdf), [[Schiavina-Relativity.pdf:file]]]

* {#CattaneoSchiavina15} [[Alberto Cattaneo]], [[Michele Schiavina]], _BV-BFV approach to General Relativity, Einstein-Hilbert action_, J. Math. Phys. **57** 023515 (2016) [[arXiv:1509.05762](https://arxiv.org/abs/1509.05762), [doi:10.1063/1.4941410](https://doi.org/10.1063/1.4941410)]

* {#CattaneoSchiavina17a} [[Alberto Cattaneo]], [[Michele Schiavina]], _The reduced phase space of Palatini-Cartan-Holst theory_, Ann. Henri Poincaré **20** (2019) 445 [[arXiv:1707.05351](https://arxiv.org/abs/1707.05351), [doi:10.1007/s00023-018-0733-z](https://doi.org/10.1007/s00023-018-0733-z)]

* {#CattaneoSchiavina17b} [[Alberto Cattaneo]], [[Michele Schiavina]], *BV-BFV approach to General Relativity: Palatini-Cartan-Holst action*, Adv. Theor. Math. Phys. **23**  (2019) 1801-1835 [[arXiv:1707.06328](https://arxiv.org/abs/1707.06328), [doi:10.4310/ATMP.2019.v23.n8.a3](https://doi.org/10.4310/ATMP.2019.v23.n8.a3)]

* [[Alberto S. Cattaneo]], *Phase space for gravity with boundaries*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]* (2023) [[arXiv:2307.04666](https://arxiv.org/abs/2307.04666)]

  > (following [Kijowski & Tulczyjew (2005)](phase+space#KijowskiTulczyjew05))

Discussion of flux-observables:

* [[Alberto S. Cattaneo]], Alejandro Perez, *A note on the Poisson bracket of 2d smeared fluxes in loop quantum gravity*, Class. Quant. Grav. **34** (2017) 107001 [[arXiv:1611.08394](https://arxiv.org/abs/1611.08394), [doi:10.1088/1361-6382/aa69b4](https://doi.org/10.1088/1361-6382/aa69b4)]

### Non-renormalizability

The result that gravity is not [[renormalizable interaction|renormalizable]] is due to:

* {#tHooftVelrman74} [[Gerard 't Hooft]], [[Martinus Veltman]], _One loop divergencies in the theory of gravitation_, Ann. Inst. Poincar é 20 (1974) 69 ([numdam:AIHPA_1974__20_1_69_0](AIHPA_1974__20_1_69_0))

Review:

* [[Zvi Bern]], _Perturbative quantum gravity and its relation to gauge theory_, Living reviews in relativity, [www.livingreviews.org/Articles/Volume5/2002-5bern](www.livingreviews.org/Articles/Volume5/2002-5bern)
.
* Assaf Shomer, _A pedagogical explanation for the non-renormalizability of gravity_ ([arXiv:0709.3555](https://arxiv.org/abs/0709.3555))

[[!redirects Einstein gravity]]

[[!redirects gravitation]]

[[!redirects gravitational field]]
[[!redirects gravitational fields]]

[[!redirects D=4 gravity]]


Gravity and Grace

\tableofcontents

\section{Introduction}

_Gravity and grace_ (_La pesanteur et la grâce_) is a collection of aphorisms written, probably without intent of publication, by [[Simone Weil]], assembled by Gustav Thibon and originally published in 1947. It contains a chapter 'Algebra', which we reproduce below.

It is possible to read (part of) this as a criticism of [[formalism]] in mathematics, and as a musing on the importance of [[semantics]] ("the relation of the sign to the thing signified") in the [[foundations of mathematics]].

\section{Algebra}

The following is directly quoted from _Gravity and grace_.

> Money, mechanisation, algebra. The three monsters of contemporary civilisation. Complete analogy.
>
>Algebra and money are essentially levellers, the ﬁrst intellectually, the second eﬀectively.
>
>About ﬁfty years ago the life of the Provençal peasants ceased to
 be like that of the Greek peasants described by Hesiod. The
 destruction of science as conceived by the Greeks took place
 at about the same period. Money and algebra triumphed
 simultaneously.
>
>The relation of the sign to the thing signiﬁed is being destroyed,
 the game of exchanges between signs is being multiplied of
 itself and for itself. And the increasing complication demands
 that there should be signs for signs....
>
>Among the characteristics of the modern world we must not forget the impossibility of thinking in concrete terms of the
 relationship between eﬀort and the result of eﬀort. There are too
 many intermediaries. As in the other cases, this relationship
 which does not lie in any thought, lies in a thing: money.
>
>As collective thought cannot exist as thought, it passes into
 things (signs, machines...). Hence the paradox: it is the thing
 which thinks and the man who is reduced to the state of a thing.
>
>There is no collective thought. On the other hand our science is
 collective like our technics. Specialisation. We inherit not only
 results but methods which we do not understand. For the matter
 of that the two are inseparable, for the results of algebra provide
 methods for the other sciences.
>
>To make an inventory or criticism of our civilisation — what does
 that mean? To try to expose in precise terms the trap which has
 made man the slave of his own inventions. How has
 unconsciousness inﬁltrated itself into methodical thought and
 action? To escape by a return to the primitive state is a lazy
 solution. We have to rediscover the original pact between the
spirit and the world in this very civilisation of which we form a
 part. But it is a task which is beyond our power on account of the
 shortness of life and the impossibility of collaboration and of
 succession. That is no reason for not undertaking it. The situation of all of us is comparable to that of Socrates when he was
 awaiting death in his prison and began to learn to play the
 lyre....At any rate we shall have lived....
>
>The spirit, overcome by the weight of quantity, has no longer
 any other criterion than eﬃciency.
>
>Modern life is given over to immoderation. Immoderation
 invades everything: actions and thought, public and private life.
>
> The decadence of art is due to it. There is no more balance
 anywhere. The Catholic movement is to some extent in reaction
 against this; the Catholic ceremonies, at least, have remained
 intact. But then they are unrelated to the rest of existence.
>
>Capitalism has brought about the emancipation of collective
 humanity with respect to nature. But this collective humanity
 has itself taken on with respect to the individual the oppressive
 function formerly exercised by nature.
>
>This is true even with material things: ﬁre, water etc. The
 community has taken possession of all these natural forces.
>
>Question: can this emancipation, won by society, be
 transferred to the individual?

\section{References}

The text of the original French can be found in

* _La Pesanteur et la grâce_ ([web](http://classiques.uqac.ca/classiques/weil_simone/pesanteur_et_grace/pesanteur_et_grace.html))

[[!redirects Gravity and grace]]


gravity as a BF theory

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#Idea#

The Palatini- or [[first-order formulation of gravity|first order formulation]] of the [[Einstein-Hilbert action]] for [[gravity]] is

$$
  \epsilon_{a b c d} \int_X R^{a b} \wedge  e^c \wedge e^d
$$

where

* $(R^{a b})$ is the curvature of an $\mathfrak{so}(n,1)$-[[connection on a bundle|connection]]

* $(e^a)$ is the vielbein.

This is reminiscent of the form of the [[action functional]] in [[BF theory]]

$$
  \int_X F^{a b} \wedge B_{a b}
  \,.
$$

Various proposals for extensions of this action functional have been made that feature $B$ as an independent field as indicated but then include some dynamical constraint which ensures that on-shell one has $B^{a b} = \epsilon_{a b c d} e^c \wedge e^d$.

This is also related to the [[Plebanski formulation of gravity]].

#References#

The blog entry

* [[Jacques Distler]], _BF_ ([blog](http://golem.ph.utexas.edu/~distler/blog/archives/000541.html))

recalls the construction of

* Laurent Freidel, Artem Starodubtsev, _Quantum gravity in terms of topological observables_ ([arXiv](http://arxiv.org/abs/hep-th/0501191))

and provides some noteworthy comments.

Approaches using the spin group instead of the rotation group include

* Jerzy Lewandowski, Andrzej Okolow, _2-Form Gravity of the Lorentzian signature_ ([arXiv](http://arxiv.org/abs/gr-qc/9911121))

and

* Han-Ying Guo, Yi Ling, Roh-Suan Tung, Yuan-Zhong Zhang, _Chern-Simons Term for BF Theory and Gravity as a Generalized Topological Field Theory in Four Dimensions_ ([arXiv](http://arxiv.org/abs/hep-th/0204059))

For that spinorial approach see also

* Roh Suan Tung, Ted Jacobson, _Spinor One-forms as Gravitational Potentials_ ([arXiv](http://golem.ph.utexas.edu/string/archives/000876.html))

See also

* R. Capovilla, M. Montesinos, V. A. Prieto, E. Rojas, _BF gravity and the Immirzi parameter_ ([arXiv](http://arxiv.org/abs/gr-qc/0102073))

Related is also the construction in

* Michael P. Reisenberger, _Classical Euclidean general relativity from "left-handed area = right-handed area"_ ([arXiv](http://arxiv.org/abs/gr-qc/9804061))

A blog discussion about this and possible interpretations in [[higher category theory]] is at

* [[Urs Schreiber]], _2-Palatini_ ([blog](http://golem.ph.utexas.edu/string/archives/000876.html))

[[!redirects gravity as a BF-theory]]


gravity contents

**[[gravity]], [[supergravity]]**

## Formalism

* [[Poincaré Lie algebra]]

  * [[super Poincaré Lie algebra]], [[supergravity Lie 3-algebra]], [[supergravity Lie 6-algebra]]

* [[differential geometry]]

* ([[super-Cartan geometry|super]]-)[[Cartan geometry]]

  * [[pseudo-Riemannian manifold|pseudo]]-[[Riemannian manifold]]

  * [[spacetime]]

## Definition

* [[Einstein-Hilbert action]], [[Einstein's equations]], [[general relativity]]

  * [[first-order formulation of gravity]], [[D'Auria-Fre formulation of supergravity]]

  * [[gravity as a BF-theory]], [[Plebanski formulation of gravity]], [[teleparallel gravity]]

## Spacetime configurations

* [[spacetime]]

* [[Minkowski spacetime]]

* [[black hole spacetime]]

  * [[Schwarzschild spacetime]]

  * [[Kerr spacetime]]

## Properties

* [[Kaluza-Klein mechanism]]

* [[Bekenstein-Hawking entropy]]

* [[generalized second law of thermodynamics]]

* [[cosmic censorship hypothesis]]

* [[weak gravity conjecture]]

## Spacetimes

* [[standard model of cosmology]]

* [[Minkowski spacetime]]

[[!include charged and rotating black holes -- table]]

* [[asymptotically flat spacetime]]

* [[gravitational wave]]

* [[anti de Sitter spacetime]]

* [[FRW spacetime]]

* [[Taub-NUT spacetime]]

* [[Einstein manifold]]

* [[pp-wave spacetime]]

* [[KK-monopole]]

* [[Malament–Hogarth spacetime]]

* [[super spacetime]]

  * [[super Minkowski spacetime]]

* [[black brane]]

## Quantum theory

* [[quantum gravity]]

  * [[graviton]], [[gravitino]]

  * [[string theory]]


Gray category > history

< [[Gray category]]
[[!redirects Gray category -- history]]


Gray group

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A **Gray-group** is a semistrict version of a [[3-group]]: it is a 3-group whose [[delooping]] is a one-object [[Gray groupoid]].

Gray-groups are modeled by [[2-crossed module]] in analogy to how [[strict 2-group]]s are modeled by [[crossed module]]s.

[[!redirects Gray-group]]

[[!redirects Gray groups]]
[[!redirects Gray-groups]]


Gray tensor product

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## Idea

The **Gray tensor product** is a "better" replacement for the cartesian product of [[strict 2-category|strict 2-categories]].  To get the idea it suffices to consider the 2-category $\mathbf{2}$ which has two objects, 0 and 1, one non-identity morphism $0\to 1$, and no nonidentity 2-cells. Then the cartesian product $\mathbf{2}\times\mathbf{2}$ is a commuting square, while the Gray tensor product $\mathbf{2}\otimes\mathbf{2}$ is a square which commutes up to isomorphism.

More generally, for any 2-categories $C$ and $D$, a 2-functor $C\times\mathbf{2} \to D$ consists of two 2-functors $C\to D$ and a strict 2-natural transformation between them, while a 2-functor $C\otimes\mathbf{2} \to D$ consists of two 2-functors $C\to D$ and a _pseudonatural_ transformation between them.

## Definition

Following up on the last comment, $B\otimes C$ can be defined by
$$
  2Cat(B\otimes C, D) \cong 2Cat(B, Ps(C,D))
$$
where $Ps(C,D)$ is the 2-category of 2-functors, pseudonatural transformations, and modifications $C\to D$.  In other words, the category $2Cat$ of strict 2-categories  and strict 2-functors is a [[closed monoidal category|closed]] [[symmetric monoidal category]], whose tensor product is $\otimes$ and whose internal hom is $Ps(-,-)$.

## Remarks

* When considered with this monoidal structure, $2Cat$ is often called $Gray$.  [[Gray-category|Gray-categories]], or categories [[enriched category|enriched]] over $Gray$, are a model for [[semi-strict infinity-category|semi-strict]] 3-categories.  Categories enriched over $2Cat$ with its cartesian product are _strict_ 3-categories, which are not as useful.  This is one precise sense in which the Gray tensor product is "more correct" than the cartesian product.

* $Gray$ is a rare example of a non-[[cartesian monoidal category|cartesian]] monoidal category whose unit object is nevertheless the terminal object --- that is, a [[semicartesian monoidal category]].

* There are also versions of the Gray tensor product in which pseudonatural transformations are replaced by lax or oplax ones.  (In fact, these were the ones originally defined by Gray.)

* $Gray$ is actually a monoidal [[model category]] (that is, a model category with a monoidal structure that interacts well with the model structure), which $2Cat$ with the cartesian product is not.  In particular, the cartesian product of two cofibrant 2-categories need not be cofibrant.  This is another precise sense in which the Gray tensor product is "more correct" than the cartesian product.

* The cartesian monoidal structure is sometimes called the "black"  product, since the square $2\times 2$ is "completely filled in" (i.e. it commutes).  There is another "white" tensor product in which the square $2\Box 2$ is "not filled in at all" (doesn't commute at all), and the "gray" tensor product is in between the two (the square commutes up to an isomorphism).  This is a pun on the name of John Gray, for whom the Gray tensor product is named.  The "white" tensor product is also called the [[funny tensor product]].

* There are generalizations to [[higher category theory|higher categories]] of the Gray tensor product. In particular there is a tensor product on [[strict omega-category|strict omega-categories]] -- the [[Crans-Gray tensor product]] -- which is such that restricted to strict 2-categories it reproduces the Gray tensor product.

* A  [[closed monoidal category|closed monoidal structure]] on [[strict omega-category|strict omega-categories]] is introduced by Al-Agl, Brown and Steiner. This uses an equivalence between the categories of strict (globular) omega categories and of strict  cubical omega categories with connections; the construction of the closed  monoidal  structure on the latter category is direct and generalises that for strict cubical omega groupoids with connections established by Brown and Higgins.

* The Gray tensor product cannot be extended to a [[2-functor]] (or 3-functor) on the [[2-category]] of 2-categories, 2-functors, [[2-natural transformations]], and [[modifications]]. See [this MathOverflow answer](https://mathoverflow.net/a/459360).

## Related entries

* see also [[generalized Gray tensor product]]

## References

Theorem 1,4.14 of:

* John W. Gray, _Formal category theory: adjointness for 2-categories_, Lecture Notes in Mathematics, Vol. 391. Springer-Verlag, Berlin-New York, 1974. xii+282 pp doi:[10.1007/BFb0061280](https://doi.org/10.1007/BFb0061280) (see also [[Adjointness for 2-Categories]])

* John W. Gray, _Coherence for the Tensor Product of 2-Categories, and Braid Groups_ , pp.62-76 in Heller, [[Myles Tierney|Tierney]] (eds.), _Algebra, Topology, and Category Theory_ , Academic Press New York 1976.

* Robert Gordon, [[John Power]], [[Ross Street]].  _Coherence for tricategories_, Mem. Amer. Math. Soc. 117 (1995), no. 558, vi+81 pp. doi:[10.1090/memo/0558](http://dx.doi.org/10.1090/memo/0558) ([AMS bookstore](https://bookstore.ams.org/memo-117-558/) incl. free sample chapter)

* [[Stephen Lack]], _A Quillen model structure for 2-categories_, K-Theory, 26(2) (2002) pp171-205, ([gzipped .ps](http://maths.mq.edu.au/~slack/papers/qmc2cat.ps.gz)) (doi:[10.1023/A:1020305604826](https://doi.org/10.1023/A:1020305604826) - requires Portico subscription)

* [[Stephen Lack]], _A Quillen model structure for bicategories_, K-theory, 33(3) (2004) pp185-197, ([gzipped .ps](http://maths.mq.edu.au/~slack/papers/qmcbicat.ps.gz)) (doi:[10.1007/s10977-004-6757-9](https://doi.org/10.1007/s10977-004-6757-9) - requires Portico subscription)

* [[Ronnie Brown]] and P.J. Higgins, Tensor products and homotopies for $\omega$-groupoids and crossed complexes,  J. Pure Appl. Alg. 47 (1987) 1-33. doi:[10.1016/0022-4049(87)90099-5](https://doi.org/10.1016/0022-4049(87%2990099-5), ([pdf](https://pdfs.semanticscholar.org/3323/d82ed1effb1953a6af573aab2e3fb0b02394.pdf))

* F.A. Al-Agl, R. Brown and   R. Steiner, _Multiple categories: the equivalence between a globular and cubical approach_, Advances in Mathematics, 170 (2002) 71-118. doi:[10.1006/aima.2001.2069](https://doi.org/10.1006/aima.2001.2069), arXiv:[math/0007009](https://arxiv.org/abs/math/0007009)

The Gray tensor product as the left Kan extension of a tensor product on the full subcategory $Cu$ of $2Cat$ is on page 16 of

* [[Ross Street]], _Gray's tensor product of 2-categories_, 22-page handwritten note, (1988) ([PDF at Macquarie](http://maths.mq.edu.au/~street/GrayTensor.pdf))

A general theory of lax tensor products, unifying Gray tensor products with the [[Crans-Gray tensor product]] is in

* [[Michael Batanin]], [[Denis-Charles Cisinski]], [[Mark Weber]], _Multitensor lifting and strictly unital higher category theory_ ([arXiv:1209.2776](http://arxiv.org/abs/1209.2776))

A proof that the Gray tensor product does form a monoidal structure, based only on its universal property, is in

* [[John Bourke]], [[Nick Gurski]], _The Gray tensor product via factorisation_, Appl Categor Struct **25** (2017) p603-624, doi:[10.1007/s10485-016-9467-6](https://doi.org/10.1007/s10485-016-9467-6), arXiv:[1508.07789](http://arxiv.org/abs/1508.07789)

[[!redirects Gray]]


Gray-category

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#Contents#
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{:toc}

## Idea

A *$Gray$-category* (or *Gray category*) is a certain type of [[semi-strict infinity-category|semi-strict]] [[3-category]], in which [[composition]] is strictly [[associativity|associative]] and [[unitality|unital]], but the [[interchange law]] holds only up to [[coherence|coherent]] [[isomorphism]].

## Definition

A **$Gray$-category** is a category [[enriched category|enriched]] over the [[symmetric monoidal category]] $Gray$, which is the category of [[strict 2-category|2-categories]] and [[strict 2-functors]] with the [[Gray tensor product]].

## Properties

### Coherence theorem

Gordon, Power, and Street proved that every [[tricategory]] (that is, weak 3-category) is equivalent to a $Gray$-category.  Not every tricategory is equivalent to a fully [[strict 3-category]]; any doubly-degenerate [[braided monoidal category]] which is not [[symmetric monoidal category|symmetric]] is an example.  So this is "the best one can do" in one sense, although there are other incomparable paths one can take, such as [[Simpson's conjecture|weakening units]] but keeping interchange strict.

The inclusion of $Gray$-categories into tricategories is not uniquely determined -- there is a left and right-hand version (from a remark in Example 9.3.9 of Leinster's book cited below).  However, the two possible ways are canonically equivalent as tricategories.

### Canonical model structure

Gray-categories support a [[canonical model structure]] ([Lack](#Lack))

## $Gray$-groupoids

A $Gray$-category that is a [[3-groupoid]] is a [[Gray-groupoid]].

## Examples

* The prototypical Gray-category is [[Gray]], which consists of [[strict 2-categories]], strict 2-functors, pseudonatural transformations, and modifications.

* A Gray-category with one object is called a Gray-monoid, and is a semi-strict version of a [[monoidal bicategory]].

* A doubly-degenerate Gray-category is the same as a category with two [[monoidal category|monoidal structures]] satisfying an [[exchange law]].  This is essentially the same as a [[braided monoidal category]]. ([Gurski & Cheng](#GurskiCheng))

* Since any tricategory is equivalent to a Gray-category, one can obtain examples of Gray-categories in this way.  For example, the tricategory [[Bicat]] of bicategories, pseudofunctors, pseudonatural transformations, and modifications is equivalent to some Gray-category.

  It is important to note that Bicat is *not* equivalent to Gray, due to the absence of pseudofunctors in the latter.  It is equivalent to the sub-Gray-category of Gray determined by the "flexible" or "cofibrant" 2-categories, however, since between such 2-categories any pseudofunctor is equivalent to a strict one.

* Since pseudofunctors between strict 2-categories compose strictly associatively, and between any 2-categories $A$ and $B$ there is a strict 2-category $Ps(A,B)$ of pseudofunctors, pseudonatural transformations, and modifications, one might hope that there is a Gray-category consisting of strict 2-categories, *pseudofunctors*, pseudonatural transformations, and modifications, despite the fact that the prototypical example $Gray$ contains only strict 2-functors.  However, this is false, because in a Gray-category the [[whiskering]] of 2-cells by a 1-cell is strictly functorial relative to composition of 2-cells along 1-cells, but this fails for whiskering of pseudonaturals by a pseudofunctor.

## Related pages

* [[intercategory]]

## References

* [[Robert Gordon]], [[John Power]], [[Ross Street]], _Coherence for tricategories_, Memoirs of the American Mathematical Society, **117**, 1995. ([doi:10.1090/memo/0558](http://dx.doi.org/10.1090/memo/0558))

* [[Nick Gurski]], _An algebraic theory of tricategories_, PhD thesis (2007) [[pdf](https://dokumen.tips/documents/an-algebraic-theory-of-tricategories.html), [[Gurski-AlgebraicTricategories.pdf:file]]]

* [[Tom Leinster]], _Higher operads, higher categories_, Cambridge University Press, 2004. ([arXiv:math/0305049](https://arxiv.org/abs/math/0305049),
[doi:10.1017/CBO9780511525896](https://doi.org/10.1017/CBO9780511525896))

*  {#Lack} [[Steve Lack]], _A Quillen model structure for Gray-categories_, Journal of K-Theory, **8**, 2011. ([arxiv:1001.2366](http://arxiv.org/abs/1001.2366), [doi:10.1017/is010008014jkt127](https://doi.org/10.1017/is010008014jkt127))

*  {#GurskiCheng} [[Nick Gurski]], [[Eugenia Cheng]], _The periodic table of n-categories II: degenerate tricategories_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, **52**, 2011. ([link](http://www.numdam.org/item/?id=CTGDC_2011__52_2_82_0), [arxiv:0706.2307](https://arxiv.org/abs/0706.2307))

* Peter Guthmann, _The tricategory of formal composites and its strictification_, [arXiv:1903.05777](https://arxiv.org/abs/1903.05777)

* {#DiVittorio2023} [[Nicola Di Vittorio]], _A Gray-categorical pasting theorem_, Theory and Applications of Categories **39** 5 (2023) 150-171 [[tac:39-05](http://www.tac.mta.ca/tac/volumes/39/5/39-05abs.html)]

[[!redirects Gray category]]
[[!redirects Gray-categories]]
[[!redirects Gray categories]]


Gray-groupoid

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A **Gray-Groupoid** is a [[semi-strict infinity-category|semistrict]] [[algebraic definition of higher categories|algebraic model]] for a [[3-groupoid]]:

it is a [[Gray-category]] -- a [[semi-strict infinity-category|semistrict]] [[3-category]] -- in which every [[k-morphism]] is invertible.

Gray-groupoids are related to [[2-crossed complex]]es and [[2-crossed module]]s as strict [[2-groupoid]]s are related to [[crossed complex]]es.

A Gray groupoid with a single object is the [[delooping]] of a [[Gray group]].

[[!redirects Gray groupoid]]
[[!redirects Gray-Groupoid]]
[[!redirects Gray-groupoids]]
[[!redirects Gray groupoids]]