gerbe (general idea)

for related entries see

* [[gerbe (as a stack)]]

* [[bundle gerbe]]

***

#Idea#

The concept _gerbe_ is a [[vertical categorification|categorification]]
of the concept of [[principal bundle]], together with a generalisation analogous to that from bundles to [[sheaf|sheave]]s.

Recall that a $G$-[[principal bundle]] (for $G$ a [[group]]) is a space $P$
equipped with a map $P \to X$ to a base space $X$,
such that each [[fiber]] of $P$ looks like $G$
in a nice way.

A _$G$-gerbe_ is similarly a "space", $P \to X$ such that
each fiber looks like $\mathbf{B}G$ in a nice way.
We can also replace $G$ with a sheaf of groups,
or even with a $2$-[[2-group|group]].

Here "space" may mean ordinary [[topological space]]. In that case
$\mathbf{B} G$ is the [[classifying space]] of the group $G$
and the above describes the construction by Stasheff and Wirth
of fibrations with fiber $B G$.

+--{: .query}
[[David Roberts]]: The list of axioms in Wirth-Stasheff about [[fibration theory|fibration theories]] is somewhat incomplete in my opinion (sorry Jim) - but only in a minor way. When it talks about the assignment of a category to each space, then goes on to talk about homotopies in that category, it seems to me we should be talking about $(\infty,1)$-categories. Even without such an extension, one needs to make sense of homotopies, and so should have the minimal structure required to talk about that - perhaps a category of fibrant objects?
=--

More generally, "space" may refer to
[[space and quantity|generalized spaces]], called
[[infinity-stack]]s: objects
in any [[(infinity,1)-topos]].

Recall from [[motivation for sheaves, cohomology and higher stacks]]
that this is just heavy terminology for a very simple idea.
The notion that a generalized space, also called an [[infinity-stack]], is an object in an $(\infty,1)$-topos
simplifies the situation conceptually by
separating

* conceptual structures (certain maps between spaces having certain fibers)

from their

* implementation (details of what is regarded as a generalized space and how).

In particular, while
gerbes are traditionally, originally by Giraud,
introduced as 1-[[stack]]s with extra
[[stuff, structure, property|properties]], one need not mention
any details of stacks for describing the concept and behaviour
of gerbes, all one needs is to remember that $\infty$-stacks are general notions of spaces, for which there is the familiar toolbox from [[homotopy theory]] of spaces, notably notions of [[homotopy limit|homotopy pullback]] and [[fibration sequence]]. This makes transparent the relation between

* gerbes
* Stasheff--Wirth fibrations
* nonabelian group extensions.

All of these are the same structure implemented in different contexts of generalized spaces.

For instance the last item here interprets [[group extension|extensions]] $G \to H \to K$ of a [[group]] $K$ by a group $G$ as a $G$-gerbe over $\mathbf{B}K$, namely as a fibration  $\mathbf{B}H \to \mathbf{B}K$ with fiber $\mathbf{B}G$.

When the group $G$
in question is [[abelian group|abelian]], the theory of gerbes is
very straightforwardly the generalization of that of [[principal bundle]]s,
because in this case the one-object [[groupoid]] $\mathbf{B}G$
obtained by shifting $G$ in categorical degree (see the discussion at [[group]])
still has itself a
group-structure: it is a [[2-group]]. Much of what makes
the discussion of nonabelian gerbes less than obvious is due to the
fact that when $G$ is not abelian then the way in which $\mathbf{B}G$
still relates to a group-like structure is slightly more involved and proceeds via the [[automorphism 2-group]] $AUT(G)$ of $G$.

Recall that an ordinary $G$-[[principal bundle]] $P \to X$
is a fibration of (generalized) spaces for which there is a
morphism $g : X \to \mathbf{B} G$ such that $P$ is the
[[homotopy pullback]] of the [[point]] along $g$:

$$
  \array{
    P &\to & {*}
    \\
    \downarrow && \downarrow
    \\
    X &\to& \mathbf{B}G
  }
  \,.
$$
So $P \to X \to \mathbf{B}G$ is a [[fibration sequence]] that extends for each
choice of point ${*} \to X$ of $X$ to the left to a fibration sequence
$G \to P \to X \to \mathbf{B}G$. This says that the fiber of $P \to X$ over each
point looks like the group $G$. General nonsense implies then
that the action of $G$ on itself induces an action of $G$ on all of $P$
and that this action is indeed [[principal bundle|principal]].

When $G$ is an abelian group, so that $\mathbf{B}G$ itself has a group
structure, the object $\mathbf{B}\mathbf{B}G$ exists and the above
statement has an immediate [[vertical categorification|categorification]]:

A $G$-gerbe for $G$ an abelian group is a fibration
$P \to X$ such that there is a morphism $g : X \to \mathbf{B}\mathbf{B}G$
such that $P$ is the [[homotopy pullback]] of the point along
this fibration.

$$
  \array{
    P &\to & {*}
    \\
    \downarrow && \downarrow
    \\
    X &\to& \mathbf{B}\mathbf{B}G
  }
  \,.
$$

In this case for each point ${*} \to X$ of $X$  this yields
a [[fibration sequence]]
$$
  \cdots \to \mathbf{B}G \to P \to X \to \mathbf{B}\mathbf{B} G
$$
which says that the fiber of $P \to X$ over each point of $X$ looks like
$\mathbf{B}G$.

As in the previous case of ordinary bundles, general nonsense implies that
$P \to X$ comes with a principal $\mathbf{B}G$-action. $P$ is therefore also
called a $\mathbf{B}G$-[[principal 2-bundle]] or a $\mathbf{B}G$-[[torsor]].
In its concrete incarnation as a [[stack]], $P$ is called a $G$-gerbe.

Moreover, since [[cohomology]] on $X$ with values in $\mathbf{B}\mathbf{B}G$ is nothing but the [[hom-set]]

$$
  H^2(X,G) = Ho(X, \mathbf{B}\mathbf{B} G)
$$

in the [[homotopy category of an (infinity,1)-category|homotopy category]] of our generalized spaces, it is a tautology that these $G$-gerbes are classified by $H^2(X,G)$.

Notice in particular that for $G = U(1)$ we have $H^2(X, U(1)) \simeq H^3(X, \mathbb{Z})$, for instance by a long exact sequence argument, so that $U(1)$-gerbes in the above sense are classified by third integral cohomology.
This classification statement was the main motivation for the study of the realization of the notion of gerbe that goes by the name [[bundle gerbe]].

In this fashion, for $G$ abelian, the entire concept of
$G$-$(n-1)$-gerbe is straightforward: it is the
$(n-1)$-[[infinity-stack|stack]] incarnation of $\mathbf{B}^n G$-[[principal infinity-bundle]]s, i.e. of fibrations $P \to X$ of (generalized) spaces
that arise as homotopy pullbacks of the form
$$
  \array{
    P &\to & {*}
    \\
    \downarrow && \downarrow
    \\
    X &\to& \mathbf{B}\mathbf{B}^n G
  }
  \,.
$$

Accordingly, such $(n-1)$-gerbes for $G$ abelian are classified in [[cohomology]] by $H^{n+2}(X,G)$.
(Another way to see why this is possible for $G$ an abelian group is that not only is $\mathbf{B}G$ a $2$-group, but $\mathbf{B}^n G$ is an $(n-1)$-[[n-group|group]], which is what we need in general for an $n$-gerbe.)

Moreover, for $A$ any pointed connected generalized space (any parameterized $\infty$-groupoid
with a single object),
we may say that $A$-principal $\infty$-bundles are fibrations $P \to X$
classified in this way by classifying morphisms $X \to A$
$$
  \array{
    P &\to & {*}
    \\
    \downarrow && \downarrow
    \\
    X &\to& A
  }
  \,.
$$
The fiber of such an $\infty$-bundle is the [[loop space object]] $\Omega A$.
The classifying morphism $X \to A$ is then called a cocycle in
[[nonabelian cohomology]].

In particular, for $H$ any [[2-group]] (not necessarily of the
form $H = \mathbf{B}G$ for $G$ an abelian group as above) an $H$-[[principal 2-bundle]] is a fibration in this sense classified
by a morphism $X \to \mathbf{B} H$. The typical fiber of
such a $2$-bundle looks like $H$.

Now, a $G$-gerbe for $G$ nonabelian is supposed to be a fibration
whose typical fiber is $\mathbf{B}G$. Since this is not a [[2-group]],
one has to say what one wants to mean by this.

This now is the crucial fact that translates between the
straightforward definition of $H$-[[principal 2-bundle]]s
as above and the notion of $G$-gerbe:

For every group $G$, there is the [[2-group]] $AUT(G)$, defined equivalently
as follows:

* $AUT(G)$ is the automorphism $2$-group of the [[groupoid]] $\mathbf{B}G$, i.e.
  $$
    AUT(G) = Aut_{Grpd}(\mathbf{B}G)
    \,;
  $$

* $AUT(G)$ is the [[2-group]] corresponding to the [[crossed module]]
  given by the sequence
  $(G \stackrel{Ad}{\to} Aut(G))$ of groups, with the canonical action
  of $Aut(G)$ on $G$.

From the second description it is manifest that one can
project out a copy of $\mathbf{B}G$ out of $AUT(G)$ (the shifted copy): there is a morphism $AUT(G) \to \mathbf{B}G$
obtained simply by identifying all objects of $AUT(G)$.
Indeed, $AUT(G)$ is a groupoid extension of $\mathbf{B}G$ by
the [[discrete category|discrete groupoid]] on $Aut(G)$ in that
$$
  Aut(G) \to AUT(G) \to \mathbf{B}G
$$
is a fibration sequence.

This means that any $AUT(G)$-[[principal 2-bundle]] with typical fiber
the groupoid $AUT(G)$ has underlying it a fibration with
typical fiber the one-object groupoid $\mathbf{B}G$. This
underlying object is the $G$-gerbe.

Notice in particular that when $G$ is abelian there is a canonical morphism

$$
  \mathbf{B}\mathbf{B} G \to  \mathbf{B} AUT(G)
$$

which however is not an equivalence when $G$ has nontrivial automorphisms. Therefore $G$-gerbes in the sense of nonabelian $G$-gerbes classified by $H(X,\mathbf{B}AUT(G))$ are even for $G$ abelian a bit more general than the things classified by just $H(X, \mathbf{B}^2 G)$, which are however also often called $G$-gerbes (in particular "[[bundle gerbe]]s").

This is the general nonsense underlying the concept of gerbe.

See also

* [[gerbe (as a stack)]]
* [[bundle gerbe]].

#References#

There is a lengthier description of gerbes, concentrating on the low-dimensional aspects, in the Menagerie notes that are available from [[Tim Porter|Tim Porter's]] home page.

Other material available online includes the following:

* J. Duskin, _An outline of non-abelian cohomology in a topos : (I) The theory of bouquets and gerbes_, Cahiers de Top. et Géom. Diff. Categ. 23 no. 2 (1982), p. 165-191 ([numdam](http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1982__23_2_165_0))

* I. Moerdijk, _Introduction to the language of stacks and gerbes_ ([arXiv](http://arxiv.org/abs/math/0212266))

* Larry Breen, _Notes on 1- and 2-gerbes_ ([arXiv](http://arxiv.org/abs/math/0611317))

See the references given there for more, in particular also the reference to the work by Jardine which relates to the discussion of gerbes in the context of [[infinity-stack]]s using the [[model structure on simplicial presheaves]].

The work by Stasheff and Wirth mentioned at the beginning is

* James Wirth & Jim Stashef, _Homotopy Transition Cocycles_ ([arXiv](http://arxiv.org/abs/math.AT/0609220) [blog](http://golem.ph.utexas.edu/category/2006/09/wirth_and_stasheff_on_homotopy.html))


gerbe (in differential geometry)

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
=--
=--

This is a sub-entry for [[gerbe]].

For related entries see

* [[bundle gerbe]]
* [[gerbe (general idea)]]
* [[gerbe (as a stack)]]

#Contents#
* table of contents
{:toc}

##Idea##

Gerbes give a nice way to group together bundle data on a smooth manifold, but gerbes also naturally define degree two cohomology. Thus the idea of using gerbes in differential geometry is to have a nice language that relates geometric concepts such as connections and curvature to cohomological classifications.

In addition, one can use the analogies above that are made precise with gerbes to define other new concepts such as 3-curvature and "local" fiberwise connections.

##Definitions##

*   Let $X$ be a smooth manifold. A *Dixmier-Douady sheaf of groupoids* over $X$ is a $\underline{\mathbb{C}}_X^*$-[[gerbe (as a stack)| gerbe]] on $X$ where $\underline{\mathbb{C}}_X^*$ is the sheaf of smooth $\mathbb{C}^*$-valued functions (not to be confused with the constant sheaf $\mathbb{C}_X^*$).

* We define $\mathbb{Z}(1)$ to be the term in the exponential sequence on $X$: $0\to \mathbb{Z}(1)\to \underline{\mathbb{C}}_X \to \underline{\mathbb{C}}_X^*\to 0$.

## Preliminaries##

Taking the associated sequence in cohomology to the exponential sequence gives us an isomorphism $H^2(X, \underline{\mathbb{C}}_X^*)\overset\sim\to H^3(X, \mathbb{Z}(1))$.

We have a canonical isomorphism between the group of equivalence classes of Dixmier-Douady sheaves of groupoids over $X$ (basically by the definition of $\mathcal{A}$-gerbe) and $H^2(X, \underline{\mathbb{C}}_X^*)\simeq H^3(X, \mathbb{Z}(1))$.

##Geometric Interpretation of $H^3(X, \mathbb{Z}(1))$##

Idea: Classes in $H^3(X, \mathbb{Z}(1))$ correspond to principal $G$-[[principal bundle| bundles]] over $X$ where $G$ is the projective linear group of a separable [[Hilbert space]], namely $C^\infty (\mathbb{T})$.

+--{: .query}

Matt: Actually, a slight issue has arisen. Most of the things I thought would go here actually already appear in other places even though they aren't grouped as coming from the same idea.

For instance, [[bundle gerbe]] contains the geometric interpretation of $H^3(X, \mathbb{Z}(1))$. Also, 3-curvature and fiber-wise connections occur at [[connection on a bundle gerbe]]. Although I think there is still a lot to say, I'm not convinced that "gerbe (in differential geometry)" is necessary anymore...
=--

##References

* [[Jean-Luc Brylinski]], _Loop Spaces, Characteristic Classes, and Geometric Quantization_, Birkhäuser (1993)  [[doi:10.1007/978-0-8176-4731-5](https://doi.org/10.1007/978-0-8176-4731-5)]

* I. Moerdijk, _Introduction to the language of stacks and gerbes_ ([arXiv](http://arxiv.org/abs/math/0212266))

* Larry Breen, _Notes on 1- and 2-gerbes_ ([arXiv](http://arxiv.org/abs/math/0611317))

Further references are given in the other entries on gerbes.


gerbe (in nonabelian cohomology)

This is a sub-entry of

* [[gerbe]] .

#Idea#

Recall that [[principal bundle]]s are classified by [[nonabelian cohomology]] in degree 1 with coefficients in a [[group]] $G$.

A central motivation for the introduction of the notion of [[gerbe]] was to find an analog of this statement for [[nonabelian cohomology]] in degree 2.

The basic theorem of Giraud's theory of [[gerbe]]s says that in the sense of [[gerbe (as a stack)]], gerbes $G$-gerbes are classified by [[nonabelian cohomology]] (usually realized as nonabelian [[?ech cohomology]], see there for more details) with coefficients in the [[strict 2-group]] $AUT(G) = Aut_{Grpd}(\mathbf{B}G)$.

#References#

For instance in

* Breen _Notes on 1- and 2-gerbes_ ([pdf](http://arxiv.org/PS_cache/math/pdf/0611/0611317v2.pdf))

this is section 5 _cocycles and coboundaries for gerbes_ .

The cocycle description itself is equation (5.1.10), the classification theorem is mentioned and referenced on the bottom of page 14.

Or

* Ieke Moerdijk _Introduction to the language of gerbes and stacks_ ([pdf](http://arxiv.org/PS_cache/math/pdf/0212/0212266v1.pdf))

discusses the Čech-cocycle description of gerbes from page 16 on, and the classification theorem appears as theorem 3.1 on p. 21.

The statement is originally due to Giraud's work

* Giraud, _Cohomologie non-abélienne

In

* Brylinski _Loop spaces, characteristic classes and geometric quantization_

the cocycle description of gerbes is extracted in chapter 5.2 _Sheaves of groupoids and gerbes_ and the classification theorem is theorem 5.2.8 on p. 200, 201.

The discussion of cocycles for gerbes is traditionally complicated by the fact that general sheaves of groups are used, instead of just a group, then there is the discussion of band, etc., all of which somewhat contributes to tending to hide a simple idea behind non-essential technical details.

Another thing that gerby tradition has is to express in linear formulas or rectangular diagrams what is intrinsically a nice geometric higher dimensional structure. The funny-looking nonabelian cocycle for a gerbe is really just a tetrahedron (the 3-simplex, since we are talking about a 2-cocycle) in $\mathbf{B} AUT(G)$.

This may be helpful, since it makes at once clear a lot of structure, such as for instance the nature of coboundaries. One can find these tetrahedra drawn in joint work [[John Baez]] and [[Urs Schreiber]], for instance the gerbe 2-cocycle tetrahedron is the title piece of

* [[John Baez]], _Namboodiri lecture slides: Higher Categories, Higher Gauge Theory, III ([pdf](http://math.ucr.edu/home/baez/namboodiri/nam3.pdf))

This recalls the theorem in question on slide 10.

Finally, gerbes, in as far as they are nonabelian, are really objects associated to [[principal 2-bundle]]s. The cocycle description of principal 2-bundles is more transparent, conceptually,  as it is the 2-bundle that is associated by abstract nonsense to the 2-cocycle, whereas the gerbe comes from that only after some fiddling (see [[gerbe (general idea)]]).

Accordingly, the nonabelian Čech cocycles in question here are discussed at length and in detail in the literature on [[principal 2-bundle]]s by Toby Bartels, Igor Baković and Christoph Wockel.

Nonabelian Čech cocycles as therefore are naturally expressed $n$-functors out of Čech $n$-groupoids.

This is described in some detail for instance in

* [[Urs Schreiber]], [[Konrad Waldorf]], _Connections on non-abelian gerbes and their cohomology_ ([arXiv](http://arxiv.org/abs/0808.1923))

Another discussion more in the style of the [[Lie groupoid]] community is in section 2 of

* Ginot, Stiénon, _$G$-gerbes, principal 2-group bundles and characteristic classes_ ([pdf](http://arxiv.org/PS_cache/arxiv/pdf/0801/0801.1238v2.pdf))

Giraud's gerbes as being the objects classified by [[nonabelian cohomology]] are recovered from general principles:

* [[Alexander Campbell]], _A higher categorical approach to Giraud's non-abelian cohomology_, PhD thesis, Macquarie University (2016) [[hdl:1959.14/1261186](http://hdl.handle.net/1959.14/1261186)]


Gerd Faltings

**Gerd Faltings** is a German mathematician, Fields medalist, currently faculty in Bonn (Max Planck Institute).

* [homepage](http://www.hcm.uni-bonn.de/old/faculty//Faltings) at University of Bonn

Some web pages on Faltings: [French wikipedia](http://fr.wikipedia.org/wiki/Gerd_Faltings), [English wikipedia](http://en.wikipedia.org/wiki/Gerd_Faltings), [history/biographies](http://www-history.mcs.st-andrews.ac.uk/Biographies/Faltings.html), [mathnet.or.kr](http://www.mathnet.or.kr/mathinfo/susang1986_2.html), [brittanica online](http://www.britannica.com/EBchecked/topic/201121/Gerd-Faltings)

## Selected writings

Proof of the [[Mordell conjecture]]:

* {#Faltings83} [[Gerd Faltings]], *Endlichkeitssätze für abelsche Varietäten über Zahlkörpern*, Inventiones Mathematicae **73** 3 (1983) 349-366  [[doi:10.1007/BF01388432](https://doi.org/10.1007/BF01388432)]

category:people

[[!redirects Faltings]]
[[!redirects G. Faltings]]


Gerd Rudolph

* [personal page](http://www.physik.uni-leipzig.de/~schmidtm/)

## Selected writings

On [[differential geometry]] and [[mathematical physics]]

specifically on [[differential geometry]] and [[Lie groups]]:

* [[Gerd Rudolph]], [[Matthias Schmidt]], *Differential Geometry and Mathematical Physics Part I. Manifolds, Lie Groups and Hamiltonian Systems*, Springer 2013 ([doi:10.1007/978-94-007-5345-7](https://link.springer.com/book/10.1007/978-94-007-5345-7))

and specifically on [[fiber bundles in physics]] ([[principal bundles]], [[principal connections]], [[classifying spaces]], [[Chern-Weil theory]], [[spin bundles]], [[Yang-Mills theory]]):

* [[Gerd Rudolph]], [[Matthias Schmidt]], *Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields*, Springer 2017 ([doi:10.1007/978-94-024-0959-8](https://link.springer.com/book/10.1007/978-94-024-0959-8))

category: people


Gereon Quick

[website](https://folk.ntnu.no/gereonq/)

## Selected writings

On [[algebraic topology]]:

* [[Gereon Quick]], _[Advanced algebraic topology](https://folk.ntnu.no/gereonq/Math231br.html)_, 2014

On the [[Hopf invariant]] and the [[Hopf invariant one problem]]:

* [[Gereon Quick]], _The Hopf invariant one problem via K-theory_, lecture notes, 2014 ([pdf](https://ncatlab.org/nlab/files/QuickHopfInvariant.pdf))

On the [[e-invariant]]:

* [[Gereon Quick]], _The $e$-invariant_, lecture notes, 2014 ([[QuickEInvariant.pdf:file]])

On the [[e-invariant]] and the [[J-homomorphism]]:

* [[Gereon Quick]], _The $e$-invariant and the $J$-homomorphism_, lecture notes, 2014 ([[QuickEInvariantAndJHomomorphism.pdf:file]])

Introducing [[Hodge-filtered differential cohomology]] and its specialization to Hodge-filtered [[complex cobordism theory]]:

* {#HopkinsQuick14} [[Michael J. Hopkins]], [[Gereon Quick]], §5 in: *Hodge filtered complex bordism*, Journal of Topology **8** 1 (2014) 147-183 [[arXiv:1212.2173](https://arxiv.org/abs/1212.2173), [doi:10.1112/jtopol/jtu021](https://doi.org/10.1112/jtopol/jtu021)]

Introduction and survey:

* [[Gereon Quick]], *Geometric Hodge filtered complex cobordism*, [talk at](Center+for+Quantum+and+Topological+Systems#QuickMar2023) *[[CQTS]]* (March 2023) [video:[YT](https://www.youtube.com/watch?v=pMu0gT5kIBo)]

Refinement of the [[Abel-Jacobi map]] to [[Hodge-filtered differential cohomology|Hodge-filtered differential]] [[MU]]-[[cobordism cohomology theory]]:

* [[Gereon Quick]], *An Abel-Jacobi invariant for cobordant cycles*, Documenta Mathematica **21** (2016) 1645–1668 [[arXiv:1503.08449](https://arxiv.org/abs/1503.08449)]

and a geometric cocycle model by actual [[cobordism]]-classes:

* [[Knut B. Haus]], [[Gereon Quick]], *Geometric Hodge filtered complex cobordism* [[arXiv:2210.13259](https://arxiv.org/abs/2210.13259)]

On [[Umkehr maps]] in [[Hodge-filtered differential cohomology|Hodge-filtered differential]] [[complex cobordism cohomology]]:

* [[Knut Bjarte Haus]], [[Gereon Quick]], *Geometric pushforward in Hodge filtered complex cobordism and secondary invariants* [[arXiv:2303.15899](https://arxiv.org/abs/2303.15899)]

## Related entries

* [[profinite homotopy theory]]

* [[intermediate Jacobian]], [[complex analytic infinity-groupoid]]

category: people

[[!redirects G. Quick]]


Gerhard Ecker

* [webpage](https://homepage.univie.ac.at/gerhard.ecker/)

## Selected writings

On [[chiral perturbation theory]]:

* [[Gerhard Ecker]], _The standard model at low energies_, Czech. J. Phys. 44:405-430, 1995 ([arXiv:hep-ph/9309268](https://arxiv.org/abs/hep-ph/9309268))

* [[Gerhard Ecker]], [[Jürg Gasser]], [[Antonio Pich]], E. DeRafael, _The role of resonances in chiral perturbation theory_, Nuclear Physics B
Volume 321, Issue 2, 24 July 1989, Pages 311-342 (doi:10.1016/0550-3213(89)90346-5)

* [[Gerhard Ecker]], _Status of chiral perturbation theory_, PoS Confinement 8:025, 2008 ([arXiv:0812.4196](https://arxiv.org/abs/0812.4196))

On [[kaon]] [[decays]]:

* V. Cirigliano, [[Gerhard Ecker]], [[Helmut Neufeld]], [[Antonio Pich]], J. Portoles, _Kaon Decays in the Standard Model_, Rev. Mod. Phys. 84 (2012) 399 ([arXiv:1107.6001](https://arxiv.org/abs/1107.6001))

and in relation to the [[chiral anomaly]]:

* [[Gerhard Ecker]], [[Helmut Neufeld]], [[Antonio Pich]], _Non-leptonic kaon decays and the chiral anomaly_, Nuclear Physics B Volume 413, Issues 1–2, 31 January 1994, Pages 321-352 (doi:10.1016/0550-3213(94)90623-8)

category: people


Gerhard Gentzen

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

## Related pages

* [[natural deduction]]

* [[sequent calculus]]

* [[Paul Hertz]]

* [[David Hilbert]]

* [[formalism]]

## References

Introducing [[proof theory]] via [[natural deduction]] and [[sequent calculus]]:

* {#Gentzen} [[Gerhard Gentzen]],  _Untersuchungen über das logische Schließen I_ _Mathematische Zeitschrift_ **39** 1 (1935) [[doi:10.1007/BF01201353](http://dx.doi.org/10.1007/BF01201353)]

which appeared in English translation as:

* [[Gerhard Gentzen]], *Investigations into Logical Deduction*, in M. E. Szabo (ed.), *The Collected Papers of Gerhard Gentzen*, Studies in Logic and the Foundations of Mathematics **55**, Springer (1969) 68-131 [[ISBN:978-0-444-53419-4](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/55), [pdf](https://logic-teaching.github.io/prop/texts/Gentzen%201969%20-%20Investigations%20into%20Logical%20Deduction.pdf)]

See also:

* Gerhard Gentzen, _Über die Existenz unabhängiger Axiomensysteme zu unendlichen Satzsystemen_, Math. Ann. **107** (1933) pp. 329-350. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002275872))

category: people

[[!redirects Gentzen]]
[[!redirects Gentzen's]]


Gerhard Hochschild

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

## Selected writings

On [[algebraic groups]] and [[Lie algebras]]:

* [[Gerhard P. Hochschild]], *Basic Theory of Algebraic Groups and Lie Algebras*, Graduate Texts in Mathematics **75**, Springer (1981) [[doi:10.1007/978-1-4613-8114-3_16](https://doi.org/10.1007/978-1-4613-8114-3_16)]

## Related entries

* [[Hochschild cohomology]]

* [[Hochschild-Serre spectral sequence]]

category: people

[[!redirects Gerhard P. Hochschild]]


Gerhard Mack

Gerhard Mack (1940-2023)

* [personal page](http://www.gerhard-mack.de/)

* [InSpire page](https://inspirehep.net/authors/999367)

* [Wikipedia entry](https://de.wikipedia.org/wiki/Gerhard_Mack) (de)

* obituary from Hamburg University [[pdf](https://www.physik.uni-hamburg.de/en/th2/documents/gerhard-mack-nachruf.pdf)]

## Selected writings

On [[non-perturbative quantum field theory]]:

* [[Gerard ’t Hooft]], [[Arthur Jaffe]], [[Gerhard Mack]], [[P. K. Mitter]], [[Raymond Stora]] (eds.) *Nonperturbative Quantum Field Theory*,  NATO Science Series B **185** (1988) [[doi:10.1007/978-1-4613-0729-7](https://doi.org/10.1007/978-1-4613-0729-7)]

category: people


Gerhard Osius

The German mathematician **Gerhard Osius** made important contributions to the topos-theoretic account of [[set theory]] and [[logic]] in the early 1970s.

## Related entries

* [[free topos]]

## References

* [[Peter Johnstone]], _Topos Theory_ , Academic Press New York 1977 (Dover reprint 2014). (sections 9.2-3)

* G. Osius, _The internal and external aspect of logic and set theory in elementary topoi_ , Cah. Top. Géom. Diff. Cat. **14** no.2 (1973) pp.199-201 . ([proceedings colloque Amiens 1973](http://archive.numdam.org/article/CTGDC_1973__14_2_153_0.pdf), 3.92 MB)

* G. Osius, _Categorical Set Theory: A Characterization of the Category of Sets_ , JPAA **4** (1974) pp.79-119.

* G. Osius, _The internal and external aspect of logic and set theory in elementary
topoi_, Cah. Top. Géom. Diff. Cat. **15** no.2 (1974) pp.157-180. ([pdf](http://archive.numdam.org/article/CTGDC_1974__15_2_157_0.pdf))

* G. Osius, _Logical and set-theoretical tools in elementary topoi_ , pp.297-346 in Lawvere, Maurer (eds.), _Model Theory and Topoi_ , LNM **445** Springer Heidelberg 1975.

* G. Osius, _A note on Kripke-Joyal semantics for the internal language of a topos_ , pp.349-354 in Lawvere, Maurer (eds.), _Model Theory and Topoi_ , LNM **445** Springer Heidelberg 1975.

category:people


Gerhart Lüders

* [Wikipedia entry](https://en.wikipedia.org/wiki/Gerhart_L%C3%BCders)

## Selected writings

On the formulation of [[wave function collapse]] in [[quantum mechanics]]/[[quantum information theory]]:

* [[Gerhart Lüders]]:

  *Über die Zustandsänderung durch den Meßprozeß*, Ann. Phys. **8** (1951) 322–328  [[doi:10.1002/andp.19504430510](https://doi.org/10.1002/andp.19504430510)]

  *Concerning the state-change due to the measurement process*, Ann. Phys. **15** 9 (2006) 663-670 [[pdf](http://myweb.rz.uni-augsburg.de/~eckern/adp/history/historic-papers/2006_518_663-670.pdf), [[Lueders-StateChange.pdf:file]]]

category: people

[[!redirects Gerhart Luders]]
[[!redirects Gerhart Lueders]]


germ

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#### Topos Theory
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A **germ** is an element of (a total space of) an [[etale space]] or equivalently an element in some [[stalk]] of a [[sheaf]] (all stalks together form the total space of the etale space of the sheaf). Exactly what this means depends on which sheaf one is considering.

This general description of what a germ of some sheaf at some point is can be extracted from [[stalk]], although that article is pretty abstract right now.

More generally, the notion of [[stalk]] makes sense in any [[topos]] that need not be a [[Grothendieck topos]] [[category of sheaves|of sheaves]] by way of the notion of [[point of a topos]]. Generally a germ is an element in the stalk of an object of a [[topos]] over some point of the topos.

## Examples

Originally, the term came from geometry, where sheaves of (continuous, smooth, holomorphic etc.) functions or more generally, sections of a (say fibre) [[bundle]] $\xi:E\to B$ were considered; germs in geometrical intuition are always germs *of* something (of a [[section]] of the sheaf in some neighborhood of a point, but also elements of a [[colimit]] at a point of local sections of an original [[presheaf]] which is not always a sheaf; though the germs a posteriori make a sheaf, they can be considered or defined in relation to an original presheaf which is not necessarily a sheaf). __Germs of sections__ are defined as the elements of the colimit sets of the appropriate sets of sections $\Gamma_U \xi$ where the colimit is over all open sets containing $x\in B$ with inverse inclusion (inverse because a presheaf of sections of a bundle is a *contravariant* functor).

In other words, germs of sections at a point $x\in B$ are equivalence classes $[U,s]$ of pairs of the form $(U,s)$, where $U\ni x$ is an open set in the base $B$ and $s:U\to E$ is a section defined over $U$; two sections $(U,s)$, $(U',s')$ in $colim_{U\ni x} \Gamma_U \xi$ are equivalent if there is a smaller $W\subset U\cap U'$ and $s|_W=s'|_W$. This construction of germs of sections of a bundle over $B$ (object of the [[slice category]] $Top/B$) leads to an [[adjoint pair of functors]] between $Top/B$ and the category of *pre*sheaves of sets over $B$ which restricts to the [[equivalence of categories|equivalence]] of the category of [[etale spaces]] $Et_B$ and the category of sheaves over $B$.

For example, take the sheaf of continuous (say, real-valued) functions on some space $X$.  Then every [[partial function]] $f$ defined on a [[neighbourhood]] of any given point $a$ in $X$ defines a germ at $a$.  Furthermore, the germ of $f$ equals the germ of $g$ if and only if $f = g$ on some neighbourhood $U$ of $a$; note that $U$ must be contained in the intersection of the domains of $f$ and $g$, but it may be smaller yet.

For a modern example of this kind, with only $1$ stalk, consider the nonarchimedean [[field]] of germs of holomorphic functions at the origin of the field $\mathbb{C}$ of [[complex number]]s, which plays an important role in mirror symmetry as the [[base field]] for the geometry of families of [[Calabi-Yau manifold]]s in the large volume limit (cf. Kontsevich, Soibelman [doi:10.1007/0-8176-4467-9](http://dx.doi.org/10.1007/0-8176-4467-9), [arxiv:math/0406564v1](http://arxiv.org/abs/math/0406564v1)).

## Related concepts

* [[synthetic differential topology]]

[[!include infinitesimal and local - table]]

## References

* R. Godement, Topologie algebrique et theorie de faisceaux, Paris 1958

* [[Masaki Kashiwara]], [[Pierre Schapira]], _[[Categories and Sheaves]]_, Grundlehren der Mathematischen Wissenschaften 332, Springer (2006)

* [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_, Springer 1992

[[!redirects germs]]


germ of a space

# Contents
* table of contents
{: toc}

## Idea

A _germ of a space_ is an [[equivalence class]] of [[pointed spaces]], where two such spaces regarded as equivalent if they are isomorphic on small [[open neighbourhoods]] of the base points.

## Definition

The category of germs of spaces has as objects pointed spaces $(X,x_0)$, where _space_ may depend on the context ([[topological space]], [[complex manifold]], ...).

Morphisms $(X,x_0) \to (Y,y_0)$ are equivalence classes of basepoint-preserving morphisms $U \to Y$ defined on arbitrary open neighbourhoods $U$ of $x_0$. Two such morphisms are considered equal if they agree on an open neighbourhood of $x_0$ contained in the intersection of their domains.

## Examples

Let $f$ be a [[germ|germ of a holomorphic function]] on a complex manifold $X$, defined on an open neighbourhood $U$ of some point $x_0$. Then the vanishing set of $f$, $V(f) \coloneqq \{ x \in U \,|\, f(x) = 0 \}$, is not well-defined as a space. However, it gives rise to a well-defined germ of a space.

The base space of a [[universal deformation]] of a complex manifold is usually considered as a germ of a space.

## As a localization

The category of germs of space is the [[localization]] of the category of pointed spaces at the class of those morphisms which restrict to isomorphisms on open neighbourhoods of the basepoints.

## Related concepts

 * [[germ]]

 * [[microbundle]]

 * [[synthetic differential topology]]

[[!include infinitesimal and local - table]]

## References

A short exposition is contained in the textbook

* {#Huybrechts04} [[Daniel Huybrechts]], _Complex geometry - an introduction_. Springer (2004). Universitext. 309 pages. ([pdf](http://www.math.uh.edu/~shanyuji/2012/Geometry/Huybrechts.pdf))

For germs in deformation theory, see for instance

* Marco Manetti, _Deformation theory via differential graded Lie algebras_ ([arXiv:0507284](http://arxiv.org/abs/math/0507284)).

[[!redirects germs of a space]]

[[!redirects germs of spaces]]


germ-determined C-infinity ring

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## Definition

+-- {: .num_defn #GermDeterminedCInfinityIdea}
###### Definition
**(germ-determined $C^\infty$-ideal)**

For $M$ a [[smooth manifold]], an [[ideal]] $I \,\subset\, C^\infty(M)$ in its [[C-infinity ring]] ([[smooth algebra]]) of [[smooth functions]] is called _germ determined_ if for any $f \in C^\infty(M)$ such that the [[germ]] of $f$ at any point $x$ in the [[zero locus]] of $I$ in $M$ coincides with the [[germ]] of some element of $I$ at $x$, then $f$ itself belongs to $I$.

=--

Equivalently, germ-determined [[ideals]] are precisely those [[ideals]]
that are closed under (possibly infinite) sums of [[locally finite]] families. One can also take sums with coefficients in a [[partition of unity]].

+-- {: .num_defn #GermDeterminedCInfinityRing}
###### Definition
**(germ-determined $C^\infty$-ring)**

A finitely generated [[C-infinity ring]] of the form $C^\infty(M)/I$ for some [[smooth manifold]] $M$ and [[ideal]] $I$ is __germ-determined__ if the [[ideal]] $I$ is a [[germ-determined ideal]] (Def. \ref{GermDeterminedCInfinityIdea}).

=--

## Related concepts

* [[Dubuc topos]]

## References

* [[Eduardo Dubuc]], _Sur les modèles de la géométrie différentielle synthétique_, [[Cahiers|Cahiers de Topologie et Geometrie Differentielle]] Vol. XX-3
(1979) ([numdam:CTGDC_1979__20_3_231_0](http://www.numdam.org/item?id=CTGDC_1979__20_3_231_0))

* [[Eduardo Dubuc]], _Schemas $C^\infty$_, "Prepublications de la Universite de Montreal" 80-81 edited by  G. Reyes (1980).

* [[Eduardo Dubuc]], _$C^\infty$ Schemes_, American Journal of Mathematics, John Hopkins University, Vol. 103-4 (1981) ([jstor:2374046](http://www.jstor.org/stable/2374046))

* [[Eduardo Dubuc]], _Open Covers and Infinitary Operations in C-inf-rings_, [[Cahiers|Cahiers de Topologie et Geometrie Differentielle]] Vol. XXII-3 (1981) ([numdam:CTGDC_1981__22_3_287_0](http://www.numdam.org/item?id=CTGDC_1981__22_3_287_0))

* [[Eduardo Dubuc]], _Integracion de campos vectoriales y Geometria Diferencial Sintetica_, Proceedings of the VII Sem. Nac. de Mat., FAMAF, Univ. Nac. de Cordoba (1984). [journal volume](http://inmabb.criba.edu.ar/revuma/revuma.php?p=toc/vol35), [pdf](http://inmabb.criba.edu.ar/revuma/pdf/v35/p151-162.pdf)

* [[Eduardo Dubuc]], _Germ representability and Local integration of vector fields in a well adapted model of SDG_, Aarhus Univ. Math. Inst. preprint series (1985/1986), published in Journal of Pure and Applied Algebra Vol. 64, (1990) (doi:10.1016/0022-4049(90)90152-8)

* [[Marta Bunge]], [[Eduardo Dubuc]], _Archimedian Local C-inf-rings and Models of SDG_, [[Cahiers|Cahiers de Topologie et Geometrie Differentielle]] Vol. XXVII-3 (1986). [NUMDAM](http://www.numdam.org/item?id=CTGDC_1986__27_3_3_0)

* [[Marta Bunge]], [[Eduardo Dubuc]], _Local concepts in SDG and germ representability_, in  D. Kueker et al. (ads), Mathematical Logic and Theoretical Computer Science, Lecture Notes in Pure and Applied Mathematics 106, Marcel Dekker (1987)

[[!redirects germ-determined C-infinity rings]]

[[!redirects germ-determined C-∞ ring]]
[[!redirects germ-determined C-∞ rings]]

[[!redirects germ determined C-infinity ring]]
[[!redirects germ determined C-infinity rings]]

[[!redirects germ determined C-∞ ring]]
[[!redirects germ determined C-∞ rings]]

[[!redirects germ-determined ideal]]
[[!redirects germ-determined ideals]]

[[!redirects germ determined ideal]]
[[!redirects germ determined ideals]]


Gershgorin circle theorem

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### Context
#### Linear algebra
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[[!include homotopy - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

While the [[eigenvalues]] of a [[diagonal matrix]] are, of course, [[equality|equal]] to its diagonal entries, *Gershgorin's circle theorem* ([Gershgorin 31](#Gershgorin31), Prop. \ref{GershgorinTheorem} below) provides upper bounds (the *Gershgorin radii*, Def. \ref{GershgorinDiscs} below) for general [[square matrices]] over the [[complex numbers]] on how far, in the [[complex plane]], the eigenvalues can be from the values of the diagonal entries.

## Statement

Let $n \in \mathbb{N}$ be a [[natural number]].

\begin{definition}\label{GershgorinDiscs}
**(Gershgorin discs)** \linebreak
 Let $A \in Mat_{n \times n}(\mathbb{C})$ be a [[square matrix]] with [[complex numbers|complex]] entries. For $j \in \{1, \cdots, n\}$ define

* the $j$th *Gershgorin radius* of $A$ to be the [[sum]] of [[absolute values]] of off-diagonal entries in the $j$th row:

  \[
    \label{GershoginRadiusFormula}
    r_j^A
    \;\coloneqq\;
    \underset{k \neq j}{\sum}
    \left\vert
      a_{j k}
    \right\vert
  \]

* the $j$th *Gershgorin disc* to be the [[closed disk]] in the [[complex plane]] centered at the $j$th diagonal entry with radius the above $j$th Gershgorin radius:

  \[
    \label{GershoginDiscFormula}
    D_j^A
    \;\coloneqq\;
    \Big\{
      z \in \mathbb{C}
      \,\Big\vert\,
      \left\vert
        z - a_{j j}
      \right\vert
      \,\leq\,
      r_j^A
    \Big\}
    \,.
  \]

\end{definition}

\begin{proposition}\label{GershgorinTheorem}
**(Gershgorin disc theorem)** \linebreak
 For $A \in Mat_{n \times n}(\mathbb{C})$ a [[square matrix]] with [[complex numbers|complex]] entries, each of its [[eigenvalues]] is contained in at least one Gershgorin disc (Def. \ref{GershgorinDiscs}):

$$
  \underset{
    c \in EV(A)
  }{\forall}
  \;\;
  \underset{
    1 \leq j \leq n
  }{\exists}
  \;\;
  \big(
    c \,\in\, D_j^A \,\subset\, \mathbb{C}
  \big)
  \,.
$$

\end{proposition}

(see, e.g., [Meckes 19, Thm. 7.1](#Meckes19))

\begin{prop}
  In the context of Prop. \ref{GershgorinTheorem}, if an [[eigenvalue]] has mulitplicity $k$, then it is contained in at least $k$ Gershgorin discs (Def. \ref{GershgorinDiscs}).
\end{prop}

This is due to [Marsli-Hall 13](#MarsliHall13).

\begin{prop}\label{StrengtheningForNonNegativeMatrices}
  If the matrix has non-negative values and an eigenvalue has multiplicity greater than 1, then it is contained within this disc of half the radius inside one of the Gershgorin discs (eq:GershoginDiscFormula).
\end{prop}

([Barany-Solymosi 16, Theorem 1](#BaranySolymosi16))

## Related theorems

* [[Cauchy interlace theorem]]

## References

The original article:

* {#Gershgorin31} [[Semyon Aronovich Gerschgorin]], *Über die Abgrenzung der Eigenwerte einer Matrix*, Bulletin de l'Académie des Sciences de l'URSS. Classe des sciences mathématiques et na, 1931, Issue 6, Pages 749–754 ([mathnet:eng/izv5235](http://mi.mathnet.ru/eng/izv5235))

Textbook account:

* [[Richard S. Varga]], *Gershgorin and His Circles*, Springer-Verlag, Berlin, 2004 ([doi:10.1007/978-3-642-17798-9](https://link.springer.com/book/10.1007/978-3-642-17798-9))

Lecture notes:

* {#Meckes19} [[Mark Meckes]], Section 7.1 of: *Lecture notes on matrix analysis*, 2019 ([pdf](https://case.edu/artsci/math/mwmeckes/matrix-analysis.pdf), [[MeckesMatrixAnalysis.pdf:file]])

Exposition:

* Zack Cramer, *The Gershgorin circle theorem*  2017 ([[CramerGershgorinCircleTheorem.pdf:file]])

See also:

* Wikipedia, *[Gershgorin circle theorem](https://en.wikipedia.org/wiki/Gershgorin_circle_theorem)*

Strengthening for eigenvalues with higher multiplicity:

* {#MarsliHall13} [[Rachid Marsli]], [[Frank J. Hall]], *Geometric Multiplicities and Geršgorin Discs*, The American Mathematical Monthly Vol. 120, No. 5 (May 2013), pp. 452-455 ([jstor:10.4169/amer.math.monthly.120.05.452](https://www.jstor.org/stable/10.4169/amer.math.monthly.120.05.452))

Further strengthening for matrices with, in addition, non-negative entries:

* {#BaranySolymosi16} Imre Bárány, József Solymosi, *Gershgorin disks for multiple eigenvalues of non-negative matrices*,  In: M. Loebl, J. Nešetřil, R. Thomas (eds.) *A Journey Through Discrete Mathematics*, Springer, Cham. 2017 ([arXiv:1609.07439](https://arxiv.org/abs/1609.07439), [doi:10.1007/978-3-319-44479-6_6](https://doi.org/10.1007/978-3-319-44479-6_6))

[[!redirects Gershgorin circle theorems]]

[[!redirects Gershgorin disc]]
[[!redirects Gershgorin discs]]

[[!redirects Gershgorin radius]]
[[!redirects Gershgorin radii]]


Gerson Goldhaber

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

* [Obituary](https://senate.universityofcalifornia.edu/_files/inmemoriam/html/gersongoldhaber.html)

## Selected writings

On observation of [[supernovae]] interpreted as indicating a [[positive number|positive]] [[cosmological constant]] ([[dark energy]], the origin of the modern [[standard model of cosmology]]):

* {#Goldhaber2001a} [[Gerson Goldhaber]], *Evidence for Dark Energy from a Study of Type Ia Supernovae*, in: *Dark Matter in Astro- and Particle Physics*, Springer (2001) [[doi:10.1007/978-3-642-56643-1_13](https://doi.org/10.1007/978-3-642-56643-1_13)]

* {#Goldhaber2001b} [[Gerson Goldhaber]], *Large Evidence for Dark Energy from a Study of Type Ia Supernovae*, in: *Recent Developments in Particle Physics and Cosmology*, Springer (2001) [[doi:10.1007/978-94-010-0676-7_12](https://doi.org/10.1007/978-94-010-0676-7_12)]

* [[Gerson Goldhaber]], *The Acceleration of the Expansion of the Universe: A Brief Early History of the Supernova Cosmology Project (SCP)*, AIP Conference Proceedings **1166** 53 (2009) [[arXiv:0907.3526](https://arxiv.org/abs/0907.3526), [doi:10.1063/1.3232196](https://doi.org/10.1063/1.3232196)]

category: people


Gersten resolution

#Contents#
* table of contents
{:toc}

## Idea

The __Gersten resolution__  or _Gersten complex_ is certain [[exact sequence]] of  [[abelian sheaves]] on a [[Zariski site]], used for the computation of higher [[algebraic K-theory]] groups. One way of deriving it is from a [[spectral sequence]] related to the study of motivic sheaves.

The exactness of the Gersten complex for [[affine scheme|affine]] [[regular scheme|regular]], equicharacteristic excellent local rings with infinite residue field (or the like) is also known as the _[[Gersten conjecture]]_ (see e.g. [Kerz 08, p. 4](#Kerz08)).

## References

Original references include

* [[Daniel Quillen]], _Algebraic K-theory, I: Higher K-theories_ (Battelle Memorial Inst., Seattle 1972), Proc. Conf., Lecture Notes in Math. __341__, 1973, pp. 85–147.

* [[Spencer Bloch]], _Algebraic cycles and higher K-theory_, Adv. in Math. __61__ (1986), 267–304.

Review includes

* Ishai Dan-Cohen, _Bloch's formula and the Gersten resolution_ ([pdf](https://www.uni-due.de/~hm0146/28JanEssen13.pdf))

* {#Kerz08} [[Moritz Kerz]], _Milnor K-theory of local rings_ Dissertation, Universität Regensburg (2008) ([web](http://epub.uni-regensburg.de/10749/))

* Thomas Geisser, _Motivic cohomology, K-theory and topological cyclic homology_, chapter II.1 in Handbook of K-theory, [pdf](http://www.math.uiuc.edu/K-theory/handbook/1-193-234.pdf)

See also

* S. O. Gorchinskii, _Adelic resolution for sheaves of K-groups_, Russ. Math. Surv. __62__, p. 184-186 (2007) [pdf](http://iopscience.iop.org/0036-0279/62/1/L07/pdf/RMS_62_1_L07.pdf)

[[!redirects Gersten resolutions]]

[[!redirects Gersten complex]]
[[!redirects Gersten complexes]]

[[!redirects Gersten conjecture]]


Gerstenhaber algebra

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{:toc}

## Idea

A _Gerstenhaber algebra_ is a [[Poisson n-algebra|Poisson 2-algebra]], a [[Poisson algebra]] in graded vector spaces with Poisson bracket of degree -1.

## Definition

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

A **Gerstenhaber algebra** is a [[chain complex]] $A$ equipped with

* a symmetric product $\cdot : A \otimes A \to A$;

* a skew-symmetric bracket $[-,-] : A \otimes A \to A[1]$;

* such that [[associativity]] of $\cdot$ and the [[Jacobi identity]] for $[-,-]$ holds and such that $[a,-]$ is a [[derivation]] of $\cdot$.

=--

## Properties

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

The [[homology]] of the  [[operad]] for Gerstenhaber algebras in [[chain complexes]] is the operad for Gerstenhaber algebras.

Accordingly the [[homology]] of an [[E2-algebra]] is a Gerstenhaber algebra.

=--

This is due to [Cohen (1976)](#Cohen).

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

A Gerstenhaber algebra equipped in addition with a certain morphism $\Delta : A \to A$ is a **[[BV-algebra]]**. This is the homology of an algebra over the [[framed little 2-disk operad]].

=--

## Related concepts

* [[Poisson n-algebra]]

* [[BV-algebra]]

## References

* {#Cohen} Cohen (1976)

* [[Murray Gerstenhaber]], _The cohomology structure of an associative ring_, Ann. of Math., 78 (1963), 267-288 [MR28:5102](http://www.ams.org/mathscinet-getitem?mr=28:5102)

* [[Ping Xu]], _Gerstenhaber algebras and BV-algebras in Poisson geometry_, Commun. Math. Phys. 200, No.3, 545-560 (1999).

[[!redirects Gerstenhaber algebras]]
[[!redirects Poisson 2-algebra]]
[[!redirects Poisson 2-algebras]]

[[!redirects Gerstenhaber bracket]]


Gerstenhaber-Schack cohomology

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#Contents#
* table of contents
{:toc}

## Idea

[[Murray Gerstenhaber|Gerstenhaber]] and Schack introduced a [[cohomology]] related to [[deformation theory|deformation]] theory of [[bialgebras]]. Teillefer has proven that this cohomology computes in fact the [[Ext]]-groups in certain [[abelian category]] of "tetramodules" over the bialgebra. A somewhat more systematic writeup of the proof is in the appendix [Shoikhet 09](#Shoikhet).

Please distinguish the Gerstenhaber-Schack bialgebra cohomology from [[Shahn Majid]]'s [[bialgebra cohomology]] which is (in full generality) nonabelian and cohomologies associated to other categories of "modules" in bialgebra theory (Hopf modules, Yetter-Drinfeld modules etc.). In the $n$Lab we choose to use the term [[bialgebra cocycle]] unadorned for the Majid's cocycles and always use Gerstenhaber-Schack cocycle/cohomology for the latter, as it is often referred nowdays.

## References

* [[Murray Gerstenhaber]], Samuel D. Schack, _Bialgebra cohomology, deformations, and quantum groups_,  Proc. Nat. Acad. Sci. U.S.A. __87__  (1990),  no. 1, 478–481,  [MR90j:16062](http://www.ams.org/mathscinet-getitem?mr=1031952), [article](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC53287)

* Rachel Taillefer, _Injective Hopf bimodules, cohomologies of infinite dimensional Hopf algebras and graded-commutativity of the Yoneda product_, J. Algebra __276__ (2004), no. 1, 259–279, [MR2005f:16067](http://www.ams.org/mathscinet-getitem?mr=2054397), [doi](http://dx.doi.org/10.1016/j.jalgebra.2004.01.024)

* Boris Shoikhet, _Hopf algebras, tetramodules, and $n$-fold monoidal categories_, [arxiv/0907.3335](http://arxiv.org/abs/0907.3335)
 {#Shoikhet}

* Brian Parshall, Jian Pan Wang, _On bialgebra cohomology. Algebra, groups and geometry_,  Bull. Soc. Math. Belg. Sér. A  __42__  (1990),  no. 3, 607–642.

[[!redirects Gerstenhaber-Schack bialgebra cohomology]]


GeV

#Contents#
* table of contents
{:toc}

### Idea

A [[physical unit]] of [[energy]]: giga-[[electronvolt]].

A [[proton]] has [[rest mass]] about a GeV:

$$
  m_{proton} \simeq 0.938 GeV
  \,.
$$

The [[Higgs particle]] has [[rest mass]] about $125 GeV$.

The [[Planck mass]] is about $10^{19} GeV$.

## Related entries

* [[MeV]]

* [[TeV]]

## References

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


gevolution

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

## Idea

_gevolution_ is a [[computer program]] for [[relativistic field theory|relativistic]] simulation of [[gravity]] ([[general relativity]]). It has been used for [[computer experiment]] in [[cosmology]], specifically for discussion of [[cosmic structure formation]] and [[inhomogeneous cosmology]].

## References

General:

* [github.com/gevolution-code/gevolution-1.1.git](https://github.com/gevolution-code/gevolution-1.1.git)

* Julian Adamek, David Daverio, Ruth Durrer, Martin Kunz, _gevolution: a cosmological N-body code based on General Relativity_, JCAP 1607 (2016) no.07, 053 ([arxiv:1604.06065](https://arxiv.org/abs/1604.06065)

In the context of [[inhomogeneous cosmology]]:

* {#Adamek18} Julian Adamek, _The Numerical Challenge -- Backreaction in Relativistic N-body Simulations of Cosmic Structure Formation_, talk at [CosmoBack 2018](https://cosmoback.sciencesconf.org/) ([[AdamekCosmoBack18.pdf:file]])


Gheorghe Nenciu

* [personal page](http://www.imar.ro/~ghnenciu/)

## Selected writings

On the [[quantum adiabatic theorem]]:

* [[Gheorghe Nenciu]], *On the adiabatic theorem of quantum mechanics*,  J. Phys. A: Math. Gen. **13** L15 ([doi:10.1088/0305-4470/13/2/002](https://iopscience.iop.org/article/10.1088/0305-4470/13/2/002))

category: people

[[!redirects G. Nenciu]]


ghost field

+-- {: .rightHandSide}
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### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[gauge theory]] the configuration space/[[phase space]] is not in general a [[smooth space]], but a [[smooth groupoid]]: the [[gauge transformations]] between [[gauge fields]] are the [[morphisms]] of this groupoid.

The [[infinity-Lie theory|infinitesimal approximation]] to this smooth groupoid is a [[Lie algebroid]]. The [[dg-algebra]] of functions on this is called the _[[BRST complex]]_ of the gauge theory. It contains in degree-0 the (duals to) the [[gauge fields]] and in degree-1 the [[cotangent space|cotangents]] to the [[gauge transformations]]. These degree-1 elements that appear here alongside the physical fields in degree 0 are called **ghost fields** in the physics literature.

If there are [[higher gauge transformations]] "[[gauge-of-gauge transformations]]" then the [[BRST complex]] has generators in higher degree, too, the cotangents to these higher gauge transformations. These are then called **ghost-of-ghost fields**.

For more details and further pointers see at _[[BRST complex]]_ and in particular at _[[BV-BRST formalism]]_.

## Related concepts

* [[field (physics)]]

  * **ghost**,

  * [[antifield]], [[antighost]]

[[!redirects ghost fields]]
[[!redirects ghost]]
[[!redirects ghosts]]

[[!redirects ghost-of-ghosts]]
[[!redirects ghosts-of-ghosts]]

[[!redirects ghost-of-ghost field]]
[[!redirects ghost-of-ghost fields]]

[[!redirects Fadeev-Popov ghost]]
[[!redirects Fadeev-Popov ghosts]]

[[!redirects ghost number]]
[[!redirects ghost numbers]]

[[!redirects ghost field number]]
[[!redirects ghost field numbers]]


Giacomo Tendas

Currently holds a postdoctoral research position at the University of Manchester.

* [personal webpage](https://gtendas.github.io/)

* [google scholar](https://scholar.google.com/citations?user=VEFYpBMAAAAJ)

## Selected writings

On [[enriched category|enriched]] [[accessible categories]]:

* [[Giacomo Tendas]], _Topics in the theory of enriched accessible categories_, PhD thesis, Macquarie University, 2022. ([doi:10.25949/21638081.v1](https://doi.org/10.25949/21638081.v1))

On [[enriched category|enriched]] [[regular categories]]:

* [[Stephen Lack]] and [[Giacomo Tendas]], _Enriched regular theories_, Journal of Pure and Applied Algebra, **224**, 2020. ([arXiv:1907.02301](https://arxiv.org/abs/1907.02301), [doi:10.1016/j.jpaa.2019.106268](https://doi.org/10.1016/j.jpaa.2019.106268))

On [[dg-categories]] and [[Cauchy completeness]]:

* Branko Nikolić, [[Ross Street]] and [[Giacomo Tendas]], _Cauchy completeness for DG-categories_, Theory and Applications of Categories, **37**, 2021. ([tac:37-28](http://www.tac.mta.ca/tac/volumes/37/28/37-28abs.html))

On [[filtered colimits]] in [[enriched category theory]]:

* [[Stephen Lack]] and [[Giacomo Tendas]], _Flat vs. filtered colimits in the enriched context_, Advances in Mathematics, **404**, 2022. ([arXiv:2107.08612](https://arxiv.org/abs/2107.08612), [doi:10.1016/j.aim.2022.108381](https://doi.org/10.1016/j.aim.2022.108381))

On [[accessible functors]]:

* [[Giacomo Tendas]], _On Continuity of Accessible Functors_ Applied Categorical Structures, **30**, 2022. ([arXiv:2110.14192](https://arxiv.org/abs/2110.14192), [doi:10.1007/s10485-022-09677-x](https://doi.org/10.1007/s10485-022-09677-x))

On ([[enriched category|enriched]]) [[accessible categories]]:

* [[Stephen Lack]], [[Giacomo Tendas]], *Virtual concepts in the theory of accessible categories*, Journal of Pure and Applied Algebra **227** 2 (2023) 107196 [[arXiv:10.1016/j.jpaa.2022.107196](https://doi.org/10.1016/j.jpaa.2022.107196), [arXiv:2205.11056](https://arxiv.org/abs/2205.11056)]

category: people


Gian Giudice

[[!redirects Gian Francesco Giudice]]

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

## Selected writings

On [[Higgs field]] [[vacuum stability]]:

* {#EEGHR09} [[John Ellis]], J.R. Espinosa, [[Gian Giudice]], A. Hoecker, A. Riotto, _The Probable Fate of the Standard Model_, Phys.Lett.B679:369-375,2009 ([arXiv:0906.0954](https://arxiv.org/abs/0906.0954))

* {#DVEEGI12} Giuseppe Degrassi, Stefano Di Vita, Joan Elias-Miró, José R. Espinosa, [[Gian Giudice]], Gino Isidori, _Higgs mass and vacuum stability in the Standard Model at NNLO_, J. High Energ. Phys. (2012) 2012: 98 ([arrXiv:1205.6497](https://arxiv.org/abs/1205.6497), [doi:10.1007/JHEP(2012)098](https://doi.org/10.1007/JHEP08%282012%29098))

* {#BDGGSSS13} Dario Buttazzo, Giuseppe Degrassi, Pier Paolo Giardino, [[Gian Giudice]], Filippo Sala, Alberto Salvio, [[Alessandro Strumia]], section 7 of _Investigating the near-criticality of the Higgs boson_, JHEP12(2013)089 ([arXiv:1307.3536](https://arxiv.org/abs/1307.3536))

* {#EGR07} J.R. Espinosa, [[Gian Giudice]], A. Riotto, _Cosmological implications of the Higgs mass measurement_, JCAP 0805:002, 2008 ([arXiv:0710.2484](https://arxiv.org/abs/0710.2484))

* {#EGMRSST15} Jose R. Espinosa, [[Gian Giudice]], Enrico Morgante, Antonio Riotto, Leonardo Senatore, [[Alessandro Strumia]], Nikolaos Tetradis, _The cosmological Higgstory of the vacuum instability_ ([arXiv:1505.04825](https://arxiv.org/abs/1505.04825))

On [[split supersymmetry]] in view of [[Higgs field]] [[vacuum stability]]:

* [[Gian Giudice]], [[Alessandro Strumia]], _Probing High-Scale and Split Supersymmetry with Higgs Mass Measurements_, Nuclear Physics B Volume 858, Issue 1, 1 May 2012, Pages 63-83 ([arXiv:1108.6077](https://arxiv.org/abs/1108.6077))

On [[naturalness]]:

* [[Gian Francesco Giudice]], _The Dawn of the Post-Naturalness Era_ ([arXiv:1710.07663](https://arxiv.org/abs/1710.07663))

## related $n$Lab entries

* [[naturalness]]

* [[split supersymmetry]]

* [[Higgs field]]

category: people


Gian Luca Cattani

* [personal page](http://www.daimi.au.dk/~luca/)

## Selected writings

On [[profunctors]] with application to [[computer science]]:

* {#Cattani99} [[Gian Luca Cattani]], *Presheaf Models for Concurrency*, PhD thesis from BRICS, University of Aarhus 1999 ([pdf](http://www.brics.dk/DS/99/1/BRICS-DS-99-1.pdf), [[Cattani_PresheafModels.pdf:file]])

* [[Gian Luca Cattani]], G. Winskel, _Profunctors, Open Maps, and Bisimulation_ , BRICS Report **04-22** (2004). ([pdf](http://www.brics.dk/RS/04/22/BRICS-RS-04-22.pdf))

category:people


Gian-Carlo Rota

> not to be confused with [[Franco Rota]]

***

__Gian-Carlo Rota__ was a great [[combinatorialist]]; in addition to numerous concrete results he deserves much credit on systematizing many methods.

* [wikipedia](http://en.wikipedia.org/wiki/Gian-Carlo_Rota)

## Selected writings

A nice reading about appreciation of systematic/foundational work in mathematics is his preface ([pdf](http://bergeron.math.uqam.ca/wp-content/uploads/2013/11/introductionrota.pdf), [ps.gz](http://bergeron.math.uqam.ca/Species/foreword.ps.gz), [bookpage](http://bergeron.math.uqam.ca/Species/especes.html)) to [[Combinatorial species and tree-like structures]] (Enc. Math. Appl., Camb. Univ. Press 1997) by F. Bergeron, G. Labelle and P. Leroux.

* Steven M. Roman, Gian-Carlo Rota, _Umbral calculus_,  Advances in Math. __27__ (1978), no. 2, 95--188, [pdf until p. 120](http://www.romanpress.com/MathArticles/Umbral1.pdf), [doi](http://dx.doi.org/10.1016/0001-8708(78)90087-7), [MR0485417](http://www.ams.org/mathscinet-getitem?mr=0485417)
* Daniel A. Klain, G-C. Rota, _Introduction to Geometric Probability_, Cambridge University Press 1997, xiv+178 pp. [gBooks](http://books.google.com/books?id=Q1ytkNM6BtAC); Book survey in Bull. AMS [pdf](http://www.ams.org/bull/2000-37-02/.../S0273-0979-99-00860-5.pdf) (from the survey:"these are notes based on Gian-Carlo Rota's 1986 "Lezioni Lincee" lectures about geometric valuations, integral geometry and geometric probability, and they are superbly polished")
* Mathematical essays in honor of Gian-Carlo Rota (Cambridge, MA, 1996), Progr. Math. __161__, Birkhäuser 1998,

## Quotes

From  _Indiscrete thoughts_:

> Mathematicians can be subdivided into two types: problem solvers and theorizers.
Most mathematicians are a mixture of the two although it is easy to find extreme examples of both types.

> To the problem solver, the supreme achievement in mathematics is the solution to a problem that has been given up as hopeless. It matters little that the solution may be clumsy; all that counts is that it should be the first and that the proof be correct. Once the problem solver finds the solution, he will permanently lose interest in it, and will listen to new simplified proofs with an air of condescension suffused with boredom. The problem solver is a conservative at heart. For him, mathematics consists of a sequence of challenges to be met, an obstacle course of problems. The mathematical concepts required to state mathematical problems are tacitly assumed to be eternal and immutable. Mathematical exposition is regarded as an inferior undertaking. New theories are viewed with deep suspicion, as intruders who must prove their worth by posing challenging problems before they can gain attention. The problem solver resents generalizations, especially those that may succeed in trivializing the solution of one of his problems. The problem solver is the role model for budding young mathematicians. When we describe to the public the conquests of mathematics, our shining heroes are the problem solvers.

> To the theorizer, the supreme achievement of mathematics is a theory that sheds sudden light on some incomprehensible phenomenon. Success in mathematics does not lie in solving problems but in their trivialization. The moment of glory comes with the discovery of a new theory that does not solve any of the old problems but renders them irrelevant. The theorizer is a revolutionary at heart. Mathematical concepts received from the past are regarded as imperfect instances of more general ones yet to be discovered. Mathematical exposition is considered a more difficult undertaking than mathematical research. To the theorizer, the only mathematics that will survive are the definitions. Great definitions are what mathematicians contribute to the world. Theorems are tolerated as a necessary evil since they play a supporting role — or rather, as the theorizer will reluctantly admit, an essential role — in the understanding of definitions. Theorizers often have trouble being recognized by the community of mathematicians. Their consolation is the certainty, which may or may not be borne out by history, that their theories will survive long after the problems of the day have been forgotten.

> If I were a space engineer looking for a mathematician to help me send a rocket into space, I would choose a problem solver. But if I were looking for a mathematician to give a good education to my child, I would unhesitatingly prefer a theorizer.

## Related entries

* [[umbral calculus]]

* [[Möbius inversion]]

* [[Rota-Baxter algebra]]

category: people

[[!redirects Gian Carlo Rota]]


Gian-Carlo Wick

* [Wikipedia entry](http://en.wikipedia.org/wiki/Gian-Carlo_Wick)

## related $n$Lab entries

* [[Wick algebra]]

* [[Wick's lemma]]

* [[Wick rotation]]

category: people


Gianfranco Mascari

Gianfranco Mascari and Jean-François Mascari seem to be the same Tunisian-Italian category theorist.  He (or they both) apply [[category theory]] to social sciences (among more common applications).

* [webpage](http://cnr-it.academia.edu/JeanFrançoisMascari)

category: people

[[!redirects Gianfranco Mascari]]
[[!redirects Jean-François Mascari]]
[[!redirects Jean-Francois Mascari]]
[[!redirects J-F Mascari]]
[[!redirects J.-F. Mascari]]
[[!redirects J F Mascari]]
[[!redirects J. F. Mascari]]
[[!redirects JF Mascari]]
[[!redirects J.F. Mascari]]


Gianguido Dall'Agata

* [webpage](http://www.pd.infn.it/~dallagat/)

## Selected writings

Formulation of ([[Lagrangian densities]] for) [[type II supergravity]] with "democratic"/"[[pregeometric RR-field|pregeometric]]" [[RR-fields]] subject to [[self-dual higher gauge theory|self-duality]]:

* [[Gianguido Dall'Agata]], [[Kurt Lechner]], [[Mario Tonin]], *$D=10$, $N=IIB$ Supergravity: Lorentz-invariant actions and duality*, JHEP 9807:017 (1998) [[arXiv:hep-th/9806140](https://arxiv.org/abs/hep-th/9806140), [doi:10.1088/1126-6708/1998/07/017](https://doi.org/10.1088/1126-6708/1998/07/017)]

## Related entries

* [[5d supergravity]]

* [[M2-brane]]

* [[intersecting D-brane model]]

category: people


Gianluca Giorgolo

* [webpage](http://www.kcl.ac.uk/artshums/depts/philosophy/people/staff/associates/research/giorgolog.aspx)

category: people


Gianluca Inverso

* [InSpire page](https://inspirehep.net/authors/1265587)

## Selected writings

On maximal [[gauged supergravity|gauged]] [[D=2 supergravity]] via [[KK-reduction]] of [[D=11 supergravity]]:

* [[Guillaume Bossard]], [[Franz Ciceri]], [[Gianluca Inverso]], [[Axel Kleinschmidt]], *Maximal $D=2$ supergravities from higher dimensions* [[arXiv:2309.07232](https://arxiv.org/abs/2309.07232)]

* [[Guillaume Bossard]], [[Franz Ciceri]], [[Gianluca Inverso]], [[Axel Kleinschmidt]], *Consistent truncation of eleven-dimensional supergravity on $S^8 \times S^1$* [[arXiv:2309.07233](https://arxiv.org/abs/2309.07233)]

category: people


Gianluca Panati

* [GoogleScholar page](https://scholar.google.com/citations?user=RPyCU1sAAAAJ&hl=en)

## Selected writings

On situations (in [[solid state physics]]) where the [[valence bundle]] ("Bloch bundle") is [[trivial bundle|trivializable]]:

* {#Panati06} [[Gianluca Panati]], *Triviality of Bloch and Bloch-Dirac bundles*, Ann. Henri Poincaré **8** (2007) 995-1011. ([arXiv:math-ph/0601034](https://arxiv.org/abs/math-ph/0601034), [doi:10.1007/s00023-007-0326-8](https://doi.org/10.1007/s00023-007-0326-8))

category: people


Gianluca Zoccarato

* [webpage](https://live-sas-physics.pantheon.sas.upenn.edu/people/post-docs/gianluca-zoccarato)

## Selected writings

On the [[cosmological constant]] in [[F-theory on Spin(7)-manifolds]] ([[Witten's Dark Fantasy]]):

* [[Jonathan Heckman]], Craig Lawrie, Ling Lin, [[Gianluca Zoccarato]], _F-theory and Dark Energy_, Fortschritte der Physik  ([arXiv:1811.01959](https://arxiv.org/abs/1811.01959), [doi:10.1002/prop.201900057]( https://doi.org/10.1002/prop.201900057))

* [[Jonathan Heckman]], Craig Lawrie, Ling Lin, Jeremy Sakstein, [[Gianluca Zoccarato]], _Pixelated Dark Energy_ ([arXiv:1901.10489](https://arxiv.org/abs/1901.10489))

Relating [[M-theory on Spin(7)-manifolds]] with [[F-theory on Spin(7)-manifolds]] via [[Higgs bundles]]:

* {#CHRTZ20} [[Mirjam Cvetic]], [[Jonathan Heckman]], Thomas B. Rochais, Ethan Torres, [[Gianluca Zoccarato]], _Geometric Unification of Higgs Bundle Vacua_ ([arXiv:2003.13682](https://arxiv.org/abs/2003.13682))

category: people


Gianluigi Catelani

member of the [Quantum Research Center](https://www.tii.ae/quantum) (QRC) at the [Technology Innovation Institute](https://www.tii.ae/) (TII), Abu Dhabi

* [webpage](https://www.tii.ae/team/prof-dr-gianluigi-catelani)

## Selected writings

On [[quasiparticle]]-excitations among [superconductive qbits](superconductivity#SuperconductingQBitsReferences):

* [[Leonid I. Glazman]], [[Gianluigi Catelani]], *Bogoliubov Quasiparticles in Superconducting Qubits*, SciPost Phys. Lect. Notes **31** (2021) [[arXiv:2003.04366](https://arxiv.org/abs/2003.04366), [doi:10.21468/SciPostPhysLectNotes.31](https://doi.org/10.21468/SciPostPhysLectNotes.31)]

category: people


giant graviton

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

#Contents#
* table of contents
{:toc}

## Idea

In [[string theory]] by a _giant graviton_ one refers to a [[D-brane]]- or [[M-brane]]-configuration which [[wrapped branes|wraps]] a contractible [[cycle]] stabilized not by topology (as for usual [[wrapped branes]]) or by [[flux]] (as for [[polarized branes]]), but by [[angular momentum]], usually along the [[equator]] of the [[n-sphere]]-factor in a [[Freund-Rubin compactification]].

## Examples

### In the BMN matrix model

The [[supersymmetry|supersymmetric]] [[classical field theory|classical]] solutions of the [[BMN matrix model]]  are configurations of  [[fuzzy 2-spheres]], corresponding to [[M2-brane]]-[[giant gravitons]] ([BMN 02 (5.4)](BMN+matrix+model#BerensteinMaldacenaNastase02),
[DSJVR 02](BMN+matrix+model#DSJVR02)).

## Related concepts

* [[polarized brane]]

* [[wrapped brane]]

* [[M2-M5 brane bound state]]

* [[dual graviton]]

## References

### General

The concept was introduced in

* [[John McGreevy]], [[Leonard Susskind]], [[Nicolaos Toumbas]], _Invasion of the Giant Gravitons from Anti-de Sitter Space_, JHEP 0006:008, 2000 ([arXiv:hep-th/0003075](https://arxiv.org/abs/hep-th/0003075))

for branes in [[anti de Sitter spacetime]] [[product space|times]] an [[n-sphere]] and generalized to other [[near horizon geometries]] in

* S. R. Das, S. P.. Trivedi, S. Vaidya, _Magnetic Moments of Branes and Giant Gravitons_, JHEP 0010 (2000) 037 ([arXiv:hep-th/0008203](https://arxiv.org/abs/hep-th/0008203))

* Andrei Mikhailov, _Giant Gravitons from Holomorphic Surfaces_, JHEP 0011 (2000) 027 ([arXiv:hep-th/0010206](https://arxiv.org/abs/hep-th/0010206))

* J. M. Camino, A.V. Ramallo, _Giant Gravitons with NSNS B field_, JHEP 0109:012,2001 ([arXiv:hep-th/0107142](https://arxiv.org/abs/hep-th/0107142))

Review:

* {#Myers03} [[Robert Myers]], Section 5 of: _Nonabelian Phenomena on D-branes_, Class. Quant. Grav. 20 (2003) S347-S372 ([arXiv:hep-th/0303072](https://arxiv.org/abs/hep-th/0303072))

Further discussion:

* Akikazu Hashimoto, Shinji Hirano, N. Itzhaki, _Large branes in AdS and their field theory dual_, JHEP 0008:051, 2000 ([arXiv:hep-th/0008016](https://arxiv.org/abs/hep-th/0008016))

* Sumit R. Das, Antal Jevicki, Samir D. Mathur, _Vibration modes of giant gravitons_, Phys.Rev. D63 (2001) 024013 ([arXiv:hep-th/0009019](https://arxiv.org/abs/hep-th/0009019))

* Sumit R. Das, Antal Jevicki, [[Samir Mathur]], _Giant Gravitons, BPS bounds and Noncommutativity_, Phys.Rev. D63 (2001) 044001 ([arXiv:hep-th/0008088](https://arxiv.org/abs/hep-th/0008088))

* Julian Lee, _Tunneling between the giant gravitons in $AdS_5 \times S^5$_,  Phys.Rev. D64 (2001) 046012 ([arXiv:hep-th/0010191](https://arxiv.org/abs/hep-th/0010191))

* Jin Young Kim, Y.S. Myung, _Vibration modes of giant gravitons in the background of dilatonic D-branes_, Phys.Lett. B509 (2001) 157-162 ([arXiv:hep-th/0103001](https://arxiv.org/abs/hep-th/0103001))

* Vijay Balasubramanian, [[Micha Berkooz]], Asad Naqvi, [[Matthew Strassler]], _Giant Gravitons in Conformal Field Theory_, JHEP 0204 (2002) 034 ([arXiv:hep-th/0107119](https://arxiv.org/abs/hep-th/0107119))

* [[Yolanda Lozano]], Jeff Murugan, [[Andrea Prinsloo]], _A giant graviton genealogy_, JHEP 08 (2013) 109 ([arXiv:1305.6932](https://arxiv.org/abs/1305.6932))

Specifically [[M2-M5 brane bound state]] giant gravitons are discussed in:

* J. M. Camino, A. V. Ramallo, _M-Theory Giant Gravitons with C field_,  Phys. Lett. B525:337-346, 2002 ([arXiv:hep-th/0110096](https://arxiv.org/abs/hep-th/0110096))

* Shinji Hirano, Yuki Sato, _Giant graviton interactions and M2-branes ending on multiple M5-branes_, JHEP 05 (2018) 065 ([arXiv:1803.04172](https://arxiv.org/abs/1803.04172))

### Relation to polarized branes

On the relation of [[polarized branes]] to giant gravitons:

* [[Iosif Bena]], Douglas Smith, _Towards the solution to the giant graviton puzzle_, Phys. Rev. D71 (2005) 025005 ([arXiv:hep-th/0401173](https://arxiv.org/abs/hep-th/0401173))

### Under AdS/CFT

Discussion of [[SYM]]-[[AdS-CFT duality|duals]] of giant gravitons and indexing of [[BPS states]] by [[Young diagrams]]:

* Steve Corley, Antal Jevicki, [[Sanjaye Ramgoolam]], *Exact Correlators of Giant Gravitons from dual $N=4$ SYM*, Adv. Theor. Math. Phys. 5 (2002) 809-839 ([arXiv:hep-th/0111222](https://arxiv.org/abs/hep-th/0111222))

* Garreth Kemp, [[Sanjaye Ramgoolam]], *BPS states, conserved charges and centres of symmetric group algebras*, JHEP 01 (2020) 146 ([arXiv:1911.11649](https://arxiv.org/abs/1911.11649))

[[!redirects giant gravitons]]


Gieseker stability

Given an ample line bundle $H$ over a projective algebraic manifold, Gieseker $H$-stability conditions are the stability conditions for torsion-free coherent sheaves over projective algebraic manifolds generalizing say Mumford stability conditions for vector bundles over curves and used for similar reasons (building moduli spaces). These are in the setup of abelian categories rather than their derived categories where notions like [[Bridgeland stability]] are more appropriate.

Introduced in

* D. Gieseker, _On the moduli of vector bundles on an algebraic surface_, Ann. Math. __106__ (1977) 45-60 [doi](https://doi.org/10.2307/1971157)

and in

* S. Kobayashi, _Differential geometry of complex vector bundles_, Iwanami Shoten Publishers and Princeton Univ. Press (1987)

The special case for algebraic surfaces has been introduced in

* F. Takemoto, _Stable vector bundles on algebraic surfaces_,
Nagoya Math. J. 47 (1972) 29-48

category: algebraic geometry


Gigel Militaru

* [personal page](https://gigelmilitaru.wordpress.com/)

* [GoogleScholar page](https://scholar.google.com/citations?user=aYspv6MAAAAJ&hl=en)

## Selected writings

On [[Frobenius functors]] (and introducing that terminology):

* {#CaenepeelMilitaruZhu97}  [[Stefaan Caenepeel]], [[Gigel Militaru]], [[Shenglin Zhu]], *Doi-Hopf modules, Yetter-Drinfel’d modules and Frobenius type properties*, Trans. Amer. Math. Soc. **349** (1997) 4311-4342 [[1997-349-11/S0002-9947-97-02004-7](https://www.ams.org/journals/tran/1997-349-11/S0002-9947-97-02004-7), [pdf](https://www.ams.org/journals/tran/1997-349-11/S0002-9947-97-02004-7/S0002-9947-97-02004-7.pdf)]

* {#CGN98} F. Castaño Iglesias, [[José Gómez-Torrecillas]], C. Nastasescu, _Frobenius functors: applications_, Comm. Alg. __27__ 10  (1998) 4879-4900 [[doi:10.1080/00927879908826735](https://doi.org/10.1080/00927879908826735)]

* {#CaenepeelDeGrooMilitaru02} [[Stefaan Caenepeel]], E. De Groot, [[Gigel Militaru]], *Frobenius Functors of the second kind*, Comm. Algebra **30** (2002) 5359-5391 [[arXiv:math/0106109](https://arxiv.org/abs/math/0106109), [doi:10.1081/AGB-120015657](https://doi.org/10.1081/AGB-120015657)]

* [[Stefaan Caenepeel]], [[Gigel Militaru]], [[Shenglin Zhu]], _Frobenius and separable functors for generalized module categories and nonlinear equations_, Springer Lec. Notes in Math. __1787__ (2002) [[gBooks](http://books.google.hr/books?id=6erTEpKIUaYC&lpg=PA89&ots=-TGKqXRstB&dq=Frobenius%20functors&hl=en&pg=PA89#v=onepage&q&f=false)]

On associative [[bialgebroids]] and in particalar about those arising as smash products of a [[Hopf algebra]] $H$ with a braided commutative Yetter-Drinfeld $H$-module algebra $A$ ([[scalar extension bialgebroid|scalar extension bialgebroids]]):

* [[Tomasz Brzeziński]], [[Gigel Militaru]], *Bialgebroids, $\times_A$-bialgebras and duality*,  J. Algebra __251__ (2002) 279-294 [[math.QA/0012164](https://arxiv.org/abs/math.QA/0012164), [doi:10.1006/jabr.2001.9101](https://doi.org/10.1006/jabr.2001.9101)]

The structure of finite-dimensional Hopf algebras can be recovered from the fundamental operator (idea of [[multiplicative unitary]]) satisfying the pentagon relation, in the framework of finite-dimensional [[Heisenberg double]]s, as shown in

* [[Gigel Militaru]], _Heisenberg double, pentagon equation, structure and classification of finite-dimensional Hopf algebras_, J. London Math. Soc. (2) 69 (2004) 44--64 [doi](https://doi.org/10.1112/S0024610703004897)

category: people


Gijs Heuts

* [webpage](http://www.math.ku.dk/english/staff/?pure=en/persons/541041)

## Selected writings

On the [[(∞,1)-Grothendieck construction]] via [[straightening and unstraightening]]:

* [[Fabian Hebestreit]], [[Gijs Heuts]], [[Jaco Ruit]], _A short proof of the straightening theorem_ ([arXiv:2111.00069](https://arxiv.org/abs/2111.00069)).

## Related entries

* [[Cartesian fibration of dendroidal sets]]

* [[model structure for dendroidal Cartesian fibrations]]

category: people


Gijs Tuynman

* [webpage](http://math.univ-lille1.fr/~gmt/)

category: people


Gijs van den Oord

* [webpage](https://www.esciencecenter.nl/profile/dr.-gijs-van-den-oord)

## related $n$Lab entries

* [[M2-brane]]

category: people


Gil Cavalcanti

* [webpage](http://www.staff.science.uu.nl/~caval101/homepage/Home.html)

category: people


Gilbert de Beauregard Robinson

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=15883)

## Selected writings

Introducing the [[hook length formula]] in the [[representation theory of the symmetric group]]:

* {#FrameRobinsonThrall54} [[James Sutherland Frame]], [[Gilbert de Beauregard Robinson]], [[Robert McDowell Thrall]], *The Hook Graphs of the Symmetric Group*, Canadian Journal of Mathematics, Volume 6, 1954, pp. 316 - 324 ([doi:10.4153/CJM-1954-030-1](https://doi.org/10.4153/CJM-1954-030-1))

category: people


Gilbert Ryle

Gilbert Ryle (1900-1976) was an Oxford philosopher, one of the early exponents of _ordinary language_ philosophy (see [Tanney](#Tanney)). He is particulary noted for his work on [[Category mistake|_category mistakes_]] (see [Magidor](#Magidor)).

> ...not only is it the case that category-propositions (namely assertions that terms belong to certain categories or types), are always philosopher’s propositions, but, I believe, the converse is also true. So we are in the dark about the nature of philosophical problems and methods if we are in the dark about types and categories. ([Ryle 1938, p. 189](#Ryle38))

### Selected writings

* {#Ryle38} _Categories_, Proceedings of the Aristotelian Society, New Series, Vol. 38 (1937 - 1938), pp. 189-206.

* _The Concept of Mind_, London: Hutchinson, 1949.

### References

* {#Magidor} Ofra Magidor, _SEP: Gilbert Ryle_, ([SEP](https://plato.stanford.edu/entries/category-mistakes/))

* {#Tanney} Julia Tanney, _SEP: Gilbert Ryle_, ([SEP](https://plato.stanford.edu/entries/ryle/))


Gilles Barthe

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

## Selected writings

On [[type equivalence]]:

* [[Gilles Barthe]], [[Olivier Pons]], *Type Isomorphisms and Proof Reuse in Dependent Type Theory*, in *Foundations of Software Science and Computation Structures. FoSSaCS 2001*, Lecture Notes in Computer Science **2030**, Springer (2001) [[doi:10.1007/3-540-45315-6_4](https://doi.org/10.1007/3-540-45315-6_4)]

On [[setoids]] in [[type theory]]:

* [[Gilles Barthe]], Venanzio Capretta, [[Olivier Pons]], *Setoids in type theory*, Journal of Functional Programming **13** 2 (2003) 261-293 [[doi:10.1017/S0956796802004501](https://doi.org/10.1017/S0956796802004501)]

Introducing the [[quantum programming language]] *[[CoqQ]]*:

* [[Li Zhou]], [[Gilles Barthe]], Pierre-Yves Strub, Junyi Liu, [[Mingsheng Ying]], *CoqQ: Foundational Verification of Quantum Programs* [[arXiv:2207.11350](https://arxiv.org/abs/2207.11350)]

  > (in [[Coq]])

Understanding the [[linear logic]]-sector of [[bunched logic]] as a kind of [[quantum logic]]:

* [[Li Zhou]], [[Gilles Barthe]], [[Justin Hsu]], [[Mingsheng Ying]], [[Nengkun Yu]], *A Quantum Interpretation of Bunched Logic for Quantum Separation Logic*, LICS '21: Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science  75 (2021) 1–14 [[arXiv:2102.00329](https://arxiv.org/abs/2102.00329), [doi:10.1109/LICS52264.2021.9470673](https://doi.org/10.1109/LICS52264.2021.9470673)]

category: people


Gilles Brassard

* [personal page](http://www.iro.umontreal.ca/~brassard/web/en/)

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

## Selected writings

Introducing the notion of [[quantum key distribution]]:

* [[Charles H. Bennett]], [[Gilles Brassard]], *Quantum cryptography: Public key distribution and coin tossing*, International Conference on Computers, Systems & Signal Processing, Bangalore, India (1984) 175-179 [scan: [arXiv:2003.06557](https://arxiv.org/abs/2003.06557)]

  reprinted in: 	Theoretical Computer Science **560** 1 (2014) 7-11 [[doi:10.1016/j.tcs.2014.05.025](https://doi.org/10.1016/j.tcs.2014.05.025)]

Introducing the notion of [[quantum teleportation]]:

* {#BBCJPW93} [[Charles H. Bennett]], [[Gilles Brassard]], [[Claude Crépeau]], [[Richard Jozsa]], [[Asher Peres]], [[William K. Wootters]]: *Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels*, Phys. Rev. Lett. **70** 1895 (1993) [[doi:10.1103/PhysRevLett.70.1895](https://doi.org/10.1103/PhysRevLett.70.1895)]

category: people


Gilles Cristol

* [webpage](http://www.math.jussieu.fr/~christol/)

category: people


Gino Isidori

* [group page](https://www.physik.uzh.ch/en/groups/isidori.html)

## Selected writings

On [[electroweak symmetry breaking]]:

* [[Gino Isidori]], _Effective theories of electroweak symmetry breaking_, PoS CD09:073, 2009 ([arXiv:0911.3219](https://arxiv.org/abs/0911.3219))

On the [[flavour problem]]:

* [[Gino Isidori]], _Kaon decays and the flavour problem_, 	Annales Henri Poincare 4 (2003) S97-S109 ([arXiv:hep-ph/0301159](https://arxiv.org/abs/hep-ph/0301159))

  (emphasis on [[kaon]] [[scattering|decays]])

* [[Gino Isidori]], _The flavor problem_, 2016 ([pdf](https://indico.cern.ch/event/504502/contributions/2147710/attachments/1287372/1915547/SWHEPPS_Talk.pdf), [[IsidoriFlavor16.pdf:file]])

On [[flavour physics]] and [[flavour anomalies]]:

* {#BordoneIsidoriPattori16} [[Marzia Bordone]], [[Gino Isidori]], Andrea Pattori, *On the Standard Model predictions for $R_K$ and $R_{K^\ast}$*, Eur. Phys. J. C 76, 440 (2016). ([arXiv:1605.07633](https://arxiv.org/abs/1605.07633), [doi:10.1140/epjc/s10052-016-4274-7](https://doi.org/10.1140/epjc/s10052-016-4274-7))

  > (argument that [[QED]]-[[radiative corrections]] are negligible for the relevant branching ratios)

* [[Gino Isidori]], *Hints of New Physics from flavor-changing processes*, talk at SPS Annual Meeting, Lausanne, Aug. 2018 ([pdf](https://indico.cern.ch/event/716246/contributions/3097683/attachments/1709141/2754825/SPS_Lausanne.pdf))

* {#Isidori19} [[Gino Isidori]], _Flavor physics & CP violation_, talk at [Symposium on the European Strategy for Particle Physics](https://indico.cern.ch/event/808335/timetable/#20190513.detailed) 2019 ([pdf](https://indico.cern.ch/event/808335/contributions/3365125/attachments/1842875/3022853/Granada_GI_13May.pdf))

On [[leptoquarks]] and [[flavour anomalies]]:

* Michael J. Baker, [[Javier Fuentes-Martin]], [[Gino Isidori]], [[Matthias König]], _High-pT Signatures in Vector-Leptoquark Models_ ([arXiv:1901.10480](https://arxiv.org/abs/1901.10480))

* [[Javier Fuentes-Martin]], [[Gino Isidori]], Matthias König, Nudzeim Selimovic, _Vector Leptoquarks Beyond Tree Level_ ([arxiv:1910.13474](https://arxiv.org/abs/1910.13474))

* Jason Aebischer, [[Gino Isidori]], Marko Pesut, Ben A. Stefanek, Felix Wilsch, *Confronting the vector leptoquark hypothesis with new low- and high-energy data* [[arXiv:2210.13422](https://arxiv.org/abs/2210.13422)]

Further on the [[flavour anomalies]]

* Barbara Gallavotti, starring [[Gino Isidori]], _How Particle Physics Works: Episode II - The New Analysis_, June 2019 ([youtube](https://www.youtube.com/watch?v=krWPeFLLBFY))

* Claudia Cornella, Darius A. Faroughy, [[Javier Fuentes-Martin]], [[Gino Isidori]], [[Matthias Neubert]], _Reading the footprints of the B-meson flavor anomalies_ ([arXiv:2103.16558](https://arxiv.org/abs/2103.16558))

> With the recent result $[$of [[LHCb]]$]$, for the first time a single observable affected by negligible theoretical uncertainties exhibits a deviation from the SM exceeding the $3 \sigma$ level.  Equally striking is the overall coherence of the picture that emerges, especially in $b \to s \ell^+ \ell^-$ −transitions.  As we shall show in this paper, combining all the $b \to s \ell^+ \ell^-$−observables in a very conservative way, the significance of the New Physics (NP) hypothesis formulated in 2014–2015 of a purely left-handed LFU-violating contact interaction has now reached a [[statistical significance|significance]] of $4.6 \sigma$.

> $[\ldots]$  A more structural way of addressing the flavor structure of the model is the idea of implementing [[GUT|Pati-Salam unification]]

* {#LIOS21} Davide Lancierini, [[Gino Isidori]], Patrick Owen, Nicola Serra, _On the significance of new physics in $b \to s \ell^+ \ell^-$ decays_ ([arXiv:2104.05631](https://arxiv.org/abs/2104.05631))

* {#IsidoriEtAl21} [[Gino Isidori]], Davide Lancierini, Abhijit Mathad, Patrick Owen, Nicola Serra, Rafael Silva Coutinho, *A general effective field theory description of $b \to s l^+ l^-$ lepton universality ratios* ([arXiv:2110.09882](https://arxiv.org/abs/2110.09882))

On [[flavour anomalies]] via [[leptoquarks]] in [[Pati-Salam GUT]]$\,$[[intersecting D-brane model|on branes]]:

* [[Javier Fuentes-Martin]], [[Gino Isidori]], Julie Pagès, Ben A. Stefanek,
*Flavor Non-universal Pati-Salam Unification and Neutrino Masses*, Physics Letters B Volume 820, 10 September 2021, 136484 ([arXiv:2012.10492](https://arxiv.org/abs/2012.10492), [talk recording](https://www.youtube.com/watch?v=ko4R8S8XYV4&t=2s))

Textbook on the [[flavour anomalies]]:

* Marina Artuso, [[Gino Isidori]], Sheldon Stone, *New Physics in $b$ Decays*, World Scientific (2020) $[$[doi:10.1142/9789811251306](https://www.worldscientific.com/doi/abs/10.1142/9789811251306_0001)$]$

category: people


Ginzburg dg-algebra

Given a quiver with potential, [[Victor Ginzburg]] associated a dg-algebra, now called Ginzburg (dg-)algebra, which supplies an example of a 3-[[Calabi-Yau algebra]] (also defined by Ginzburg). They may be viewed as a derived thickening...

* Victor Ginzburg, _Calabi-Yau algebras_, [arXiv:math.AG/0612139](https://arxiv.org/abs/math/0612139)

* [[Bernhard Keller]] (with an appendix by [[Michel van den Bergh]]), _Deformed Calabi-Yau completions_, J. Reine Angew. Math. 654 (2011) 125–180 [doi](https://doi.org/10.1515/crelle.2011.031)

* Stephen Hermes, _Minimal model of Ginzburg algebras_, Journal of Algebra __459__ (2016) 389-436 [doi](https://doi.org/10.1016/j.jalgebra.2016.03.041); _On the Homology of the Ginzburg Algebra_, slides (2013) [pdf](https://web.northeastern.edu/martsinkovsky/p/MADL/2013/slides-hermes.pdf)

* M. Christ, _Ginzburg algebras of triangulated surfaces and perverse schobers_, Forum of Mathematics, Sigma (2022), Vol. 10:e8 1–72 [doi](https://doi.org/10.1017/fms.2022.1)


Giordano Bruno

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

## Related entries

* [[monad terminology]]

* [[multiverse]]

category: people


Giorgi Dvali

Giorgi (Gia) Dvali

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

## related $n$Lab entries

* [[axion]]

category: people

[[!redirects Gia Dvali]]


Giovanni Caviglia

* [webpage](https://www.math.ru.nl/~gcaviglia/)

## Selected writings

On a [[model structure on operads]] for [[colored operads]], generalizing a corresponding [[model structure on enriched categories]]:

* {#Caviglia14} [[Giovanni Caviglia]], _A Model Structure for Enriched Coloured Operads_ ([arXiv:1401.6983](http://arxiv.org/abs/1401.6983))

On [[PROPs]] and the [[biequivalence]] between [[Feynman categories]] and [[colored operads]]

* [[Giovanni Caviglia]], _The Dwyer-Kan model structure for enriched coloured PROPs_ ([arXiv:1510.01289](https://arxiv.org/abs/1510.01289))

See also

* [[Giovanni Caviglia]], [[Geoffroy Horel]], _Rigidification of higher categorical structures_, Algebraic & Geometric Topology, Vol. 16 (2016), pp. 3533–3562 ([doi:10.2140/agt.2016.16.3533](https://msp.org/agt/2016/16-6/p13.xhtml), [euclid:agt/1510841268](https://projecteuclid.org/euclid.agt/1510841268) [arXiv:1511.01119](https://arxiv.org/abs/1511.01119))

category: people


Giovanni Collini

* [Institute page](http://home.uni-leipzig.de/tet/?page_id=40)

## related $n$Lab entries

* [[deformation quantization]]

* [[locally covariant perturbative quantum field theory]]

category: people


Giovanni Curi

* [personal page](http://www.giovannicuri.com/)

## Selected writings

On [[uniform locales]]:

* [[Giovanni Curi]], *On the collection of points of a formal space*, Annals of Pure and Applied Logic* **137** 1-3 (2006)  126-146 $[$[doi:10.1016/j.apal.2005.05.019](https://doi.org/10.1016/j.apal.2005.05.019)$]$

category: people


Giovanni Faonte

* [webpage](http://giovannifaonte.weebly.com/about.html)

* [GoogleScholar page](https://scholar.google.com/citations?user=Xd26B4UAAAAJ&hl=en)

## Selected writings

Introducing [[dg-nerves]] for [[A-infinity categories|$A_\infty$-categories]] and proving that it takes [[pretriangulated category|pretriangulated]] [[dg-categories]] to [[stable (infinity,1)-categories|stable $\infty$-categories]]:

* {#Faonte13} [[Giovanni Faonte]], *Simplicial nerve of an $\mathcal{A}_\infty$-category*, Theory and Applications of Categories **32** 2 (2017) 31-52 [[arXiv:1312.2127](http://arxiv.org/abs/1312.2127), [tac:32-02](http://www.tac.mta.ca/tac/volumes/32/2/32-02abs.html)]

category: people

[[!redirects Faonte]]


Giovanni Felder

* [webpage](http://www.math.ethz.ch/u/felder)

## Some publications

* G. Felder, L. Stevens, [[A. Varchenko]], _Elliptic Selberg integrals and conformal blocks_, Math. Res. Lett. __10__ (2003), no. 5-6, 671–684, [MR2004i:32024](http://www.ams.org/mathscinet-getitem?mr=2024724), [pdf](http://www.mrlonline.org/mrl/2003-010-005/2003-010-005-010.pdf)

## Related entries

* [[Poisson sigma-model]]

* [[deformation quantization]]

* [[BV-quantization]]

category: people

[[!redirects G. Felder]]
[[!redirects Felder]]


Giovanni Marchetti

* [personal page](https://www.kth.se/profile/glma)

## Selected writings

On [[t-structures]] and [[reflective factorization systems]]:

* [[Domenico Fiorenza]], [[Fosco Loregian]], [[Giovanni Marchetti]], *Hearts and towers in stable $\infty$-categories*,  J. Homotopy Relat. Struct. **14**  (2019) 993–1042 [[arXiv:1501.04658](https://arxiv.org/abs/1501.04658), [doi:10.1007/s40062-019-00237-0](https://doi.org/10.1007/s40062-019-00237-0)]

category: people

[[!redirects Giovanni L. Marcheti]]
[[!redirects Giovanni Luca Marchetti]]


Giovanni Morchio

* [InSpire page](https://inspirehep.net/authors/996877)

* [[Sergio Albeverio]], Sergio Albeverio, *Giovanni Morchio (1948–2021)*, in *Trails in Modern Theoretical and Mathematical*, Springer (2023) 3-7 [[doi:10.1007/978-3-031-44988-8_1](https://doi.org/10.1007/978-3-031-44988-8_1)]

## Selected writings

On the [[Gauss law]] in relation to non-[[local field theory|locality]] of charged states in [[quantum field theory]]:

* [[Giovanni Morchio]], [[Franco Strocchi]], *Localization and symmetries*, J. Phys. A **40** (2007) 3173-3188 [[arXiv:math-ph/0607015](https://arxiv.org/abs/math-ph/0607015), [doi:10.1088/1751-8113/40/12/S17](https://doi.org/10.1088/1751-8113/40/12/S17)]

On the [[theta vacuum]] ([[Yang-Mills instanton]] vacuum) from [[chiral symmetry breaking]]:

* [[Giovanni Morchio]], [[Franco Strocchi]], _Chiral symmetry breaking and theta vacuum structure in QCD_, Annals Phys. **324** (2009) 2236-2254 [[arXiv:0907.2522](https://arxiv.org/abs/0907.2522), [doi:10.1016/j.aop.2009.07.005](https://doi.org/10.1016/j.aop.2009.07.005)]

category: people


Giovanni Sambin

__Giovanni Sambin__ is a mathematician working in [[formal topology]], [[constructive mathematics]], [[intuitionistic logic]]. He is a professor at Università di Padova.

* [personal page](http://www.math.unipd.it/~sambin)

## Selected writings

On [[Martin-Löf type theory]]:

* [[Per Martin-Löf]], _Intuitionistic type theory_, Lecture notes Padua 1984 (notes by [[Giovanni Sambin]]), Bibliopolis, Napoli (1984) ([pdf](https://archive-pml.github.io/martin-lof/pdfs/Bibliopolis-Book-retypeset-1984.pdf), [[MartinLofIntuitionisticTypeTheory.pdf:file]])


Giraud's theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### $(\infty,1)$-Topos Theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A classical theorem of [[Jean Giraud|Giraud]] characterizes [[sheaf toposes]] abstractly as [[categories]] with certain [[properties]] known as *Giraud's axioms*.

In [[higher topos theory]] there are corresponding [[higher category theory|higher]] analogs .

## Statement

\begin{proposition}
**(Giraud's theorem)**
\linebreak
A [[category]] is a [[Grothendieck topos]] if and only if it satisfies the following conditions:

1. it is [[locally presentable category|locally presentable]];

1. it has [[universal colimits]];

1. it has [[disjoint coproducts]];

1. it has [[effective quotients]].

\end{proposition}
(e.g. [Lurie 2009, Prop. 6.1.0.1](#Lurie09));

Similarly:

\begin{proposition}
**(Giraud-Rezk-Lurie theorem)**
\linebreak
An [[(infinity,1)-category|$(\infty,1)$-category]] is an [[(infinity,1)-category of (infinity,1)-sheaves|Grothendieck-Rezk-Lurie]]-[[(infinity,1)-topos]] if and only if it satisfies the following conditions:

1. it is [[locally presentable (infinity,1)-category|locally presentable]];

1. it has [[universal colimits|universal]] [[(infinity,1)-colimits|$\infty$-colimits]];

1. it has [[disjoint coproducts]];

1. it has [[groupoid objects in an (∞,1)-topos are effective|effective groupoid objects]].

\end{proposition}
(e.g. [Lurie 2009, Prop. 6.1.0.6](#Lurie09))

Similarly, in [[2-category theory]] there is the analogous *[[2-Giraud theorem]]* for Grothendieck [[2-toposes]].

## Related concepts

* [[fundamental theorem of topos theory]]

## References

Textbook accounts:

* {#Lurie09} [[Jacob Lurie]], *[[Higher Topos Theory]]*, Annals of Mathematics Studies 170, Princeton University Press 2009 ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf))

following:

* {#Lurie03} [[Jacob Lurie]], Section 2.2 of: *On $\infty$-Topoi* ([arXiv:math/0306109](https://arxiv.org/abs/math/0306109))

Lecture notes:

* [[Jacob Lurie]], *Giraud's theorem*, Lecture 10 in: *[Categorical Logic](https://www.math.ias.edu/~lurie/278x.html)*, 2018 ([pdf](https://www.math.ias.edu/~lurie/278xnotes/Lecture10-Giraud.pdf))

[[!redirects Giraud theorem]]

[[!redirects Giraud theorems]]

[[!redirects Giraud-Rezk-Lurie axiom]]
[[!redirects Giraud-Rezk-Lurie axioms]]

[[!redirects Giraud-Rezk-Lurie theorem]]
[[!redirects Giraud-Rezk-Lurie theorem]]