In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a gerbe (; ) is a construct in
homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topol ...
and
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. Gerbes were introduced by
Jean Giraud
Jean Henri Gaston Giraud (; 8 May 1938 – 10 March 2012) was a French artist, cartoonist, and writer who worked in the Franco-Belgian ''bandes dessinées'' (BD) tradition. Giraud garnered worldwide acclaim under the pseudonym Mœbius (; ) ...
following ideas of
Alexandre Grothendieck as a tool for non-commutative
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
in degree 2. They can be seen as an analogue of
fibre bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
s where the fibre is the
classifying stack In algebraic geometry, a quotient stack is a stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a scheme or a variety by a group: a quotient variety, say, would be a coarse approximation of a quotient stack.
T ...
of a group. Gerbes provide a convenient, if highly abstract, language for dealing with many types of
deformation
Deformation can refer to:
* Deformation (engineering), changes in an object's shape or form due to the application of a force or forces.
** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies.
* Def ...
questions especially in modern
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. In addition, special cases of gerbes have been used more recently in
differential topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
and
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mul ...
to give alternative descriptions to certain
cohomology classes and additional structures attached to them.
"Gerbe" is a French (and archaic English) word that literally means
wheat
Wheat is a grass widely cultivated for its seed, a cereal grain that is a worldwide staple food. The many species of wheat together make up the genus ''Triticum'' ; the most widely grown is common wheat (''T. aestivum''). The archaeologica ...
sheaf.
Definitions
Gerbes on a topological space
A gerbe on a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
is a
stack of
groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
*'' Group'' with a partial funct ...
s over
which is ''locally non-empty'' (each point
has an open neighbourhood
over which the
section category of the gerbe is not empty) and ''transitive'' (for any two objects
and
of
for any open set
, there is an open covering
of
such that the restrictions of
and
to each
are connected by at least one morphism).
A canonical example is the gerbe
of
principal bundles
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
with a fixed
structure group
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
: the section category over an open set
is the category of principal
-bundles on
with isomorphism as morphisms (thus the category is a groupoid). As principal bundles glue together (satisfy the descent condition), these groupoids form a stack. The trivial bundle
shows that the local non-emptiness condition is satisfied, and finally as principal bundles are locally trivial, they become isomorphic when restricted to sufficiently small open sets; thus the transitivity condition is satisfied as well.
Gerbes on a site
The most general definition of gerbes are defined over a
site
Site most often refers to:
* Archaeological site
* Campsite, a place used for overnight stay in an outdoor area
* Construction site
* Location, a point or an area on the Earth's surface or elsewhere
* Website, a set of related web pages, typicall ...
. Given a site
a
-gerbe
is a category fibered in groupoids
such that
# There exists a refinement
of
such that for every object
the associated fibered category
is non-empty
# For every
any two objects in the fibered category
are locally isomorphic
Note that for a site
with a final object
, a category fibered in groupoids
is a
-gerbe admits a local section, meaning satisfies the first axiom, if
.
Motivation for gerbes on a site
One of the main motivations for considering gerbes on a site is to consider the following naive question: if the Cech cohomology group
for a suitable covering
of a space
gives the isomorphism classes of principal
-bundles over
, what does the iterated cohomology functor
represent? Meaning, we are gluing together the groups
via some one cocycle. Gerbes are a technical response for this question: they give geometric representations of elements in the higher cohomology group
. It is expected this intuition should hold for
higher gerbes.
Cohomological classification
One of the main theorems concerning gerbes is their cohomological classification whenever they have automorphism groups given by a fixed sheaf of abelian groups
,
called a band. For a gerbe
on a site
, an object
, and an object
, the automorphism group of a gerbe is defined as the automorphism group
. Notice this is well defined whenever the automorphism group is always the same. Given a covering
, there is an associated class
representing the isomorphism class of the gerbe
banded by
.
For example, in topology, many examples of gerbes can be constructed by considering gerbes banded by the group
. As the classifying space
is the second
Eilenberg-Maclane space for the integers, a bundle gerbe banded by
on a topological space
is constructed from a homotopy class of maps in
which is exactly the third singular homology group
. It has been found that all gerbes representing torsion cohomology classes in
are represented by a bundle of finite dimensional algebras
for a fixed complex vector space
. In addition, the non-torsion classes are represented as infinite-dimensional principal bundles
of the projective group of unitary operators on a fixed infinite dimensional
separable Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. Note this is well defined because all separable Hilbert spaces are isomorphic to the space of square-summable sequences
.
The homotopy-theoretic interpretation of gerbes comes from looking at the
homotopy fiber squareanalogous to how a line bundle comes from the homotopy fiber square
where
, giving
as the group of isomorphism classes of line bundles on
.
Examples
C*-algebras
There are natural examples of Gerbes which arise from studying the algebra of compactly supported complex valued functions on a paracompact space
pg 3. Given a cover
of
there is the Cech groupoid defined as
with source and target maps given by the inclusions
and the space of composable arrows is just
Then a degree 2 cohomology class
is just a map
We can then form a non-commutative
C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuo ...
which is associated to the set of compact supported complex valued functions of the space
It has a non-commutative product given by
where the cohomology class
twists the multiplication of the standard
-algebra product.
Algebraic geometry
Let
be a
variety
Variety may refer to:
Arts and entertainment Entertainment formats
* Variety (radio)
* Variety show, in theater and television
Films
* ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont
* ''Variety'' (1935 film) ...
over an
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
,
an
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Ma ...
, for example
. Recall that a
''G''-torsor over
is an
algebraic space In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively,
schemes are given by gluing together affine schemes using the Zariski topology, ...
with an action of
and a map
, such that locally on
(in
étale topology In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic. The étale t ...
or
fppf topology)
is a direct product
. A ''G''-gerbe over ''M'' may be defined in a similar way. It is an
Artin stack
In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to construct fine moduli stacks when fine moduli ...
with a map
, such that locally on ''M'' (in étale or fppf topology)
is a direct product
. Here
denotes the
classifying stack In algebraic geometry, a quotient stack is a stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a scheme or a variety by a group: a quotient variety, say, would be a coarse approximation of a quotient stack.
T ...
of
, i.e. a quotient