In
category theory, a branch of
mathematics, a groupoid object is both a generalization of a
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 fun ...
which is built on richer structures than sets, and a generalization of a
group objects when the multiplication is only partially defined.
Definition
A groupoid object in a
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
''C'' admitting finite
fiber product
In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is ofte ...
s consists of a pair of
objects together with five
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphis ...
s
:
satisfying the following groupoid axioms
#
where the
are the two projections,
# (associativity)
# (unit)
# (inverse)
,
,
.
Examples
Group objects
A
group object is a special case of a groupoid object, where
and
. One recovers therefore
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two ...
s by taking the
category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again con ...
, or
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...
s by taking the
category of manifolds, etc.
Groupoids
A groupoid object in the
category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition ...
is precisely a
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 fun ...
in the usual sense: a category in which every morphism is an isomorphism. Indeed, given such a category ''C'', take ''U'' to be the set of all objects in ''C'', ''R'' the set of all arrows in ''C'', the five morphisms given by
,
,
and
. When the term "groupoid" can naturally refer to a groupoid object in some particular category in mind, the term groupoid set is used to refer to a groupoid object in the category of sets.
However, unlike in the previous example with Lie groups, a groupoid object in the category of manifolds is not necessarily a
Lie groupoid, since the maps s and t fail to satisfy further requirements (they are not necessarily
submersions).
Groupoid schemes
A groupoid ''S''-scheme is a groupoid object in the category of
schemes over some fixed base scheme ''S''. If
, then a groupoid scheme (where
are necessarily the structure map) is the same as a
group scheme
In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups ha ...
. A groupoid scheme is also called an algebraic groupoid, for example in , to convey the idea it is a generalization of
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.
...
s and their actions.
For example, suppose 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.
...
''G''
acts
The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
from the right on a scheme ''U''. Then take
, ''s'' the projection, ''t'' the given action. This determines a groupoid scheme.
Constructions
Given a groupoid object (''R'', ''U''), the
equalizer of
, if any, is a group object called the inertia group of the groupoid. The
coequalizer
In category theory, a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. It is the categorical construction dual to the equalizer.
Definition
A coequalizer is a co ...
of the same diagram, if any, is the quotient of the groupoid.
Each groupoid object in a category ''C'' (if any) may be thought of as a
contravariant functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...
from ''C'' to the category of groupoids. This way, each groupoid object determines a
prestack in groupoids. This prestack is not a
stack
Stack may refer to:
Places
* Stack Island, an island game reserve in Bass Strait, south-eastern Australia, in Tasmania’s Hunter Island Group
* Blue Stack Mountains, in Co. Donegal, Ireland
People
* Stack (surname) (including a list of people ...
but it can be
stackified to yield a stack.
The main use of the notion is that it provides an
atlas
An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth.
Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geograp ...
for a stack. More specifically, let