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 func ...
which is built on richer structures than sets, and a generalization of a
group object In category theory, a branch of mathematics, group objects are certain generalizations of groups that are built on more complicated structures than sets. A typical example of a group object is a topological group, a group whose underlying set is ...
s 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 often w ...
s consists of a pair of
objects
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ...
together with five
morphisms
:
satisfying the following groupoid axioms
#
where the
are the two projections,
# (associativity)
# (unit)
# (inverse)
,
,
.
Examples
Group objects
A
group object In category theory, a branch of mathematics, group objects are certain generalizations of groups that are built on more complicated structures than sets. A typical example of a group object is a topological group, a group whose underlying set is ...
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 st ...
s by taking the
category of topological spaces, or
Lie groups by taking the
category of manifolds In mathematics, the category of manifolds, often denoted Man''p'', is the category whose objects are manifolds of smoothness class ''C'p'' and whose morphisms are ''p''-times continuously differentiable maps. This is a category because the ...
, 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 o ...
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 func ...
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 In mathematics, a Lie groupoid is a groupoid where the set \operatorname of objects and the set \operatorname of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smo ...
, 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 have ...
. 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.
Ma ...
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.
Ma ...
''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 algebraic geometry, a prestack ''F'' over a category ''C'' equipped with some Grothendieck topology is a category together with a functor ''p'': ''F'' → ''C'' satisfying a certain lifting condition and such that (when the fibers are groupoids ...
in groupoids. This prestack is not a
stack 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