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 ...
, especially in
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
and
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolo ...
, 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 function replacing the
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
;
*''
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) ...
'' in which every
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, morphisms ...
is invertible. A category of this sort can be viewed as augmented with a
unary operation
In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation o ...
on the morphisms, called ''inverse'' by analogy with
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
.
A groupoid where there is only one object is a usual group.
In the presence of
dependent typing, a category in general can be viewed as a typed
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoid ...
, and similarly, a groupoid can be viewed as simply a typed group. The morphisms take one from one object to another, and form a dependent family of types, thus morphisms might be typed
,
, say. Composition is then a total function:
, so that
.
Special cases include:
*''
Setoids'':
sets that come with an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
,
*''
G-set
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphis ...
s'': sets equipped with an
action of a group
.
Groupoids are often used to reason about
geometrical
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
objects such as
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s. introduced groupoids implicitly via
Brandt semigroup In mathematics, Brandt semigroups are completely 0-simple inverse semigroups. In other words, they are semigroups without proper ideals and which are also inverse semigroups. They are built in the same way as completely 0-simple semigroups:
Let '' ...
s.
Definitions
A groupoid is an algebraic structure
consisting of a non-empty set
and a binary
partial function '
' defined on
.
Algebraic
A groupoid is a set
with a
unary operation
In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation o ...
and a
partial function . Here * is not a
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
because it is not necessarily defined for all pairs of elements of
. The precise conditions under which
is defined are not articulated here and vary by situation.
The operations
and
−1 have the following axiomatic properties: For all
,
, and
in
,
# ''
Associativity
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
'': If
and
are defined, then
and
are defined and are equal. Conversely, if one of
and
is defined, then so are both
and
as well as
=
.
# ''
Inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when a ...
'':
and
are always defined.
# ''
Identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), an ...
'': If
is defined, then
, and
. (The previous two axioms already show that these expressions are defined and unambiguous.)
Two easy and convenient properties follow from these axioms:
*
,
* If
is defined, then
.
Category theoretic
A groupoid is a
small category in which every
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, morphisms ...
is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
, i.e., invertible.
More explicitly, a groupoid ''G'' is:
* A set ''G''
0 of ''objects'';
* For each pair of objects ''x'' and ''y'' in ''G''
0, there exists a (possibly empty) set ''G''(''x'',''y'') of ''morphisms'' (or ''arrows'') from ''x'' to ''y''. We write ''f'' : ''x'' → ''y'' to indicate that ''f'' is an element of ''G''(''x'',''y'').
* For every object ''x'', a designated element
of ''G''(''x'',''x'');
* For each triple of objects ''x'', ''y'', and ''z'', a
function ;
* For each pair of objects ''x'', ''y'' a function
;
satisfying, for any ''f'' : ''x'' → ''y'', ''g'' : ''y'' → ''z'', and ''h'' : ''z'' → ''w'':
*
and
;
*
;
*
and
.
If ''f'' is an element of ''G''(''x'',''y'') then ''x'' is called the source of ''f'', written ''s''(''f''), and ''y'' is called the target of ''f'', written ''t''(''f''). A groupoid ''G'' is sometimes denoted as
, where
is the set of all morphisms, and the two arrows
represent the source and the target.
More generally, one can consider a
groupoid object In category theory, a branch of mathematics, a groupoid object is both a generalization of a groupoid which is built on richer structures than sets, and a generalization of a group objects when the multiplication is only partially defined.
Defi ...
in an arbitrary category admitting finite fiber products.
Comparing the definitions
The algebraic and category-theoretic definitions are equivalent, as we now show. Given a groupoid in the category-theoretic sense, let ''G'' be the
disjoint union of all of the sets ''G''(''x'',''y'') (i.e. the sets of morphisms from ''x'' to ''y''). Then
and
become partial operations on ''G'', and
will in fact be defined everywhere. We define ∗ to be
and
−1 to be
, which gives a groupoid in the algebraic sense. Explicit reference to ''G''
0 (and hence to
) can be dropped.
Conversely, given a groupoid ''G'' in the algebraic sense, define an equivalence relation
on its elements by
iff ''a'' ∗ ''a''
−1 = ''b'' ∗ ''b''
−1. Let ''G''
0 be the set of equivalence classes of
, i.e.
. Denote ''a'' ∗ ''a''
−1 by
if
with
.
Now define
as the set of all elements ''f'' such that
exists. Given
and
their composite is defined as
. To see that this is well defined, observe that since
and
exist, so does
. The identity morphism on ''x'' is then
, and the category-theoretic inverse of ''f'' is ''f''
−1.
''Sets'' in the definitions above may be replaced with
class
Class or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used differently ...
es, as is generally the case in category theory.
Vertex groups and orbits
Given a groupoid ''G'', the vertex groups or isotropy groups or object groups in ''G'' are the subsets of the form ''G''(''x'',''x''), where ''x'' is any object of ''G''. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.
The orbit of a groupoid ''G'' at a point
is given by the set
containing every point that can be joined to x by an morphism in G. If two points
and
are in the same orbits, their vertex groups
and
are
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
: if
is any morphism from
to
, then the isomorphism is given by the mapping
.
Orbits form a partition of the set X, and a groupoid is called transitive if it has only one orbit (equivalently, if it is
connected as a category). In that case, all the vertex groups are isomorphic (on the other hand, this is not a sufficient condition for transitivity; see the section
below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
for counterexamples).
Subgroupoids and morphisms
A subgroupoid of
is a
subcategory
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, ...
that is itself a groupoid. It is called wide or full if it is
wide
WIDE or Wide may refer to:
*Wide (cricket)
*Wide and narrow data, terms used to describe two different presentations for tabular data
*WIDE Project, Widely Integrated Distributed Environment
*Wide-angle Infinity Display Equipment
*WIDE-LP, a radio ...
or
full as a subcategory i.e., respectively, if
or
for every
.
A groupoid morphism is simply a functor between two (category-theoretic) groupoids.
Particular kinds of morphisms of groupoids are of interest. A morphism
of groupoids is called a
fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.
Fibrations are used, for example, in postnikov-systems or obstruction theory.
In this article, all ma ...
if for each object
of
and each morphism
of
starting at
there is a morphism
of
starting at
such that
. A fibration is called a
covering morphism or
covering of groupoids if further such an
is unique. The covering morphisms of groupoids are especially useful because they can be used to model
covering maps of spaces.
It is also true that the category of covering morphisms of a given groupoid
is equivalent to the category of actions of the groupoid
on sets.
Examples
Topology
Given 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 poin ...
, let
be the set
. The morphisms from the point
to the point
are
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es of
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
paths from
to
, with two paths being equivalent if they are
homotopic
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
.
Two such morphisms are composed by first following the first path, then the second; the homotopy equivalence guarantees that this composition is
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
. This groupoid is called the
fundamental groupoid In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a ...
of
, denoted
(or sometimes,
). The usual fundamental group
is then the vertex group for the point
.
The orbits of the fundamental groupoid
are the path-connected components of
. Accordingly, the fundamental groupoid of a
path-connected space is transitive, and we recover the known fact that the fundamental groups at any base point are isomorphic. Moreover, in this case, the fundamental groupoid and the fundamental groups are
equivalent as categories (see the section
below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
for the general theory).
An important extension of this idea is to consider the fundamental groupoid
where
is a chosen set of "base points". Here
is a (wide) subgroupoid of
, where one considers only paths whose endpoints belong to
. The set
may be chosen according to the geometry of the situation at hand.
Equivalence relation
If
is a
setoid, i.e. a set with an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
, then a groupoid "representing" this equivalence relation can be formed as follows:
* The objects of the groupoid are the elements of
;
*For any two elements
and
in
, there is a single morphism from
to
(denote by
) if and only if
;
*The composition of
and
is
.
The vertex groups of this groupoid are always trivial; moreover, this groupoid is in general not transitive and its orbits are precisely the equivalence classes. There are two extreme examples:
* If every element of
is in relation with every other element of
, we obtain the pair groupoid of
, which has the entire
as set of arrows, and which is transitive.
* If every element of
is only in relation with itself, one obtains the unit groupoid, which has
as set of arrows,
, and which is completely intransitive (every singleton
is an orbit).
Examples
*If
is a smooth
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
submersion of
smooth manifolds, then
is an equivalence relation
since
has a topology isomorphic to the
quotient topology of
under the surjective map of topological spaces. If we write,
then we get a groupoid
which is sometimes called the banal groupoid of a surjective submersion of smooth manifolds.
*If we relax the reflexivity requirement and consider ''partial equivalence relations'', then it becomes possible to consider
semidecidable notions of equivalence on computable realisers for sets. This allows groupoids to be used as a computable approximation to set theory, called ''PER models''. Considered as a category, PER models are a cartesian closed category with natural numbers object and subobject classifier, giving rise to the
effective topos introduced by
Martin Hyland
(John) Martin Elliott Hyland is professor of mathematical logic at the University of Cambridge and a fellow of King's College, Cambridge. His interests include mathematical logic, category theory, and theoretical computer science.
Education
Hy ...
.
Čech groupoid
A Čech groupoid
pg 5 is a special kind of groupoid associated to an equivalence relation given by an open cover
of some manifold
. It's objects are given by the disjoint union
and its arrows are the intersections
The source and target maps are then given by the induced maps
and the inclusion map
giving the structure of a groupoid. In fact, this can be further extended by setting
as the
-iterated fiber product where the
represents
-tuples of composable arrows. The structure map of the fiber product is implicitly the target map, since
is a cartesian diagram where the maps to
are the target maps. This construction can be seen as a model for some
∞-groupoids. Also, another artifact of this construction is
k-cocyclesfor some constant
sheaf of abelian groups
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
can be represented as a function
giving an explicit representation of cohomology classes.
Group action
If the
group acts on the set
, then we can form the action groupoid (or transformation groupoid) representing this
group action as follows:
*The objects are the elements of
;
*For any two elements
and
in
, the
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, morphisms ...
s from
to
correspond to the elements
of
such that
;
*
Composition of morphisms interprets the
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
of
.
More explicitly, the ''action groupoid'' is a small category with
and
and with source and target maps
and
. It is often denoted
(or
for a right action). Multiplication (or composition) in the groupoid is then
which is defined provided
.
For
in
, the vertex group consists of those
with
, which is just the
isotropy subgroup at
for the given action (which is why vertex groups are also called isotropy groups). Similarly, the orbits of the action groupoid are the
orbit
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
of the group action, and the groupoid is transitive if and only if the group action is
transitive.
Another way to describe
-sets is the
functor category