In
mathematics, especially in
category theory and
homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
in several equivalent ways. A groupoid can be seen as a:
*''
Group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
'' with a
partial function
In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is de ...
replacing the
binary operation;
*''
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 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 group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
.
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:
*''
Setoid
In mathematics, a setoid (''X'', ~) is a set (or type) ''X'' equipped with an equivalence relation ~. A setoid may also be called E-set, Bishop set, or extensional set.
Setoids are studied especially in proof theory and in type-theoretic fou ...
s'':
sets that come with an
equivalence relation,
*''
G-sets'': sets equipped with an
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
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 ...
objects such as
manifolds. 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
In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is de ...
'
' 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
In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is de ...
. Here * is not a
binary operation 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'':
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), ...
'': 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 mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows asso ...
in which every
morphism 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
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
;
* 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 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
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...
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 differentl ...
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: 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
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
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 for counterexamples).
Subgroupoids and morphisms
A subgroupoid of
is a
subcategory 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 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 map A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
s 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 po ...
, let
be the set
. The morphisms from the point
to the point
are
equivalence classes 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 ...
path
A path is a route for physical travel – see Trail.
Path or PATH may also refer to:
Physical paths of different types
* Bicycle path
* Bridle path, used by people on horseback
* Course (navigation), the intended path of a vehicle
* Desire p ...
s 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. 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 to ...
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
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties ...
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
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
* Equivalence class (music)
*'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*''Equiva ...
as categories (see the section
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
In mathematics, a setoid (''X'', ~) is a set (or type) ''X'' equipped with an equivalence relation ~. A setoid may also be called E-set, Bishop set, or extensional set.
Setoids are studied especially in proof theory and in type-theoretic fou ...
, i.e. a set with an
equivalence relation , 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 submersion of
smooth manifolds
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, then
is an equivalence relation
since
has a topology isomorphic to the
quotient topology
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...
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
In logic, a true/false decision problem is decidable if there exists an effective method for deriving the correct answer. Zeroth-order logic (propositional logic) is decidable, whereas first-order and higher-order logic are not. Logical systems a ...
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 In mathematics, the effective topos is a topos
In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much ...
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
H ...
.
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
∞-groupoid
In category theory, a branch of mathematics, an ∞-groupoid is an abstract homotopical model for topological spaces. One model uses Kan complexes which are fibrant objects in the category of simplicial sets (with the standard model structure). I ...
s. 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
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
acts on the set
, then we can form the action groupoid (or transformation groupoid) representing this
group action
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 automorphism ...
as follows:
*The objects are the elements of
;
*For any two elements
and
in
, the
morphisms from
to
correspond to the elements
of
such that
;
*
Composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
of morphisms interprets the
binary operation 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
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 automorphism g ...
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 In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object in t ...