In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, especially in
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
and
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipli ...
, 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 iden ...
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 iden ...
'' with a
partial function
In mathematics, a partial function from a set to a set is a function from a subset of (possibly the whole itself) to . The subset , that is, the '' domain'' of viewed as a function, is called the domain of definition or natural domain ...
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, a binary operation ...
;
* ''
Category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
'' in which every
morphism
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
is invertible. A category of this sort can be viewed as augmented with a
unary operation
In mathematics, a 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 ...
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
In computer science and logic, a dependent type is a type whose definition depends on a value. It is an overlapping feature of type theory and type systems. In intuitionistic type theory, dependent types are used to encode logic's quantifiers lik ...
, a category in general can be viewed as a typed
monoid
In abstract algebra, 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 .
Monoids are semigroups with identity ...
, 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 foun ...
s'':
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. A simpler example is equ ...
,
* ''
G-set
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S.
Many sets of transformations form a group under func ...
s'': sets equipped with an
action
Action may refer to:
* Action (philosophy), something which is done by a person
* Action principles the heart of fundamental physics
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video gam ...
of a group .
Groupoids are often used to reason about
geometrical
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
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
Algebraic
A groupoid can be viewed as an algebraic structure consisting of a set with a binary
partial function
In mathematics, a partial function from a set to a set is a function from a subset of (possibly the whole itself) to . The subset , that is, the '' domain'' of viewed as a function, is called the domain of definition or natural domain ...
.
Precisely, it is a non-empty set
with a
unary operation
In mathematics, a 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 ...
, and a
partial function
In mathematics, a partial function from a set to a set is a function from a subset of (possibly the whole itself) to . The subset , that is, the '' domain'' of viewed as a function, is called the domain of definition or natural domain ...
. 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, 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 that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replaceme ...
'': If
and
are defined, then
and
are defined and are equal. Conversely, if one of
or
is defined, then they are both defined (and they are equal to each other), and
and
are also defined.
# ''
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, the inverse of a number that, when added to the ...
'':
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 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
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism 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 the ...
, i.e., invertible.
More explicitly, a groupoid
is a set
of ''objects'' with
* for each pair of objects ''x'' and ''y'', 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-orie ...
;
* 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.
Defini ...
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, the disjoint union (or discriminated union) A \sqcup B of the sets and is the set formed from the elements of and labelled (indexed) with the name of the set from which they come. So, an element belonging to both and appe ...
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, Classes, 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 d ...
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 a 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 or morphism 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 the ...
: 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
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 Belo ...
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), a type of illegal delivery to a batter
*Wide and narrow data Wide and narrow (sometimes un-stacked and stacked, or wide and tall) are terms used to describe two different presentations for tabular data ...
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 map
In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphisms ...
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
Fundamental groupoid
Given a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, 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 ...
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 ...
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
* Desir ...
s from
to , with two paths being equivalent if they are
homotopic
In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ...
.
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 that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...
. 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 (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Conne ...
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
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 Belo ...
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 (full) 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 foun ...
, 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. A simpler example is equ ...
, 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 such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
submersion
Submersion may refer to:
*Being or going underwater, as via submarine, underwater diving, or scuba diving
*Submersion (coastal management), the sustainable cyclic portion of foreshore erosion
*Submersion (mathematics)
* Submersion (Stargate Atlanti ...
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 (topology ...
, 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 computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if:
*There is an algorithm such that the ...
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 introduced by captures the mathematical idea of effectivity within the category theoretical framework.
Definition Preliminaries Kleene realizability
The topos is based on the partial combinatory algebra give ...
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 ...
.
ÄŒech groupoid
A ÄŒech groupoid
p. 5 is a special kind of groupoid associated to an equivalence relation given by an open cover
of some manifold . Its 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 (mathematics), category of simplicial sets (with the standa ...
s. Also, another artifact of this construction is
k-cocyclesfor some constant
sheaf of abelian groups
In mathematics, a sheaf (: sheaves) 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 d ...
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 iden ...
acts on the set , then we can form the
action groupoid In mathematics, an action groupoid or a transformation groupoid is a groupoid that expresses a group action. Namely, given a (right) group action
:X \times G \to X,
we get the groupoid \mathcal (= a category whose morphisms are all invertible) where ...
(or transformation groupoid) representing this
group action
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S.
Many sets of transformations form a group under ...
as follows:
* The objects are the elements of ;
* For any two elements
and
in , the
morphism
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
s 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 ...
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, a 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 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 (also known as orbital revolution) 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 ...
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 i ...
, where
is the groupoid (category) with one element and
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism 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 the ...
to the group . Indeed, every functor
of this category defines a set
and for every
in
(i.e. for every morphism in ) induces a
bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
: . The categorical structure of the functor
assures us that
defines a
-action on the set . The (unique)
representable functor
In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets an ...
is the
Cayley representation of . In fact, this functor is isomorphic to
and so sends
to the set
which is by definition the "set"
and the morphism
of
(i.e. the element
of ) to the permutation
of the set . We deduce from the
Yoneda embedding
In mathematics, the Yoneda lemma is a fundamental result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a ...
that the group
is isomorphic to the group , a
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation  ...
of the group of
permutation
In mathematics, a permutation of a set can mean one of two different things:
* an arrangement of its members in a sequence or linear order, or
* the act or process of changing the linear order of an ordered set.
An example of the first mean ...
s of .
Finite set
Consider the group action of
on the finite set
where 1 acts by taking each number to its negative, so
and . The quotient groupoid