Editing Groupoid (section)

== Examples ==

=== Fundamental groupoid ===
{{main|Fundamental groupoid}}
Given a [[topological space]] {{tmath|1= X }}, let <math>G_0</math> be the set {{tmath|1= X }}. The morphisms from the point <math>p</math> to the point <math>q</math> are [[equivalence class]]es of [[continuous function (topology)|continuous]] [[path (topology)|path]]s from <math>p</math> to {{tmath|1= q }}, with two paths being equivalent if they are [[homotopic]].
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]] of {{tmath|1= X }}, denoted <math>\pi_1(X)</math> (or sometimes, {{tmath|1= \Pi_1(X) }}).<ref>{{cite web |url=https://ncatlab.org/nlab/show/fundamental+groupoid |title=fundamental groupoid in nLab |website=ncatlab.org |access-date=2017-09-17 }}</ref> The usual fundamental group <math>\pi_1(X,x)</math> is then the vertex group for the point {{tmath|1= x }}.

The orbits of the fundamental groupoid <math>\pi_1(X)</math> are the path-connected components of {{tmath|1= X }}. 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 [[Equivalence of categories|equivalent]] as categories (see the section [[Groupoid#Relation to groups|below]] for the general theory).

An important extension of this idea is to consider the fundamental groupoid <math>\pi_1(X,A)</math> where <math>A\subset X</math> is a chosen set of "base points". Here <math>\pi_1(X,A)</math> is a (full) subgroupoid of {{tmath|1= \pi_1(X) }}, where one considers only paths whose endpoints belong to {{tmath|1= A }}. The set <math>A</math> may be chosen according to the geometry of the situation at hand.

=== Equivalence relation ===
If <math>X</math> is a [[setoid]], i.e. a set with an [[equivalence relation]] {{tmath|1= \sim }}, then a groupoid "representing" this equivalence relation can be formed as follows:
* The objects of the groupoid are the elements of {{tmath|1= X }};
*For any two elements <math>x</math> and <math>y</math> in {{tmath|1= X }}, there is a single morphism from <math>x</math> to <math>y</math> (denote by {{tmath|1= (y,x) }}) if and only if {{tmath|1= x\sim y }};
*The composition of <math>(z,y)</math> and <math>(y,x)</math> is {{tmath|1= (z,x) }}.
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 <math>X</math> is in relation with every other element of {{tmath|1= X }}, we obtain the '''pair groupoid''' of {{tmath|1= X }}, which has the entire <math>X \times X</math> as set of arrows, and which is transitive.
* If every element of <math>X</math> is only in relation with itself, one obtains the '''unit groupoid''', which has <math>X</math> as set of arrows, {{tmath|1= s = t = \mathrm{id}_X }}, and which is completely intransitive (every singleton <math>\{x\}</math> is an orbit).

==== Examples ====
* If <math>f: X_0 \to Y</math> is a smooth [[Surjective function|surjective]] [[Submersion (mathematics)|submersion]] of [[smooth manifolds]], then <math>X_0\times_YX_0 \subset X_0\times X_0</math> is an equivalence relation<ref name=":0" /> since <math>Y</math> has a topology isomorphic to the [[quotient topology]] of <math>X_0</math> under the surjective map of topological spaces. If we write, <math>X_1 = X_0\times_YX_0</math> then we get a groupoid <math display=block>X_1 \rightrightarrows X_0,</math> 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]].

=== Čech groupoid ===
{{See also|Simplicial manifold|Nerve of a covering}}
A Čech groupoid<ref name=":0">{{cite arXiv|last1=Block|first1=Jonathan|last2=Daenzer|first2=Calder|date=2009-01-09|title=Mukai duality for gerbes with connection|class=math.QA|eprint=0803.1529}}</ref><sup>p.&nbsp;5</sup> is a special kind of groupoid associated to an equivalence relation given by an open cover <math>\mathcal{U} = \{U_i\}_{i\in I}</math> of some manifold {{tmath|1= X }}. Its objects are given by the disjoint union
<math display="block">\mathcal{G}_0 = \coprod U_i ,</math>
and its arrows are the intersections <math display=block>\mathcal{G}_1 = \coprod U_{ij} .</math>
The source and target maps are then given by the induced maps<blockquote><math>\begin{align}
s = \phi_j: U_{ij} \to U_j\\
t = \phi_i: U_{ij} \to U_i
\end{align}</math></blockquote>and the inclusion map<blockquote><math>\varepsilon: U_i \to U_{ii}</math></blockquote>giving the structure of a groupoid. In fact, this can be further extended by setting<blockquote><math>\mathcal{G}_n = \mathcal{G}_1\times_{\mathcal{G}_0} \cdots \times_{\mathcal{G}_0}\mathcal{G}_1</math></blockquote>as the <math>n</math>-iterated fiber product where the <math>\mathcal{G}_n</math> represents <math>n</math>-tuples of composable arrows. The structure map of the fiber product is implicitly the target map, since<blockquote><math>\begin{matrix}
U_{ijk} & \to & U_{ij} \\
\downarrow & & \downarrow \\
U_{ik} & \to & U_{i}
\end{matrix}</math></blockquote>is a cartesian diagram where the maps to <math>U_i</math> are the target maps. This construction can be seen as a model for some [[∞-groupoid]]s. Also, another artifact of this construction is [[Čech cohomology|k-cocycles]]<blockquote><math>[\sigma] \in \check{H}^k(\mathcal{U},\underline{A})</math></blockquote>for some constant [[sheaf of abelian groups]] can be represented as a function<blockquote><math>\sigma:\coprod U_{i_1\cdots i_k} \to A</math></blockquote>giving an explicit representation of cohomology classes.

=== Group action ===
{{main|action groupoid}}
If the [[group (mathematics)|group]] <math>G</math> acts on the set {{tmath|1= X }}, then we can form the '''[[action groupoid]]''' (or '''transformation groupoid''') representing this [[Group action (mathematics)|group action]] as follows:
* The objects are the elements of {{tmath|1= X }};
* For any two elements <math>x</math> and <math>y</math> in {{tmath|1= X }}, the [[morphism]]s from <math>x</math> to <math>y</math> correspond to the elements <math>g</math> of <math>G</math> such that {{tmath|1= gx = y }};
* [[Function composition|Composition]] of morphisms interprets the [[binary operation]] of {{tmath|1= G }}.

More explicitly, the ''action groupoid'' is a small category with <math>\mathrm{ob}(C)=X</math> and <math>\mathrm{hom}(C)=G\times X</math> and with source and target maps <math>s(g,x) = x</math> and {{tmath|1= t(g,x) = gx }}. It is often denoted <math>G \ltimes X</math> (or <math>X\rtimes G</math> for a right action). Multiplication (or composition) in the groupoid is then {{tmath|1= (h,y)(g,x) = (hg,x) }}, which is defined provided {{tmath|1= y=gx }}.

For <math>x</math> in {{tmath|1= X }}, the vertex group consists of those <math>(g,x)</math> with {{tmath|1= gx=x }}, which is just the [[isotropy subgroup]] at <math>x</math> for the given action (which is why vertex groups are also called isotropy groups). Similarly, the orbits of the action groupoid are the [[Orbit (group theory)|orbit]] of the group action, and the groupoid is transitive if and only if the group action is [[Transitive group action|transitive]].

Another way to describe <math>G</math>-sets is the [[functor category]] {{tmath|1= [\mathrm{Gr},\mathrm{Set}] }}, where <math>\mathrm{Gr}</math> is the groupoid (category) with one element and [[isomorphism|isomorphic]] to the group {{tmath|1= G }}. Indeed, every functor <math>F</math> of this category defines a set <math>X=F(\mathrm{Gr})</math> and for every <math>g</math> in <math>G</math> (i.e. for every morphism in {{tmath|1= \mathrm{Gr} }}) induces a [[bijection]] <math>F_g</math> : {{tmath|1= X\to X }}. The categorical structure of the functor <math>F</math> assures us that <math>F</math> defines a <math>G</math>-action on the set {{tmath|1= G }}. The (unique) [[representable functor]] <math>F : \mathrm{Gr} \to \mathrm{Set}</math> is the [[Cayley's theorem|Cayley representation]] of {{tmath|1= G }}. In fact, this functor is isomorphic to <math>\mathrm{Hom}(\mathrm{Gr},-)</math> and so sends <math>\mathrm{ob}(\mathrm{Gr})</math> to the set <math>\mathrm{Hom}(\mathrm{Gr},\mathrm{Gr})</math> which is by definition the "set" <math>G</math> and the morphism <math>g</math> of <math>\mathrm{Gr}</math> (i.e. the element <math>g</math> of {{tmath|1= G }}) to the permutation <math>F_g</math> of the set {{tmath|1= G }}. We deduce from the [[Yoneda embedding]] that the group <math>G</math> is isomorphic to the group {{tmath|1= \{F_g\mid g\in G\} }}, a [[subgroup]] of the group of [[permutation group|permutation]]s of {{tmath|1= G }}.

==== Finite set ====
Consider the group action of <math>\mathbb{Z}/2</math> on the finite set <math>X = \{-2, -1, 0, 1, 2\}</math> where 1 acts by taking each number to its negative, so <math>-2 \mapsto 2</math> and {{tmath|1= 1 \mapsto -1 }}. The quotient groupoid <math>[X/G]</math> is the set of equivalence classes from this group action {{tmath|1= \{[0],[1],[2]\} }}, and <math>[0]</math> has a group action of <math>\mathbb{Z}/2</math> on it.{{fact|date=May 2025}}

==== Quotient variety ====
Any finite group <math>
G
</math> that maps to <math>
GL(n)
</math> gives a group action on the [[affine space]] <math>
\mathbb{A}^n
</math> (since this is the group of automorphisms). Then, a quotient groupoid can be of the form {{tmath|1= [\mathbb{A}^n/G] }}, which has one point with stabilizer <math> G </math> at the origin. Examples like these form the basis for the theory of [[orbifold]]s. Another commonly studied family of orbifolds are [[weighted projective space]]s <math>\mathbb{P}(n_1,\ldots, n_k)</math> and subspaces of them, such as [[Calabi–Yau manifold|Calabi–Yau orbifold]]s.

=== Inertia groupoid ===
{{main|Inertia groupoid}}
The inertia groupoid of a groupoid is roughly a groupoid of loops in the given groupoid.

=== Fiber product of groupoids ===
Given a diagram of groupoids with groupoid morphisms
: <math>
\begin{align}
 & & X \\
 & & \downarrow \\
Y &\rightarrow & Z 
\end{align}
</math>
where <math>f:X\to Z</math> and {{tmath|1= g:Y\to Z }}, we can form the groupoid <math>X\times_ZY</math> whose objects are triples {{tmath|1= (x,\phi,y) }}, where {{tmath|1= x \in \text{Ob}(X) }}, {{tmath|1= y \in \text{Ob}(Y) }}, and <math>\phi: f(x) \to g(y)</math> in {{tmath|1= Z }}. Morphisms can be defined as a pair of morphisms <math>(\alpha,\beta)</math> where <math>\alpha: x \to x'</math> and 
<math>\beta: y \to y'</math> such that for triples {{tmath|1= (x,\phi,y), (x',\phi',y') }}, there is a commutative diagram in <math>Z</math> of {{tmath|1= f(\alpha):f(x) \to f(x') }}, <math>g(\beta):g(y) \to g(y')</math> and the {{tmath|1= \phi,\phi' }}.<ref>{{Cite web|url=https://www.math.ubc.ca/~behrend/cet.pdf|title=Localization and Gromov-Witten Invariants|page=9|url-status=live|archive-url=https://web.archive.org/web/20200212202830/https://www.math.ubc.ca/~behrend/cet.pdf|archive-date=February 12, 2020}}</ref>

=== Homological algebra ===
A two term complex
: <math>C_1 ~\overset{d}{\rightarrow}~ C_0</math>
of objects in a [[Concrete category|concrete]] [[Abelian category]] can be used to form a groupoid. It has as objects the set <math>C_0</math> and as arrows the set {{tmath|1= C_1\oplus C_0 }}; the source morphism is just the projection onto <math>C_0</math> while the target morphism is the addition of projection onto <math>C_1</math> composed with <math>d</math> and projection onto {{tmath|1= C_0 }}. That is, given {{tmath|1= c_1 + c_0 \in C_1\oplus C_0 }}, we have
: <math>t(c_1 + c_0) = d(c_1) + c_0.</math>
Of course, if the abelian category is the category of [[coherent sheaves]] on a scheme, then this construction can be used to form a [[presheaf]] of groupoids.

=== Puzzles ===

While puzzles such as the [[Rubik's Cube]] can be modeled using group theory (see [[Rubik's Cube group]]), certain puzzles are better modeled as groupoids.<ref>[https://www.crcpress.com/An-Introduction-to-Groups-Groupoids-and-Their-Representations/Ibort-Rodriguez/p/book/9781138035867 An Introduction to Groups, Groupoids and Their Representations: An Introduction]; Alberto Ibort, Miguel A. Rodriguez; CRC Press, 2019.</ref>

The transformations of the [[fifteen puzzle]] form a groupoid (not a group, as not all moves can be composed).<ref>Jim Belk (2008) [https://cornellmath.wordpress.com/2008/01/27/puzzles-groups-and-groupoids/ Puzzles, Groups, and Groupoids], The Everything Seminar</ref><ref>[http://www.neverendingbooks.org/the-15-puzzle-groupoid-1 The 15-puzzle groupoid (1)] {{Webarchive|url=https://web.archive.org/web/20151225220110/http://www.neverendingbooks.org/the-15-puzzle-groupoid-1 |date=2015-12-25 }}, Never Ending Books</ref><ref>[http://www.neverendingbooks.org/the-15-puzzle-groupoid-2 The 15-puzzle groupoid (2)] {{Webarchive|url=https://web.archive.org/web/20151225210035/http://www.neverendingbooks.org/the-15-puzzle-groupoid-2 |date=2015-12-25 }}, Never Ending Books</ref> This [[Group action (mathematics)#Variants and generalizations|groupoid acts]] on configurations.

=== Mathieu groupoid ===

The [[Mathieu groupoid]] is a groupoid introduced by [[John Horton Conway]] acting on 13 points such that the elements fixing a point form a copy of the [[Mathieu group]] M<sub>12</sub>.