Editing Groupoid

{{Short description|Category where every morphism is invertible; generalization of a group}}
{{About|groupoids in category theory|the algebraic structure with a single binary operation|magma (algebra)}}
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In [[mathematics]], especially in [[category theory]] and [[homotopy theory]], a '''groupoid''' (less often '''Brandt groupoid''' or '''virtual group''') generalises the notion of [[group (mathematics)|group]] in several equivalent ways. A groupoid can be seen as a:
* ''[[group (mathematics)|Group]]'' with a [[partial function]] replacing the [[binary operation]];
* ''[[category theory|Category]]'' in which every [[morphism]] is invertible. A category of this sort can be viewed as augmented with a [[unary operation]] on the morphisms, called ''inverse'' by analogy with [[group theory]].<ref name="dicks-ventura-96">{{cite book|author=Dicks & Ventura|year=1996|title=The Group Fixed by a Family of Injective Endomorphisms of a Free Group|url={{Google books|plainurl=y|id=3sWSRRfNFKgC|page=6|text=G has the structure of a graph}}|page=6}}</ref> A groupoid where there is only one object is a usual group.

In the presence of [[Dependent type|dependent typing]], a category in general can be viewed as a typed [[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 {{tmath|1= g:A \rightarrow B }},  {{tmath|1= h:B \rightarrow C }}, say. Composition is then a total function: {{tmath|1= \circ : (B \rightarrow C) \rightarrow (A \rightarrow B) \rightarrow (A \rightarrow C) }}, so that {{tmath|1= h \circ g : A \rightarrow C }}.

Special cases include:
* ''[[Setoid]]s'': [[Set (mathematics)|sets]] that come with an [[equivalence relation]],
* ''[[G-set]]s'': sets equipped with an [[Group action (mathematics)|action]] of a group {{tmath|1= G }}.

Groupoids are often used to reason about [[geometrical]] objects such as [[manifold]]s. {{harvs|txt|first=Heinrich |last=Brandt|authorlink=Heinrich Brandt|year=1927}} introduced groupoids implicitly via [[Brandt semigroup]]s.<ref>{{SpringerEOM|title=Brandt semi-group|ISBN=1-4020-0609-8}}</ref>

== Definitions ==

=== Algebraic ===

A groupoid can be viewed as an algebraic structure  consisting of a set with a binary [[partial function]] {{Citation needed|reason=appears to contradict prominent sources such as MathWorld|date=July 2024}}.
Precisely, it is a non-empty set <math>G</math> with a [[unary operation]] {{tmath|1= {}^{-1} : G\to G }}, and a [[partial function]] {{tmath|1= *:G\times G \rightharpoonup G }}.  Here <math>*</math> is not a [[binary operation]] because it is not necessarily defined for all pairs of elements of {{tmath|1= G }}.  The precise conditions under which <math>*</math> is defined are not articulated here and vary by situation.

The operations <math>\ast</math> and <sup>−1</sup> have the following axiomatic properties:  For all {{tmath|1= a }}, {{tmath|1= b }}, and <math>c</math> in {{tmath|1= G }},
# ''[[Associativity]]'': If <math>a*b</math> and <math>b*c</math> are defined, then <math>(a * b) * c</math> and <math>a * (b * c)</math> are defined and are equal.  Conversely, if one of <math>(a * b) * c</math> or <math>a * (b * c)</math> is defined, then they are both defined (and they are equal to each other), and <math>a*b</math> and <math>b * c</math> are also defined.
# ''[[Multiplicative inverse|Inverse]]'': <math>a^{-1} * a</math> and <math>a*{a^{-1}}</math> are always defined.
# ''[[Identity element|Identity]]'': If <math>a * b</math> is defined, then {{tmath|1= a * b * {b^{-1} } = a }}, and {{tmath|1= {a^{-1} } * a * b = b }}.  (The previous two axioms already show that these expressions are defined and unambiguous.)

Two easy and convenient properties follow from these axioms:
* {{tmath|1= (a^{-1} )^{-1} = a }},
* If <math>a * b</math> is defined, then {{tmath|1= (a * b)^{-1} = b^{-1} * a^{-1} }}.<ref>

Proof of first property: from 2. and 3. we obtain ''a''<sup>−1</sup> = ''a''<sup>−1</sup> * ''a'' * ''a''<sup>−1</sup> and (''a''<sup>−1</sup>)<sup>−1</sup> = (''a''<sup>−1</sup>)<sup>−1</sup> * ''a''<sup>−1</sup> * (''a''<sup>−1</sup>)<sup>−1</sup>. Substituting the first into the second and applying 3. two more times yields (''a''<sup>−1</sup>)<sup>−1</sup> = (''a''<sup>−1</sup>)<sup>−1</sup> * ''a''<sup>−1</sup> * ''a'' * ''a''<sup>−1</sup> * (''a''<sup>−1</sup>)<sup>−1</sup> = (''a''<sup>−1</sup>)<sup>−1</sup> * ''a''<sup>−1</sup> * ''a'' = ''a''.  ✓ <br />
Proof of second property: since ''a'' * ''b'' is defined, so is (''a'' * ''b'')<sup>−1</sup> * ''a'' * ''b''.  Therefore (''a'' * ''b'')<sup>−1</sup> * ''a'' * ''b'' * ''b''<sup>−1</sup> = (''a'' * ''b'')<sup>−1</sup> * ''a'' is also defined.  Moreover since ''a'' * ''b'' is defined, so is ''a'' * ''b'' * ''b''<sup>−1</sup> = ''a''.  Therefore ''a'' * ''b'' * ''b''<sup>−1</sup> * ''a''<sup>−1</sup> is also defined.  From 3. we obtain (''a'' * ''b'')<sup>−1</sup> = (''a'' * ''b'')<sup>−1</sup> * ''a'' * ''a''<sup>−1</sup> = (''a'' * ''b'')<sup>−1</sup> * ''a'' * ''b'' * ''b''<sup>−1</sup> * ''a''<sup>−1</sup> = ''b''<sup>−1</sup> * ''a''<sup>−1</sup>.  ✓</ref>

=== Category-theoretic ===
A groupoid is a [[category (mathematics)#Small and large categories|small category]] in which every [[morphism]] is an [[isomorphism]], i.e., invertible.<ref name="dicks-ventura-96"/> More explicitly, a groupoid <math>G</math> is a set <math>G_0</math> 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 <math>\mathrm{id}_x</math> of ''G''(''x'', ''x'');
* for each triple of objects ''x'', ''y'', and ''z'', a [[function (mathematics)|function]] {{tmath|1= \mathrm{comp}_{x,y,z} : G(y, z)\times G(x, y) \rightarrow G(x, z): (g, f) \mapsto gf }};
* for each pair of objects ''x'', ''y'', a function <math>\mathrm{inv}: G(x, y) \rightarrow G(y, x): f \mapsto f^{-1}</math> satisfying, for any ''f'' : ''x'' → ''y'', ''g'' : ''y'' → ''z'', and ''h'' : ''z'' → ''w'':
** {{tmath|1= f\ \mathrm{id}_x = f }} and {{tmath|1= \mathrm{id}_y\ f = f }};
** {{tmath|1= (h g) f = h (g f) }};
** <math>f f^{-1} = \mathrm{id}_y</math> and {{tmath|1= f^{-1} f = \mathrm{id}_x }}.

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 {{tmath|1= G_1 \rightrightarrows G_0 }}, where <math>G_1</math> is the set of all morphisms, and the two arrows <math>G_1 \to G_0</math> 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]] of all of the sets ''G''(''x'',''y'') (i.e. the sets of morphisms from ''x'' to ''y''). Then <math>\mathrm{comp}</math> and <math>\mathrm{inv}</math> become partial operations on ''G'', and <math>\mathrm{inv}</math> will in fact be defined everywhere. We define ∗ to be <math>\mathrm{comp}</math> and <sup>−1</sup> to be {{tmath|1= \mathrm{inv} }}, which gives a groupoid in the algebraic sense. Explicit reference to ''G''<sub>0</sub> (and hence to {{tmath|1= \mathrm{id} }}) can be dropped.

Conversely, given a groupoid ''G'' in the algebraic sense, define an equivalence relation <math>\sim</math> on its elements by
<math>a \sim b</math> iff ''a'' ∗ ''a''<sup>−1</sup> = ''b'' ∗ ''b''<sup>−1</sup>. Let ''G''<sub>0</sub> be the set of equivalence classes of {{tmath|1= \sim }}, i.e. {{tmath|1= G_0:=G/\!\!\sim }}. Denote  ''a'' ∗ ''a''<sup>−1</sup> by <math>1_x</math> if <math>a\in G</math> with {{tmath|1= x\in G_0 }}.

Now define <math>G(x, y)</math> as the set of all elements ''f'' such that <math>1_x*f*1_y</math> exists. Given <math>f \in G(x,y)</math> and {{tmath|1= g \in G(y, z) }}, their composite is defined as {{tmath|1= gf:=f*g \in G(x,z) }}. To see that this is well defined, observe that since <math>(1_x*f)*1_y</math> and <math>1_y*(g*1_z)</math> exist, so does {{tmath|1= (1_x*f*1_y)*(g*1_z)=f*g }}. The identity morphism on ''x'' is then {{tmath|1= 1_x }}, and the category-theoretic inverse of ''f'' is ''f''<sup>−1</sup>.

Sets in the definitions above may be replaced with [[class (set theory)|class]]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 <math>x \in X</math> is given by the set <math>s(t^{-1}(x)) \subseteq X</math> containing every point that can be joined to x by a morphism in G. If two points <math>x</math> and <math>y</math> are in the same orbits, their vertex groups <math>G(x)</math> and <math>G(y)</math> are [[group isomorphism|isomorphic]]: if <math>f</math> is any morphism from <math>x</math> to {{tmath|1= y }}, then the isomorphism is given by the mapping {{tmath|1= g\to fgf^{-1} }}.

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 (category theory)|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 [[Groupoid#Examples|below]] for counterexamples).

=== Subgroupoids and morphisms ===
A '''subgroupoid''' of <math>G \rightrightarrows X</math> is a [[subcategory]] <math>H \rightrightarrows Y</math> that is itself a groupoid. It is called '''wide''' or '''full''' if it is [[Wide subcategory|wide]] or [[Full subcategory|full]] as a subcategory, i.e., respectively, if <math>X = Y</math> or <math>G(x,y)=H(x,y)</math> for every {{tmath|1= x,y \in Y }}.

A '''groupoid morphism''' is simply a functor between two (category-theoretic) groupoids.

Particular kinds of morphisms of groupoids are of interest. A morphism <math>p: E \to B</math> of groupoids is called a [[fibration]] if for each object <math>x</math> of <math>E</math> and each morphism <math>b</math> of <math>B</math> starting at <math>p(x)</math> there is a morphism <math>e</math> of <math>E</math> starting at <math>x</math> such that {{tmath|1= p(e)=b }}. A fibration is called a [[covering morphism]] or [[covering of groupoids]] if further such an <math>e</math> is unique. The covering morphisms of groupoids are especially useful because they can be used to model [[covering map]]s of spaces.<ref>J.P. May, ''A Concise Course in Algebraic Topology'', 1999, The University of Chicago Press {{ISBN|0-226-51183-9}} (''see chapter 2'')</ref>

It is also true that the category of covering morphisms of a given groupoid <math>B</math> is equivalent to the category of actions of the groupoid <math>B</math> on sets.

== 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>.

== Relation to groups ==
{{Group-like structures}}
If a groupoid has only one object, then the set of its morphisms forms a [[group (algebra)|group]]. Using the algebraic definition, such a groupoid is literally just a group.<ref>Mapping a group to the corresponding groupoid with one object is sometimes called delooping, especially in the context of [[Homotopy|homotopy theory]], see {{cite web |url=https://ncatlab.org/nlab/show/delooping#delooping_of_a_group_to_a_groupoid |title=delooping in nLab |website=ncatlab.org |access-date=2017-10-31 }}.</ref> Many concepts of [[group theory]] generalize to groupoids, with the notion of [[functor]] replacing that of [[group homomorphism]].

Every transitive/connected groupoid - that is, as explained above, one in which any two objects are connected by at least one morphism - is isomorphic to an action groupoid (as defined above) {{tmath|1= (G, X) }}. By transitivity, there will only be one [[orbit (group theory)|orbit]] under the action.

Note that the isomorphism just mentioned is not unique, and there is no [[natural equivalence|natural]] choice. Choosing such an isomorphism for a transitive groupoid essentially amounts to picking one object {{tmath|1= x_0 }}, a [[group isomorphism]] <math>h</math> from <math>G(x_0)</math> to {{tmath|1= G }}, and for each <math>x</math> other than {{tmath|1= x_0 }}, a morphism in <math>G</math> from <math>x_0</math> to {{tmath|1= x }}.

If a groupoid is not transitive, then it is isomorphic to a [[disjoint union]] of groupoids of the above type, also called its '''connected components''' (possibly with different groups <math>G</math> and sets <math>X</math> for each connected component).

In category-theoretic terms, each connected component of a groupoid is [[equivalent categories|equivalent]] (but not [[isomorphic categories|isomorphic]]) to a groupoid with a single object, that is, a single group. Thus any groupoid is equivalent to a [[multiset]] of unrelated groups. In other words, for equivalence instead of isomorphism, one does not need to specify the sets {{tmath|1= X }}, but only the groups {{tmath|1= G }}. For example,

*The fundamental groupoid of <math>X</math> is equivalent to the collection of the [[fundamental group]]s of each [[path-connected component]] of {{tmath|1= X }}, but an isomorphism requires specifying the set of points in each component;
*The set <math>X</math> with the equivalence relation <math>\sim</math> is equivalent (as a groupoid) to one copy of the [[trivial group]] for each [[equivalence class]], but an isomorphism requires specifying what each equivalence class is;
*The set <math>X</math> equipped with an [[Group action (mathematics)|action]] of the group <math>G</math> is equivalent (as a groupoid) to one copy of <math>G</math> for each [[orbit (group theory)|orbit]] of the action, but an [[isomorphism]] requires specifying what set each orbit is.

The collapse of a groupoid into a mere collection of groups loses some information, even from a category-theoretic point of view, because it is not [[natural (category theory)|natural]]. Thus when groupoids arise in terms of other structures, as in the above examples, it can be helpful to maintain the entire groupoid. Otherwise, one must choose a way to view each <math>G(x)</math> in terms of a single group, and this choice can be arbitrary. In the example from [[topology]], one would have to make a coherent choice of paths (or equivalence classes of paths) from each point <math>p</math> to each point <math>q</math> in the same path-connected component.

As a more illuminating example, the classification of groupoids with one [[endomorphism]] does not reduce to purely group theoretic considerations. This is analogous to the fact that the classification of [[vector space]]s with one endomorphism is nontrivial.

Morphisms of groupoids come in more kinds than those of groups: we have, for example, [[fibration]]s, [[covering morphism]]s, [[universal morphism]]s, and [[quotient morphism]]s. Thus a subgroup <math>H</math> of a group <math>G</math> yields an action of <math>G</math> on the set of [[coset]]s of <math>H</math> in <math>G</math> and hence a covering morphism <math>p</math> from, say, <math>K</math> to {{tmath|1= G }}, where <math>K</math> is a groupoid with [[#Vertex groups and orbits|vertex group]]s isomorphic to {{tmath|1= H }}. In this way, presentations of the group <math>G</math> can be "lifted" to presentations of the groupoid {{tmath|1= K }}, and this is a useful way of obtaining information about presentations of the subgroup {{tmath|1= H }}. For further information, see the books by Higgins and by Brown in the References.

== Category of groupoids ==
The category whose objects are groupoids and whose morphisms are groupoid morphisms is called the '''groupoid category''', or the '''category of groupoids''', and is denoted by '''Grpd'''.

The category '''Grpd''' is, like the category of small categories, [[Cartesian closed]]: for any groupoids <math>H,K</math> we can construct a groupoid <math>\operatorname{GPD}(H,K)</math> whose objects are the morphisms <math> H \to K </math> and whose arrows are the natural equivalences of morphisms. Thus if <math> H,K </math> are just groups, then such arrows are the conjugacies of morphisms. The main result is that for any groupoids <math> G,H,K </math> there is a natural bijection
: <math>\operatorname{Grpd}(G \times H, K) \cong \operatorname{Grpd}(G, \operatorname{GPD}(H,K)).</math>

This result is of interest even if all the groupoids <math> G,H,K </math> are just groups.

Another important property of '''Grpd''' is that it is both [[Complete category|complete]] and [[Cocomplete category|cocomplete]].

=== Relation to [[Category of small categories|Cat]] ===

The inclusion <math>i : \mathbf{Grpd} \to \mathbf{Cat}</math> has both a left and a right [[Adjoint functors|adjoint]]:
: <math> \hom_{\mathbf{Grpd}}(C[C^{-1}], G) \cong \hom_{\mathbf{Cat}}(C, i(G)) </math>
: <math> \hom_{\mathbf{Cat}}(i(G), C) \cong \hom_{\mathbf{Grpd}}(G, \mathrm{Core}(C)) </math>

Here, <math>C[C^{-1}]</math> denotes the [[localization of a category]] that inverts every morphism, and <math>\mathrm{Core}(C)</math> denotes the subcategory of all isomorphisms.

=== Relation to [[Simplicial set|sSet]] ===

The [[Nerve (category theory)|nerve functor]] <math>N :  \mathbf{Grpd} \to \mathbf{sSet}</math> embeds '''Grpd''' as a full subcategory of the category of simplicial sets. The nerve of a groupoid is always a [[Kan complex]].

The nerve has a left adjoint
: <math> \hom_{\mathbf{Grpd}}(\pi_1(X), G) \cong \hom_{\mathbf{sSet}}(X, N(G)) </math>
Here, <math>\pi_1(X)</math> denotes the fundamental groupoid of the simplicial set {{tmath|1= X }}.

=== Groupoids in Grpd ===
{{Main|Double groupoid}}
There is an additional structure which can be derived from groupoids internal to the category of groupoids, '''double-groupoids'''.<ref>{{cite arXiv|last1=Cegarra|first1=Antonio M.|last2=Heredia|first2=Benjamín A.|last3=Remedios|first3=Josué|date=2010-03-19|title=Double groupoids and homotopy 2-types|class=math.AT|eprint=1003.3820}}</ref><ref>{{Cite journal|last=Ehresmann|first=Charles|date=1964|title=Catégories et structures : extraits|url=http://www.numdam.org/item/?id=SE_1964__6__A8_0|journal=Séminaire Ehresmann. Topologie et géométrie différentielle|language=en|volume=6|pages=1–31}}</ref> Because '''Grpd''' is a 2-category, these objects form a 2-category instead of a 1-category since there is extra structure. Essentially, these are groupoids <math>\mathcal{G}_1,\mathcal{G}_0</math> with functors<blockquote><math>s,t: \mathcal{G}_1 \to \mathcal{G}_0</math></blockquote>and an embedding given by an identity functor<blockquote><math>i:\mathcal{G}_0 \to\mathcal{G}_1</math></blockquote>One way to think about these 2-groupoids is they contain objects, morphisms, and squares which can compose together vertically and horizontally. For example, given squares<blockquote><math>\begin{matrix}
\bullet & \to & \bullet \\
\downarrow & & \downarrow \\
\bullet & \xrightarrow{a} & \bullet
\end{matrix}
</math> and <math>\begin{matrix}
\bullet & \xrightarrow{a} & \bullet \\
\downarrow & & \downarrow \\
\bullet & \to & \bullet
\end{matrix}</math></blockquote>with <math>a</math> the same morphism, they can be vertically conjoined giving a diagram<blockquote><math>\begin{matrix}
\bullet & \to & \bullet \\
\downarrow & & \downarrow \\
\bullet & \xrightarrow{a} & \bullet \\
\downarrow & & \downarrow \\
\bullet & \to & \bullet
\end{matrix}</math></blockquote>which can be converted into another square by composing the vertical arrows. There is a similar composition law for horizontal attachments of squares.

== Groupoids with geometric structures ==

When studying geometrical objects, the arising groupoids often carry a [[Topological space|topology]], turning them into [[topological groupoid]]s, or even some [[differentiable structure]], turning them into [[Lie groupoid]]s. These last objects can be also studied in terms of their associated [[Lie algebroid]]s, in analogy to the relation between [[Lie group]]s and [[Lie algebra]]s.

Groupoids arising from geometry often possess further structures which interact with the groupoid multiplication. For instance, in [[Poisson geometry]] one has the notion of a [[symplectic groupoid]], which is a Lie groupoid endowed with a compatible [[Symplectic manifold|symplectic form]]. Similarly, one can have groupoids with a compatible [[Riemannian metric]], or [[Complex manifold|complex structure]], etc.

== See also ==
* [[∞-groupoid]]
* [[2-group]]
* [[Homotopy type theory]]
* [[Inverse category]]
* [[Groupoid algebra]] (not to be confused with [[algebraic groupoid]])
* [[R-algebroid]]

== Notes ==
{{reflist}}

== References ==
* {{citation |title=Über eine Verallgemeinerung des Gruppenbegriffes |journal=Mathematische Annalen |volume=96 |issue=1 |pages=360–366 |year=1927 |doi=10.1007/BF01209171 |first=H |last=Brandt |s2cid=119597988 
}}
* Brown, Ronald, 1987, "[https://groupoids.org.uk/pdffiles/groupoidsurvey.pdf From groups to groupoids: a brief survey]", ''Bull. London Math. Soc.'' '''19''': 113–34. Reviews the history of groupoids up to 1987, starting with the work of Brandt on quadratic forms. The downloadable version updates the many references.
* &mdash;, 2006. ''[http://arquivo.pt/wayback/20160514115224/http://www.bangor.ac.uk/r.brown/topgpds.html Topology and groupoids.]'' Booksurge. Revised and extended edition of a book previously published in 1968 and 1988. Groupoids are introduced in the context of their topological application.
* &mdash;, [https://groupoids.org.uk/hdaweb2.html Higher dimensional group theory.]  Explains how the groupoid concept has led to higher-dimensional homotopy groupoids, having applications in [[homotopy theory]] and in group [[cohomology]]. Many references.
* {{citation |last1=Dicks |first1=Warren  |last2=Ventura |first2=Enric |title=The group fixed by a family of injective endomorphisms of a free group |series=Mathematical Surveys and Monographs |volume=195 |year=1996 |publisher=AMS Bookstore |isbn=978-0-8218-0564-0 }}
* {{cite journal |last1=Dokuchaev |first1=M. |last2=Exel |first2=R. |last3=Piccione |first3=P. |year=2000 |title=Partial Representations and Partial Group Algebras |journal=Journal of Algebra |volume=226 |pages=505–532 |publisher=Elsevier |issn=0021-8693 |doi= 10.1006/jabr.1999.8204|arxiv= math/9903129
|s2cid=14622598 }}
* F. Borceux, G. Janelidze, 2001, ''[https://archive.today/20121223050454/http://www.cup.cam.ac.uk/catalogue/catalogue.asp?isbn=9780521803090 Galois theories.]'' Cambridge Univ. Press. Shows how generalisations of [[Galois theory]] lead to [[Galois groupoid]]s.
* [[Ana Cannas da Silva|Cannas da Silva, A.]], and [[Alan Weinstein|A. Weinstein]], ''[http://www.math.ist.utl.pt/~acannas/Books/models_final.pdf Geometric Models for Noncommutative Algebras.]'' Especially Part VI.
* [[Marty Golubitsky|Golubitsky, M.]], Ian Stewart, 2006, "[https://www.ams.org/bull/2006-43-03/S0273-0979-06-01108-6/S0273-0979-06-01108-6.pdf Nonlinear dynamics of networks: the groupoid formalism]", ''Bull. Amer. Math. Soc.'' '''43''': 305–64
* {{springer|title=Groupoid|id=p/g045360}}
* Higgins, P. J., "The fundamental groupoid of a [[graph of groups]]", J. London Math. Soc. (2) 13 (1976) 145–149.
* Higgins, P. J. and Taylor, J., "The fundamental groupoid and the homotopy crossed complex of   an [[orbit space]]",  in Category theory (Gummersbach, 1981), Lecture Notes   in Math., Volume 962. Springer, Berlin (1982), 115–122.
*Higgins, P. J., 1971.  ''Categories and groupoids''. Van Nostrand Notes in Mathematics. Republished in ''Reprints in Theory and Applications of Categories'', No. 7 (2005) pp.&nbsp;1–195; [http://www.tac.mta.ca/tac/reprints/articles/7/tr7abs.html freely downloadable]. Substantial introduction to [[category theory]] with special emphasis on groupoids. Presents applications of groupoids in group theory, for example to a generalisation of [[Grushko's theorem]], and in topology, e.g. [[fundamental groupoid]].
* Mackenzie, K. C. H., 2005. ''[https://web.archive.org/web/20050310034123/http://www.shef.ac.uk/~pm1kchm/gt.html General theory of Lie groupoids and Lie algebroids]''. Cambridge Univ. Press.
* Weinstein, Alan, "[https://www.ams.org/notices/199607/weinstein.pdf Groupoids: unifying internal and external symmetry &ndash; A tour through some examples]". Also available in [http://math.berkeley.edu/~alanw/Groupoids.ps Postscript], Notices of the AMS, July 1996, pp.&nbsp;744–752.
* Weinstein, Alan, "[https://arxiv.org/abs/math/0208108 The Geometry of Momentum]" (2002)
* R.T. Zivaljevic. "Groupoids in combinatorics&mdash;applications of a theory of local symmetries". In ''Algebraic and geometric combinatorics'', volume 423 of  ''Contemp. Math''., 305–324.  Amer. Math. Soc., Providence, RI (2006)
* {{nlab|id=fundamental+groupoid|title=fundamental groupoid}}
* {{nlab|id=core|title=core}}

[[Category:Algebraic structures]]
[[Category:Category theory]]
[[Category:Homotopy theory]]