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{{Short description|Category where every morphism is invertible; generalization of a group}}
{{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)}}
{{about|groupoids in category theory|the algebraic structure with a single binary operation|magma (algebra)}}
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Special cases include:
Special cases include:
* ''[[Setoid]]s'': [[Set (mathematics)|sets]] that come with an [[equivalence relation]],
* ''[[Setoid]]'': a [[Set (mathematics)|set]] that comes with an [[equivalence relation]],
* ''[[G-set]]s'': sets equipped with an [[Group action (mathematics)|action]] of a group {{tmath|1= G }}.
* ''[[G-set]]'': a set 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>
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 ==
== Definitions ==
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=== Algebraic ===
=== 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}}.
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.
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 }},
The operations <math>\ast</math> and {{tmath| {}^{-1} }} have the following axiomatic properties:  For all {{tmath| a }}, {{tmath| 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.
# ''[[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.
# ''[[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.)
# ''[[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:
Two convenient properties follow from these axioms:
* {{tmath|1= (a^{-1} )^{-1} = a }},
* {{tmath|1= (a^{-1} )^{-1} = a }},
* If <math>a * b</math> is defined, then {{tmath|1= (a * b)^{-1} = b^{-1} * a^{-1} }}.<ref>
* 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 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>
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>
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=== Category-theoretic ===
=== 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
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 each pair of objects {{tmath| x }} and {{tmath| y }}, a (possibly empty) set {{tmath| G(x, y) }} of ''morphisms'' (or ''arrows'') from {{tmath| x }} to {{tmath| y}}; we write {{tmath| f : x \to y }} to indicate that {{tmath| f }} is an element of {{tmath| G(x, y) }};
* for every object ''x'', a designated element <math>\mathrm{id}_x</math> of ''G''(''x'', ''x'');
* for each triple of objects {{tmath| x }}, {{tmath| y }}, and {{tmath| z }}, a [[function (mathematics)|function]] <math>\mathrm{comp}_{x,y,z} : G(y, z)\times G(x, y) \rightarrow G(x, z) </math> <math> (g, f) \mapsto gf </math> that is [[associative]]. That is, for every four objects {{tmath| x }}, {{tmath| y }}, {{tmath| z }}, {{tmath| w }} and functions {{tmath| f : x \to y, g : y \to z, h : z \to w }}
* 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 }};
** <math> h(g f) = (h g)f </math>
* 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'':
* for every object {{tmath| x }}, a designated element <math>\mathrm{id}_x</math> of {{tmath| G(x,x) }} satisfying, for any morphism {{tmath| f : x \to y }}
** {{tmath|1= f\ \mathrm{id}_x = f }} and {{tmath|1= \mathrm{id}_y\ f = f }};
** {{tmath|1= f\ \mathrm{id}_x = f }} and {{tmath|1= \mathrm{id}_y\ f = f }};
** {{tmath|1= (h g) f = h (g f) }};
* for each pair of objects {{tmath| x }}, {{tmath| y }}, a function <math>\mathrm{inv}: G(x, y) \rightarrow G(y, x): f \mapsto f^{-1}</math> satisfying, for any {{tmath| f : x \to y }}:
** <math>f f^{-1} = \mathrm{id}_y</math> and {{tmath|1= f^{-1} f = \mathrm{id}_x }}.
** <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'').
If the requirement that inverses exist is removed while keeping everything else, this is then the definition of a category. Thus, a groupoid is a category in which every morphism has an inverse.
 
If {{tmath| f }} is an element of {{tmath| G(x, y) }}, then {{tmath| x }} is called the '''source''' of {{tmath| f }}, written {{tmath| s(f) }}, and {{tmath| y }} is called the '''target''' of {{tmath| f }}, written {{tmath| 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.
A groupoid {{tmath| G }} is sometimes denoted as {{tmath| 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.
More generally, one can consider a [[groupoid object]] in an arbitrary category admitting finite fiber products.


=== Comparing the definitions ===
=== 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.
The algebraic and category-theoretic definitions are equivalent, as we now show. Given a groupoid in the category-theoretic sense, let {{tmath| G }} be the [[disjoint union]] of all of the sets {{tmath| G(x, y) }} (i.e. the sets of morphisms from {{tmath| x }} to {{tmath| y }}). Then <math>\mathrm{comp}</math> and <math>\mathrm{inv}</math> become partial operations on {{tmath| G }}, and <math>\mathrm{inv}</math> will in fact be defined everywhere. We define {{tmath| * }} to be <math>\mathrm{comp}</math> and {{tmath| {}^{-1} }} to be {{tmath| \mathrm{inv} }}, which gives a groupoid in the algebraic sense. Explicit reference to {{tmath| G_0 }} (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
Conversely, given a groupoid {{tmath| 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 }}.
<math>a \sim b</math> iff {{tmath|1= a * a^{-1} = b * b^{-1} }}. Let {{tmath| G_0 }} be the set of equivalence classes of {{tmath|1= \sim }}, i.e. {{tmath|1= G_0:=G/\!\!\sim }}. Denote  {{tmath| a * a^{-1} }} 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>.
Now define <math>G(x, y)</math> as the set of all elements {{tmath| 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 {{tmath| x }} is then {{tmath|1= 1_x }}, and the category-theoretic inverse of {{tmath| f }} is {{tmath| f^{-1} }}.


Sets in the definitions above may be replaced with [[class (set theory)|class]]es, as is generally the case in category theory.
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 ===
=== 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.
Given a groupoid {{tmath| G }}, the '''vertex groups''' or '''isotropy groups''' or '''object groups''' in {{tmath| G }} are the subsets of the form {{tmath| G(x, x) }}, where {{tmath| x }} is any object of {{tmath| 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} }}.
The '''orbit''' of a groupoid {{tmath| 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 {{tmath| x }} by a morphism in {{tmath| 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).
Orbits form a partition of the set {{tmath| 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 ===
=== Subgroupoids and morphisms ===
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A '''groupoid morphism''' is simply a functor between two (category-theoretic) groupoids.
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>
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>{{citation| first1=J.P. |last1=May |author-link1=J. Peter May |title=A Concise Course in Algebraic Topology |date=1999 |publisher=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.
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 ==
== Examples ==
Every [[Group (mathematics)|group]] is a groupoid.


=== Fundamental groupoid ===
=== Fundamental groupoid ===
{{main|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]].
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 }}.
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 }}.
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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:
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 }};
* 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 }};
* 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 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:
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 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).
* 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).
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=== Čech groupoid ===
=== Čech groupoid ===
{{See also|Simplicial manifold|Nerve of a covering}}
{{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
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>
<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>
and its arrows are the intersections <math display=block>\mathcal{G}_1 = \coprod U_{ij} .</math>
Line 113: Line 115:
\downarrow & & \downarrow \\
\downarrow & & \downarrow \\
U_{ik} & \to & U_{i}
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.
\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|{{tmath| \infty }}-groupoid]]s. Also, another artifact of this construction is [[Čech cohomology|{{tmath| 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 ===
=== Group action ===
{{main|action groupoid}}
{{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:
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 }};
* The objects are the elements of {{tmath|1= X }};
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==== Finite set ====
==== 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}}
Consider the group action of <math>\Z/2\Z</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>\Z/2\Z</math> on it.{{fact|date=May 2025}}


==== Quotient variety ====
==== Quotient variety ====
Any finite group <math>
Any finite group <math>G</math> that maps to <math>\mathrm{GL}(n)</math> gives a group action on the [[affine space]] <math>\mathbb{A}^n
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.
</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 ===
=== Inertia groupoid ===
{{main|Inertia groupoid}}
{{main|Inertia groupoid}}
The inertia groupoid of a groupoid is roughly a groupoid of loops in the given groupoid.
The inertia groupoid of a groupoid is roughly a groupoid of loops in the given groupoid.


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=== Puzzles ===
=== 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>
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.
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.
Line 174: Line 173:


== Relation to groups ==
== Relation to groups ==
{{Group-like structures}}
{{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]].
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]].


Line 215: Line 214:
=== Relation to [[Simplicial set|sSet]] ===
=== 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 (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
The nerve has a left adjoint
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=== Groupoids in Grpd ===
=== Groupoids in Grpd ===
{{Main|Double groupoid}}
{{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}
 
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=https://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 \\
\bullet & \to & \bullet \\
\downarrow & & \downarrow \\
\downarrow & & \downarrow \\
Line 260: Line 260:
* {{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  
* {{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.
* {{citation |last1=Brown |first1=Ronald |date=1987 |url=https://groupoids.org.uk/pdffiles/groupoidsurvey.pdf |title=From groups to groupoids: a brief survey |journal=Bull. London Math. Soc. |volume=19 |pages=113–134 }} – 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;, 2006. ''[https://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.
* &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 }}
* {{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
* {{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 }}
|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.
* {{citation |first1=F. |last1=Borceux |first2=G. |last2=Janelidze |date=2001 |url=https://archive.today/20121223050454/http://www.cup.cam.ac.uk/catalogue/catalogue.asp?isbn=9780521803090 |title=Galois theories |publisher=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.
* {{citation |author1-link=Ana Cannas da Silva |last1=Cannas da Silva |first1=A. |author2-link=Alan Weinstein |first2=A. |last2=Weinstein |url=https://www.math.ist.utl.pt/~acannas/Books/models_final.pdf |title=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
* {{citation |author1-link=Marty Golubitsky |last1=Golubitsky |first1=M. |first2=Ian |last2=Stewart |date=2006 |url=https://www.ams.org/bull/2006-43-03/S0273-0979-06-01108-6/S0273-0979-06-01108-6.pdf |title=Nonlinear dynamics of networks: the groupoid formalism |journal=Bull. Amer. Math. Soc. |volume=43 |pages=305–364 }}
* {{springer|title=Groupoid|id=p/g045360}}
* {{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.
* {{citation |last1=Higgins |first1=P. J. |title=The fundamental groupoid of a [[graph of groups]] |journal=J. London Math. Soc. |volume=2 |number=13 |date=1976 |pages=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. and Taylor, J. (1982), "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, 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]].
*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.
* {{citation |last1=Mackenzie |first1=K. C. H. |date=2005 |url=https://web.archive.org/web/20050310034123/http://www.shef.ac.uk/~pm1kchm/gt.html |title=General theory of Lie groupoids and Lie algebroids |publisher=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.
* {{citation |last1=Weinstein |first1=Alan |url=https://www.ams.org/notices/199607/weinstein.pdf |title=Groupoids: unifying internal and external symmetry &ndash; A tour through some examples }} – Also available in [https://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)
* Weinstein, Alan (2002), "[https://arxiv.org/abs/math/0208108 The Geometry of Momentum]"
* 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)
* R.T. Zivaljevic (2006). "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
* {{nlab|id=fundamental+groupoid|title=fundamental groupoid}}
* {{nlab|id=fundamental+groupoid|title=fundamental groupoid}}
* {{nlab|id=core|title=core}}
* {{nlab|id=core|title=core}}

Latest revision as of 05:37, 17 May 2026

This article is about groupoids in category theory. For the algebraic structure with a single binary operation, see magma (algebra).

In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:

In the presence of 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 Template:Tmath, Template:Tmath, say. Composition is then a total function: Template:Tmath, so that Template:Tmath.

Special cases include:

Groupoids are often used to reason about geometrical objects such as manifolds. Template:Harvs introduced groupoids implicitly via Brandt semigroups.[2]

Definitions

Algebraic

A groupoid can be viewed as an algebraic structure consisting of a set with a binary partial function.[citation needed] Precisely, it is a non-empty set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} with a unary operation Template:Tmath, and a partial function Template:Tmath. Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle *} is not a binary operation because it is not necessarily defined for all pairs of elements of Template:Tmath. The precise conditions under which Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle *} is defined are not articulated here and vary by situation.

The operations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ast} and Template:Tmath have the following axiomatic properties: For all Template:Tmath, Template:Tmath, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} in Template:Tmath,

  1. Associativity: If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a*b} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b*c} are defined, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a * b) * c} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a * (b * c)} are defined and are equal. Conversely, if one of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a * b) * c} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a * (b * c)} is defined, then they are both defined (and they are equal to each other), and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a*b} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b * c} are also defined.
  2. Inverse: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1} * a} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a*{a^{-1}}} are always defined.
  3. Identity: If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a * b} is defined, then Template:Tmath, and Template:Tmath. (The previous two axioms already show that these expressions are defined and unambiguous.)

Two convenient properties follow from these axioms:

  • Template:Tmath,
  • If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a * b} is defined, then Template:Tmath.[3]

Category-theoretic

A groupoid is a small category in which every morphism is an isomorphism, i.e., invertible.[1] More explicitly, a groupoid Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is a set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_0} of objects with

  • for each pair of objects Template:Tmath and Template:Tmath, a (possibly empty) set Template:Tmath of morphisms (or arrows) from Template:Tmath to Template:Tmath; we write Template:Tmath to indicate that Template:Tmath is an element of Template:Tmath;
  • for each triple of objects Template:Tmath, Template:Tmath, and Template:Tmath, a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{comp}_{x,y,z} : G(y, z)\times G(x, y) \rightarrow G(x, z) } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (g, f) \mapsto gf } that is associative. That is, for every four objects Template:Tmath, Template:Tmath, Template:Tmath, Template:Tmath and functions Template:Tmath
    • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(g f) = (h g)f }
  • for every object Template:Tmath, a designated element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{id}_x} of Template:Tmath satisfying, for any morphism Template:Tmath
  • for each pair of objects Template:Tmath, Template:Tmath, a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{inv}: G(x, y) \rightarrow G(y, x): f \mapsto f^{-1}} satisfying, for any Template:Tmath:
    • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f f^{-1} = \mathrm{id}_y} and Template:Tmath.

If the requirement that inverses exist is removed while keeping everything else, this is then the definition of a category. Thus, a groupoid is a category in which every morphism has an inverse.

If Template:Tmath is an element of Template:Tmath, then Template:Tmath is called the source of Template:Tmath, written Template:Tmath, and Template:Tmath is called the target of Template:Tmath, written Template:Tmath.

A groupoid Template:Tmath is sometimes denoted as Template:Tmath, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_1} is the set of all morphisms, and the two arrows Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_1 \to G_0} 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 Template:Tmath be the disjoint union of all of the sets Template:Tmath (i.e. the sets of morphisms from Template:Tmath to Template:Tmath). Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{comp}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{inv}} become partial operations on Template:Tmath, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{inv}} will in fact be defined everywhere. We define Template:Tmath to be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{comp}} and Template:Tmath to be Template:Tmath, which gives a groupoid in the algebraic sense. Explicit reference to Template:Tmath (and hence to Template:Tmath) can be dropped.

Conversely, given a groupoid Template:Tmath in the algebraic sense, define an equivalence relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim} on its elements by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \sim b} iff Template:Tmath. Let Template:Tmath be the set of equivalence classes of Template:Tmath, i.e. Template:Tmath. Denote Template:Tmath by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1_x} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\in G} with Template:Tmath.

Now define Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(x, y)} as the set of all elements Template:Tmath such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1_x*f*1_y} exists. Given Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f \in G(x,y)} and Template:Tmath, their composite is defined as Template:Tmath. To see that this is well defined, observe that since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1_x*f)*1_y} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1_y*(g*1_z)} exist, so does Template:Tmath. The identity morphism on Template:Tmath is then Template:Tmath, and the category-theoretic inverse of Template:Tmath is Template:Tmath.

Sets in the definitions above may be replaced with classes, as is generally the case in category theory.

Vertex groups and orbits

Given a groupoid Template:Tmath, the vertex groups or isotropy groups or object groups in Template:Tmath are the subsets of the form Template:Tmath, where Template:Tmath is any object of Template:Tmath. 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 Template:Tmath at a point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} is given by the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s(t^{-1}(x)) \subseteq X} containing every point that can be joined to Template:Tmath by a morphism in Template:Tmath. If two points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} are in the same orbits, their vertex groups Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(x)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(y)} are isomorphic: if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is any morphism from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} to Template:Tmath, then the isomorphism is given by the mapping Template:Tmath.

Orbits form a partition of the set Template:Tmath, and a groupoid is called transitive if it has only one orbit (equivalently, if it is connected as a category). In that case, all the vertex groups are isomorphic (on the other hand, this is not a sufficient condition for transitivity; see the section below for counterexamples).

Subgroupoids and morphisms

A subgroupoid of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G \rightrightarrows X} is a subcategory Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H \rightrightarrows Y} that is itself a groupoid. It is called wide or full if it is wide or full as a subcategory, i.e., respectively, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = Y} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(x,y)=H(x,y)} for every Template:Tmath.

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

Particular kinds of morphisms of groupoids are of interest. A morphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p: E \to B} of groupoids is called a fibration if for each object Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} and each morphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} starting at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)} there is a morphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} starting at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} such that Template:Tmath. A fibration is called a covering morphism or covering of groupoids if further such an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e} is unique. The covering morphisms of groupoids are especially useful because they can be used to model covering maps of spaces.[4]

It is also true that the category of covering morphisms of a given groupoid Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} is equivalent to the category of actions of the groupoid Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} on sets.

Examples

Every group is a groupoid.

Fundamental groupoid

Main article: Fundamental groupoid

Given a topological space Template:Tmath, let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_0} be the set Template:Tmath. The morphisms from the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} to the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} are equivalence classes of continuous paths from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} to Template:Tmath, 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 Template:Tmath, denoted Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi_1(X)} (or sometimes, Template:Tmath).[5] The usual fundamental group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi_1(X,x)} is then the vertex group for the point Template:Tmath.

The orbits of the fundamental groupoid Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi_1(X)} are the path-connected components of Template:Tmath. Accordingly, the fundamental groupoid of a path-connected space is transitive, and we recover the known fact that the fundamental groups at any base point are isomorphic. Moreover, in this case, the fundamental groupoid and the fundamental groups are equivalent as categories (see the section below for the general theory).

An important extension of this idea is to consider the fundamental groupoid Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi_1(X,A)} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\subset X} is a chosen set of "base points". Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi_1(X,A)} is a (full) subgroupoid of Template:Tmath, where one considers only paths whose endpoints belong to Template:Tmath. The set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} may be chosen according to the geometry of the situation at hand.

Equivalence relation

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a setoid, i.e. a set with an equivalence relation Template:Tmath, then a groupoid "representing" this equivalence relation can be formed as follows:

  • The objects of the groupoid are the elements of Template:Tmath;
  • For any two elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} in Template:Tmath, there is a single morphism from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} (denote by Template:Tmath) if and only if Template:Tmath;
  • The composition of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (z,y)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (y,x)} is Template:Tmath.

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is in relation with every other element of Template:Tmath, we obtain the pair groupoid of Template:Tmath, which has the entire Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \times X} as set of arrows, and which is transitive.
  • If every element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is only in relation with itself, one obtains the unit groupoid, which has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} as set of arrows, Template:Tmath, and which is completely intransitive (every singleton Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{x\}} is an orbit).

Examples

  • If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f: X_0 \to Y} is a smooth surjective submersion of smooth manifolds, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_0\times_YX_0 \subset X_0\times X_0} is an equivalence relation[6] since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} has a topology isomorphic to the quotient topology of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_0} under the surjective map of topological spaces. If we write, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1 = X_0\times_YX_0} then we get a groupoid Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1 \rightrightarrows X_0,} 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 and Nerve of a covering

A Čech groupoid[6]p. 5 is a special kind of groupoid associated to an equivalence relation given by an open cover Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{U} = \{U_i\}_{i\in I}} of some manifold Template:Tmath. Its objects are given by the disjoint union Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{G}_0 = \coprod U_i ,} and its arrows are the intersections Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{G}_1 = \coprod U_{ij} .}

The source and target maps are then given by the induced maps

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} s = \phi_j: U_{ij} \to U_j\\ t = \phi_i: U_{ij} \to U_i \end{align}}

and the inclusion map

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon: U_i \to U_{ii}}

giving the structure of a groupoid. In fact, this can be further extended by setting

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{G}_n = \mathcal{G}_1\times_{\mathcal{G}_0} \cdots \times_{\mathcal{G}_0}\mathcal{G}_1}

as the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -iterated fiber product where the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{G}_n} represents Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -tuples of composable arrows. The structure map of the fiber product is implicitly the target map, since

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} U_{ijk} & \to & U_{ij} \\ \downarrow & & \downarrow \\ U_{ik} & \to & U_{i} \end{matrix}}

is a cartesian diagram where the maps to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_i} are the target maps. This construction can be seen as a model for some [[∞-groupoid|Template:Tmath-groupoid]]s. Also, another artifact of this construction is [[Čech cohomology|Template:Tmath-cocycles]]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\sigma] \in \check{H}^k(\mathcal{U}, \underline{A})}

for some constant sheaf of abelian groups can be represented as a function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma:\coprod U_{i_1\cdots i_k} \to A}

giving an explicit representation of cohomology classes.

Group action

Main article: action groupoid

If the group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} acts on the set Template:Tmath, then we can form the action groupoid (or transformation groupoid) representing this group action as follows:

  • The objects are the elements of Template:Tmath;
  • For any two elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} in Template:Tmath, the morphisms from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} correspond to the elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} such that Template:Tmath;
  • Composition of morphisms interprets the binary operation of Template:Tmath.

More explicitly, the action groupoid is a small category with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ob}(C)=X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{hom}(C)=G\times X} and with source and target maps Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s(g,x) = x} and Template:Tmath. It is often denoted Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G \ltimes X} (or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X\rtimes G} for a right action). Multiplication (or composition) in the groupoid is then Template:Tmath, which is defined provided Template:Tmath.

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} in Template:Tmath, the vertex group consists of those Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (g,x)} with Template:Tmath, which is just the isotropy subgroup at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} for the given action (which is why vertex groups are also called isotropy groups). Similarly, the orbits of the action groupoid are the orbit of the group action, and the groupoid is transitive if and only if the group action is transitive.

Another way to describe Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} -sets is the functor category Template:Tmath, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Gr}} is the groupoid (category) with one element and isomorphic to the group Template:Tmath. Indeed, every functor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} of this category defines a set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X=F(\mathrm{Gr})} and for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} (i.e. for every morphism in Template:Tmath) induces a bijection Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_g}  : Template:Tmath. The categorical structure of the functor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} assures us that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} defines a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} -action on the set Template:Tmath. The (unique) representable functor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F : \mathrm{Gr} \to \mathrm{Set}} is the Cayley representation of Template:Tmath. In fact, this functor is isomorphic to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Hom}(\mathrm{Gr},-)} and so sends Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ob}(\mathrm{Gr})} to the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Hom}(\mathrm{Gr},\mathrm{Gr})} which is by definition the "set" Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} and the morphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Gr}} (i.e. the element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} of Template:Tmath) to the permutation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_g} of the set Template:Tmath. We deduce from the Yoneda embedding that the group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is isomorphic to the group Template:Tmath, a subgroup of the group of permutations of Template:Tmath.

Finite set

Consider the group action of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Z/2\Z} on the finite set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = \{-2, -1, 0, 1, 2\}} where 1 acts by taking each number to its negative, so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 \mapsto 2} and Template:Tmath. The quotient groupoid Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [X/G]} is the set of equivalence classes from this group action Template:Tmath, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0]} has a group action of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Z/2\Z} on it.[citation needed]

Quotient variety

Any finite group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} that maps to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{GL}(n)} gives a group action on the affine space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{A}^n } (since this is the group of automorphisms). Then, a quotient groupoid can be of the form Template:Tmath, which has one point with stabilizer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G } at the origin. Examples like these form the basis for the theory of orbifolds. Another commonly studied family of orbifolds are weighted projective spaces Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{P}(n_1,\ldots, n_k)} and subspaces of them, such as Calabi–Yau orbifolds.

Inertia groupoid

Main article: 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} & & X \\ & & \downarrow \\ Y &\rightarrow & Z \end{align} }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:X\to Z} and Template:Tmath, we can form the groupoid Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X\times_ZY} whose objects are triples Template:Tmath, where Template:Tmath, Template:Tmath, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi: f(x) \to g(y)} in Template:Tmath. Morphisms can be defined as a pair of morphisms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\alpha,\beta)} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha: x \to x'} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta: y \to y'} such that for triples Template:Tmath, there is a commutative diagram in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z} of Template:Tmath, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(\beta):g(y) \to g(y')} and the Template:Tmath.[7]

Homological algebra

A two term complex

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_1 ~\overset{d}{\rightarrow}~ C_0}

of objects in a concrete Abelian category can be used to form a groupoid. It has as objects the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_0} and as arrows the set Template:Tmath; the source morphism is just the projection onto Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_0} while the target morphism is the addition of projection onto Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_1} composed with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} and projection onto Template:Tmath. That is, given Template:Tmath, we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t(c_1 + c_0) = d(c_1) + c_0.}

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.[8]

The transformations of the fifteen puzzle form a groupoid (not a group, as not all moves can be composed).[9][10][11] This 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 M12.

Relation to groups

Template:Group-like structures If a groupoid has only one object, then the set of its morphisms forms a group. Using the algebraic definition, such a groupoid is literally just a group.[12] 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) Template:Tmath. By transitivity, there will only be one orbit under the action.

Note that the isomorphism just mentioned is not unique, and there is no natural choice. Choosing such an isomorphism for a transitive groupoid essentially amounts to picking one object Template:Tmath, a group isomorphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h} from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(x_0)} to Template:Tmath, and for each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} other than Template:Tmath, a morphism in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} to Template:Tmath.

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} and sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} for each connected component).

In category-theoretic terms, each connected component of a groupoid is equivalent (but not 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 Template:Tmath, but only the groups Template:Tmath. For example,

  • The fundamental groupoid of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is equivalent to the collection of the fundamental groups of each path-connected component of Template:Tmath, but an isomorphism requires specifying the set of points in each component;
  • The set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} with the equivalence relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} equipped with an action of the group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is equivalent (as a groupoid) to one copy of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} for each 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. 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(x)} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} to each point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} 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 spaces with one endomorphism is nontrivial.

Morphisms of groupoids come in more kinds than those of groups: we have, for example, fibrations, covering morphisms, universal morphisms, and quotient morphisms. Thus a subgroup Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} of a group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} yields an action of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} on the set of cosets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} and hence a covering morphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} from, say, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} to Template:Tmath, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} is a groupoid with vertex groups isomorphic to Template:Tmath. In this way, presentations of the group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} can be "lifted" to presentations of the groupoid Template:Tmath, and this is a useful way of obtaining information about presentations of the subgroup Template:Tmath. 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H,K} we can construct a groupoid Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{GPD}(H,K)} whose objects are the morphisms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H \to K } and whose arrows are the natural equivalences of morphisms. Thus if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H,K } are just groups, then such arrows are the conjugacies of morphisms. The main result is that for any groupoids Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G,H,K } there is a natural bijection

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Grpd}(G \times H, K) \cong \operatorname{Grpd}(G, \operatorname{GPD}(H,K)).}

This result is of interest even if all the groupoids Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G,H,K } are just groups.

Another important property of Grpd is that it is both complete and cocomplete.

Relation to Cat

The inclusion Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i : \mathbf{Grpd} \to \mathbf{Cat}} has both a left and a right adjoint:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hom_{\mathbf{Grpd}}(C[C^{-1}], G) \cong \hom_{\mathbf{Cat}}(C, i(G)) }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hom_{\mathbf{Cat}}(i(G), C) \cong \hom_{\mathbf{Grpd}}(G, \mathrm{Core}(C)) }

Here, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C[C^{-1}]} denotes the localization of a category that inverts every morphism, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Core}(C)} denotes the subcategory of all isomorphisms.

Relation to sSet

The nerve functor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N : \mathbf{Grpd} \to \mathbf{sSet}} 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hom_{\mathbf{Grpd}}(\pi_1(X), G) \cong \hom_{\mathbf{sSet}}(X, N(G)) }

Here, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi_1(X)} denotes the fundamental groupoid of the simplicial set Template:Tmath.

Groupoids in Grpd

Main article: Double groupoid

There is an additional structure which can be derived from groupoids internal to the category of groupoids, double-groupoids.[13][14] 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{G}_1,\mathcal{G}_0} with functors

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s,t: \mathcal{G}_1 \to \mathcal{G}_0}

and an embedding given by an identity functor

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i:\mathcal{G}_0 \to\mathcal{G}_1}

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \bullet & \to & \bullet \\ \downarrow & & \downarrow \\ \bullet & \xrightarrow{a} & \bullet \end{matrix} } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \bullet & \xrightarrow{a} & \bullet \\ \downarrow & & \downarrow \\ \bullet & \to & \bullet \end{matrix}}

with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} the same morphism, they can be vertically conjoined giving a diagram

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \bullet & \to & \bullet \\ \downarrow & & \downarrow \\ \bullet & \xrightarrow{a} & \bullet \\ \downarrow & & \downarrow \\ \bullet & \to & \bullet \end{matrix}}

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 topology, turning them into topological groupoids, or even some differentiable structure, turning them into Lie groupoids. These last objects can be also studied in terms of their associated Lie algebroids, in analogy to the relation between Lie groups and Lie algebras.

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 form. Similarly, one can have groupoids with a compatible Riemannian metric, or complex structure, etc.

See also

Notes

  1. ↑ 1.0 1.1 Dicks & Ventura (1996). [[[:Template:Google books]] The Group Fixed by a Family of Injective Endomorphisms of a Free Group] Check |url= value (help). p. 6.
  2. ↑ Template:SpringerEOM
  3. ↑ Proof of first property: from 2. and 3. we obtain a−1 = a−1 * a * a−1 and (a−1)−1 = (a−1)−1 * a−1 * (a−1)−1. Substituting the first into the second and applying 3. two more times yields (a−1)−1 = (a−1)−1 * a−1 * a * a−1 * (a−1)−1 = (a−1)−1 * a−1 * a = a. ✓
    Proof of second property: since a * b is defined, so is (a * b)−1 * a * b. Therefore (a * b)−1 * a * b * b−1 = (a * b)−1 * a is also defined. Moreover since a * b is defined, so is a * b * b−1 = a. Therefore a * b * b−1 * a−1 is also defined. From 3. we obtain (a * b)−1 = (a * b)−1 * a * a−1 = (a * b)−1 * a * b * b−1 * a−1 = b−1 * a−1. ✓
  4. ↑ May, J.P. (1999), A Concise Course in Algebraic Topology, The University of Chicago Press, ISBN 0-226-51183-9 (see chapter 2)
  5. ↑ "fundamental groupoid in nLab". ncatlab.org. Retrieved 2017-09-17.
  6. ↑ 6.0 6.1 Block, Jonathan; Daenzer, Calder (2009-01-09). "Mukai duality for gerbes with connection". arXiv:0803.1529 [math.QA].
  7. ↑ "Localization and Gromov-Witten Invariants" (PDF). p. 9. Archived (PDF) from the original on February 12, 2020.
  8. ↑ An Introduction to Groups, Groupoids and Their Representations: An Introduction; Alberto Ibort, Miguel A. Rodriguez; CRC Press, 2019.
  9. ↑ Jim Belk (2008) Puzzles, Groups, and Groupoids, The Everything Seminar
  10. ↑ The 15-puzzle groupoid (1) Archived 2015-12-25 at the Wayback Machine, Never Ending Books
  11. ↑ The 15-puzzle groupoid (2) Archived 2015-12-25 at the Wayback Machine, Never Ending Books
  12. ↑ Mapping a group to the corresponding groupoid with one object is sometimes called delooping, especially in the context of homotopy theory, see "delooping in nLab". ncatlab.org. Retrieved 2017-10-31..
  13. ↑ Cegarra, Antonio M.; Heredia, Benjamín A.; Remedios, Josué (2010-03-19). "Double groupoids and homotopy 2-types". arXiv:1003.3820 [math.AT].
  14. ↑ Ehresmann, Charles (1964). "Catégories et structures: extraits". Séminaire Ehresmann. Topologie et géométrie différentielle. 6: 1–31.

References

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