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| {{About|groupoids in category theory|the algebraic structure with a single binary operation|magma (algebra)}}
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| <!-- Please see Talk page ("Disambiguation revisited") before modifying -->
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| In [[mathematics]], especially in [[category theory]] and [[homotopy theory]], a '''groupoid''' (less often '''Brandt groupoid''' or '''virtual group''') generalises the notion of [[group (mathematics)|group]] in several equivalent ways. A groupoid can be seen as a:
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| *''[[group (mathematics)|Group]]'' with a [[partial function]] replacing the [[binary operation]];
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| *''[[category theory|Category]]'' in which every [[morphism]] is invertible. A category of this sort can be viewed as augmented with a [[unary operation]], called ''inverse'' by analogy with [[group theory]].<ref name="dicks-ventura-96">Dicks & Ventura (1996), {{Google books quote|id=3sWSRRfNFKgC|page=6|text=G has the structure of a graph|p. 6}}</ref>
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| *''[[Graph (mathematics)|Oriented graph]]'' <ref name="dicks-ventura-96"/><ref>Dokuchaev, Exel & Piccione (2000), p. 16</ref>
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| Special cases include:
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| *''[[Setoid]]s'', that is: [[Set (mathematics)|sets]] that come with an [[equivalence relation]];
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| *''[[G-set]]s'', sets equipped with an [[group action|action]] of a group ''G''.
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| 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 semigroups.<ref>[http://eom.springer.de/b/b017600.htm Brandt semigroup] in Springer Encyclopaedia of Mathematics - ISBN 1-4020-0609-8</ref>
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| == Definitions ==
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| A groupoid is an algebraic structure (G,<math>\ast</math>) consisting of a non-empty set G and a binary operation '<math>\ast</math>' defined on G.
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| === Algebraic ===
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| A groupoid is a set ''G'' with a [[unary operation]] <math>^{-1}:G\to G,</math> and a [[partial function]] <math>*:G\times G \rightharpoonup G.</math>. Here * is not a [[binary operation]] because it is not necessarily defined for all possible pairs of ''G''-elements. The precise conditions under which * is defined are not articulated here and vary by situation.
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| <math>\ast</math> and <sup>−1</sup> have the following axiomatic properties. Let ''a'', ''b'', and ''c'' be elements of ''G''. Then:
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| # ''[[Associativity]]'': If ''a'' * ''b'' and ''b'' * ''c'' are defined, then (''a'' * ''b'') * ''c'' and ''a'' * (''b'' * ''c'') are defined and equal. Conversely, if either of these last two expressions is defined, then so is the other (and again they are equal).
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| # ''[[Inverse (mathematics)|Inverse]]'': ''a''<sup>−1</sup> * ''a'' and ''a'' * ''a''<sup>−1</sup> are always defined.
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| # ''[[Identity (mathematics)|Identity]]'': If ''a'' * ''b'' is defined, then ''a'' * ''b'' * ''b''<sup>−1</sup> = ''a'', and ''a''<sup>−1</sup> * ''a'' * ''b'' = ''b''. (The previous two axioms already show that these expressions are defined and unambiguous.)
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| From these axioms, two easy and convenient properties follow:
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| * (''a''<sup>−1</sup>)<sup>−1</sup> = ''a'';
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| * If ''a'' * ''b'' is defined, then (''a'' * ''b'')<sup>−1</sup> = ''b''<sup>−1</sup> * ''a''<sup>−1</sup>.
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| Proof of first property: from 2. and 3. we obtain (''a''<sup>−1</sup>)<sup>−1</sup> = (''a''<sup>−1</sup>)<sup>−1</sup> * ''a''<sup>−1</sup> * ''a'' = ''a''. ✓ <br />
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| 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>. ✓
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| ===Category theoretic===
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| A groupoid is a [[category (mathematics)|small category]] in which every [[morphism]] is an [[isomorphism]], and hence invertible.<ref name="dicks-ventura-96"/> More precisely, a groupoid ''G'' is:
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| * A set ''G''<sub>0</sub> of ''objects'';
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| * For each pair of objects ''x'' and ''y'' in ''G''<sub>0</sub>, there exists a (possibly empty) set ''G''(''x'',''y'') of ''morphisms'' (or ''arrows'') from ''x'' to ''y''. We write ''f'' : ''x'' → ''y'' to indicate that ''f'' is an element of ''G''(''x'',''y'').
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| The objects and morphisms have the properties:
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| * For every object ''x'', there exists the element <math>\mathrm{id}_x</math> of ''G''(''x'',''x'');
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| * For each triple of objects ''x'', ''y'', and ''z'', there exists the [[function (mathematics)|function]] <math>\mathrm{comp}_{x,y,z} :</math> ''G''(''x'',''y'')<math>\times</math>''G''(''y'',''z'') → ''G''(''x'',''z''). We write ''gf'' for <math>\mathrm{comp}_{x,y,z}(f,g)</math>, where ''f''<math>\in</math>''G''(''x'',''y''), and ''g''<math>\in</math>''G''(''y'',''z'');
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| * There exists the [[function (mathematics)|function]] <math>\mathrm{inv}_{x,y} :</math> ''G''(''x'',''y'') → ''G''(''y'',''x'').
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| Moreover, if ''f'' : ''x'' → ''y'', ''g'' : ''y'' → ''z'', and ''h'' : ''z'' → ''w'', then:
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| * <math>f \mathrm{id}_x = f</math> and <math>\mathrm{id}_y f = f</math>;
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| * (''hg'')''f'' = ''h''(''gf'');
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| * <math>f \mathrm{inv}(f) = \mathrm{id}_y</math> and <math>\mathrm{inv}(f)f = \mathrm{id}_x</math>.
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| If ''f'' is an element of ''G''(''x'',''y'') then ''x'' is called the '''source''' of ''f'', written ''s''(''f''), and ''y'' the '''target''' of ''f'' (written ''t''(''f'')).
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| ===Comparing the definitions===
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| The algebraic and category-theoretic definitions are equivalent, as follows. 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 partially defined operations on ''G'', and <math>\mathrm{inv}</math> will in fact be defined everywhere; so we define * to be <math>\mathrm{comp}</math> and <math>^{-1}</math> to be <math>\mathrm{inv}</math>. Thus we have a groupoid in the algebraic sense. Explicit reference to ''G''<sub>0</sub> (and hence to <math>\mathrm{id}</math>) can be dropped.
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| Conversely, given a groupoid ''G'' in the algebraic sense, let ''G''<sub>0</sub> be the set of all elements of the form ''x * x''<sup>−1</sup> with ''x'' varying through ''G'' and define ''G(x*x<sup> -1</sup>,y*y<sup> -1</sup>)'' as the set of all elements ''f'' such that ''y * y<sup> -1</sup> * f * x * x<sup> -1</sup>'' exists. Given ''f∈G(x*x<sup>-1</sup>,y*y<sup> -1</sup>)'' and ''g∈G(y*y<sup> -1</sup>,z*z<sup> -1</sup>)'', their composite is defined as ''g * f ∈ G(x*x<sup> -1</sup>,z*z<sup> -1</sup>)''. To see this is well defined, observe that since ''z*z<sup> -1</sup> * g * y*y<sup> -1</sup>'' and ''y*y<sup>-1</sup> * f * x*x<sup>-1</sup>'' exist, so does ''z*z<sup>-1</sup> * g * y*y<sup>-1</sup> * y*y<sup>-1</sup> * f * x*x<sup> -1</sup> = z*z<sup>-1</sup> * g*f * x*x<sup> -1</sup>''. The identity morphism on ''x*x''<sup>−1</sup> is then ''x*x''<sup>−1</sup> itself, and the category-theoretic inverse of ''f'' is ''f<sup>-1</sup>''.
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| ''Sets'' in the definitions above may be replaced with [[class (set theory)|class]]es, as is generally the case in [[category theory]].
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| ===Vertex groups===
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| 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.
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| ===Category of groupoids===
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| A '''subgroupoid''' is a [[subcategory]] that is itself a groupoid. A '''groupoid morphism''' is simply a functor between two (category-theoretic) groupoids. The category whose objects are groupoids and whose morphisms are groupoid morphisms is called the '''groupoid category''', or the '''category of groupoids''', denoted '''Grpd'''.
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| It is useful that this category is, like the category of small categories, [[cartesian closed]]. That is, we can construct for any groupoids <math>H,K</math> a groupoid <math>GPD(H,K)</math> whose objects are the morphisms <math> H \to K </math> and whose arrows are the natural equivalences of morphisms. Thus if <math> H,K </math> are just groups, then such arrows are the conjugacies of morphisms. The main result is that for any groupoids <math> G,H,K </math> there is a natural bijection
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| <math> Grpd(G \times H, K) \cong Grpd(G, GPD(H,K)).</math>
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| This result is of interest even if all the groupoids <math> G,H,K </math> are just groups.
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| ===Fibrations, Coverings===
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| 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 <math>p(e)=b</math>. 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>
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| 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.
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| == Examples ==
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| ===Linear algebra===
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| Given a [[field (algebra)|field]] ''K'', the corresponding '''general linear groupoid''' ''GL''<sub>*</sub>(''K'') consists of all [[matrix inversion|invertible]] [[matrix (mathematics)|matrices]] whose entries range over ''K''. [[Matrix multiplication]] interprets composition. If ''G'' = ''GL''<sub>*</sub>(''K''), then the set of [[natural number]]s is a [[proper subset]] of ''G''<sub>0</sub>, since for each [[natural number]] ''n'', there is a corresponding [[identity matrix]] of dimension ''n''. ''G''(''m'',''n'') is [[empty set|empty]] unless ''m''=''n'', in which case it is the set of all ''n''x''n'' invertible matrices.
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| ===Topology===
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| Given a [[topological space]] ''X'', let ''G''<sub>0</sub> be the set ''X''. The morphisms from the point ''p'' to the point ''q'' are [[equivalence class]]es of [[continuous function (topology)|continuous]] [[path (topology)|path]]s from ''p'' to ''q'', with two paths being equivalent if they are [[homotopic]].
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| 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 ''X'', denoted <math>\pi_1</math>(''X''). The usual fundamental group <math>\pi_1(X,x)</math> is then the vertex group for the point ''x''.
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| An important extension of this idea is to consider the fundamental groupoid <math>\pi_1</math>(''X'',''A'') where ''A'' is a set of "base points" and a subset of ''X''. Here, one considers only paths whose endpoints belong to ''A''. <math>\pi_1</math>(''X'',''A'') is a sub-groupoid of <math>\pi_1</math>(''X''). The set ''A'' may be chosen according to the geometry of the situation at hand.
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| ===Equivalence relation===
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| If ''X'' is a set with an [[equivalence relation]] denoted by [[infix]] <math>\sim</math>, then a groupoid "representing" this equivalence relation can be formed as follows:
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| * The objects of the groupoid are the elements of ''X'';
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| *For any two elements ''x'' and ''y'' in ''X'', there is a single morphism from ''x'' to ''y'' [[if and only if]] ''x''~''y''.
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| ===Group action===
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| If the [[group (mathematics)|group]] ''G'' acts on the set ''X'', then we can form the '''action groupoid''' representing this [[group action]] as follows:
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| *The objects are the elements of ''X'';
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| *For any two elements ''x'' and ''y'' in ''X'', there is a [[morphism]] from ''x'' to ''y'' corresponding to every element ''g'' of ''G'' such that ''gx'' = ''y'';
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| *[[Function composition|Composition]] of morphisms interprets the [[binary operation]] of ''G''.
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| More explicitly, the ''action groupoid'' is the set <math>G\times X</math> with source and target maps ''s''(''g'',''x'') = ''x'' and ''t''(''g'',''x'') = ''gx''. It is often denoted <math>G \ltimes X</math> (or <math>X\rtimes G</math>). Multiplication (or composition) in the groupoid is then <math>(h,y)(g,x) = (hg,x)</math> which is defined provided ''y=gx''.
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| For ''x'' in ''X'', the vertex group consists of those (''g'',''x'') with ''gx'' = ''x'', which is just the isotropy subgroup at ''x'' for the given action (which is why vertex groups are also called isotropy groups).
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| Another way to describe ''G''-sets is the [[functor category]] <math>[\mathrm{Gr},\mathrm{Set}]</math>, where <math>\mathrm{Gr}</math> is the groupoid (category) with one element and [[isomorphism|isomorphic]] to the group ''G''. Indeed, every functor ''F'' of this category defines a set ''X''=''F''<math>(\mathrm{Gr})</math> and for every ''g'' in ''G'' (i.e. for every morphism in <math>\mathrm{Gr}</math>) induces a [[bijection]] ''F''<sub>''g''</sub> : ''X''→''X''. The categorical structure of the functor ''F'' assures us that ''F'' defines a ''G''-action on the set ''X''. The (unique) [[representable functor]] ''F'' : <math>\mathrm{Gr}</math>→<math>\mathrm{Set}</math> is the [[Cayley's theorem|Cayley representation]] of ''G''. In fact, this functor is isomorphic to <math>\mathrm{Hom}(\mathrm{Gr},-)</math> and so sends <math>\mathrm{ob}(\mathrm{Gr})</math> to the set <math>\mathrm{Hom}(\mathrm{Gr},\mathrm{Gr})</math> which is by definition the "set" ''G'' and the morphism ''g'' of <math>\mathrm{Gr}</math> (i.e. the element ''g'' of ''G'') to the permutation ''F''<sub>''g''</sub> of the set ''G''. We deduce from the [[Yoneda embedding]] that the group ''G'' is isomorphic to the group {''F''<sub>''g''</sub> | ''g''∈''G''}, a [[subgroup]] of the group of [[permutation group|permutation]]s of ''G''.
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| ===Fifteen puzzle===
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| The symmetries of the [[fifteen puzzle]] form a groupoid (not a group, as not all moves can be composed).<ref>[http://www.neverendingbooks.org/index.php/the-15-puzzle-groupoid-1.html The 15-puzzle groupoid (1)], Never Ending Books</ref><ref>[http://www.neverendingbooks.org/index.php/the-15-puzzle-groupoid-2.html The 15-puzzle groupoid (2)], Never Ending Books</ref> This [[Group_action#Generalizations|groupoid acts]] on configurations.
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| ===Mathieu groupoid===
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| The [[Mathieu groupoid]] is a groupoid introduced by [[John Horton Conway]] acting on 13 points such that the elements fixing a point form a copy of the [[Mathieu group]] M<sub>12</sub>.
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| == Relation to groups ==
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| {{Group-like structures}}
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| 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. Many concepts of [[group theory]] generalize to groupoids, with the notion of [[functor]] replacing that of [[group homomorphism]].
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| If ''x'' is an object of the groupoid ''G'', then the set of all morphisms from ''x'' to ''x'' forms a group ''G''(''x''). If there is a morphism ''f'' from ''x'' to ''y'', then the groups ''G''(''x'') and ''G''(''y'') are [[group isomorphism|isomorphic]], with an isomorphism given by the [[map (mathematics)|mapping]] ''g'' → ''fgf''<sup> −1</sup>.
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| Every [[connected (category theory)|connected]] groupoid (that is, one in which any two objects are connected by at least one morphism) is isomorphic to an action groupoid (as defined above) (''G'', ''X'') [by connectedness, there will only be one [[orbit (group theory)|orbit]] under the action]. If the groupoid is not connected, then it is isomorphic to a [[disjoint union]] of groupoids of the above type (possibly with different groups ''G'' and sets ''X'' for each connected component).
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| Note that the isomorphism described above is not unique, and there is no [[natural equivalence|natural]] choice. Choosing such an isomorphism for a connected groupoid essentially amounts to picking one object ''x''<sub>0</sub>, a [[group isomorphism]] ''h'' from ''G''(''x''<sub>0</sub>) to ''G'', and for each ''x'' other than ''x''<sub>0</sub>, a morphism in ''G'' from ''x''<sub>0</sub> to ''x''.
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| In category-theoretic terms, each connected component of a groupoid is [[equivalent categories|equivalent]] (but not [[isomorphic categories|isomorphic]]) to a groupoid with a single object, that is, a single group. Thus any groupoid is equivalent to a [[multiset]] of unrelated groups. In other words, for equivalence instead of isomorphism, one need not specify the sets ''X'', only the groups ''G''.
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| Consider the examples in the previous section. The general linear groupoid is both equivalent and isomorphic to the disjoint union of the various [[general linear group]]s ''GL''<sub>n</sub>(''F''). On the other hand:
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| *The fundamental groupoid of ''X'' is equivalent to the collection of the [[fundamental group]]s of each [[path-connected component]] of ''X'', but an isomorphism requires specifying the set of points in each component;
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| *The set ''X'' with the equivalence relation <math>\sim</math> is equivalent (as a groupoid) to one copy of the [[trivial group]] for each [[equivalence class]], but an isomorphism requires specifying what each equivalence class is:
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| *The set ''X'' equipped with an [[group action|action]] of the group ''G'' is equivalent (as a groupoid) to one copy of ''G'' for each [[orbit (group theory)|orbit]] of the action, but an [[isomorphism]] requires specifying what set each orbit is.
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| The collapse of a groupoid into a mere collection of groups loses some information, even from a category-theoretic point of view, because it is not [[natural (category theory)|natural]]. Thus when groupoids arise in terms of other structures, as in the above examples, it can be helpful to maintain the full groupoid. Otherwise, one must choose a way to view each ''G''(''x'') in terms of a single group, and this choice can be arbitrary. In our example from [[topology]], you would have to make a coherent choice of paths (or equivalence classes of paths) from each point ''p'' to each point ''q'' in the same path-connected component.
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| As a more illuminating example, the classification of groupoids with one [[endomorphism]] does not reduce to purely group theoretic considerations. This is analogous to the fact that the classification of [[vector space]]s with one endomorphism is nontrivial.
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| Morphisms of groupoids come in more kinds than those of groups: we have, for example, [[fibration]]s, [[covering morphism]]s, [[universal morphism]]s, and [[quotient morphism]]s. Thus a subgroup ''H'' of a group ''G'' yields an action of ''G'' on the set of [[coset]]s of ''H'' in ''G'' and hence a covering morphism ''p'' from, say, ''K'' to ''G'', where ''K'' is a groupoid with [[#Vertex groups|vertex group]]s isomorphic to ''H''. In this way, presentations of the group ''G'' can be "lifted" to presentations of the groupoid ''K'', and this is a useful way of obtaining information about presentations of the subgroup ''H''. For further information, see the books by Higgins and by Brown in the References.
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| == Lie groupoids and Lie algebroids ==
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| When studying geometrical objects, the arising groupoids often carry some [[differentiable structure]], turning them into [[Lie groupoid]]s.
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| These can be studied in terms of [[Lie algebroid]]s, in analogy to the relation between [[Lie group]]s and [[Lie algebra]]s.
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| == Groupoid actions ==
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| {{Empty section|date=September 2011}}
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| == Groupoid representations ==
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| {{Empty section|date=September 2011}}
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| == See also ==
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| *[[∞-groupoid]]
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| == Notes ==
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| {{reflist}}
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| ==References==
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| *{{citation|title=Über eine Verallgemeinerung des Gruppenbegriffes|journal=Mathematische Annalen
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| |volume=96|issue=1 |pages=360–366|year=1927|doi=10.1007/BF01209171|first=H|last=Brandt}}
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| *Brown, Ronald, 1987, "[http://www.bangor.ac.uk/r.brown/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.
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| * —, 2006. ''[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.
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| * —, [http://www.bangor.ac.uk/r.brown/hdaweb2.htm Higher dimensional group theory ] Explains how the groupoid concept has to led to higher dimensional homotopy groupoids, having applications in [[homotopy theory]] and in group [[cohomology]]. Many references.
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| * {{citation |last1=Dicks |first1=Warren |authorlink1= |last2=Ventura |first2=Enric |authorlink2= |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 |location= |isbn=978-0-8218-0564-0 }}
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| * {{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 |issue= |pages=505–532 |publisher=Elsevier |issn=0021-8693 |doi= 10.1006/jabr.1999.8204|url=|arxiv= math/9903129
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| }}
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| *F. Borceux, G. Janelidze, 2001, ''[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.
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| * Cannas da Silva, A., and A. Weinstein, ''[http://www.math.ist.utl.pt/~acannas/Books/models_final.pdf Geometric Models for Noncommutative Algebras.]'' Especially Part VI.
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| *[[Marty Golubitsky|Golubitsky, M.]], Ian Stewart, 2006, "[http://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
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| * {{springer|title=Groupoid|id=p/g045360}}
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| * Higgins, P. J., "The fundamental groupoid of a [[graph of groups]]", J. London Math. Soc. (2) 13 (1976) 145—149.
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| * 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.
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| *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. 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]].
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| *Mackenzie, K. C. H., 2005. ''[http://www.shef.ac.uk/~pm1kchm/gt.html General theory of Lie groupoids and Lie algebroids.]'' Cambridge Univ. Press.
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| *Weinstein, Alan, "[http://www.ams.org/notices/199607/weinstein.pdf Groupoids: unifying internal and external symmetry — A tour through some examples.]" Also available in [http://math.berkeley.edu/~alanw/Groupoids.ps Postscript.], Notices of AMS, July 1996, pp. 744–752.
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| * Weinstein, Alan, "[http://arxiv.org/pdf/math.SG/0208108.pdf The Geometry of Momentum]" (2002)
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| * R.T. Zivaljevic. "Groupoids in combinatorics—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)
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| * {{nlab|id=fundamental+groupoid|title=fundamental groupoid}}
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| [[Category:Algebraic structures]]
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| [[Category:Category theory]]
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| [[Category:Homotopy theory]]
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