Editing Groupoid (section)

== Definitions ==

=== Algebraic ===

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

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

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

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

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

If ''f'' is an element of ''G''(''x'',''y''), then ''x'' is called the '''source''' of ''f'', written ''s''(''f''), and ''y'' is called the '''target''' of ''f'', written ''t''(''f'').

A groupoid ''G'' is sometimes denoted as {{tmath|1= G_1 \rightrightarrows G_0 }}, where <math>G_1</math> is the set of all morphisms, and the two arrows <math>G_1 \to G_0</math> represent the source and the target.

More generally, one can consider a [[groupoid object]] in an arbitrary category admitting finite fiber products.

=== Comparing the definitions ===
The algebraic and category-theoretic definitions are equivalent, as we now show. Given a groupoid in the category-theoretic sense, let ''G'' be the [[disjoint union]] of all of the sets ''G''(''x'',''y'') (i.e. the sets of morphisms from ''x'' to ''y''). Then <math>\mathrm{comp}</math> and <math>\mathrm{inv}</math> become partial operations on ''G'', and <math>\mathrm{inv}</math> will in fact be defined everywhere. We define ∗ to be <math>\mathrm{comp}</math> and <sup>−1</sup> to be {{tmath|1= \mathrm{inv} }}, which gives a groupoid in the algebraic sense. Explicit reference to ''G''<sub>0</sub> (and hence to {{tmath|1= \mathrm{id} }}) can be dropped.

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

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

Sets in the definitions above may be replaced with [[class (set theory)|class]]es, as is generally the case in category theory.

=== Vertex groups and orbits ===
Given a groupoid ''G'', the '''vertex groups''' or '''isotropy groups''' or '''object groups''' in ''G'' are the subsets of the form ''G''(''x'',''x''), where ''x'' is any object of ''G''. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

The '''orbit''' of a groupoid ''G'' at a point <math>x \in X</math> is given by the set <math>s(t^{-1}(x)) \subseteq X</math> containing every point that can be joined to x by a morphism in G. If two points <math>x</math> and <math>y</math> are in the same orbits, their vertex groups <math>G(x)</math> and <math>G(y)</math> are [[group isomorphism|isomorphic]]: if <math>f</math> is any morphism from <math>x</math> to {{tmath|1= y }}, then the isomorphism is given by the mapping {{tmath|1= g\to fgf^{-1} }}.

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

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

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

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

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