Groupoid

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

Inmathematics,especially incategory theoryandhomotopy theory,agroupoid(less oftenBrandt groupoidorvirtual group) generalises the notion ofgroupin several equivalent ways. A groupoid can be seen as a:

• Groupwith apartial functionreplacing thebinary operation;
• Categoryin which everymorphismis invertible. A category of this sort can be viewed as augmented with aunary operationon the morphisms, calledinverseby analogy withgroup theory.A groupoid where there is only one object is a usual group.

In the presence ofdependent typing,a category in general can be viewed as a typedmonoid,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$g:A\rightarrow B$,$h:B\rightarrow C$,say. Composition is then a total function:$\circ:(B\rightarrow C)\rightarrow (A\rightarrow B)\rightarrow A\rightarrow C$,so that$h\circ g:A\rightarrow C$.

Special cases include:

• Setoids:setsthat come with anequivalence relation,
• G-sets:sets equipped with anactionof a group$G$.

Groupoids are often used to reason aboutgeometricalobjects such asmanifolds.Heinrich Brandt(1927) introduced groupoids implicitly viaBrandt semigroups.

Definitions

Algebraic

A groupoid can be viewed as an algebraic structure consisting of a set with a binarypartial function. Precisely, it is a non-empty set$G$with aunary operation${}^{-1}:G\to G,$and apartial function$*:G\times G\rightharpoonup G$.Here * is not abinary operationbecause it is not necessarily defined for all pairs of elements of$G$.The precise conditions under which$*$is defined are not articulated here and vary by situation.

The operations$\ast$and⁻¹have the following axiomatic properties: For all$a$,$b$,and$c$in$G$,

1. Associativity:If$a*b$and$b*c$are defined, then$(a*b)*c$and$a*(b*c)$are defined and are equal. Conversely, if one of$(a*b)*c$or$a*(b*c)$is defined, then they are both defined (and they are equal to each other), and$a*b$and$b*c$are also defined.
2. Inverse:$a^{-1}*a$and$a*{a^{-1}}$are always defined.
3. Identity:If$a*b$is defined, then$a*b*{b^{-1}}=a$,and${a^{-1}}*a*b=b$.(The previous two axioms already show that these expressions are defined and unambiguous.)

Two easy and convenient properties follow from these axioms:

• $(a^{-1})^{-1}=a$,
• If$a*b$is defined, then$(a*b)^{-1}=b^{-1}*a^{-1}$.

Category theoretic

A groupoid is asmall categoryin which everymorphismis anisomorphism,i.e., invertible.More explicitly, a groupoidGis a setG₀ofobjectswith

• for each pair of objectsxandya (possibly empty) setG(x,y) ofmorphisms(orarrows) fromxtoy;we writef:x→yto indicate thatfis an element ofG(x,y);
• for every objectxa designated element$\mathrm {id} _{x}$ofG(x,x);
• for each triple of objectsx,y,andzafunction$\mathrm {comp} _{x,y,z}:G(y,z)\times G(x,y)\rightarrow G(x,z):(g,f)\mapsto gf$;
• for each pair of objectsx,ya function$\mathrm {inv}:G(x,y)\rightarrow G(y,x):f\mapsto f^{-1}$satisfying, for anyf:x→y,g:y→z,andh:z→w:
  • $f\ \mathrm {id} _{x}=f$and$\mathrm {id} _{y}\ f=f$;
  • $(hg)f=h(gf)$;
  • $ff^{-1}=\mathrm {id} _{y}$and$f^{-1}f=\mathrm {id} _{x}$.

Iffis an element ofG(x,y) thenxis called thesourceoff,writtens(f), andyis called thetargetoff,writtent(f).

A groupoidGis sometimes denoted as$G_{1}\rightrightarrows G_{0}$,where$G_{1}$is the set of all morphisms, and the two arrows$G_{1}\to G_{0}$represent the source and the target.

More generally, one can consider agroupoid objectin 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, letGbe thedisjoint unionof all of the setsG(x,y) (i.e. the sets of morphisms fromxtoy). Then$\mathrm {comp}$and$\mathrm {inv}$become partial operations onG,and$\mathrm {inv}$will in fact be defined everywhere. We define ∗ to be$\mathrm {comp}$and⁻¹to be$\mathrm {inv}$,which gives a groupoid in the algebraic sense. Explicit reference toG₀(and hence to$\mathrm {id}$) can be dropped.

Conversely, given a groupoidGin the algebraic sense, define an equivalence relation$\sim$on its elements by $a\sim b$iffa∗a⁻¹=b∗b⁻¹.LetG₀be the set of equivalence classes of$\sim$,i.e.$G_{0}:=G/\!\!\sim$.Denotea∗a⁻¹by$1_{x}$if$a\in G$with$x\in G_{0}$.

Now define$G(x,y)$as the set of all elementsfsuch that$1_{x}*f*1_{y}$exists. Given$f\in G(x,y)$and$g\in G(y,z),$their composite is defined as$gf:=f*g\in G(x,z)$.To see that this is well defined, observe that since$(1_{x}*f)*1_{y}$and$1_{y}*(g*1_{z})$exist, so does$(1_{x}*f*1_{y})*(g*1_{z})=f*g$.The identity morphism onxis then$1_{x}$,and the category-theoretic inverse offisf⁻¹.

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

Vertex groups and orbits

Given a groupoidG,thevertex groupsorisotropy groupsorobject groupsinGare the subsets of the formG(x,x), wherexis any object ofG.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.

Theorbitof a groupoidGat a point$x\in X$is given by the set$s(t^{-1}(x))\subseteq X$containing every point that can be joined to x by a morphism in G. If two points$x$and$y$are in the same orbits, their vertex groups$G(x)$and$G(y)$areisomorphic:if$f$is any morphism from$x$to$y$,then the isomorphism is given by the mapping$g\to fgf^{-1}$.

Orbits form a partition of the set X, and a groupoid is calledtransitiveif it has only one orbit (equivalently, if it isconnectedas 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 sectionbelowfor counterexamples).

Subgroupoids and morphisms

Asubgroupoidof$G\rightrightarrows X$is asubcategory$H\rightrightarrows Y$that is itself a groupoid. It is calledwideorfullif it iswideorfullas a subcategory, i.e., respectively, if$X=Y$or$G(x,y)=H(x,y)$for every$x,y\in Y$.

Agroupoid morphismis simply a functor between two (category-theoretic) groupoids.

Particular kinds of morphisms of groupoids are of interest. A morphism$p:E\to B$of groupoids is called afibrationif for each object$x$of$E$and each morphism$b$of$B$starting at$p(x)$there is a morphism$e$of$E$starting at$x$such that$p(e)=b$.A fibration is called acovering morphismorcovering of groupoidsif further such an$e$is unique. The covering morphisms of groupoids are especially useful because they can be used to modelcovering mapsof spaces.

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

Examples

Topology

Main article:Fundamental groupoid

Given atopological space$X$,let$G_{0}$be the set$X$.The morphisms from the point$p$to the point$q$areequivalence classesofcontinuouspathsfrom$p$to$q$,with two paths being equivalent if they arehomotopic. Two such morphisms are composed by first following the first path, then the second; the homotopy equivalence guarantees that this composition isassociative.This groupoid is called thefundamental groupoidof$X$,denoted$\pi _{1}(X)$(or sometimes,$\Pi _{1}(X)$).The usual fundamental group$\pi _{1}(X,x)$is then the vertex group for the point$x$.

The orbits of the fundamental groupoid$\pi _{1}(X)$are the path-connected components of$X$.Accordingly, the fundamental groupoid of apath-connected spaceis 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 areequivalentas categories (see the sectionbelowfor the general theory).

An important extension of this idea is to consider the fundamental groupoid$\pi _{1}(X,A)$where$A\subset X$is a chosen set of "base points". Here$\pi _{1}(X,A)$is a (wide) subgroupoid of$\pi _{1}(X)$,where one considers only paths whose endpoints belong to$A$.The set$A$may be chosen according to the geometry of the situation at hand.

Equivalence relation

If$X$is asetoid,i.e. a set with anequivalence relation$\sim$,then a groupoid "representing" this equivalence relation can be formed as follows:

• The objects of the groupoid are the elements of$X$;
• For any two elements$x$and$y$in$X$,there is a single morphism from$x$to$y$(denote by$(y,x)$) if and only if$x\sim y$;
• The composition of$(z,y)$and$(y,x)$is$(z,x)$.

The vertex groups of this groupoid are always trivial; moreover, this groupoid is in general not transitive and its orbits are precisely the equivalence classes. There are two extreme examples:

• If every element of$X$is in relation with every other element of$X$,we obtain thepair groupoidof$X$,which has the entire$X\times X$as set of arrows, and which is transitive.
• If every element of$X$is only in relation with itself, one obtains theunit groupoid,which has$X$as set of arrows,$s=t=id_{X}$,and which is completely intransitive (every singleton$\{x\}$is an orbit).

Examples

• If$f:X_{0}\to Y$is a smoothsurjectivesubmersionofsmooth manifolds,then$X_{0}\times _{Y}X_{0}\subset X_{0}\times X_{0}$is an equivalence relationsince$Y$has a topology isomorphic to thequotient topologyof$X_{0}$under the surjective map of topological spaces. If we write,$X_{1}=X_{0}\times _{Y}X_{0}$then we get a groupoid

  $X_{1}\rightrightarrows X_{0}$

  which is sometimes called thebanal groupoidof a surjective submersion of smooth manifolds.
• If we relax the reflexivity requirement and considerpartial equivalence relations,then it becomes possible to considersemidecidablenotions of equivalence on computable realisers for sets. This allows groupoids to be used as a computable approximation to set theory, calledPER models.Considered as a category, PER models are a cartesian closed category with natural numbers object and subobject classifier, giving rise to theeffective toposintroduced byMartin Hyland.

Čech groupoid

See also:Simplicial manifoldandNerve of a covering

A Čech groupoid^{p. 5}is a special kind of groupoid associated to an equivalence relation given by an open cover${\mathcal {U}}=\{U_{i}\}_{i\in I}$of some manifold$X$.Its objects are given by the disjoint union

${\mathcal {G}}_{0}=\coprod U_{i}$,

and its arrows are the intersections

${\mathcal {G}}_{1}=\coprod U_{ij}$.

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

${\begin{aligned}s=\phi _{j}:U_{ij}\to U_{j}\\t=\phi _{i}:U_{ij}\to U_{i}\end{aligned}}$

and the inclusion map

$\varepsilon:U_{i}\to U_{ii}$

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

${\mathcal {G}}_{n}={\mathcal {G}}_{1}\times _{{\mathcal {G}}_{0}}\cdots \times _{{\mathcal {G}}_{0}}{\mathcal {G}}_{1}$

as the$n$-iterated fiber product where the${\mathcal {G}}_{n}$represents$n$-tuples of composable arrows. The structure map of the fiber product is implicitly the target map, since

${\begin{matrix}U_{ijk}&\to &U_{ij}\\\downarrow &&\downarrow \\U_{ik}&\to &U_{i}\end{matrix}}$

is a cartesian diagram where the maps to$U_{i}$are the target maps. This construction can be seen as a model for some∞-groupoids.Also, another artifact of this construction isk-cocycles

$[\sigma ]\in {\check {H}}^{k}({\mathcal {U}},{\underline {A}})$

for some constantsheaf of abelian groupscan be represented as a function

$\sigma:\coprod U_{i_{1}\cdots i_{k}}\to A$

giving an explicit representation of cohomology classes.

Group action

If thegroup$G$acts on the set$X$,then we can form theaction groupoid(ortransformation groupoid) representing thisgroup actionas follows:

• The objects are the elements of$X$;
• For any two elements$x$and$y$in$X$,themorphismsfrom$x$to$y$correspond to the elements$g$of$G$such that$gx=y$;
• Compositionof morphisms interprets thebinary operationof$G$.

More explicitly, theaction groupoidis a small category with$\mathrm {ob} (C)=X$and$\mathrm {hom} (C)=G\times X$and with source and target maps$s(g,x)=x$and$t(g,x)=gx$.It is often denoted$G\ltimes X$(or$X\rtimes G$for a right action). Multiplication (or composition) in the groupoid is then$(h,y)(g,x)=(hg,x)$which is defined provided$y=gx$.

For$x$in$X$,the vertex group consists of those$(g,x)$with$gx=x$,which is just theisotropy subgroupat$x$for the given action (which is why vertex groups are also called isotropy groups). Similarly, the orbits of the action groupoid are theorbitof the group action, and the groupoid is transitive if and only if the group action istransitive.

Another way to describe$G$-sets is thefunctor category$[\mathrm {Gr},\mathrm {Set} ]$,where$\mathrm {Gr}$is the groupoid (category) with one element andisomorphicto the group$G$.Indeed, every functor$F$of this category defines a set$X=F(\mathrm {Gr} )$and for every$g$in$G$(i.e. for every morphism in$\mathrm {Gr}$) induces abijection$F_{g}$:$X\to X$.The categorical structure of the functor$F$assures us that$F$defines a$G$-action on the set$G$.The (unique)representable functor$F$:$\mathrm {Gr} \to \mathrm {Set}$is theCayley representationof$G$.In fact, this functor is isomorphic to$\mathrm {Hom} (\mathrm {Gr},-)$and so sends$\mathrm {ob} (\mathrm {Gr} )$to the set$\mathrm {Hom} (\mathrm {Gr},\mathrm {Gr} )$which is by definition the "set"$G$and the morphism$g$of$\mathrm {Gr}$(i.e. the element$g$of$G$) to the permutation$F_{g}$of the set$G$.We deduce from theYoneda embeddingthat the group$G$is isomorphic to the group$\{F_{g}\mid g\in G\}$,asubgroupof the group ofpermutationsof$G$.

Finite set

Consider the group action of$\mathbb {Z} /2$on the finite set$X=\{-2,-1,0,1,2\}$which takes each number to its negative, so$-2\mapsto 2$and$1\mapsto -1$.The quotient groupoid$[X/G]$is the set of equivalence classes from this group action$\{[0],[1],[2]\}$,and$[0]$has a group action of$\mathbb {Z} /2$on it.

Quotient variety

Any finite group$G$that maps to$GL(n)$gives a group action on theaffine space$\mathbb {A} ^{n}$(since this is the group of automorphisms). Then, a quotient groupoid can be of the form$[\mathbb {A} ^{n}/G]$,which has one point with stabilizer$G$at the origin. Examples like these form the basis for the theory oforbifolds.Another commonly studied family of orbifolds areweighted projective spaces$\mathbb {P} (n_{1},\ldots,n_{k})$and subspaces of them, such asCalabi–Yau orbifolds.

Fiber product of groupoids

Given a diagram of groupoids with groupoid morphisms

${\begin{aligned}&&X\\&&\downarrow \\Y&\rightarrow &Z\end{aligned}}$

where$f:X\to Z$and$g:Y\to Z$,we can form the groupoid$X\times _{Z}Y$whose objects are triples$(x,\phi,y)$,where$x\in {\text{Ob}}(X)$,$y\in {\text{Ob}}(Y)$,and$\phi:f(x)\to g(y)$in$Z$.Morphisms can be defined as a pair of morphisms$(\alpha,\beta )$where$\alpha:x\to x'$and$\beta:y\to y'$such that for triples$(x,\phi,y),(x',\phi ',y')$,there is a commutative diagram in$Z$of$f(\alpha ):f(x)\to f(x')$,$g(\beta ):g(y)\to g(y')$and the$\phi,\phi '$.

Homological algebra

A two term complex

$C_{1}{\overset {d}{\rightarrow }}C_{0}$

of objects in aconcreteAbelian categorycan be used to form a groupoid. It has as objects the set$C_{0}$and as arrows the set$C_{1}\oplus C_{0}$;the source morphism is just the projection onto$C_{0}$while the target morphism is the addition of projection onto$C_{1}$composed with$d$and projection onto$C_{0}$.That is, given$c_{1}+c_{0}\in C_{1}\oplus C_{0}$,we have

$t(c_{1}+c_{0})=d(c_{1})+c_{0}.$

Of course, if the abelian category is the category ofcoherent sheaveson a scheme, then this construction can be used to form apresheafof groupoids.

Puzzles

While puzzles such as theRubik's Cubecan be modeled using group theory (seeRubik's Cube group), certain puzzles are better modeled as groupoids.

The transformations of thefifteen puzzleform a groupoid (not a group, as not all moves can be composed).Thisgroupoid actson configurations.

Mathieu groupoid

TheMathieu groupoidis a groupoid introduced byJohn Horton Conwayacting on 13 points such that the elements fixing a point form a copy of theMathieu groupM₁₂.

Relation to groups

Group-like structures

	Total	Associative	Identity	Divisible	Commutative
Partial magma	Unneeded	Unneeded	Unneeded	Unneeded	Unneeded
Semigroupoid	Unneeded	Required	Unneeded	Unneeded	Unneeded
Small category	Unneeded	Required	Required	Unneeded	Unneeded
Groupoid	Unneeded	Required	Required	Required	Unneeded
Commutativegroupoid	Unneeded	Required	Required	Required	Required
Magma	Required	Unneeded	Unneeded	Unneeded	Unneeded
Commutativemagma	Required	Unneeded	Unneeded	Unneeded	Required
Quasigroup	Required	Unneeded	Unneeded	Required	Unneeded
Commutativequasigroup	Required	Unneeded	Unneeded	Required	Required
Unital magma	Required	Unneeded	Required	Unneeded	Unneeded
Commutativeunital magma	Required	Unneeded	Required	Unneeded	Required
Loop	Required	Unneeded	Required	Required	Unneeded
Commutativeloop	Required	Unneeded	Required	Required	Required
Semigroup	Required	Required	Unneeded	Unneeded	Unneeded
Commutativesemigroup	Required	Required	Unneeded	Unneeded	Required
Associativequasigroup	Required	Required	Unneeded	Required	Unneeded
Commutative-and-associativequasigroup	Required	Required	Unneeded	Required	Required
Monoid	Required	Required	Required	Unneeded	Unneeded
Commutative monoid	Required	Required	Required	Unneeded	Required
Group	Required	Required	Required	Required	Unneeded
Abelian group	Required	Required	Required	Required	Required

If a groupoid has only one object, then the set of its morphisms forms agroup.Using the algebraic definition, such a groupoid is literally just a group.Many concepts ofgroup theorygeneralize to groupoids, with the notion offunctorreplacing that ofgroup 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)$(G,X)$.By transitivity, there will only be oneorbitunder the action.

Note that the isomorphism just mentioned is not unique, and there is nonaturalchoice. Choosing such an isomorphism for a transitive groupoid essentially amounts to picking one object$x_{0}$,agroup isomorphism$h$from$G(x_{0})$to$G$,and for each$x$other than$x_{0}$,a morphism in$G$from$x_{0}$to$x$.

If a groupoid is not transitive, then it is isomorphic to adisjoint unionof groupoids of the above type, also called itsconnected components(possibly with different groups$G$and sets$X$for each connected component).

In category-theoretic terms, each connected component of a groupoid isequivalent(but notisomorphic) to a groupoid with a single object, that is, a single group. Thus any groupoid is equivalent to amultisetof unrelated groups. In other words, for equivalence instead of isomorphism, one does not need to specify the sets$X$,but only the groups$G.$For example,

• The fundamental groupoid of$X$is equivalent to the collection of thefundamental groupsof eachpath-connected componentof$X$,but an isomorphism requires specifying the set of points in each component;
• The set$X$with the equivalence relation$\sim$is equivalent (as a groupoid) to one copy of thetrivial groupfor eachequivalence class,but an isomorphism requires specifying what each equivalence class is:
• The set$X$equipped with anactionof the group$G$is equivalent (as a groupoid) to one copy of$G$for eachorbitof the action, but anisomorphismrequires 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 notnatural.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$G(x)$in terms of a single group, and this choice can be arbitrary. In the example fromtopology,one 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.

As a more illuminating example, the classification of groupoids with oneendomorphismdoes not reduce to purely group theoretic considerations. This is analogous to the fact that the classification ofvector spaceswith one endomorphism is nontrivial.

Morphisms of groupoids come in more kinds than those of groups: we have, for example,fibrations,covering morphisms,universal morphisms,andquotient morphisms.Thus a subgroup$H$of a group$G$yields an action of$G$on the set ofcosetsof$H$in$G$and hence a covering morphism$p$from, say,$K$to$G$,where$K$is a groupoid withvertex groupsisomorphic 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.

Category of groupoids

The category whose objects are groupoids and whose morphisms are groupoid morphisms is called thegroupoid category,or thecategory of groupoids,and is denoted byGrpd.

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

$\operatorname {Grpd} (G\times H,K)\cong \operatorname {Grpd} (G,\operatorname {GPD} (H,K)).$

This result is of interest even if all the groupoids$G,H,K$are just groups.

Another important property ofGrpdis that it is bothcompleteandcocomplete.

Relation toCat

The inclusion$i:\mathbf {Grpd} \to \mathbf {Cat}$has both a left and a rightadjoint:

$\hom _{\mathbf {Grpd} }(C[C^{-1}],G)\cong \hom _{\mathbf {Cat} }(C,i(G))$

$\hom _{\mathbf {Cat} }(i(G),C)\cong \hom _{\mathbf {Grpd} }(G,\mathrm {Core} (C))$

Here,$C[C^{-1}]$denotes thelocalization of a categorythat inverts every morphism, and$\mathrm {Core} (C)$denotes the subcategory of all isomorphisms.

Relation tosSet

Thenerve functor$N:\mathbf {Grpd} \to \mathbf {sSet}$embedsGrpdas a full subcategory of the category of simplicial sets. The nerve of a groupoid is always aKan complex.

The nerve has a left adjoint

$\hom _{\mathbf {Grpd} }(\pi _{1}(X),G)\cong \hom _{\mathbf {sSet} }(X,N(G))$

Here,$\pi _{1}(X)$denotes the fundamental groupoid of the simplicial set X.

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.BecauseGrpdis a 2-category, these objects form a 2-category instead of a 1-category since there is extra structure. Essentially, these are groupoids${\mathcal {G}}_{1},{\mathcal {G}}_{0}$with functors

$s,t:{\mathcal {G}}_{1}\to {\mathcal {G}}_{0}$

and an embedding given by an identity functor

$i:{\mathcal {G}}_{0}\to {\mathcal {G}}_{1}$

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

${\begin{matrix}\bullet &\to &\bullet \\\downarrow &&\downarrow \\\bullet &\xrightarrow {a} &\bullet \end{matrix}}$and${\begin{matrix}\bullet &\xrightarrow {a} &\bullet \\\downarrow &&\downarrow \\\bullet &\to &\bullet \end{matrix}}$

with$a$the same morphism, they can be vertically conjoined giving a diagram

${\begin{matrix}\bullet &\to &\bullet \\\downarrow &&\downarrow \\\bullet &\xrightarrow {a} &\bullet \\\downarrow &&\downarrow \\\bullet &\to &\bullet \end{matrix}}$

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 atopology,turning them intotopological groupoids,or even somedifferentiable structure,turning them intoLie groupoids.These last objects can be also studied in terms of their associatedLie algebroids,in analogy to the relation betweenLie groupsandLie algebras.

Groupoids arising from geometry often possess further structures which interact with the groupoid multiplication. For instance, inPoisson geometryone has the notion of asymplectic groupoid,which is a Lie groupoid endowed with a compatiblesymplectic form.Similarly, one can have groupoids with a compatibleRiemannian metric,orcomplex structure,etc.

See also

• ∞-groupoid
• 2-group
• Homotopy type theory
• Inverse category
• Groupoid algebra(not to be confused withalgebraic groupoid)
• R-algebroid

References

• Brandt, H (1927), "Über eine Verallgemeinerung des Gruppenbegriffes",Mathematische Annalen,96(1): 360–366,doi:10.1007/BF01209171,S2CID119597988
• Brown, Ronald, 1987, "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.
• —, 2006.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.
• —,Higher dimensional group theory.Explains how the groupoid concept has led to higher-dimensional homotopy groupoids, having applications inhomotopy theoryand in groupcohomology.Many references.
• Dicks, Warren; Ventura, Enric (1996),The group fixed by a family of injective endomorphisms of a free group,Mathematical Surveys and Monographs, vol. 195, AMS Bookstore,ISBN978-0-8218-0564-0
• Dokuchaev, M.; Exel, R.; Piccione, P. (2000). "Partial Representations and Partial Group Algebras".Journal of Algebra.226.Elsevier: 505–532.arXiv:math/9903129.doi:10.1006/jabr.1999.8204.ISSN0021-8693.S2CID14622598.
• F. Borceux, G. Janelidze, 2001,Galois theories.Cambridge Univ. Press. Shows how generalisations ofGalois theorylead toGalois groupoids.
• Cannas da Silva, A.,andA. Weinstein,Geometric Models for Noncommutative Algebras.Especially Part VI.
• Golubitsky, M.,Ian Stewart, 2006, "Nonlinear dynamics of networks: the groupoid formalism",Bull. Amer. Math. Soc.43:305–64
• "Groupoid",Encyclopedia of Mathematics,EMS Press,2001 [1994]
• Higgins, P. J., "The fundamental groupoid of agraph of groups",J. London Math. Soc. (2) 13 (1976) 145–149.
• Higgins, P. J. and Taylor, J., "The fundamental groupoid and the homotopy crossed complex of anorbit space",in Category theory (Gummersbach, 1981), Lecture Notes in Math., Volume 962. Springer, Berlin (1982), 115–122.
• Higgins, P. J., 1971.Categories and groupoids.Van Nostrand Notes in Mathematics. Republished inReprints in Theory and Applications of Categories,No. 7 (2005) pp. 1–195;freely downloadable.Substantial introduction tocategory theorywith special emphasis on groupoids. Presents applications of groupoids in group theory, for example to a generalisation ofGrushko's theorem,and in topology, e.g.fundamental groupoid.
• Mackenzie, K. C. H., 2005.General theory of Lie groupoids and Lie algebroids.Cambridge Univ. Press.
• Weinstein, Alan, "Groupoids: unifying internal and external symmetry — A tour through some examples."Also available inPostscript.,Notices of the AMS, July 1996, pp. 744–752.
• Weinstein, Alan, "The Geometry of Momentum"(2002)
• R.T. Zivaljevic. "Groupoids in combinatorics—applications of a theory of local symmetries". InAlgebraic and geometric combinatorics,volume 423 ofContemp. Math., 305–324. Amer. Math. Soc., Providence, RI (2006)
• fundamental groupoidat thenLab
• coreat thenLab