Groupoid

In mathematics, especiallycategory theoryhomotopy theory,groupoid isconcept that simultaneously generalises groupss, equivalence relations on sets,actionssgroups on sets. Theyoften usedcapture information about geometrical objects such as manifolds.

The term "groupoid"also used formagma:setany sortbinary operation on it. We do not usetermthat conceptthis encyclopedia.

Tablecontents

1 Definitions 2 Examples 3 Relationgroups 4 Lie groupoidsLie algebroids

Definitions

From one pointview,groupoidsimplycategorywhich every morphisman isomorphism (that is, invertible). To be explicit,groupoid G is:

• A set G₀objects;
• For each pairobjects xyG₀,set G(x,y)morphisms\ (or arrows) from xy -- we write f: x -> yindicate that fan elementG(x,y);

equipped with:

• An element id_xG(x,x);
• For each tripleobjects x, y,z,binary function comp_x,y,z from G(x,y)G(y,z)G(x,z) -- we write gfcomp_x,y,z(f,g);
• A function inv_x,y from G(x,y)G(y,x) -- we write f^-1inv_x,y(f);

such that:

• If f: x -> y, then fid_x = fid_yf = f;
• If f: x -> y, g: y -> z,h: z -> w, then (hg)f = h(gf);
• If f: x -> y, then ff^-1 = id_yf^-1f = id_x.

One can also definegroupoid ascertain algebraic structure. To be specific, let G besetlet comp bepartially defined binary operation on G. That is, given elements fgG, comp(f,g) may be an elementG, ormay be undefined. We write gfcomp(f,g). Therealsototal (everywhere defined) function inv on G. We write f^-1 forinverse inv(f)f. Then G isgroupoid if:

• Whenever fgghboth defined, then (fg)hf(gh)also defined,theyequal;
• f^-1fff^-1always defined;
• Whenever fgdefined, then fgg^-1 = ff^-1fg = g -- we already know that these expressionsunambiguously defined byprevious conditions.

The relation between this definitionsas follows: Givengroupoid incategory-theoretic sense, let G bedisjoint unionall ofsets G(x,y). Then invcomp become partially defined operations on G,inv willfact be defined everywhere. Explicit referenceG₀ (and henceid) can be dropped.

Onother hand, givengroupoid inalgebraic sense, let G₀ besetall elements ofform ff^-1some element fG. In other words,objectsidentified withidentity morphisms,id_xjust x. Let G(x,y) beset all elements f such that yfxdefined. Then invcomp break up into several functions onvarious G(x,y).

While we have referredsets indefinitions above, one may instead wantuse classeses, insame way asother categories.

Examples

From linear algebra: Givenfield K,general linear groupoid GL_*(K) consistsall invertible matricesentries from K,composition given by matrix multiplication. If G = GL_*(K), then G₀ can be identified withsetnatural numbers, since thereone identity matrixeach natural number. G(m,n)empty unless m = n,which case it'ssetn by n matrices.

From topology: Start withtopological space Xlet G₀ beset X. The morphisms frompoint p topoint qequivalence classescontinuous paths from pq,two paths being considered equivalent if theyhomotopic. Two such morphismscomposed by first followingfirst path, thensecond;homotopy equivalence guarantees that this compositionassociative. This groupoidcalledfundamental groupoidX, denoted Π₁(X).

If X isset~an equivalence relation on X, then we can formgroupoid representing this equivalence relation as follows: The objects areelementsX,for any two elements xyX, there issingle morphism from xy ifonly if x ~ y.

Ifgroup G actss onset X, then we can formgroupoid representing this group action as follows: The objects areelementsX,for any two elements xyX, there ismorphism from xyevery element gG such that g.x = y. Compositionmorphismsgiven bygroup operationG.

Relationgroups

Ifgroupoid has only one object, thensetits morphisms formsgroup. Usingalgebraic definition, suchgroupoidliterally justgroup. Many conceptsgroup theory can be generalizedgroupoids, withnotiongroup homomorphism being replaced by thatfunctor.

If xan object ofgroupoid G, thensetall morphisms from xx formsgroup G(x). If there ismorphism f from xy, thengroups G(x)G(y)isomorphic,an isomorphism given by mapping gfgf^-1.

Every connected groupoid (that is, onewhich any two objectsconnected by at least one morphism)isomorphic togroupoid offollowing form: Pickgroup G andset (or class) X. Letobjects ofgroupoid beelementsX. For elements xyX, letsetmorphisms from xy be G. Compositionmorphisms isgroup operationG. Ifgroupoidnot connected, then itisomorphic todisjoint uniongroupoids ofabove type (possiblydifferent groups G per connected component). Thus, any groupoid may be given (upisomorphism) bysetordered pairs (X,G).

Note thatisomorphism described abovenot unique,thereno natural choice. Choosing such an isomorphism forconnected groupoid essentially amountspicking one object x₀,group isomorphism h from G(x₀)G,for each x other than x₀morphismG from x₀x.

In category-theoretic terms, each connected component ofgroupoidequivalent (but not isomorphic) togroupoid withsingle object, that is,single group. Thus any groupoidequivalent tomultisetunrelated groups. In other words,equivalence insteadisomorphism, you don't havespecifysets X, onlygroups G.

Considerexamples inprevious section. The general linear groupoidboth equivalentisomorphic todisjoint union ofvarious general linear groups GL_n(F). Onother hand,fundamental groupoidXequivalent tocollection offundamental groupseach path-connected componentX, butan isomorphism you must also specifysetpointseach component. The set X withequivalence relation ~equivalent (asgroupoid)one copy oftrivial groupeach equivalence class, butan isomorphism you must also specify what each equivalence class is. Finally,set X equippedan action ofgroup Gequivalent (asgroupoid)one copyGeach orbit ofaction, butan isomorphism you must also specify what set each orbit is.

The collapse ofgroupoid intomere collectiongroups loses some information, even fromcategory-theoretic pointview, because it's not natural. Thus when groupoids arisetermsother structures, as inabove examples,can be helpfulmaintainfull groupoid. If you don't, then you must choosewayview each G(x)terms ofsingle group,this can be rather arbitrary. In our example from topology, you would havemakecoherent choicepaths (or equivalence classespaths) from each point peach point q insame path-connected component.

An examplethis phenomenon thatwell knownphysicscovariancespecial relativity. Working withsingle group correspondspickingspecific framereference,you can do allphysicsthis fashion. But it's more naturaldescribe physics inway that makes no mentionany particular framereference,this correspondsusingentire groupoid. (I needgo into more detail about this. It really isprecise correspondence --particular group involved isPoincaré group -- but I'm not sure how bestexplainyet.)

Lie groupoidsLie algebroids

When studying geometrical objects,arising groupoids often carry some differentiable structure, turning them into Lie groupoids. These can be studiedtermsLie algebroids,analogy torelation between Lie groupsLie algebras.

Explain this