PlanetPhysics/Category of Groupoids 2

Category of Groupoids

Properties

The category of groupoids, $G_{pd}$, has several important properties not available for [[../TrivialGroupoid/|groups]], although it does contain the [[../Cod/|category]] of groups as a full subcategory. One such important property is that $Gpd$ is cartesian closed . Thus, if $J$ and $K$ are two [[../QuantumOperatorAlgebra5/|groupoids]], one can form a groupoid $GPD(J,K)$ such that if $G$ also is a groupoid then there exists a [[../IsomorphismClass/|natural equivalence]] $Gpd(G\times J,K)\to Gpd(G,GPD(J,K))$.

Other important properties of $G_{pd}$ are:

1. The category $G_{pd}$ also has a unit interval [[../TrivialGroupoid/|object]] $I$, which is the groupoid with two objects $0,1$ and exactly one arrow $0\to 1$;

1. The groupoid $I$ has allowed the development of a useful

Homotopy Theory for groupoids that leads to analogies between groupoids and spaces or [[../NoncommutativeGeometry4/|manifolds]]; effectively, groupoids may be viewed as "adding the spatial notion of a `place' or location" to that of a group. In this context, the [[../HomotopyCategory/|homotopy category]] plays an important role;

1. Groupoids extend the notion of invertible [[../Cod/|operation]] by comparison with that available for groups; such invertible operations also occur in the theory of inverse [[../TrivialGroupoid/|semigroups]]. Moreover, there are interesting [[../Bijective/|relations]] beteen inverse semigroups and ordered groupoids. Such [[../PreciseIdea/|concepts]] are thus applicable to [[../AAT/|sequential machines]] and automata whose [[../StableAutomaton/|state spaces]] are semigroups. Interestingly, the category of finite automata, just like $G_{pd}$ is also cartesian closed ;

1. The category $G_{pd}$ has a variety of [[../Bijective/|types]] of [[../TrivialGroupoid/|morphisms]], such as: quotient morphisms, [[../IsomorphicObjectsUnderAnIsomorphism/|retractions]], [[../CubicalHigherHomotopyGroupoid/|covering]] morphisms, fibrations, universal morphisms, (in contrast to only the [[../IsomorphicObjectsUnderAnIsomorphism/|epimorphisms]] and [[../InjectiveMap/|monomorphisms]] of group theory);

1. A [[../TrivialGroupoid/|monoid]] object, $END(J)=GPD(J,J)$, also exists in the category of groupoids, that contains a maximal subgroup object denoted here as $AUT(J)$. Regarded as a group object in the category groupoids, $AUT(J)$ is equivalent to a [[../CubicalHigherHomotopyGroupoid/|crossed module]] $C_M$, which in the case when $J$ is a group is the traditional crossed module $J\to Aut(J)$, defined by the inner automorphisms.

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