PlanetPhysics/Category of Groupoids 2

Category of Groupoids

Properties

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

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

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

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

Homotopy Theory for groupoids that leads to analogies between groupoids and spaces or 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 homotopy category plays an important role;

1. Groupoids extend the notion of invertible operation by comparison with that available for groups; such invertible operations also occur in the theory of inverse semigroups. Moreover, there are interesting relations beteen inverse semigroups and ordered groupoids. Such concepts are thus applicable to sequential machines and automata whose 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 types of morphisms, such as: quotient morphisms, retractions, covering morphisms, fibrations, universal morphisms, (in contrast to only the epimorphisms and monomorphisms of group theory);

1. A 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 crossed module $C_{M}$, which in the case when $J$ is a group is the traditional crossed module $J\rightarrow Aut(J)$, defined by the inner automorphisms.

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