category of groupoids

1 Category of Groupoids

1.1 Properties

The category of groupoids, $G_{pd}$, has several important properties distinct from those of the category of groups,$G_p$, although it does contain the category of groups as a full subcategory. One such important property is that $G_{pd}$ 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

$$G_{pd}(G\times J,K)\to G_{pd}(G,GPD(J,K))$$

.

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 $0\to 1$;

2. The groupoid $I$ has allowed the development of a useful Homotopy Theory (http://planetmath.org/http://planetmath.org/encyclopedia/HomotopyCategory2.html) 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;

3. 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;

4. 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);

5. 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 $J\to Aut(J)$, defined by the inner automorphisms.

Title	category of groupoids
Canonical name	CategoryOfGroupoids
Date of creation	2013-03-22 19:15:54
Last modified on	2013-03-22 19:15:54
Owner	bci1 (20947)
Last modified by	bci1 (20947)
Numerical id	16
Author	bci1 (20947)
Entry type	Topic
Classification	msc 55U05
Classification	msc 55U35
Classification	msc 55U40
Classification	msc 18G55
Classification	msc 18B40
Related topic	GroupoidCategory
Related topic	HomotopyCategory