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^{}.
References
 1 May, J.P. 1999, A Concise Course in Algebraic Topology., The University of Chicago Press: Chicago
 2 R. Brown and G. Janelidze.(2004). Galois theory and a new homotopy double groupoid^{} of a map of spaces.(2004). Applied Categorical Structures^{},12: 6380. Pdf file in arxiv: math.AT/0208211
 3 P. J. Higgins. 1971. Categories and Groupoids., Originally published by: Van Nostrand Reinhold, 1971 Republished in: Reprints in Theory and Applications of Categories, No. 7 (2005) pp 1195: http://www.tac.mta.ca/tac/reprints/articles/7/tr7.pdf
Title  category of groupoids 
Canonical name  CategoryOfGroupoids 
Date of creation  20130322 19:15:54 
Last modified on  20130322 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 