The category of groupoids, G p d {\displaystyle 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 G p d {\displaystyle Gpd} is cartesian closed . Thus, if J {\displaystyle J} and K {\displaystyle K} are two groupoids, one can form a groupoid G P D ( J , K ) {\displaystyle GPD(J,K)} such that if G {\displaystyle G} also is a groupoid then there exists a natural equivalence G p d ( G × J , K ) → G p d ( G , G P D ( J , K ) ) {\displaystyle Gpd(G\times J,K)\rightarrow Gpd(G,GPD(J,K))} . Other important properties of G p d {\displaystyle G_{pd}} are: The category G p d {\displaystyle G_{pd}} also has a unit interval object I {\displaystyle I} , which is the groupoid with two objects 0 , 1 {\displaystyle 0,1} and exactly one arrow 0 → 1 {\displaystyle 0\rightarrow 1} ; The groupoid I {\displaystyle 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; 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 p d {\displaystyle G_{pd}} is also cartesian closed ; The category G p d {\displaystyle 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); A monoid object, E N D ( J ) = G P D ( J , J ) {\displaystyle END(J)=GPD(J,J)} , also exists in the category of groupoids, that contains a maximal subgroup object denoted here as A U T ( J ) {\displaystyle AUT(J)} . Regarded as a group object in the category groupoids, A U T ( J ) {\displaystyle AUT(J)} is equivalent to a crossed module C M {\displaystyle C_{M}} , which in the case when J {\displaystyle J} is a group is the traditional crossed module J → A u t ( J ) {\displaystyle J\rightarrow Aut(J)} , defined by the inner automorphisms. \begin{thebibliography} {9}  May, J.P. 1999, A Concise Course in Algebraic Topology. , The University of Chicago Press: Chicago R. Brown and G. Janelidze.(2004). Galois theory and a new homotopy double groupoid of a map of spaces.(2004). Applied Categorical Structures ,12 : 63-80. Pdf file in arxiv: math.AT/0208211 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 1-195: http://www.tac.mta.ca/tac/reprints/articles/7/tr7.pdf
{'wiki_source': 'wikiversity', 'other_info': "{'content_model': 'wikitext', 'language': 'en', 'title': 'PlanetPhysics/Category of Groupoids 2', 'type': 'text'}"}