PlanetPhysics/Category of Groupoids 2

Category of Groupoids edit

Properties edit

The category of groupoids,  , 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   is cartesian closed . Thus, if   and   are two groupoids, one can form a groupoid   such that if   also is a groupoid then there exists a natural equivalence  .

Other important properties of   are:

  1. The category   also has a unit interval object  , which is the groupoid with two objects   and exactly one arrow  ;
  1. The groupoid   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   is also cartesian closed ;
  1. The category   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,  , also exists in the category of groupoids, that contains a maximal subgroup object denoted here as  . Regarded as a group object in the category groupoids,   is equivalent to a crossed module  , which in the case when   is a group is the traditional crossed module  , defined by the inner automorphisms.

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  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 : 63-80. 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 1-195: http://www.tac.mta.ca/tac/reprints/articles/7/tr7.pdf