category of groupoids

1 Category of Groupoids

1.1 Properties

The category of groupoidsPlanetmathPlanetmath, Gpd, has several important properties distinct from those of the category of groups,Gp, 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 groupoidsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, one can form a groupoid GPD(J,K) such that if G also is a groupoid then there exists a natural equivalence



Other important properties of Gpd are:

  1. 1.

    The categoryMathworldPlanetmath Gpd also has a unit interval object I, which is the groupoid with two objects 0,1 and exactly one arrow 01;

  2. 2.

    The groupoid I has allowed the development of a useful Homotopy Theory ( for groupoids that leads to analogiesMathworldPlanetmath 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. 3.

    Groupoids extend the notion of invertible operationMathworldPlanetmath by comparison with that available for groups; such invertible operations also occur in the theory of inverse semigroups. Moreover, there are interesting relationsMathworldPlanetmathPlanetmathPlanetmath beteen inverse semigroups and ordered groupoids. Such concepts are thus applicable to sequential machines and automata whose state spacesMathworldPlanetmath are semigroupsPlanetmathPlanetmath. Interestingly, the category of finite automata, just like Gpd is also cartesian closed;

  4. 4.

    The category Gpd has a varietyMathworldPlanetmathPlanetmath of types of morphisms, such as: quotient morphismsMathworldPlanetmath, retractionsMathworldPlanetmathPlanetmathPlanetmath, covering morphisms, fibrationsMathworldPlanetmath, universal morphisms, (in contrast to only the epimorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath and monomorphisms of group theory);

  5. 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 equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to a crossed module CM, which in the case when J is a group is the traditional crossed module JAut(J), defined by the inner automorphismsMathworldPlanetmath.


  • 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 groupoidPlanetmathPlanetmath of a map of spaces.(2004). Applied Categorical StructuresMathworldPlanetmath,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:
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