category of groupoids
1 Category of Groupoids
1.1 Properties
The category of groupoids, 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 groupoids
, one can form a groupoid GPD(J,K) such that if G also is a groupoid then there exists a natural equivalence
Gpd(G×J,K)→Gpd(G,GPD(J,K)) |
.
Other important properties of Gpd are:
-
1.
The category
Gpd also has a unit interval object I, which is the groupoid with two objects 0,1 and exactly one arrow 0→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 Gpd is also cartesian closed;
-
4.
The category Gpd 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 CM, which in the case when J is a group is the traditional crossed module J→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: 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
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 |