category of groupoids
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:
- The category
also has a unit interval object
, which is the groupoid with two objects
and exactly one arrow
;
- 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;
- 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;
- 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);
- 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.
-
- 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
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