categories of groupoids
Definition 0.1
Groupoid categories, or
categories of groupoids, can be defined
simply by considering a
groupoid
as a
category

with all invertible
morphisms, and
objects
defined by the groupoid class or set of groupoid elements; then, the groupoid category,
,
is defined as the
-category whose objects are
categories (groupoids), and whose morphisms are
functors
of
categories consistent with the definition of
groupoid homomorphisms, or in the case of
topological groupoids, consistent as well with topological groupoid
homeomorphisms. The
2-category
of groupoids
, plays a central role in the generalised, categorical Galois theory involving
fundamental groupoid functors.
Definition 0.2
Let

and

be two groupoids considered as two distinct categories with all invertible morphisms between their objects (or `elements'), respectively,

and

. A
groupoid homomorphism is then defined as a functor

.
A composition
of groupoid homomorphisms is naturally a homomorphism, and natural transformations of groupoid homomorphisms (as defined above by groupoid functors) preserve groupoid structure(s), i.e., both the algebraic
and the topological structure
of groupoids. Thus, in the case of topological groupoids,
, one also has the associated topological
space homeomorphisms that naturally preserve topological structure.
Remark:
Note that the morphisms in the category of groupoids,
, are, of course, groupoid homomorphisms, and
that groupoid homomorphisms also form (groupoid) functor categories
defined in the standard manner for
categories.
Contributors to this entry (in most recent order):
As of this snapshot date, this entry was owned by bci1.