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(1.1) |
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(1.2) |
Thus, as it is well kown, a topological groupoid is just a groupoid internal to the category of topological spaces and continuous maps. The notion of internal groupoid has proved significant in a number of fields, since groupoids generalise bundles of groups, group actions, and equivalence relations. For a further study of groupoids we refer the reader to Brown (2006).
Several examples of groupoids are:
As a simple, helpful example of a groupoid, consider (b) above. Thus, let R be an equivalence relation on a set X. Then R is a groupoid under the following operations:
. Here,
, (the diagonal of
) and
.
Therefore,
=
.
When
, R is called a trivial groupoid. A special case of a trivial groupoid is
. (So every i is equivalent to every j). Identify
with the matrix unit
. Then the groupoid
is just matrix multiplication except that we only multiply
when
, and
. We do not really lose anything by restricting the multiplication, since the pairs
excluded from groupoid multiplication just give the 0 product in normal algebra anyway.
For a groupoid
to be a locally compact groupoid
means that
is required to be a (second countable) locally compact Hausdorff space, and the product and also inversion maps are required to be continuous. Each
as well as the unit space
is closed in
. What replaces the left Haar measure
on
is a system
of measures
(
), where
is a positive regular Borel measure on
with dense support. In addition, the
's are required to vary continuously (when integrated against
and to form an invariant family in the sense that for each x, the map
is a measure preserving homeomorphism from
onto
. Such a system
is called a left Haar system for the locally compact groupoid
.
As of this snapshot date, this entry was owned by bci1.