Let
be a Hausdorff space. Also consider the HDA
concept
of a
double groupoid,
and how it can be completely specified for a Hausdorff space,
. Thus, in
ref. [2] Brown et al. associated to
a double groupoid,
, called the homotopy double groupoid of X which is completely defined by the data specified in Definitions 0.1 to 0.3 in this entry and related objects.
Generally, the geometry of squares and their compositions leads to a common representation of a double groupoid in the following form:
where
is a set of `points',
are `horizontal' and `vertical' groupoids, and
is a set of
`squares' with two compositions.
The laws for a double groupoid are also defined, more generally, for any topological
space
, and make it also describable as a groupoid internal to the category of groupoids. Further details of this general definition are provided next.
Given two groupoids
over a set
, there is a double groupoid
with
as
horizontal and vertical edge groupoids, and squares given by
quadruples
![]() |
(0.2) |
![]() |
(0.3) |
Alternatively, the data for the above double groupoid
can be specified as a triple of groupoid structures:
where:
and
Then, as a first step, consider this data for the homotopy double groupoid specified in the following definition; in order to specify completely such data one also needs to define the related concepts of thin equivalence and the relation of cubically thin homotopy, as provided in the two definitions following the homotopy double groupoid data specified above and in the (main) Definition 0.1.
Here
denotes the path groupoid of
from ref. [1] where it was defined as follows. The objects of
are the points of
. The morphisms
of
are the equivalence classes of paths in
with respect to the following (thin) equivalence relation
, defined as follows. The data for
is defined last; furthermore, the symbols specified after the thin square
symbol specify both the sides (or the groupoid `dimensions') of the square which are involved (i.e., 1 and 2, respectively), and also the order in which the shown operations
(
,
... , etc) are to be performed relative to the thin square specified for each groupoid,
; moreover, all such symbols are explicitly and precisely defined in the related entries of the concepts involved in this definition. These two groupoids can also be pictorially represented as the
pair depicted in the large diagram
(0.1), or
, shown at the top of this page.
Let
be paths in
. Then
is thinly equivalent to
, denoted
, if
there is a thin relative homotopy between
and
.
We note that
is an equivalence relation, see
[2]. We use
to denote
the
class of a path
and call
the semitrack of
. The groupoid
structure of
is induced by concatenation,
+, of paths. Here one makes use of the fact that if
are paths then
there are canonical thin relative homotopies
The source and target maps
of
are given by
if
At the next step, in order to construct the groupoid
data in Definition 0.1, R. Brown et al. defined as follows a
relation
of cubically thin homotopy on the set
of squares.
Let
be squares in
with common vertices.
(i)
is a homotopy between
and
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