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# variable groupoid

###### Definition 0.1.

A *variable groupoid* is defined as a family of groupoids
$\{\mathsf{G}_{{\lambda}}\}$ indexed by a parameter $\lambda\in T$ , with $T$ being either an index set or a class (which may be a time parameter, for *time-dependent or dynamic groupoids*). If $\lambda$ belongs to a set $M$, then we may consider simply a projection $\mathsf{G}\times M{\longrightarrow}M$, which is an
example of a trivial fibration. More generally, one can consider a *fibration of groupoids* $\mathsf{G}\hookrightarrow Z{\longrightarrow}M$ (Higgins and Mackenzie, 1990) as defining a non-trivial *variable groupoid*.

Remarks
An indexed family or class of topological groupoids $[\mathsf{G}_{i}]$ with $i\in I$ in the category Grpd of groupoids
with additional axioms, rules, or properties of the underlying topological groupoids,
that specify an indexed family of topological groupoid homomorphisms for each variable groupoid
structure.

Besides systems modelled in terms of a *fibration of groupoids*,
one may consider a *multiple groupoid* defined as a set of $N$
groupoid structures, any distinct pair of which satisfy an
interchange law which can be formulated as follows.
There exists a unique expression with the following content:

$\begin{bmatrix}x&y\\ z&w\end{bmatrix}\quad\objectmargin={0pt}\xy(0,4)*+{}="a",(0,-2)*+{\rule{0.0pt}% {6.45pt}i}="b",(7,4)*+{\;j}="c"\ar@{->}"a";"b"\ar@{->}"a";"c"\endxy,$ | (0.1) |

where $i$ and $j$ must be distinct for this concept to be well defined. This uniqueness can also be represented by the equation

$(x\circ_{j}y)\circ_{i}(z\circ_{j}w)=(x\circ_{i}z)\circ_{j}(y\circ_{i}w).$ | (0.2) |

Remarks This illustrates the principle that a 2-dimensional formula may be more comprehensible than a linear one.

Brown and Higgins, 1981a, showed that certain multiple groupoids
equipped with an extra structure called *connections* were
equivalent to another structure called a *crossed complex*
which had already occurred in homotopy theory. such as
*double, or multiple* groupoids (Brown, 2004; 2005).
For example, the notion of an *atlas* of structures should,
in principle, apply to a lot of interesting, topological and/or
algebraic, structures: groupoids, multiple groupoids, Heyting
algebras, $n$-valued logic algebras and $C^{*}$-convolution
-algebras. Such examples occur frequently in *Higher Dimensional Algebra*
(HDA).

## Mathematics Subject Classification

55U05*no label found*55U35

*no label found*55U40

*no label found*18G55

*no label found*18B40

*no label found*

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