variable groupoid

curve

Definition 0.1.

A variable groupoid is defined as a family of groupoids$\{{𝖦}_{\lambda }\}$ 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 $𝖦\times M⟶M$, which is anexample of a trivial fibration. More generally, one can consider a fibration of groupoids $𝖦\hookrightarrow Z⟶M$ (Higgins and Mackenzie, 1990) as defining a non-trivial variable groupoid.

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

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 aninterchange law which can be formulated as follows.There exists a unique expression with the following content:

(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)

RemarksThis illustrates the principle that a 2-dimensional formula may bemore comprehensible than a linear one.

Brown and Higgins, 1981a, showed that certain multiple groupoidsequipped with an extra structure called connections wereequivalent to another structure called a crossed complexwhich had already occurred in homotopy theory. such asdouble, 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/oralgebraic, structures: groupoids, multiple groupoids, Heytingalgebras, $n$-valued logic algebras and $C^{*}$-convolution-algebras. Such examples occur frequently in Higher Dimensional Algebra(HDA).