PlanetPhysics/Variable Groupoid

A variable groupoid is defined as a family of [[../QuantumOperatorAlgebra5/|groupoids]] $\{{𝖦}_{\lambda }\}$ indexed by a [[../Parameter/|parameter]] $\lambda \in T$ , with $T$ being either an index set or a class (which may be a time parameter, for time-dependent or \htmladdnormallink{dynamic {http://planetphysics.us/encyclopedia/NewtonianMechanics.html} groupoids}). If $\lambda$ belongs to a set $M$, then we may consider simply a projection Failed to parse (unknown function "\lra"): {\displaystyle \mathsf{G} \times M \lra M} , which is an example of a trivial fibration. More generally, one can consider a fibration of groupoids Failed to parse (unknown function "\lra"): {\displaystyle \mathsf{G} \hookrightarrow Z \lra M} (Higgins and Mackenzie, 1990) as defining a non-trivial variable groupoid .

Remarks An indexed family or class of [[../GroupoidHomomorphism2/|topological groupoids]] Failed to parse (unknown function "\grp"): {\displaystyle [\grp_i]} with $i\in I$ in the [[../Cod/|category]] Grpd of groupoids with additional axioms, rules, or properties of the underlying topological groupoids, that specify an indexed family of topological groupoid [[../TrivialGroupoid/|homomorphisms]] for each variable groupoid structure.

Besides [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|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 {\it interchange law} which can be formulated as follows. There exists a unique expression with the following content:

Failed to parse (unknown function "\directs"): {\displaystyle \begin{bmatrix} x&y \\z&w \end{bmatrix}\quad \directs{j}{i}, }

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

$(x{\circ }_{j}y){\circ }_i(z{\circ }_{j}w)=(x{\circ }_{i}z){\circ }_j(y{\circ }_{i}w).$

Remarks This illustrates the principle that a 2-dimensional [[../Formula/|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 [[../SingularComplexOfASpace/|crossed complex]] which had already occurred in [[../CubicalHigherHomotopyGroupoid/|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, [[../CoIntersections/|topological]] and/or [[../CoIntersections/|algebraic]], structures: groupoids, multiple groupoids, Heyting algebras, $n$-valued logic algebras and $C^{*}$-convolution -algebras. Such examples occur frequently in [[../HigherDimensionalAlgebra2/|higher dimensional algebra]] ([[../2Groupoid2/|HDA]]).