variable groupoid



Definition 0.1.

A variable groupoid is defined as a family of groupoids {𝖦λ} indexed by a parameter λT , with T being either an index setMathworldPlanetmath or a class (which may be a time parameter, for time-dependent or dynamic groupoidsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath). If λ belongs to a set M, then we may consider simply a projectionPlanetmathPlanetmath 𝖦×MM, which is an example of a trivial fibrationMathworldPlanetmath. More generally, one can consider a fibration of groupoids 𝖦ZM (Higgins and Mackenzie, 1990) as defining a non-trivial variable groupoid.

Remarks An indexed family or class of topological groupoidsPlanetmathPlanetmathPlanetmathPlanetmath [𝖦i] with iI in the categoryMathworldPlanetmath Grpd of groupoids with additional axioms, rules, or properties of the underlying topological groupoids, that specify an indexed family of topological groupoid homomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath for each variable groupoid structureMathworldPlanetmath.

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:

[xyzw]\objectmargin=0pt\xy(0,4)*+="a",(0,-2)*+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

(xjy)i(zjw)=(xiz)j(yiw). (0.2)

Remarks This illustrates the principle that a 2-dimensional formulaMathworldPlanetmathPlanetmath 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 equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath 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 algebrasPlanetmathPlanetmath and C*-convolution -algebrasMathworldPlanetmathPlanetmath. Such examples occur frequently in Higher Dimensional AlgebraPlanetmathPlanetmath (HDA).

Title variable groupoid
Canonical name VariableGroupoid
Date of creation 2013-03-22 18:15:45
Last modified on 2013-03-22 18:15:45
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 17
Author bci1 (20947)
Entry type Definition
Classification msc 55U05
Classification msc 55U35
Classification msc 55U40
Classification msc 18G55
Classification msc 18B40
Synonym variable topology
Related topic VariableCategory
Related topic HigherDimensionalAlgebra
Related topic GroupoidCDynamicalSystem
Related topic HDA
Related topic VariableTopology
Related topic HigherDimensionalAlgebraHDA
Related topic Supercategories3
Related topic 2Category2
Defines family of groupoids
Defines GroupoidCDynamicalSystem