Definition 0.1.

If 𝖦 is a groupoidPlanetmathPlanetmathPlanetmathPlanetmath (for example, regarded as a categoryMathworldPlanetmath with all morphismsMathworldPlanetmath invertible) then we can construct an R-algebroid, R𝖦 as follows. Let us consider first a module over a ring R, also called a R-module, that is, a module (http://planetmath.org/Module) MR that takes its coefficients in a ring R. Then, the object set of R𝖦 is the same as that of 𝖦 and R𝖦(b,c) is the free R-module on the set 𝖦(b,c), with composition given by the usual bilinear rule, extending the composition of 𝖦.

Definition 0.2.

Alternatively, one can define R¯𝖦(b,c) to be the set of functions 𝖦(b,c)R with finite support, and then one defines the convolution productPlanetmathPlanetmath as follows:

(f*g)(z)={(fx)(gy)z=xy}. (0.1)
Remark 0.1.

As it is very well known, only the second construction is natural for the topological case, when one needs to replace the general concept of ‘function’ by the topological-analytical concept of ‘continuous functionMathworldPlanetmathPlanetmath with compact support’ (or alternatively, with ‘locally compact supportMathworldPlanetmath’) for all quantum field theory (QFT) extended symmetryPlanetmathPlanetmath sectors; in this case, one has that R . The point made here is that to carry out the usual construction and end up with only an algebra rather than an algebroid, is a procedure analogous to replacing a groupoid 𝖦 by a semigroup G=G{0} in which the compositions not defined in G are defined to be 0 in G. We argue that this construction removes the main advantage of groupoids, namely the presence of the spatial component given by the set of objects of the groupoid.

More generally, a R-category (http://planetmath.org/RCategory) is similarly defined as an extensionPlanetmathPlanetmath to this R-algebroid concept.


  • 1 R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales–Bangor, Maths Preprint, 1986.
  • 2 G. H. Mosa: Higher dimensional algebroids and Crossed complexes, PhD thesis, University of Wales, Bangor, (1986). (supervised by R. Brown).
Title R-algebroid
Canonical name Ralgebroid
Date of creation 2013-03-22 18:14:19
Last modified on 2013-03-22 18:14:19
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 25
Author bci1 (20947)
Entry type Definition
Classification msc 81T10
Classification msc 81P05
Classification msc 81T05
Classification msc 81R10
Classification msc 81R50
Synonym groupoid-derived algebroids
Synonym double groupoidPlanetmathPlanetmathPlanetmath dual of an algebroid
Related topic Module
Related topic RCategory
Related topic Algebroids
Related topic HamiltonianAlgebroids
Related topic RSupercategory
Related topic SuperalgebroidsAndHigherDimensionalAlgebroids
Related topic CategoricalAlgebras
Defines R-module
Defines convolution product
Defines R-algebroid