PlanetPhysics/R Algebroid

R-algebroidEdit

If is a groupoid (for example, considered as a category with all morphisms invertible) then we can construct an -algebroid , as follows. The object set of is the same as that of and is the free -module on the set , with composition given by the usual bilinear rule, extending the composition of .

Alternatively, one can define to be the set of functions Failed to parse (unknown function "\lra"): {\displaystyle \mathsf{G}(b,c)\lra R} with finite support, and then we define the \htmladdnormallink{convolution {http://planetphysics.us/encyclopedia/AssociatedGroupoidAlgebraRepresentations.html} product} as follows:

  • As it is very well known, only the second construction is natural for the topological case, when one needs to replace 'function' by 'continuous function with compact support' (or \emph{locally compact support} for the QFT extended symmetry sectors), and in this case ~. 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 by a semigroup in which the compositions not defined in are defined to be in . We argue that this construction removes the main advantage of groupoids, namely the spatial component given by the set of objects.
  • More generally, an R-category is similarly defined as an extension to this R- algebroid concept.

All SourcesEdit

[1][2]

ReferencesEdit

  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: \emph{Higher dimensional algebroids and Crossed complexes}, PhD thesis, University of Wales, Bangor, (1986). (supervised by R. Brown).