# 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

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