PlanetPhysics/R Algebroid

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

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

$$ (f*g)(z)= \sum \{(fx)(gy)\mid z=x\circ y \} ~. $$


 * 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 $$R \cong \mathbb{C}$$~. 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 $$\mathsf{G}$$ by a semigroup $$G'=G\cup \{0\}$$ 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 spatial component given by the set of objects.
 * More generally, an R-category is similarly defined as an extension to this R- algebroid concept.