# 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 $\displaystyle \mathsf{G}(b,c)\lra R$ with finite support, and then we define the \htmladdnormallink{convolution {http://planetphysics.us/encyclopedia/AssociatedGroupoidAlgebraRepresentations.html} product} as follows:

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