# PlanetPhysics/R Algebroid

### R-algebroid

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

Alternatively, one can define ${\displaystyle {\bar {R}}{\mathsf {G}}(b,c)}$ 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:

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

• 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 ${\displaystyle R\cong \mathbb {C} }$~. 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 ${\displaystyle {\mathsf {G}}}$ by a semigroup ${\displaystyle G'=G\cup \{0\}}$ in which the compositions not defined in ${\displaystyle G}$ are defined to be ${\displaystyle 0}$ in ${\displaystyle G'}$. 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.

## References

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).