R-algebroidRAlgebroid2009-01-31 18:59:222009-01-31 18:59:22DefinitionringoidR-algebroid% almost certainly you want these
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\newcommand{\hr}{{\hookrightarrow}}\subsection{R-algebroid}
\begin{definition}
If $\mathsf{G}$ is a groupoid (for example, considered as a category with all morphisms invertible)
then we can construct an \emph{$R$-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 $\mathsf{G}$.
\end{definition}
\begin{definition}
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 \emph{convolution product} as follows:
\end{definition}
\begin{equation}
(f*g)(z)= \sum \{(fx)(gy)\mid z=x\circ y \} ~.
\end{equation}
\begin{remark}
\begin{itemize}
\item 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.
\item More generally, an R-category is similarly defined as an extension to this R- algebroid concept.
\end{itemize}
\begin{thebibliography}{9}
\bibitem{BMos86}
R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales--Bangor, Maths Preprint, 1986.
\bibitem{Mo86}
G. H. Mosa: \emph{Higher dimensional algebroids and Crossed
complexes}, PhD thesis, University of Wales, Bangor, (1986). (supervised by R. Brown).
\end{thebibliography}