R-algebroidRAlgebroid2008-07-19 07:21:482009-02-03 12:31:02DefinitionR-category$R$-moduleconvolution productdefintions of R-algebroidR-CategoryGroupoid-derived algebroid% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\hr}{{\hookrightarrow}}\begin{definition}
If $\mathsf{G}$ is a groupoid (for example, regarded as a category with all morphisms invertible)
then we can construct an $R$-algebroid, $R\mathsf{G}$ as follows. Let us consider first a module over a ring $R$, also called a {\em $R$-module}, that is, a \PMlinkname{module}{Module} $M_R$ that takes its coefficients in a ring $R$. Then, 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 one defines the \emph{convolution product} as follows:
\end{definition}
\med
\begin{equation}
(f*g)(z)= \sum \{(fx)(gy)\mid z=x\circ y \} ~.
\end{equation}
\begin{remark}
As it is very well known, only the second construction is natural
for the topological case, when one needs to replace the general concept of `function' by
the topological-analytical concept of `continuous function with compact support' (or alternatively, with `locally
compact support') for all quantum field theory (QFT) extended symmetry sectors; in this case, one has that $R \cong \mathbb{C}$~.
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 presence of the {\em spatial component} given by the set of objects of the groupoid.
More generally, a \PMlinkname{R-category}{RCategory} is similarly defined as an extension to this R-algebroid
concept.
\end{remark}
\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}