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%%% Primary Title: R-category
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%%% Owner: bci1
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\begin{document}
\section{R-category definition}
\begin{definition}
An \textit{$R$-category }$A$ is a \emph{\htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} equipped with an $R$-module structure on each \textit{hom} set such that the \htmladdnormallink{composition}{http://planetphysics.us/encyclopedia/Cod.html} is $R$-bilinear}. More precisely, let us assume for instance that we are given a \htmladdnormallink{commutative ring}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} $R$ with \htmladdnormallink{identity}{http://planetphysics.us/encyclopedia/Cod.html}. Then a small $R$-category--or equivalently an \emph{$R$-algebroid}-- will be defined as a category enriched in the monoidal category of $R$-modules, with respect to the monoidal structure of \htmladdnormallink{tensor}{http://planetphysics.us/encyclopedia/Tensor.html} product. This means simply that for all \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $b,c$ of $A$, the set $A(b,c)$ is given the structure of an $R$-module, and composition $A(b,c) \times A(c,d) \lra A(b,d)$ is $R$--bilinear, or is a \htmladdnormallink{morphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of $R$-modules $A(b,c) \otimes_R A(c,d) \lra A(b,d)$.
\end{definition}
\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}
\end{document}
