PlanetPhysics/R Category

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