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 [[../Cod/|composition]] is $R$-bilinear}. More precisely, let us assume for instance that we are given a [[../AbelianCategory3/|commutative ring]] $R$ with [[../Cod/|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/|tensor]] product. This means simply that for all [[../TrivialGroupoid/|objects]] $b,c$ of $A$, the set $A(b,c)$ is given the structure of an $R$-module, and composition Failed to parse (unknown function "\lra"): {\displaystyle A(b,c) \times A(c,d) \lra A(b,d)} is $R$--bilinear, or is a [[../TrivialGroupoid/|morphism]] of $R$-modules Failed to parse (unknown function "\lra"): {\displaystyle A(b,c) \otimes_R A(c,d) \lra A(b,d)} .