PlanetPhysics/R Category

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R-category definition

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

References

1. R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales--Bangor, Maths Preprint, 1986.
2. G. H. Mosa: \emph{Higher dimensional algebroids and Crossed complexes}, PhD thesis, University of Wales, Bangor, (1986). (supervised by R. Brown).