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 Failed to parse (unknown function "\lra"): {\displaystyle A(b,c) \times A(c,d) \lra A(b,d)} is $R$ --bilinear, or is a morphism of $R$ -modules Failed to parse (unknown function "\lra"): {\displaystyle A(b,c) \otimes_R A(c,d) \lra A(b,d)} .

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References

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

[BMos86-1] R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales--Bangor, Maths Preprint, 1986.

[Mo86-2] G. H. Mosa: \emph{Higher dimensional algebroids and Crossed complexes}, PhD thesis, University of Wales, Bangor, (1986). (supervised by R. Brown).

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