R-category

R-category definition

Definition 1.1  

An $ R$ -category $ A$ is a category 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) {\longrightarrow}A(b,d)$ is $ R$ -bilinear, or is a morphism of $ R$ -modules $ A(b,c) \otimes_R A(c,d) {\longrightarrow}A(b,d)$ .

Bibliography

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: Higher dimensional algebroids and Crossed complexes, PhD thesis, University of Wales, Bangor, (1986). (supervised by R. Brown).



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