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R-category (Definition)

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).



"R-category" is owned by bci1.

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See Also: R-algebroid, R-supercategory

Keywords:  R-category definition

Cross-references: morphism, objects, tensor, identity, commutative ring, composition, category
There are 7 references to this object.

This is version 2 of R-category, born on 2009-01-31, modified 2009-01-31.
Object id is 457, canonical name is RCategory.
Accessed 436 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
Pending Errata and Addenda
None.
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