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R-category (Definition)
Definition 0.1  

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: algebroid structures and extended symmetries, Hamiltonian algebroids, R-algebroid, $R$-supercategories

Other names:  R-module category
Also defines:  morphism of R-modules, extension of R-algebroids over rings
Keywords:  R-category, extensions of R-algebroids over rings

Attachments:
R-algebroid (Definition) by bci1
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Cross-references: morphism, objects, tensor product, monoidal category, identity, commutative ring, composition, structure, category
There are 5 references to this entry.

This is version 10 of R-category, born on 2008-07-19, modified 2008-08-26.
Object id is 10826, canonical name is RCategory.
Accessed 548 times total.

Classification:
AMS MSC: 81R50 (Quantum theory :: Groups and algebras in quantum theory :: Quantum groups and related algebraic methods)
 81R10 (Quantum theory :: Groups and algebras in quantum theory :: Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $W$-algebras and other current alg)
 18B40 (Category theory; homological algebra :: Special categories :: Groupoids, semigroupoids, semigroups, groups )
 18G55 (Category theory; homological algebra :: Homological algebra :: Homotopical algebra)
 55U40 (Algebraic topology :: Applied homological algebra and category theory :: Topological categories, foundations of homotopy theory)
 55U35 (Algebraic topology :: Applied homological algebra and category theory :: Abstract and axiomatic homotopy theory)
 55U05 (Algebraic topology :: Applied homological algebra and category theory :: Abstract complexes)
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