# R-category

###### 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 $A(b,c)\otimes_{R}A(c,d){\longrightarrow}A(b,d)$.

## 0.1 Note:

See also the extension of the R-category to the concept of http://planetphysics.org/?op=getobj&from=objects&id=756R-supercategory.

## 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: Higher dimensional algebroids and Crossed complexes, PhD thesis, University of Wales, Bangor, (1986). (supervised by R. Brown).
• 3 I. C. Baianu , James F. Glazebrook, and Ronald Brown. 2009. Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review. SIGMA 5 (2009), 051, 70 pages. $arXiv:0904.3644$, $doi:10.3842/SIGMA.2009.051$, http://www.emis.de/journals/SIGMA/2009/051/Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
 Title R-category Canonical name Rcategory Date of creation 2013-03-22 18:14:15 Last modified on 2013-03-22 18:14:15 Owner bci1 (20947) Last modified by bci1 (20947) Numerical id 16 Author bci1 (20947) Entry type Definition Classification msc 55U05 Classification msc 55U35 Classification msc 55U40 Classification msc 18G55 Classification msc 18B40 Classification msc 81R10 Classification msc 81R50 Synonym R-module category Related topic Algebroids Related topic HamiltonianAlgebroids Related topic RAlgebroid Defines morphism of R-modules Defines extension of R-algebroids over rings