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