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 $A(b,c)\times A(c,d)\u27f6A(b,d)$ is $R$–bilinear, or is a morphism of $R$-modules $A(b,c){\otimes}_{R}A(c,d)\u27f6A(b,d)$.
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. $arXiv:0904.3644$, $doi:10.3842/SIGMA\mathrm{.2009.051}$, 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 |