Definition 0.1.

An R-categoryMathworldPlanetmath 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 identityPlanetmathPlanetmath. 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 productPlanetmathPlanetmathPlanetmath. 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)×A(c,d)A(b,d) is R–bilinear, or is a morphism of R-modules A(b,c)RA(c,d)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.


  • 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/SymmetryPlanetmathPlanetmath, 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