## You are here

HomeR-category

## Primary tabs

# 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 R-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. $arXiv:0904.3644$, $doi:10.3842/SIGMA.2009.051$, Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

## Mathematics Subject Classification

55U05*no label found*55U35

*no label found*55U40

*no label found*18G55

*no label found*18B40

*no label found*81R10

*no label found*81R50

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections