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 Crossedcomplexes, 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)

单词	Rcategory
释义	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 Crossedcomplexes, 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)
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