PlanetPhysics/R Category

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }

R-category definitionEdit

An  -category   is a \htmladdnormallink{category {http://planetphysics.us/encyclopedia/Cod.html} equipped with an  -module structure on each hom set such that the composition is  -bilinear}. More precisely, let us assume for instance that we are given a commutative ring   with identity. Then a small  -category--or equivalently an  -algebroid -- will be defined as a category enriched in the monoidal category of  -modules, with respect to the monoidal structure of tensor product. This means simply that for all objects   of  , the set   is given the structure of an  -module, and composition Failed to parse (unknown function "\lra"): {\displaystyle A(b,c) \times A(c,d) \lra A(b,d)} is  --bilinear, or is a morphism of  -modules Failed to parse (unknown function "\lra"): {\displaystyle A(b,c) \otimes_R A(c,d) \lra A(b,d)} .

All SourcesEdit

[1][2]

ReferencesEdit

  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: \emph{Higher dimensional algebroids and Crossed complexes}, PhD thesis, University of Wales, Bangor, (1986). (supervised by R. Brown).