1 R-category definition
Definition 1.1.
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) × 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).
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).
