The category of linear
representations of an affine group is isomorphic to the category of comodules over a
k-Hopf-algebra where k denotes a commutative ring. The category of C-comodules
Comod-C over an arbitrary k-coalgebra C is comonadic over the category k-Mod of
k-modules. It is complete, cocomplete and has a cogenerator. The C-comodules
whose cardinality ≦max (cardk, ℵ0) generate the category Comod-C. Comod-C is in
general not abelian but can nicely be embedded into an AB-4 category.
Comod-C is a tensored and cotensored k-Mod-category (enriched over k-Mod)
with a canonical (E,M)-factorization which is the factorization in k-mod if
and only if C is flat. Comod-C has free C-comodules if and only if C is
finitely generated and projective. Furthermore I give numerous examples and
counterexamples as well as the explicit description of all constructions, in particular
of the limits in Comod-C which was not known even for coalgebras over
fields.
