Grothendieck Categorical Representations
To each object R of R we associate a Grothendieck category Rep(R). For every f ∈ HomR(S, R), f : S → R, we are given an exact functor f o = F : Rep(R) → Rep(S), which commutes with products and coproducts and satisfies the following conditions:
i. (1R)o = IRep(R) for every R ∈ R ii. For g : T → S, f : S → R inR, ( f ◦ g)o = go ◦ f o. We did not demand that
for R = S in R necessarily Rep(R) = Rep(S) If G is the class consisting of objects Rep(R), R ∈ R, we let HomG(Rep(R), Rep(S)) consist of functors of type ho provided these go from Rep(R) to Rep(S). Note that if Rep is separating objects of R, then we may write HomG(Rep(R), Rep(S)) = HomR(S, R)o. In any case G as defined above becomes a category.