2 - Representable Equivalences

Summary

We are concerned with equivalences and dualities between subcategories of the categories of modules over rings. Henceforth, by “subcategory” we shall mean full subcategory that is closed under isomorphic images, and all functors between categories of modules are assumed to be additive.

Suppose C and D are subcategories of Mod-R and Mod-S, respectively. A functor H : C → D is an equivalence if there is a functor T : D → C such that T ∘ H and H ∘ T are naturally isomorphic to the identity functors 1_C and 1_D, respectively. When this is the case we write C ≈ D. By Theorem A.3.4 these natural isomorphisms can be taken to be of the form μ : T H → 1_C and θ : 1_D → H T where H μ ∘ θ H = 1_H and μT ∘ T θ = 1_T. That is, μ and θ, an arrow of adjunction and its quasi-inverse, establish T as a left adjoint of H (see Appendix A). If _SV_R is a bimodule, then we have functors

Hom_R(V, _-) : Mod-R ⇄ Mod-S : (_- ⊗_S V),

and we say that the equivalence H : C ⇄ D : T is representable by _SV_R if H and T are naturally isomorphic to the restrictions of these functors, that is,

H ≅ Hom_R(V, _-)|C and T ≅ (_- ⊗_S V)|D.

In this case we shall make the identifications

and then by Theorem A.2.2 the canonical natural transformations ν and η defined beloware natural isomorphisms when restricted to C and D, respectively.