On Rational Pairings of Functors

In the theory of coalgebras C over a ring R, the rational functor relates the category C∗M of modules over the algebra C∗ (with convolution product) with the category CM of comodules over C. This is based on the pairing of the algebra C∗ with the coalgebra C provided by the evaluation map ev : C∗ ⊗R C → R. The (rationality) condition under consideration ensures that CM becomes a coreflective full subcategory of C∗M. We generalise this situation by defining a pairing between endofunctors T and G on any category A as a map, natural in a, b ∈ A, βa,b : A(a,G(b))→ A(T (a), b), and we call it rational if these all are injective. In case T = (T,mT , eT ) is a monad and G = (G, δG, εG) is a comonad on A, additional compatibility conditions are imposed on a pairing between T and G. If such a pairing is given and is rational, and T has a right adjoint monad T , we construct a rational functor as the functor-part of an idempotent comonad on the T-modules AT which generalises the crucial properties of the rational functor for coalgebras. As a special case we consider pairings on monoidal categories.