On Rational Pairings of Functors

In the theory of coalgebras C over a ring R, the rational functor relates the category $_{C^*}{\mathbb{M}}$ of modules over the algebra C ^* (with convolution product) with the category $^C{\mathbb{M}}$ of comodules over C. This is based on the pairing of the algebra C ^* with the coalgebra C provided by the evaluation map ${\rm ev}:C^*\otimes_R C\to R$ . The (rationality) condition under consideration ensures that $^C{\mathbb{M}}$ becomes a coreflective full subcategory of $_{C^*}{\mathbb{M}}$ . We generalise this situation by defining a pairing between endofunctors T and G on any category ${\mathbb{A}}$ as a map, natural in $a,b\in {\mathbb{A}}$ , $$ \beta_{a,b}:{\mathbb{A}}(a, G(b)) \to {\mathbb{A}}(T(a),b), $$ and we call it rational if these all are injective. In case T?=?(T, m _T , e _T ) is a monad and G?=?(G, δ _G , ε _G ) is a comonad on ${\mathbb{A}}$ , 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 ${\mathbb{A}}_{T}$ which generalises the crucial properties of the rational functor for coalgebras. As a special case we consider pairings on monoidal categories.