On Rational Pairings of Functors

Abstract

In the theory of coalgebras C over a ring R, the rational functor relates the category of modules over the algebra C * (with convolution product) with the category of comodules over C. It is based on the pairing of the algebra C * with the coalgebra C provided by the evaluation map ev :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,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 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 A T which generalises the crucial properties of the rational functor for coalgebras. As a special case we consider pairings on monoidal categories.

Coalgebraic structures in module theory

“…The constructions of this section can also be formulated in a general categorical context and are investigated in [12].…”