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 $\ev:C^*\ot_R C\to R$. We generalise this situation by defining a {\em pairing} between endofunctors $T$ and $G$ on any category $\A$ as a map, natural in $a,b\in \A$, $$\beta_{a,b}:\A(a, G(b)) \to \A(T(a),b),$$ and we call it {\em rational} if these all are injective. In case $\bT=(T,m_T,e_T)$ is a monad and $\bG=(G,\delta_G,\ve_G)$ is a comonad on $\A$, additional compatibility conditions are imposed on a pairing between $\bT$ and $\bG$. If such a pairing is given and is rational, and $\bT$ has a right adjoint monad $\bT^\di$, we construct a {\em rational functor} as the functor-part of an idempotent comonad on the $\bT$-modules $\A_{\rT}$ which generalises the crucial properties of the rational functor for coalgebras. As a special case we consider pairings on monoidal categories.