On Rational Pairings of Functors

TL;DR

This paper generalizes the concept of rational functors from coalgebra theory to arbitrary categories by defining and analyzing pairings between endofunctors, leading to new constructions of rational functors with broad applicability.

Contribution

It introduces a general framework for rational pairings of endofunctors on categories, extending the classical coalgebra rational functor to a more abstract setting.

Findings

Defined rational pairings between endofunctors on categories.

Constructed a rational functor as an idempotent comonad on module categories.

Extended the concept to pairings on monoidal categories.

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.