Regular pairings of functors and weak (co)monads

Abstract

For functors \(L:\mathbb{A}\to \mathbb{B}\) and \(R:\mathbb{B}\to \mathbb{A}\) between

any categories \(\mathbb{A}\) and \(\mathbb{B}\), a  pairing is defined by maps, natural in 

\(A\in \mathbb{A}\) and  \(B\in \mathbb{B}\),

\[\xymatrix{{\rm Mor}_\mathbb{B} (L(A),B) \ar@<0.5ex>[r]^{\alpha} & {\rm Mor}_\mathbb{A} (A,R(B))\ar@<0.5ex>[l]^{\beta}}.\]

 
\((L,R)\) is an adjoint pair provided \(\alpha\) (or \(\beta\)) is a bijection. In this case the composition \(RL\) defines a monad on the category \(\mathbb{A}\),  \(LR\) defines a comonad on the category \(\mathbb{B}\), and there is a well-known correspondence between monads (or comonads) and adjoint pairs of functors.

For various applications it was observed that the conditions for a unit of a monad was too restrictive and weakening it still allowed for a useful generalised notion of a monad. This led to the introduction of weak monads and weak comonads and the definitions needed were made without referring to this kind of adjunction. The motivation for the present paper is to show that these notions can be naturally derived from pairings of functors  \((L,R,\alpha,\beta)\) with  \(\alpha = \alpha\cdot \beta\cdot \alpha\) and \(\beta = \beta \cdot\alpha\cdot\beta\).  Following closely the constructions known for monads (and unital modules) and comonads (and counital comodules), we show that any weak (co)monad on \(\mathbb{A}\) gives rise to a regular pairing between \(\mathbb{A}\) and the category of compatible (co)modules.