Regular pairings of functors and weak (co)monads

• Дата: 2018
• Автор: Wisbauer, Robert
• Формат: Стаття
• Мова: English
• Опубліковано: Lugansk National Taras Shevchenko University 2018
• Теми: pairing of functors; adjoint functors; weak monads and comonads; \(r\)-unital monads; \(r\)-counital comonads; 18A40; 18C20; 16T15
• Онлайн доступ: https://admjournal.luguniv.edu.ua/index.php/adm/article/view/738
• Назва журналу: Algebra and Discrete Mathematics

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