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Pure Mathematics

Adjunction contexts and regular quasi-monads

Profile image of Robert WisbauerRobert Wisbauer

2011

Abstract

For functors $L:\A\to \B$ and $R:\B\to \A$ between any categories $\A$ and $\B$, a {\em pairing} is defined by maps, natural in $A\in \A$ and $B\in \B$, $$\xymatrix{\Mor_\B (L(A),B) \ar@<0.5ex>[r]^{\alpha} & \Mor_\A (A,R(B))\ar@<0.5ex>[l]^{\beta}}.$$ $(L,R)$ is an {\em adjoint pair} provided $\alpha$ (or $\beta$) is a bijection. In this case the composition $RL$ defines a monad on the category $\A$, $LR$ defines a comonad on the category $\B$, 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 {\em weak monads} and {\em 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 $\alpha = \alpha\dcirc \beta\dcirc \alpha$ and $\beta = \beta \dcirc\alpha\dcirc\beta$. Following closely the constructions known for monads (and unital modules) and comonads (and counital comodules), we show that any weak (co)monad on $\A$ gives rise to a regular pairing between $\A$ and the category of {\em compatible (co)modules}.

Figures (4)
The top and bottom diagrams are commutative by compatibility of the G-modules, the right trapezium is commutative since T(f) is a G-morphism, and the outer paths commute by symmetry of 7’. Thus the inner diagram is commutative showing naturality of p. oO
The top and bottom diagrams are commutative by compatibility of the G-modules, the right trapezium is commutative since T(f) is a G-morphism, and the outer paths commute by symmetry of 7’. Thus the inner diagram is commutative showing naturality of p. oO
is commutative where the U’s denote the forgetful functors. 5.5. Lifting of functors to comodules. Let (G,4d,<) and (H,6’,¢’) be r-unital comonads on the categories A and B, respectively, and AS, BH the corresponding categories of the compatible comodules (ees 4.2). Given a functor JT: A > B, a functor T: AG > BY i is said to be is a lifting of T if the diagram
is commutative where the U’s denote the forgetful functors. 5.5. Lifting of functors to comodules. Let (G,4d,<) and (H,6’,¢’) be r-unital comonads on the categories A and B, respectively, and AS, BH the corresponding categories of the compatible comodules (ees 4.2). Given a functor JT: A > B, a functor T: AG > BY i is said to be is a lifting of T if the diagram
6.2. Weak crossed products. Given (F),7) and T :A— A, the composition TF may have a weak monad structure without requiring such a structure on T. For example, replacing the natural transformations j1F' and X- F'y) in tions 6.1{c) by some natural transforma-
6.2. Weak crossed products. Given (F),7) and T :A— A, the composition TF may have a weak monad structure without requiring such a structure on T. For example, replacing the natural transformations j1F' and X- F'y) in tions 6.1{c) by some natural transforma-
7.1. Liftings of monads and comonads. Consider the diagrams In both cases the lifting properties are related to a natural transformation Throughout this section let (F’, 41,7) denote a weak monad and (G, 6, ¢) a weak comonad on any category A. In this section we investigate the lifting properties to compatible F-modules and compatible G-comodules, respectively.
7.1. Liftings of monads and comonads. Consider the diagrams In both cases the lifting properties are related to a natural transformation Throughout this section let (F’, 41,7) denote a weak monad and (G, 6, ¢) a weak comonad on any category A. In this section we investigate the lifting properties to compatible F-modules and compatible G-comodules, respectively.

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