category theory

# Contents

## Idea

The notion of a comonadic functor is the dual of that of a monadic functor. See there for more background.

## Definition

Given a pair $L⊣R$ of adjoint functors, $L:A\to B:R$, with counit $ϵ$ and unit $\eta$, one forms a comonad $\Omega =\left(\Omega ,\delta ,ϵ\right)$ by $\Omega ≔L\circ R$, $\delta ≔L\eta R$. $\Omega$ comodules form a category ${B}_{\Omega }$ and there is a natural comparison functor $K={K}_{\Omega }:A\to {B}_{\Omega }$ given by $A↦\left(LA,LA\stackrel{L\left({\eta }_{A}\right)}{\to }LRLA\right)$.

A functor $L:A\to B$ is comonadic if it has a right adjoint $R$ and the corresponding comparison functor $K$ is an equivalence of categories. The adjunction $L⊣R$ is said to be a comonadic adjunction.

## Properties

Beck’s monadicity theorem has its dual, comonadic analogue. To discuss it, observe that for every $\Omega$-comodule $\left(N,\rho \right)$,

manifestly exhibits a split equalizer sequence.

Revised on May 18, 2011 15:33:18 by Urs Schreiber (131.211.238.127)