nLab
comonadic functor

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 $\Omega =(\Omega ,\delta ,ϵ)$ 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 $A\mapsto (LA,LA\stackrel{L({\eta }_A)}{\to }LRLA)$.

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 $(N,\rho )$,

manifestly exhibits a split equalizer sequence.

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Related concepts

• monadic functor, comonadic functor

• monadicity theorem