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Stream: learning: questions

Topic: does every comonad arise from a monadic adjunction?

Patrick Nicodemus (Jul 18 2022 at 08:07):

If $C$ is a category and $T$ is a monad on $C$ we can always construct a category $C^T$ of $T$-algebras such that the adjunction between $C$ and $C^T$ is monadic.

I am interested in the dual question. If $G$ is a comonad on $D$, is $D$ always equivalent to $C^T$ for some category $C$ and monad $T$ such that $G$ is the comonad arising from the Eilenberg-Moore adjunction?

Patrick Nicodemus (Jul 18 2022 at 08:09):

My first guess would be that we could take $C$ to be the category of cofree $G$-coalgebras. I'm working through this trying to see if it gives me what I want.

Mike Shulman (Jul 22 2022 at 06:27):

Seems unlikely to me. Consider for instance the case when $G$ is a coreflection into a very small subcategory, like the subcategory containing only the initial object.