CatDat

comonadic

A functor $F : \mathcal{C} \to \mathcal{D}$ is comonadic when there is a comonad $T$ on $\mathcal{D}$ such that $F$ is equivalent to the forgetful functor $U^T : \mathbf{CoAlg}(T) \to \mathcal{D}$.

• Dual property: monadic
• nLab Link

Relevant implications

• comonadic andfull implies equivalence
• comonadic implies conservative andfaithful andleft adjoint
• equivalence implies cocontinuous andcomonadic andleft adjoint

*Those implications also require assumptions on the source or target category.

Examples

There is 1 functor with this property.

• identity functor on the category of sets

Counterexamples

There are 4 functors without this property.

• abelianization functor for groups
• contravariant power set functor
• covariant power set functor
• forgetful functor for vector spaces

Unknown

There is 1 functor for which the database has no information on whether it satisfies this property. Please help us fill in the gaps by contributing to this project.

• free group functor