CatDat

monadic

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

• Dual property: comonadic
• nLab Link

Relevant implications

• equivalence implies continuous andmonadic andright adjoint
• full andmonadic implies equivalence
• monadic implies conservative andfaithful andright adjoint

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

Examples

There are 3 functors with this property.

• contravariant power set functor
• forgetful functor for vector spaces
• identity functor on the category of sets

Counterexamples

There are 3 functors without this property.

• abelianization functor for groups
• covariant power set functor
• free group functor

Unknown

There are 0 functors for which the database has no information on whether they satisfy this property.

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