CatDat

comonadic

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

• Dual property: monadic
• Related properties: faithful, left adjoint
• nLab Link

Relevant implications

• comonadic implies conservative andfaithful andleft adjoint
• conservative andleft adjoint andpreserves coreflexive equalizers () implies comonadic
• equivalence implies comonadic

Examples

There are 9 functors with this property.

• binary coproduct functor on sets
• binary diagonal functor on the category of sets
• countable copower functor on sets
• discrete topology functor
• doubling functor on sets
• forgetful functor from groups to monoids
• free group functor
• identity functor on the category of sets
• monoid ring functor

Counterexamples

There are 18 functors without this property.

• abelianization functor for groups
• binary product functor on sets
• contravariant power set functor
• covariant power set functor
• enveloping group functor
• forgetful functor for groups
• forgetful functor for rings
• forgetful functor for topological spaces
• forgetful functor for vector spaces
• forgetful functor from abelian groups to groups
• forgetful functor from rings to monoids
• functor of continuous functions
• group of units functor
• modulo p functor
• p-torsion functor
• sequences functor on sets
• squaring functor on sets
• torsion functor

Unknown

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

—