CatDat

right adjoint

A functor $F : \mathcal{C} \to \mathcal{D}$ is a right adjoint when there is a functor $G : \mathcal{D} \to \mathcal{C}$ such that there are natural bijections $\hom(G(A),B) \cong \hom(A,F(B))$.

• Dual property: left adjoint
• nLab Link

Relevant implications

• continuous implies right adjoint *
• equivalence implies continuous andmonadic andright adjoint
• monadic implies conservative andfaithful andright adjoint
• right adjoint implies continuous

*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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