Skip to main content

For questions about adjoint functors from category theory. Use in conjunction with the tag (category-theory).

In category theory, two functors $F : \mathsf{C} \to \mathsf{D}$ and $G : \mathsf{D} \to \mathsf{C}$ are said to be adjoint, denoted $F \dashv G$, if there is a natural bijection: $$\hom_{\mathsf{D}}(F(X), Y) \cong \hom_{\mathsf{C}}(X, G(Y)), \; \forall X \in \mathsf{C}, Y \in \mathsf{D}.$$ The functor $F$ is called the left adjoint, and the functor $G$ is the right adjoint. There are several reformulations of this property, in terms of universal morphisms, unit-counit adjunction, and hom-set adjunction as above.

This concept, initially introduced in the context of , is ubiquitous in mathematics, and many problems can be reformulated in terms of finding an adjoint to some functor. Use this tag if you have questions related to adjoint functors, in conjunction with the tag .