adjuncions

An adjunction is a fundamental concept in category theory that describes a specific relationship between two functors. When two functors, say F and G, exist between categories C and D, an adjunction means that F maps objects and morphisms in C to objects and morphisms in D in a way that is "best approximated" by G mapping back from D to C. More formally, an adjunction between functors F: C -> D and G: D -> C is defined by a natural isomorphism between the set of morphisms from an object X in C to G(Y) in D, and the set of morphisms from F(X) in D to Y in D. This can be written as Hom_D(F(X), Y) ≅ Hom_C(X, G(Y)) for all objects X in C and Y in D.

This isomorphism, known as the adjunction map or unit and counit, provides a deep connection between the two categories and their respective functors. The functor F is called the left adjoint to G, and G is called the right adjoint to F. Adjunctions are ubiquitous in mathematics and computer science, appearing in areas such as abstract algebra, topology, logic, and programming language semantics. They capture essential dualities and constructions, providing a unifying framework for understanding many mathematical structures and their relationships. For example, the relationship between free groups and the forgetful functor from the category of groups to the category of sets is an adjunction. Recognizing an adjunction often reveals underlying structure and simplifies proofs.