Adjunctions in Full
There are hence numerous functors and natural transformations associated with every adjunction, and only a small portion is sufficient to determine the rest.
An adjunction between categories C and D consists of
- A functor F : C ← D called the left adjoint
- A functor G : C → D called the right adjoint
- A natural isomorphism Φ : homC(F–,–) → homD(–,G–)
- A natural transformation ε : FG → 1C called the counit
- A natural transformation η : 1D → GF called the unit
An equivalent formulation, where X denotes any object of C and Y denotes any object of D:
For every C-morphism there is a unique D-morphism such that the diagrams below commute, and for every D-morphism there is a unique C-morphism in C such that the diagrams below commute:
From this assertion, one can recover that:
- The transformations ε, η, and Φ are related by the equations
- The transformations ε, η satisfy the counit-unit equations
- Each pair is a terminal morphism from F to X in C
- Each pair is an initial morphism from Y to G in D
In particular, the equations above allow one to define Φ, ε, and η in terms of any one of the three. However, the adjoint functors F and G alone are in general not sufficient to determine the adjunction. We will demonstrate the equivalence of these situations below.
Read more about this topic: Adjoint Functors
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