Adjunctions between functors #

F ⊣ G represents the data of an adjunction between two functors F : C ⥤ D and G : D ⥤ C. F is the left adjoint and G is the right adjoint.

We provide various useful constructors:

mkOfHomEquiv
mk': construct an adjunction from the data of a hom set equivalence, unit and counit natural transformations together with proofs of the equalities homEquiv_unit and homEquiv_counit relating them to each other.
leftAdjointOfEquiv / rightAdjointOfEquiv construct a left/right adjoint of a given functor given the action on objects and the relevant equivalence of morphism spaces.
adjunctionOfEquivLeft / adjunctionOfEquivRight witness that these constructions give adjunctions.

There are also typeclasses IsLeftAdjoint / IsRightAdjoint, which asserts the existence of a adjoint functor. Given [F.IsLeftAdjoint], a chosen right adjoint can be obtained as F.rightAdjoint.

Adjunction.comp composes adjunctions.

toEquivalence upgrades an adjunction to an equivalence, given witnesses that the unit and counit are pointwise isomorphisms. Conversely Equivalence.toAdjunction recovers the underlying adjunction from an equivalence.

Overview of the directory `CategoryTheory.Adjunction` #

Adjoint lifting theorems are in the directory Lifting.
The file AdjointFunctorTheorems proves the adjoint functor theorems.
The file Comma shows that for a functor G : D ⥤ C the data of an initial object in each StructuredArrow category on G is equivalent to a left adjoint to G, as well as the dual.
The file Evaluation shows that products and coproducts are adjoint to evaluation of functors.
The file FullyFaithful characterizes when adjoints are full or faithful in terms of the unit and counit.
The file Limits proves that left adjoints preserve colimits and right adjoints preserve limits.
The file Mates establishes the bijection between the 2-cells
```
        L₁                  R₁
      C --→ D             C ←-- D
    G ↓  ↗  ↓ H         G ↓  ↘  ↓ H
      E --→ F             E ←-- F
        L₂                  R₂
```
where L₁ ⊣ R₁ and L₂ ⊣ R₂. Specializing to a pair of adjoints L₁ L₂ : C ⥤ D, R₁ R₂ : D ⥤ C, it provides equivalences (L₂ ⟶ L₁) ≃ (R₁ ⟶ R₂) and (L₂ ≅ L₁) ≃ (R₁ ≅ R₂).
The file Opposites contains constructions to relate adjunctions of functors to adjunctions of their opposites.
The file Reflective defines reflective functors, i.e. fully faithful right adjoints. Note that many facts about reflective functors are proved in the earlier file FullyFaithful.
The file Restrict defines the restriction of an adjunction along fully faithful functors.
The file Triple proves that in an adjoint triple, the left adjoint is fully faithful if and only if the right adjoint is.
The file Unique proves uniqueness of adjoints.
The file Whiskering proves that functors F : D ⥤ E and G : E ⥤ D with an adjunction F ⊣ G, induce adjunctions between the functor categories C ⥤ D and C ⥤ E, and the functor categories E ⥤ C and D ⥤ C.

The file CategoryTheory.Monad.Adjunction develops the basic relationship between adjunctions and (co)monads. There it is also shown that given an adjunction L ⊣ R and an isomorphism L ⋙ R ≅ 𝟭 C, the unit is an isomorphism, and similarly for the counit.

Adjunctions between functors #

Overview of the directory CategoryTheory.Adjunction #

Other files related to adjunctions #

Overview of the directory `CategoryTheory.Adjunction` #