Adjoint lifting #
This file gives two constructions for building left adjoints: the adjoint triangle theorem and the
adjoint lifting theorem.
The adjoint triangle theorem concerns a functor U : B ⥤ C with a left adjoint F such
that ε_X : FUX ⟶ X is a regular epi. Then for any category A with coequalizers of reflexive
pairs, a functor R : A ⥤ B has a left adjoint if (and only if) the composite R ⋙ U does.
Note that the condition on U regarding ε_X is automatically satisfied in the case when U is
a monadic functor, giving the corollary: monadicAdjointTriangleLift, i.e. if U is monadic,
A has reflexive coequalizers then R : A ⥤ B has a left adjoint provided R ⋙ U does.
The adjoint lifting theorem says that given a commutative square of functors (up to isomorphism):
Q
A → B
U ↓ ↓ V C → D R
where U and V are monadic and A has reflexive coequalizers, then if R has a left adjoint
then Q has a left adjoint.
Implementation #
It is more convenient to prove this theorem by assuming we are given the explicit adjunction rather
than just a functor known to be a right adjoint. In docstrings, we write (η, ε) for the unit
and counit of the adjunction adj₁ : F ⊣ U and (ι, δ) for the unit and counit of the adjunction
adj₂ : F' ⊣ R ⋙ U.
TODO #
Dualise to lift right adjoints through comonads (by reversing 1-cells) and dualise to lift right adjoints through monads (by reversing 2-cells), and the combination.
References #
- https://ncatlab.org/nlab/show/adjoint+triangle+theorem
- https://ncatlab.org/nlab/show/adjoint+lifting+theorem
- Adjoint Lifting Theorems for Categories of Algebras (PT Johnstone, 1975)
- A unified approach to the lifting of adjoints (AJ Power, 1988)
To show that ε_X is a coequalizer for (FUε_X, ε_FUX), it suffices to assume it's always a
coequalizer of something (i.e. a regular epi).
Instances For
(Implementation)
To construct the left adjoint, we use the coequalizer of F' U ε_Y with the composite
F' U F U X ⟶ F' U F U R F U' X ⟶ F' U R F' U X ⟶ F' U X
where the first morphism is F' U F ι_UX, the second is F' U ε_RF'UX, and the third is δ_F'UX.
We will show that this coequalizer exists and that it forms the object map for a left adjoint to
R.
Instances For
(F'Uε_X, otherMap X) is a reflexive pair: in particular if A has reflexive coequalizers then
it has a coequalizer.
Construct the object part of the desired left adjoint as the coequalizer of F'Uε_Y with
otherMap.
Instances For
The homset equivalence which helps show that R is a right adjoint.
Instances For
Construct the left adjoint to R, with object map constructLeftAdjointObj.
Instances For
The adjoint triangle theorem: Suppose U : B ⥤ C has a left adjoint F such that each counit
ε_X : FUX ⟶ X is a regular epimorphism. Then if a category A has coequalizers of reflexive
pairs, then a functor R : A ⥤ B has a left adjoint if the composite R ⋙ U does.
Note the converse is true (with weaker assumptions), by Adjunction.comp.
See https://ncatlab.org/nlab/show/adjoint+triangle+theorem
Instances For
If R ⋙ U has a left adjoint, the domain of R has reflexive coequalizers and U is a monadic
functor, then R has a left adjoint.
This is a special case of adjointTriangleLift which is often more useful in practice.
Instances For
Suppose we have a commutative square of functors
Q
A → B
U ↓ ↓ V C → D R
where U has a left adjoint, A has reflexive coequalizers and V has a left adjoint such that
each component of the counit is a regular epi.
Then Q has a left adjoint if R has a left adjoint.
See https://ncatlab.org/nlab/show/adjoint+lifting+theorem
Instances For
Suppose we have a commutative square of functors
Q
A → B
U ↓ ↓ V C → D R
where U has a left adjoint, A has reflexive coequalizers and V is monadic.
Then Q has a left adjoint if R has a left adjoint.
See https://ncatlab.org/nlab/show/adjoint+lifting+theorem