# Documentation

This file proves the (general) adjoint functor theorem, in the form:

• If G : D ⥤ C preserves limits and D has limits, and satisfies the solution set condition, then it has a left adjoint: isRightAdjointOfPreservesLimitsOfIsCoseparating.

We show that the converse holds, i.e. that if G has a left adjoint then it satisfies the solution set condition, see solutionSetCondition_of_isRightAdjoint (the file CategoryTheory/Adjunction/Limits already shows it preserves limits).

We define the solution set condition for the functor G : D ⥤ C to mean, for every object A : C, there is a set-indexed family ${f_i : A ⟶ G (B_i)}$ such that any morphism A ⟶ G X factors through one of the f_i.

This file also proves the special adjoint functor theorem, in the form:

• If G : D ⥤ C preserves limits and D is complete, well-powered and has a small coseparating set, then G has a left adjoint: isRightAdjointOfPreservesLimitsOfIsCoseparating

Finally, we prove the following corollary of the special adjoint functor theorem:

• If C is complete, well-powered and has a small coseparating set, then it is cocomplete: hasColimits_of_hasLimits_of_isCoseparating

The functor G : D ⥤ C satisfies the solution set condition if for every A : C, there is a family of morphisms {f_i : A ⟶ G (B_i) // i ∈ ι} such that given any morphism h : A ⟶ G X, there is some i ∈ ι such that h factors through f_i.

The key part of this definition is that the indexing set ι lives in Type v, where v is the universe of morphisms of the category: this is the "smallness" condition which allows the general adjoint functor theorem to go through.

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• One or more equations did not get rendered due to their size.
Instances For
theorem CategoryTheory.solutionSetCondition_of_isRightAdjoint {C : Type u} {D : Type u} (G : ) [G.IsRightAdjoint] :

If G : D ⥤ C is a right adjoint it satisfies the solution set condition.

The general adjoint functor theorem says that if G : D ⥤ C preserves limits and D has them, if G satisfies the solution set condition then G is a right adjoint.

theorem CategoryTheory.isRightAdjoint_of_preservesLimits_of_isCoseparating {C : Type u} {D : Type u'} [] {𝒢 : Set D} [] (h𝒢 : ) (G : ) :
The special adjoint functor theorem: if G : D ⥤ C preserves limits and D is complete, well-powered and has a small coseparating set, then G has a left adjoint.
The special adjoint functor theorem: if F : C ⥤ D preserves colimits and C is cocomplete, well-copowered and has a small separating set, then F has a right adjoint.
A consequence of the special adjoint functor theorem: if C is complete, well-powered and has a small coseparating set, then it is cocomplete.
A consequence of the special adjoint functor theorem: if C is cocomplete, well-copowered and has a small separating set, then it is complete.