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category_theory.adjunction.adjoint_functor_theorems

Adjoint functor theorem #

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

We show that the converse holds, i.e. that if G has a left adjoint then it satisfies the solution set condition, see solution_set_condition_of_is_right_adjoint (the file category_theory/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.

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def category_theory.solution_set_condition {C : Type u} [category_theory.category C] {D : Type u} [category_theory.category D] (G : D ⥤ C) :
Prop

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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theorem category_theory.solution_set_condition_of_is_right_adjoint {C : Type u} [category_theory.category C] {D : Type u} [category_theory.category D] (G : D ⥤ C) [category_theory.is_right_adjoint G] :
category_theory.solution_set_condition G

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

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def category_theory.is_right_adjoint_of_preserves_limits_of_solution_set_condition {C : Type u} [category_theory.category C] {D : Type u} [category_theory.category D] (G : D ⥤ C) [category_theory.limits.has_limits D] [category_theory.limits.preserves_limits G] (hG : category_theory.solution_set_condition G) :
category_theory.is_right_adjoint G

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.

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