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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.

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

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

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.

Equations

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

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.

Equations

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.

Equations

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.