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Mathlib.CategoryTheory.Adjunction.AdjointFunctorTheorems

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 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:

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

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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Instances For

    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.

    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.