Morphisms of localizers #

A morphism of localizers consists of a functor F : C₁ ⥤ C₂ between two categories equipped with morphism properties W₁ and W₂ such that F sends morphisms in W₁ to morphisms in W₂.

If Φ : LocalizerMorphism W₁ W₂, and that L₁ : C₁ ⥤ D₁ and L₂ : C₂ ⥤ D₂ are localization functors for W₁ and W₂, the induced functor D₁ ⥤ D₂ is denoted Φ.localizedFunctor L₁ L₂; we introduce the condition Φ.IsLocalizedEquivalence which expresses that this functor is an equivalence of categories. This condition is independent of the choice of the localized categories.

References #

structure CategoryTheory.LocalizerMorphism {C₁ : Type u₁} {C₂ : Type u₂} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] (W₁ : CategoryTheory.MorphismProperty C₁) (W₂ : CategoryTheory.MorphismProperty C₂) :
Type (max (max (max u₁ u₂) v₁) v₂)

If W₁ : MorphismProperty C₁ and W₂ : MorphismProperty C₂, a LocalizerMorphism W₁ W₂ is the datum of a functor C₁ ⥤ C₂ which sends morphisms in W₁ to morphisms in W₂

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    The identity functor as a morphism of localizers.

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      When Φ : LocalizerMorphism W₁ W₂ and that L₁ and L₂ are localization functors for W₁ and W₂, then Φ.localizedFunctor L₁ L₂ is the induced functor on the localized categories. -

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        The 2-commutative square expressing that Φ.localizedFunctor L₁ L₂ lifts the functor Φ.functor

        If a localizer morphism induces an equivalence on some choice of localized categories, it will be so for any choice of localized categoriees.

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          Condition that a LocalizerMorphism induces an equivalence on the localized categories


            If a LocalizerMorphism is a localized equivalence, then any compatible functor between the localized categories is an equivalence.

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              When the underlying functor Φ.functor of Φ : LocalizerMorphism W₁ W₂ is an equivalence of categories and that W₁ and W₂ essentially correspond to each other via this equivalence, then Φ is a localized equivalence.