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category_theory.localization.construction

Construction of the localized category #

This file constructs the localized category, obtained by formally inverting a class of maps W : morphism_property C in a category C.

We first construct a quiver loc_quiver W whose objects are the same as those of C and whose maps are the maps in C and placeholders for the formal inverses of the maps in W.

The localized category W.localization is obtained by taking the quotient of the path category of loc_quiver W by the congruence generated by four types of relations.

The obvious functor Q W : C ⥤ W.localization satisfies the universal property of the localization. Indeed, if G : C ⥤ D sends morphisms in W to isomorphisms in D (i.e. we have hG : W.is_inverted_by G), then there exists a unique functor G' : W.localization ⥤ D such that Q W ≫ G' = G. This G' is lift G hG. The expected property of lift G hG if expressed by the lemma fac and the uniqueness is expressed by uniq.

TODO:

  1. implement a constructor for is_localization L W which would take as an input a strict universal property (lift/fac/uniq) similar to what is obtained here for W.localization. (Practically speaking, this is the easiest way to show that a functor is a localization.)

  2. when we have is_localization L W, then show that D ⥤ E identifies to the full subcategory of C ⥤ E consisting of W-inverting functors.

  3. provide an API for the lifting of functors C ⥤ E, for which fac/uniq assertions would be expressed as isomorphisms rather than by equalities of functors.

References #

@[nolint]
  • obj : C

If W : morphism_property C, loc_quiver W is a quiver with the same objects as C, and whose morphisms are those in C and placeholders for formal inverses of the morphisms in W.

Instances for category_theory.localization.construction.loc_quiver

The object in the path category of loc_quiver W attached to an object in the category C

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The relations by which we take the quotient in order to get the localized category.

@[nolint]

The localized category obtained by formally inverting the morphisms in W : morphism_property C

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Instances for category_theory.morphism_property.localization
def category_theory.localization.construction.Wiso {C : Type u_1} [category_theory.category C] {W : category_theory.morphism_property C} {X Y : C} (w : X Y) (hw : W w) :
W.Q.obj X W.Q.obj Y

The isomorphism in W.localization associated to a morphism w in W

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@[reducible]
def category_theory.localization.construction.Winv {C : Type u_1} [category_theory.category C] {W : category_theory.morphism_property C} {X Y : C} (w : X Y) (hw : W w) :
(category_theory.morphism_property.Q (λ {X Y : C} (w : X Y), W w)).obj Y (category_theory.morphism_property.Q (λ {X Y : C} (w : X Y), W w)).obj X

The formal inverse in W.localization of a morphism w in W.

The lifting of a functor to the path category of loc_quiver W

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theorem category_theory.localization.construction.uniq {C : Type u_1} [category_theory.category C] {W : category_theory.morphism_property C} {D : Type u_3} [category_theory.category D] (G₁ G₂ : W.localization D) (h : W.Q G₁ = W.Q G₂) :
G₁ = G₂

The canonical bijection between objects in a category and its localization with respect to a morphism_property W

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theorem category_theory.localization.construction.morphism_property_is_top {C : Type u_1} [category_theory.category C] {W : category_theory.morphism_property C} (P : category_theory.morphism_property W.localization) (hP₁ : ∀ ⦃X Y : C⦄ (f : X Y), P (W.Q.map f)) (hP₂ : ∀ ⦃X Y : C⦄ (w : X Y) (hw : W w), P (category_theory.localization.construction.Winv w hw)) (hP₃ : P.stable_under_composition) :
P =

A morphism_property in W.localization is satisfied by all morphisms in the localized category if it contains the image of the morphisms in the original category, the inverses of the morphisms in W and if it is stable under composition

theorem category_theory.localization.construction.morphism_property_is_top' {C : Type u_1} [category_theory.category C] {W : category_theory.morphism_property C} (P : category_theory.morphism_property W.localization) (hP₁ : ∀ ⦃X Y : C⦄ (f : X Y), P (W.Q.map f)) (hP₂ : ∀ ⦃X Y : W.localization⦄ (e : X Y), P e.homP e.inv) (hP₃ : P.stable_under_composition) :
P =

A morphism_property in W.localization is satisfied by all morphisms in the localized category if it contains the image of the morphisms in the original category, if is stable under composition and if the property is stable by passing to inverses.

def category_theory.localization.construction.nat_trans_extension.app {C : Type u_1} [category_theory.category C] {W : category_theory.morphism_property C} {D : Type u_3} [category_theory.category D] {F₁ F₂ : W.localization D} (τ : W.Q F₁ W.Q F₂) (X : W.localization) :
F₁.obj X F₂.obj X

If F₁ and F₂ are functors W.localization ⥤ D and if we have τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂, we shall define a natural transformation F₁ ⟶ F₂. This is the app field of this natural transformation.

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def category_theory.localization.construction.nat_trans_extension {C : Type u_1} [category_theory.category C] {W : category_theory.morphism_property C} {D : Type u_3} [category_theory.category D] {F₁ F₂ : W.localization D} (τ : W.Q F₁ W.Q F₂) :
F₁ F₂

If F₁ and F₂ are functors W.localization ⥤ D, a natural transformation F₁ ⟶ F₂ can be obtained from a natural transformation W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂.

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theorem category_theory.localization.construction.nat_trans_hcomp_injective {C : Type u_1} [category_theory.category C] {W : category_theory.morphism_property C} {D : Type u_3} [category_theory.category D] {F G : W.localization D} {τ₁ τ₂ : F G} (h : 𝟙 W.Q τ₁ = 𝟙 W.Q τ₂) :
τ₁ = τ₂

The functor (W.localization ⥤ D) ⥤ (W.functors_inverting D) induced by the composition with W.Q : C ⥤ W.localization.

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