Projective objects and categories with enough projectives #

An object P is called projective if every morphism out of P factors through every epimorphism.

A category C has enough projectives if every object admits an epimorphism from some projective object.

projective.over X picks an arbitrary such projective object, and projective.π X : projective.over X ⟶ X is the corresponding epimorphism.

Given a morphism f : X ⟶ Y, projective.left f is a projective object over kernel f, and projective.d f : projective.left f ⟶ X is the morphism π (kernel f) ≫ kernel.ι f.

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@[class]

structure category_theory.projective {C : Type u} [category_theory.category C] (P : C) :

Prop

factors : ∀ {E X : C} (f : P ⟶ X) (e : E ⟶ X) [_inst_2 : category_theory.epi e], ∃ (f' : P ⟶ E), f' ≫ e = f

An object P is called projective if every morphism out of P factors through every epimorphism.

Instances of this typeclass

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@[nolint]

structure category_theory.projective_presentation {C : Type u} [category_theory.category C] (X : C) :

Type (max u v)

P : C
projective : category_theory.projective self.P . "apply_instance"
f : self.P ⟶ X
epi : category_theory.epi self.f . "apply_instance"

A projective presentation of an object X consists of an epimorphism f : P ⟶ X from some projective object P.

Instances for category_theory.projective_presentation

category_theory.projective_presentation.has_sizeof_inst

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@[class]

structure category_theory.enough_projectives (C : Type u) [category_theory.category C] :

Prop

presentation : ∀ (X : C), nonempty (category_theory.projective_presentation X)

A category "has enough projectives" if for every object X there is a projective object P and an epimorphism P ↠ X.

Instances of this typeclass

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noncomputable def category_theory.projective.factor_thru {C : Type u} [category_theory.category C] {P X E : C} [category_theory.projective P] (f : P ⟶ X) (e : E ⟶ X) [category_theory.epi e] :

P ⟶ E

An arbitrarily chosen factorisation of a morphism out of a projective object through an epimorphism.

Equations

category_theory.projective.factor_thru f e = _.some

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@[simp]

theorem category_theory.projective.factor_thru_comp {C : Type u} [category_theory.category C] {P X E : C} [category_theory.projective P] (f : P ⟶ X) (e : E ⟶ X) [category_theory.epi e] :

category_theory.projective.factor_thru f e ≫ e = f

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@[protected, instance]

def category_theory.projective.zero_projective {C : Type u} [category_theory.category C] [category_theory.limits.has_zero_object C] [category_theory.limits.has_zero_morphisms C] :

category_theory.projective 0

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theorem category_theory.projective.of_iso {C : Type u} [category_theory.category C] {P Q : C} (i : P ≅ Q) (hP : category_theory.projective P) :

category_theory.projective Q

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theorem category_theory.projective.iso_iff {C : Type u} [category_theory.category C] {P Q : C} (i : P ≅ Q) :

category_theory.projective P ↔ category_theory.projective Q

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@[protected, instance]

def category_theory.projective.projective (X : Type u) :

category_theory.projective X

The axiom of choice says that every type is a projective object in Type.

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@[protected, instance]

def category_theory.projective.Type.enough_projectives :

category_theory.enough_projectives (Type u)

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@[protected, instance]

def category_theory.projective.category_theory.limits.coprod.projective {C : Type u} [category_theory.category C] {P Q : C} [category_theory.limits.has_binary_coproduct P Q] [category_theory.projective P] [category_theory.projective Q] :

category_theory.projective (P ⨿ Q)

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@[protected, instance]

def category_theory.projective.category_theory.limits.sigma_obj.projective {C : Type u} [category_theory.category C] {β : Type v} (g : β → C) [category_theory.limits.has_coproduct g] [∀ (b : β), category_theory.projective (g b)] :

category_theory.projective (∐ g)

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@[protected, instance]

def category_theory.projective.category_theory.limits.biprod.projective {C : Type u} [category_theory.category C] {P Q : C} [category_theory.limits.has_zero_morphisms C] [category_theory.limits.has_binary_biproduct P Q] [category_theory.projective P] [category_theory.projective Q] :

category_theory.projective (P ⊞ Q)

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@[protected, instance]

def category_theory.projective.category_theory.limits.biproduct.projective {C : Type u} [category_theory.category C] {β : Type v} (g : β → C) [category_theory.limits.has_zero_morphisms C] [category_theory.limits.has_biproduct g] [∀ (b : β), category_theory.projective (g b)] :

category_theory.projective (⨁ g)

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theorem category_theory.projective.projective_iff_preserves_epimorphisms_coyoneda_obj {C : Type u} [category_theory.category C] (P : C) :

category_theory.projective P ↔ (category_theory.coyoneda.obj (opposite.op P)).preserves_epimorphisms

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theorem category_theory.projective.projective_iff_preserves_epimorphisms_preadditive_coyoneda_obj {C : Type u} [category_theory.category C] [category_theory.preadditive C] (P : C) :

category_theory.projective P ↔ (category_theory.preadditive_coyoneda.obj (opposite.op P)).preserves_epimorphisms

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theorem category_theory.projective.projective_iff_preserves_epimorphisms_preadditive_coyoneda_obj' {C : Type u} [category_theory.category C] [category_theory.preadditive C] (P : C) :

category_theory.projective P ↔ (category_theory.preadditive_coyoneda_obj (opposite.op P)).preserves_epimorphisms

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noncomputable def category_theory.projective.over {C : Type u} [category_theory.category C] [category_theory.enough_projectives C] (X : C) :

projective.over X provides an arbitrarily chosen projective object equipped with an epimorphism projective.π : projective.over X ⟶ X.

Equations

category_theory.projective.over X = _.some.P

Instances for category_theory.projective.over

category_theory.projective.projective_over

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@[protected, instance]

def category_theory.projective.projective_over {C : Type u} [category_theory.category C] [category_theory.enough_projectives C] (X : C) :

category_theory.projective (category_theory.projective.over X)

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noncomputable def category_theory.projective.π {C : Type u} [category_theory.category C] [category_theory.enough_projectives C] (X : C) :

category_theory.projective.over X ⟶ X

The epimorphism projective.π : projective.over X ⟶ X from the arbitrarily chosen projective object over X.

Equations

category_theory.projective.π X = _.some.f

Instances for category_theory.projective.π

category_theory.projective.π_epi

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@[protected, instance]

def category_theory.projective.π_epi {C : Type u} [category_theory.category C] [category_theory.enough_projectives C] (X : C) :

category_theory.epi (category_theory.projective.π X)

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@[protected, instance]

def category_theory.projective.syzygies.projective {C : Type u} [category_theory.category C] [category_theory.enough_projectives C] [category_theory.limits.has_zero_morphisms C] {X Y : C} (f : X ⟶ Y) [category_theory.limits.has_kernel f] :

category_theory.projective (category_theory.projective.syzygies f)

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noncomputable def category_theory.projective.syzygies {C : Type u} [category_theory.category C] [category_theory.enough_projectives C] [category_theory.limits.has_zero_morphisms C] {X Y : C} (f : X ⟶ Y) [category_theory.limits.has_kernel f] :

When C has enough projectives, the object projective.syzygies f is an arbitrarily chosen projective object over kernel f.

Equations

category_theory.projective.syzygies f = category_theory.projective.over (category_theory.limits.kernel f)

Instances for category_theory.projective.syzygies

category_theory.projective.syzygies.projective

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@[reducible]

noncomputable def category_theory.projective.d {C : Type u} [category_theory.category C] [category_theory.enough_projectives C] [category_theory.limits.has_zero_morphisms C] {X Y : C} (f : X ⟶ Y) [category_theory.limits.has_kernel f] :

category_theory.projective.syzygies f ⟶ X

When C has enough projectives, projective.d f : projective.syzygies f ⟶ X is the composition π (kernel f) ≫ kernel.ι f.

(When C is abelian, we have exact (projective.d f) f.)

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noncomputable def category_theory.exact.lift {C : Type u} [category_theory.category C] [category_theory.limits.has_zero_morphisms C] [category_theory.limits.has_equalizers C] [category_theory.limits.has_images C] {P Q R S : C} [category_theory.projective P] (h : P ⟶ R) (f : Q ⟶ R) (g : R ⟶ S) (hfg : category_theory.exact f g) (w : h ≫ g = 0) :

P ⟶ Q

Given a projective object P mapping via h into the middle object R of a pair of exact morphisms f : Q ⟶ R and g : R ⟶ S, such that h ≫ g = 0, there is a lift of h to Q.

Equations

category_theory.exact.lift h f g hfg w = category_theory.projective.factor_thru (category_theory.projective.factor_thru (category_theory.limits.factor_thru_kernel_subobject g h w) (image_to_kernel f g _)) (category_theory.limits.factor_thru_image_subobject f)

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@[simp]

theorem category_theory.exact.lift_comp {C : Type u} [category_theory.category C] [category_theory.limits.has_zero_morphisms C] [category_theory.limits.has_equalizers C] [category_theory.limits.has_images C] {P Q R S : C} [category_theory.projective P] (h : P ⟶ R) (f : Q ⟶ R) (g : R ⟶ S) (hfg : category_theory.exact f g) (w : h ≫ g = 0) :

category_theory.exact.lift h f g hfg w ≫ f = h