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.

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

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

class CategoryTheory.Projective {C : Type u} [CategoryTheory.Category.{v, u} C] (P : C) : Prop

• factors : ∀ {E X : C} (f : P ⟶ X) (e : E ⟶ X) [inst : CategoryTheory.Epi e], ∃ f', CategoryTheory.CategoryStruct.comp f' e = f

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

structure CategoryTheory.ProjectivePresentation {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) : Type (max u v)

• p : C
• projective : CategoryTheory.Projective s.p
• f : s.p ⟶ X
• epi : CategoryTheory.Epi s.f

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

class CategoryTheory.EnoughProjectives (C : Type u) [CategoryTheory.Category.{v, u} C] : Prop

• presentation : ∀ (X : C), Nonempty (CategoryTheory.ProjectivePresentation X)

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

def CategoryTheory.Projective.factorThru {C : Type u} [CategoryTheory.Category.{v, u} C] {P : C} {X : C} {E : C} [CategoryTheory.Projective P] (f : P ⟶ X) (e : E ⟶ X) [CategoryTheory.Epi e] : P ⟶ E

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

@[simp]

theorem CategoryTheory.Projective.factorThru_comp {C : Type u} [CategoryTheory.Category.{v, u} C] {P : C} {X : C} {E : C} [CategoryTheory.Projective P] (f : P ⟶ X) (e : E ⟶ X) [CategoryTheory.Epi e] : CategoryTheory.CategoryStruct.comp (CategoryTheory.Projective.factorThru f e) e = f

instance CategoryTheory.Projective.zero_projective {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroMorphisms C] : CategoryTheory.Projective 0

theorem CategoryTheory.Projective.of_iso {C : Type u} [CategoryTheory.Category.{v, u} C] {P : C} {Q : C} (i : P ≅ Q) (hP : CategoryTheory.Projective P) : CategoryTheory.Projective Q

theorem CategoryTheory.Projective.iso_iff {C : Type u} [CategoryTheory.Category.{v, u} C] {P : C} {Q : C} (i : P ≅ Q) : CategoryTheory.Projective P ↔ CategoryTheory.Projective Q

instance CategoryTheory.Projective.instProjectiveTypeTypes (X : Type u) : CategoryTheory.Projective X

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

instance CategoryTheory.Projective.Type.enoughProjectives : CategoryTheory.EnoughProjectives (Type u)

instance CategoryTheory.Projective.instProjectiveCoprod {C : Type u} [CategoryTheory.Category.{v, u} C] {P : C} {Q : C} [CategoryTheory.Limits.HasBinaryCoproduct P Q] [CategoryTheory.Projective P] [CategoryTheory.Projective Q] : CategoryTheory.Projective (P ⨿ Q)

instance CategoryTheory.Projective.instProjectiveSigmaObj {C : Type u} [CategoryTheory.Category.{v, u} C] {β : Type v} (g : β → C) [CategoryTheory.Limits.HasCoproduct g] [∀ (b : β), CategoryTheory.Projective (g b)] : CategoryTheory.Projective (∐ g)

instance CategoryTheory.Projective.instProjectiveBiprod {C : Type u} [CategoryTheory.Category.{v, u} C] {P : C} {Q : C} [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasBinaryBiproduct P Q] [CategoryTheory.Projective P] [CategoryTheory.Projective Q] : CategoryTheory.Projective (P ⊞ Q)

instance CategoryTheory.Projective.instProjectiveBiproduct {C : Type u} [CategoryTheory.Category.{v, u} C] {β : Type v} (g : β → C) [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasBiproduct g] [∀ (b : β), CategoryTheory.Projective (g b)] : CategoryTheory.Projective (⨁ g)

theorem CategoryTheory.Projective.projective_iff_preservesEpimorphisms_coyoneda_obj {C : Type u} [CategoryTheory.Category.{v, u} C] (P : C) : CategoryTheory.Projective P ↔ CategoryTheory.Functor.PreservesEpimorphisms (CategoryTheory.coyoneda.obj (Opposite.op P))

def CategoryTheory.Projective.over {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.EnoughProjectives C] (X : C) : C

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

instance CategoryTheory.Projective.projective_over {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.EnoughProjectives C] (X : C) : CategoryTheory.Projective (CategoryTheory.Projective.over X)

def CategoryTheory.Projective.π {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.EnoughProjectives C] (X : C) : CategoryTheory.Projective.over X ⟶ X

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

instance CategoryTheory.Projective.π_epi {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.EnoughProjectives C] (X : C) : CategoryTheory.Epi (CategoryTheory.Projective.π X)

def CategoryTheory.Projective.syzygies {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.EnoughProjectives C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} (f : X ⟶ Y) [CategoryTheory.Limits.HasKernel f] : C

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

instance CategoryTheory.Projective.instProjectiveSyzygies {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.EnoughProjectives C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} (f : X ⟶ Y) [CategoryTheory.Limits.HasKernel f] : CategoryTheory.Projective (CategoryTheory.Projective.syzygies f)

@[inline, reducible]

abbrev CategoryTheory.Projective.d {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.EnoughProjectives C] [CategoryTheory.Limits.HasZeroMorphisms C] {X : C} {Y : C} (f : X ⟶ Y) [CategoryTheory.Limits.HasKernel f] : CategoryTheory.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.)

theorem CategoryTheory.Adjunction.map_projective {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F ⊣ G) [CategoryTheory.Functor.PreservesEpimorphisms G] (P : C) (hP : CategoryTheory.Projective P) : CategoryTheory.Projective (F.obj P)

theorem CategoryTheory.Adjunction.projective_of_map_projective {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F ⊣ G) [CategoryTheory.Full F] [CategoryTheory.Faithful F] (P : C) (hP : CategoryTheory.Projective (F.obj P)) : CategoryTheory.Projective P

def CategoryTheory.Adjunction.mapProjectivePresentation {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F ⊣ G) [CategoryTheory.Functor.PreservesEpimorphisms G] (X : C) (Y : CategoryTheory.ProjectivePresentation X) : CategoryTheory.ProjectivePresentation (F.obj X)

Given an adjunction F ⊣ G such that G preserves epis, F maps a projective presentation of X to a projective presentation of F(X).

theorem CategoryTheory.Equivalence.map_projective_iff {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] (F : C ≌ D) (P : C) : CategoryTheory.Projective (F.functor.obj P) ↔ CategoryTheory.Projective P

def CategoryTheory.Equivalence.projectivePresentationOfMapProjectivePresentation {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] (F : C ≌ D) (X : C) (Y : CategoryTheory.ProjectivePresentation (F.functor.obj X)) : CategoryTheory.ProjectivePresentation X

Given an equivalence of categories F, a projective presentation of F(X) induces a projective presentation of X.

theorem CategoryTheory.Equivalence.enoughProjectives_iff {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] (F : C ≌ D) : CategoryTheory.EnoughProjectives C ↔ CategoryTheory.EnoughProjectives D

def CategoryTheory.Exact.lift {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasEqualizers C] [CategoryTheory.Limits.HasImages C] {P : C} {Q : C} {R : C} {S : C} [CategoryTheory.Projective P] (h : P ⟶ R) (f : Q ⟶ R) (g : R ⟶ S) (hfg : CategoryTheory.Exact f g) (w : CategoryTheory.CategoryStruct.comp 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.

@[simp]

theorem CategoryTheory.Exact.lift_comp {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasEqualizers C] [CategoryTheory.Limits.HasImages C] {P : C} {Q : C} {R : C} {S : C} [CategoryTheory.Projective P] (h : P ⟶ R) (f : Q ⟶ R) (g : R ⟶ S) (hfg : CategoryTheory.Exact f g) (w : CategoryTheory.CategoryStruct.comp h g = 0) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Exact.lift h f g hfg w) f = h