Factoring through subobjects

The predicate h : P.Factors f, for P : Subobject Y and f : X ⟶ Y asserts the existence of some P.factorThru f : X ⟶ (P : C) making the obvious diagram commute.

def CategoryTheory.MonoOver.Factors {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (P : CategoryTheory.MonoOver Y) (f : X ⟶ Y) : Prop

When f : X ⟶ Y and P : MonoOver Y, P.Factors f expresses that there exists a factorisation of f through P. Given h : P.Factors f, you can recover the morphism as P.factorThru f h.

Equations

• P.Factors f = ∃ (g : X ⟶ P.obj.left), CategoryTheory.CategoryStruct.comp g P.arrow = f

theorem CategoryTheory.MonoOver.factors_congr {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {f g : CategoryTheory.MonoOver X} {Y : C} (h : Y ⟶ X) (e : f ≅ g) : f.Factors h ↔ g.Factors h

def CategoryTheory.MonoOver.factorThru {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (P : CategoryTheory.MonoOver Y) (f : X ⟶ Y) (h : P.Factors f) : X ⟶ P.obj.left

P.factorThru f h provides a factorisation of f : X ⟶ Y through some P : MonoOver Y, given the evidence h : P.Factors f that such a factorisation exists.

Equations

• P.factorThru f h = Classical.choose h

def CategoryTheory.Subobject.Factors {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (P : CategoryTheory.Subobject Y) (f : X ⟶ Y) : Prop

When f : X ⟶ Y and P : Subobject Y, P.Factors f expresses that there exists a factorisation of f through P. Given h : P.Factors f, you can recover the morphism as P.factorThru f h.

Equations

• P.Factors f = Quotient.liftOn' P (fun (P : CategoryTheory.MonoOver Y) => P.Factors f) ⋯

@[simp]

theorem CategoryTheory.Subobject.mk_factors_iff {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y Z : C} (f : Y ⟶ X) [CategoryTheory.Mono f] (g : Z ⟶ X) : (CategoryTheory.Subobject.mk f).Factors g ↔ (CategoryTheory.MonoOver.mk' f).Factors g

theorem CategoryTheory.Subobject.mk_factors_self {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (f : X ⟶ Y) [CategoryTheory.Mono f] : (CategoryTheory.Subobject.mk f).Factors f

theorem CategoryTheory.Subobject.factors_iff {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (P : CategoryTheory.Subobject Y) (f : X ⟶ Y) : P.Factors f ↔ (CategoryTheory.Subobject.representative.obj P).Factors f

theorem CategoryTheory.Subobject.factors_self {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} (P : CategoryTheory.Subobject X) : P.Factors P.arrow

theorem CategoryTheory.Subobject.factors_comp_arrow {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} {P : CategoryTheory.Subobject Y} (f : X ⟶ CategoryTheory.Subobject.underlying.obj P) : P.Factors (CategoryTheory.CategoryStruct.comp f P.arrow)

theorem CategoryTheory.Subobject.factors_of_factors_right {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y Z : C} {P : CategoryTheory.Subobject Z} (f : X ⟶ Y) {g : Y ⟶ Z} (h : P.Factors g) : P.Factors (CategoryTheory.CategoryStruct.comp f g)

theorem CategoryTheory.Subobject.factors_zero {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X Y : C} {P : CategoryTheory.Subobject Y} : P.Factors 0

theorem CategoryTheory.Subobject.factors_of_le {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {Y Z : C} {P Q : CategoryTheory.Subobject Y} (f : Z ⟶ Y) (h : P ≤ Q) : P.Factors f → Q.Factors f

def CategoryTheory.Subobject.factorThru {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (P : CategoryTheory.Subobject Y) (f : X ⟶ Y) (h : P.Factors f) : X ⟶ CategoryTheory.Subobject.underlying.obj P

P.factorThru f h provides a factorisation of f : X ⟶ Y through some P : Subobject Y, given the evidence h : P.Factors f that such a factorisation exists.

Equations

• P.factorThru f h = Classical.choose ⋯

@[simp]

theorem CategoryTheory.Subobject.factorThru_arrow {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (P : CategoryTheory.Subobject Y) (f : X ⟶ Y) (h : P.Factors f) : CategoryTheory.CategoryStruct.comp (P.factorThru f h) P.arrow = f

@[simp]

theorem CategoryTheory.Subobject.factorThru_arrow_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (P : CategoryTheory.Subobject Y) (f : X ⟶ Y) (h : P.Factors f) {Z : C} (h✝ : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp (P.factorThru f h) (CategoryTheory.CategoryStruct.comp P.arrow h✝) = CategoryTheory.CategoryStruct.comp f h✝

@[simp]

theorem CategoryTheory.Subobject.factorThru_self {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} (P : CategoryTheory.Subobject X) (h : P.Factors P.arrow) : P.factorThru P.arrow h = CategoryTheory.CategoryStruct.id (CategoryTheory.Subobject.underlying.obj P)

@[simp]

theorem CategoryTheory.Subobject.factorThru_mk_self {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (f : X ⟶ Y) [CategoryTheory.Mono f] : (CategoryTheory.Subobject.mk f).factorThru f ⋯ = (CategoryTheory.Subobject.underlyingIso f).inv

@[simp]

theorem CategoryTheory.Subobject.factorThru_comp_arrow {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y : C} {P : CategoryTheory.Subobject Y} (f : X ⟶ CategoryTheory.Subobject.underlying.obj P) (h : P.Factors (CategoryTheory.CategoryStruct.comp f P.arrow)) : P.factorThru (CategoryTheory.CategoryStruct.comp f P.arrow) h = f

@[simp]

theorem CategoryTheory.Subobject.factorThru_eq_zero {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X Y : C} {P : CategoryTheory.Subobject Y} {f : X ⟶ Y} {h : P.Factors f} : P.factorThru f h = 0 ↔ f = 0

theorem CategoryTheory.Subobject.factorThru_right {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X Y Z : C} {P : CategoryTheory.Subobject Z} (f : X ⟶ Y) (g : Y ⟶ Z) (h : P.Factors g) : CategoryTheory.CategoryStruct.comp f (P.factorThru g h) = P.factorThru (CategoryTheory.CategoryStruct.comp f g) ⋯

@[simp]

theorem CategoryTheory.Subobject.factorThru_zero {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X Y : C} {P : CategoryTheory.Subobject Y} (h : P.Factors 0) : P.factorThru 0 h = 0

theorem CategoryTheory.Subobject.factorThru_ofLE {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {Y Z : C} {P Q : CategoryTheory.Subobject Y} {f : Z ⟶ Y} (h : P ≤ Q) (w : P.Factors f) : Q.factorThru f ⋯ = CategoryTheory.CategoryStruct.comp (P.factorThru f w) (P.ofLE Q h)

theorem CategoryTheory.Subobject.factors_add {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] {X Y : C} {P : CategoryTheory.Subobject Y} (f g : X ⟶ Y) (wf : P.Factors f) (wg : P.Factors g) : P.Factors (f + g)

theorem CategoryTheory.Subobject.factorThru_add {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] {X Y : C} {P : CategoryTheory.Subobject Y} (f g : X ⟶ Y) (w : P.Factors (f + g)) (wf : P.Factors f) (wg : P.Factors g) : P.factorThru (f + g) w = P.factorThru f wf + P.factorThru g wg

theorem CategoryTheory.Subobject.factors_left_of_factors_add {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] {X Y : C} {P : CategoryTheory.Subobject Y} (f g : X ⟶ Y) (w : P.Factors (f + g)) (wg : P.Factors g) : P.Factors f

@[simp]

theorem CategoryTheory.Subobject.factorThru_add_sub_factorThru_right {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] {X Y : C} {P : CategoryTheory.Subobject Y} (f g : X ⟶ Y) (w : P.Factors (f + g)) (wg : P.Factors g) : P.factorThru (f + g) w - P.factorThru g wg = P.factorThru f ⋯

theorem CategoryTheory.Subobject.factors_right_of_factors_add {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] {X Y : C} {P : CategoryTheory.Subobject Y} (f g : X ⟶ Y) (w : P.Factors (f + g)) (wf : P.Factors f) : P.Factors g

@[simp]

theorem CategoryTheory.Subobject.factorThru_add_sub_factorThru_left {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Preadditive C] {X Y : C} {P : CategoryTheory.Subobject Y} (f g : X ⟶ Y) (w : P.Factors (f + g)) (wf : P.Factors f) : P.factorThru (f + g) w - P.factorThru f wf = P.factorThru g ⋯