Theorems · Theorem · category theory

CategoryTheory.Functor.relativelyRepresentable.hom_ext

∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
  {F : CategoryTheory.Functor C D} {X Y : D} {f : X ⟶ Y} (hf : F.relativelyRepresentable f) {a : C} {g : F.obj a ⟶ Y}
  [F.Faithful] {c : C} {a_1 b : c ⟶ hf.pullback g},
  CategoryTheory.CategoryStruct.comp (F.map a_1) (hf.fst g) = CategoryTheory.CategoryStruct.comp (F.map b) (hf.fst g) →
    CategoryTheory.CategoryStruct.comp a_1 (hf.snd g) = CategoryTheory.CategoryStruct.comp b (hf.snd g) → a_1 = b

Two morphisms a b : c ⟶ hf.pullback g are equal if * Their compositions (in C) with hf.snd g : hf.pullback ⟶ X are equal. * The compositions of F.map a and F.map b with hf.fst g are equal.

Defined in

Mathlib.CategoryTheory.MorphismProperty.Representable

Cited by

2 results in Mathlib

Foundations

Depth 28 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.Faithful