Theorems · Theorem · category theory

CategoryTheory.CostructuredArrow.lift_projectQuotient

∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
  {S : CategoryTheory.Functor C D} {T : D} [inst_2 : CategoryTheory.Limits.HasFiniteColimits C]
  [inst_3 : CategoryTheory.Limits.PreservesFiniteColimits S] {A : CategoryTheory.CostructuredArrow S T}
  (P : CategoryTheory.Subobject (Opposite.op A))
  {q :
    S.obj
        (Opposite.unop (CategoryTheory.Subobject.underlying.obj (CategoryTheory.CostructuredArrow.projectQuotient P))) ⟶
      T}
  (hq :
    CategoryTheory.CategoryStruct.comp (S.map (CategoryTheory.CostructuredArrow.projectQuotient P).arrow.unop) q =
      A.hom),
  CategoryTheory.CostructuredArrow.liftQuotient (CategoryTheory.CostructuredArrow.projectQuotient P) hq = P

Projecting and then lifting a quotient recovers the original quotient, because there is at most one morphism making the projected quotient into a costructured arrow.

Defined in

Mathlib.CategoryTheory.Subobject.Comma

Cited by

0 results in Mathlib

Foundations

Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasFiniteColimitsCategoryTheory.Limits.PreservesFiniteColimits