Theorems · Theorem · category theory

CategoryTheory.Abelian.Preradical.isIso_toColon_hom_left_app_iff

∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_2, u_1} C] [inst_1 : CategoryTheory.Abelian C]
  {Φ Ψ : CategoryTheory.Abelian.Preradical C} {X : C},
  CategoryTheory.IsIso ((CategoryTheory.Over.Hom.left (Φ.toColon Ψ).hom).app X) ↔
    CategoryTheory.Limits.IsZero (Ψ.r.obj (Φ.quotient.obj X))

For X : C, the morphism (toColon Φ Ψ) is an isomorphism if and only if (Ψ.r.obj (Φ.quotient.obj X)) is the zero object.

Defined in

Mathlib.CategoryTheory.Abelian.Preradical.Colon

Cited by

1 results in Mathlib

Foundations

Depth 97 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Abelian