Theorems · Theorem · category theory

CategoryTheory.Abelian.Preradical.isIso_toColon_iff

∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_2, u_1} C] [inst_1 : CategoryTheory.Abelian C]
  {Φ Ψ : CategoryTheory.Abelian.Preradical C},
  CategoryTheory.IsIso (Φ.toColon Ψ) ↔ CategoryTheory.Limits.IsZero (Φ.quotient.comp Ψ.r)

The morphism (toColon Φ Ψ) is an isomorphism if and only if Φ.quotient ⋙ Ψ.r is the zero object.

Defined in

Mathlib.CategoryTheory.Abelian.Preradical.Colon

Cited by

1 results in Mathlib

Foundations

Depth 98 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Abelian