Theorems · Definition · category theory

CategoryTheory.SubobjectRepresentableBy.iso

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    [inst_1 : CategoryTheory.Limits.HasPullbacks C] →
      {Ω : C} →
        (h : CategoryTheory.SubobjectRepresentableBy Ω) →
          {U X : C} →
            (m : U ⟶ X) →
              [inst_2 : CategoryTheory.Mono m] →
                CategoryTheory.MonoOver.mk m ≅
                  CategoryTheory.Subobject.representative.obj ((CategoryTheory.Subobject.pullback (h.χ m)).obj h.Ω₀)

h.iso m is the isomorphism between m and the pullback of Ω₀ along the characteristic map of m.

Defined in

Mathlib.CategoryTheory.Subobject.Classifier.Defs

Cited by

7 results in Mathlib

Foundations

Depth 47 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasPullbacksCategoryTheory.Mono