Theorems · Definition · category theory

CategoryTheory.Limits.image.compIso

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    {X Y : C} →
      (f : X ⟶ Y) →
        {Z : C} →
          (g : Y ⟶ Z) →
            [CategoryTheory.Limits.HasEqualizers C] →
              [inst_2 : CategoryTheory.Limits.HasImage f] →
                [inst_3 : CategoryTheory.IsIso g] →
                  CategoryTheory.Limits.image f ≅ CategoryTheory.Limits.image (CategoryTheory.CategoryStruct.comp f g)

Postcomposing by an isomorphism induces an isomorphism on the image.

Defined in: Mathlib.CategoryTheory.Limits.Shapes.Images
Cited by: 7 results in Mathlib
Foundations: Depth 32 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Limits.HasEqualizers CategoryTheory.Limits.HasImage CategoryTheory.IsIso

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Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites12

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

CategoryTheory.Categorystatement and proof · cited by 32,673
Quiver.Homstatement and proof · cited by 32,603
CategoryTheory.CategoryStruct.compstatement and proof · cited by 17,999
CategoryTheory.Isostatement · cited by 3,963
CategoryTheory.IsIsostatement and proof · cited by 1,156
CategoryTheory.Limits.imagestatement · cited by 124
CategoryTheory.Limits.HasImagestatement and proof · cited by 107
CategoryTheory.Limits.HasEqualizersstatement and proof · cited by 60
CategoryTheory.Limits.image.liftproof · cited by 16
CategoryTheory.Limits.Image.monoFactorisationproof · cited by 15
CategoryTheory.Limits.MonoFactorisation.ofCompIsoproof · cited by 4
CategoryTheory.Limits.MonoFactorisation.compMonoproof · cited by 4

Cited by8

Results whose statement or proof uses this declaration.

CategoryTheory.Limits.imageSubobjectCompIsoproof · cited by 6
CategoryTheory.Limits.image.compIso_hom_comp_image_ιstatement · cited by 2
CategoryTheory.Limits.image.compIso_inv_comp_image_ιstatement · cited by 2
CategoryTheory.Limits.imageSubobjectCompIso_hom_arrowproof · cited by 1
CategoryTheory.Limits.imageSubobjectCompIso_inv_arrowproof · cited by 1
CategoryTheory.Limits.image.compIso_hom_comp_image_ι_assocstatement and proof · cited by 0
CategoryTheory.Limits.image.compIso_inv_comp_image_ι_assocstatement and proof · cited by 0
CategoryTheory.Limits.image.compIso.congr_simpstatement and proof · cited by 0