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[Merged by Bors] - feat(CategoryTheory): transport a NormalMono structure via an isomorphism of arrows #34496

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[Merged by Bors] - feat(CategoryTheory): transport a NormalMono structure via an isomorphism of arrows #34496

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5 changes: 3 additions & 2 deletions Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -298,8 +298,9 @@ theorem parallelPairHom_app_one {X' Y' : C} (f g : X ⟶ Y) (f' g' : X' ⟶ Y')
its components. -/
@[simps!]
def parallelPair.ext {F G : WalkingParallelPair ⥤ C} (zero : F.obj zero ≅ G.obj zero)
(one : F.obj one ≅ G.obj one) (left : F.map left ≫ one.hom = zero.hom ≫ G.map left)
(right : F.map right ≫ one.hom = zero.hom ≫ G.map right) : F ≅ G :=
(one : F.obj one ≅ G.obj one)
(left : F.map left ≫ one.hom = zero.hom ≫ G.map left := by cat_disch)
(right : F.map right ≫ one.hom = zero.hom ≫ G.map right := by cat_disch) : F ≅ G :=
NatIso.ofComponents
(by
rintro ⟨j⟩
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31 changes: 31 additions & 0 deletions Mathlib/CategoryTheory/Limits/Shapes/NormalMono/Basic.lean
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Expand Up @@ -115,6 +115,21 @@ def normalOfIsPullbackFstOfNormal {P Q R S : C} {f : P ⟶ Q} {g : P ⟶ R} {h :
NormalMono f :=
normalOfIsPullbackSndOfNormal comm.symm (PullbackCone.flipIsLimit t)

/-- Transport a `NormalMono` structure via an isomorphism of arrows. -/
def NormalMono.ofArrowIso {X Y : C} {f : X ⟶ Y}
(hf : NormalMono f) {X' Y' : C} {f' : X' ⟶ Y'} (e : Arrow.mk f ≅ Arrow.mk f') :
NormalMono f' where
Z := hf.Z
g := e.inv.right ≫ hf.g
w := by
have := Arrow.w e.inv
dsimp at this
rw [← reassoc_of% this, hf.w, comp_zero]
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@metakunt metakunt Jan 27, 2026

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I have often seen this % operator, but I can't figure out what it does.

isLimit := by
refine (IsLimit.equivOfNatIsoOfIso ?_ _ _ ?_).1 hf.isLimit
· exact parallelPair.ext (Arrow.rightFunc.mapIso e) (Iso.refl _)
· exact Fork.ext (Arrow.leftFunc.mapIso e)

section

variable (C)
Expand Down Expand Up @@ -219,6 +234,22 @@ open Opposite

variable [HasZeroMorphisms C]

/-- Transport a `NormalEpi` structure via an isomorphism of arrows. -/
def NormalEpi.ofArrowIso [HasZeroMorphisms C] {X Y : C} {f : X ⟶ Y}
(hf : NormalEpi f) {X' Y' : C} {f' : X' ⟶ Y'} (e : Arrow.mk f ≅ Arrow.mk f') :
NormalEpi f' where
W := hf.W
g := hf.g ≫ e.hom.left
w := by
have := Arrow.w e.hom
dsimp at this
rw [Category.assoc, this, reassoc_of% hf.w, zero_comp]
isColimit := by
refine (IsColimit.equivOfNatIsoOfIso ?_ _ _ ?_).1 hf.isColimit
· exact parallelPair.ext (Iso.refl _) (Arrow.leftFunc.mapIso e)
· exact Cofork.ext (Arrow.rightFunc.mapIso e) (by simp [Cofork.π])


/-- A normal mono becomes a normal epi in the opposite category. -/
def normalEpiOfNormalMonoUnop {X Y : Cᵒᵖ} (f : X ⟶ Y) (m : NormalMono f.unop) : NormalEpi f where
W := op m.Z
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