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[Merged by Bors] - feat: trivial morphisms in StructuredArrow #10244

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[Merged by Bors] - feat: trivial morphisms in StructuredArrow #10244

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332029e
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TwoFX Feb 4, 2024
a866a82
Fix typo
TwoFX Feb 4, 2024
abd55c8
Remove unnecessary eqToHom as pointed out in code review
TwoFX Feb 4, 2024
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35 changes: 28 additions & 7 deletions Mathlib/CategoryTheory/Comma/StructuredArrow.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -112,7 +112,7 @@ and to check that the triangle commutes.
@[simps]
def homMk {f f' : StructuredArrow S T} (g : f.right ⟶ f'.right)
(w : f.hom ≫ T.map g = f'.hom := by aesop_cat) : f ⟶ f' where
left := eqToHom (by ext)
left := 𝟙 _
right := g
w := by
dsimp
Expand All @@ -127,7 +127,7 @@ attribute [-simp, nolint simpNF] homMk_left
structured arrows given by `(X ⟶ T(Y)) ⟶ (X ⟶ T(Y) ⟶ T(Y'))`. -/
@[simps]
def homMk' (f : StructuredArrow S T) (g : f.right ⟶ Y') : f ⟶ mk (f.hom ≫ T.map g) where
left := eqToHom (by ext)
left := 𝟙 _
right := g
#align category_theory.structured_arrow.hom_mk' CategoryTheory.StructuredArrow.homMk'

Expand All @@ -147,6 +147,17 @@ lemma homMk'_mk_comp (f : S ⟶ T.obj Y) (g : Y ⟶ Y') (g' : Y' ⟶ Y'') :
homMk' (mk f) (g ≫ g') = homMk' (mk f) g ≫ homMk' (mk (f ≫ T.map g)) g' ≫ eqToHom (by simp) :=
homMk'_comp _ _ _

/-- Variant of `homMk'` where both objects are applications of `mk`. -/
@[simps]
def mkPostcomp (f : S ⟶ T.obj Y) (g : Y ⟶ Y') : mk f ⟶ mk (f ≫ T.map g) where
left := 𝟙 _
right := g

lemma mkPostcomp_id (f : S ⟶ T.obj Y) : mkPostcomp f (𝟙 Y) = eqToHom (by aesop_cat) := by aesop_cat
lemma mkPostcomp_comp (f : S ⟶ T.obj Y) (g : Y ⟶ Y') (g' : Y' ⟶ Y'') :
mkPostcomp f (g ≫ g') = mkPostcomp f g ≫ mkPostcomp (f ≫ T.map g) g' ≫ eqToHom (by simp) := by
aesop_cat

/-- To construct an isomorphism of structured arrows,
we need an isomorphism of the objects underlying the target,
and to check that the triangle commutes.
Expand All @@ -155,7 +166,7 @@ and to check that the triangle commutes.
def isoMk {f f' : StructuredArrow S T} (g : f.right ≅ f'.right)
(w : f.hom ≫ T.map g.hom = f'.hom := by aesop_cat) :
f ≅ f' :=
Comma.isoMk (eqToIso (by ext)) g (by simpa [eqToHom_map] using w.symm)
Comma.isoMk (eqToIso (by ext)) g (by simpa using w.symm)
#align category_theory.structured_arrow.iso_mk CategoryTheory.StructuredArrow.isoMk

/- Porting note : it appears the simp lemma is not getting generated but the linter
Expand Down Expand Up @@ -445,8 +456,7 @@ and to check that the triangle commutes.
def homMk {f f' : CostructuredArrow S T} (g : f.left ⟶ f'.left)
(w : S.map g ≫ f'.hom = f.hom := by aesop_cat) : f ⟶ f' where
left := g
right := eqToHom (by ext)
w := by simpa [eqToHom_map] using w
right := 𝟙 _
#align category_theory.costructured_arrow.hom_mk CategoryTheory.CostructuredArrow.homMk

/- Porting note : it appears the simp lemma is not getting generated but the linter
Expand All @@ -458,7 +468,7 @@ attribute [-simp, nolint simpNF] homMk_right_down_down
@[simps]
def homMk' (f : CostructuredArrow S T) (g : Y' ⟶ f.left) : mk (S.map g ≫ f.hom) ⟶ f where
left := g
right := eqToHom (by ext)
right := 𝟙 _

lemma homMk'_id (f : CostructuredArrow S T) : homMk' f (𝟙 f.left) = eqToHom (by aesop_cat) := by
ext
Expand All @@ -476,14 +486,25 @@ lemma homMk'_mk_comp (f : S.obj Y ⟶ T) (g : Y' ⟶ Y) (g' : Y'' ⟶ Y') :
homMk' (mk f) (g' ≫ g) = eqToHom (by simp) ≫ homMk' (mk (S.map g ≫ f)) g' ≫ homMk' (mk f) g :=
homMk'_comp _ _ _

/-- Variant of `homMk'` where both objects are applications of `mk`. -/
@[simps]
def mkPrecomp (f : S.obj Y ⟶ T) (g : Y' ⟶ Y) : mk (S.map g ≫ f) ⟶ mk f where
left := g
right := 𝟙 _

lemma mkPrecomp_id (f : S.obj Y ⟶ T) : mkPrecomp f (𝟙 Y) = eqToHom (by aesop_cat) := by aesop_cat
lemma mkPrecomp_comp (f : S.obj Y ⟶ T) (g : Y' ⟶ Y) (g' : Y'' ⟶ Y') :
mkPrecomp f (g' ≫ g) = eqToHom (by simp) ≫ mkPrecomp (S.map g ≫ f) g' ≫ mkPrecomp f g := by
aesop_cat

/-- To construct an isomorphism of costructured arrows,
we need an isomorphism of the objects underlying the source,
and to check that the triangle commutes.
-/
@[simps!]
def isoMk {f f' : CostructuredArrow S T} (g : f.left ≅ f'.left)
(w : S.map g.hom ≫ f'.hom = f.hom := by aesop_cat) : f ≅ f' :=
Comma.isoMk g (eqToIso (by ext)) (by simpa [eqToHom_map] using w)
Comma.isoMk g (eqToIso (by ext)) (by simpa using w)
#align category_theory.costructured_arrow.iso_mk CategoryTheory.CostructuredArrow.isoMk

/- Porting note : it appears the simp lemma is not getting generated but the linter
Expand Down