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[Merged by Bors] - feat: sufficient condition for StructuredArrow.pre to be an equivalence #6248

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[Merged by Bors] - feat: sufficient condition for StructuredArrow.pre to be an equivalence #6248

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2b1cfd2
feat: sufficient condition for StructuredArrow.pre to be an equivalence
TwoFX Jul 30, 2023
6f3da35
Merge remote-tracking branch 'origin/master' into pre-equivalence
TwoFX Aug 4, 2023
f6436bb
Naming
TwoFX Aug 4, 2023
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36 changes: 36 additions & 0 deletions Mathlib/CategoryTheory/Comma.lean
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Expand Up @@ -300,6 +300,24 @@ def preLeft (F : C ⥤ A) (L : A ⥤ T) (R : B ⥤ T) : Comma (F ⋙ L) R ⥤ Co
w := by simpa using f.w }
#align category_theory.comma.pre_left CategoryTheory.Comma.preLeft

instance (F : C ⥤ A) (L : A ⥤ T) (R : B ⥤ T) [Faithful F] : Faithful (preLeft F L R) where
map_injective {X Y} f g h := hom_ext _ _ (F.map_injective (congrArg CommaMorphism.left h))
(by apply congrArg CommaMorphism.right h)

instance (F : C ⥤ A) (L : A ⥤ T) (R : B ⥤ T) [Full F] : Full (preLeft F L R) where
preimage {X Y} f := CommaMorphism.mk (F.preimage f.left) f.right (by simpa using f.w)

instance (F : C ⥤ A) (L : A ⥤ T) (R : B ⥤ T) [EssSurj F] : EssSurj (preLeft F L R) where
mem_essImage Y :=
⟨Comma.mk (F.objPreimage Y.left) Y.right ((L.mapIso (F.objObjPreimageIso _)).hom ≫ Y.hom),
⟨isoMk (F.objObjPreimageIso _) (Iso.refl _) (by simp)⟩⟩

/-- If `F` is an equivalence, then so is `preLeft F L R`. -/
noncomputable def isEquivalencePreLeft (F : C ⥤ A) (L : A ⥤ T) (R : B ⥤ T) [IsEquivalence F] :
IsEquivalence (preLeft F L R) :=
have := Equivalence.essSurj_of_equivalence F
Equivalence.ofFullyFaithfullyEssSurj _

/-- The functor `(F ⋙ L, R) ⥤ (L, R)` -/
@[simps]
def preRight (L : A ⥤ T) (F : C ⥤ B) (R : B ⥤ T) : Comma L (F ⋙ R) ⥤ Comma L R where
Expand All @@ -312,6 +330,24 @@ def preRight (L : A ⥤ T) (F : C ⥤ B) (R : B ⥤ T) : Comma L (F ⋙ R) ⥤ C
right := F.map f.right }
#align category_theory.comma.pre_right CategoryTheory.Comma.preRight

instance (L : A ⥤ T) (F : C ⥤ B) (R : B ⥤ T) [Faithful F] : Faithful (preRight L F R) where
map_injective {X Y } f g h := hom_ext _ _ (by apply congrArg CommaMorphism.left h)
(F.map_injective (congrArg CommaMorphism.right h))

instance (L : A ⥤ T) (F : C ⥤ B) (R : B ⥤ T) [Full F] : Full (preRight L F R) where
preimage {X Y} f := CommaMorphism.mk f.left (F.preimage f.right) (by simpa using f.w)

instance (L : A ⥤ T) (F : C ⥤ B) (R : B ⥤ T) [EssSurj F] : EssSurj (preRight L F R) where
mem_essImage Y :=
⟨Comma.mk Y.left (F.objPreimage Y.right) (Y.hom ≫ (R.mapIso (F.objObjPreimageIso _)).inv),
⟨isoMk (Iso.refl _) (F.objObjPreimageIso _) (by simp [← R.map_comp])⟩⟩

/-- If `F` is an equivalence, then so is `preRight L F R`. -/
noncomputable def isEquivalencePreRight (L : A ⥤ T) (F : C ⥤ B) (R : B ⥤ T) [IsEquivalence F] :
IsEquivalence (preRight L F R) :=
have := Equivalence.essSurj_of_equivalence F
Equivalence.ofFullyFaithfullyEssSurj _

/-- The functor `(L, R) ⥤ (L ⋙ F, R ⋙ F)` -/
@[simps]
def post (L : A ⥤ T) (R : B ⥤ T) (F : T ⥤ C) : Comma L R ⥤ Comma (L ⋙ F) (R ⋙ F) where
Expand Down
28 changes: 28 additions & 0 deletions Mathlib/CategoryTheory/StructuredArrow.lean
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Expand Up @@ -244,6 +244,20 @@ def pre (S : D) (F : B ⥤ C) (G : C ⥤ D) : StructuredArrow S (F ⋙ G) ⥤ St
Comma.preRight _ F G
#align category_theory.structured_arrow.pre CategoryTheory.StructuredArrow.pre

instance (S : D) (F : B ⥤ C) (G : C ⥤ D) [Faithful F] : Faithful (pre S F G) :=
show Faithful (Comma.preRight _ _ _) from inferInstance

instance (S : D) (F : B ⥤ C) (G : C ⥤ D) [Full F] : Full (pre S F G) :=
show Full (Comma.preRight _ _ _) from inferInstance

instance (S : D) (F : B ⥤ C) (G : C ⥤ D) [EssSurj F] : EssSurj (pre S F G) :=
show EssSurj (Comma.preRight _ _ _) from inferInstance

/-- If `F` is an equivalence, then so is the functor `(S, F ⋙ G) ⥤ (S, G)`. -/
noncomputable def isEquivalencePre (S : D) (F : B ⥤ C) (G : C ⥤ D) [IsEquivalence F] :
IsEquivalence (pre S F G) :=
Comma.isEquivalencePreRight _ _ _

/-- The functor `(S, F) ⥤ (G(S), F ⋙ G)`. -/
@[simps]
def post (S : C) (F : B ⥤ C) (G : C ⥤ D) : StructuredArrow S F ⥤ StructuredArrow (G.obj S) (F ⋙ G)
Expand Down Expand Up @@ -468,6 +482,20 @@ def pre (F : B ⥤ C) (G : C ⥤ D) (S : D) : CostructuredArrow (F ⋙ G) S ⥤
Comma.preLeft F G _
#align category_theory.costructured_arrow.pre CategoryTheory.CostructuredArrow.pre

instance (F : B ⥤ C) (G : C ⥤ D) (S : D) [Faithful F] : Faithful (pre F G S) :=
show Faithful (Comma.preLeft _ _ _) from inferInstance

instance (F : B ⥤ C) (G : C ⥤ D) (S : D) [Full F] : Full (pre F G S) :=
show Full (Comma.preLeft _ _ _) from inferInstance

instance (F : B ⥤ C) (G : C ⥤ D) (S : D) [EssSurj F] : EssSurj (pre F G S) :=
show EssSurj (Comma.preLeft _ _ _) from inferInstance

/-- If `F` is an equivalence, then so is the functor `(F ⋙ G, S) ⥤ (G, S)`. -/
noncomputable def isEquivalencePre (F : B ⥤ C) (G : C ⥤ D) (S : D) [IsEquivalence F] :
IsEquivalence (pre F G S) :=
Comma.isEquivalencePreLeft _ _ _

/-- The functor `(F, S) ⥤ (F ⋙ G, G(S))`. -/
@[simps]
def post (F : B ⥤ C) (G : C ⥤ D) (S : C) :
Expand Down
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