fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
noncomputable equivSmallModel (C : Type u) [Category.{v} C] [EssentiallySmall.{w} C] :
C ≌ SmallModel C :=
Nonempty.some
(Classical.choose_spec (Classical.choose_spec (@EssentiallySmall.equiv_smallCategory C _ _))) | def | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | equivSmallModel | The (noncomputable) categorical equivalence between
an essentially small category and its small model. |
essentiallySmall_congr {C : Type u} [Category.{v} C] {D : Type u'} [Category.{v'} D]
(e : C ≌ D) : EssentiallySmall.{w} C ↔ EssentiallySmall.{w} D := by
fconstructor
· rintro ⟨S, 𝒮, ⟨f⟩⟩
exact EssentiallySmall.mk' (e.symm.trans f)
· rintro ⟨S, 𝒮, ⟨f⟩⟩
exact EssentiallySmall.mk' (e.trans f) | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | essentiallySmall_congr | null |
Discrete.essentiallySmallOfSmall {α : Type u} [Small.{w} α] :
EssentiallySmall.{w} (Discrete α) :=
⟨⟨Discrete (Shrink α), ⟨inferInstance, ⟨Discrete.equivalence (equivShrink _)⟩⟩⟩⟩ | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | Discrete.essentiallySmallOfSmall | null |
essentiallySmallSelf : EssentiallySmall.{max w v u} C :=
EssentiallySmall.mk' (AsSmall.equiv : C ≌ AsSmall.{w} C) | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | essentiallySmallSelf | null |
@[pp_with_univ]
LocallySmall (C : Type u) [Category.{v} C] : Prop where
/-- A locally small category has small hom-types. -/
hom_small : ∀ X Y : C, Small.{w} (X ⟶ Y) := by infer_instance | class | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | LocallySmall | A category is `w`-locally small if every hom set is `w`-small.
See `ShrinkHoms C` for a category instance where every hom set has been replaced by a small model. |
locallySmall_of_faithful {C : Type u} [Category.{v} C] {D : Type u'} [Category.{v'} D]
(F : C ⥤ D) [F.Faithful] [LocallySmall.{w} D] : LocallySmall.{w} C where
hom_small {_ _} := small_of_injective F.map_injective | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | locallySmall_of_faithful | null |
locallySmall_congr {C : Type u} [Category.{v} C] {D : Type u'} [Category.{v'} D]
(e : C ≌ D) : LocallySmall.{w} C ↔ LocallySmall.{w} D :=
⟨fun _ => locallySmall_of_faithful e.inverse, fun _ => locallySmall_of_faithful e.functor⟩ | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | locallySmall_congr | null |
locallySmall_max {C : Type u} [Category.{v} C] : LocallySmall.{max v w} C where
hom_small _ _ := small_max.{w} _ | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | locallySmall_max | null |
@[pp_with_univ]
ShrinkHoms (C : Type u) :=
C | def | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | ShrinkHoms | We define a type alias `ShrinkHoms C` for `C`. When we have `LocallySmall.{w} C`,
we'll put a `Category.{w}` instance on `ShrinkHoms C`. |
toShrinkHoms {C' : Type*} (X : C') : ShrinkHoms C' :=
X | def | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | toShrinkHoms | Help the typechecker by explicitly translating from `C` to `ShrinkHoms C`. |
fromShrinkHoms {C' : Type*} (X : ShrinkHoms C') : C' :=
X
@[simp] | def | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | fromShrinkHoms | Help the typechecker by explicitly translating from `ShrinkHoms C` to `C`. |
to_from (X : C') : fromShrinkHoms (toShrinkHoms X) = X :=
rfl
@[simp] | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | to_from | null |
from_to (X : ShrinkHoms C') : toShrinkHoms (fromShrinkHoms X) = X :=
rfl | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | from_to | null |
@[simps]
noncomputable functor : C ⥤ ShrinkHoms C where
obj X := toShrinkHoms X
map {X Y} f := equivShrink (X ⟶ Y) f | def | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | functor | Implementation of `ShrinkHoms.equivalence`. |
@[simps]
noncomputable inverse : ShrinkHoms C ⥤ C where
obj X := fromShrinkHoms X
map {X Y} f := (equivShrink (fromShrinkHoms X ⟶ fromShrinkHoms Y)).symm f | def | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | inverse | Implementation of `ShrinkHoms.equivalence`. |
@[simps]
noncomputable equivalence : C ≌ ShrinkHoms C where
functor := functor C
inverse := inverse C
unitIso := NatIso.ofComponents (fun _ ↦ Iso.refl _)
counitIso := NatIso.ofComponents (fun _ ↦ Iso.refl _) | def | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | equivalence | The categorical equivalence between `C` and `ShrinkHoms C`, when `C` is locally small. |
noncomputable equivalence [Small.{w} C] : C ≌ Shrink.{w} C :=
(Equivalence.induced _).symm | def | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | equivalence | The categorical equivalence between `C` and `Shrink C`, when `C` is small. |
essentiallySmall_iff (C : Type u) [Category.{v} C] :
EssentiallySmall.{w} C ↔ Small.{w} (Skeleton C) ∧ LocallySmall.{w} C := by
fconstructor
· intro h
fconstructor
· rcases h with ⟨S, 𝒮, ⟨e⟩⟩
refine ⟨⟨Skeleton S, ⟨?_⟩⟩⟩
exact e.skeletonEquiv
· infer_instance
· rintro ⟨⟨S, ⟨e⟩⟩, L⟩
let e' := (ShrinkHoms.equivalence C).skeletonEquiv.symm
letI : Category S := InducedCategory.category (e'.trans e).symm
refine ⟨⟨S, this, ⟨?_⟩⟩⟩
refine (ShrinkHoms.equivalence C).trans <|
(skeletonEquivalence (ShrinkHoms C)).symm.trans
((inducedFunctor (e'.trans e).symm).asEquivalence.symm) | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | essentiallySmall_iff | A category is essentially small if and only if
the underlying type of its skeleton (i.e. the "set" of isomorphism classes) is small,
and it is locally small. |
essentiallySmall_of_small_of_locallySmall [Small.{w} C] [LocallySmall.{w} C] :
EssentiallySmall.{w} C :=
(essentiallySmall_iff C).2 ⟨small_of_surjective Quotient.exists_rep, by infer_instance⟩ | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | essentiallySmall_of_small_of_locallySmall | null |
small_skeleton_of_essentiallySmall [h : EssentiallySmall.{w} C] : Small.{w} (Skeleton C) :=
essentiallySmall_iff C |>.1 h |>.1
variable {C} in | instance | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | small_skeleton_of_essentiallySmall | null |
essentiallySmall_of_fully_faithful {D : Type u'} [Category.{v'} D] (F : C ⥤ D)
[F.Full] [F.Faithful] [EssentiallySmall.{w} D] : EssentiallySmall.{w} C :=
(essentiallySmall_iff C).2 ⟨small_of_injective F.mapSkeleton_injective,
locallySmall_of_faithful F⟩ | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | essentiallySmall_of_fully_faithful | null |
locallySmall_fullSubcategory [LocallySmall.{w} C] (P : ObjectProperty C) :
LocallySmall.{w} P.FullSubcategory :=
locallySmall_of_faithful <| P.ι | instance | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | locallySmall_fullSubcategory | null |
essentiallySmall_fullSubcategory_mem (s : Set C) [Small.{w} s] [LocallySmall.{w} C] :
EssentiallySmall.{w} (ObjectProperty.FullSubcategory (· ∈ s)) :=
suffices Small.{w} (ObjectProperty.FullSubcategory (· ∈ s)) from
essentiallySmall_of_small_of_locallySmall _
small_of_injective (f := fun x => (⟨x.1, x.2⟩ : s)) (by cat_disch) | instance | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | essentiallySmall_fullSubcategory_mem | null |
essentiallySmall_iff_of_thin {C : Type u} [Category.{v} C] [Quiver.IsThin C] :
EssentiallySmall.{w} C ↔ Small.{w} (Skeleton C) := by
simp [essentiallySmall_iff, CategoryTheory.locallySmall_of_thin] | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Category.ULift",
"Mathlib.CategoryTheory.EqToHom",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.Logic.UnivLE",
"Mathlib.Logic.Small.Basic"
] | Mathlib/CategoryTheory/EssentiallySmall.lean | essentiallySmall_iff_of_thin | Any thin category is locally small.
-/
instance (priority := 100) locallySmall_of_thin {C : Type u} [Category.{v} C] [Quiver.IsThin C] :
LocallySmall.{w} C where
/--
A thin category is essentially small if and only if the underlying type of its skeleton is small. |
HasPullbacksOfInclusions (C : Type u) [Category.{v} C] [HasBinaryCoproducts C] : Prop where
[hasPullbackInl : ∀ {X Y Z : C} (f : Z ⟶ X ⨿ Y), HasPullback coprod.inl f]
attribute [instance] HasPullbacksOfInclusions.hasPullbackInl | class | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | HasPullbacksOfInclusions | A category has pullback of inclusions if it has all pullbacks along coproduct injections. |
PreservesPullbacksOfInclusions {C : Type*} [Category C] {D : Type*} [Category D]
(F : C ⥤ D) [HasBinaryCoproducts C] where
[preservesPullbackInl : ∀ {X Y Z : C} (f : Z ⟶ X ⨿ Y), PreservesLimit (cospan coprod.inl f) F]
attribute [instance] PreservesPullbacksOfInclusions.preservesPullbackInl | class | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | PreservesPullbacksOfInclusions | A functor preserves pullback of inclusions if it preserves all pullbacks along coproduct injections. |
FinitaryPreExtensive (C : Type u) [Category.{v} C] : Prop where
[hasFiniteCoproducts : HasFiniteCoproducts C]
[hasPullbacksOfInclusions : HasPullbacksOfInclusions C]
/-- In a finitary extensive category, all coproducts are van Kampen -/
universal' : ∀ {X Y : C} (c : BinaryCofan X Y), IsColimit c → IsUniversalColimit c
attribute [instance] FinitaryPreExtensive.hasFiniteCoproducts
attribute [instance] FinitaryPreExtensive.hasPullbacksOfInclusions | class | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | FinitaryPreExtensive | A category is (finitary) pre-extensive if it has finite coproducts,
and binary coproducts are universal. |
FinitaryExtensive (C : Type u) [Category.{v} C] : Prop where
[hasFiniteCoproducts : HasFiniteCoproducts C]
[hasPullbacksOfInclusions : HasPullbacksOfInclusions C]
/-- In a finitary extensive category, all coproducts are van Kampen -/
van_kampen' : ∀ {X Y : C} (c : BinaryCofan X Y), IsColimit c → IsVanKampenColimit c
attribute [instance] FinitaryExtensive.hasFiniteCoproducts
attribute [instance] FinitaryExtensive.hasPullbacksOfInclusions | class | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | FinitaryExtensive | A category is (finitary) extensive if it has finite coproducts,
and binary coproducts are van Kampen. |
FinitaryExtensive.vanKampen [FinitaryExtensive C] {F : Discrete WalkingPair ⥤ C}
(c : Cocone F) (hc : IsColimit c) : IsVanKampenColimit c := by
let X := F.obj ⟨WalkingPair.left⟩
let Y := F.obj ⟨WalkingPair.right⟩
have : F = pair X Y := by
apply Functor.hext
· rintro ⟨⟨⟩⟩ <;> rfl
· rintro ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩ <;> simp [X, Y]
clear_value X Y
subst this
exact FinitaryExtensive.van_kampen' c hc | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | FinitaryExtensive.vanKampen | null |
preservesPullbackInl' :
HasPullback f coprod.inl :=
hasPullback_symmetry _ _ | instance | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | preservesPullbackInl' | null |
hasPullbackInr' :
HasPullback f coprod.inr := by
have : IsPullback (𝟙 _) (f ≫ (coprod.braiding X Y).hom) f (coprod.braiding Y X).hom :=
IsPullback.of_horiz_isIso ⟨by simp⟩
have := (IsPullback.of_hasPullback (f ≫ (coprod.braiding X Y).hom) coprod.inl).paste_horiz this
simp only [coprod.braiding_hom, Category.comp_id, colimit.ι_desc, BinaryCofan.mk_pt,
BinaryCofan.ι_app_left, BinaryCofan.mk_inl] at this
exact ⟨⟨⟨_, this.isLimit⟩⟩⟩ | instance | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | hasPullbackInr' | null |
hasPullbackInr :
HasPullback coprod.inr f :=
hasPullback_symmetry _ _ | instance | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | hasPullbackInr | null |
noncomputable
preservesPullbackInl' :
PreservesLimit (cospan f coprod.inl) F :=
preservesPullback_symmetry _ _ _ | instance | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | preservesPullbackInl' | null |
noncomputable
preservesPullbackInr' :
PreservesLimit (cospan f coprod.inr) F := by
apply preservesLimit_of_iso_diagram (K₁ := cospan (f ≫ (coprod.braiding X Y).hom) coprod.inl)
apply cospanExt (Iso.refl _) (Iso.refl _) (coprod.braiding X Y).symm <;> simp | instance | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | preservesPullbackInr' | null |
noncomputable
preservesPullbackInr :
PreservesLimit (cospan coprod.inr f) F :=
preservesPullback_symmetry _ _ _ | instance | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | preservesPullbackInr | null |
FinitaryExtensive.mono_inr_of_isColimit [FinitaryExtensive C] {c : BinaryCofan X Y}
(hc : IsColimit c) : Mono c.inr :=
BinaryCofan.mono_inr_of_isVanKampen (FinitaryExtensive.vanKampen c hc) | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | FinitaryExtensive.mono_inr_of_isColimit | null |
FinitaryExtensive.mono_inl_of_isColimit [FinitaryExtensive C] {c : BinaryCofan X Y}
(hc : IsColimit c) : Mono c.inl :=
FinitaryExtensive.mono_inr_of_isColimit (BinaryCofan.isColimitFlip hc) | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | FinitaryExtensive.mono_inl_of_isColimit | null |
FinitaryExtensive.isPullback_initial_to_binaryCofan [FinitaryExtensive C]
{c : BinaryCofan X Y} (hc : IsColimit c) :
IsPullback (initial.to _) (initial.to _) c.inl c.inr :=
BinaryCofan.isPullback_initial_to_of_isVanKampen (FinitaryExtensive.vanKampen c hc) | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | FinitaryExtensive.isPullback_initial_to_binaryCofan | null |
finitaryExtensive_iff_of_isTerminal (C : Type u) [Category.{v} C] [HasFiniteCoproducts C]
[HasPullbacksOfInclusions C]
(T : C) (HT : IsTerminal T) (c₀ : BinaryCofan T T) (hc₀ : IsColimit c₀) :
FinitaryExtensive C ↔ IsVanKampenColimit c₀ := by
refine ⟨fun H => H.van_kampen' c₀ hc₀, fun H => ?_⟩
constructor
simp_rw [BinaryCofan.isVanKampen_iff] at H ⊢
intro X Y c hc X' Y' c' αX αY f hX hY
obtain ⟨d, hd, hd'⟩ :=
Limits.BinaryCofan.IsColimit.desc' hc (HT.from _ ≫ c₀.inl) (HT.from _ ≫ c₀.inr)
rw [H c' (αX ≫ HT.from _) (αY ≫ HT.from _) (f ≫ d) (by rw [← reassoc_of% hX, hd, Category.assoc])
(by rw [← reassoc_of% hY, hd', Category.assoc])]
obtain ⟨hl, hr⟩ := (H c (HT.from _) (HT.from _) d hd.symm hd'.symm).mp ⟨hc⟩
rw [hl.paste_vert_iff hX.symm, hr.paste_vert_iff hY.symm] | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | finitaryExtensive_iff_of_isTerminal | null |
types.finitaryExtensive : FinitaryExtensive (Type u) := by
classical
rw [finitaryExtensive_iff_of_isTerminal (Type u) PUnit Types.isTerminalPunit _
(Types.binaryCoproductColimit _ _)]
apply BinaryCofan.isVanKampen_mk _ _ (fun X Y => Types.binaryCoproductColimit X Y) _
fun f g => (Limits.Types.pullbackLimitCone f g).2
· intro _ _ _ _ f hαX hαY
constructor
· refine ⟨⟨hαX.symm⟩, ⟨PullbackCone.isLimitAux' _ ?_⟩⟩
intro s
have : ∀ x, ∃! y, s.fst x = Sum.inl y := by
intro x
rcases h : s.fst x with val | val
· simp only [Types.binaryCoproductCocone_pt, Functor.const_obj_obj, Sum.inl.injEq,
existsUnique_eq']
· apply_fun f at h
cases ((congr_fun s.condition x).symm.trans h).trans (congr_fun hαY val :).symm
delta ExistsUnique at this
choose l hl hl' using this
exact ⟨l, (funext hl).symm, Types.isTerminalPunit.hom_ext _ _,
fun {l'} h₁ _ => funext fun x => hl' x (l' x) (congr_fun h₁ x).symm⟩
· refine ⟨⟨hαY.symm⟩, ⟨PullbackCone.isLimitAux' _ ?_⟩⟩
intro s
have : ∀ x, ∃! y, s.fst x = Sum.inr y := by
intro x
rcases h : s.fst x with val | val
· apply_fun f at h
cases ((congr_fun s.condition x).symm.trans h).trans (congr_fun hαX val :).symm
· simp only [Types.binaryCoproductCocone_pt, Functor.const_obj_obj, Sum.inr.injEq,
existsUnique_eq']
delta ExistsUnique at this
choose l hl hl' using this
exact ⟨l, (funext hl).symm, Types.isTerminalPunit.hom_ext _ _,
fun {l'} h₁ _ => funext fun x => hl' x (l' x) (congr_fun h₁ x).symm⟩
· intro Z f
dsimp [Limits.Types.binaryCoproductCocone]
delta Types.PullbackObj
have : ∀ x, f x = Sum.inl PUnit.unit ∨ f x = Sum.inr PUnit.unit := by
intro x
rcases f x with (⟨⟨⟩⟩ | ⟨⟨⟩⟩)
exacts [Or.inl rfl, Or.inr rfl]
let eX : { p : Z × PUnit // f p.fst = Sum.inl p.snd } ≃ { x : Z // f x = Sum.inl PUnit.unit } :=
⟨fun p => ⟨p.1.1, by convert p.2⟩, fun x => ⟨⟨_, _⟩, x.2⟩, fun _ => by ext; rfl,
fun _ => by ext; rfl⟩
let eY : { p : Z × PUnit // f p.fst = Sum.inr p.snd } ≃ { x : Z // f x = Sum.inr PUnit.unit } :=
⟨fun p => ⟨p.1.1, p.2.trans (congr_arg Sum.inr <| Subsingleton.elim _ _)⟩,
fun x => ⟨⟨_, _⟩, x.2⟩, fun _ => by ext; rfl, fun _ => by ext; rfl⟩
fapply BinaryCofan.isColimitMk
· exact fun s x => dite _ (fun h => s.inl <| eX.symm ⟨x, h⟩)
fun h => s.inr <| eY.symm ⟨x, (this x).resolve_left h⟩
· intro s
... | instance | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | types.finitaryExtensive | null |
noncomputable finitaryExtensiveTopCatAux (Z : TopCat.{u})
(f : Z ⟶ TopCat.of (PUnit.{u + 1} ⊕ PUnit.{u + 1})) :
IsColimit (BinaryCofan.mk
(TopCat.pullbackFst f (TopCat.binaryCofan (TopCat.of PUnit) (TopCat.of PUnit)).inl)
(TopCat.pullbackFst f (TopCat.binaryCofan (TopCat.of PUnit) (TopCat.of PUnit)).inr)) := by
have h₁ : Set.range (TopCat.pullbackFst f (TopCat.binaryCofan (.of PUnit) (.of PUnit)).inl) =
f ⁻¹' Set.range Sum.inl := by
apply le_antisymm
· rintro _ ⟨x, rfl⟩; exact ⟨PUnit.unit, x.2.symm⟩
· rintro x ⟨⟨⟩, hx⟩; refine ⟨⟨⟨x, PUnit.unit⟩, hx.symm⟩, rfl⟩
have h₂ : Set.range (TopCat.pullbackFst f (TopCat.binaryCofan (.of PUnit) (.of PUnit)).inr) =
f ⁻¹' Set.range Sum.inr := by
apply le_antisymm
· rintro _ ⟨x, rfl⟩; exact ⟨PUnit.unit, x.2.symm⟩
· rintro x ⟨⟨⟩, hx⟩; refine ⟨⟨⟨x, PUnit.unit⟩, hx.symm⟩, rfl⟩
refine ((TopCat.binaryCofan_isColimit_iff _).mpr ⟨?_, ?_, ?_⟩).some
· refine ⟨(Homeomorph.prodPUnit Z).isEmbedding.comp .subtypeVal, ?_⟩
convert f.hom.2.1 _ isOpen_range_inl
· refine ⟨(Homeomorph.prodPUnit Z).isEmbedding.comp .subtypeVal, ?_⟩
convert f.hom.2.1 _ isOpen_range_inr
· convert Set.isCompl_range_inl_range_inr.preimage f | def | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | finitaryExtensiveTopCatAux | (Implementation) An auxiliary lemma for the proof that `TopCat` is finitary extensive. |
finitaryExtensive_TopCat : FinitaryExtensive TopCat.{u} := by
rw [finitaryExtensive_iff_of_isTerminal TopCat.{u} _ TopCat.isTerminalPUnit _
(TopCat.binaryCofanIsColimit _ _)]
apply BinaryCofan.isVanKampen_mk _ _ (fun X Y => TopCat.binaryCofanIsColimit X Y) _
fun f g => TopCat.pullbackConeIsLimit f g
· intro X' Y' αX αY f hαX hαY
constructor
· refine ⟨⟨hαX.symm⟩, ⟨PullbackCone.isLimitAux' _ ?_⟩⟩
intro s
have : ∀ x, ∃! y, s.fst x = Sum.inl y := by
intro x
rcases h : s.fst x with val | val
· exact ⟨val, rfl, fun y h => Sum.inl_injective h.symm⟩
· apply_fun f at h
cases ((ConcreteCategory.congr_hom s.condition x).symm.trans h).trans
(ConcreteCategory.congr_hom hαY val :).symm
delta ExistsUnique at this
choose l hl hl' using this
refine ⟨TopCat.ofHom ⟨l, ?_⟩, TopCat.ext fun a => (hl a).symm,
TopCat.isTerminalPUnit.hom_ext _ _,
fun {l'} h₁ _ => TopCat.ext fun x =>
hl' x (l' x) (ConcreteCategory.congr_hom h₁ x).symm⟩
apply (IsEmbedding.inl (X := X') (Y := Y')).isInducing.continuous_iff.mpr
convert s.fst.hom.2 using 1
exact (funext hl).symm
· refine ⟨⟨hαY.symm⟩, ⟨PullbackCone.isLimitAux' _ ?_⟩⟩
intro s
have : ∀ x, ∃! y, s.fst x = Sum.inr y := by
intro x
rcases h : s.fst x with val | val
· apply_fun f at h
cases ((ConcreteCategory.congr_hom s.condition x).symm.trans h).trans
(ConcreteCategory.congr_hom hαX val :).symm
· exact ⟨val, rfl, fun y h => Sum.inr_injective h.symm⟩
delta ExistsUnique at this
choose l hl hl' using this
refine ⟨TopCat.ofHom ⟨l, ?_⟩, TopCat.ext fun a => (hl a).symm,
TopCat.isTerminalPUnit.hom_ext _ _,
fun {l'} h₁ _ =>
TopCat.ext fun x => hl' x (l' x) (ConcreteCategory.congr_hom h₁ x).symm⟩
apply (IsEmbedding.inr (X := X') (Y := Y')).isInducing.continuous_iff.mpr
convert s.fst.hom.2 using 1
exact (funext hl).symm
· intro Z f
exact finitaryExtensiveTopCatAux Z f | instance | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | finitaryExtensive_TopCat | null |
finitaryExtensive_of_reflective
[HasFiniteCoproducts D] [HasPullbacksOfInclusions D] [FinitaryExtensive C]
{Gl : C ⥤ D} {Gr : D ⥤ C} (adj : Gl ⊣ Gr) [Gr.Full] [Gr.Faithful]
[∀ X Y (f : X ⟶ Gl.obj Y), HasPullback (Gr.map f) (adj.unit.app Y)]
[∀ X Y (f : X ⟶ Gl.obj Y), PreservesLimit (cospan (Gr.map f) (adj.unit.app Y)) Gl]
[PreservesPullbacksOfInclusions Gl] :
FinitaryExtensive D := by
have : PreservesColimitsOfSize Gl := adj.leftAdjoint_preservesColimits
constructor
intro X Y c hc
apply (IsVanKampenColimit.precompose_isIso_iff
(Functor.isoWhiskerLeft _ (asIso adj.counit) ≪≫ Functor.rightUnitor _).hom).mp
have : ∀ (Z : C) (i : Discrete WalkingPair) (f : Z ⟶ (colimit.cocone (pair X Y ⋙ Gr)).pt),
PreservesLimit (cospan f ((colimit.cocone (pair X Y ⋙ Gr)).ι.app i)) Gl := by
have : pair X Y ⋙ Gr = pair (Gr.obj X) (Gr.obj Y) := by
apply Functor.hext
· rintro ⟨⟨⟩⟩ <;> rfl
· rintro ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩ <;> simp
rw [this]
rintro Z ⟨_ | _⟩ f <;> dsimp <;> infer_instance
refine ((FinitaryExtensive.vanKampen _ (colimit.isColimit <| pair X Y ⋙ _)).map_reflective
adj).of_iso (IsColimit.uniqueUpToIso ?_ ?_)
· exact isColimitOfPreserves Gl (colimit.isColimit _)
· exact (IsColimit.precomposeHomEquiv _ _).symm hc | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | finitaryExtensive_of_reflective | null |
finitaryExtensive_functor [HasPullbacks C] [FinitaryExtensive C] :
FinitaryExtensive (D ⥤ C) :=
haveI : HasFiniteCoproducts (D ⥤ C) := ⟨fun _ => Limits.functorCategoryHasColimitsOfShape⟩
⟨fun c hc => isVanKampenColimit_of_evaluation _ c fun _ =>
FinitaryExtensive.vanKampen _ <| isColimitOfPreserves _ hc⟩ | instance | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | finitaryExtensive_functor | null |
finitaryExtensive_of_preserves_and_reflects (F : C ⥤ D) [FinitaryExtensive D]
[HasFiniteCoproducts C] [HasPullbacksOfInclusions C]
[PreservesPullbacksOfInclusions F]
[ReflectsLimitsOfShape WalkingCospan F] [PreservesColimitsOfShape (Discrete WalkingPair) F]
[ReflectsColimitsOfShape (Discrete WalkingPair) F] : FinitaryExtensive C := by
constructor
intro X Y c hc
refine IsVanKampenColimit.of_iso ?_ (hc.uniqueUpToIso (coprodIsCoprod X Y)).symm
have (i : Discrete WalkingPair) (Z : C) (f : Z ⟶ X ⨿ Y) :
PreservesLimit (cospan f ((BinaryCofan.mk coprod.inl coprod.inr).ι.app i)) F := by
rcases i with ⟨_ | _⟩ <;> dsimp <;> infer_instance
refine (FinitaryExtensive.vanKampen _
(isColimitOfPreserves F (coprodIsCoprod X Y))).of_mapCocone F | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | finitaryExtensive_of_preserves_and_reflects | null |
finitaryExtensive_of_preserves_and_reflects_isomorphism (F : C ⥤ D) [FinitaryExtensive D]
[HasFiniteCoproducts C] [HasPullbacks C] [PreservesLimitsOfShape WalkingCospan F]
[PreservesColimitsOfShape (Discrete WalkingPair) F] [F.ReflectsIsomorphisms] :
FinitaryExtensive C := by
haveI : ReflectsLimitsOfShape WalkingCospan F := reflectsLimitsOfShape_of_reflectsIsomorphisms
haveI : ReflectsColimitsOfShape (Discrete WalkingPair) F :=
reflectsColimitsOfShape_of_reflectsIsomorphisms
exact finitaryExtensive_of_preserves_and_reflects F | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | finitaryExtensive_of_preserves_and_reflects_isomorphism | null |
FinitaryPreExtensive.isUniversal_finiteCoproducts_Fin [FinitaryPreExtensive C] {n : ℕ}
{F : Discrete (Fin n) ⥤ C} {c : Cocone F} (hc : IsColimit c) : IsUniversalColimit c := by
let f : Fin n → C := F.obj ∘ Discrete.mk
have : F = Discrete.functor f :=
Functor.hext (fun _ ↦ rfl) (by rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩; simp [f])
clear_value f
subst this
induction n with
| zero => exact (isVanKampenColimit_of_isEmpty _ hc).isUniversal
| succ n IH =>
refine IsUniversalColimit.of_iso (@isUniversalColimit_extendCofan _ _ _ _ _ _
(IH _ (coproductIsCoproduct _)) (FinitaryPreExtensive.universal' _ (coprodIsCoprod _ _)) ?_)
((extendCofanIsColimit f (coproductIsCoproduct _) (coprodIsCoprod _ _)).uniqueUpToIso hc)
· dsimp
infer_instance | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | FinitaryPreExtensive.isUniversal_finiteCoproducts_Fin | null |
FinitaryPreExtensive.isUniversal_finiteCoproducts [FinitaryPreExtensive C] {ι : Type*}
[Finite ι] {F : Discrete ι ⥤ C} {c : Cocone F} (hc : IsColimit c) : IsUniversalColimit c := by
obtain ⟨n, ⟨e⟩⟩ := Finite.exists_equiv_fin ι
apply (IsUniversalColimit.whiskerEquivalence_iff (Discrete.equivalence e).symm).mp
apply FinitaryPreExtensive.isUniversal_finiteCoproducts_Fin
exact (IsColimit.whiskerEquivalenceEquiv (Discrete.equivalence e).symm) hc | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | FinitaryPreExtensive.isUniversal_finiteCoproducts | null |
FinitaryExtensive.isVanKampen_finiteCoproducts_Fin [FinitaryExtensive C] {n : ℕ}
{F : Discrete (Fin n) ⥤ C} {c : Cocone F} (hc : IsColimit c) : IsVanKampenColimit c := by
let f : Fin n → C := F.obj ∘ Discrete.mk
have : F = Discrete.functor f :=
Functor.hext (fun _ ↦ rfl) (by rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩; simp [f])
clear_value f
subst this
induction n with
| zero => exact isVanKampenColimit_of_isEmpty _ hc
| succ n IH =>
apply IsVanKampenColimit.of_iso _
((extendCofanIsColimit f (coproductIsCoproduct _) (coprodIsCoprod _ _)).uniqueUpToIso hc)
apply @isVanKampenColimit_extendCofan _ _ _ _ _ _ _ _ ?_
· apply IH
exact coproductIsCoproduct _
· apply FinitaryExtensive.van_kampen'
exact coprodIsCoprod _ _
· dsimp
infer_instance | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | FinitaryExtensive.isVanKampen_finiteCoproducts_Fin | null |
FinitaryExtensive.isVanKampen_finiteCoproducts [FinitaryExtensive C] {ι : Type*}
[Finite ι] {F : Discrete ι ⥤ C} {c : Cocone F} (hc : IsColimit c) : IsVanKampenColimit c := by
obtain ⟨n, ⟨e⟩⟩ := Finite.exists_equiv_fin ι
apply (IsVanKampenColimit.whiskerEquivalence_iff (Discrete.equivalence e).symm).mp
apply FinitaryExtensive.isVanKampen_finiteCoproducts_Fin
exact (IsColimit.whiskerEquivalenceEquiv (Discrete.equivalence e).symm) hc | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | FinitaryExtensive.isVanKampen_finiteCoproducts | null |
FinitaryPreExtensive.hasPullbacks_of_is_coproduct [FinitaryPreExtensive C] {ι : Type*}
[Finite ι] {F : Discrete ι ⥤ C} {c : Cocone F} (hc : IsColimit c) (i : Discrete ι) {X : C}
(g : X ⟶ _) : HasPullback g (c.ι.app i) := by
classical
let f : ι → C := F.obj ∘ Discrete.mk
have : F = Discrete.functor f :=
Functor.hext (fun i ↦ rfl) (by rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩; simp [f])
clear_value f
subst this
change Cofan f at c
obtain ⟨i⟩ := i
let e : ∐ f ≅ f i ⨿ (∐ fun j : ({i}ᶜ : Set ι) ↦ f j) :=
{ hom := Sigma.desc (fun j ↦ if h : j = i then eqToHom (congr_arg f h) ≫ coprod.inl else
Sigma.ι (fun j : ({i}ᶜ : Set ι) ↦ f j) ⟨j, h⟩ ≫ coprod.inr)
inv := coprod.desc (Sigma.ι f i) (Sigma.desc fun j ↦ Sigma.ι f j)
hom_inv_id := by cat_disch
inv_hom_id := by
ext j
· simp
· simp only [coprod.desc_comp, colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app,
eqToHom_refl, Category.id_comp, dite_true, BinaryCofan.mk_pt, BinaryCofan.ι_app_right,
BinaryCofan.mk_inr, colimit.ι_desc_assoc, Discrete.functor_obj, Category.comp_id]
exact dif_neg j.prop }
let e' : c.pt ≅ f i ⨿ (∐ fun j : ({i}ᶜ : Set ι) ↦ f j) :=
hc.coconePointUniqueUpToIso (getColimitCocone _).2 ≪≫ e
have : coprod.inl ≫ e'.inv = c.ι.app ⟨i⟩ := by
simp only [e, e', Iso.trans_inv, coprod.desc_comp, colimit.ι_desc, BinaryCofan.mk_pt,
BinaryCofan.ι_app_left, BinaryCofan.mk_inl]
exact colimit.comp_coconePointUniqueUpToIso_inv _ _
clear_value e'
rw [← this]
have : IsPullback (𝟙 _) (g ≫ e'.hom) g e'.inv := IsPullback.of_horiz_isIso ⟨by simp⟩
exact ⟨⟨⟨_, ((IsPullback.of_hasPullback (g ≫ e'.hom) coprod.inl).paste_horiz this).isLimit⟩⟩⟩ | lemma | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | FinitaryPreExtensive.hasPullbacks_of_is_coproduct | null |
FinitaryExtensive.mono_ι [FinitaryExtensive C] {ι : Type*} [Finite ι] {F : Discrete ι ⥤ C}
{c : Cocone F} (hc : IsColimit c) (i : Discrete ι) :
Mono (c.ι.app i) :=
mono_of_cofan_isVanKampen (isVanKampen_finiteCoproducts hc) _ | lemma | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | FinitaryExtensive.mono_ι | null |
FinitaryExtensive.isPullback_initial_to [FinitaryExtensive C]
{ι : Type*} [Finite ι] {F : Discrete ι ⥤ C}
{c : Cocone F} (hc : IsColimit c) (i j : Discrete ι) (e : i ≠ j) :
IsPullback (initial.to _) (initial.to _) (c.ι.app i) (c.ι.app j) :=
isPullback_initial_to_of_cofan_isVanKampen (isVanKampen_finiteCoproducts hc) i j e | lemma | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | FinitaryExtensive.isPullback_initial_to | null |
FinitaryExtensive.isPullback_initial_to_sigma_ι [FinitaryExtensive C] {ι : Type*} [Finite ι]
(X : ι → C) (i j : ι) (e : i ≠ j) :
IsPullback (initial.to _) (initial.to _) (Sigma.ι X i) (Sigma.ι X j) :=
FinitaryExtensive.isPullback_initial_to (coproductIsCoproduct _) ⟨i⟩ ⟨j⟩
(ne_of_apply_ne Discrete.as e) | lemma | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | FinitaryExtensive.isPullback_initial_to_sigma_ι | null |
FinitaryPreExtensive.hasPullbacks_of_inclusions [FinitaryPreExtensive C] {X Z : C}
{α : Type*} (f : X ⟶ Z) {Y : (a : α) → C} (i : (a : α) → Y a ⟶ Z) [Finite α]
[hi : IsIso (Sigma.desc i)] (a : α) : HasPullback f (i a) := by
apply FinitaryPreExtensive.hasPullbacks_of_is_coproduct (c := Cofan.mk Z i)
exact @IsColimit.ofPointIso (t := Cofan.mk Z i) (P := _) (i := hi) | instance | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | FinitaryPreExtensive.hasPullbacks_of_inclusions | null |
FinitaryPreExtensive.isIso_sigmaDesc_fst [FinitaryPreExtensive C] {α : Type} [Finite α]
{X : C} {Z : α → C} (π : (a : α) → Z a ⟶ X) {Y : C} (f : Y ⟶ X) (hπ : IsIso (Sigma.desc π)) :
IsIso (Sigma.desc ((fun _ ↦ pullback.fst _ _) : (a : α) → pullback f (π a) ⟶ _)) := by
let c := (Cofan.mk _ ((fun _ ↦ pullback.fst _ _) : (a : α) → pullback f (π a) ⟶ _))
apply c.isColimit_iff_isIso_sigmaDesc.mpr
have hau : IsUniversalColimit (Cofan.mk X π) := FinitaryPreExtensive.isUniversal_finiteCoproducts
((Cofan.isColimit_iff_isIso_sigmaDesc _).mp hπ).some
refine hau.nonempty_isColimit_of_pullbackCone_left _ (𝟙 _) _ _ (fun i ↦ ?_)
(PullbackCone.mk (𝟙 _) f (by simp)) (IsPullback.id_horiz f).isLimit _ (Iso.refl _)
(by simp) (by simp [c]) (by simp [pullback.condition, c])
exact pullback.isLimit _ _
@[deprecated (since := "2025-06-20")]
alias FinitaryPreExtensive.sigma_desc_iso := FinitaryPreExtensive.isIso_sigmaDesc_fst | lemma | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | FinitaryPreExtensive.isIso_sigmaDesc_fst | null |
FinitaryPreExtensive.isIso_sigmaDesc_map [HasPullbacks C] [FinitaryPreExtensive C]
{ι ι' : Type*} [Finite ι] [Finite ι'] {S : C} {X : ι → C} {Y : ι' → C}
(f : ∀ i, X i ⟶ S) (g : ∀ i, Y i ⟶ S) :
IsIso (Sigma.desc fun (p : ι × ι') ↦
pullback.map (f p.1) (g p.2) (Sigma.desc f) (Sigma.desc g) (Sigma.ι _ p.1)
(Sigma.ι _ p.2) (𝟙 S) (by simp) (by simp)) := by
let c : Cofan _ := Cofan.mk _ <| fun (p : ι × ι') ↦
pullback.map (f p.1) (g p.2) (Sigma.desc f) (Sigma.desc g) (Sigma.ι _ p.1)
(Sigma.ι _ p.2) (𝟙 S) (by simp) (by simp)
apply c.isColimit_iff_isIso_sigmaDesc.mpr
refine IsUniversalColimit.nonempty_isColimit_prod_of_pullbackCone
(a := Cofan.mk _ <| fun i ↦ Sigma.ι _ i) (b := Cofan.mk _ <| fun i ↦ Sigma.ι _ i)
?_ ?_ f g (Sigma.desc f) (Sigma.desc g) (fun i j ↦ (pullback.cone (f i) (g j)))
(fun i j ↦ pullback.isLimit (f i) (g j)) (pullback.cone _ _) ?_ (Iso.refl _)
· exact FinitaryPreExtensive.isUniversal_finiteCoproducts (coproductIsCoproduct X)
· exact FinitaryPreExtensive.isUniversal_finiteCoproducts (coproductIsCoproduct Y)
· exact pullback.isLimit (Sigma.desc f) (Sigma.desc g) | instance | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | FinitaryPreExtensive.isIso_sigmaDesc_map | If `C` has pullbacks and is finitary (pre-)extensive, pullbacks distribute over finite
coproducts, i.e., `∐ (Xᵢ ×[S] Xⱼ) ≅ (∐ Xᵢ) ×[S] (∐ Xⱼ)`.
For an `IsPullback` version, see `FinitaryPreExtensive.isPullback_sigmaDesc`. |
FinitaryPreExtensive.isPullback_sigmaDesc [HasPullbacks C] [FinitaryPreExtensive C]
{ι ι' : Type*} [Finite ι] [Finite ι'] {S : C} {X : ι → C} {Y : ι' → C}
(f : ∀ i, X i ⟶ S) (g : ∀ i, Y i ⟶ S) :
IsPullback
(Limits.Sigma.desc fun (p : ι × ι') ↦ pullback.fst (f p.1) (g p.2) ≫ Sigma.ι X p.1)
(Limits.Sigma.desc fun (p : ι × ι') ↦ pullback.snd (f p.1) (g p.2) ≫ Sigma.ι Y p.2)
(Limits.Sigma.desc f) (Limits.Sigma.desc g) := by
let c : Cofan _ := Cofan.mk _ <| fun (p : ι × ι') ↦
pullback.map (f p.1) (g p.2) (Sigma.desc f) (Sigma.desc g) (Sigma.ι _ p.1)
(Sigma.ι _ p.2) (𝟙 S) (by simp) (by simp)
convert IsUniversalColimit.isPullback_prod_of_isColimit
(d := Cofan.mk _ (Sigma.ι fun (p : ι × ι') ↦ pullback (f p.1) (g p.2)))
(hd := coproductIsCoproduct (fun (p : ι × ι') ↦ pullback (f p.1) (g p.2)))
(a := Cofan.mk _ <| fun i ↦ Sigma.ι _ i) (b := Cofan.mk _ <| fun i ↦ Sigma.ι _ i)
?_ ?_ f g (Sigma.desc f) (Sigma.desc g) (fun i j ↦ IsPullback.of_hasPullback (f i) (g j))
· ext
simp [Cofan.IsColimit.desc, Sigma.ι, coproductIsCoproduct]
· ext
simp [Cofan.IsColimit.desc, Sigma.ι, coproductIsCoproduct]
· exact FinitaryPreExtensive.isUniversal_finiteCoproducts (coproductIsCoproduct X)
· exact FinitaryPreExtensive.isUniversal_finiteCoproducts (coproductIsCoproduct Y) | lemma | CategoryTheory | [
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.Limits.Shapes.StrictInitial",
"Mathlib.CategoryTheory.Limits.Types.Shapes",
"Mathlib.Topology.Category.TopCat.Limits.Pullbacks",
"Mathlib.CategoryTheory.Limits.FunctorCategory.Basic",
"Mathlib.CategoryTheory.Limits.Constructio... | Mathlib/CategoryTheory/Extensive.lean | FinitaryPreExtensive.isPullback_sigmaDesc | If `C` has pullbacks and is finitary (pre-)extensive, pullbacks distribute over finite
coproducts, i.e., `∐ (Xᵢ ×[S] Xⱼ) ≅ (∐ Xᵢ) ×[S] (∐ Xⱼ)`.
For a variant, see `FinitaryPreExtensive.isIso_sigmaDesc_map`. |
FintypeCat where
/-- The underlying type. -/
carrier : Type*
[str : Fintype carrier]
attribute [instance] FintypeCat.str | structure | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | FintypeCat | The category of finite types. |
instCoeSort : CoeSort FintypeCat Type* :=
⟨carrier⟩ | instance | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | instCoeSort | null |
of (X : Type*) [Fintype X] : FintypeCat where
carrier := X | abbrev | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | of | Construct a bundled `FintypeCat` from the underlying type and typeclass. |
@[simps!]
incl : FintypeCat ⥤ Type* :=
inducedFunctor _ | def | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | incl | The fully faithful embedding of `FintypeCat` into the category of types. |
concreteCategoryFintype : ConcreteCategory FintypeCat (· ⟶ ·) where
hom f := f
ofHom f := f
/- Help typeclass inference infer fullness of forgetful functor. -/ | instance | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | concreteCategoryFintype | null |
@[simp]
id_apply (X : FintypeCat) (x : X) : (𝟙 X : X → X) x = x :=
rfl
@[simp] | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | id_apply | null |
comp_apply {X Y Z : FintypeCat} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) : (f ≫ g) x = g (f x) :=
rfl | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | comp_apply | null |
hom_inv_id_apply {X Y : FintypeCat} (f : X ≅ Y) (x : X) : f.inv (f.hom x) = x :=
congr_fun f.hom_inv_id x | lemma | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | hom_inv_id_apply | null |
inv_hom_id_apply {X Y : FintypeCat} (f : X ≅ Y) (y : Y) : f.hom (f.inv y) = y :=
congr_fun f.inv_hom_id y
@[ext] | lemma | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | inv_hom_id_apply | null |
hom_ext {X Y : FintypeCat} (f g : X ⟶ Y) (h : ∀ x, f x = g x) : f = g := by
funext
apply h | lemma | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | hom_ext | null |
@[simps]
equivEquivIso {A B : FintypeCat} : A ≃ B ≃ (A ≅ B) where
toFun e :=
{ hom := e
inv := e.symm }
invFun i :=
{ toFun := i.hom
invFun := i.inv
left_inv := congr_fun i.hom_inv_id
right_inv := congr_fun i.inv_hom_id }
left_inv := by cat_disch
right_inv := by cat_disch | def | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | equivEquivIso | Equivalences between finite types are the same as isomorphisms in `FintypeCat`. |
Skeleton : Type u :=
ULift ℕ | def | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | Skeleton | The "standard" skeleton for `FintypeCat`. This is the full subcategory of `FintypeCat`
spanned by objects of the form `ULift (Fin n)` for `n : ℕ`. We parameterize the objects
of `Fintype.Skeleton` directly as `ULift ℕ`, as the type `ULift (Fin m) ≃ ULift (Fin n)`
is nonempty if and only if `n = m`. Specifying universes, `Skeleton : Type u` is a small
skeletal category equivalent to `Fintype.{u}`. |
mk : ℕ → Skeleton :=
ULift.up | def | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | mk | Given any natural number `n`, this creates the associated object of `Fintype.Skeleton`. |
len : Skeleton → ℕ :=
ULift.down
@[ext] | def | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | len | Given any object of `Fintype.Skeleton`, this returns the associated natural number. |
ext (X Y : Skeleton) : X.len = Y.len → X = Y :=
ULift.ext _ _ | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | ext | null |
is_skeletal : Skeletal Skeleton.{u} := fun X Y ⟨h⟩ =>
ext _ _ <|
Fin.equiv_iff_eq.mp <|
Nonempty.intro <|
{ toFun := fun x => (h.hom ⟨x⟩).down
invFun := fun x => (h.inv ⟨x⟩).down
left_inv := by
intro a
change ULift.down _ = _
rw [ULift.up_down]
change ((h.hom ≫ h.inv) _).down = _
simp
rfl
right_inv := by
intro a
change ULift.down _ = _
rw [ULift.up_down]
change ((h.inv ≫ h.hom) _).down = _
simp
rfl } | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | is_skeletal | null |
incl : Skeleton.{u} ⥤ FintypeCat.{u} where
obj X := FintypeCat.of (ULift (Fin X.len))
map f := f | def | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | incl | The canonical fully faithful embedding of `Fintype.Skeleton` into `FintypeCat`. |
noncomputable equivalence : Skeleton ≌ FintypeCat :=
incl.asEquivalence
@[simp] | def | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | equivalence | The equivalence between `Fintype.Skeleton` and `Fintype`. |
incl_mk_nat_card (n : ℕ) : Fintype.card (incl.obj (mk n)) = n := by
convert Finset.card_fin n
apply Fintype.ofEquiv_card | theorem | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | incl_mk_nat_card | null |
isSkeleton : IsSkeletonOf FintypeCat Skeleton Skeleton.incl where
skel := Skeleton.is_skeletal
eqv := by infer_instance | lemma | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | isSkeleton | `Fintype.Skeleton` is a skeleton of `Fintype`. |
noncomputable uSwitch : FintypeCat.{u} ⥤ FintypeCat.{v} where
obj X := FintypeCat.of <| ULift.{v} (Fin (Fintype.card X))
map {X Y} f x := ULift.up <| (Fintype.equivFin Y) (f ((Fintype.equivFin X).symm x.down))
map_comp {X Y Z} f g := by funext; simp | def | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | uSwitch | If `u` and `v` are two arbitrary universes, we may construct a functor
`uSwitch.{u, v} : FintypeCat.{u} ⥤ FintypeCat.{v}` by sending
`X : FintypeCat.{u}` to `ULift.{v} (Fin (Fintype.card X))`. |
noncomputable uSwitchEquiv (X : FintypeCat.{u}) :
uSwitch.{u, v}.obj X ≃ X :=
Equiv.ulift.trans (Fintype.equivFin X).symm | def | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | uSwitchEquiv | Switching the universe of an object `X : FintypeCat.{u}` does not change `X` up to equivalence
of types. This is natural in the sense that it commutes with `uSwitch.map f` for
any `f : X ⟶ Y` in `FintypeCat.{u}`. |
uSwitchEquiv_naturality {X Y : FintypeCat.{u}} (f : X ⟶ Y)
(x : uSwitch.{u, v}.obj X) :
f (X.uSwitchEquiv x) = Y.uSwitchEquiv (uSwitch.map f x) := by
simp only [uSwitch, uSwitchEquiv, Equiv.trans_apply, Equiv.ulift_apply, Equiv.symm_apply_apply] | lemma | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | uSwitchEquiv_naturality | null |
uSwitchEquiv_symm_naturality {X Y : FintypeCat.{u}} (f : X ⟶ Y) (x : X) :
uSwitch.map f (X.uSwitchEquiv.symm x) = Y.uSwitchEquiv.symm (f x) := by
rw [← Equiv.apply_eq_iff_eq_symm_apply, ← uSwitchEquiv_naturality f,
Equiv.apply_symm_apply] | lemma | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | uSwitchEquiv_symm_naturality | null |
uSwitch_map_uSwitch_map {X Y : FintypeCat.{u}} (f : X ⟶ Y) :
uSwitch.map (uSwitch.map f) =
(equivEquivIso ((uSwitch.obj X).uSwitchEquiv.trans X.uSwitchEquiv)).hom ≫
f ≫ (equivEquivIso ((uSwitch.obj Y).uSwitchEquiv.trans
Y.uSwitchEquiv)).inv := rfl | lemma | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | uSwitch_map_uSwitch_map | null |
noncomputable uSwitchEquivalence : FintypeCat.{u} ≌ FintypeCat.{v} where
functor := uSwitch
inverse := uSwitch
unitIso := NatIso.ofComponents (fun X ↦ (equivEquivIso <|
(uSwitch.obj X).uSwitchEquiv.trans X.uSwitchEquiv).symm) <| by
simp [uSwitch_map_uSwitch_map]
counitIso := NatIso.ofComponents (fun X ↦ equivEquivIso <|
(uSwitch.obj X).uSwitchEquiv.trans X.uSwitchEquiv) <| by
simp [uSwitch_map_uSwitch_map]
functor_unitIso_comp X := by
ext x
simp [← uSwitchEquiv_naturality] | def | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | uSwitchEquivalence | `uSwitch.{u, v}` is an equivalence of categories with quasi-inverse `uSwitch.{v, u}`. |
naturality (σ : F ⟶ G) (f : X ⟶ Y) (x : F.obj X) :
σ.app Y (F.map f x) = G.map f (σ.app X x) :=
congr_fun (σ.naturality f) x | lemma | CategoryTheory | [
"Mathlib.CategoryTheory.ConcreteCategory.Basic",
"Mathlib.CategoryTheory.Endomorphism",
"Mathlib.CategoryTheory.Skeletal",
"Mathlib.Data.Finite.Prod"
] | Mathlib/CategoryTheory/FintypeCat.lean | naturality | null |
GlueData where
/-- The index type `J` of a gluing datum -/
J : Type v
/-- For each `i : J`, an object `U i` -/
U : J → C
/-- For each `i j : J`, an object `V i j` -/
V : J × J → C
/-- For each `i j : J`, a monomorphism `f i j : V i j ⟶ U i` -/
f : ∀ i j, V (i, j) ⟶ U i
f_mono : ∀ i j, Mono (f i j) := by infer_instance
f_hasPullback : ∀ i j k, HasPullback (f i j) (f i k) := by infer_instance
f_id : ∀ i, IsIso (f i i) := by infer_instance
/-- For each `i j : J`, a transition map `t i j : V i j ⟶ V j i` -/
t : ∀ i j, V (i, j) ⟶ V (j, i)
t_id : ∀ i, t i i = 𝟙 _
/-- The morphism via which `V i j ×[U i] V i k ⟶ V i j ⟶ V j i` factors through
`V j k ×[U j] V j i ⟶ V j i` -/
t' : ∀ i j k, pullback (f i j) (f i k) ⟶ pullback (f j k) (f j i)
t_fac : ∀ i j k, t' i j k ≫ pullback.snd _ _ = pullback.fst _ _ ≫ t i j
cocycle : ∀ i j k, t' i j k ≫ t' j k i ≫ t' k i j = 𝟙 _
attribute [simp] GlueData.t_id
attribute [instance] GlueData.f_id GlueData.f_mono GlueData.f_hasPullback
attribute [reassoc] GlueData.t_fac GlueData.cocycle | structure | CategoryTheory | [
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer",
"Mathlib.CategoryTheory.Limits.Constructions.EpiMono",
"Mathlib.CategoryTheory.Limits.Preserves.Limits",
"Mathlib.CategoryTheory.Limits.Types.Shapes"
] | Mathlib/CategoryTheory/GlueData.lean | GlueData | A gluing datum consists of
1. An index type `J`
2. An object `U i` for each `i : J`.
3. An object `V i j` for each `i j : J`.
4. A monomorphism `f i j : V i j ⟶ U i` for each `i j : J`.
5. A transition map `t i j : V i j ⟶ V j i` for each `i j : J`.
such that
6. `f i i` is an isomorphism.
7. `t i i` is the identity.
8. The pullback for `f i j` and `f i k` exists.
9. `V i j ×[U i] V i k ⟶ V i j ⟶ V j i` factors through `V j k ×[U j] V j i ⟶ V j i` via some
`t' : V i j ×[U i] V i k ⟶ V j k ×[U j] V j i`.
10. `t' i j k ≫ t' j k i ≫ t' k i j = 𝟙 _`. |
@[simp]
t'_iij (i j : D.J) : D.t' i i j = (pullbackSymmetry _ _).hom := by
have eq₁ := D.t_fac i i j
have eq₂ := (IsIso.eq_comp_inv (D.f i i)).mpr (@pullback.condition _ _ _ _ _ _ (D.f i j) _)
rw [D.t_id, Category.comp_id, eq₂] at eq₁
have eq₃ := (IsIso.eq_comp_inv (D.f i i)).mp eq₁
rw [Category.assoc, ← pullback.condition, ← Category.assoc] at eq₃
exact
Mono.right_cancellation _ _
((Mono.right_cancellation _ _ eq₃).trans (pullbackSymmetry_hom_comp_fst _ _).symm) | theorem | CategoryTheory | [
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer",
"Mathlib.CategoryTheory.Limits.Constructions.EpiMono",
"Mathlib.CategoryTheory.Limits.Preserves.Limits",
"Mathlib.CategoryTheory.Limits.Types.Shapes"
] | Mathlib/CategoryTheory/GlueData.lean | t'_iij | null |
t'_jii (i j : D.J) : D.t' j i i = pullback.fst _ _ ≫ D.t j i ≫ inv (pullback.snd _ _) := by
rw [← Category.assoc, ← D.t_fac]
simp | theorem | CategoryTheory | [
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer",
"Mathlib.CategoryTheory.Limits.Constructions.EpiMono",
"Mathlib.CategoryTheory.Limits.Preserves.Limits",
"Mathlib.CategoryTheory.Limits.Types.Shapes"
] | Mathlib/CategoryTheory/GlueData.lean | t'_jii | null |
t'_iji (i j : D.J) : D.t' i j i = pullback.fst _ _ ≫ D.t i j ≫ inv (pullback.snd _ _) := by
rw [← Category.assoc, ← D.t_fac]
simp
@[reassoc, elementwise (attr := simp)] | theorem | CategoryTheory | [
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer",
"Mathlib.CategoryTheory.Limits.Constructions.EpiMono",
"Mathlib.CategoryTheory.Limits.Preserves.Limits",
"Mathlib.CategoryTheory.Limits.Types.Shapes"
] | Mathlib/CategoryTheory/GlueData.lean | t'_iji | null |
t_inv (i j : D.J) : D.t i j ≫ D.t j i = 𝟙 _ := by
have eq : (pullbackSymmetry (D.f i i) (D.f i j)).hom =
pullback.snd _ _ ≫ inv (pullback.fst _ _) := by simp
have := D.cocycle i j i
rw [D.t'_iij, D.t'_jii, D.t'_iji, fst_eq_snd_of_mono_eq, eq] at this
simp only [Category.assoc, IsIso.inv_hom_id_assoc] at this
rw [← IsIso.eq_inv_comp, ← Category.assoc, IsIso.comp_inv_eq] at this
simpa using this | theorem | CategoryTheory | [
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer",
"Mathlib.CategoryTheory.Limits.Constructions.EpiMono",
"Mathlib.CategoryTheory.Limits.Preserves.Limits",
"Mathlib.CategoryTheory.Limits.Types.Shapes"
] | Mathlib/CategoryTheory/GlueData.lean | t_inv | null |
t'_inv (i j k : D.J) :
D.t' i j k ≫ (pullbackSymmetry _ _).hom ≫ D.t' j i k ≫ (pullbackSymmetry _ _).hom = 𝟙 _ := by
rw [← cancel_mono (pullback.fst (D.f i j) (D.f i k))]
simp [t_fac, t_fac_assoc] | theorem | CategoryTheory | [
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer",
"Mathlib.CategoryTheory.Limits.Constructions.EpiMono",
"Mathlib.CategoryTheory.Limits.Preserves.Limits",
"Mathlib.CategoryTheory.Limits.Types.Shapes"
] | Mathlib/CategoryTheory/GlueData.lean | t'_inv | null |
t_isIso (i j : D.J) : IsIso (D.t i j) :=
⟨⟨D.t j i, D.t_inv _ _, D.t_inv _ _⟩⟩ | instance | CategoryTheory | [
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer",
"Mathlib.CategoryTheory.Limits.Constructions.EpiMono",
"Mathlib.CategoryTheory.Limits.Preserves.Limits",
"Mathlib.CategoryTheory.Limits.Types.Shapes"
] | Mathlib/CategoryTheory/GlueData.lean | t_isIso | null |
t'_isIso (i j k : D.J) : IsIso (D.t' i j k) :=
⟨⟨D.t' j k i ≫ D.t' k i j, D.cocycle _ _ _, by simpa using D.cocycle _ _ _⟩⟩
@[reassoc] | instance | CategoryTheory | [
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer",
"Mathlib.CategoryTheory.Limits.Constructions.EpiMono",
"Mathlib.CategoryTheory.Limits.Preserves.Limits",
"Mathlib.CategoryTheory.Limits.Types.Shapes"
] | Mathlib/CategoryTheory/GlueData.lean | t'_isIso | null |
t'_comp_eq_pullbackSymmetry (i j k : D.J) :
D.t' j k i ≫ D.t' k i j =
(pullbackSymmetry _ _).hom ≫ D.t' j i k ≫ (pullbackSymmetry _ _).hom := by
trans inv (D.t' i j k)
· exact IsIso.eq_inv_of_hom_inv_id (D.cocycle _ _ _)
· rw [← cancel_mono (pullback.fst (D.f i j) (D.f i k))]
simp [t_fac, t_fac_assoc] | theorem | CategoryTheory | [
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer",
"Mathlib.CategoryTheory.Limits.Constructions.EpiMono",
"Mathlib.CategoryTheory.Limits.Preserves.Limits",
"Mathlib.CategoryTheory.Limits.Types.Shapes"
] | Mathlib/CategoryTheory/GlueData.lean | t'_comp_eq_pullbackSymmetry | null |
sigmaOpens [HasCoproduct D.U] : C :=
∐ D.U | def | CategoryTheory | [
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer",
"Mathlib.CategoryTheory.Limits.Constructions.EpiMono",
"Mathlib.CategoryTheory.Limits.Preserves.Limits",
"Mathlib.CategoryTheory.Limits.Types.Shapes"
] | Mathlib/CategoryTheory/GlueData.lean | sigmaOpens | (Implementation) The disjoint union of `U i`. |
diagram : MultispanIndex (.prod D.J) C where
left := D.V
right := D.U
fst := fun ⟨i, j⟩ => D.f i j
snd := fun ⟨i, j⟩ => D.t i j ≫ D.f j i
@[simp] | def | CategoryTheory | [
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer",
"Mathlib.CategoryTheory.Limits.Constructions.EpiMono",
"Mathlib.CategoryTheory.Limits.Preserves.Limits",
"Mathlib.CategoryTheory.Limits.Types.Shapes"
] | Mathlib/CategoryTheory/GlueData.lean | diagram | (Implementation) The diagram to take colimit of. |
diagram_fst (i j : D.J) : D.diagram.fst ⟨i, j⟩ = D.f i j :=
rfl
@[simp] | theorem | CategoryTheory | [
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer",
"Mathlib.CategoryTheory.Limits.Constructions.EpiMono",
"Mathlib.CategoryTheory.Limits.Preserves.Limits",
"Mathlib.CategoryTheory.Limits.Types.Shapes"
] | Mathlib/CategoryTheory/GlueData.lean | diagram_fst | null |
diagram_snd (i j : D.J) : D.diagram.snd ⟨i, j⟩ = D.t i j ≫ D.f j i :=
rfl
@[simp] | theorem | CategoryTheory | [
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer",
"Mathlib.CategoryTheory.Limits.Constructions.EpiMono",
"Mathlib.CategoryTheory.Limits.Preserves.Limits",
"Mathlib.CategoryTheory.Limits.Types.Shapes"
] | Mathlib/CategoryTheory/GlueData.lean | diagram_snd | null |
diagram_left : D.diagram.left = D.V :=
rfl
@[simp] | theorem | CategoryTheory | [
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer",
"Mathlib.CategoryTheory.Limits.Constructions.EpiMono",
"Mathlib.CategoryTheory.Limits.Preserves.Limits",
"Mathlib.CategoryTheory.Limits.Types.Shapes"
] | Mathlib/CategoryTheory/GlueData.lean | diagram_left | null |
diagram_right : D.diagram.right = D.U :=
rfl | theorem | CategoryTheory | [
"Mathlib.Tactic.CategoryTheory.Elementwise",
"Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer",
"Mathlib.CategoryTheory.Limits.Constructions.EpiMono",
"Mathlib.CategoryTheory.Limits.Preserves.Limits",
"Mathlib.CategoryTheory.Limits.Types.Shapes"
] | Mathlib/CategoryTheory/GlueData.lean | diagram_right | null |
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