feat(CategoryTheory/Sites): local sites

Mathlib.lean

@@ -3374,6 +3374,7 @@ public import Mathlib.CategoryTheory.Sites.JointlySurjective
 public import Mathlib.CategoryTheory.Sites.LeftExact
 public import Mathlib.CategoryTheory.Sites.Limits
 public import Mathlib.CategoryTheory.Sites.LocalProperties
+public import Mathlib.CategoryTheory.Sites.LocalSite
 public import Mathlib.CategoryTheory.Sites.Localization
 public import Mathlib.CategoryTheory.Sites.LocallyBijective
 public import Mathlib.CategoryTheory.Sites.LocallyFullyFaithful

Mathlib/CategoryTheory/Limits/Shapes/Products.lean

@@ -826,10 +826,21 @@ instance (priority := 100) hasProduct_unique [Nonempty β] [Subsingleton β] (f
   let ⟨_⟩ := nonempty_unique β; HasLimit.mk (limitConeOfUnique f)
 
 /-- A product over an index type with exactly one term is just the object over that term. -/
-@[simps!]
 def productUniqueIso [Unique β] (f : β → C) : ∏ᶜ f ≅ f default :=
   IsLimit.conePointUniqueUpToIso (limit.isLimit _) (limitConeOfUnique f).isLimit
 
+@[simp]
+lemma productUniqueIso_hom [Unique β] (f : β → C) : (productUniqueIso f).hom = Pi.π f default :=
+  rfl
+
+@[reassoc (attr := simp)]
+lemma productUniqueIso_inv_π [Unique β] (f : β → C) (b : β) :
+    (productUniqueIso f).inv ≫ Pi.π f b = eqToHom (congrArg _ <| Subsingleton.allEq _ _) := by
+  obtain rfl := Subsingleton.allEq b default
+  simp [Iso.inv_comp_eq]
+
+@[deprecated (since := "2026-06-30")] alias productUniqueIso_inv := productUniqueIso_inv_π
+
 set_option backward.defeqAttrib.useBackward true in
 /-- Any isomorphism is the projection from a single object product. -/
 def Fan.isLimitMkOfUnique {X Y : C} (e : X ≅ Y) (J : Type*) [Unique J] :
@@ -863,10 +874,23 @@ instance (priority := 100) hasCoproduct_unique [Nonempty β] [Subsingleton β] (
   let ⟨_⟩ := nonempty_unique β; HasColimit.mk (colimitCoconeOfUnique f)
 
 /-- A coproduct over an index type with exactly one term is just the object over that term. -/
-@[simps!]
 def coproductUniqueIso [Unique β] (f : β → C) : ∐ f ≅ f default :=
   IsColimit.coconePointUniqueUpToIso (colimit.isColimit _) (colimitCoconeOfUnique f).isColimit
 
+@[simp]
+lemma coproductUniqueIso_inv [Unique β] (f : β → C) :
+    (coproductUniqueIso f).inv = Sigma.ι f default :=
+  rfl
+
+@[reassoc (attr := simp)]
+lemma ι_coproductUniqueIso_hom [Unique β] (f : β → C) (b : β) :
+    Sigma.ι f b ≫ (coproductUniqueIso f).hom = eqToHom (congrArg _ <| Subsingleton.allEq _ _) := by
+  obtain rfl := Subsingleton.allEq b default
+  symm
+  simp [← Iso.comp_inv_eq]
+
+@[deprecated (since := "2026-06-30")] alias coproductUniqueIso_hom := ι_coproductUniqueIso_hom
+
 set_option backward.defeqAttrib.useBackward true in
 /-- Any isomorphism is the projection from a single object product. -/
 def Cofan.isColimitMkOfUnique {X Y : C} (e : X ≅ Y) (J : Type*) [Unique J] :

Mathlib/CategoryTheory/ShrinkYoneda.lean

@@ -87,13 +87,16 @@ noncomputable def shrinkYonedaObjObjEquiv {X : C} {Y : Cᵒᵖ} :
     ((shrinkYoneda.{w}.obj X).obj Y) ≃ (Y.unop ⟶ X) :=
   (equivShrink _).symm
 
-set_option backward.defeqAttrib.useBackward true in
-set_option backward.isDefEq.respectTransparency false in
+lemma shrinkYoneda_obj_map {X : C} {Y Y' : Cᵒᵖ} (g : Y ⟶ Y') (f : (shrinkYoneda.obj X).obj Y) :
+    (shrinkYoneda.obj _).map g f =
+      shrinkYonedaObjObjEquiv.symm (g.unop ≫ shrinkYonedaObjObjEquiv f) :=
+  rfl
+
 lemma shrinkYoneda_obj_map_shrinkYonedaObjObjEquiv_symm
     {X : C} {Y Y' : Cᵒᵖ} (g : Y ⟶ Y') (f : Y.unop ⟶ X) :
     (shrinkYoneda.obj _).map g (shrinkYonedaObjObjEquiv.symm f) =
       shrinkYonedaObjObjEquiv.symm (g.unop ≫ f) := by
-  simp [shrinkYoneda, shrinkYonedaObjObjEquiv]
+  simp [shrinkYoneda_obj_map]
 
 lemma shrinkYonedaObjObjEquiv_symm_comp {X Y Y' : C} (g : Y' ⟶ Y) (f : Y ⟶ X) :
     shrinkYonedaObjObjEquiv.symm (g ≫ f) =
@@ -274,6 +277,11 @@ noncomputable abbrev shrinkCoyonedaObjObjEquiv {X : Cᵒᵖ} {Y : C} :
     ((shrinkCoyoneda.{w}.obj X).obj Y) ≃ (X.unop ⟶ Y) :=
   shrinkYonedaObjObjEquiv
 
+lemma shrinkCoyoneda_obj_map {X : Cᵒᵖ} {Y Y' : C} (g : Y ⟶ Y') (f : (shrinkCoyoneda.obj X).obj Y) :
+    (shrinkCoyoneda.obj _).map g f =
+      shrinkCoyonedaObjObjEquiv.symm (shrinkCoyonedaObjObjEquiv f ≫ g) :=
+  rfl
+
 lemma shrinkCoyoneda_obj_map_shrinkCoyonedaObjObjEquiv_symm
     {X : Cᵒᵖ} {Y Y' : C} (g : Y ⟶ Y') (f : X.unop ⟶ Y) :
     (shrinkCoyoneda.obj _).map g (shrinkCoyonedaObjObjEquiv.symm f) =

Mathlib/CategoryTheory/Sites/LocalSite.lean

@@ -0,0 +1,213 @@
+/-
+Copyright (c) 2025 Ben Eltschig. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Ben Eltschig
+-/
+module
+
+public import Mathlib.CategoryTheory.Adjunction.Triple
+public import Mathlib.CategoryTheory.Limits.Elements
+public import Mathlib.CategoryTheory.Sites.GlobalSections
+public import Mathlib.CategoryTheory.Sites.Point.Skyscraper
+
+/-!
+# Local sites
+
+A site is called local if it has a terminal object whose only covering sieve is trivial -
+this makes it possible to define coconstant sheaves on it, giving its sheaf topos the structure
+of a local topos. This is one of the conditions of cohesive sites.
+
+Sheaves of types on any local site form a local topos (i.e. a topos whose global sections functor
+has a fully faithful right adjoint), and a subcanonical site is local if and only if its topos of
+sheaves of types is (see TODOs).
+
+## Main definitions / results
+
+* `J.IsLocalSite`: typeclass stating that `J` makes the category it is defined on into a local site.
+* `IsLocalSite.point J`: the canonical point of any local site, whose fibre functor is given by
+  the coyoneda embedding of the terminal object and extends to the global sections functors on
+  presheaves and sheaves.
+* `coconstantSheaf J A`: the coconstant sheaf functor `A ⥤ Sheaf J A` for any local site and
+  sufficiently nice target category `A`, defined as the skyscraper sheaf functor of the canonical
+  point.
+* `ΓCoconstantSheafAdj J A`: the adjunction between the global sections functor `Γ J A` and
+  `coconstantSheaf J A`.
+* `fullyFaithfulCoconstantSheaf`: `coconstantSheaf` is fully faithful.
+* `fullyFaithfulConstantSheaf`: on local sites, the constant sheaf functor is fully faithful.
+
+## References
+
+* https://ncatlab.org/nlab/show/local+site
+
+## TODO
+
+* Define local topoi and prove that sheaves on any local site form a local topos
+* Show that a subcanonical site is local if and only if its global sections functor has a fully
+  faithful right adjoint
+-/
+
+universe w u v u' v'
+
+@[expose] public section
+
+open CategoryTheory Limits Sheaf Opposite GrothendieckTopology
+
+namespace CategoryTheory
+
+variable {C : Type u} [Category.{v} C] (J : GrothendieckTopology C)
+
+/-- A local site is a site that has a terminal object with only a single covering sieve. -/
+class GrothendieckTopology.IsLocalSite extends HasTerminal C where
+  /-- The only covering sieve of the terminal object is the trivial sieve. -/
+  eq_top_of_mem : ∀ S ∈ J (⊤_ C), S = ⊤
+
+namespace GrothendieckTopology.IsLocalSite
+
+/-- On a local site, every covering sieve contains every morphism from the terminal object. -/
+lemma from_terminal_mem_of_mem [J.IsLocalSite] {X : C} (f : ⊤_ C ⟶ X) {S : Sieve X}
+    (hS : S ∈ J X) : S.arrows f :=
+  (S.mem_iff_pullback_eq_top f).mpr <| eq_top_of_mem _ <| J.pullback_stable f hS
+
+/-- Every category with a terminal object becomes a local site with the trivial topology. -/
+instance {C : Type u} [Category.{v} C] [HasTerminal C] : (trivial C).IsLocalSite where
+  eq_top_of_mem _ := trivial_covering.mp
+
+/-- Every local site has a canonical point, given as a fibre functor by the coyoneda embedding of
+the terminal object `⊤_ C`. -/
+noncomputable def point [LocallySmall.{w} C] [J.IsLocalSite] : Point.{w} J where
+  fiber := shrinkCoyoneda.obj (op (⊤_ C))
+  jointly_surjective R hR x :=
+    ⟨(⊤_ C), shrinkCoyonedaObjObjEquiv x,
+      (from_terminal_mem_of_mem J (shrinkCoyonedaObjObjEquiv x) hR),
+          shrinkCoyonedaObjObjEquiv.symm (𝟙 _), by
+        rw [shrinkCoyoneda_obj_map_shrinkCoyonedaObjObjEquiv_symm]
+        simp⟩
+
+variable [LocallySmall.{w} C] [J.IsLocalSite] (A : Type u') [Category.{v', u'} A]
+
+/-- The right adjoint to the global sections functor that exists over any local site. This is
+implemented as the skyscraper functor associated to `point.{w} J`, but can be thought of
+as taking any object `X : A` to the sheaf that sends each `Y : C` to the product over copies of `A`
+indexed by the points `⊤_ C ⟶ Y` of `Y`.
+
+Note this takes in an extra universe parameter `w` that does not appear in the output type
+`A ⥤ Sheaf J A` but is required for the construction; it should always be given explicitly when
+referring to this functor, as in e.g. `coconstantSheaf.{w} J A`. -/
+noncomputable def coconstantSheaf [HasProducts.{w} A] : A ⥤ Sheaf J A :=
+  (point.{w} J).skyscraperSheafFunctor
+
+variable [HasColimitsOfSize.{w, w, v', u'} A]
+
+variable {A} in
+/-- The fibre of any presheaf `P : Cᵒᵖ ⥤ A` at `point J` is just `P` evaluated at
+the terminal object. -/
+noncomputable def pointPresheafFiberIso (P : Cᵒᵖ ⥤ A) :
+    (point J).presheafFiber.obj P ≅ P.obj (op (⊤_ C)) :=
+  (colimit.isColimit _).coconePointUniqueUpToIso
+    (colimitOfDiagramTerminal (Functor.Elements.isInitialOfCorepresentableBy
+      <| shrinkCoyonedaCorepresentableBy <| op (⊤_ C)).op _)
+
+variable {A} in
+set_option backward.isDefEq.respectTransparency false in
+set_option backward.defeqAttrib.useBackward true in
+@[reassoc (attr := simp)]
+lemma toPresheafFiber_pointPresheafFiberIso_hom {P : Cᵒᵖ ⥤ A} (X : C) (x : (point J).fiber.obj X) :
+    (point J).toPresheafFiber  _ x _ ≫ (pointPresheafFiberIso J P).hom =
+      P.map (.op <| shrinkCoyonedaObjObjEquiv x) := by
+  simp [Point.toPresheafFiber, pointPresheafFiberIso]
+  rfl
+
+variable {A} in
+@[reassoc (attr := simp)]
+lemma pointPresheafFiberIso_naturality {P P' : Cᵒᵖ ⥤ A} (F : P ⟶ P') :
+    (point J).presheafFiber.map F ≫ (pointPresheafFiberIso J P').hom =
+      (pointPresheafFiberIso J P).hom ≫ F.app (op (⊤_ C)) := by
+  cat_disch
+
+/-- The presheaf fibre functor of `point J` is given by evaluation at the terminal
+object. -/
+noncomputable def pointPresheafFiberNatIso :
+    ((point J).presheafFiber : _ ⥤ A) ≅ (evaluation _ _).obj (op (⊤_ C)) :=
+  NatIso.ofComponents (pointPresheafFiberIso J) fun F ↦ pointPresheafFiberIso_naturality J F
+
+/-- The sheaf fibre functor of `point J` is the global sections functor. -/
+noncomputable def pointSheafFiberIso
+    [HasWeakSheafify J A] : (point J).sheafFiber ≅ Γ J A :=
+  ((sheafToPresheaf J A).isoWhiskerLeft (pointPresheafFiberNatIso J A)).trans
+    (ΓNatIsoSheafSections J A terminalIsTerminal).symm
+
+variable [HasProducts.{w} A] [HasWeakSheafify J A]
+
+/-- On local sites, the global sections functor `Γ` is left-adjoint to the coconstant functor. -/
+noncomputable def ΓCoconstantSheafAdj : Γ J A ⊣ coconstantSheaf.{w} J A :=
+  (point.{w} J).skyscraperSheafAdjunction.ofNatIsoLeft (pointSheafFiberIso J A)
+
+/-- On any locally `w`-small local site, the global sections functor to any category with colimits
+and products of size `w` is a left adulljoint. A variant of this without the universe parameter `w`
+is registered as an instance. -/
+lemma Γ_isLeftAdjoint : (Γ J A).IsLeftAdjoint :=
+  ⟨coconstantSheaf.{w} J A, ⟨ΓCoconstantSheafAdj J A⟩⟩
+
+/-- On any local site with morphism types in `Type v`, the global sections functor to any category
+with colimits and products of size `v` is a left adjoint. See `ΓIsLeftAdjoint` for a
+version for `w`-locally small sites that can't be registered as an instance because of the extra
+universe parameter `w`. -/
+instance (A : Type u') [Category.{v', u'} A] [HasColimitsOfSize.{v, v, v', u'} A]
+    [HasProducts.{v} A] [HasWeakSheafify J A] : (Γ J A).IsLeftAdjoint :=
+  Γ_isLeftAdjoint.{v} J A
+
+instance : (coconstantSheaf.{w} J A).IsRightAdjoint :=
+  ⟨Γ J A, ⟨ΓCoconstantSheafAdj J A⟩⟩
+
+set_option backward.defeqAttrib.useBackward true in
+/-- The global sections of the coconstant sheaf on a type are naturally isomorphic to that type. -/
+noncomputable def coconstantSheafΓNatIsoId :
+    IsLocalSite.coconstantSheaf.{w} J A ⋙ Γ J A ≅ 𝟭 A :=
+  letI : Unique (unop ((IsLocalSite.point J).fiber.op.obj (op (⊤_ C)))) :=
+    (equivShrink (⊤_ C ⟶ ⊤_ C)).symm.unique
+  (Functor.isoWhiskerLeft _ (ΓNatIsoSheafSections J _ terminalIsTerminal)) ≪≫
+    NatIso.ofComponents (fun X ↦ productUniqueIso _) (by simp [IsLocalSite.coconstantSheaf])
+
+/-- `coconstantSheaf` is fully faithful. -/
+noncomputable def fullyFaithfulCoconstantSheaf :
+    (coconstantSheaf.{w} J A).FullyFaithful :=
+  (ΓCoconstantSheafAdj J A).fullyFaithfulROfCompIsoId (coconstantSheafΓNatIsoId J A)
+
+instance : (coconstantSheaf.{w} J A).Full :=
+  (fullyFaithfulCoconstantSheaf J A).full
+
+instance : (coconstantSheaf.{w} J A).Faithful :=
+  (fullyFaithfulCoconstantSheaf J A).faithful
+
+/-- The adjoint triple `constantSheaf J A ⊣ Γ J A ⊣ coconstantSheaf J A` on any local site. -/
+noncomputable abbrev constantΓCoconstantTriple :
+    Adjunction.Triple (constantSheaf J A) (Γ J A) (coconstantSheaf.{w} J A) where
+  adj₁ := constantSheafΓAdj J A
+  adj₂ := ΓCoconstantSheafAdj J A
+
+/-- On local sites, the constant sheaf functor is fully faithful. -/
+noncomputable def fullyFaithfulConstantSheaf : (constantSheaf J A).FullyFaithful :=
+  (constantΓCoconstantTriple J A).fullyFaithfulEquiv.symm <|
+    fullyFaithfulCoconstantSheaf.{w} J A
+
+lemma full_constantSheaf : (constantSheaf J A).Full :=
+  (fullyFaithfulConstantSheaf.{w} J A).full
+
+lemma faithful_constantSheaf : (constantSheaf J A).Faithful :=
+  (fullyFaithfulConstantSheaf.{w} J A).faithful
+
+/-- See `IsLocalSite.full_constantSheaf` for a version for `w`-locally small sites. -/
+instance {C : Type u} [Category.{v} C] (J : GrothendieckTopology C) [J.IsLocalSite]
+    (A : Type u') [Category.{v', u'} A] [HasColimitsOfSize.{v, v, v', u'} A]
+    [HasProducts.{v} A] [HasWeakSheafify J A] : (constantSheaf J A).Full :=
+  full_constantSheaf.{v} J A
+
+/-- See `IsLocalSite.faithful_constantSheaf` for a version for `w`-locally small sites. -/
+instance {C : Type u} [Category.{v} C] (J : GrothendieckTopology C) [J.IsLocalSite]
+    (A : Type u') [Category.{v', u'} A] [HasColimitsOfSize.{v, v, v', u'} A]
+    [HasProducts.{v} A] [HasWeakSheafify J A] : (constantSheaf J A).Faithful :=
+  faithful_constantSheaf.{v} J A
+
+end GrothendieckTopology.IsLocalSite
+
+end CategoryTheory