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Result

Found 86 declarations mentioning CategoryTheory.Over.map.

CategoryTheory.Over.map 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {X Y : T} (f : X ⟶ Y) : CategoryTheory.Functor (CategoryTheory.Over X) (CategoryTheory.Over Y)
CategoryTheory.Over.mapId_eq 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] (Y : T) : CategoryTheory.Over.map (CategoryTheory.CategoryStruct.id Y) = CategoryTheory.Functor.id (CategoryTheory.Over Y)
CategoryTheory.Over.instIsEquivalenceMapOfIsIso 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {X Y : T} {f : X ⟶ Y} [CategoryTheory.IsIso f] : (CategoryTheory.Over.map f).IsEquivalence
CategoryTheory.Over.mapFunctor_map 📋 Mathlib.CategoryTheory.Comma.Over.Basic
(T : Type u₁) [CategoryTheory.Category.{v₁, u₁} T] {X✝ Y✝ : T} (f : X✝ ⟶ Y✝) : (CategoryTheory.Over.mapFunctor T).map f = CategoryTheory.Over.map f
CategoryTheory.Over.mapForget_eq 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {X Y : T} (f : X ⟶ Y) : (CategoryTheory.Over.map f).comp (CategoryTheory.Over.forget Y) = CategoryTheory.Over.forget X
CategoryTheory.Over.mapId 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] (Y : T) : CategoryTheory.Over.map (CategoryTheory.CategoryStruct.id Y) ≅ CategoryTheory.Functor.id (CategoryTheory.Over Y)
CategoryTheory.Over.mapIso_functor 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {X Y : T} (f : X ≅ Y) : (CategoryTheory.Over.mapIso f).functor = CategoryTheory.Over.map f.hom
CategoryTheory.Over.mapIso_inverse 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {X Y : T} (f : X ≅ Y) : (CategoryTheory.Over.mapIso f).inverse = CategoryTheory.Over.map f.inv
CategoryTheory.Over.mapForget 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {X Y : T} (f : X ⟶ Y) : (CategoryTheory.Over.map f).comp (CategoryTheory.Over.forget Y) ≅ CategoryTheory.Over.forget X
CategoryTheory.Over.map.eq_1 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {X Y : T} (f : X ⟶ Y) : CategoryTheory.Over.map f = CategoryTheory.Comma.mapRight (CategoryTheory.Functor.id T) (CategoryTheory.Discrete.natTrans fun x => f)
CategoryTheory.Over.map_obj_left 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {X Y : T} {f : X ⟶ Y} {U : CategoryTheory.Over X} : ((CategoryTheory.Over.map f).obj U).left = U.left
CategoryTheory.Over.mapCongr 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {X Y : T} (f g : X ⟶ Y) (h : f = g) : CategoryTheory.Over.map f ≅ CategoryTheory.Over.map g
CategoryTheory.Over.mapComp_eq 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {X Y Z : T} (f : X ⟶ Y) (g : Y ⟶ Z) : CategoryTheory.Over.map (CategoryTheory.CategoryStruct.comp f g) = (CategoryTheory.Over.map f).comp (CategoryTheory.Over.map g)
CategoryTheory.Over.mapComp 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {X Y Z : T} (f : X ⟶ Y) (g : Y ⟶ Z) : CategoryTheory.Over.map (CategoryTheory.CategoryStruct.comp f g) ≅ (CategoryTheory.Over.map f).comp (CategoryTheory.Over.map g)
CategoryTheory.Over.mapCongr_rfl 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {X Y : T} (f : X ⟶ Y) : CategoryTheory.Over.mapCongr f f ⋯ = CategoryTheory.Iso.refl (CategoryTheory.Over.map f)
CategoryTheory.Over.postCongr 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {X : T} {F G : CategoryTheory.Functor T D} (e : F ≅ G) : (CategoryTheory.Over.post F).comp (CategoryTheory.Over.map (e.hom.app X)) ≅ CategoryTheory.Over.post G
CategoryTheory.Over.postAdjunctionRight 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {Y : D} {F : CategoryTheory.Functor T D} {G : CategoryTheory.Functor D T} (a : F ⊣ G) : (CategoryTheory.Over.post F).comp (CategoryTheory.Over.map (a.counit.app Y)) ⊣ CategoryTheory.Over.post G
CategoryTheory.Over.map_map_left 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {X Y : T} {f : X ⟶ Y} {U V : CategoryTheory.Over X} {g : U ⟶ V} : ((CategoryTheory.Over.map f).map g).left = g.left
CategoryTheory.Over.map_obj_hom 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {X Y : T} {f : X ⟶ Y} {U : CategoryTheory.Over X} : ((CategoryTheory.Over.map f).obj U).hom = CategoryTheory.CategoryStruct.comp U.hom f
CategoryTheory.Over.postMap 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {X : T} {F G : CategoryTheory.Functor T D} (e : F ⟶ G) : (CategoryTheory.Over.post F).comp (CategoryTheory.Over.map (e.app X)) ⟶ CategoryTheory.Over.post G
CategoryTheory.Over.mapId_hom_app_left 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] (Y : T) (X : CategoryTheory.Over Y) : ((CategoryTheory.Over.mapId Y).hom.app X).left = CategoryTheory.CategoryStruct.id X.left
CategoryTheory.Over.mapId_inv_app_left 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] (Y : T) (X : CategoryTheory.Over Y) : ((CategoryTheory.Over.mapId Y).inv.app X).left = CategoryTheory.CategoryStruct.id X.left
CategoryTheory.Over.mapCongr_hom_app_left 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {X Y : T} (f g : X ⟶ Y) (h : f = g) (X✝ : CategoryTheory.Over X) : ((CategoryTheory.Over.mapCongr f g h).hom.app X✝).left = CategoryTheory.CategoryStruct.id X✝.left
CategoryTheory.Over.mapCongr_inv_app_left 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {X Y : T} (f g : X ⟶ Y) (h : f = g) (X✝ : CategoryTheory.Over X) : ((CategoryTheory.Over.mapCongr f g h).inv.app X✝).left = CategoryTheory.CategoryStruct.id X✝.left
CategoryTheory.Over.postEquiv_inverse 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (X : T) (F : T ≌ D) : (CategoryTheory.Over.postEquiv X F).inverse = (CategoryTheory.Over.post F.inverse).comp (CategoryTheory.Over.map (F.unitIso.inv.app X))
CategoryTheory.Over.mapComp_hom_app_left 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {X Y Z : T} (f : X ⟶ Y) (g : Y ⟶ Z) (X✝ : CategoryTheory.Over X) : ((CategoryTheory.Over.mapComp f g).hom.app X✝).left = CategoryTheory.CategoryStruct.id X✝.left
CategoryTheory.Over.mapComp_inv_app_left 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {X Y Z : T} (f : X ⟶ Y) (g : Y ⟶ Z) (X✝ : CategoryTheory.Over X) : ((CategoryTheory.Over.mapComp f g).inv.app X✝).left = CategoryTheory.CategoryStruct.id X✝.left
CategoryTheory.Over.postMap_app 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {X : T} {F G : CategoryTheory.Functor T D} (e : F ⟶ G) (Y : CategoryTheory.Over X) : (CategoryTheory.Over.postMap e).app Y = CategoryTheory.Over.homMk (e.app Y.left) ⋯
CategoryTheory.Over.postCongr_hom_app_left 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {X : T} {F G : CategoryTheory.Functor T D} (e : F ≅ G) (X✝ : CategoryTheory.Over X) : ((CategoryTheory.Over.postCongr e).hom.app X✝).left = e.hom.app X✝.left
CategoryTheory.Over.postCongr_inv_app_left 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {X : T} {F G : CategoryTheory.Functor T D} (e : F ≅ G) (X✝ : CategoryTheory.Over X) : ((CategoryTheory.Over.postCongr e).inv.app X✝).left = e.inv.app X✝.left
CategoryTheory.Over.postEquiv_unitIso 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (X : T) (F : T ≌ D) : (CategoryTheory.Over.postEquiv X F).unitIso = CategoryTheory.NatIso.ofComponents (fun A => CategoryTheory.Over.isoMk (F.unitIso.app A.left) ⋯) ⋯
CategoryTheory.Over.postAdjunctionRight_unit_app 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {Y : D} {F : CategoryTheory.Functor T D} {G : CategoryTheory.Functor D T} (a : F ⊣ G) (A : CategoryTheory.Over (G.obj Y)) : (CategoryTheory.Over.postAdjunctionRight a).unit.app A = CategoryTheory.Over.homMk (a.unit.app A.left) ⋯
CategoryTheory.Over.postAdjunctionRight_counit_app 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {Y : D} {F : CategoryTheory.Functor T D} {G : CategoryTheory.Functor D T} (a : F ⊣ G) (A : CategoryTheory.Over ((CategoryTheory.Functor.id D).obj Y)) : (CategoryTheory.Over.postAdjunctionRight a).counit.app A = CategoryTheory.Over.homMk (a.counit.app A.left) ⋯
CategoryTheory.Over.postEquiv_counitIso 📋 Mathlib.CategoryTheory.Comma.Over.Basic
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (X : T) (F : T ≌ D) : (CategoryTheory.Over.postEquiv X F).counitIso = CategoryTheory.NatIso.ofComponents (fun A => CategoryTheory.Over.isoMk (F.counitIso.app A.left) ⋯) ⋯
CategoryTheory.Over.mapPullbackAdj 📋 Mathlib.CategoryTheory.Comma.Over.Pullback
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : C} (f : X ⟶ Y) [CategoryTheory.Limits.HasPullbacksAlong f] : CategoryTheory.Over.map f ⊣ CategoryTheory.Over.pullback f
CategoryTheory.Over.forgetMapTerminal 📋 Mathlib.CategoryTheory.Comma.Over.Pullback
{C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) {T : C} (hT : CategoryTheory.Limits.IsTerminal T) : CategoryTheory.Over.forget X ≅ (CategoryTheory.Over.map (hT.from X)).comp (CategoryTheory.Over.equivalenceOfIsTerminal hT).functor
CategoryTheory.Over.forgetMapTerminal_hom_app 📋 Mathlib.CategoryTheory.Comma.Over.Pullback
{C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) {T : C} (hT : CategoryTheory.Limits.IsTerminal T) (X✝ : CategoryTheory.Over X) : (CategoryTheory.Over.forgetMapTerminal X hT).hom.app X✝ = CategoryTheory.CategoryStruct.id X✝.left
CategoryTheory.Over.forgetMapTerminal_inv_app 📋 Mathlib.CategoryTheory.Comma.Over.Pullback
{C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) {T : C} (hT : CategoryTheory.Limits.IsTerminal T) (X✝ : CategoryTheory.Over X) : (CategoryTheory.Over.forgetMapTerminal X hT).inv.app X✝ = CategoryTheory.CategoryStruct.id X✝.left
CategoryTheory.Over.mapPullbackAdj_unit_app 📋 Mathlib.CategoryTheory.Comma.Over.Pullback
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : C} (f : X ⟶ Y) [CategoryTheory.Limits.HasPullbacksAlong f] (X✝ : CategoryTheory.Over X) : (CategoryTheory.Over.mapPullbackAdj f).unit.app X✝ = CategoryTheory.Over.homMk (CategoryTheory.Limits.pullback.lift (CategoryTheory.CategoryStruct.id X✝.left) X✝.hom ⋯) ⋯
CategoryTheory.Over.mapPullbackAdj_counit_app 📋 Mathlib.CategoryTheory.Comma.Over.Pullback
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : C} (f : X ⟶ Y) [CategoryTheory.Limits.HasPullbacksAlong f] (Y✝ : CategoryTheory.Over Y) : (CategoryTheory.Over.mapPullbackAdj f).counit.app Y✝ = CategoryTheory.Over.homMk (CategoryTheory.Limits.pullback.fst Y✝.hom f) ⋯
CategoryTheory.Over.postAdjunctionLeft_counit_app_left 📋 Mathlib.CategoryTheory.Comma.Over.Pullback
{C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.Limits.HasPullbacks C] {X : C} {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (a : F ⊣ G) (Y : CategoryTheory.Over ((CategoryTheory.Functor.id D).obj (F.obj X))) : ((CategoryTheory.Over.postAdjunctionLeft a).counit.app Y).left = ((((CategoryTheory.Over.mapPullbackAdj (a.unit.app X)).comp (CategoryTheory.Over.postAdjunctionRight a)).homEquiv ((CategoryTheory.Over.pullback (a.unit.app X)).obj (CategoryTheory.Over.mk (G.map Y.hom))) Y).symm (CategoryTheory.CategoryStruct.id ((CategoryTheory.Over.pullback (a.unit.app X)).obj (CategoryTheory.Over.mk (G.map Y.hom))))).left
CategoryTheory.Over.postAdjunctionLeft_unit_app_left 📋 Mathlib.CategoryTheory.Comma.Over.Pullback
{C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.Limits.HasPullbacks C] {X : C} {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (a : F ⊣ G) (X✝ : CategoryTheory.Over ((CategoryTheory.Functor.id C).obj X)) : ((CategoryTheory.Over.postAdjunctionLeft a).unit.app X✝).left = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pullback.lift (CategoryTheory.CategoryStruct.id X✝.left) X✝.hom ⋯) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pullback.lift (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pullback.fst (CategoryTheory.CategoryStruct.comp X✝.hom (a.unit.app X)) (a.unit.app X)) (a.unit.app X✝.left)) (CategoryTheory.Limits.pullback.snd (CategoryTheory.CategoryStruct.comp X✝.hom (a.unit.app X)) (a.unit.app X)) ⋯) (CategoryTheory.Limits.pullback.lift (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pullback.fst (G.map (CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.CategoryStruct.comp X✝.hom (a.unit.app X))) (a.counit.app (F.obj X)))) (a.unit.app X)) (G.map ((CategoryTheory.Adjunction.equivHomsetLeftOfNatIso (CategoryTheory.NatIso.ofComponents (fun Y => CategoryTheory.Over.isoMk (CategoryTheory.Iso.refl (F.obj Y.left)) ⋯) ⋯).symm) (CategoryTheory.CategoryStruct.id (CategoryTheory.Over.mk (F.map X✝.hom)))).left)) (CategoryTheory.Limits.pullback.snd (G.map (CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.CategoryStruct.comp X✝.hom (a.unit.app X))) (a.counit.app (F.obj X)))) (a.unit.app X)) ⋯))
CategoryTheory.Over.preservesColimitsOfSize_map 📋 Mathlib.CategoryTheory.Limits.Over
{C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} [CategoryTheory.Limits.HasColimitsOfSize.{w, w', v, u} C] {Y : C} (f : X ⟶ Y) : CategoryTheory.Limits.PreservesColimitsOfSize.{w, w', v, v, max u v, max u v} (CategoryTheory.Over.map f)
CategoryTheory.Over.createsColimitsOfSizeMapCompForget 📋 Mathlib.CategoryTheory.Limits.Over
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : C} (f : X ⟶ Y) : CategoryTheory.CreatesColimitsOfSize.{w, w', v, v, max u v, u} ((CategoryTheory.Over.map f).comp (CategoryTheory.Over.forget Y))
CategoryTheory.GrothendieckTopology.over_map_compatiblePreserving 📋 Mathlib.CategoryTheory.Sites.Over
{C : Type u} [CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) {X Y : C} (f : X ⟶ Y) : CategoryTheory.CompatiblePreserving (J.over Y) (CategoryTheory.Over.map f)
CategoryTheory.GrothendieckTopology.instIsContinuousOverMapOver 📋 Mathlib.CategoryTheory.Sites.Over
{C : Type u} [CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) {X Y : C} (f : X ⟶ Y) : (CategoryTheory.Over.map f).IsContinuous (J.over X) (J.over Y)
CategoryTheory.GrothendieckTopology.over_map_coverPreserving 📋 Mathlib.CategoryTheory.Sites.Over
{C : Type u} [CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) {X Y : C} (f : X ⟶ Y) : CategoryTheory.CoverPreserving (J.over X) (J.over Y) (CategoryTheory.Over.map f)
CategoryTheory.Over.mapCongr.congr_simp 📋 Mathlib.CategoryTheory.Sites.Over
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {X Y : T} (f g : X ⟶ Y) (h : f = g) : CategoryTheory.Over.mapCongr f g h = CategoryTheory.Over.mapCongr f g h
CategoryTheory.Sieve.functorPushforward_over_map 📋 Mathlib.CategoryTheory.Sites.Over
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : C} (f : X ⟶ Y) (Z : CategoryTheory.Over X) (S : CategoryTheory.Sieve Z.left) : CategoryTheory.Sieve.functorPushforward (CategoryTheory.Over.map f) ((CategoryTheory.Sieve.overEquiv Z).symm S) = (CategoryTheory.Sieve.overEquiv ((CategoryTheory.Over.map f).obj Z)).symm S
CategoryTheory.GrothendieckTopology.overMapPullbackCongr_hom_app_val_app 📋 Mathlib.CategoryTheory.Sites.Over
{C : Type u} [CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) (A : Type u') [CategoryTheory.Category.{v', u'} A] {X Y : C} {f g : X ⟶ Y} (h : f = g) (M : CategoryTheory.Sheaf (J.over Y) A) (X✝ : (CategoryTheory.Over X)ᵒᵖ) : ((J.overMapPullbackCongr A h).hom.app M).val.app X✝ = M.val.map ((CategoryTheory.Over.mapCongr f g h).inv.app (Opposite.unop X✝)).op
CategoryTheory.GrothendieckTopology.overMapPullbackCongr_inv_app_val_app 📋 Mathlib.CategoryTheory.Sites.Over
{C : Type u} [CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) (A : Type u') [CategoryTheory.Category.{v', u'} A] {X Y : C} {f g : X ⟶ Y} (h : f = g) (M : CategoryTheory.Sheaf (J.over Y) A) (X✝ : (CategoryTheory.Over X)ᵒᵖ) : ((J.overMapPullbackCongr A h).inv.app M).val.app X✝ = M.val.map ((CategoryTheory.Over.mapCongr f g h).hom.app (Opposite.unop X✝)).op
CategoryTheory.GrothendieckTopology.overMapPullbackId_hom_app_val_app 📋 Mathlib.CategoryTheory.Sites.Over
{C : Type u} [CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) (A : Type u') [CategoryTheory.Category.{v', u'} A] (X : C) (X✝ : CategoryTheory.Sheaf (J.over X) A) (X✝¹ : (CategoryTheory.Over X)ᵒᵖ) : ((J.overMapPullbackId A X).hom.app X✝).val.app X✝¹ = X✝.val.map ((CategoryTheory.Over.mapId X).inv.app (Opposite.unop X✝¹)).op
CategoryTheory.GrothendieckTopology.overMapPullbackId_inv_app_val_app 📋 Mathlib.CategoryTheory.Sites.Over
{C : Type u} [CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) (A : Type u') [CategoryTheory.Category.{v', u'} A] (X : C) (X✝ : CategoryTheory.Sheaf (J.over X) A) (X✝¹ : (CategoryTheory.Over X)ᵒᵖ) : ((J.overMapPullbackId A X).inv.app X✝).val.app X✝¹ = X✝.val.map ((CategoryTheory.Over.mapId X).hom.app (Opposite.unop X✝¹)).op
CategoryTheory.GrothendieckTopology.overMapPullbackComp_hom_app_val_app 📋 Mathlib.CategoryTheory.Sites.Over
{C : Type u} [CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) (A : Type u') [CategoryTheory.Category.{v', u'} A] {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) (X✝ : CategoryTheory.Sheaf (J.over Z) A) (X✝¹ : (CategoryTheory.Over X)ᵒᵖ) : ((J.overMapPullbackComp A f g).hom.app X✝).val.app X✝¹ = X✝.val.map ((CategoryTheory.Over.mapComp f g).hom.app (Opposite.unop X✝¹)).op
CategoryTheory.GrothendieckTopology.overMapPullbackComp_inv_app_val_app 📋 Mathlib.CategoryTheory.Sites.Over
{C : Type u} [CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) (A : Type u') [CategoryTheory.Category.{v', u'} A] {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) (X✝ : CategoryTheory.Sheaf (J.over Z) A) (X✝¹ : (CategoryTheory.Over X)ᵒᵖ) : ((J.overMapPullbackComp A f g).inv.app X✝).val.app X✝¹ = X✝.val.map ((CategoryTheory.Over.mapComp f g).inv.app (Opposite.unop X✝¹)).op
CategoryTheory.ChosenPullbacksAlong.mapPullbackAdj 📋 Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong
{C : Type u₁} {inst✝ : CategoryTheory.Category.{v₁, u₁} C} {Y X : C} (f : Y ⟶ X) [self : CategoryTheory.ChosenPullbacksAlong f] : CategoryTheory.Over.map f ⊣ CategoryTheory.ChosenPullbacksAlong.pullback f
CategoryTheory.ChosenPullbacksAlong.mk 📋 Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {Y X : C} {f : Y ⟶ X} (pullback : CategoryTheory.Functor (CategoryTheory.Over X) (CategoryTheory.Over Y)) (mapPullbackAdj : CategoryTheory.Over.map f ⊣ pullback) : CategoryTheory.ChosenPullbacksAlong f
CategoryTheory.ChosenPullbacksAlong.ofHasPullbacksAlong_mapPullbackAdj 📋 Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {Y X : C} (f : Y ⟶ X) [CategoryTheory.Limits.HasPullbacksAlong f] : CategoryTheory.ChosenPullbacksAlong.mapPullbackAdj f = CategoryTheory.Over.mapPullbackAdj f
CategoryTheory.ChosenPullbacksAlong.fst' 📋 Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {Y Z X : C} (f : Y ⟶ X) (g : Z ⟶ X) [CategoryTheory.ChosenPullbacksAlong g] : (CategoryTheory.Over.map g).obj ((CategoryTheory.ChosenPullbacksAlong.pullback g).obj (CategoryTheory.Over.mk f)) ⟶ CategoryTheory.Over.mk f
CategoryTheory.ChosenPullbacksAlong.snd' 📋 Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {Y Z X : C} (f : Y ⟶ X) (g : Z ⟶ X) [CategoryTheory.ChosenPullbacksAlong g] : (CategoryTheory.Over.map g).obj ((CategoryTheory.ChosenPullbacksAlong.pullback g).obj (CategoryTheory.Over.mk f)) ⟶ CategoryTheory.Over.mk g
CategoryTheory.ChosenPullbacksAlong.id_mapPullbackAdj 📋 Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (X : C) : CategoryTheory.ChosenPullbacksAlong.mapPullbackAdj (CategoryTheory.CategoryStruct.id X) = CategoryTheory.Adjunction.id.ofNatIsoLeft (CategoryTheory.Over.mapId X).symm
CategoryTheory.ChosenPullbacksAlong.fst'_left 📋 Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {Y Z X : C} (f : Y ⟶ X) (g : Z ⟶ X) [CategoryTheory.ChosenPullbacksAlong g] : (CategoryTheory.ChosenPullbacksAlong.fst' f g).left = CategoryTheory.ChosenPullbacksAlong.fst f g
CategoryTheory.ChosenPullbacksAlong.snd'_left 📋 Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {Y Z X : C} (f : Y ⟶ X) (g : Z ⟶ X) [CategoryTheory.ChosenPullbacksAlong g] : (CategoryTheory.ChosenPullbacksAlong.snd' f g).left = CategoryTheory.ChosenPullbacksAlong.snd f g
CategoryTheory.ChosenPullbacksAlong.comp_mapPullbackAdj 📋 Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {Z Y X : C} (f : Y ⟶ X) (g : Z ⟶ Y) [CategoryTheory.ChosenPullbacksAlong f] [CategoryTheory.ChosenPullbacksAlong g] : CategoryTheory.ChosenPullbacksAlong.mapPullbackAdj (CategoryTheory.CategoryStruct.comp g f) = ((CategoryTheory.ChosenPullbacksAlong.mapPullbackAdj g).comp (CategoryTheory.ChosenPullbacksAlong.mapPullbackAdj f)).ofNatIsoLeft (CategoryTheory.Over.mapComp g f).symm
CategoryTheory.ChosenPullbacksAlong.lift.eq_1 📋 Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {Y Z X : C} {f : Y ⟶ X} {g : Z ⟶ X} [CategoryTheory.ChosenPullbacksAlong g] {W : C} (a : W ⟶ Y) (b : W ⟶ Z) (h : CategoryTheory.CategoryStruct.comp a f = CategoryTheory.CategoryStruct.comp b g) : CategoryTheory.ChosenPullbacksAlong.lift a b h = (((CategoryTheory.ChosenPullbacksAlong.mapPullbackAdj g).homEquiv (CategoryTheory.Over.mk b) (CategoryTheory.Over.mk f)) (CategoryTheory.Over.homMk a ⋯)).left
CategoryTheory.ChosenPullbacksAlong.isoInv_mapPullbackAdj_unit_app_left 📋 Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {Y X : C} (f : Y ≅ X) (T : CategoryTheory.Over X) : ((CategoryTheory.ChosenPullbacksAlong.mapPullbackAdj f.inv).unit.app T).left = CategoryTheory.CategoryStruct.id T.left
CategoryTheory.ChosenPullbacksAlong.iso_mapPullbackAdj_unit_app 📋 Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {Y X : C} (f : Y ≅ X) (T : CategoryTheory.Over Y) : (CategoryTheory.ChosenPullbacksAlong.mapPullbackAdj f.hom).unit.app T = CategoryTheory.Over.homMk (CategoryTheory.CategoryStruct.id T.left) ⋯
CategoryTheory.ChosenPullbacksAlong.iso_mapPullbackAdj_counit_app 📋 Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {Y X : C} (f : Y ≅ X) (U : CategoryTheory.Over X) : (CategoryTheory.ChosenPullbacksAlong.mapPullbackAdj f.hom).counit.app U = CategoryTheory.Over.homMk (CategoryTheory.CategoryStruct.id (({ obj := fun Z => CategoryTheory.Over.mk (CategoryTheory.CategoryStruct.comp Z.hom f.inv), map := fun {Y_1 Z} g => CategoryTheory.Over.homMk g.left ⋯, map_id := ⋯, map_comp := ⋯ }.comp (CategoryTheory.Over.map f.hom)).obj U).left) ⋯
CategoryTheory.ChosenPullbacksAlong.isoInv_mapPullbackAdj_counit_app_left 📋 Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {Y X : C} (f : Y ≅ X) (U : CategoryTheory.Over Y) : ((CategoryTheory.ChosenPullbacksAlong.mapPullbackAdj f.inv).counit.app U).left = CategoryTheory.CategoryStruct.id U.left
CategoryTheory.ChosenPullbacksAlong.cartesianMonoidalCategorySnd_mapPullbackAdj_counit_app 📋 Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.CartesianMonoidalCategory C] (X Y : C) (U : CategoryTheory.Over Y) : (CategoryTheory.ChosenPullbacksAlong.mapPullbackAdj (CategoryTheory.SemiCartesianMonoidalCategory.snd X Y)).counit.app U = CategoryTheory.Over.homMk (CategoryTheory.SemiCartesianMonoidalCategory.snd X ((CategoryTheory.Functor.id C).obj U.left)) ⋯
CategoryTheory.ChosenPullbacksAlong.cartesianMonoidalCategoryToUnit_mapPullbackAdj_unit_app 📋 Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.CartesianMonoidalCategory C] {X : C} (f : X ⟶ CategoryTheory.MonoidalCategoryStruct.tensorUnit C) (T : CategoryTheory.Over X) : (CategoryTheory.ChosenPullbacksAlong.mapPullbackAdj f).unit.app T = CategoryTheory.Over.homMk (CategoryTheory.CartesianMonoidalCategory.lift (CategoryTheory.CategoryStruct.id ((CategoryTheory.Functor.id (CategoryTheory.Over X)).obj T).left) T.hom) ⋯
CategoryTheory.ChosenPullbacksAlong.cartesianMonoidalCategoryFst_mapPullbackAdj_counit_app 📋 Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.CartesianMonoidalCategory C] (X Y : C) (U : CategoryTheory.Over X) : (CategoryTheory.ChosenPullbacksAlong.mapPullbackAdj (CategoryTheory.SemiCartesianMonoidalCategory.fst X Y)).counit.app U = CategoryTheory.Over.homMk (CategoryTheory.SemiCartesianMonoidalCategory.fst ((CategoryTheory.Functor.id C).obj U.left) Y) ⋯
CategoryTheory.ChosenPullbacksAlong.cartesianMonoidalCategoryToUnit_mapPullbackAdj_counit_app 📋 Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.CartesianMonoidalCategory C] {X : C} (f : X ⟶ CategoryTheory.MonoidalCategoryStruct.tensorUnit C) (U : CategoryTheory.Over (CategoryTheory.MonoidalCategoryStruct.tensorUnit C)) : (CategoryTheory.ChosenPullbacksAlong.mapPullbackAdj f).counit.app U = CategoryTheory.Over.homMk (CategoryTheory.SemiCartesianMonoidalCategory.fst U.left X) ⋯
CategoryTheory.ChosenPullbacksAlong.cartesianMonoidalCategorySnd_mapPullbackAdj_unit_app 📋 Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.CartesianMonoidalCategory C] (X Y : C) (T : CategoryTheory.Over (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y)) : (CategoryTheory.ChosenPullbacksAlong.mapPullbackAdj (CategoryTheory.SemiCartesianMonoidalCategory.snd X Y)).unit.app T = CategoryTheory.Over.homMk (CategoryTheory.CartesianMonoidalCategory.lift (CategoryTheory.CategoryStruct.comp T.hom (CategoryTheory.SemiCartesianMonoidalCategory.fst X Y)) (CategoryTheory.CategoryStruct.id ((CategoryTheory.Functor.id (CategoryTheory.Over (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y))).obj T).left)) ⋯
CategoryTheory.ChosenPullbacksAlong.cartesianMonoidalCategoryFst_mapPullbackAdj_unit_app 📋 Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.CartesianMonoidalCategory C] (X Y : C) (T : CategoryTheory.Over (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y)) : (CategoryTheory.ChosenPullbacksAlong.mapPullbackAdj (CategoryTheory.SemiCartesianMonoidalCategory.fst X Y)).unit.app T = CategoryTheory.Over.homMk (CategoryTheory.CartesianMonoidalCategory.lift (CategoryTheory.CategoryStruct.id ((CategoryTheory.Functor.id (CategoryTheory.Over (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y))).obj T).left) (CategoryTheory.CategoryStruct.comp T.hom (CategoryTheory.SemiCartesianMonoidalCategory.snd X Y))) ⋯
CategoryTheory.Pseudofunctor.overMapCompPresheafHomIso 📋 Mathlib.CategoryTheory.Sites.Descent.IsPrestack
{C : Type u} [CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Pseudofunctor (CategoryTheory.LocallyDiscrete Cᵒᵖ) CategoryTheory.Cat) {S : C} (M N : ↑(F.obj { as := Opposite.op S })) {S' : C} (q : S' ⟶ S) : (CategoryTheory.Over.map q).op.comp (F.presheafHom M N) ≅ F.presheafHom ((F.map q.op.toLoc).obj M) ((F.map q.op.toLoc).obj N)
CategoryTheory.Over.mapId.eq_1 📋 Mathlib.CategoryTheory.Sites.SheafHom
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] (Y : T) : CategoryTheory.Over.mapId Y = CategoryTheory.NatIso.ofComponents (fun x => CategoryTheory.Over.isoMk (CategoryTheory.Iso.refl ((CategoryTheory.Over.map (CategoryTheory.CategoryStruct.id Y)).obj x).left) ⋯) ⋯
CategoryTheory.Over.mapComp.eq_1 📋 Mathlib.CategoryTheory.Sites.SheafHom
{T : Type u₁} [CategoryTheory.Category.{v₁, u₁} T] {X Y Z : T} (f : X ⟶ Y) (g : Y ⟶ Z) : CategoryTheory.Over.mapComp f g = CategoryTheory.NatIso.ofComponents (fun x => CategoryTheory.Over.isoMk (CategoryTheory.Iso.refl ((CategoryTheory.Over.map (CategoryTheory.CategoryStruct.comp f g)).obj x).left) ⋯) ⋯
CategoryTheory.presheafHom.eq_1 📋 Mathlib.CategoryTheory.Sites.Monoidal
{C : Type u} [CategoryTheory.Category.{v, u} C] {A : Type u'} [CategoryTheory.Category.{v', u'} A] (F G : CategoryTheory.Functor Cᵒᵖ A) : CategoryTheory.presheafHom F G = { obj := fun X => (CategoryTheory.Over.forget (Opposite.unop X)).op.comp F ⟶ (CategoryTheory.Over.forget (Opposite.unop X)).op.comp G, map := fun {X Y} f => (CategoryTheory.Over.map f.unop).op.whiskerLeft, map_id := ⋯, map_comp := ⋯ }
CategoryTheory.GrothendieckTopology.pseudofunctorOver_toPrelaxFunctor_toPrelaxFunctorStruct_toPrefunctor_map_obj_val_obj 📋 Mathlib.CategoryTheory.Sites.PseudofunctorSheafOver
{C : Type u} [CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) (A : Type u') [CategoryTheory.Category.{v', u'} A] {X✝ Y✝ : CategoryTheory.LocallyDiscrete Cᵒᵖ} (f : X✝ ⟶ Y✝) (ℱ : CategoryTheory.Sheaf (J.over (Opposite.unop X✝.as)) A) (X : (CategoryTheory.Over (Opposite.unop Y✝.as))ᵒᵖ) : (((J.pseudofunctorOver A).map f).obj ℱ).val.obj X = ℱ.val.obj (Opposite.op ((CategoryTheory.Over.map f.as.unop).obj (Opposite.unop X)))
CategoryTheory.GrothendieckTopology.pseudofunctorOver_toPrelaxFunctor_toPrelaxFunctorStruct_toPrefunctor_map_obj_val_map 📋 Mathlib.CategoryTheory.Sites.PseudofunctorSheafOver
{C : Type u} [CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) (A : Type u') [CategoryTheory.Category.{v', u'} A] {X✝ Y✝ : CategoryTheory.LocallyDiscrete Cᵒᵖ} (f : X✝ ⟶ Y✝) (ℱ : CategoryTheory.Sheaf (J.over (Opposite.unop X✝.as)) A) {X✝¹ Y✝¹ : (CategoryTheory.Over (Opposite.unop Y✝.as))ᵒᵖ} (f✝ : X✝¹ ⟶ Y✝¹) : (((J.pseudofunctorOver A).map f).obj ℱ).val.map f✝ = ℱ.val.map ((CategoryTheory.Over.map f.as.unop).map f✝.unop).op
CategoryTheory.GrothendieckTopology.pseudofunctorOver_mapId_hom_app_val_app 📋 Mathlib.CategoryTheory.Sites.PseudofunctorSheafOver
{C : Type u} [CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) (A : Type u') [CategoryTheory.Category.{v', u'} A] (x✝ : CategoryTheory.LocallyDiscrete Cᵒᵖ) (X : CategoryTheory.Sheaf (J.over (Opposite.unop x✝.as)) A) (X✝ : (CategoryTheory.Over (Opposite.unop x✝.as))ᵒᵖ) : (((J.pseudofunctorOver A).mapId x✝).hom.app X).val.app X✝ = X.val.map ((CategoryTheory.Over.mapId (Opposite.unop x✝.as)).inv.app (Opposite.unop X✝)).op
CategoryTheory.GrothendieckTopology.pseudofunctorOver_mapId_inv_app_val_app 📋 Mathlib.CategoryTheory.Sites.PseudofunctorSheafOver
{C : Type u} [CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) (A : Type u') [CategoryTheory.Category.{v', u'} A] (x✝ : CategoryTheory.LocallyDiscrete Cᵒᵖ) (X : CategoryTheory.Sheaf (J.over (Opposite.unop x✝.as)) A) (X✝ : (CategoryTheory.Over (Opposite.unop x✝.as))ᵒᵖ) : (((J.pseudofunctorOver A).mapId x✝).inv.app X).val.app X✝ = X.val.map ((CategoryTheory.Over.mapId (Opposite.unop x✝.as)).hom.app (Opposite.unop X✝)).op
CategoryTheory.GrothendieckTopology.pseudofunctorOver_toPrelaxFunctor_toPrelaxFunctorStruct_toPrefunctor_map_map_val_app 📋 Mathlib.CategoryTheory.Sites.PseudofunctorSheafOver
{C : Type u} [CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) (A : Type u') [CategoryTheory.Category.{v', u'} A] {X✝ Y✝ : CategoryTheory.LocallyDiscrete Cᵒᵖ} (f : X✝ ⟶ Y✝) {X✝¹ Y✝¹ : CategoryTheory.Sheaf (J.over (Opposite.unop X✝.as)) A} (f✝ : X✝¹ ⟶ Y✝¹) (X : (CategoryTheory.Over (Opposite.unop Y✝.as))ᵒᵖ) : (((J.pseudofunctorOver A).map f).map f✝).val.app X = f✝.val.app (Opposite.op ((CategoryTheory.Over.map f.as.unop).obj (Opposite.unop X)))
CategoryTheory.GrothendieckTopology.pseudofunctorOver_mapComp_hom_app_val_app 📋 Mathlib.CategoryTheory.Sites.PseudofunctorSheafOver
{C : Type u} [CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) (A : Type u') [CategoryTheory.Category.{v', u'} A] {a✝ b✝ c✝ : CategoryTheory.LocallyDiscrete Cᵒᵖ} (x✝ : a✝ ⟶ b✝) (x✝¹ : b✝ ⟶ c✝) (X : CategoryTheory.Sheaf (J.over (Opposite.unop a✝.as)) A) (X✝ : (CategoryTheory.Over (Opposite.unop c✝.as))ᵒᵖ) : (((J.pseudofunctorOver A).mapComp x✝ x✝¹).hom.app X).val.app X✝ = X.val.map ((CategoryTheory.Over.mapComp x✝¹.as.unop x✝.as.unop).inv.app (Opposite.unop X✝)).op
CategoryTheory.GrothendieckTopology.pseudofunctorOver_mapComp_inv_app_val_app 📋 Mathlib.CategoryTheory.Sites.PseudofunctorSheafOver
{C : Type u} [CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) (A : Type u') [CategoryTheory.Category.{v', u'} A] {a✝ b✝ c✝ : CategoryTheory.LocallyDiscrete Cᵒᵖ} (x✝ : a✝ ⟶ b✝) (x✝¹ : b✝ ⟶ c✝) (X : CategoryTheory.Sheaf (J.over (Opposite.unop a✝.as)) A) (X✝ : (CategoryTheory.Over (Opposite.unop c✝.as))ᵒᵖ) : (((J.pseudofunctorOver A).mapComp x✝ x✝¹).inv.app X).val.app X✝ = X.val.map ((CategoryTheory.Over.mapComp x✝¹.as.unop x✝.as.unop).hom.app (Opposite.unop X✝)).op

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using (by default) Ctrl-K Ctrl-S. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

By constant:
🔍 Real.sin
finds all lemmas whose statement somehow mentions the sine function.
By lemma name substring:
🔍 "differ"
finds all lemmas that have "differ" somewhere in their lemma name.
By subexpression:
🔍 _ * (_ ^ _)
finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

The pattern can also be non-linear, as in
🔍 Real.sqrt ?a * Real.sqrt ?a

If the pattern has parameters, they are matched in any order. Both of these will find List.map:
🔍 (?a -> ?b) -> List ?a -> List ?b
🔍 List ?a -> (?a -> ?b) -> List ?b
By main conclusion:
🔍 |- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
🔍 |- _ < _ → tsum _ < tsum _
will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO.

This is Loogle revision 6ff4759 serving mathlib revision edaf32c