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Found 92 declarations mentioning CategoryTheory.Functor.isoWhiskerLeft.

CategoryTheory.Functor.isoWhiskerLeft 📋 Mathlib.CategoryTheory.Whiskering
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] (F : CategoryTheory.Functor C D) {G H : CategoryTheory.Functor D E} (α : G ≅ H) : F.comp G ≅ F.comp H
CategoryTheory.Functor.isoWhiskerLeft_refl 📋 Mathlib.CategoryTheory.Whiskering
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] (F : CategoryTheory.Functor C D) (G : CategoryTheory.Functor D E) : F.isoWhiskerLeft (CategoryTheory.Iso.refl G) = CategoryTheory.Iso.refl (F.comp G)
CategoryTheory.Functor.isoWhiskerLeft_symm 📋 Mathlib.CategoryTheory.Whiskering
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] (F : CategoryTheory.Functor C D) {G H : CategoryTheory.Functor D E} (α : G ≅ H) : (F.isoWhiskerLeft α).symm = F.isoWhiskerLeft α.symm
CategoryTheory.Functor.isoWhiskerLeft_hom 📋 Mathlib.CategoryTheory.Whiskering
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] (F : CategoryTheory.Functor C D) {G H : CategoryTheory.Functor D E} (α : G ≅ H) : (F.isoWhiskerLeft α).hom = F.whiskerLeft α.hom
CategoryTheory.Functor.isoWhiskerLeft_inv 📋 Mathlib.CategoryTheory.Whiskering
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] (F : CategoryTheory.Functor C D) {G H : CategoryTheory.Functor D E} (α : G ≅ H) : (F.isoWhiskerLeft α).inv = F.whiskerLeft α.inv
CategoryTheory.Functor.isoWhiskerLeft_trans 📋 Mathlib.CategoryTheory.Whiskering
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] (F : CategoryTheory.Functor C D) {G H K : CategoryTheory.Functor D E} (α : G ≅ H) (β : H ≅ K) : F.isoWhiskerLeft (α ≪≫ β) = F.isoWhiskerLeft α ≪≫ F.isoWhiskerLeft β
CategoryTheory.Functor.triangleIso 📋 Mathlib.CategoryTheory.Whiskering
{A : Type u₁} [CategoryTheory.Category.{v₁, u₁} A] {B : Type u₂} [CategoryTheory.Category.{v₂, u₂} B] {C : Type u₃} [CategoryTheory.Category.{v₃, u₃} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor B C) : F.associator (CategoryTheory.Functor.id B) G ≪≫ F.isoWhiskerLeft G.leftUnitor = CategoryTheory.Functor.isoWhiskerRight F.rightUnitor G
CategoryTheory.Functor.isoWhiskerLeft_trans_isoWhiskerRight 📋 Mathlib.CategoryTheory.Whiskering
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] {F G : CategoryTheory.Functor C D} {H K : CategoryTheory.Functor D E} (α : F ≅ G) (β : H ≅ K) : F.isoWhiskerLeft β ≪≫ CategoryTheory.Functor.isoWhiskerRight α K = CategoryTheory.Functor.isoWhiskerRight α H ≪≫ G.isoWhiskerLeft β
CategoryTheory.Functor.isoWhiskerLeft_trans_assoc 📋 Mathlib.CategoryTheory.Whiskering
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] (F : CategoryTheory.Functor C D) {G H K : CategoryTheory.Functor D E} (α : G ≅ H) (β : H ≅ K) {Z : CategoryTheory.Functor C E} (h : F.comp K ≅ Z) : F.isoWhiskerLeft (α ≪≫ β) ≪≫ h = F.isoWhiskerLeft α ≪≫ F.isoWhiskerLeft β ≪≫ h
CategoryTheory.Functor.isoWhiskerLeft_trans_isoWhiskerRight_assoc 📋 Mathlib.CategoryTheory.Whiskering
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] {F G : CategoryTheory.Functor C D} {H K : CategoryTheory.Functor D E} (α : F ≅ G) (β : H ≅ K) {Z : CategoryTheory.Functor C E} (h : G.comp K ≅ Z) : F.isoWhiskerLeft β ≪≫ CategoryTheory.Functor.isoWhiskerRight α K ≪≫ h = CategoryTheory.Functor.isoWhiskerRight α H ≪≫ G.isoWhiskerLeft β ≪≫ h
CategoryTheory.Functor.triangleIso_assoc 📋 Mathlib.CategoryTheory.Whiskering
{A : Type u₁} [CategoryTheory.Category.{v₁, u₁} A] {B : Type u₂} [CategoryTheory.Category.{v₂, u₂} B] {C : Type u₃} [CategoryTheory.Category.{v₃, u₃} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor B C) {Z : CategoryTheory.Functor A C} (h : F.comp G ≅ Z) : F.associator (CategoryTheory.Functor.id B) G ≪≫ F.isoWhiskerLeft G.leftUnitor ≪≫ h = CategoryTheory.Functor.isoWhiskerRight F.rightUnitor G ≪≫ h
CategoryTheory.Functor.isoWhiskerLeft_twice 📋 Mathlib.CategoryTheory.Whiskering
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] {B : Type u₄} [CategoryTheory.Category.{v₄, u₄} B] (F : CategoryTheory.Functor B C) (G : CategoryTheory.Functor C D) {H K : CategoryTheory.Functor D E} (α : H ≅ K) : F.isoWhiskerLeft (G.isoWhiskerLeft α) = (F.associator G H).symm ≪≫ (F.comp G).isoWhiskerLeft α ≪≫ F.associator G K
CategoryTheory.Functor.isoWhiskerLeft_right 📋 Mathlib.CategoryTheory.Whiskering
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] {B : Type u₄} [CategoryTheory.Category.{v₄, u₄} B] (F : CategoryTheory.Functor B C) {G H : CategoryTheory.Functor C D} (α : G ≅ H) (K : CategoryTheory.Functor D E) : F.isoWhiskerLeft (CategoryTheory.Functor.isoWhiskerRight α K) = (F.associator G K).symm ≪≫ CategoryTheory.Functor.isoWhiskerRight (F.isoWhiskerLeft α) K ≪≫ F.associator H K
CategoryTheory.Functor.isoWhiskerRight_left 📋 Mathlib.CategoryTheory.Whiskering
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] {B : Type u₄} [CategoryTheory.Category.{v₄, u₄} B] (F : CategoryTheory.Functor B C) {G H : CategoryTheory.Functor C D} (α : G ≅ H) (K : CategoryTheory.Functor D E) : CategoryTheory.Functor.isoWhiskerRight (F.isoWhiskerLeft α) K = F.associator G K ≪≫ F.isoWhiskerLeft (CategoryTheory.Functor.isoWhiskerRight α K) ≪≫ (F.associator H K).symm
CategoryTheory.Functor.isoWhiskerLeft_right_assoc 📋 Mathlib.CategoryTheory.Whiskering
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] {B : Type u₄} [CategoryTheory.Category.{v₄, u₄} B] (F : CategoryTheory.Functor B C) {G H : CategoryTheory.Functor C D} (α : G ≅ H) (K : CategoryTheory.Functor D E) {Z : CategoryTheory.Functor B E} (h : F.comp (H.comp K) ≅ Z) : F.isoWhiskerLeft (CategoryTheory.Functor.isoWhiskerRight α K) ≪≫ h = (F.associator G K).symm ≪≫ CategoryTheory.Functor.isoWhiskerRight (F.isoWhiskerLeft α) K ≪≫ F.associator H K ≪≫ h
CategoryTheory.Functor.isoWhiskerRight_left_assoc 📋 Mathlib.CategoryTheory.Whiskering
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] {B : Type u₄} [CategoryTheory.Category.{v₄, u₄} B] (F : CategoryTheory.Functor B C) {G H : CategoryTheory.Functor C D} (α : G ≅ H) (K : CategoryTheory.Functor D E) {Z : CategoryTheory.Functor B E} (h : (F.comp H).comp K ≅ Z) : CategoryTheory.Functor.isoWhiskerRight (F.isoWhiskerLeft α) K ≪≫ h = F.associator G K ≪≫ F.isoWhiskerLeft (CategoryTheory.Functor.isoWhiskerRight α K) ≪≫ (F.associator H K).symm ≪≫ h
CategoryTheory.Functor.pentagonIso 📋 Mathlib.CategoryTheory.Whiskering
{A : Type u₁} [CategoryTheory.Category.{v₁, u₁} A] {B : Type u₂} [CategoryTheory.Category.{v₂, u₂} B] {C : Type u₃} [CategoryTheory.Category.{v₃, u₃} C] {D : Type u₄} [CategoryTheory.Category.{v₄, u₄} D] {E : Type u₅} [CategoryTheory.Category.{v₅, u₅} E] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor B C) (H : CategoryTheory.Functor C D) (K : CategoryTheory.Functor D E) : CategoryTheory.Functor.isoWhiskerRight (F.associator G H) K ≪≫ F.associator (G.comp H) K ≪≫ F.isoWhiskerLeft (G.associator H K) = (F.comp G).associator H K ≪≫ F.associator G (H.comp K)
CategoryTheory.Functor.pentagonIso_assoc 📋 Mathlib.CategoryTheory.Whiskering
{A : Type u₁} [CategoryTheory.Category.{v₁, u₁} A] {B : Type u₂} [CategoryTheory.Category.{v₂, u₂} B] {C : Type u₃} [CategoryTheory.Category.{v₃, u₃} C] {D : Type u₄} [CategoryTheory.Category.{v₄, u₄} D] {E : Type u₅} [CategoryTheory.Category.{v₅, u₅} E] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor B C) (H : CategoryTheory.Functor C D) (K : CategoryTheory.Functor D E) {Z : CategoryTheory.Functor A E} (h : F.comp (G.comp (H.comp K)) ≅ Z) : CategoryTheory.Functor.isoWhiskerRight (F.associator G H) K ≪≫ F.associator (G.comp H) K ≪≫ F.isoWhiskerLeft (G.associator H K) ≪≫ h = (F.comp G).associator H K ≪≫ F.associator G (H.comp K) ≪≫ h
CategoryTheory.Equivalence.trans_counitIso 📋 Mathlib.CategoryTheory.Equivalence
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] (e : C ≌ D) (f : D ≌ E) : (e.trans f).counitIso = ((f.inverse.comp e.inverse).associator e.functor f.functor).symm ≪≫ CategoryTheory.Functor.isoWhiskerRight (f.inverse.associator e.inverse e.functor ≪≫ f.inverse.isoWhiskerLeft e.counitIso ≪≫ f.inverse.rightUnitor) f.functor ≪≫ f.counitIso
CategoryTheory.Equivalence.trans_unitIso 📋 Mathlib.CategoryTheory.Equivalence
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] (e : C ≌ D) (f : D ≌ E) : (e.trans f).unitIso = e.unitIso ≪≫ CategoryTheory.Functor.isoWhiskerRight (e.functor.rightUnitor.symm ≪≫ e.functor.isoWhiskerLeft f.unitIso ≪≫ (e.functor.associator f.functor f.inverse).symm) e.inverse ≪≫ (e.functor.comp f.functor).associator f.inverse e.inverse
CategoryTheory.NatIso.op_isoWhiskerLeft 📋 Mathlib.CategoryTheory.Opposites
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F G : CategoryTheory.Functor C D} {E : Type u_1} [CategoryTheory.Category.{v_1, u_1} E] {H : CategoryTheory.Functor E C} (α : F ≅ G) : CategoryTheory.NatIso.op (H.isoWhiskerLeft α) = H.opComp G ≪≫ H.op.isoWhiskerLeft (CategoryTheory.NatIso.op α) ≪≫ (H.opComp F).symm
CategoryTheory.NatIso.unop_rightUnitor 📋 Mathlib.CategoryTheory.Opposites
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ} : CategoryTheory.NatIso.unop F.rightUnitor = F.unop.rightUnitor.symm ≪≫ F.unop.isoWhiskerLeft (CategoryTheory.Functor.unopId D).symm ≪≫ (F.unopComp (CategoryTheory.Functor.id Dᵒᵖ)).symm
CategoryTheory.NatIso.unop_whiskerLeft 📋 Mathlib.CategoryTheory.Opposites
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F G : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ} {E : Type u_1} [CategoryTheory.Category.{v_1, u_1} E] {H : CategoryTheory.Functor Eᵒᵖ Cᵒᵖ} (α : F ≅ G) : CategoryTheory.NatIso.unop (H.isoWhiskerLeft α) = H.unopComp G ≪≫ H.unop.isoWhiskerLeft (CategoryTheory.NatIso.unop α) ≪≫ (H.unopComp F).symm
CategoryTheory.NatIso.op_rightUnitor 📋 Mathlib.CategoryTheory.Opposites
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} : CategoryTheory.NatIso.op F.rightUnitor = F.op.rightUnitor.symm ≪≫ F.op.isoWhiskerLeft (CategoryTheory.Functor.opId D).symm ≪≫ (F.opComp (CategoryTheory.Functor.id D)).symm
CategoryTheory.NatIso.unop_associator 📋 Mathlib.CategoryTheory.Opposites
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u_1} {E' : Type u_2} [CategoryTheory.Category.{v_1, u_1} E] [CategoryTheory.Category.{v_2, u_2} E'] {F : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ} {G : CategoryTheory.Functor Dᵒᵖ Eᵒᵖ} {H : CategoryTheory.Functor Eᵒᵖ E'ᵒᵖ} : CategoryTheory.NatIso.unop (F.associator G H) = F.unopComp (G.comp H) ≪≫ F.unop.isoWhiskerLeft (G.unopComp H) ≪≫ (F.unop.associator G.unop H.unop).symm ≪≫ CategoryTheory.Functor.isoWhiskerRight (F.unopComp G).symm H.unop ≪≫ ((F.comp G).unopComp H).symm
CategoryTheory.NatIso.op_associator 📋 Mathlib.CategoryTheory.Opposites
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u_1} {E' : Type u_2} [CategoryTheory.Category.{v_1, u_1} E] [CategoryTheory.Category.{v_2, u_2} E'] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D E} {H : CategoryTheory.Functor E E'} : CategoryTheory.NatIso.op (F.associator G H) = F.opComp (G.comp H) ≪≫ F.op.isoWhiskerLeft (G.opComp H) ≪≫ (F.op.associator G.op H.op).symm ≪≫ CategoryTheory.Functor.isoWhiskerRight (F.opComp G).symm H.op ≪≫ ((F.comp G).opComp H).symm
CategoryTheory.Pi.equivalenceOfEquiv_unitIso 📋 Mathlib.CategoryTheory.Pi.Basic
{I : Type w₀} {J : Type w₁} (C : I → Type u₁) [(i : I) → CategoryTheory.Category.{v₁, u₁} (C i)] (e : J ≃ I) : (CategoryTheory.Pi.equivalenceOfEquiv C e).unitIso = CategoryTheory.NatIso.pi' fun i' => (CategoryTheory.Pi.eval (fun i => C (e i)) i').leftUnitor ≪≫ (CategoryTheory.Pi.evalCompEqToEquivalenceFunctor (fun j => C (e j)) ⋯).symm ≪≫ (CategoryTheory.Pi.eval (fun j => C (e j)) (e.symm (e i'))).isoWhiskerLeft (CategoryTheory.Pi.eqToEquivalenceFunctorIso C ⇑e ⋯).symm ≪≫ (CategoryTheory.Functor.pi'CompEval (CategoryTheory.Pi.eval (fun j => C (e j)) (e.symm (e i'))).comp (CategoryTheory.Pi.eqToEquivalence C ⋯).functor).symm ≪≫ (CategoryTheory.Functor.pi' (CategoryTheory.Pi.eval (fun j => C (e j)) (e.symm (e i'))).comp).isoWhiskerLeft (CategoryTheory.Functor.pi'CompEval (CategoryTheory.Pi.eval fun i => C (e i')) (CategoryTheory.Pi.eqToEquivalence C ⋯).functor).symm ≪≫ ((CategoryTheory.Functor.pi' (CategoryTheory.Pi.eval (fun j => C (e j)) (e.symm (e i'))).comp).associator (CategoryTheory.Functor.pi' (CategoryTheory.Pi.eval fun i => C (e i'))) (CategoryTheory.Pi.eval (fun i => C (e i')) (CategoryTheory.Pi.eqToEquivalence C ⋯).functor)).symm
CategoryTheory.Limits.Cocone.functorialityEquivalence_inverse 📋 Mathlib.CategoryTheory.Limits.Cones
{J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {C : Type u₃} [CategoryTheory.Category.{v₃, u₃} C] {D : Type u₄} [CategoryTheory.Category.{v₄, u₄} D] (F : CategoryTheory.Functor J C) (e : C ≌ D) : (CategoryTheory.Limits.Cocone.functorialityEquivalence F e).inverse = (CategoryTheory.Limits.Cocone.functoriality (F.comp e.functor) e.inverse).comp (CategoryTheory.Limits.Cocone.precomposeEquivalence (F.associator e.functor e.inverse ≪≫ F.isoWhiskerLeft e.unitIso.symm ≪≫ F.rightUnitor)).functor
CategoryTheory.Limits.Cone.functorialityEquivalence_inverse 📋 Mathlib.CategoryTheory.Limits.Cones
{J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {C : Type u₃} [CategoryTheory.Category.{v₃, u₃} C] {D : Type u₄} [CategoryTheory.Category.{v₄, u₄} D] (F : CategoryTheory.Functor J C) (e : C ≌ D) : (CategoryTheory.Limits.Cone.functorialityEquivalence F e).inverse = (CategoryTheory.Limits.Cone.functoriality (F.comp e.functor) e.inverse).comp (CategoryTheory.Limits.Cone.postcomposeEquivalence (F.associator e.functor e.inverse ≪≫ F.isoWhiskerLeft e.unitIso.symm ≪≫ F.rightUnitor)).functor
CategoryTheory.Limits.Cone.functorialityEquivalence_unitIso 📋 Mathlib.CategoryTheory.Limits.Cones
{J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {C : Type u₃} [CategoryTheory.Category.{v₃, u₃} C] {D : Type u₄} [CategoryTheory.Category.{v₄, u₄} D] (F : CategoryTheory.Functor J C) (e : C ≌ D) : (CategoryTheory.Limits.Cone.functorialityEquivalence F e).unitIso = CategoryTheory.NatIso.ofComponents (fun c => CategoryTheory.Limits.Cone.ext (e.unitIso.app ((CategoryTheory.Functor.id (CategoryTheory.Limits.Cone F)).obj c).1) ⋯) ⋯
CategoryTheory.Limits.Cocone.functorialityEquivalence_unitIso 📋 Mathlib.CategoryTheory.Limits.Cones
{J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {C : Type u₃} [CategoryTheory.Category.{v₃, u₃} C] {D : Type u₄} [CategoryTheory.Category.{v₄, u₄} D] (F : CategoryTheory.Functor J C) (e : C ≌ D) : (CategoryTheory.Limits.Cocone.functorialityEquivalence F e).unitIso = CategoryTheory.NatIso.ofComponents' (fun c => CategoryTheory.Limits.Cocone.extInv (e.unitIso.app ((CategoryTheory.Functor.id (CategoryTheory.Limits.Cocone F)).obj c).pt) ⋯) ⋯
CategoryTheory.Limits.Cocone.functorialityEquivalence_counitIso 📋 Mathlib.CategoryTheory.Limits.Cones
{J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {C : Type u₃} [CategoryTheory.Category.{v₃, u₃} C] {D : Type u₄} [CategoryTheory.Category.{v₄, u₄} D] (F : CategoryTheory.Functor J C) (e : C ≌ D) : (CategoryTheory.Limits.Cocone.functorialityEquivalence F e).counitIso = CategoryTheory.NatIso.ofComponents' (fun c => CategoryTheory.Limits.Cocone.extInv (e.counitIso.app c.pt) ⋯) ⋯
CategoryTheory.Limits.Cone.functorialityEquivalence_counitIso 📋 Mathlib.CategoryTheory.Limits.Cones
{J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {C : Type u₃} [CategoryTheory.Category.{v₃, u₃} C] {D : Type u₄} [CategoryTheory.Category.{v₄, u₄} D] (F : CategoryTheory.Functor J C) (e : C ≌ D) : (CategoryTheory.Limits.Cone.functorialityEquivalence F e).counitIso = CategoryTheory.NatIso.ofComponents (fun c => CategoryTheory.Limits.Cone.ext (e.counitIso.app c.pt) ⋯) ⋯
CategoryTheory.Limits.Cocone.equivalenceOfReindexing_counitIso 📋 Mathlib.CategoryTheory.Limits.Cones
{J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [CategoryTheory.Category.{v₂, u₂} K] {C : Type u₃} [CategoryTheory.Category.{v₃, u₃} C] {F : CategoryTheory.Functor J C} {G : CategoryTheory.Functor K C} (e : K ≌ J) (α : e.functor.comp F ≅ G) : (CategoryTheory.Limits.Cocone.equivalenceOfReindexing e α).counitIso = (((CategoryTheory.Limits.Cocone.precompose α.hom).comp ((CategoryTheory.Limits.Cocone.whiskering e.inverse).comp (CategoryTheory.Limits.Cocone.precompose (e.invFunIdAssoc F).inv))).associator (CategoryTheory.Limits.Cocone.whiskering e.functor) (CategoryTheory.Limits.Cocone.precompose α.inv)).symm ≪≫ CategoryTheory.Functor.isoWhiskerRight ((CategoryTheory.Limits.Cocone.precompose α.hom).associator ((CategoryTheory.Limits.Cocone.whiskering e.inverse).comp (CategoryTheory.Limits.Cocone.precompose (e.invFunIdAssoc F).inv)) (CategoryTheory.Limits.Cocone.whiskering e.functor)) (CategoryTheory.Limits.Cocone.precompose α.inv) ≪≫ CategoryTheory.Functor.isoWhiskerRight ((CategoryTheory.Limits.Cocone.precompose α.hom).isoWhiskerLeft (CategoryTheory.NatIso.ofComponents' (fun s => CategoryTheory.Limits.Cocone.extInv (CategoryTheory.Iso.refl s.pt) ⋯) ⋯)) (CategoryTheory.Limits.Cocone.precompose α.inv) ≪≫ CategoryTheory.Functor.isoWhiskerRight (CategoryTheory.Limits.Cocone.precompose α.hom).rightUnitor (CategoryTheory.Limits.Cocone.precompose α.inv) ≪≫ CategoryTheory.NatIso.ofComponents' (fun s => CategoryTheory.Limits.Cocone.extInv (CategoryTheory.Iso.refl s.pt) ⋯) ⋯
CategoryTheory.Limits.Cone.equivalenceOfReindexing_counitIso 📋 Mathlib.CategoryTheory.Limits.Cones
{J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [CategoryTheory.Category.{v₂, u₂} K] {C : Type u₃} [CategoryTheory.Category.{v₃, u₃} C] {F : CategoryTheory.Functor J C} {G : CategoryTheory.Functor K C} (e : K ≌ J) (α : e.functor.comp F ≅ G) : (CategoryTheory.Limits.Cone.equivalenceOfReindexing e α).counitIso = (((CategoryTheory.Limits.Cone.postcompose α.inv).comp ((CategoryTheory.Limits.Cone.whiskering e.inverse).comp (CategoryTheory.Limits.Cone.postcompose (e.invFunIdAssoc F).hom))).associator (CategoryTheory.Limits.Cone.whiskering e.functor) (CategoryTheory.Limits.Cone.postcompose α.hom)).symm ≪≫ CategoryTheory.Functor.isoWhiskerRight ((CategoryTheory.Limits.Cone.postcompose α.inv).associator ((CategoryTheory.Limits.Cone.whiskering e.inverse).comp (CategoryTheory.Limits.Cone.postcompose (e.invFunIdAssoc F).hom)) (CategoryTheory.Limits.Cone.whiskering e.functor)) (CategoryTheory.Limits.Cone.postcompose α.hom) ≪≫ CategoryTheory.Functor.isoWhiskerRight ((CategoryTheory.Limits.Cone.postcompose α.inv).isoWhiskerLeft (CategoryTheory.NatIso.ofComponents (fun s => CategoryTheory.Limits.Cone.ext (CategoryTheory.Iso.refl s.pt) ⋯) ⋯)) (CategoryTheory.Limits.Cone.postcompose α.hom) ≪≫ CategoryTheory.Functor.isoWhiskerRight (CategoryTheory.Limits.Cone.postcompose α.inv).rightUnitor (CategoryTheory.Limits.Cone.postcompose α.hom) ≪≫ CategoryTheory.NatIso.ofComponents (fun s => CategoryTheory.Limits.Cone.ext (CategoryTheory.Iso.refl s.pt) ⋯) ⋯
CategoryTheory.Limits.Cocone.equivalenceOfReindexing_unitIso 📋 Mathlib.CategoryTheory.Limits.Cones
{J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [CategoryTheory.Category.{v₂, u₂} K] {C : Type u₃} [CategoryTheory.Category.{v₃, u₃} C] {F : CategoryTheory.Functor J C} {G : CategoryTheory.Functor K C} (e : K ≌ J) (α : e.functor.comp F ≅ G) : (CategoryTheory.Limits.Cocone.equivalenceOfReindexing e α).unitIso = CategoryTheory.NatIso.ofComponents' (fun s => CategoryTheory.Limits.Cocone.extInv (CategoryTheory.Iso.refl s.pt) ⋯) ⋯ ≪≫ CategoryTheory.Functor.isoWhiskerRight (CategoryTheory.Limits.Cocone.whiskering e.functor).rightUnitor.symm ((CategoryTheory.Limits.Cocone.whiskering e.inverse).comp (CategoryTheory.Limits.Cocone.precompose (e.invFunIdAssoc F).inv)) ≪≫ CategoryTheory.Functor.isoWhiskerRight ((CategoryTheory.Limits.Cocone.whiskering e.functor).isoWhiskerLeft (CategoryTheory.NatIso.ofComponents' (fun s => CategoryTheory.Limits.Cocone.extInv (CategoryTheory.Iso.refl s.pt) ⋯) ⋯)) ((CategoryTheory.Limits.Cocone.whiskering e.inverse).comp (CategoryTheory.Limits.Cocone.precompose (e.invFunIdAssoc F).inv)) ≪≫ CategoryTheory.Functor.isoWhiskerRight ((CategoryTheory.Limits.Cocone.whiskering e.functor).associator (CategoryTheory.Limits.Cocone.precompose α.inv) (CategoryTheory.Limits.Cocone.precompose α.hom)).symm ((CategoryTheory.Limits.Cocone.whiskering e.inverse).comp (CategoryTheory.Limits.Cocone.precompose (e.invFunIdAssoc F).inv)) ≪≫ ((CategoryTheory.Limits.Cocone.whiskering e.functor).comp (CategoryTheory.Limits.Cocone.precompose α.inv)).associator (CategoryTheory.Limits.Cocone.precompose α.hom) ((CategoryTheory.Limits.Cocone.whiskering e.inverse).comp (CategoryTheory.Limits.Cocone.precompose (e.invFunIdAssoc F).inv))
CategoryTheory.Limits.Cone.equivalenceOfReindexing_unitIso 📋 Mathlib.CategoryTheory.Limits.Cones
{J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [CategoryTheory.Category.{v₂, u₂} K] {C : Type u₃} [CategoryTheory.Category.{v₃, u₃} C] {F : CategoryTheory.Functor J C} {G : CategoryTheory.Functor K C} (e : K ≌ J) (α : e.functor.comp F ≅ G) : (CategoryTheory.Limits.Cone.equivalenceOfReindexing e α).unitIso = CategoryTheory.NatIso.ofComponents (fun s => CategoryTheory.Limits.Cone.ext (CategoryTheory.Iso.refl s.pt) ⋯) ⋯ ≪≫ CategoryTheory.Functor.isoWhiskerRight (CategoryTheory.Limits.Cone.whiskering e.functor).rightUnitor.symm ((CategoryTheory.Limits.Cone.whiskering e.inverse).comp (CategoryTheory.Limits.Cone.postcompose (e.invFunIdAssoc F).hom)) ≪≫ CategoryTheory.Functor.isoWhiskerRight ((CategoryTheory.Limits.Cone.whiskering e.functor).isoWhiskerLeft (CategoryTheory.NatIso.ofComponents (fun s => CategoryTheory.Limits.Cone.ext (CategoryTheory.Iso.refl s.pt) ⋯) ⋯)) ((CategoryTheory.Limits.Cone.whiskering e.inverse).comp (CategoryTheory.Limits.Cone.postcompose (e.invFunIdAssoc F).hom)) ≪≫ CategoryTheory.Functor.isoWhiskerRight ((CategoryTheory.Limits.Cone.whiskering e.functor).associator (CategoryTheory.Limits.Cone.postcompose α.hom) (CategoryTheory.Limits.Cone.postcompose α.inv)).symm ((CategoryTheory.Limits.Cone.whiskering e.inverse).comp (CategoryTheory.Limits.Cone.postcompose (e.invFunIdAssoc F).hom)) ≪≫ ((CategoryTheory.Limits.Cone.whiskering e.functor).comp (CategoryTheory.Limits.Cone.postcompose α.hom)).associator (CategoryTheory.Limits.Cone.postcompose α.inv) ((CategoryTheory.Limits.Cone.whiskering e.inverse).comp (CategoryTheory.Limits.Cone.postcompose (e.invFunIdAssoc F).hom))
CategoryTheory.Limits.IsColimit.coconePointsIsoOfEquivalence_inv 📋 Mathlib.CategoryTheory.Limits.IsLimit
{J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [CategoryTheory.Category.{v₂, u₂} K] {C : Type u₃} [CategoryTheory.Category.{v₃, u₃} C] {F : CategoryTheory.Functor J C} {s : CategoryTheory.Limits.Cocone F} {G : CategoryTheory.Functor K C} {t : CategoryTheory.Limits.Cocone G} (P : CategoryTheory.Limits.IsColimit s) (Q : CategoryTheory.Limits.IsColimit t) (e : J ≌ K) (w : e.functor.comp G ≅ F) : (P.coconePointsIsoOfEquivalence Q e w).inv = Q.desc ((CategoryTheory.Limits.Cocone.equivalenceOfReindexing e.symm ((e.inverse.isoWhiskerLeft w).symm ≪≫ e.invFunIdAssoc G)).functor.obj s)
CategoryTheory.Limits.IsLimit.conePointsIsoOfEquivalence_hom 📋 Mathlib.CategoryTheory.Limits.IsLimit
{J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [CategoryTheory.Category.{v₂, u₂} K] {C : Type u₃} [CategoryTheory.Category.{v₃, u₃} C] {F : CategoryTheory.Functor J C} {s : CategoryTheory.Limits.Cone F} {G : CategoryTheory.Functor K C} {t : CategoryTheory.Limits.Cone G} (P : CategoryTheory.Limits.IsLimit s) (Q : CategoryTheory.Limits.IsLimit t) (e : J ≌ K) (w : e.functor.comp G ≅ F) : (P.conePointsIsoOfEquivalence Q e w).hom = Q.lift ((CategoryTheory.Limits.Cone.equivalenceOfReindexing e.symm ((e.inverse.isoWhiskerLeft w).symm ≪≫ e.invFunIdAssoc G)).functor.obj s)
CategoryTheory.Equivalence.congrFullSubcategory_counitIso 📋 Mathlib.CategoryTheory.ObjectProperty.Equivalence
{C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] {P : CategoryTheory.ObjectProperty C} {Q : CategoryTheory.ObjectProperty D} (e : C ≌ D) [Q.IsClosedUnderIsomorphisms] (h : Q.inverseImage e.functor = P) : (e.congrFullSubcategory h).counitIso = (Q.fullyFaithfulι.whiskeringRight Q.FullSubcategory).preimageIso (Q.ι.isoWhiskerLeft e.counitIso)
CategoryTheory.Equivalence.congrFullSubcategory_unitIso 📋 Mathlib.CategoryTheory.ObjectProperty.Equivalence
{C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] {P : CategoryTheory.ObjectProperty C} {Q : CategoryTheory.ObjectProperty D} (e : C ≌ D) [Q.IsClosedUnderIsomorphisms] (h : Q.inverseImage e.functor = P) : (e.congrFullSubcategory h).unitIso = (P.fullyFaithfulι.whiskeringRight P.FullSubcategory).preimageIso (P.ι.isoWhiskerLeft e.unitIso)
CategoryTheory.shiftFunctorAdd'_add_zero 📋 Mathlib.CategoryTheory.Shift.Basic
(C : Type u) {A : Type u_1} [CategoryTheory.Category.{v, u} C] [AddMonoid A] [CategoryTheory.HasShift C A] (a : A) : CategoryTheory.shiftFunctorAdd' C a 0 a ⋯ = (CategoryTheory.shiftFunctor C a).rightUnitor.symm ≪≫ (CategoryTheory.shiftFunctor C a).isoWhiskerLeft (CategoryTheory.shiftFunctorZero C A).symm
CategoryTheory.shiftFunctorAdd'_assoc 📋 Mathlib.CategoryTheory.Shift.Basic
(C : Type u) {A : Type u_1} [CategoryTheory.Category.{v, u} C] [AddMonoid A] [CategoryTheory.HasShift C A] (a₁ a₂ a₃ a₁₂ a₂₃ a₁₂₃ : A) (h₁₂ : a₁ + a₂ = a₁₂) (h₂₃ : a₂ + a₃ = a₂₃) (h₁₂₃ : a₁ + a₂ + a₃ = a₁₂₃) : CategoryTheory.shiftFunctorAdd' C a₁₂ a₃ a₁₂₃ ⋯ ≪≫ CategoryTheory.Functor.isoWhiskerRight (CategoryTheory.shiftFunctorAdd' C a₁ a₂ a₁₂ h₁₂) (CategoryTheory.shiftFunctor C a₃) ≪≫ (CategoryTheory.shiftFunctor C a₁).associator (CategoryTheory.shiftFunctor C a₂) (CategoryTheory.shiftFunctor C a₃) = CategoryTheory.shiftFunctorAdd' C a₁ a₂₃ a₁₂₃ ⋯ ≪≫ (CategoryTheory.shiftFunctor C a₁).isoWhiskerLeft (CategoryTheory.shiftFunctorAdd' C a₂ a₃ a₂₃ h₂₃)
CategoryTheory.shiftFunctorAdd_assoc 📋 Mathlib.CategoryTheory.Shift.Basic
(C : Type u) {A : Type u_1} [CategoryTheory.Category.{v, u} C] [AddMonoid A] [CategoryTheory.HasShift C A] (a₁ a₂ a₃ : A) : CategoryTheory.shiftFunctorAdd C (a₁ + a₂) a₃ ≪≫ CategoryTheory.Functor.isoWhiskerRight (CategoryTheory.shiftFunctorAdd C a₁ a₂) (CategoryTheory.shiftFunctor C a₃) ≪≫ (CategoryTheory.shiftFunctor C a₁).associator (CategoryTheory.shiftFunctor C a₂) (CategoryTheory.shiftFunctor C a₃) = CategoryTheory.shiftFunctorAdd' C a₁ (a₂ + a₃) (a₁ + a₂ + a₃) ⋯ ≪≫ (CategoryTheory.shiftFunctor C a₁).isoWhiskerLeft (CategoryTheory.shiftFunctorAdd C a₂ a₃)
CategoryTheory.Functor.ShiftSequence.leftComp_isoZero 📋 Mathlib.CategoryTheory.Shift.ShiftSequence
{C : Type u_1} {D : Type u_2} {A : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} A] {F : CategoryTheory.Functor C A} {π : CategoryTheory.Functor C D} {H : CategoryTheory.Functor D A} (e : π.comp H ≅ F) (M : Type u_4) [AddMonoid M] [CategoryTheory.HasShift C M] [CategoryTheory.HasShift D M] [π.CommShift M] [H.ShiftSequence M] : CategoryTheory.Functor.ShiftSequence.isoZero = π.isoWhiskerLeft (H.isoShiftZero M) ≪≫ e
CategoryTheory.Functor.shiftIso_add' 📋 Mathlib.CategoryTheory.Shift.ShiftSequence
{C : Type u_1} {A : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} A] (F : CategoryTheory.Functor C A) {M : Type u_4} [AddMonoid M] [CategoryTheory.HasShift C M] [F.ShiftSequence M] (n m mn : M) (hnm : m + n = mn) (a a' a'' : M) (ha' : n + a = a') (ha'' : m + a' = a'') : F.shiftIso mn a a'' ⋯ = CategoryTheory.Functor.isoWhiskerRight (CategoryTheory.shiftFunctorAdd' C m n mn hnm) (F.shift a) ≪≫ (CategoryTheory.shiftFunctor C m).associator (CategoryTheory.shiftFunctor C n) (F.shift a) ≪≫ (CategoryTheory.shiftFunctor C m).isoWhiskerLeft (F.shiftIso n a a' ha') ≪≫ F.shiftIso m a' a'' ha''
CategoryTheory.Functor.shiftIso_add 📋 Mathlib.CategoryTheory.Shift.ShiftSequence
{C : Type u_1} {A : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} A] (F : CategoryTheory.Functor C A) {M : Type u_4} [AddMonoid M] [CategoryTheory.HasShift C M] [F.ShiftSequence M] (n m a a' a'' : M) (ha' : n + a = a') (ha'' : m + a' = a'') : F.shiftIso (m + n) a a'' ⋯ = CategoryTheory.Functor.isoWhiskerRight (CategoryTheory.shiftFunctorAdd C m n) (F.shift a) ≪≫ (CategoryTheory.shiftFunctor C m).associator (CategoryTheory.shiftFunctor C n) (F.shift a) ≪≫ (CategoryTheory.shiftFunctor C m).isoWhiskerLeft (F.shiftIso n a a' ha') ≪≫ F.shiftIso m a' a'' ha''
CategoryTheory.Functor.ShiftSequence.shiftIso_add 📋 Mathlib.CategoryTheory.Shift.ShiftSequence
{C : Type u_1} {A : Type u_3} {inst✝ : CategoryTheory.Category.{v_1, u_1} C} {inst✝¹ : CategoryTheory.Category.{v_3, u_3} A} {F : CategoryTheory.Functor C A} {M : Type u_4} {inst✝² : AddMonoid M} {inst✝³ : CategoryTheory.HasShift C M} [self : F.ShiftSequence M] (n m a a' a'' : M) (ha' : n + a = a') (ha'' : m + a' = a'') : CategoryTheory.Functor.ShiftSequence.shiftIso (m + n) a a'' ⋯ = CategoryTheory.Functor.isoWhiskerRight (CategoryTheory.shiftFunctorAdd C m n) (CategoryTheory.Functor.ShiftSequence.sequence F a) ≪≫ (CategoryTheory.shiftFunctor C m).associator (CategoryTheory.shiftFunctor C n) (CategoryTheory.Functor.ShiftSequence.sequence F a) ≪≫ (CategoryTheory.shiftFunctor C m).isoWhiskerLeft (CategoryTheory.Functor.ShiftSequence.shiftIso n a a' ha') ≪≫ CategoryTheory.Functor.ShiftSequence.shiftIso m a' a'' ha''
CategoryTheory.Functor.ShiftSequence.mk 📋 Mathlib.CategoryTheory.Shift.ShiftSequence
{C : Type u_1} {A : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} A] {F : CategoryTheory.Functor C A} {M : Type u_4} [AddMonoid M] [CategoryTheory.HasShift C M] (sequence : M → CategoryTheory.Functor C A) (isoZero : sequence 0 ≅ F) (shiftIso : (n a a' : M) → n + a = a' → ((CategoryTheory.shiftFunctor C n).comp (sequence a) ≅ sequence a')) (shiftIso_zero : ∀ (a : M), shiftIso 0 a a ⋯ = CategoryTheory.Functor.isoWhiskerRight (CategoryTheory.shiftFunctorZero C M) (sequence a) ≪≫ (sequence a).leftUnitor) (shiftIso_add : ∀ (n m a a' a'' : M) (ha' : n + a = a') (ha'' : m + a' = a''), shiftIso (m + n) a a'' ⋯ = CategoryTheory.Functor.isoWhiskerRight (CategoryTheory.shiftFunctorAdd C m n) (sequence a) ≪≫ (CategoryTheory.shiftFunctor C m).associator (CategoryTheory.shiftFunctor C n) (sequence a) ≪≫ (CategoryTheory.shiftFunctor C m).isoWhiskerLeft (shiftIso n a a' ha') ≪≫ shiftIso m a' a'' ha'') : F.ShiftSequence M
CategoryTheory.Functor.ShiftSequence.leftComp_shiftIso 📋 Mathlib.CategoryTheory.Shift.ShiftSequence
{C : Type u_1} {D : Type u_2} {A : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} A] {F : CategoryTheory.Functor C A} {π : CategoryTheory.Functor C D} {H : CategoryTheory.Functor D A} (e : π.comp H ≅ F) (M : Type u_4) [AddMonoid M] [CategoryTheory.HasShift C M] [CategoryTheory.HasShift D M] [π.CommShift M] [H.ShiftSequence M] (n a a' : M) (ha' : n + a = a') : CategoryTheory.Functor.ShiftSequence.shiftIso n a a' ha' = ((CategoryTheory.shiftFunctor C n).associator π (H.shift a)).symm ≪≫ CategoryTheory.Functor.isoWhiskerRight (CategoryTheory.Functor.commShiftIso π n) (H.shift a) ≪≫ π.associator (CategoryTheory.shiftFunctor D n) (H.shift a) ≪≫ π.isoWhiskerLeft (H.shiftIso n a a' ha')
CategoryTheory.Localization.Lifting.ofIsos_iso 📋 Mathlib.CategoryTheory.Localization.Predicate
{C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) {E : Type u_3} [CategoryTheory.Category.{v_3, u_3} E] {F₁ F₂ : CategoryTheory.Functor C E} {F₁' F₂' : CategoryTheory.Functor D E} (e : F₁ ≅ F₂) (e' : F₁' ≅ F₂') [CategoryTheory.Localization.Lifting L W F₁ F₁'] : CategoryTheory.Localization.Lifting.iso L W F₂ F₂' = L.isoWhiskerLeft e'.symm ≪≫ CategoryTheory.Localization.Lifting.iso L W F₁ F₁' ≪≫ e
CategoryTheory.SingleFunctors.shiftIso_zero 📋 Mathlib.CategoryTheory.Shift.SingleFunctors
{C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] {A : Type u_5} [AddMonoid A] [CategoryTheory.HasShift D A] (self : CategoryTheory.SingleFunctors C D A) (a : A) : self.shiftIso 0 a a ⋯ = (self.functor a).isoWhiskerLeft (CategoryTheory.shiftFunctorZero D A)
CategoryTheory.SingleFunctors.shiftIso_add' 📋 Mathlib.CategoryTheory.Shift.SingleFunctors
{C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] {A : Type u_5} [AddMonoid A] [CategoryTheory.HasShift D A] (F : CategoryTheory.SingleFunctors C D A) (n m mn : A) (hnm : m + n = mn) (a a' a'' : A) (ha' : n + a = a') (ha'' : m + a' = a'') : F.shiftIso mn a a'' ⋯ = (F.functor a'').isoWhiskerLeft (CategoryTheory.shiftFunctorAdd' D m n mn hnm) ≪≫ ((F.functor a'').associator (CategoryTheory.shiftFunctor D m) (CategoryTheory.shiftFunctor D n)).symm ≪≫ CategoryTheory.Functor.isoWhiskerRight (F.shiftIso m a' a'' ha'') (CategoryTheory.shiftFunctor D n) ≪≫ F.shiftIso n a a' ha'
CategoryTheory.SingleFunctors.mk 📋 Mathlib.CategoryTheory.Shift.SingleFunctors
{C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] {A : Type u_5} [AddMonoid A] [CategoryTheory.HasShift D A] (functor : A → CategoryTheory.Functor C D) (shiftIso : (n a a' : A) → n + a = a' → ((functor a').comp (CategoryTheory.shiftFunctor D n) ≅ functor a)) (shiftIso_zero : ∀ (a : A), shiftIso 0 a a ⋯ = (functor a).isoWhiskerLeft (CategoryTheory.shiftFunctorZero D A)) (shiftIso_add : ∀ (n m a a' a'' : A) (ha' : n + a = a') (ha'' : m + a' = a''), shiftIso (m + n) a a'' ⋯ = (functor a'').isoWhiskerLeft (CategoryTheory.shiftFunctorAdd D m n) ≪≫ ((functor a'').associator (CategoryTheory.shiftFunctor D m) (CategoryTheory.shiftFunctor D n)).symm ≪≫ CategoryTheory.Functor.isoWhiskerRight (shiftIso m a' a'' ha'') (CategoryTheory.shiftFunctor D n) ≪≫ shiftIso n a a' ha') : CategoryTheory.SingleFunctors C D A
CategoryTheory.SingleFunctors.shiftIso_add 📋 Mathlib.CategoryTheory.Shift.SingleFunctors
{C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] {A : Type u_5} [AddMonoid A] [CategoryTheory.HasShift D A] (self : CategoryTheory.SingleFunctors C D A) (n m a a' a'' : A) (ha' : n + a = a') (ha'' : m + a' = a'') : self.shiftIso (m + n) a a'' ⋯ = (self.functor a'').isoWhiskerLeft (CategoryTheory.shiftFunctorAdd D m n) ≪≫ ((self.functor a'').associator (CategoryTheory.shiftFunctor D m) (CategoryTheory.shiftFunctor D n)).symm ≪≫ CategoryTheory.Functor.isoWhiskerRight (self.shiftIso m a' a'' ha'') (CategoryTheory.shiftFunctor D n) ≪≫ self.shiftIso n a a' ha'
CategoryTheory.Functor.commShiftPullback_iso_eq 📋 Mathlib.CategoryTheory.Shift.Pullback
{C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] {A : Type u_2} {B : Type u_3} [AddMonoid A] [AddMonoid B] [CategoryTheory.HasShift C B] (φ : A →+ B) {D : Type u_4} [CategoryTheory.Category.{v_2, u_4} D] [CategoryTheory.HasShift D B] (F : CategoryTheory.Functor C D) [F.CommShift B] (a : A) (b : B) (h : b = φ a) : CategoryTheory.Functor.commShiftIso (CategoryTheory.PullbackShift.functor φ F) a = CategoryTheory.Functor.isoWhiskerRight (CategoryTheory.pullbackShiftIso C φ a b h) F ≪≫ CategoryTheory.Functor.commShiftIso F b ≪≫ F.isoWhiskerLeft (CategoryTheory.pullbackShiftIso D φ a b h).symm
PresheafOfModules.pushforward_comp_id 📋 Mathlib.Algebra.Category.ModuleCat.Presheaf.Pushforward
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {R : CategoryTheory.Functor Dᵒᵖ RingCat} {S : CategoryTheory.Functor Cᵒᵖ RingCat} (φ : S ⟶ F.op.comp R) : PresheafOfModules.pushforwardComp (CategoryTheory.CategoryStruct.id S) φ = (PresheafOfModules.pushforward φ).isoWhiskerLeft (PresheafOfModules.pushforwardId S) ≪≫ (PresheafOfModules.pushforward φ).rightUnitor
PresheafOfModules.pushforward_assoc 📋 Mathlib.Algebra.Category.ModuleCat.Presheaf.Pushforward
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] {E' : Type u₄} [CategoryTheory.Category.{v₄, u₄} E'] {F : CategoryTheory.Functor C D} {R : CategoryTheory.Functor Dᵒᵖ RingCat} {S : CategoryTheory.Functor Cᵒᵖ RingCat} (φ : S ⟶ F.op.comp R) {T : CategoryTheory.Functor Eᵒᵖ RingCat} {G : CategoryTheory.Functor D E} (ψ : R ⟶ G.op.comp T) {T' : CategoryTheory.Functor E'ᵒᵖ RingCat} {G' : CategoryTheory.Functor E E'} (ψ' : T ⟶ G'.op.comp T') : (PresheafOfModules.pushforward ψ').isoWhiskerLeft (PresheafOfModules.pushforwardComp φ ψ) ≪≫ PresheafOfModules.pushforwardComp (CategoryTheory.CategoryStruct.comp φ (F.op.whiskerLeft ψ)) ψ' = ((PresheafOfModules.pushforward ψ').associator (PresheafOfModules.pushforward ψ) (PresheafOfModules.pushforward φ)).symm ≪≫ CategoryTheory.Functor.isoWhiskerRight (PresheafOfModules.pushforwardComp ψ ψ') (PresheafOfModules.pushforward φ) ≪≫ PresheafOfModules.pushforwardComp φ (CategoryTheory.CategoryStruct.comp ψ (G.op.whiskerLeft ψ'))
CategoryTheory.Adjunction.leftAdjointCompIso_comp_id 📋 Mathlib.CategoryTheory.Adjunction.CompositionIso
{C₀ : Type u_1} {C₁ : Type u_2} [CategoryTheory.Category.{v_1, u_1} C₀] [CategoryTheory.Category.{v_2, u_2} C₁] {F₀₁ : CategoryTheory.Functor C₀ C₁} {F₁₁' : CategoryTheory.Functor C₁ C₁} {G₁₀ : CategoryTheory.Functor C₁ C₀} {G₁'₁ : CategoryTheory.Functor C₁ C₁} (adj₀₁ : F₀₁ ⊣ G₁₀) (adj₁₁' : F₁₁' ⊣ G₁'₁) (e₀₁₁' : G₁'₁.comp G₁₀ ≅ G₁₀) (e₁'₁ : G₁'₁ ≅ CategoryTheory.Functor.id C₁) (h : e₀₁₁' = CategoryTheory.Functor.isoWhiskerRight e₁'₁ G₁₀ ≪≫ G₁₀.leftUnitor) : adj₀₁.leftAdjointCompIso adj₁₁' adj₀₁ e₀₁₁' = F₀₁.isoWhiskerLeft (adj₁₁'.leftAdjointIdIso e₁'₁) ≪≫ F₀₁.rightUnitor
CategoryTheory.Adjunction.leftAdjointCompIso_id_comp 📋 Mathlib.CategoryTheory.Adjunction.CompositionIso
{C₀ : Type u_1} {C₁ : Type u_2} [CategoryTheory.Category.{v_1, u_1} C₀] [CategoryTheory.Category.{v_2, u_2} C₁] {F₀₀' : CategoryTheory.Functor C₀ C₀} {F₀'₁ : CategoryTheory.Functor C₀ C₁} {G₀'₀ : CategoryTheory.Functor C₀ C₀} {G₁₀' : CategoryTheory.Functor C₁ C₀} (adj₀₀' : F₀₀' ⊣ G₀'₀) (adj₀'₁ : F₀'₁ ⊣ G₁₀') (e₀₀'₁ : G₁₀'.comp G₀'₀ ≅ G₁₀') (e₀'₀ : G₀'₀ ≅ CategoryTheory.Functor.id C₀) (h : e₀₀'₁ = G₁₀'.isoWhiskerLeft e₀'₀ ≪≫ G₁₀'.rightUnitor) : adj₀₀'.leftAdjointCompIso adj₀'₁ adj₀'₁ e₀₀'₁ = CategoryTheory.Functor.isoWhiskerRight (adj₀₀'.leftAdjointIdIso e₀'₀) F₀'₁ ≪≫ F₀'₁.leftUnitor
CategoryTheory.Adjunction.leftAdjointCompIso_assoc 📋 Mathlib.CategoryTheory.Adjunction.CompositionIso
{C₀ : Type u_1} {C₁ : Type u_2} {C₂ : Type u_3} {C₃ : Type u_4} [CategoryTheory.Category.{v_1, u_1} C₀] [CategoryTheory.Category.{v_2, u_2} C₁] [CategoryTheory.Category.{v_3, u_3} C₂] [CategoryTheory.Category.{v_4, u_4} C₃] {F₀₁ : CategoryTheory.Functor C₀ C₁} {F₁₂ : CategoryTheory.Functor C₁ C₂} {F₂₃ : CategoryTheory.Functor C₂ C₃} {F₀₂ : CategoryTheory.Functor C₀ C₂} {F₁₃ : CategoryTheory.Functor C₁ C₃} {F₀₃ : CategoryTheory.Functor C₀ C₃} {G₁₀ : CategoryTheory.Functor C₁ C₀} {G₂₁ : CategoryTheory.Functor C₂ C₁} {G₃₂ : CategoryTheory.Functor C₃ C₂} {G₂₀ : CategoryTheory.Functor C₂ C₀} {G₃₁ : CategoryTheory.Functor C₃ C₁} {G₃₀ : CategoryTheory.Functor C₃ C₀} (adj₀₁ : F₀₁ ⊣ G₁₀) (adj₁₂ : F₁₂ ⊣ G₂₁) (adj₂₃ : F₂₃ ⊣ G₃₂) (adj₀₂ : F₀₂ ⊣ G₂₀) (adj₁₃ : F₁₃ ⊣ G₃₁) (adj₀₃ : F₀₃ ⊣ G₃₀) (e₀₁₂ : G₂₁.comp G₁₀ ≅ G₂₀) (e₁₂₃ : G₃₂.comp G₂₁ ≅ G₃₁) (e₀₁₃ : G₃₁.comp G₁₀ ≅ G₃₀) (e₀₂₃ : G₃₂.comp G₂₀ ≅ G₃₀) (h : G₃₂.isoWhiskerLeft e₀₁₂ ≪≫ e₀₂₃ = (G₃₂.associator G₂₁ G₁₀).symm ≪≫ CategoryTheory.Functor.isoWhiskerRight e₁₂₃ G₁₀ ≪≫ e₀₁₃) : F₀₁.isoWhiskerLeft (adj₁₂.leftAdjointCompIso adj₂₃ adj₁₃ e₁₂₃) ≪≫ adj₀₁.leftAdjointCompIso adj₁₃ adj₀₃ e₀₁₃ = (F₀₁.associator F₁₂ F₂₃).symm ≪≫ CategoryTheory.Functor.isoWhiskerRight (adj₀₁.leftAdjointCompIso adj₁₂ adj₀₂ e₀₁₂) F₂₃ ≪≫ adj₀₂.leftAdjointCompIso adj₂₃ adj₀₃ e₀₂₃
PresheafOfModules.pullback_comp_id 📋 Mathlib.Algebra.Category.ModuleCat.Presheaf.Pullback
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {R : CategoryTheory.Functor Dᵒᵖ RingCat} {S : CategoryTheory.Functor Cᵒᵖ RingCat} (φ : S ⟶ F.op.comp R) [(PresheafOfModules.pushforward φ).IsRightAdjoint] : PresheafOfModules.pullbackComp φ (CategoryTheory.CategoryStruct.id R) = (PresheafOfModules.pullback φ).isoWhiskerLeft (PresheafOfModules.pullbackId R) ≪≫ (PresheafOfModules.pullback φ).rightUnitor
PresheafOfModules.pullback_assoc 📋 Mathlib.Algebra.Category.ModuleCat.Presheaf.Pullback
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] {E' : Type u₄} [CategoryTheory.Category.{v₄, u₄} E'] {F : CategoryTheory.Functor C D} {R : CategoryTheory.Functor Dᵒᵖ RingCat} {S : CategoryTheory.Functor Cᵒᵖ RingCat} (φ : S ⟶ F.op.comp R) {G : CategoryTheory.Functor D E} {T : CategoryTheory.Functor Eᵒᵖ RingCat} (ψ : R ⟶ G.op.comp T) [(PresheafOfModules.pushforward φ).IsRightAdjoint] [(PresheafOfModules.pushforward ψ).IsRightAdjoint] {T' : CategoryTheory.Functor E'ᵒᵖ RingCat} {G' : CategoryTheory.Functor E E'} (ψ' : T ⟶ G'.op.comp T') [(PresheafOfModules.pushforward ψ').IsRightAdjoint] : (PresheafOfModules.pullback φ).isoWhiskerLeft (PresheafOfModules.pullbackComp ψ ψ') ≪≫ PresheafOfModules.pullbackComp φ (CategoryTheory.CategoryStruct.comp ψ (G.op.whiskerLeft ψ')) = ((PresheafOfModules.pullback φ).associator (PresheafOfModules.pullback ψ) (PresheafOfModules.pullback ψ')).symm ≪≫ CategoryTheory.Functor.isoWhiskerRight (PresheafOfModules.pullbackComp φ ψ) (PresheafOfModules.pullback ψ') ≪≫ PresheafOfModules.pullbackComp (CategoryTheory.CategoryStruct.comp φ (F.op.whiskerLeft ψ)) ψ'
SheafOfModules.pushforward_comp_id 📋 Mathlib.Algebra.Category.ModuleCat.Sheaf.PushforwardContinuous
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {J : CategoryTheory.GrothendieckTopology C} {K : CategoryTheory.GrothendieckTopology D} {F : CategoryTheory.Functor C D} {S : CategoryTheory.Sheaf J RingCat} {R : CategoryTheory.Sheaf K RingCat} [F.IsContinuous J K] (φ : S ⟶ (F.sheafPushforwardContinuous RingCat J K).obj R) : SheafOfModules.pushforwardComp (CategoryTheory.CategoryStruct.id S) φ = (SheafOfModules.pushforward φ).isoWhiskerLeft (SheafOfModules.pushforwardId S) ≪≫ (SheafOfModules.pushforward φ).rightUnitor
SheafOfModules.pushforward_assoc 📋 Mathlib.Algebra.Category.ModuleCat.Sheaf.PushforwardContinuous
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {D' : Type u₃} [CategoryTheory.Category.{v₃, u₃} D'] {D'' : Type u₄} [CategoryTheory.Category.{v₄, u₄} D''] {J : CategoryTheory.GrothendieckTopology C} {K : CategoryTheory.GrothendieckTopology D} {F : CategoryTheory.Functor C D} {S : CategoryTheory.Sheaf J RingCat} {R : CategoryTheory.Sheaf K RingCat} [F.IsContinuous J K] (φ : S ⟶ (F.sheafPushforwardContinuous RingCat J K).obj R) {K' : CategoryTheory.GrothendieckTopology D'} {K'' : CategoryTheory.GrothendieckTopology D''} {G : CategoryTheory.Functor D D'} {R' : CategoryTheory.Sheaf K' RingCat} [G.IsContinuous K K'] (ψ : R ⟶ (G.sheafPushforwardContinuous RingCat K K').obj R') {G' : CategoryTheory.Functor D' D''} {R'' : CategoryTheory.Sheaf K'' RingCat} [G'.IsContinuous K' K''] [(G.comp G').IsContinuous K K''] [(F.comp G).IsContinuous J K'] (ψ' : R' ⟶ (G'.sheafPushforwardContinuous RingCat K' K'').obj R'') : (SheafOfModules.pushforward ψ').isoWhiskerLeft (SheafOfModules.pushforwardComp φ ψ) ≪≫ SheafOfModules.pushforwardComp (CategoryTheory.CategoryStruct.comp φ ((F.sheafPushforwardContinuous RingCat J K).map ψ)) ψ' = ((SheafOfModules.pushforward ψ').associator (SheafOfModules.pushforward ψ) (SheafOfModules.pushforward φ)).symm ≪≫ CategoryTheory.Functor.isoWhiskerRight (SheafOfModules.pushforwardComp ψ ψ') (SheafOfModules.pushforward φ) ≪≫ SheafOfModules.pushforwardComp φ (CategoryTheory.CategoryStruct.comp ψ ((G.sheafPushforwardContinuous RingCat K K').map ψ'))
SheafOfModules.pullback_comp_id 📋 Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackContinuous
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {J : CategoryTheory.GrothendieckTopology C} {K : CategoryTheory.GrothendieckTopology D} {F : CategoryTheory.Functor C D} {S : CategoryTheory.Sheaf J RingCat} {R : CategoryTheory.Sheaf K RingCat} [F.IsContinuous J K] (φ : S ⟶ (F.sheafPushforwardContinuous RingCat J K).obj R) [(SheafOfModules.pushforward φ).IsRightAdjoint] : SheafOfModules.pullbackComp φ (CategoryTheory.CategoryStruct.id R) = (SheafOfModules.pullback φ).isoWhiskerLeft (SheafOfModules.pullbackId R) ≪≫ (SheafOfModules.pullback φ).rightUnitor
SheafOfModules.pullback_assoc 📋 Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackContinuous
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {D' : Type u₃} [CategoryTheory.Category.{v₃, u₃} D'] {D'' : Type u₄} [CategoryTheory.Category.{v₄, u₄} D''] {J : CategoryTheory.GrothendieckTopology C} {K : CategoryTheory.GrothendieckTopology D} {F : CategoryTheory.Functor C D} {S : CategoryTheory.Sheaf J RingCat} {R : CategoryTheory.Sheaf K RingCat} [F.IsContinuous J K] (φ : S ⟶ (F.sheafPushforwardContinuous RingCat J K).obj R) [(SheafOfModules.pushforward φ).IsRightAdjoint] {K' : CategoryTheory.GrothendieckTopology D'} {K'' : CategoryTheory.GrothendieckTopology D''} {G : CategoryTheory.Functor D D'} {R' : CategoryTheory.Sheaf K' RingCat} [G.IsContinuous K K'] [(F.comp G).IsContinuous J K'] (ψ : R ⟶ (G.sheafPushforwardContinuous RingCat K K').obj R') [(SheafOfModules.pushforward ψ).IsRightAdjoint] {G' : CategoryTheory.Functor D' D''} {R'' : CategoryTheory.Sheaf K'' RingCat} [G'.IsContinuous K' K''] [(G.comp G').IsContinuous K K''] [((F.comp G).comp G').IsContinuous J K''] [(F.comp (G.comp G')).IsContinuous J K''] (ψ' : R' ⟶ (G'.sheafPushforwardContinuous RingCat K' K'').obj R'') [(SheafOfModules.pushforward ψ').IsRightAdjoint] : (SheafOfModules.pullback φ).isoWhiskerLeft (SheafOfModules.pullbackComp ψ ψ') ≪≫ SheafOfModules.pullbackComp φ (CategoryTheory.CategoryStruct.comp ψ ((G.sheafPushforwardContinuous RingCat K K').map ψ')) = ((SheafOfModules.pullback φ).associator (SheafOfModules.pullback ψ) (SheafOfModules.pullback ψ')).symm ≪≫ CategoryTheory.Functor.isoWhiskerRight (SheafOfModules.pullbackComp φ ψ) (SheafOfModules.pullback ψ') ≪≫ SheafOfModules.pullbackComp (CategoryTheory.CategoryStruct.comp φ ((F.sheafPushforwardContinuous RingCat J K).map ψ)) ψ'
CategoryTheory.Iso.coreRightUnitor 📋 Mathlib.CategoryTheory.Core
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} : F.rightUnitor.core = F.coreComp (CategoryTheory.Functor.id D) ≪≫ F.core.isoWhiskerLeft (CategoryTheory.Functor.coreId D) ≪≫ F.core.rightUnitor
CategoryTheory.Iso.coreWhiskerLeft 📋 Mathlib.CategoryTheory.Core
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] (F : CategoryTheory.Functor C D) {G H : CategoryTheory.Functor D E} (η : G ≅ H) : (F.isoWhiskerLeft η).core = F.coreComp G ≪≫ F.core.isoWhiskerLeft η.core ≪≫ (F.coreComp H).symm
CategoryTheory.Iso.coreAssociator 📋 Mathlib.CategoryTheory.Core
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] {E' : Type u₄} [CategoryTheory.Category.{v₄, u₄} E'] (F : CategoryTheory.Functor C D) (G : CategoryTheory.Functor D E) (H : CategoryTheory.Functor E E') : (F.associator G H).core = (F.comp G).coreComp H ≪≫ CategoryTheory.Functor.isoWhiskerRight (F.coreComp G) H.core ≪≫ F.core.associator G.core H.core ≪≫ (F.core.isoWhiskerLeft (G.coreComp H)).symm ≪≫ (F.coreComp (G.comp H)).symm
CategoryTheory.Limits.fiberwiseColimitLimitIso_inv_app 📋 Mathlib.CategoryTheory.Limits.Preserves.Grothendieck
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {H : Type u₂} [CategoryTheory.Category.{v₂, u₂} H] {J : Type u₃} [CategoryTheory.Category.{v₃, u₃} J] {F : CategoryTheory.Functor C CategoryTheory.Cat} (K : CategoryTheory.Functor J (CategoryTheory.Functor (CategoryTheory.Grothendieck F) H)) [∀ (c : C), CategoryTheory.Limits.HasColimitsOfShape (↑(F.obj c)) H] [CategoryTheory.Limits.HasLimitsOfShape J H] [∀ (c : C), CategoryTheory.Limits.PreservesLimitsOfShape J CategoryTheory.Limits.colim] (X : C) : (CategoryTheory.Limits.fiberwiseColimitLimitIso K).inv.app X = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation (K.comp (CategoryTheory.Limits.fiberwiseColim F H)) X).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.HasLimit.isoOfNatIso (K.associator ((CategoryTheory.Functor.whiskeringLeft (↑(F.obj X)) (CategoryTheory.Grothendieck F) H).obj (CategoryTheory.Grothendieck.ι F X)) CategoryTheory.Limits.colim ≪≫ K.isoWhiskerLeft (CategoryTheory.Limits.fiberwiseColimCompEvaluationIso X).symm ≪≫ (K.associator (CategoryTheory.Limits.fiberwiseColim F H) ((CategoryTheory.evaluation C H).obj X)).symm)).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.preservesLimitIso CategoryTheory.Limits.colim (K.comp ((CategoryTheory.Functor.whiskeringLeft (↑(F.obj X)) (CategoryTheory.Grothendieck F) H).obj (CategoryTheory.Grothendieck.ι F X)))).inv (CategoryTheory.Limits.HasColimit.isoOfNatIso (CategoryTheory.Limits.limitCompWhiskeringLeftIsoCompLimit K (CategoryTheory.Grothendieck.ι F X)).symm).inv))
CategoryTheory.Limits.fiberwiseColimitLimitIso_hom_app 📋 Mathlib.CategoryTheory.Limits.Preserves.Grothendieck
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {H : Type u₂} [CategoryTheory.Category.{v₂, u₂} H] {J : Type u₃} [CategoryTheory.Category.{v₃, u₃} J] {F : CategoryTheory.Functor C CategoryTheory.Cat} (K : CategoryTheory.Functor J (CategoryTheory.Functor (CategoryTheory.Grothendieck F) H)) [∀ (c : C), CategoryTheory.Limits.HasColimitsOfShape (↑(F.obj c)) H] [CategoryTheory.Limits.HasLimitsOfShape J H] [∀ (c : C), CategoryTheory.Limits.PreservesLimitsOfShape J CategoryTheory.Limits.colim] (X : C) : (CategoryTheory.Limits.fiberwiseColimitLimitIso K).hom.app X = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.HasColimit.isoOfNatIso (CategoryTheory.Limits.limitCompWhiskeringLeftIsoCompLimit K (CategoryTheory.Grothendieck.ι F X)).symm).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.preservesLimitIso CategoryTheory.Limits.colim (K.comp ((CategoryTheory.Functor.whiskeringLeft (↑(F.obj X)) (CategoryTheory.Grothendieck F) H).obj (CategoryTheory.Grothendieck.ι F X)))).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.HasLimit.isoOfNatIso (K.associator ((CategoryTheory.Functor.whiskeringLeft (↑(F.obj X)) (CategoryTheory.Grothendieck F) H).obj (CategoryTheory.Grothendieck.ι F X)) CategoryTheory.Limits.colim ≪≫ K.isoWhiskerLeft (CategoryTheory.Limits.fiberwiseColimCompEvaluationIso X).symm ≪≫ (K.associator (CategoryTheory.Limits.fiberwiseColim F H) ((CategoryTheory.evaluation C H).obj X)).symm)).hom (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation (K.comp (CategoryTheory.Limits.fiberwiseColim F H)) X).inv))
CategoryTheory.Join.mkNatIso 📋 Mathlib.CategoryTheory.Join.Basic
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] {F G : CategoryTheory.Functor (CategoryTheory.Join C D) E} (eₗ : (CategoryTheory.Join.inclLeft C D).comp F ≅ (CategoryTheory.Join.inclLeft C D).comp G) (eᵣ : (CategoryTheory.Join.inclRight C D).comp F ≅ (CategoryTheory.Join.inclRight C D).comp G) (h : CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) F) ((CategoryTheory.Prod.snd C D).isoWhiskerLeft eᵣ).hom = CategoryTheory.CategoryStruct.comp ((CategoryTheory.Prod.fst C D).isoWhiskerLeft eₗ).hom (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) G) := by cat_disch) : F ≅ G
CategoryTheory.Join.mkNatIso_hom 📋 Mathlib.CategoryTheory.Join.Basic
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] {F G : CategoryTheory.Functor (CategoryTheory.Join C D) E} (eₗ : (CategoryTheory.Join.inclLeft C D).comp F ≅ (CategoryTheory.Join.inclLeft C D).comp G) (eᵣ : (CategoryTheory.Join.inclRight C D).comp F ≅ (CategoryTheory.Join.inclRight C D).comp G) (h : CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) F) ((CategoryTheory.Prod.snd C D).isoWhiskerLeft eᵣ).hom = CategoryTheory.CategoryStruct.comp ((CategoryTheory.Prod.fst C D).isoWhiskerLeft eₗ).hom (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) G) := by cat_disch) : (CategoryTheory.Join.mkNatIso eₗ eᵣ h).hom = CategoryTheory.Join.mkNatTrans eₗ.hom eᵣ.hom h
CategoryTheory.Join.mkNatIso_inv 📋 Mathlib.CategoryTheory.Join.Basic
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] {F G : CategoryTheory.Functor (CategoryTheory.Join C D) E} (eₗ : (CategoryTheory.Join.inclLeft C D).comp F ≅ (CategoryTheory.Join.inclLeft C D).comp G) (eᵣ : (CategoryTheory.Join.inclRight C D).comp F ≅ (CategoryTheory.Join.inclRight C D).comp G) (h : CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) F) ((CategoryTheory.Prod.snd C D).isoWhiskerLeft eᵣ).hom = CategoryTheory.CategoryStruct.comp ((CategoryTheory.Prod.fst C D).isoWhiskerLeft eₗ).hom (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) G) := by cat_disch) : (CategoryTheory.Join.mkNatIso eₗ eᵣ h).inv = CategoryTheory.Join.mkNatTrans eₗ.inv eᵣ.inv ⋯
CategoryTheory.Join.InclLeftCompRightOpOpEquivFunctor_hom_app 📋 Mathlib.CategoryTheory.Join.Opposites
(C : Type u₁) (D : Type u₂) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (X : C) : (CategoryTheory.Join.InclLeftCompRightOpOpEquivFunctor C D).hom.app X = CategoryTheory.CategoryStruct.comp (((CategoryTheory.Join.inclLeft C D).isoWhiskerLeft (CategoryTheory.Join.mkFunctor (CategoryTheory.Join.inclRight Dᵒᵖ Cᵒᵖ).rightOp (CategoryTheory.Join.inclLeft Dᵒᵖ Cᵒᵖ).rightOp { app := fun x => (CategoryTheory.Join.edge (Opposite.op x.2) (Opposite.op x.1)).op, naturality := ⋯ }).leftOpRightOpIso).hom.app X) (CategoryTheory.CategoryStruct.id (Opposite.op (CategoryTheory.Join.right (Opposite.op X))))
CategoryTheory.Join.InclLeftCompRightOpOpEquivFunctor_inv_app 📋 Mathlib.CategoryTheory.Join.Opposites
(C : Type u₁) (D : Type u₂) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (X : C) : (CategoryTheory.Join.InclLeftCompRightOpOpEquivFunctor C D).inv.app X = CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.id (Opposite.op (CategoryTheory.Join.right (Opposite.op X)))) (((CategoryTheory.Join.inclLeft C D).isoWhiskerLeft (CategoryTheory.Join.mkFunctor (CategoryTheory.Join.inclRight Dᵒᵖ Cᵒᵖ).rightOp (CategoryTheory.Join.inclLeft Dᵒᵖ Cᵒᵖ).rightOp { app := fun x => (CategoryTheory.Join.edge (Opposite.op x.2) (Opposite.op x.1)).op, naturality := ⋯ }).leftOpRightOpIso).inv.app X)
CategoryTheory.Join.InclRightCompRightOpOpEquivFunctor_hom_app 📋 Mathlib.CategoryTheory.Join.Opposites
(C : Type u₁) (D : Type u₂) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (X : D) : (CategoryTheory.Join.InclRightCompRightOpOpEquivFunctor C D).hom.app X = CategoryTheory.CategoryStruct.comp (((CategoryTheory.Join.inclRight C D).isoWhiskerLeft (CategoryTheory.Join.mkFunctor (CategoryTheory.Join.inclRight Dᵒᵖ Cᵒᵖ).rightOp (CategoryTheory.Join.inclLeft Dᵒᵖ Cᵒᵖ).rightOp { app := fun x => (CategoryTheory.Join.edge (Opposite.op x.2) (Opposite.op x.1)).op, naturality := ⋯ }).leftOpRightOpIso).hom.app X) (CategoryTheory.CategoryStruct.id (Opposite.op (CategoryTheory.Join.left (Opposite.op X))))
CategoryTheory.Join.InclRightCompRightOpOpEquivFunctor_inv_app 📋 Mathlib.CategoryTheory.Join.Opposites
(C : Type u₁) (D : Type u₂) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (X : D) : (CategoryTheory.Join.InclRightCompRightOpOpEquivFunctor C D).inv.app X = CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.id (Opposite.op (CategoryTheory.Join.left (Opposite.op X)))) (((CategoryTheory.Join.inclRight C D).isoWhiskerLeft (CategoryTheory.Join.mkFunctor (CategoryTheory.Join.inclRight Dᵒᵖ Cᵒᵖ).rightOp (CategoryTheory.Join.inclLeft Dᵒᵖ Cᵒᵖ).rightOp { app := fun x => (CategoryTheory.Join.edge (Opposite.op x.2) (Opposite.op x.1)).op, naturality := ⋯ }).leftOpRightOpIso).inv.app X)
CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose_obj_map_fst_app 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) {X : Type u₄} {Y : Type u₅} [CategoryTheory.Category.{v₄, u₄} X] [CategoryTheory.Category.{v₅, u₅} Y] (U : CategoryTheory.Functor X Y) {S S' : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G Y} (φ : S ⟶ S') (X✝ : X) : (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj U).map φ).fst.app X✝ = φ.fst.app (U.obj X✝)
CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose_obj_map_snd_app 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) {X : Type u₄} {Y : Type u₅} [CategoryTheory.Category.{v₄, u₄} X] [CategoryTheory.Category.{v₅, u₅} Y] (U : CategoryTheory.Functor X Y) {S S' : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G Y} (φ : S ⟶ S') (X✝ : X) : (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj U).map φ).snd.app X✝ = φ.snd.app (U.obj X✝)
CategoryTheory.Limits.CategoricalPullback.toCatCommSqOver_map_fst_app 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) (X : Type u₄) [CategoryTheory.Category.{v₄, u₄} X] {J J' : CategoryTheory.Functor X (CategoryTheory.Limits.CategoricalPullback F G)} (F✝ : J ⟶ J') (X✝ : X) : ((CategoryTheory.Limits.CategoricalPullback.toCatCommSqOver F G X).map F✝).fst.app X✝ = (F✝.app X✝).fst
CategoryTheory.Limits.CategoricalPullback.toCatCommSqOver_map_snd_app 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) (X : Type u₄) [CategoryTheory.Category.{v₄, u₄} X] {J J' : CategoryTheory.Functor X (CategoryTheory.Limits.CategoricalPullback F G)} (F✝ : J ⟶ J') (X✝ : X) : ((CategoryTheory.Limits.CategoricalPullback.toCatCommSqOver F G X).map F✝).snd.app X✝ = (F✝.app X✝).snd
CategoryTheory.Limits.CategoricalPullback.functorEquiv_functor_map_fst_app 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) (X : Type u₄) [CategoryTheory.Category.{v₄, u₄} X] {J J' : CategoryTheory.Functor X (CategoryTheory.Limits.CategoricalPullback F G)} (F✝ : J ⟶ J') (X✝ : X) : ((CategoryTheory.Limits.CategoricalPullback.functorEquiv F G X).functor.map F✝).fst.app X✝ = (F✝.app X✝).fst
CategoryTheory.Limits.CategoricalPullback.functorEquiv_functor_map_snd_app 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) (X : Type u₄) [CategoryTheory.Category.{v₄, u₄} X] {J J' : CategoryTheory.Functor X (CategoryTheory.Limits.CategoricalPullback F G)} (F✝ : J ⟶ J') (X✝ : X) : ((CategoryTheory.Limits.CategoricalPullback.functorEquiv F G X).functor.map F✝).snd.app X✝ = (F✝.app X✝).snd
CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform_obj_map_fst_app 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {A₁ : Type u₄} {B₁ : Type u₅} {C₁ : Type u₆} [CategoryTheory.Category.{v₄, u₄} A₁] [CategoryTheory.Category.{v₅, u₅} B₁] [CategoryTheory.Category.{v₆, u₆} C₁] {F₁ : CategoryTheory.Functor A₁ B₁} {G₁ : CategoryTheory.Functor C₁ B₁} (X : Type u₇) [CategoryTheory.Category.{v₇, u₇} X] (ψ : CategoryTheory.Limits.CatCospanTransform F G F₁ G₁) {x y : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X} (f : x ⟶ y) (X✝ : X) : (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ψ).map f).fst.app X✝ = ψ.left.map (f.fst.app X✝)
CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform_obj_map_snd_app 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {A₁ : Type u₄} {B₁ : Type u₅} {C₁ : Type u₆} [CategoryTheory.Category.{v₄, u₄} A₁] [CategoryTheory.Category.{v₅, u₅} B₁] [CategoryTheory.Category.{v₆, u₆} C₁] {F₁ : CategoryTheory.Functor A₁ B₁} {G₁ : CategoryTheory.Functor C₁ B₁} (X : Type u₇) [CategoryTheory.Category.{v₇, u₇} X] (ψ : CategoryTheory.Limits.CatCospanTransform F G F₁ G₁) {x y : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X} (f : x ⟶ y) (X✝ : X) : (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ψ).map f).snd.app X✝ = ψ.right.map (f.snd.app X✝)
CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose_map_app_fst_app 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) {X : Type u₄} {Y : Type u₅} [CategoryTheory.Category.{v₄, u₄} X] [CategoryTheory.Category.{v₅, u₅} Y] {U V : CategoryTheory.Functor X Y} (α : U ⟶ V) (x : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G Y) (X✝ : X) : (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).map α).app x).fst.app X✝ = x.fst.map (α.app X✝)
CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose_map_app_snd_app 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) {X : Type u₄} {Y : Type u₅} [CategoryTheory.Category.{v₄, u₄} X] [CategoryTheory.Category.{v₅, u₅} Y] {U V : CategoryTheory.Functor X Y} (α : U ⟶ V) (x : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G Y) (X✝ : X) : (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).map α).app x).snd.app X✝ = x.snd.map (α.app X✝)
CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform_map_app_fst_app 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {A₁ : Type u₄} {B₁ : Type u₅} {C₁ : Type u₆} [CategoryTheory.Category.{v₄, u₄} A₁] [CategoryTheory.Category.{v₅, u₅} B₁] [CategoryTheory.Category.{v₆, u₆} C₁] {F₁ : CategoryTheory.Functor A₁ B₁} {G₁ : CategoryTheory.Functor C₁ B₁} (X : Type u₇) [CategoryTheory.Category.{v₇, u₇} X] {ψ ψ' : CategoryTheory.Limits.CatCospanTransform F G F₁ G₁} (η : ψ ⟶ ψ') (S : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X) (y : X) : (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).map η).app S).fst.app y = η.left.app (S.fst.obj y)
CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform_map_app_snd_app 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {A₁ : Type u₄} {B₁ : Type u₅} {C₁ : Type u₆} [CategoryTheory.Category.{v₄, u₄} A₁] [CategoryTheory.Category.{v₅, u₅} B₁] [CategoryTheory.Category.{v₆, u₆} C₁] {F₁ : CategoryTheory.Functor A₁ B₁} {G₁ : CategoryTheory.Functor C₁ B₁} (X : Type u₇) [CategoryTheory.Category.{v₇, u₇} X] {ψ ψ' : CategoryTheory.Limits.CatCospanTransform F G F₁ G₁} (η : ψ ⟶ ψ') (S : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X) (y : X) : (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).map η).app S).snd.app y = η.right.app (S.snd.obj y)
CategoryTheory.MorphismProperty.hasLocalizationOfLocallySmall'_def 📋 Mathlib.CategoryTheory.Localization.LocallySmall
{C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] (W : CategoryTheory.MorphismProperty C) {D : Type u_3} [CategoryTheory.Category.{u_4, u_3} D] [CategoryTheory.LocallySmall.{u_5, u_4, u_3} D] (L : CategoryTheory.Functor C D) [L.IsLocalization W] : W.hasLocalizationOfLocallySmall' L = have this := ⋯; let L' := { obj := fun X => X, map := fun {X Y} f => CategoryTheory.InducedCategory.homMk (L.map f), map_id := ⋯, map_comp := ⋯ }; have this_1 := ⋯; have this_2 := ⋯; have this_3 := ⋯; have e := (CategoryTheory.inducedFunctor L.obj).asEquivalence; have e' := L'.associator e.functor e.inverse ≪≫ L'.isoWhiskerLeft e.unitIso.symm ≪≫ L'.rightUnitor; have this_4 := ⋯; W.hasLocalizationOfLocallySmall L'

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. 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.
You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

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. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision 69fae59