Loogle!

Result

Found 68 declarations mentioning CategoryTheory.Prod.fst.

CategoryTheory.Prod.fst 📋 Mathlib.CategoryTheory.Products.Basic
(C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] : CategoryTheory.Functor (C × D) C
CategoryTheory.Prod.fst_obj 📋 Mathlib.CategoryTheory.Products.Basic
(C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] (X : C × D) : (CategoryTheory.Prod.fst C D).obj X = X.1
CategoryTheory.Functor.prod'CompFst 📋 Mathlib.CategoryTheory.Products.Basic
{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 A C) : (F.prod' G).comp (CategoryTheory.Prod.fst B C) ≅ F
CategoryTheory.Prod.fst_map 📋 Mathlib.CategoryTheory.Products.Basic
(C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] {X✝ Y✝ : C × D} (f : X✝ ⟶ Y✝) : (CategoryTheory.Prod.fst C D).map f = f.1
CategoryTheory.functorProdToProdFunctor_obj 📋 Mathlib.CategoryTheory.Products.Basic
(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 × C)) : (CategoryTheory.functorProdToProdFunctor A B C).obj F = (F.comp (CategoryTheory.Prod.fst B C), F.comp (CategoryTheory.Prod.snd B C))
CategoryTheory.Functor.prod'CompFst_hom_app 📋 Mathlib.CategoryTheory.Products.Basic
{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 A C) (X : A) : (F.prod'CompFst G).hom.app X = CategoryTheory.CategoryStruct.id (F.obj X)
CategoryTheory.Functor.prod'CompFst_inv_app 📋 Mathlib.CategoryTheory.Products.Basic
{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 A C) (X : A) : (F.prod'CompFst G).inv.app X = CategoryTheory.CategoryStruct.id (F.obj X)
CategoryTheory.functorProdToProdFunctor_map 📋 Mathlib.CategoryTheory.Products.Basic
(A : Type u₁) [CategoryTheory.Category.{v₁, u₁} A] (B : Type u₂) [CategoryTheory.Category.{v₂, u₂} B] (C : Type u₃) [CategoryTheory.Category.{v₃, u₃} C] {X✝ Y✝ : CategoryTheory.Functor A (B × C)} (α : X✝ ⟶ Y✝) : (CategoryTheory.functorProdToProdFunctor A B C).map α = CategoryTheory.Prod.mkHom (CategoryTheory.Functor.whiskerRight α (CategoryTheory.Prod.fst B C)) (CategoryTheory.Functor.whiskerRight α (CategoryTheory.Prod.snd B C))
CategoryTheory.functorProdFunctorEquivUnitIso_hom_app 📋 Mathlib.CategoryTheory.Products.Basic
(A : Type u₁) [CategoryTheory.Category.{v₁, u₁} A] (B : Type u₂) [CategoryTheory.Category.{v₂, u₂} B] (C : Type u₃) [CategoryTheory.Category.{v₃, u₃} C] (X : CategoryTheory.Functor A B × CategoryTheory.Functor A C) : (CategoryTheory.functorProdFunctorEquivUnitIso A B C).hom.app X = CategoryTheory.Prod.mkHom (X.1.prod'CompFst X.2).inv (X.1.prod'CompSnd X.2).inv
CategoryTheory.functorProdFunctorEquivUnitIso_inv_app 📋 Mathlib.CategoryTheory.Products.Basic
(A : Type u₁) [CategoryTheory.Category.{v₁, u₁} A] (B : Type u₂) [CategoryTheory.Category.{v₂, u₂} B] (C : Type u₃) [CategoryTheory.Category.{v₃, u₃} C] (X : CategoryTheory.Functor A B × CategoryTheory.Functor A C) : (CategoryTheory.functorProdFunctorEquivUnitIso A B C).inv.app X = CategoryTheory.Prod.mkHom (X.1.prod'CompFst X.2).hom (X.1.prod'CompSnd X.2).hom
CategoryTheory.Pi.optionEquivalence_inverse 📋 Mathlib.CategoryTheory.Pi.Basic
{J : Type w₁} (C' : Option J → Type u₁) [(i : Option J) → CategoryTheory.Category.{v₁, u₁} (C' i)] : (CategoryTheory.Pi.optionEquivalence C').inverse = CategoryTheory.Functor.pi' fun i => match i with | none => CategoryTheory.Prod.fst (C' none) ((j : J) → C' (some j)) | some i => (CategoryTheory.Prod.snd (C' none) ((j : J) → C' (some j))).comp (CategoryTheory.Pi.eval (fun j => C' (some j)) i)
CategoryTheory.Pi.optionEquivalence_unitIso 📋 Mathlib.CategoryTheory.Pi.Basic
{J : Type w₁} (C' : Option J → Type u₁) [(i : Option J) → CategoryTheory.Category.{v₁, u₁} (C' i)] : (CategoryTheory.Pi.optionEquivalence C').unitIso = CategoryTheory.NatIso.pi' fun i => match i with | none => CategoryTheory.Iso.refl ((CategoryTheory.Functor.id ((i : Option J) → C' i)).comp (CategoryTheory.Pi.eval C' none)) | some val => CategoryTheory.Iso.refl ((CategoryTheory.Functor.id ((i : Option J) → C' i)).comp (CategoryTheory.Pi.eval C' (some val)))
CategoryTheory.Pi.optionEquivalence_counitIso 📋 Mathlib.CategoryTheory.Pi.Basic
{J : Type w₁} (C' : Option J → Type u₁) [(i : Option J) → CategoryTheory.Category.{v₁, u₁} (C' i)] : (CategoryTheory.Pi.optionEquivalence C').counitIso = CategoryTheory.Iso.refl ((CategoryTheory.Functor.pi' fun i => match i with | none => CategoryTheory.Prod.fst (C' none) ((j : J) → C' (some j)) | some i => (CategoryTheory.Prod.snd (C' none) ((j : J) → C' (some j))).comp (CategoryTheory.Pi.eval (fun j => C' (some j)) i)).comp ((CategoryTheory.Pi.eval C' none).prod' (CategoryTheory.Functor.pi' fun i => CategoryTheory.Pi.eval C' (some i))))
CategoryTheory.final_fst 📋 Mathlib.CategoryTheory.Limits.Final.Connected
{C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] [CategoryTheory.IsConnected D] : (CategoryTheory.Prod.fst C D).Final
CategoryTheory.initial_fst 📋 Mathlib.CategoryTheory.Limits.Final.Connected
{C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] [CategoryTheory.IsConnected D] : (CategoryTheory.Prod.fst C D).Initial
CategoryTheory.prod.functorProdToProdFunctorAssociator_hom_app 📋 Mathlib.CategoryTheory.Products.Associator
(C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] (E : Type u₃) [CategoryTheory.Category.{v₃, u₃} E] (A : Type u₄) [CategoryTheory.Category.{v₄, u₄} A] (X : CategoryTheory.Functor A ((C × D) × E)) : (CategoryTheory.prod.functorProdToProdFunctorAssociator C D E A).hom.app X = CategoryTheory.Prod.mkHom (CategoryTheory.CategoryStruct.id ((X.comp (CategoryTheory.prod.associator C D E)).comp (CategoryTheory.Prod.fst C (D × E)))) (CategoryTheory.Prod.mkHom (CategoryTheory.CategoryStruct.id (((X.comp (CategoryTheory.prod.associator C D E)).comp (CategoryTheory.Prod.snd C (D × E))).comp (CategoryTheory.Prod.fst D E))) (CategoryTheory.CategoryStruct.id (((X.comp (CategoryTheory.prod.associator C D E)).comp (CategoryTheory.Prod.snd C (D × E))).comp (CategoryTheory.Prod.snd D E))))
CategoryTheory.prod.functorProdToProdFunctorAssociator_inv_app 📋 Mathlib.CategoryTheory.Products.Associator
(C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] (E : Type u₃) [CategoryTheory.Category.{v₃, u₃} E] (A : Type u₄) [CategoryTheory.Category.{v₄, u₄} A] (X : CategoryTheory.Functor A ((C × D) × E)) : (CategoryTheory.prod.functorProdToProdFunctorAssociator C D E A).inv.app X = CategoryTheory.Prod.mkHom (CategoryTheory.CategoryStruct.id ((X.comp (CategoryTheory.prod.associator C D E)).comp (CategoryTheory.Prod.fst C (D × E)))) (CategoryTheory.Prod.mkHom (CategoryTheory.CategoryStruct.id (((X.comp (CategoryTheory.prod.associator C D E)).comp (CategoryTheory.Prod.snd C (D × E))).comp (CategoryTheory.Prod.fst D E))) (CategoryTheory.CategoryStruct.id (((X.comp (CategoryTheory.prod.associator C D E)).comp (CategoryTheory.Prod.snd C (D × E))).comp (CategoryTheory.Prod.snd D E))))
CategoryTheory.IsFiltered.final_fst 📋 Mathlib.CategoryTheory.Filtered.Final
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.IsFiltered D] : (CategoryTheory.Prod.fst C D).Final
CategoryTheory.IsFiltered.initial_fst 📋 Mathlib.CategoryTheory.Filtered.Final
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.IsCofiltered D] : (CategoryTheory.Prod.fst C D).Initial
CategoryTheory.Monoidal.rightUnitor_hom 📋 Mathlib.CategoryTheory.Monoidal.Cartesian.Cat
(C : CategoryTheory.Cat) : (CategoryTheory.MonoidalCategoryStruct.rightUnitor C).hom = (CategoryTheory.Prod.fst ↑C ↑(CategoryTheory.MonoidalCategoryStruct.tensorUnit CategoryTheory.Cat)).toCatHom
CategoryTheory.Monoidal.whiskerLeft_fst 📋 Mathlib.CategoryTheory.Monoidal.Cartesian.Cat
(X : CategoryTheory.Cat) {A B : CategoryTheory.Cat} (f : A ⟶ B) : (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X f).toFunctor.comp (CategoryTheory.Prod.fst ↑X ↑B) = CategoryTheory.Prod.fst ↑X ↑A
CategoryTheory.Monoidal.whiskerRight_fst 📋 Mathlib.CategoryTheory.Monoidal.Cartesian.Cat
{A B : CategoryTheory.Cat} (f : A ⟶ B) (X : CategoryTheory.Cat) : (CategoryTheory.MonoidalCategoryStruct.whiskerRight f X).toFunctor.comp (CategoryTheory.Prod.fst ↑B ↑X) = (CategoryTheory.Prod.fst ↑A ↑X).comp f.toFunctor
CategoryTheory.Monoidal.associator_hom 📋 Mathlib.CategoryTheory.Monoidal.Cartesian.Cat
(X Y Z : CategoryTheory.Cat) : (CategoryTheory.MonoidalCategoryStruct.associator X Y Z).hom = (((CategoryTheory.Prod.fst (↑X × ↑Y) ↑Z).comp (CategoryTheory.Prod.fst ↑X ↑Y)).prod' (((CategoryTheory.Prod.fst (↑X × ↑Y) ↑Z).comp (CategoryTheory.Prod.snd ↑X ↑Y)).prod' (CategoryTheory.Prod.snd (↑X × ↑Y) ↑Z))).toCatHom
CategoryTheory.Monoidal.associator_inv 📋 Mathlib.CategoryTheory.Monoidal.Cartesian.Cat
(X Y Z : CategoryTheory.Cat) : (CategoryTheory.MonoidalCategoryStruct.associator X Y Z).inv = (((CategoryTheory.Prod.fst (↑X) (↑Y × ↑Z)).prod' ((CategoryTheory.Prod.snd (↑X) (↑Y × ↑Z)).comp (CategoryTheory.Prod.fst ↑Y ↑Z))).prod' ((CategoryTheory.Prod.snd (↑X) (↑Y × ↑Z)).comp (CategoryTheory.Prod.snd ↑Y ↑Z))).toCatHom
SSet.Truncated.HomotopyCategory.BinaryProduct.inverse_comp_mapHomotopyCategory_fst 📋 Mathlib.AlgebraicTopology.SimplicialSet.HoFunctorMonoidal
(X Y : SSet.Truncated 2) : (SSet.Truncated.HomotopyCategory.BinaryProduct.inverse X Y).comp (SSet.Truncated.mapHomotopyCategory (CategoryTheory.SemiCartesianMonoidalCategory.fst X Y)) = CategoryTheory.Prod.fst X.HomotopyCategory Y.HomotopyCategory
SSet.Truncated.HomotopyCategory.BinaryProduct.inverseCompMapHomotopyCategoryFstIso 📋 Mathlib.AlgebraicTopology.SimplicialSet.HoFunctorMonoidal
(X Y : SSet.Truncated 2) : (SSet.Truncated.HomotopyCategory.BinaryProduct.inverse X Y).comp (SSet.Truncated.mapHomotopyCategory (CategoryTheory.SemiCartesianMonoidalCategory.fst X Y)) ≅ CategoryTheory.Prod.fst X.HomotopyCategory Y.HomotopyCategory
SSet.Truncated.HomotopyCategory.BinaryProduct.right_unitality 📋 Mathlib.AlgebraicTopology.SimplicialSet.HoFunctorMonoidal
(X Y : SSet.Truncated 2) [Unique (Y.obj (Opposite.op { obj := { len := 0 }, property := SSet.Truncated.Edge.tensor._proof_1 }))] [Subsingleton (Y.obj (Opposite.op { obj := { len := 1 }, property := SSet.Truncated.HomotopyCategory.isoTerminal._proof_1 }))] : CategoryTheory.Prod.fst X.HomotopyCategory ↑CategoryTheory.Cat.chosenTerminal = ((CategoryTheory.Functor.id X.HomotopyCategory).prod (SSet.Truncated.HomotopyCategory.isoTerminal Y).inv.toFunctor).comp ((SSet.Truncated.HomotopyCategory.BinaryProduct.inverse X Y).comp (SSet.Truncated.mapHomotopyCategory (CategoryTheory.SemiCartesianMonoidalCategory.fst X Y)))
CategoryTheory.Join.mkFunctor 📋 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 : CategoryTheory.Functor C E) (G : CategoryTheory.Functor D E) (α : (CategoryTheory.Prod.fst C D).comp F ⟶ (CategoryTheory.Prod.snd C D).comp G) : CategoryTheory.Functor (CategoryTheory.Join C D) E
CategoryTheory.Join.mkFunctor_obj_left 📋 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 : CategoryTheory.Functor C E) (G : CategoryTheory.Functor D E) (α : (CategoryTheory.Prod.fst C D).comp F ⟶ (CategoryTheory.Prod.snd C D).comp G) (c : C) : (CategoryTheory.Join.mkFunctor F G α).obj (CategoryTheory.Join.left c) = F.obj c
CategoryTheory.Join.mkFunctor_obj_right 📋 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 : CategoryTheory.Functor C E) (G : CategoryTheory.Functor D E) (α : (CategoryTheory.Prod.fst C D).comp F ⟶ (CategoryTheory.Prod.snd C D).comp G) (d : D) : (CategoryTheory.Join.mkFunctor F G α).obj (CategoryTheory.Join.right d) = G.obj d
CategoryTheory.Join.mkFunctorLeft 📋 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 : CategoryTheory.Functor C E) (G : CategoryTheory.Functor D E) (α : (CategoryTheory.Prod.fst C D).comp F ⟶ (CategoryTheory.Prod.snd C D).comp G) : (CategoryTheory.Join.inclLeft C D).comp (CategoryTheory.Join.mkFunctor F G α) ≅ F
CategoryTheory.Join.mkFunctorRight 📋 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 : CategoryTheory.Functor C E) (G : CategoryTheory.Functor D E) (α : (CategoryTheory.Prod.fst C D).comp F ⟶ (CategoryTheory.Prod.snd C D).comp G) : (CategoryTheory.Join.inclRight C D).comp (CategoryTheory.Join.mkFunctor F G α) ≅ G
CategoryTheory.Join.edgeTransform 📋 Mathlib.CategoryTheory.Join.Basic
(C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] : (CategoryTheory.Prod.fst C D).comp (CategoryTheory.Join.inclLeft C D) ⟶ (CategoryTheory.Prod.snd C D).comp (CategoryTheory.Join.inclRight C D)
CategoryTheory.Join.isoMkFunctor 📋 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 : CategoryTheory.Functor (CategoryTheory.Join C D) E) : F ≅ CategoryTheory.Join.mkFunctor ((CategoryTheory.Join.inclLeft C D).comp F) ((CategoryTheory.Join.inclRight C D).comp F) (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) F)
CategoryTheory.Join.edgeTransform_app 📋 Mathlib.CategoryTheory.Join.Basic
(C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] (x✝ : C × D) : (CategoryTheory.Join.edgeTransform C D).app x✝ = CategoryTheory.Join.edge x✝.1 x✝.2
CategoryTheory.Join.mkFunctor_map_edge 📋 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 : CategoryTheory.Functor C E) (G : CategoryTheory.Functor D E) (α : (CategoryTheory.Prod.fst C D).comp F ⟶ (CategoryTheory.Prod.snd C D).comp G) (c : C) (d : D) : (CategoryTheory.Join.mkFunctor F G α).map (CategoryTheory.Join.edge c d) = α.app (c, d)
CategoryTheory.Join.mkFunctor_map_inclLeft 📋 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 : CategoryTheory.Functor C E) (G : CategoryTheory.Functor D E) (α : (CategoryTheory.Prod.fst C D).comp F ⟶ (CategoryTheory.Prod.snd C D).comp G) {c c' : C} (f : c ⟶ c') : (CategoryTheory.Join.mkFunctor F G α).map ((CategoryTheory.Join.inclLeft C D).map f) = F.map f
CategoryTheory.Join.mkFunctor_map_inclRight 📋 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 : CategoryTheory.Functor C E) (G : CategoryTheory.Functor D E) (α : (CategoryTheory.Prod.fst C D).comp F ⟶ (CategoryTheory.Prod.snd C D).comp G) {d d' : D} (f : d ⟶ d') : (CategoryTheory.Join.mkFunctor F G α).map ((CategoryTheory.Join.inclRight C D).map f) = G.map f
CategoryTheory.Join.mkFunctorLeft_hom_app 📋 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 : CategoryTheory.Functor C E) (G : CategoryTheory.Functor D E) (α : (CategoryTheory.Prod.fst C D).comp F ⟶ (CategoryTheory.Prod.snd C D).comp G) (X : C) : (CategoryTheory.Join.mkFunctorLeft F G α).hom.app X = CategoryTheory.CategoryStruct.id (F.obj X)
CategoryTheory.Join.mkFunctorLeft_inv_app 📋 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 : CategoryTheory.Functor C E) (G : CategoryTheory.Functor D E) (α : (CategoryTheory.Prod.fst C D).comp F ⟶ (CategoryTheory.Prod.snd C D).comp G) (X : C) : (CategoryTheory.Join.mkFunctorLeft F G α).inv.app X = CategoryTheory.CategoryStruct.id (F.obj X)
CategoryTheory.Join.mkFunctorRight_hom_app 📋 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 : CategoryTheory.Functor C E) (G : CategoryTheory.Functor D E) (α : (CategoryTheory.Prod.fst C D).comp F ⟶ (CategoryTheory.Prod.snd C D).comp G) (X : D) : (CategoryTheory.Join.mkFunctorRight F G α).hom.app X = CategoryTheory.CategoryStruct.id (G.obj X)
CategoryTheory.Join.mkFunctorRight_inv_app 📋 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 : CategoryTheory.Functor C E) (G : CategoryTheory.Functor D E) (α : (CategoryTheory.Prod.fst C D).comp F ⟶ (CategoryTheory.Prod.snd C D).comp G) (X : D) : (CategoryTheory.Join.mkFunctorRight F G α).inv.app X = CategoryTheory.CategoryStruct.id (G.obj X)
CategoryTheory.Join.mkFunctor_edgeTransform 📋 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 : CategoryTheory.Functor C E) (G : CategoryTheory.Functor D E) (α : (CategoryTheory.Prod.fst C D).comp F ⟶ (CategoryTheory.Prod.snd C D).comp G) : CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) (CategoryTheory.Join.mkFunctor F G α) = α
CategoryTheory.Join.isoMkFunctor_hom_app 📋 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 : CategoryTheory.Functor (CategoryTheory.Join C D) E) (x : CategoryTheory.Join C D) : (CategoryTheory.Join.isoMkFunctor F).hom.app x = match x with | CategoryTheory.Join.left x => CategoryTheory.CategoryStruct.id (F.obj (CategoryTheory.Join.left x)) | CategoryTheory.Join.right x => CategoryTheory.CategoryStruct.id (F.obj (CategoryTheory.Join.right x))
CategoryTheory.Join.isoMkFunctor_inv_app 📋 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 : CategoryTheory.Functor (CategoryTheory.Join C D) E) (x : CategoryTheory.Join C D) : (CategoryTheory.Join.isoMkFunctor F).inv.app x = match x with | CategoryTheory.Join.left x => CategoryTheory.CategoryStruct.id (F.obj (CategoryTheory.Join.left x)) | CategoryTheory.Join.right x => CategoryTheory.CategoryStruct.id (F.obj (CategoryTheory.Join.right x))
CategoryTheory.Join.mkNatTrans 📋 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 F' : CategoryTheory.Functor (CategoryTheory.Join C D) E} (αₗ : (CategoryTheory.Join.inclLeft C D).comp F ⟶ (CategoryTheory.Join.inclLeft C D).comp F') (αᵣ : (CategoryTheory.Join.inclRight C D).comp F ⟶ (CategoryTheory.Join.inclRight C D).comp F') (h : CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) F) ((CategoryTheory.Prod.snd C D).whiskerLeft αᵣ) = CategoryTheory.CategoryStruct.comp ((CategoryTheory.Prod.fst C D).whiskerLeft αₗ) (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) F') := by cat_disch) : F ⟶ F'
CategoryTheory.Join.whiskerLeft_inclLeft_mkNatTrans 📋 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 F' : CategoryTheory.Functor (CategoryTheory.Join C D) E} (αₗ : (CategoryTheory.Join.inclLeft C D).comp F ⟶ (CategoryTheory.Join.inclLeft C D).comp F') (αᵣ : (CategoryTheory.Join.inclRight C D).comp F ⟶ (CategoryTheory.Join.inclRight C D).comp F') (h : CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) F) ((CategoryTheory.Prod.snd C D).whiskerLeft αᵣ) = CategoryTheory.CategoryStruct.comp ((CategoryTheory.Prod.fst C D).whiskerLeft αₗ) (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) F') := by cat_disch) : (CategoryTheory.Join.inclLeft C D).whiskerLeft (CategoryTheory.Join.mkNatTrans αₗ αᵣ h) = αₗ
CategoryTheory.Join.whiskerLeft_inclRight_mkNatTrans 📋 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 F' : CategoryTheory.Functor (CategoryTheory.Join C D) E} (αₗ : (CategoryTheory.Join.inclLeft C D).comp F ⟶ (CategoryTheory.Join.inclLeft C D).comp F') (αᵣ : (CategoryTheory.Join.inclRight C D).comp F ⟶ (CategoryTheory.Join.inclRight C D).comp F') (h : CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) F) ((CategoryTheory.Prod.snd C D).whiskerLeft αᵣ) = CategoryTheory.CategoryStruct.comp ((CategoryTheory.Prod.fst C D).whiskerLeft αₗ) (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) F') := by cat_disch) : (CategoryTheory.Join.inclRight C D).whiskerLeft (CategoryTheory.Join.mkNatTrans αₗ αᵣ h) = αᵣ
CategoryTheory.Join.mkNatTrans_app_left 📋 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 F' : CategoryTheory.Functor (CategoryTheory.Join C D) E} (αₗ : (CategoryTheory.Join.inclLeft C D).comp F ⟶ (CategoryTheory.Join.inclLeft C D).comp F') (αᵣ : (CategoryTheory.Join.inclRight C D).comp F ⟶ (CategoryTheory.Join.inclRight C D).comp F') (h : CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) F) ((CategoryTheory.Prod.snd C D).whiskerLeft αᵣ) = CategoryTheory.CategoryStruct.comp ((CategoryTheory.Prod.fst C D).whiskerLeft αₗ) (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) F') := by cat_disch) (c : C) : (CategoryTheory.Join.mkNatTrans αₗ αᵣ h).app (CategoryTheory.Join.left c) = αₗ.app c
CategoryTheory.Join.mkNatTrans_app_right 📋 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 F' : CategoryTheory.Functor (CategoryTheory.Join C D) E} (αₗ : (CategoryTheory.Join.inclLeft C D).comp F ⟶ (CategoryTheory.Join.inclLeft C D).comp F') (αᵣ : (CategoryTheory.Join.inclRight C D).comp F ⟶ (CategoryTheory.Join.inclRight C D).comp F') (h : CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) F) ((CategoryTheory.Prod.snd C D).whiskerLeft αᵣ) = CategoryTheory.CategoryStruct.comp ((CategoryTheory.Prod.fst C D).whiskerLeft αₗ) (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) F') := by cat_disch) (d : D) : (CategoryTheory.Join.mkNatTrans αₗ αᵣ h).app (CategoryTheory.Join.right d) = αᵣ.app d
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.mapPairLeft_hom_app 📋 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] {E' : Type u₄} [CategoryTheory.Category.{v₄, u₄} E'] (Fₗ : CategoryTheory.Functor C E) (Fᵣ : CategoryTheory.Functor D E') (X : C) : (CategoryTheory.Join.mapPairLeft Fₗ Fᵣ).hom.app X = CategoryTheory.CategoryStruct.id (CategoryTheory.Join.left (Fₗ.obj X))
CategoryTheory.Join.mapPairLeft_inv_app 📋 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] {E' : Type u₄} [CategoryTheory.Category.{v₄, u₄} E'] (Fₗ : CategoryTheory.Functor C E) (Fᵣ : CategoryTheory.Functor D E') (X : C) : (CategoryTheory.Join.mapPairLeft Fₗ Fᵣ).inv.app X = CategoryTheory.CategoryStruct.id (CategoryTheory.Join.left (Fₗ.obj X))
CategoryTheory.Join.mapPairRight_hom_app 📋 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] {E' : Type u₄} [CategoryTheory.Category.{v₄, u₄} E'] (Fₗ : CategoryTheory.Functor C E) (Fᵣ : CategoryTheory.Functor D E') (X : D) : (CategoryTheory.Join.mapPairRight Fₗ Fᵣ).hom.app X = CategoryTheory.CategoryStruct.id (CategoryTheory.Join.right (Fᵣ.obj X))
CategoryTheory.Join.mapPairRight_inv_app 📋 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] {E' : Type u₄} [CategoryTheory.Category.{v₄, u₄} E'] (Fₗ : CategoryTheory.Functor C E) (Fᵣ : CategoryTheory.Functor D E') (X : D) : (CategoryTheory.Join.mapPairRight Fₗ Fᵣ).inv.app X = CategoryTheory.CategoryStruct.id (CategoryTheory.Join.right (Fᵣ.obj X))
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.mkNatTransComp 📋 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 F' F'' : CategoryTheory.Functor (CategoryTheory.Join C D) E} (αₗ : (CategoryTheory.Join.inclLeft C D).comp F ⟶ (CategoryTheory.Join.inclLeft C D).comp F') (αᵣ : (CategoryTheory.Join.inclRight C D).comp F ⟶ (CategoryTheory.Join.inclRight C D).comp F') (βₗ : (CategoryTheory.Join.inclLeft C D).comp F' ⟶ (CategoryTheory.Join.inclLeft C D).comp F'') (βᵣ : (CategoryTheory.Join.inclRight C D).comp F' ⟶ (CategoryTheory.Join.inclRight C D).comp F'') (h : CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) F) ((CategoryTheory.Prod.snd C D).whiskerLeft αᵣ) = CategoryTheory.CategoryStruct.comp ((CategoryTheory.Prod.fst C D).whiskerLeft αₗ) (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) F') := by cat_disch) (h' : CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) F') ((CategoryTheory.Prod.snd C D).whiskerLeft βᵣ) = CategoryTheory.CategoryStruct.comp ((CategoryTheory.Prod.fst C D).whiskerLeft βₗ) (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) F'') := by cat_disch) : CategoryTheory.Join.mkNatTrans (CategoryTheory.CategoryStruct.comp αₗ βₗ) (CategoryTheory.CategoryStruct.comp αᵣ βᵣ) ⋯ = CategoryTheory.CategoryStruct.comp (CategoryTheory.Join.mkNatTrans αₗ αᵣ h) (CategoryTheory.Join.mkNatTrans βₗ βᵣ h')
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.Sum.functorEquivFunctorCompFstIso 📋 Mathlib.CategoryTheory.Sums.Products
{A : Type u_1} [CategoryTheory.Category.{v_1, u_1} A] {A' : Type u_2} [CategoryTheory.Category.{v_2, u_2} A'] {B : Type u} [CategoryTheory.Category.{v, u} B] : (CategoryTheory.Sum.functorEquiv A A' B).functor.comp (CategoryTheory.Prod.fst (CategoryTheory.Functor A B) (CategoryTheory.Functor A' B)) ≅ (CategoryTheory.Functor.whiskeringLeft A (A ⊕ A') B).obj (CategoryTheory.Sum.inl_ A A')
CategoryTheory.Sum.functorEquivInverseCompWhiskeringLeftInlIso 📋 Mathlib.CategoryTheory.Sums.Products
{A : Type u_1} [CategoryTheory.Category.{v_1, u_1} A] {A' : Type u_2} [CategoryTheory.Category.{v_2, u_2} A'] {B : Type u} [CategoryTheory.Category.{v, u} B] : (CategoryTheory.Sum.functorEquiv A A' B).inverse.comp ((CategoryTheory.Functor.whiskeringLeft A (A ⊕ A') B).obj (CategoryTheory.Sum.inl_ A A')) ≅ CategoryTheory.Prod.fst (CategoryTheory.Functor A B) (CategoryTheory.Functor A' B)
CategoryTheory.Sum.functorEquivInverseCompWhiskeringLeftInlIso_hom_app_app 📋 Mathlib.CategoryTheory.Sums.Products
{A : Type u_1} [CategoryTheory.Category.{v_1, u_1} A] {A' : Type u_2} [CategoryTheory.Category.{v_2, u_2} A'] {B : Type u} [CategoryTheory.Category.{v, u} B] (X : CategoryTheory.Functor A B × CategoryTheory.Functor A' B) (X✝ : A) : (CategoryTheory.Sum.functorEquivInverseCompWhiskeringLeftInlIso.hom.app X).app X✝ = CategoryTheory.CategoryStruct.id (X.1.obj X✝)
CategoryTheory.Sum.functorEquivInverseCompWhiskeringLeftInlIso_inv_app_app 📋 Mathlib.CategoryTheory.Sums.Products
{A : Type u_1} [CategoryTheory.Category.{v_1, u_1} A] {A' : Type u_2} [CategoryTheory.Category.{v_2, u_2} A'] {B : Type u} [CategoryTheory.Category.{v, u} B] (X : CategoryTheory.Functor A B × CategoryTheory.Functor A' B) (X✝ : A) : (CategoryTheory.Sum.functorEquivInverseCompWhiskeringLeftInlIso.inv.app X).app X✝ = CategoryTheory.CategoryStruct.id (X.1.obj X✝)
CategoryTheory.Sum.functorEquivFunctorCompFstIso_hom_app_app 📋 Mathlib.CategoryTheory.Sums.Products
{A : Type u_1} [CategoryTheory.Category.{v_1, u_1} A] {A' : Type u_2} [CategoryTheory.Category.{v_2, u_2} A'] {B : Type u} [CategoryTheory.Category.{v, u} B] (X : CategoryTheory.Functor (A ⊕ A') B) (X✝ : A) : (CategoryTheory.Sum.functorEquivFunctorCompFstIso.hom.app X).app X✝ = CategoryTheory.CategoryStruct.id (X.obj (Sum.inl X✝))
CategoryTheory.Sum.functorEquivFunctorCompFstIso_inv_app_app 📋 Mathlib.CategoryTheory.Sums.Products
{A : Type u_1} [CategoryTheory.Category.{v_1, u_1} A] {A' : Type u_2} [CategoryTheory.Category.{v_2, u_2} A'] {B : Type u} [CategoryTheory.Category.{v, u} B] (X : CategoryTheory.Functor (A ⊕ A') B) (X✝ : A) : (CategoryTheory.Sum.functorEquivFunctorCompFstIso.inv.app X).app X✝ = CategoryTheory.CategoryStruct.id (X.obj (Sum.inl X✝))

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 ce5dd8c