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Result

Found 43 declarations mentioning CategoryTheory.Prod.swap.

• CategoryTheory.Prod.swap 📋 Mathlib.CategoryTheory.Products.Basic
  (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] : CategoryTheory.Functor (C × D) (D × C)
• CategoryTheory.Prod.swapIsEquivalence 📋 Mathlib.CategoryTheory.Products.Basic
  (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] : (CategoryTheory.Prod.swap C D).IsEquivalence
• CategoryTheory.Prod.swap_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.swap C D).obj X = (X.2, X.1)
• CategoryTheory.Prod.braiding_functor 📋 Mathlib.CategoryTheory.Products.Basic
  (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] : (CategoryTheory.Prod.braiding C D).functor = CategoryTheory.Prod.swap C D
• CategoryTheory.Prod.braiding_inverse 📋 Mathlib.CategoryTheory.Products.Basic
  (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] : (CategoryTheory.Prod.braiding C D).inverse = CategoryTheory.Prod.swap D C
• CategoryTheory.Prod.symmetry 📋 Mathlib.CategoryTheory.Products.Basic
  (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] : (CategoryTheory.Prod.swap C D).comp (CategoryTheory.Prod.swap D C) ≅ CategoryTheory.Functor.id (C × D)
• CategoryTheory.Prod.swap_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.swap C D).map f = CategoryTheory.Prod.mkHom f.2 f.1
• CategoryTheory.Prod.symmetry_hom_app 📋 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.symmetry C D).hom.app X = CategoryTheory.CategoryStruct.id X
• CategoryTheory.Prod.symmetry_inv_app 📋 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.symmetry C D).inv.app X = CategoryTheory.CategoryStruct.id X
• CategoryTheory.Prod.braiding_counitIso 📋 Mathlib.CategoryTheory.Products.Basic
  (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] : (CategoryTheory.Prod.braiding C D).counitIso = CategoryTheory.Iso.refl ((CategoryTheory.Prod.swap D C).comp (CategoryTheory.Prod.swap C D))
• CategoryTheory.Functor.uncurryObjFlip 📋 Mathlib.CategoryTheory.Functor.Currying
  {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) : CategoryTheory.Functor.uncurry.obj F.flip ≅ (CategoryTheory.Prod.swap D C).comp (CategoryTheory.Functor.uncurry.obj F)
• CategoryTheory.Functor.flipIsoCurrySwapUncurry 📋 Mathlib.CategoryTheory.Functor.Currying
  {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) : F.flip ≅ CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap D C).comp (CategoryTheory.Functor.uncurry.obj F))
• CategoryTheory.Functor.uncurryObjFlip_hom_app 📋 Mathlib.CategoryTheory.Functor.Currying
  {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : D × C) : F.uncurryObjFlip.hom.app X = CategoryTheory.CategoryStruct.id ((F.obj X.2).obj X.1)
• CategoryTheory.Functor.uncurryObjFlip_inv_app 📋 Mathlib.CategoryTheory.Functor.Currying
  {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : D × C) : F.uncurryObjFlip.inv.app X = CategoryTheory.CategoryStruct.id ((F.obj X.2).obj X.1)
• CategoryTheory.Functor.flipIsoCurrySwapUncurry_hom_app_app 📋 Mathlib.CategoryTheory.Functor.Currying
  {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : D) (X✝ : C) : (F.flipIsoCurrySwapUncurry.hom.app X).app X✝ = CategoryTheory.CategoryStruct.id ((F.obj X✝).obj X)
• CategoryTheory.Functor.flipIsoCurrySwapUncurry_inv_app_app 📋 Mathlib.CategoryTheory.Functor.Currying
  {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : D) (X✝ : C) : (F.flipIsoCurrySwapUncurry.inv.app X).app X✝ = CategoryTheory.CategoryStruct.id ((F.obj X✝).obj X)
• CategoryTheory.Limits.colimitIsoSwapCompColim 📋 Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [CategoryTheory.Category.{v₂, u₂} K] [CategoryTheory.Limits.HasColimitsOfShape J C] (G : CategoryTheory.Functor J (CategoryTheory.Functor K C)) : CategoryTheory.Limits.colimit G ≅ (CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp (CategoryTheory.Functor.uncurry.obj G))).comp CategoryTheory.Limits.colim
• CategoryTheory.Limits.limitIsoSwapCompLim 📋 Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [CategoryTheory.Category.{v₂, u₂} K] [CategoryTheory.Limits.HasLimitsOfShape J C] (G : CategoryTheory.Functor J (CategoryTheory.Functor K C)) : CategoryTheory.Limits.limit G ≅ (CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp (CategoryTheory.Functor.uncurry.obj G))).comp CategoryTheory.Limits.lim
• CategoryTheory.Limits.colimitIsoSwapCompColim_hom_app 📋 Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [CategoryTheory.Category.{v₂, u₂} K] [CategoryTheory.Limits.HasColimitsOfShape J C] (G : CategoryTheory.Functor J (CategoryTheory.Functor K C)) (X : K) : (CategoryTheory.Limits.colimitIsoSwapCompColim G).hom.app X = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation G X).hom (CategoryTheory.Limits.colimMap (G.flipIsoCurrySwapUncurry.hom.app X))
• CategoryTheory.Limits.colimitIsoSwapCompColim_inv_app 📋 Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [CategoryTheory.Category.{v₂, u₂} K] [CategoryTheory.Limits.HasColimitsOfShape J C] (G : CategoryTheory.Functor J (CategoryTheory.Functor K C)) (X : K) : (CategoryTheory.Limits.colimitIsoSwapCompColim G).inv.app X = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimMap (G.flipIsoCurrySwapUncurry.inv.app X)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation G X).inv
• CategoryTheory.Limits.limitIsoSwapCompLim_hom_app 📋 Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [CategoryTheory.Category.{v₂, u₂} K] [CategoryTheory.Limits.HasLimitsOfShape J C] (G : CategoryTheory.Functor J (CategoryTheory.Functor K C)) (X : K) : (CategoryTheory.Limits.limitIsoSwapCompLim G).hom.app X = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation G X).hom (CategoryTheory.Limits.limMap (G.flipIsoCurrySwapUncurry.hom.app X))
• CategoryTheory.Limits.limitIsoSwapCompLim_inv_app 📋 Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [CategoryTheory.Category.{v₂, u₂} K] [CategoryTheory.Limits.HasLimitsOfShape J C] (G : CategoryTheory.Functor J (CategoryTheory.Functor K C)) (X : K) : (CategoryTheory.Limits.limitIsoSwapCompLim G).inv.app X = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limMap (G.flipIsoCurrySwapUncurry.inv.app X)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation G X).inv
• CategoryTheory.MonoidalOpposite.tensorIso 📋 Mathlib.CategoryTheory.Monoidal.Opposite
  (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] : CategoryTheory.MonoidalCategory.tensor Cᴹᵒᵖ ≅ ((CategoryTheory.unmopFunctor C).prod (CategoryTheory.unmopFunctor C)).comp ((CategoryTheory.Prod.swap C C).comp ((CategoryTheory.MonoidalCategory.tensor C).comp (CategoryTheory.mopFunctor C)))
• CategoryTheory.MonoidalOpposite.tensorIso_hom_app_unmop 📋 Mathlib.CategoryTheory.Monoidal.Opposite
  (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : Cᴹᵒᵖ × Cᴹᵒᵖ) : ((CategoryTheory.MonoidalOpposite.tensorIso C).hom.app X).unmop = CategoryTheory.CategoryStruct.id (CategoryTheory.MonoidalCategoryStruct.tensorObj X.2.unmop X.1.unmop)
• CategoryTheory.MonoidalOpposite.tensorIso_inv_app_unmop 📋 Mathlib.CategoryTheory.Monoidal.Opposite
  (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : Cᴹᵒᵖ × Cᴹᵒᵖ) : ((CategoryTheory.MonoidalOpposite.tensorIso C).inv.app X).unmop = CategoryTheory.CategoryStruct.id (CategoryTheory.MonoidalCategoryStruct.tensorObj X.2.unmop X.1.unmop)
• CategoryTheory.MonoidalCategory.externalProductSwap 📋 Mathlib.CategoryTheory.Monoidal.ExternalProduct.Basic
  (J₁ : Type u₁) (J₂ : Type u₂) (C : Type u₃) [CategoryTheory.Category.{v₁, u₁} J₁] [CategoryTheory.Category.{v₂, u₂} J₂] [CategoryTheory.Category.{v₃, u₃} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] : (CategoryTheory.MonoidalCategory.externalProductBifunctor J₁ J₂ C).comp ((CategoryTheory.Functor.whiskeringLeft (J₂ × J₁) (J₁ × J₂) C).obj (CategoryTheory.Prod.swap J₂ J₁)) ≅ (CategoryTheory.Prod.swap (CategoryTheory.Functor J₁ C) (CategoryTheory.Functor J₂ C)).comp (CategoryTheory.MonoidalCategory.externalProductBifunctor J₂ J₁ C)
• CategoryTheory.MonoidalCategory.externalProductSwap_hom_app_app 📋 Mathlib.CategoryTheory.Monoidal.ExternalProduct.Basic
  (J₁ : Type u₁) (J₂ : Type u₂) (C : Type u₃) [CategoryTheory.Category.{v₁, u₁} J₁] [CategoryTheory.Category.{v₂, u₂} J₂] [CategoryTheory.Category.{v₃, u₃} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : CategoryTheory.Functor J₁ C × CategoryTheory.Functor J₂ C) (X✝ : J₂ × J₁) : ((CategoryTheory.MonoidalCategory.externalProductSwap J₁ J₂ C).hom.app X).app X✝ = (β_ (X.1.obj X✝.2) (X.2.obj X✝.1)).hom
• CategoryTheory.MonoidalCategory.externalProductSwap_inv_app_app 📋 Mathlib.CategoryTheory.Monoidal.ExternalProduct.Basic
  (J₁ : Type u₁) (J₂ : Type u₂) (C : Type u₃) [CategoryTheory.Category.{v₁, u₁} J₁] [CategoryTheory.Category.{v₂, u₂} J₂] [CategoryTheory.Category.{v₃, u₃} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : CategoryTheory.Functor J₁ C × CategoryTheory.Functor J₂ C) (X✝ : J₂ × J₁) : ((CategoryTheory.MonoidalCategory.externalProductSwap J₁ J₂ C).inv.app X).app X✝ = (β_ (X.1.obj X✝.2) (X.2.obj X✝.1)).inv
• CategoryTheory.Limits.colimitCurrySwapCompColimIsoColimitCurryCompColim 📋 Mathlib.CategoryTheory.Limits.Fubini
  {J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] (G : CategoryTheory.Functor (J × K) C) [CategoryTheory.Limits.HasColimitsOfShape K C] [CategoryTheory.Limits.HasColimitsOfShape J C] [CategoryTheory.Limits.HasColimit ((CategoryTheory.Functor.curry.obj G).comp CategoryTheory.Limits.colim)] : CategoryTheory.Limits.colimit ((CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp G)).comp CategoryTheory.Limits.colim) ≅ CategoryTheory.Limits.colimit ((CategoryTheory.Functor.curry.obj G).comp CategoryTheory.Limits.colim)
• CategoryTheory.Limits.limitCurrySwapCompLimIsoLimitCurryCompLim 📋 Mathlib.CategoryTheory.Limits.Fubini
  {J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] (G : CategoryTheory.Functor (J × K) C) [CategoryTheory.Limits.HasLimitsOfShape K C] [CategoryTheory.Limits.HasLimitsOfShape J C] [CategoryTheory.Limits.HasLimit ((CategoryTheory.Functor.curry.obj G).comp CategoryTheory.Limits.lim)] : CategoryTheory.Limits.limit ((CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp G)).comp CategoryTheory.Limits.lim) ≅ CategoryTheory.Limits.limit ((CategoryTheory.Functor.curry.obj G).comp CategoryTheory.Limits.lim)
• CategoryTheory.Limits.limitCurrySwapCompLimIsoLimitCurryCompLim_inv_π_π 📋 Mathlib.CategoryTheory.Limits.Fubini
  {J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] (G : CategoryTheory.Functor (J × K) C) [CategoryTheory.Limits.HasLimitsOfShape K C] [CategoryTheory.Limits.HasLimitsOfShape J C] [CategoryTheory.Limits.HasLimit ((CategoryTheory.Functor.curry.obj G).comp CategoryTheory.Limits.lim)] {j : J} {k : K} : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limitCurrySwapCompLimIsoLimitCurryCompLim G).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limit.π ((CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp G)).comp CategoryTheory.Limits.lim) k) (CategoryTheory.Limits.limit.π ((CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp G)).obj k) j)) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limit.π ((CategoryTheory.Functor.curry.obj G).comp CategoryTheory.Limits.lim) j) (CategoryTheory.Limits.limit.π ((CategoryTheory.Functor.curry.obj G).obj j) k)
• CategoryTheory.Limits.limitCurrySwapCompLimIsoLimitCurryCompLim_hom_π_π 📋 Mathlib.CategoryTheory.Limits.Fubini
  {J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] (G : CategoryTheory.Functor (J × K) C) [CategoryTheory.Limits.HasLimitsOfShape K C] [CategoryTheory.Limits.HasLimitsOfShape J C] [CategoryTheory.Limits.HasLimit ((CategoryTheory.Functor.curry.obj G).comp CategoryTheory.Limits.lim)] {j : J} {k : K} : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limitCurrySwapCompLimIsoLimitCurryCompLim G).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limit.π ((CategoryTheory.Functor.curry.obj G).comp CategoryTheory.Limits.lim) j) (CategoryTheory.Limits.limit.π ((CategoryTheory.Functor.curry.obj G).obj j) k)) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limit.π ((CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp G)).comp CategoryTheory.Limits.lim) k) (CategoryTheory.Limits.limit.π ((CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp G)).obj k) j)
• CategoryTheory.Limits.colimitCurrySwapCompColimIsoColimitCurryCompColim_ι_ι_hom 📋 Mathlib.CategoryTheory.Limits.Fubini
  {J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] (G : CategoryTheory.Functor (J × K) C) [CategoryTheory.Limits.HasColimitsOfShape K C] [CategoryTheory.Limits.HasColimitsOfShape J C] [CategoryTheory.Limits.HasColimit ((CategoryTheory.Functor.curry.obj G).comp CategoryTheory.Limits.colim)] {j : J} {k : K} : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι ((CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp G)).obj k) j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι ((CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp G)).comp CategoryTheory.Limits.colim) k) (CategoryTheory.Limits.colimitCurrySwapCompColimIsoColimitCurryCompColim G).hom) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι ((CategoryTheory.Functor.curry.obj G).obj j) k) (CategoryTheory.Limits.colimit.ι ((CategoryTheory.Functor.curry.obj G).comp CategoryTheory.Limits.colim) j)
• CategoryTheory.Limits.colimitCurrySwapCompColimIsoColimitCurryCompColim_ι_ι_inv 📋 Mathlib.CategoryTheory.Limits.Fubini
  {J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] (G : CategoryTheory.Functor (J × K) C) [CategoryTheory.Limits.HasColimitsOfShape K C] [CategoryTheory.Limits.HasColimitsOfShape J C] [CategoryTheory.Limits.HasColimit ((CategoryTheory.Functor.curry.obj G).comp CategoryTheory.Limits.colim)] {j : J} {k : K} : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι ((CategoryTheory.Functor.curry.obj G).obj j) k) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι ((CategoryTheory.Functor.curry.obj G).comp CategoryTheory.Limits.colim) j) (CategoryTheory.Limits.colimitCurrySwapCompColimIsoColimitCurryCompColim G).inv) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι ((CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp G)).obj k) j) (CategoryTheory.Limits.colimit.ι ((CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp G)).comp CategoryTheory.Limits.colim) k)
• CategoryTheory.Limits.map_id_right_eq_curry_swap_map 📋 Mathlib.CategoryTheory.Limits.ColimitLimit
  {J : Type u₁} {K : Type u₂} [CategoryTheory.Category.{v₁, u₁} J] [CategoryTheory.Category.{v₂, u₂} K] {C : Type u} [CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Functor (J × K) C) {j j' : J} {f : j ⟶ j'} {k : K} : F.map (CategoryTheory.Prod.mkHom f (CategoryTheory.CategoryStruct.id k)) = ((CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp F)).obj k).map f
• CategoryTheory.Limits.colimitLimitToLimitColimit 📋 Mathlib.CategoryTheory.Limits.ColimitLimit
  {J : Type u₁} {K : Type u₂} [CategoryTheory.Category.{v₁, u₁} J] [CategoryTheory.Category.{v₂, u₂} K] {C : Type u} [CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Functor (J × K) C) [CategoryTheory.Limits.HasLimitsOfShape J C] [CategoryTheory.Limits.HasColimitsOfShape K C] : CategoryTheory.Limits.colimit ((CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp F)).comp CategoryTheory.Limits.lim) ⟶ CategoryTheory.Limits.limit ((CategoryTheory.Functor.curry.obj F).comp CategoryTheory.Limits.colim)
• CategoryTheory.Limits.ι_colimitLimitToLimitColimit_π 📋 Mathlib.CategoryTheory.Limits.ColimitLimit
  {J : Type u₁} {K : Type u₂} [CategoryTheory.Category.{v₁, u₁} J] [CategoryTheory.Category.{v₂, u₂} K] {C : Type u} [CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Functor (J × K) C) [CategoryTheory.Limits.HasLimitsOfShape J C] [CategoryTheory.Limits.HasColimitsOfShape K C] (j : J) (k : K) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι ((CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp F)).comp CategoryTheory.Limits.lim) k) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimitLimitToLimitColimit F) (CategoryTheory.Limits.limit.π ((CategoryTheory.Functor.curry.obj F).comp CategoryTheory.Limits.colim) j)) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limit.π ((CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp F)).obj k) j) (CategoryTheory.Limits.colimit.ι ((CategoryTheory.Functor.curry.obj F).obj j) k)
• CategoryTheory.Limits.ι_colimitLimitToLimitColimit_π_assoc 📋 Mathlib.CategoryTheory.Limits.ColimitLimit
  {J : Type u₁} {K : Type u₂} [CategoryTheory.Category.{v₁, u₁} J] [CategoryTheory.Category.{v₂, u₂} K] {C : Type u} [CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Functor (J × K) C) [CategoryTheory.Limits.HasLimitsOfShape J C] [CategoryTheory.Limits.HasColimitsOfShape K C] (j : J) (k : K) {Z : C} (h : CategoryTheory.Limits.colim.obj ((CategoryTheory.Functor.curry.obj F).obj j) ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι ((CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp F)).comp CategoryTheory.Limits.lim) k) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimitLimitToLimitColimit F) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limit.π ((CategoryTheory.Functor.curry.obj F).comp CategoryTheory.Limits.colim) j) h)) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limit.π ((CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp F)).obj k) j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι ((CategoryTheory.Functor.curry.obj F).obj j) k) h)
• CategoryTheory.Limits.colimitLimitToLimitColimitCone_hom 📋 Mathlib.CategoryTheory.Limits.ColimitLimit
  {J : Type u₁} {K : Type u₂} [CategoryTheory.Category.{v₁, u₁} J] [CategoryTheory.Category.{v₂, u₂} K] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasLimitsOfShape J C] [CategoryTheory.Limits.HasColimitsOfShape K C] (G : CategoryTheory.Functor J (CategoryTheory.Functor K C)) [CategoryTheory.Limits.HasLimit G] : (CategoryTheory.Limits.colimitLimitToLimitColimitCone G).hom = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colim.map (CategoryTheory.Limits.limitIsoSwapCompLim G).hom) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimitLimitToLimitColimit (CategoryTheory.Functor.uncurry.obj G)) (CategoryTheory.Limits.lim.map (CategoryTheory.Functor.whiskerRight (CategoryTheory.Functor.currying.unitIso.app G).inv CategoryTheory.Limits.colim)))
• CategoryTheory.Limits.ι_colimitLimitToLimitColimit_π_apply 📋 Mathlib.CategoryTheory.Limits.ColimitLimit
  {J : Type u₁} {K : Type u₂} [CategoryTheory.Category.{v₁, u₁} J] [CategoryTheory.Category.{v₂, u₂} K] {C : Type u} [CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Functor (J × K) C) [CategoryTheory.Limits.HasLimitsOfShape J C] [CategoryTheory.Limits.HasColimitsOfShape K C] (j : J) (k : K) {F✝ : C → C → Type uF} {carrier : C → Type w} {instFunLike : (X Y : C) → FunLike (F✝ X Y) (carrier X) (carrier Y)} [inst : CategoryTheory.ConcreteCategory C F✝] (x : carrier (((CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp F)).comp CategoryTheory.Limits.lim).obj k)) : (CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.limit.π ((CategoryTheory.Functor.curry.obj F).comp CategoryTheory.Limits.colim) j)) ((CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.colimitLimitToLimitColimit F)) ((CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.colimit.ι ((CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp F)).comp CategoryTheory.Limits.lim) k)) x)) = (CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.colimit.ι ((CategoryTheory.Functor.curry.obj F).obj j) k)) ((CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.limit.π ((CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp F)).obj k) j)) x)
• CategoryTheory.Limits.colimitLimitToLimitColimit_isIso 📋 Mathlib.CategoryTheory.Limits.FilteredColimitCommutesFiniteLimit
  {J : Type u₁} {K : Type u₂} [CategoryTheory.SmallCategory J] [CategoryTheory.Category.{v₂, u₂} K] [Small.{v, u₂} K] [CategoryTheory.FinCategory J] (F : CategoryTheory.Functor (J × K) (Type v)) [CategoryTheory.IsFiltered K] : CategoryTheory.IsIso (CategoryTheory.Limits.colimitLimitToLimitColimit F)
• CategoryTheory.Limits.colimitLimitToLimitColimit_injective 📋 Mathlib.CategoryTheory.Limits.FilteredColimitCommutesFiniteLimit
  {J : Type u₁} {K : Type u₂} [CategoryTheory.Category.{v₁, u₁} J] [CategoryTheory.Category.{v₂, u₂} K] [Small.{v, u₂} K] (F : CategoryTheory.Functor (J × K) (Type v)) [CategoryTheory.IsFiltered K] [Finite J] : Function.Injective ⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.colimitLimitToLimitColimit F))
• CategoryTheory.Limits.colimitLimitToLimitColimit_surjective 📋 Mathlib.CategoryTheory.Limits.FilteredColimitCommutesFiniteLimit
  {J : Type u₁} {K : Type u₂} [CategoryTheory.SmallCategory J] [CategoryTheory.Category.{v₂, u₂} K] [Small.{v, u₂} K] [CategoryTheory.FinCategory J] (F : CategoryTheory.Functor (J × K) (Type v)) [CategoryTheory.IsFiltered K] : Function.Surjective ⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.colimitLimitToLimitColimit F))

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:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. 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

4. 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.

5. 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