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Found 55 declarations mentioning CategoryTheory.Limits.coprod.desc.

• CategoryTheory.Limits.coprod.desc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y : C} [CategoryTheory.Limits.HasBinaryCoproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) : X ⨿ Y ⟶ W
• CategoryTheory.Limits.coprod.epi_desc_of_epi_left 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y : C} [CategoryTheory.Limits.HasBinaryCoproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) [CategoryTheory.Epi f] : CategoryTheory.Epi (CategoryTheory.Limits.coprod.desc f g)
• CategoryTheory.Limits.coprod.epi_desc_of_epi_right 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y : C} [CategoryTheory.Limits.HasBinaryCoproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) [CategoryTheory.Epi g] : CategoryTheory.Epi (CategoryTheory.Limits.coprod.desc f g)
• CategoryTheory.Limits.coprod.diag_comp 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : C} [CategoryTheory.Limits.HasBinaryCoproduct X X] (f : X ⟶ Y) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.codiag X) f = CategoryTheory.Limits.coprod.desc f f
• CategoryTheory.Limits.coprod.desc_inl_inr 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : C} [CategoryTheory.Limits.HasBinaryCoproduct X Y] : CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inl CategoryTheory.Limits.coprod.inr = CategoryTheory.CategoryStruct.id (X ⨿ Y)
• CategoryTheory.Limits.coprod.inl_desc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y : C} [CategoryTheory.Limits.HasBinaryCoproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.Limits.coprod.desc f g) = f
• CategoryTheory.Limits.coprod.inr_desc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y : C} [CategoryTheory.Limits.HasBinaryCoproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inr (CategoryTheory.Limits.coprod.desc f g) = g
• CategoryTheory.Limits.coprod.leftUnitor_hom 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasBinaryCoproducts C] [CategoryTheory.Limits.HasInitial C] (P : C) : (CategoryTheory.Limits.coprod.leftUnitor P).hom = CategoryTheory.Limits.coprod.desc (CategoryTheory.Limits.initial.to P) (CategoryTheory.CategoryStruct.id P)
• CategoryTheory.Limits.coprod.rightUnitor_hom 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasBinaryCoproducts C] [CategoryTheory.Limits.HasInitial C] (P : C) : (CategoryTheory.Limits.coprod.rightUnitor P).hom = CategoryTheory.Limits.coprod.desc (CategoryTheory.CategoryStruct.id P) (CategoryTheory.Limits.initial.to P)
• CategoryTheory.Over.coprodObj_obj 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasBinaryCoproducts C] {A : C} (a✝ g : CategoryTheory.Over A) : a✝.coprodObj.obj g = CategoryTheory.Over.mk (CategoryTheory.Limits.coprod.desc a✝.hom g.hom)
• CategoryTheory.Limits.coprod.inl_desc_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y : C} [CategoryTheory.Limits.HasBinaryCoproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) {Z : C} (h : W ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.desc f g) h) = CategoryTheory.CategoryStruct.comp f h
• CategoryTheory.Limits.coprod.inr_desc_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y : C} [CategoryTheory.Limits.HasBinaryCoproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) {Z : C} (h : W ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inr (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.desc f g) h) = CategoryTheory.CategoryStruct.comp g h
• CategoryTheory.Limits.coprod.desc_comp 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {V W X Y : C} [CategoryTheory.Limits.HasBinaryCoproduct X Y] (f : V ⟶ W) (g : X ⟶ V) (h : Y ⟶ V) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.desc g h) f = CategoryTheory.Limits.coprod.desc (CategoryTheory.CategoryStruct.comp g f) (CategoryTheory.CategoryStruct.comp h f)
• CategoryTheory.Limits.coprod.desc_comp_inl_comp_inr 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y Z : C} [CategoryTheory.Limits.HasBinaryCoproduct W Y] [CategoryTheory.Limits.HasBinaryCoproduct X Z] (g : W ⟶ X) (g' : Y ⟶ Z) : CategoryTheory.Limits.coprod.desc (CategoryTheory.CategoryStruct.comp g CategoryTheory.Limits.coprod.inl) (CategoryTheory.CategoryStruct.comp g' CategoryTheory.Limits.coprod.inr) = CategoryTheory.Limits.coprod.map g g'
• CategoryTheory.Limits.coprod.braiding_hom 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasBinaryCoproducts C] (P Q : C) : (CategoryTheory.Limits.coprod.braiding P Q).hom = CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inr CategoryTheory.Limits.coprod.inl
• CategoryTheory.Limits.coprod.braiding_inv 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasBinaryCoproducts C] (P Q : C) : (CategoryTheory.Limits.coprod.braiding P Q).inv = CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inr CategoryTheory.Limits.coprod.inl
• CategoryTheory.Limits.coprod.map_desc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {S T U V W : C} [CategoryTheory.Limits.HasBinaryCoproduct U W] [CategoryTheory.Limits.HasBinaryCoproduct T V] (f : U ⟶ S) (g : W ⟶ S) (h : T ⟶ U) (k : V ⟶ W) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.map h k) (CategoryTheory.Limits.coprod.desc f g) = CategoryTheory.Limits.coprod.desc (CategoryTheory.CategoryStruct.comp h f) (CategoryTheory.CategoryStruct.comp k g)
• CategoryTheory.Limits.coprod.desc_comp_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {V W X Y : C} [CategoryTheory.Limits.HasBinaryCoproduct X Y] (f : V ⟶ W) (g : X ⟶ V) (h : Y ⟶ V) {Z : C} (h✝ : W ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.desc g h) (CategoryTheory.CategoryStruct.comp f h✝) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.desc (CategoryTheory.CategoryStruct.comp g f) (CategoryTheory.CategoryStruct.comp h f)) h✝
• CategoryTheory.Limits.coprod.map_desc_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {S T U V W : C} [CategoryTheory.Limits.HasBinaryCoproduct U W] [CategoryTheory.Limits.HasBinaryCoproduct T V] (f : U ⟶ S) (g : W ⟶ S) (h : T ⟶ U) (k : V ⟶ W) {Z : C} (h✝ : S ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.map h k) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.desc f g) h✝) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.desc (CategoryTheory.CategoryStruct.comp h f) (CategoryTheory.CategoryStruct.comp k g)) h✝
• CategoryTheory.Limits.coprod.symmetry' 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasBinaryCoproducts C] (P Q : C) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inr CategoryTheory.Limits.coprod.inl) (CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inr CategoryTheory.Limits.coprod.inl) = CategoryTheory.CategoryStruct.id (P ⨿ Q)
• CategoryTheory.Limits.coprod.symmetry'_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasBinaryCoproducts C] (P Q : C) {Z : C} (h : P ⨿ Q ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inr CategoryTheory.Limits.coprod.inl) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inr CategoryTheory.Limits.coprod.inl) h) = h
• CategoryTheory.Over.coprodObj_map 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasBinaryCoproducts C] {A : C} (a✝ : CategoryTheory.Over A) {X✝ Y✝ : CategoryTheory.Over A} (k : X✝ ⟶ Y✝) : a✝.coprodObj.map k = CategoryTheory.Over.homMk (CategoryTheory.Limits.coprod.map (CategoryTheory.CategoryStruct.id a✝.left) (CategoryTheory.Over.Hom.left k)) ⋯
• CategoryTheory.Limits.coprod.associator_hom 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasBinaryCoproducts C] (P Q R : C) : (CategoryTheory.Limits.coprod.associator P Q R).hom = CategoryTheory.Limits.coprod.desc (CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl CategoryTheory.Limits.coprod.inr)) (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inr CategoryTheory.Limits.coprod.inr)
• CategoryTheory.Limits.coprod.associator_inv 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasBinaryCoproducts C] (P Q R : C) : (CategoryTheory.Limits.coprod.associator P Q R).inv = CategoryTheory.Limits.coprod.desc (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl CategoryTheory.Limits.coprod.inl) (CategoryTheory.Limits.coprod.desc (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inr CategoryTheory.Limits.coprod.inl) CategoryTheory.Limits.coprod.inr)
• CategoryTheory.Limits.epi_coprod_to_pushout 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) [CategoryTheory.Limits.HasPushout f g] [CategoryTheory.Limits.HasBinaryCoproduct Y Z] : CategoryTheory.Epi (CategoryTheory.Limits.coprod.desc (CategoryTheory.Limits.pushout.inl f g) (CategoryTheory.Limits.pushout.inr f g))
• CategoryTheory.Limits.biprod.isoCoprod_inv 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
  {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] : (CategoryTheory.Limits.biprod.isoCoprod X Y).inv = CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.biprod.inl CategoryTheory.Limits.biprod.inr
• CategoryTheory.underToAlgebra_obj_a 📋 Mathlib.CategoryTheory.Monad.Products
  {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryCoproducts C] (f : CategoryTheory.Under X) : ((CategoryTheory.underToAlgebra X).obj f).a = CategoryTheory.Limits.coprod.desc f.hom (CategoryTheory.CategoryStruct.id f.right)
• CategoryTheory.coprodMonad_μ_app 📋 Mathlib.CategoryTheory.Monad.Products
  {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryCoproducts C] (x✝ : C) : (CategoryTheory.coprodMonad X).μ.app x✝ = CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.id (X ⨿ x✝))
• CategoryTheory.underToAlgebra_map_f 📋 Mathlib.CategoryTheory.Monad.Products
  {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryCoproducts C] {X✝ Y✝ : CategoryTheory.Under X} (g : X✝ ⟶ Y✝) : ((CategoryTheory.underToAlgebra X).map g).f = CategoryTheory.Under.Hom.right g
• CategoryTheory.Under.costar_obj_hom 📋 Mathlib.CategoryTheory.Comma.Over.Pullback
  {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryCoproducts C] (X✝ : C) : ((CategoryTheory.Under.costar X).obj X✝).hom = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.id (X ⨿ X✝)))
• CategoryTheory.Limits.HasCoequalizersOfHasPushoutsAndBinaryCoproducts.pushoutInl_eq_pushout_inr 📋 Mathlib.CategoryTheory.Limits.Constructions.Equalizers
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasBinaryCoproducts C] [CategoryTheory.Limits.HasPushouts C] (F : CategoryTheory.Functor CategoryTheory.Limits.WalkingParallelPair C) : CategoryTheory.Limits.HasCoequalizersOfHasPushoutsAndBinaryCoproducts.pushoutInl F = CategoryTheory.Limits.pushout.inr (CategoryTheory.Limits.coprod.desc (CategoryTheory.CategoryStruct.id (F.obj CategoryTheory.Limits.WalkingParallelPair.one)) (F.map CategoryTheory.Limits.WalkingParallelPairHom.left)) (CategoryTheory.Limits.coprod.desc (CategoryTheory.CategoryStruct.id (F.obj CategoryTheory.Limits.WalkingParallelPair.one)) (F.map CategoryTheory.Limits.WalkingParallelPairHom.right))
• HomotopicalAlgebra.CofibrantBrownFactorization.mk' 📋 Mathlib.AlgebraicTopology.ModelCategory.BrownLemma
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] [HomotopicalAlgebra.ModelCategory C] {X Y : C} (f : X ⟶ Y) [HomotopicalAlgebra.IsCofibrant X] [HomotopicalAlgebra.IsCofibrant Y] (h : (HomotopicalAlgebra.cofibrations C).MapFactorizationData (HomotopicalAlgebra.trivialFibrations C) (CategoryTheory.Limits.coprod.desc f (CategoryTheory.CategoryStruct.id Y))) : HomotopicalAlgebra.CofibrantBrownFactorization f
• HomotopicalAlgebra.CofibrantBrownFactorization.mk'_Z 📋 Mathlib.AlgebraicTopology.ModelCategory.BrownLemma
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] [HomotopicalAlgebra.ModelCategory C] {X Y : C} (f : X ⟶ Y) [HomotopicalAlgebra.IsCofibrant X] [HomotopicalAlgebra.IsCofibrant Y] (h : (HomotopicalAlgebra.cofibrations C).MapFactorizationData (HomotopicalAlgebra.trivialFibrations C) (CategoryTheory.Limits.coprod.desc f (CategoryTheory.CategoryStruct.id Y))) : (HomotopicalAlgebra.CofibrantBrownFactorization.mk' f h).Z = h.Z
• HomotopicalAlgebra.CofibrantBrownFactorization.mk'_p 📋 Mathlib.AlgebraicTopology.ModelCategory.BrownLemma
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] [HomotopicalAlgebra.ModelCategory C] {X Y : C} (f : X ⟶ Y) [HomotopicalAlgebra.IsCofibrant X] [HomotopicalAlgebra.IsCofibrant Y] (h : (HomotopicalAlgebra.cofibrations C).MapFactorizationData (HomotopicalAlgebra.trivialFibrations C) (CategoryTheory.Limits.coprod.desc f (CategoryTheory.CategoryStruct.id Y))) : (HomotopicalAlgebra.CofibrantBrownFactorization.mk' f h).p = h.p
• HomotopicalAlgebra.CofibrantBrownFactorization.mk'_s 📋 Mathlib.AlgebraicTopology.ModelCategory.BrownLemma
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] [HomotopicalAlgebra.ModelCategory C] {X Y : C} (f : X ⟶ Y) [HomotopicalAlgebra.IsCofibrant X] [HomotopicalAlgebra.IsCofibrant Y] (h : (HomotopicalAlgebra.cofibrations C).MapFactorizationData (HomotopicalAlgebra.trivialFibrations C) (CategoryTheory.Limits.coprod.desc f (CategoryTheory.CategoryStruct.id Y))) : (HomotopicalAlgebra.CofibrantBrownFactorization.mk' f h).s = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inr h.i
• HomotopicalAlgebra.CofibrantBrownFactorization.mk'_i 📋 Mathlib.AlgebraicTopology.ModelCategory.BrownLemma
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] [HomotopicalAlgebra.ModelCategory C] {X Y : C} (f : X ⟶ Y) [HomotopicalAlgebra.IsCofibrant X] [HomotopicalAlgebra.IsCofibrant Y] (h : (HomotopicalAlgebra.cofibrations C).MapFactorizationData (HomotopicalAlgebra.trivialFibrations C) (CategoryTheory.Limits.coprod.desc f (CategoryTheory.CategoryStruct.id Y))) : (HomotopicalAlgebra.CofibrantBrownFactorization.mk' f h).i = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl h.i
• CategoryTheory.Limits.isPushout_coequalizer_coprod 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equalizer
  {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : C} (f g : X ⟶ Y) [CategoryTheory.Limits.HasCoequalizer f g] [CategoryTheory.Limits.HasBinaryCoproduct X X] : CategoryTheory.IsPushout (CategoryTheory.Limits.coprod.desc f g) (CategoryTheory.Limits.coprod.desc (CategoryTheory.CategoryStruct.id X) (CategoryTheory.CategoryStruct.id X)) (CategoryTheory.Limits.coequalizer.π f g) (CategoryTheory.CategoryStruct.comp f (CategoryTheory.Limits.coequalizer.π f g))
• AlgebraicGeometry.instIsAffineHomDescScheme 📋 Mathlib.AlgebraicGeometry.Morphisms.Affine
  {U V X : AlgebraicGeometry.Scheme} (f : U ⟶ X) (g : V ⟶ X) [AlgebraicGeometry.IsAffineHom f] [AlgebraicGeometry.IsAffineHom g] : AlgebraicGeometry.IsAffineHom (CategoryTheory.Limits.coprod.desc f g)
• AlgebraicGeometry.HasAffineProperty.coprodDesc_affineAnd 📋 Mathlib.AlgebraicGeometry.Morphisms.AffineAnd
  {Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop} {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} (hP : AlgebraicGeometry.HasAffineProperty P (AlgebraicGeometry.affineAnd fun {R S} [CommRing R] [CommRing S] => Q)) (hQi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] => Q) (hQ : ∀ {R S T : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] (f : R →+* S) (g : R →+* T), Q f → Q g → Q (f.prod g)) {U V X : AlgebraicGeometry.Scheme} (f : U ⟶ X) (g : V ⟶ X) (hf : P f) (hg : P g) : P (CategoryTheory.Limits.coprod.desc f g)
• AlgebraicGeometry.IsIntegralHom.instDescScheme 📋 Mathlib.AlgebraicGeometry.Morphisms.Integral
  {U V X : AlgebraicGeometry.Scheme} (f : U ⟶ X) (g : V ⟶ X) [AlgebraicGeometry.IsIntegralHom f] [AlgebraicGeometry.IsIntegralHom g] : AlgebraicGeometry.IsIntegralHom (CategoryTheory.Limits.coprod.desc f g)
• AlgebraicGeometry.IsFinite.instDescScheme 📋 Mathlib.AlgebraicGeometry.Morphisms.Finite
  {U V X : AlgebraicGeometry.Scheme} (f : U ⟶ X) (g : V ⟶ X) [AlgebraicGeometry.IsFinite f] [AlgebraicGeometry.IsFinite g] : AlgebraicGeometry.IsFinite (CategoryTheory.Limits.coprod.desc f g)
• AlgebraicGeometry.Scheme.Hom.normalizationCoprodIso_inv_coprodDesc_fromNormalization 📋 Mathlib.AlgebraicGeometry.Normalization
  {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.QuasiCompact f] [AlgebraicGeometry.QuasiSeparated f] {U V : AlgebraicGeometry.Scheme} {iU : U ⟶ X} {iV : V ⟶ X} (e : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.BinaryCofan.mk iU iV)) [AlgebraicGeometry.QuasiCompact iU] [AlgebraicGeometry.QuasiSeparated iU] [AlgebraicGeometry.QuasiCompact iV] [AlgebraicGeometry.QuasiSeparated iV] : CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.normalizationCoprodIso f e).inv (CategoryTheory.Limits.coprod.desc (AlgebraicGeometry.Scheme.Hom.fromNormalization (CategoryTheory.CategoryStruct.comp iU f)) (AlgebraicGeometry.Scheme.Hom.fromNormalization (CategoryTheory.CategoryStruct.comp iV f))) = AlgebraicGeometry.Scheme.Hom.fromNormalization f
• AlgebraicGeometry.Scheme.Hom.normalizationCoprodIso_inv_coprodDesc_fromNormalization_assoc 📋 Mathlib.AlgebraicGeometry.Normalization
  {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.QuasiCompact f] [AlgebraicGeometry.QuasiSeparated f] {U V : AlgebraicGeometry.Scheme} {iU : U ⟶ X} {iV : V ⟶ X} (e : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.BinaryCofan.mk iU iV)) [AlgebraicGeometry.QuasiCompact iU] [AlgebraicGeometry.QuasiSeparated iU] [AlgebraicGeometry.QuasiCompact iV] [AlgebraicGeometry.QuasiSeparated iV] {Z : AlgebraicGeometry.Scheme} (h : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.normalizationCoprodIso f e).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.desc (AlgebraicGeometry.Scheme.Hom.fromNormalization (CategoryTheory.CategoryStruct.comp iU f)) (AlgebraicGeometry.Scheme.Hom.fromNormalization (CategoryTheory.CategoryStruct.comp iV f))) h) = CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.fromNormalization f) h
• CategoryTheory.leftDistrib_hom 📋 Mathlib.CategoryTheory.Distributive.Monoidal
  {C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasBinaryCoproducts C] [CategoryTheory.IsMonoidalLeftDistrib C] {X Y Z : C} : (CategoryTheory.leftDistrib X Y Z).hom = CategoryTheory.Limits.coprod.desc (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X CategoryTheory.Limits.coprod.inl) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X CategoryTheory.Limits.coprod.inr)
• CategoryTheory.rightDistrib_hom 📋 Mathlib.CategoryTheory.Distributive.Monoidal
  {C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasBinaryCoproducts C] [CategoryTheory.IsMonoidalRightDistrib C] {X Y Z : C} : (∂R X Y Z).hom = CategoryTheory.Limits.coprod.desc (CategoryTheory.MonoidalCategoryStruct.whiskerRight CategoryTheory.Limits.coprod.inl X) (CategoryTheory.MonoidalCategoryStruct.whiskerRight CategoryTheory.Limits.coprod.inr X)
• CategoryTheory.MonoidalClosed.leftDistrib_inv 📋 Mathlib.CategoryTheory.Distributive.Monoidal
  {C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasBinaryCoproducts C] [CategoryTheory.MonoidalClosed C] {X Y Z : C} : (CategoryTheory.leftDistrib X Y Z).inv = CategoryTheory.MonoidalClosed.uncurry (CategoryTheory.Limits.coprod.desc (CategoryTheory.MonoidalClosed.curry CategoryTheory.Limits.coprod.inl) (CategoryTheory.MonoidalClosed.curry CategoryTheory.Limits.coprod.inr))
• CategoryTheory.Limits.CokernelCofork.mapBifunctor 📋 Mathlib.CategoryTheory.Limits.Preserves.BifunctorCokernel
  {C₁ : Type u_1} {C₂ : Type u_2} {C : Type u_3} [CategoryTheory.Category.{v_1, u_1} C₁] [CategoryTheory.Category.{v_2, u_2} C₂] [CategoryTheory.Category.{v_3, u_3} C] [CategoryTheory.Limits.HasZeroMorphisms C₁] [CategoryTheory.Limits.HasZeroMorphisms C₂] [CategoryTheory.Limits.HasZeroMorphisms C] {X₁ Y₁ : C₁} {f₁ : X₁ ⟶ Y₁} (c₁ : CategoryTheory.Limits.CokernelCofork f₁) {X₂ Y₂ : C₂} {f₂ : X₂ ⟶ Y₂} (c₂ : CategoryTheory.Limits.CokernelCofork f₂) (F : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [(F.obj c₁.pt).PreservesZeroMorphisms] [F.PreservesZeroMorphisms] [CategoryTheory.Limits.HasBinaryCoproduct ((F.obj X₁).obj Y₂) ((F.obj Y₁).obj X₂)] : CategoryTheory.Limits.CokernelCofork (CategoryTheory.Limits.coprod.desc ((F.map f₁).app Y₂) ((F.obj Y₁).map f₂))
• CategoryTheory.Limits.CokernelCofork.isColimitMapBifunctor 📋 Mathlib.CategoryTheory.Limits.Preserves.BifunctorCokernel
  {C₁ : Type u_1} {C₂ : Type u_2} {C : Type u_3} [CategoryTheory.Category.{v_1, u_1} C₁] [CategoryTheory.Category.{v_2, u_2} C₂] [CategoryTheory.Category.{v_3, u_3} C] [CategoryTheory.Limits.HasZeroMorphisms C₁] [CategoryTheory.Limits.HasZeroMorphisms C₂] [CategoryTheory.Limits.HasZeroMorphisms C] {X₁ Y₁ : C₁} {f₁ : X₁ ⟶ Y₁} {c₁ : CategoryTheory.Limits.CokernelCofork f₁} (hc₁ : CategoryTheory.Limits.IsColimit c₁) {X₂ Y₂ : C₂} {f₂ : X₂ ⟶ Y₂} {c₂ : CategoryTheory.Limits.CokernelCofork f₂} (hc₂ : CategoryTheory.Limits.IsColimit c₂) (F : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [(F.obj c₁.pt).PreservesZeroMorphisms] [F.PreservesZeroMorphisms] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.parallelPair f₂ 0) (F.obj c₁.pt)] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.parallelPair f₁ 0) (F.flip.obj Y₂)] [CategoryTheory.Limits.HasBinaryCoproduct ((F.obj X₁).obj Y₂) ((F.obj Y₁).obj X₂)] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.parallelPair f₁ 0) (F.flip.obj X₂)] : CategoryTheory.Limits.IsColimit (c₁.mapBifunctor c₂ F)
• CategoryTheory.Limits.CokernelCofork.isColimitMapBifunctor.exists_desc 📋 Mathlib.CategoryTheory.Limits.Preserves.BifunctorCokernel
  {C₁ : Type u_1} {C₂ : Type u_2} {C : Type u_3} [CategoryTheory.Category.{v_1, u_1} C₁] [CategoryTheory.Category.{v_2, u_2} C₂] [CategoryTheory.Category.{v_3, u_3} C] [CategoryTheory.Limits.HasZeroMorphisms C₁] [CategoryTheory.Limits.HasZeroMorphisms C₂] [CategoryTheory.Limits.HasZeroMorphisms C] {X₁ Y₁ : C₁} {f₁ : X₁ ⟶ Y₁} {c₁ : CategoryTheory.Limits.CokernelCofork f₁} (hc₁ : CategoryTheory.Limits.IsColimit c₁) {X₂ Y₂ : C₂} {f₂ : X₂ ⟶ Y₂} {c₂ : CategoryTheory.Limits.CokernelCofork f₂} (hc₂ : CategoryTheory.Limits.IsColimit c₂) (F : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [(F.obj c₁.pt).PreservesZeroMorphisms] [F.PreservesZeroMorphisms] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.parallelPair f₂ 0) (F.obj c₁.pt)] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.parallelPair f₁ 0) (F.flip.obj Y₂)] [CategoryTheory.Limits.HasBinaryCoproduct ((F.obj X₁).obj Y₂) ((F.obj Y₁).obj X₂)] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.parallelPair f₁ 0) (F.flip.obj X₂)] (s : CategoryTheory.Limits.CokernelCofork (CategoryTheory.Limits.coprod.desc ((F.map f₁).app Y₂) ((F.obj Y₁).map f₂))) : ∃ l, CategoryTheory.CategoryStruct.comp ((F.map (CategoryTheory.Limits.Cofork.π c₁)).app Y₂) (CategoryTheory.CategoryStruct.comp ((F.obj c₁.pt).map (CategoryTheory.Limits.Cofork.π c₂)) l) = CategoryTheory.Limits.Cofork.π s
• CategoryTheory.Limits.CokernelCofork.tensor 📋 Mathlib.CategoryTheory.Monoidal.Limits.Cokernels
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalPreadditive C] {X₁ Y₁ : C} {f₁ : X₁ ⟶ Y₁} (c₁ : CategoryTheory.Limits.CokernelCofork f₁) {X₂ Y₂ : C} {f₂ : X₂ ⟶ Y₂} (c₂ : CategoryTheory.Limits.CokernelCofork f₂) [CategoryTheory.Limits.HasBinaryCoproduct (CategoryTheory.MonoidalCategoryStruct.tensorObj X₁ Y₂) (CategoryTheory.MonoidalCategoryStruct.tensorObj Y₁ X₂)] : CategoryTheory.Limits.CokernelCofork (CategoryTheory.Limits.coprod.desc (CategoryTheory.MonoidalCategoryStruct.whiskerRight f₁ Y₂) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft Y₁ f₂))
• CategoryTheory.Limits.CokernelCofork.isColimitTensor 📋 Mathlib.CategoryTheory.Monoidal.Limits.Cokernels
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalPreadditive C] {X₁ Y₁ : C} {f₁ : X₁ ⟶ Y₁} {c₁ : CategoryTheory.Limits.CokernelCofork f₁} (hc₁ : CategoryTheory.Limits.IsColimit c₁) {X₂ Y₂ : C} {f₂ : X₂ ⟶ Y₂} {c₂ : CategoryTheory.Limits.CokernelCofork f₂} (hc₂ : CategoryTheory.Limits.IsColimit c₂) [CategoryTheory.Limits.HasBinaryCoproduct (CategoryTheory.MonoidalCategoryStruct.tensorObj X₁ Y₂) (CategoryTheory.MonoidalCategoryStruct.tensorObj Y₁ X₂)] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.parallelPair f₂ 0) (CategoryTheory.MonoidalCategory.tensorLeft c₁.pt)] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.parallelPair f₁ 0) (CategoryTheory.MonoidalCategory.tensorRight Y₂)] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.parallelPair f₁ 0) (CategoryTheory.MonoidalCategory.tensorRight X₂)] : CategoryTheory.Limits.IsColimit (c₁.tensor c₂)
• CategoryTheory.monoidalOfHasFiniteCoproducts.leftUnitor_hom 📋 Mathlib.CategoryTheory.Monoidal.OfHasFiniteProducts
  (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasInitial C] [CategoryTheory.Limits.HasBinaryCoproducts C] (X : C) : (CategoryTheory.MonoidalCategoryStruct.leftUnitor X).hom = CategoryTheory.Limits.coprod.desc (CategoryTheory.Limits.initial.to X) (CategoryTheory.CategoryStruct.id X)
• CategoryTheory.monoidalOfHasFiniteCoproducts.rightUnitor_hom 📋 Mathlib.CategoryTheory.Monoidal.OfHasFiniteProducts
  (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasInitial C] [CategoryTheory.Limits.HasBinaryCoproducts C] (X : C) : (CategoryTheory.MonoidalCategoryStruct.rightUnitor X).hom = CategoryTheory.Limits.coprod.desc (CategoryTheory.CategoryStruct.id X) (CategoryTheory.Limits.initial.to X)
• CategoryTheory.monoidalOfHasFiniteCoproducts.associator_hom 📋 Mathlib.CategoryTheory.Monoidal.OfHasFiniteProducts
  (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasInitial C] [CategoryTheory.Limits.HasBinaryCoproducts C] (X Y Z : C) : (CategoryTheory.MonoidalCategoryStruct.associator X Y Z).hom = CategoryTheory.Limits.coprod.desc (CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl CategoryTheory.Limits.coprod.inr)) (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inr CategoryTheory.Limits.coprod.inr)
• CategoryTheory.monoidalOfHasFiniteCoproducts.associator_inv 📋 Mathlib.CategoryTheory.Monoidal.OfHasFiniteProducts
  (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasInitial C] [CategoryTheory.Limits.HasBinaryCoproducts C] (X Y Z : C) : (CategoryTheory.MonoidalCategoryStruct.associator X Y Z).inv = CategoryTheory.Limits.coprod.desc (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl CategoryTheory.Limits.coprod.inl) (CategoryTheory.Limits.coprod.desc (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inr CategoryTheory.Limits.coprod.inl) CategoryTheory.Limits.coprod.inr)

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