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Found 74 declarations mentioning CategoryTheory.Limits.biprod.inr.

• CategoryTheory.Limits.biprod.inr 📋 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] : Y ⟶ X ⊞ Y
• CategoryTheory.Limits.biprod.inr_mono 📋 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.IsSplitMono CategoryTheory.Limits.biprod.inr
• CategoryTheory.Limits.instHasCokernelInr 📋 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.HasCokernel CategoryTheory.Limits.biprod.inr
• CategoryTheory.Limits.biprod.inrCokernelCofork 📋 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.CokernelCofork CategoryTheory.Limits.biprod.inr
• CategoryTheory.Limits.cokernelBiprodInrIso 📋 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.cokernel CategoryTheory.Limits.biprod.inr ≅ X
• CategoryTheory.Limits.BinaryBiproduct.bicone_inr 📋 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.BinaryBiproduct.bicone X Y).inr = CategoryTheory.Limits.biprod.inr
• CategoryTheory.Limits.biprod.inr_snd 📋 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.CategoryStruct.comp CategoryTheory.Limits.biprod.inr CategoryTheory.Limits.biprod.snd = CategoryTheory.CategoryStruct.id Y
• CategoryTheory.Limits.isoZeroBiprod_hom 📋 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] (hY : CategoryTheory.Limits.IsZero X) : (CategoryTheory.Limits.isoZeroBiprod hY).hom = CategoryTheory.Limits.biprod.inr
• CategoryTheory.Limits.biprod.inr_desc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
  {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {W X Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr (CategoryTheory.Limits.biprod.desc f g) = g
• CategoryTheory.Limits.biprod.inr_snd_assoc 📋 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] {Z : C} (h : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.snd h) = h
• CategoryTheory.Limits.biprod.inr_fst 📋 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.CategoryStruct.comp CategoryTheory.Limits.biprod.inr CategoryTheory.Limits.biprod.fst = 0
• CategoryTheory.Limits.biprod.isCokernelInrCokernelFork 📋 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.IsColimit (CategoryTheory.Limits.biprod.inrCokernelCofork X Y)
• 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.Limits.biprod.inr_desc_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
  {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {W X Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) {Z : C} (h : W ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.desc f g) h) = CategoryTheory.CategoryStruct.comp g h
• CategoryTheory.Limits.biprod.inr_fst_assoc 📋 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] {Z : C} (h : X ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.fst h) = CategoryTheory.CategoryStruct.comp 0 h
• CategoryTheory.Limits.biprod.inr_map 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
  {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {W X Y Z : C} [CategoryTheory.Limits.HasBinaryBiproduct W X] [CategoryTheory.Limits.HasBinaryBiproduct Y Z] (f : W ⟶ Y) (g : X ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr (CategoryTheory.Limits.biprod.map f g) = CategoryTheory.CategoryStruct.comp g CategoryTheory.Limits.biprod.inr
• CategoryTheory.Limits.biprod.braiding'_hom 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
  {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasBinaryBiproducts C] (P Q : C) : (CategoryTheory.Limits.biprod.braiding' P Q).hom = CategoryTheory.Limits.biprod.desc CategoryTheory.Limits.biprod.inr CategoryTheory.Limits.biprod.inl
• CategoryTheory.Limits.biprod.braiding'_inv 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
  {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasBinaryBiproducts C] (P Q : C) : (CategoryTheory.Limits.biprod.braiding' P Q).inv = CategoryTheory.Limits.biprod.desc CategoryTheory.Limits.biprod.inr CategoryTheory.Limits.biprod.inl
• CategoryTheory.Limits.biprod.inr_map_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
  {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {W X Y Z : C} [CategoryTheory.Limits.HasBinaryBiproduct W X] [CategoryTheory.Limits.HasBinaryBiproduct Y Z] (f : W ⟶ Y) (g : X ⟶ Z) {Z✝ : C} (h : Y ⊞ Z ⟶ Z✝) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.map f g) h) = CategoryTheory.CategoryStruct.comp g (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr h)
• CategoryTheory.Limits.biprod.hom_ext' 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
  {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X Y Z : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] (f g : X ⊞ Y ⟶ Z) (h₀ : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inl f = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inl g) (h₁ : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr f = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr g) : f = g
• CategoryTheory.Limits.biprod.opIso_hom_snd 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
  {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] (P Q : C) [CategoryTheory.Limits.HasBinaryBiproduct P Q] : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.opIso P Q).hom CategoryTheory.Limits.biprod.snd = CategoryTheory.Limits.biprod.inr.op
• CategoryTheory.Limits.biprod.hom_ext'_iff 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
  {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X Y Z : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] {f g : X ⊞ Y ⟶ Z} : f = g ↔ CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inl f = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inl g ∧ CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr f = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr g
• CategoryTheory.Limits.biprod.inr_opIso_inv 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
  {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] (P Q : C) [CategoryTheory.Limits.HasBinaryBiproduct P Q] : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr (CategoryTheory.Limits.biprod.opIso P Q).inv = CategoryTheory.Limits.biprod.snd.op
• CategoryTheory.Limits.biprod.fstKernelFork_ι 📋 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.Fork.ι (CategoryTheory.Limits.biprod.fstKernelFork X Y) = CategoryTheory.Limits.biprod.inr
• CategoryTheory.Limits.biprod.inrCokernelCofork_π 📋 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.Cofork.π (CategoryTheory.Limits.biprod.inrCokernelCofork X Y) = CategoryTheory.Limits.biprod.fst
• CategoryTheory.Limits.biprod.opIso_inv_inr_op 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
  {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] (P Q : C) [CategoryTheory.Limits.HasBinaryBiproduct P Q] : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.opIso P Q).inv CategoryTheory.Limits.biprod.inr.op = CategoryTheory.Limits.biprod.snd
• CategoryTheory.Limits.biprod.snd_op_opIso_hom 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
  {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] (P Q : C) [CategoryTheory.Limits.HasBinaryBiproduct P Q] : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.snd.op (CategoryTheory.Limits.biprod.opIso P Q).hom = CategoryTheory.Limits.biprod.inr
• CategoryTheory.Limits.biprod.inr_opIso_inv_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
  {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] (P Q : C) [CategoryTheory.Limits.HasBinaryBiproduct P Q] {Z : Cᵒᵖ} (h : Opposite.op (P ⊞ Q) ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.opIso P Q).inv h) = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.snd.op h
• CategoryTheory.Limits.biprod.opIso_hom_snd_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
  {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] (P Q : C) [CategoryTheory.Limits.HasBinaryBiproduct P Q] {Z : Cᵒᵖ} (h : Opposite.op Q ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.opIso P Q).hom (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.snd h) = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr.op h
• CategoryTheory.Limits.biprod.opIso_inv_inr_op_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
  {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] (P Q : C) [CategoryTheory.Limits.HasBinaryBiproduct P Q] {Z : Cᵒᵖ} (h : Opposite.op Q ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.opIso P Q).inv (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr.op h) = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.snd h
• CategoryTheory.Limits.biprod.snd_op_opIso_hom_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
  {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] (P Q : C) [CategoryTheory.Limits.HasBinaryBiproduct P Q] {Z : Cᵒᵖ} (h : Opposite.op P ⊞ Opposite.op Q ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.snd.op (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.opIso P Q).hom h) = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr h
• CategoryTheory.Limits.cokernelBiprodInrIso_hom 📋 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.cokernelBiprodInrIso.hom = CategoryTheory.Limits.colimit.desc (CategoryTheory.Limits.parallelPair CategoryTheory.Limits.biprod.inr 0) (CategoryTheory.Limits.biprod.inrCokernelCofork X Y)
• CategoryTheory.Limits.cokernelBiprodInrIso_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.cokernelBiprodInrIso.inv = (CategoryTheory.Limits.biprod.isCokernelInrCokernelFork X Y).desc (CategoryTheory.Limits.colimit.cocone (CategoryTheory.Limits.parallelPair CategoryTheory.Limits.biprod.inr 0))
• CategoryTheory.Functor.inr_biprodComparison' 📋 Mathlib.CategoryTheory.Limits.Preserves.Shapes.Biproducts
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] (F : CategoryTheory.Functor C D) (X Y : C) [CategoryTheory.Limits.HasBinaryBiproduct X Y] [CategoryTheory.Limits.HasBinaryBiproduct (F.obj X) (F.obj Y)] : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr (F.biprodComparison' X Y) = F.map CategoryTheory.Limits.biprod.inr
• CategoryTheory.Functor.inr_biprodComparison'_assoc 📋 Mathlib.CategoryTheory.Limits.Preserves.Shapes.Biproducts
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] (F : CategoryTheory.Functor C D) (X Y : C) [CategoryTheory.Limits.HasBinaryBiproduct X Y] [CategoryTheory.Limits.HasBinaryBiproduct (F.obj X) (F.obj Y)] {Z : D} (h : F.obj (X ⊞ Y) ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr (CategoryTheory.CategoryStruct.comp (F.biprodComparison' X Y) h) = CategoryTheory.CategoryStruct.comp (F.map CategoryTheory.Limits.biprod.inr) h
• CategoryTheory.Functor.mapBiprod_inv 📋 Mathlib.CategoryTheory.Limits.Preserves.Shapes.Biproducts
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms D] (F : CategoryTheory.Functor C D) (X Y : C) [CategoryTheory.Limits.HasBinaryBiproduct X Y] [F.PreservesZeroMorphisms] [CategoryTheory.Limits.PreservesBinaryBiproduct X Y F] : (F.mapBiprod X Y).inv = CategoryTheory.Limits.biprod.desc (F.map CategoryTheory.Limits.biprod.inl) (F.map CategoryTheory.Limits.biprod.inr)
• CategoryTheory.Limits.biprod.ext_from_iff 📋 Mathlib.CategoryTheory.Preadditive.Biproducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {X Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] {Z : C} {f g : X ⊞ Y ⟶ Z} : f = g ↔ CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inl f = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inl g ∧ CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr f = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr g
• CategoryTheory.Limits.biprod.lift_eq 📋 Mathlib.CategoryTheory.Preadditive.Biproducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {X Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] {T : C} {f : T ⟶ X} {g : T ⟶ Y} : CategoryTheory.Limits.biprod.lift f g = CategoryTheory.CategoryStruct.comp f CategoryTheory.Limits.biprod.inl + CategoryTheory.CategoryStruct.comp g CategoryTheory.Limits.biprod.inr
• CategoryTheory.Limits.biprod.decomp_hom_to 📋 Mathlib.CategoryTheory.Preadditive.Biproducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {X Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] {Z : C} (f : Z ⟶ X ⊞ Y) : ∃ f₁ f₂, f = CategoryTheory.CategoryStruct.comp f₁ CategoryTheory.Limits.biprod.inl + CategoryTheory.CategoryStruct.comp f₂ CategoryTheory.Limits.biprod.inr
• CategoryTheory.Limits.biprod.total 📋 Mathlib.CategoryTheory.Preadditive.Biproducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {X Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.fst CategoryTheory.Limits.biprod.inl + CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.snd CategoryTheory.Limits.biprod.inr = CategoryTheory.CategoryStruct.id (X ⊞ Y)
• CategoryTheory.Biprod.inl_ofComponents 📋 Mathlib.CategoryTheory.Preadditive.Biproducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {X₁ X₂ Y₁ Y₂ : C} (f₁₁ : X₁ ⟶ Y₁) (f₁₂ : X₁ ⟶ Y₂) (f₂₁ : X₂ ⟶ Y₁) (f₂₂ : X₂ ⟶ Y₂) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inl (CategoryTheory.Biprod.ofComponents f₁₁ f₁₂ f₂₁ f₂₂) = CategoryTheory.CategoryStruct.comp f₁₁ CategoryTheory.Limits.biprod.inl + CategoryTheory.CategoryStruct.comp f₁₂ CategoryTheory.Limits.biprod.inr
• CategoryTheory.Biprod.inr_ofComponents 📋 Mathlib.CategoryTheory.Preadditive.Biproducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {X₁ X₂ Y₁ Y₂ : C} (f₁₁ : X₁ ⟶ Y₁) (f₁₂ : X₁ ⟶ Y₂) (f₂₁ : X₂ ⟶ Y₁) (f₂₂ : X₂ ⟶ Y₂) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr (CategoryTheory.Biprod.ofComponents f₁₁ f₁₂ f₂₁ f₂₂) = CategoryTheory.CategoryStruct.comp f₂₁ CategoryTheory.Limits.biprod.inl + CategoryTheory.CategoryStruct.comp f₂₂ CategoryTheory.Limits.biprod.inr
• CategoryTheory.Biprod.ofComponents_eq 📋 Mathlib.CategoryTheory.Preadditive.Biproducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {X₁ X₂ Y₁ Y₂ : C} (f : X₁ ⊞ X₂ ⟶ Y₁ ⊞ Y₂) : CategoryTheory.Biprod.ofComponents (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inl (CategoryTheory.CategoryStruct.comp f CategoryTheory.Limits.biprod.fst)) (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inl (CategoryTheory.CategoryStruct.comp f CategoryTheory.Limits.biprod.snd)) (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr (CategoryTheory.CategoryStruct.comp f CategoryTheory.Limits.biprod.fst)) (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr (CategoryTheory.CategoryStruct.comp f CategoryTheory.Limits.biprod.snd)) = f
• CategoryTheory.Limits.biprod.map_eq 📋 Mathlib.CategoryTheory.Preadditive.Biproducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {W X Y Z : C} {f : W ⟶ Y} {g : X ⟶ Z} : CategoryTheory.Limits.biprod.map f g = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.fst (CategoryTheory.CategoryStruct.comp f CategoryTheory.Limits.biprod.inl) + CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.snd (CategoryTheory.CategoryStruct.comp g CategoryTheory.Limits.biprod.inr)
• HomologicalComplex.biprod_inr_snd_f 📋 Mathlib.Algebra.Homology.HomologicalComplexBiprod
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Preadditive C] {c : ComplexShape ι} (K L : HomologicalComplex C c) [∀ (i : ι), CategoryTheory.Limits.HasBinaryBiproduct (K.X i) (L.X i)] (i : ι) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.inr.f i) (CategoryTheory.Limits.biprod.snd.f i) = CategoryTheory.CategoryStruct.id (L.X i)
• HomologicalComplex.biprod_inr_snd_f_assoc 📋 Mathlib.Algebra.Homology.HomologicalComplexBiprod
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Preadditive C] {c : ComplexShape ι} (K L : HomologicalComplex C c) [∀ (i : ι), CategoryTheory.Limits.HasBinaryBiproduct (K.X i) (L.X i)] (i : ι) {Z : C} (h : L.X i ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.inr.f i) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.snd.f i) h) = h
• HomologicalComplex.biprod_inr_fst_f 📋 Mathlib.Algebra.Homology.HomologicalComplexBiprod
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Preadditive C] {c : ComplexShape ι} (K L : HomologicalComplex C c) [∀ (i : ι), CategoryTheory.Limits.HasBinaryBiproduct (K.X i) (L.X i)] (i : ι) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.inr.f i) (CategoryTheory.Limits.biprod.fst.f i) = 0
• HomologicalComplex.inr_biprodXIso_inv 📋 Mathlib.Algebra.Homology.HomologicalComplexBiprod
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Preadditive C] {c : ComplexShape ι} (K L : HomologicalComplex C c) [∀ (i : ι), CategoryTheory.Limits.HasBinaryBiproduct (K.X i) (L.X i)] (i : ι) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr (K.biprodXIso L i).inv = CategoryTheory.Limits.biprod.inr.f i
• HomologicalComplex.biprod_inr_desc_f 📋 Mathlib.Algebra.Homology.HomologicalComplexBiprod
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Preadditive C] {c : ComplexShape ι} {K L : HomologicalComplex C c} [∀ (i : ι), CategoryTheory.Limits.HasBinaryBiproduct (K.X i) (L.X i)] {M : HomologicalComplex C c} (α : K ⟶ M) (β : L ⟶ M) (i : ι) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.inr.f i) ((CategoryTheory.Limits.biprod.desc α β).f i) = β.f i
• HomologicalComplex.biprod_inr_fst_f_assoc 📋 Mathlib.Algebra.Homology.HomologicalComplexBiprod
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Preadditive C] {c : ComplexShape ι} (K L : HomologicalComplex C c) [∀ (i : ι), CategoryTheory.Limits.HasBinaryBiproduct (K.X i) (L.X i)] (i : ι) {Z : C} (h : K.X i ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.inr.f i) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.fst.f i) h) = CategoryTheory.CategoryStruct.comp 0 h
• HomologicalComplex.biprod_inr_desc_f_assoc 📋 Mathlib.Algebra.Homology.HomologicalComplexBiprod
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Preadditive C] {c : ComplexShape ι} {K L : HomologicalComplex C c} [∀ (i : ι), CategoryTheory.Limits.HasBinaryBiproduct (K.X i) (L.X i)] {M : HomologicalComplex C c} (α : K ⟶ M) (β : L ⟶ M) (i : ι) {Z : C} (h : M.X i ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.inr.f i) (CategoryTheory.CategoryStruct.comp ((CategoryTheory.Limits.biprod.desc α β).f i) h) = CategoryTheory.CategoryStruct.comp (β.f i) h
• HomologicalComplex.inr_biprodXIso_inv_assoc 📋 Mathlib.Algebra.Homology.HomologicalComplexBiprod
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Preadditive C] {c : ComplexShape ι} (K L : HomologicalComplex C c) [∀ (i : ι), CategoryTheory.Limits.HasBinaryBiproduct (K.X i) (L.X i)] (i : ι) {Z : C} (h : (K ⊞ L).X i ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr (CategoryTheory.CategoryStruct.comp (K.biprodXIso L i).inv h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.inr.f i) h
• HomologicalComplex.biprodX_ext_from 📋 Mathlib.Algebra.Homology.HomologicalComplexBiprod
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Preadditive C] {c : ComplexShape ι} {K L : HomologicalComplex C c} [∀ (i : ι), CategoryTheory.Limits.HasBinaryBiproduct (K.X i) (L.X i)] {A : C} {i : ι} {f g : (K ⊞ L).X i ⟶ A} (h₁ : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.inl.f i) f = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.inl.f i) g) (h₂ : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.inr.f i) f = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.inr.f i) g) : f = g
• HomologicalComplex.biprodX_ext_from_iff 📋 Mathlib.Algebra.Homology.HomologicalComplexBiprod
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Preadditive C] {c : ComplexShape ι} {K L : HomologicalComplex C c} [∀ (i : ι), CategoryTheory.Limits.HasBinaryBiproduct (K.X i) (L.X i)] {A : C} {i : ι} {f g : (K ⊞ L).X i ⟶ A} : f = g ↔ CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.inl.f i) f = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.inl.f i) g ∧ CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.inr.f i) f = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.inr.f i) g
• HomologicalComplex.biprod_total_f 📋 Mathlib.Algebra.Homology.HomologicalComplexBiprod
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Preadditive C] {c : ComplexShape ι} (K L : HomologicalComplex C c) [∀ (i : ι), CategoryTheory.Limits.HasBinaryBiproduct (K.X i) (L.X i)] (i : ι) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.fst.f i) (CategoryTheory.Limits.biprod.inl.f i) + CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.snd.f i) (CategoryTheory.Limits.biprod.inr.f i) = CategoryTheory.CategoryStruct.id ((K ⊞ L).X i)
• HomologicalComplex.homotopyCofiber.inr_XIsoBiprod_inv 📋 Mathlib.Algebra.Homology.HomotopyCofiber
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} {F G : HomologicalComplex C c} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] [DecidableRel c.Rel] (i j : ι) (hij : c.Rel j i) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr (HomologicalComplex.homotopyCofiber.XIsoBiprod φ j i hij).inv = HomologicalComplex.homotopyCofiber.inrX φ j
• HomologicalComplex.homotopyCofiber.inrX_XIsoBiprod_hom 📋 Mathlib.Algebra.Homology.HomotopyCofiber
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} {F G : HomologicalComplex C c} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] [DecidableRel c.Rel] (i j : ι) (hij : c.Rel j i) : CategoryTheory.CategoryStruct.comp (HomologicalComplex.homotopyCofiber.inrX φ j) (HomologicalComplex.homotopyCofiber.XIsoBiprod φ j i hij).hom = CategoryTheory.Limits.biprod.inr
• HomologicalComplex.homotopyCofiber.inr_XIsoBiprod_inv_assoc 📋 Mathlib.Algebra.Homology.HomotopyCofiber
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} {F G : HomologicalComplex C c} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] [DecidableRel c.Rel] (i j : ι) (hij : c.Rel j i) {Z : C} (h : HomologicalComplex.homotopyCofiber.X φ j ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr (CategoryTheory.CategoryStruct.comp (HomologicalComplex.homotopyCofiber.XIsoBiprod φ j i hij).inv h) = CategoryTheory.CategoryStruct.comp (HomologicalComplex.homotopyCofiber.inrX φ j) h
• HomologicalComplex.homotopyCofiber.inrX_XIsoBiprod_hom_assoc 📋 Mathlib.Algebra.Homology.HomotopyCofiber
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Preadditive C] {ι : Type u_2} {c : ComplexShape ι} {F G : HomologicalComplex C c} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] [DecidableRel c.Rel] (i j : ι) (hij : c.Rel j i) {Z : C} (h : F.X i ⊞ G.X j ⟶ Z) : CategoryTheory.CategoryStruct.comp (HomologicalComplex.homotopyCofiber.inrX φ j) (CategoryTheory.CategoryStruct.comp (HomologicalComplex.homotopyCofiber.XIsoBiprod φ j i hij).hom h) = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr h
• CategoryTheory.Abelian.Ext.biprod_ext 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {Y : C} {n : ℕ} {X₁ X₂ : C} {α β : CategoryTheory.Abelian.Ext (X₁ ⊞ X₂) Y n} (h₁ : (CategoryTheory.Abelian.Ext.mk₀ CategoryTheory.Limits.biprod.inl).comp α ⋯ = (CategoryTheory.Abelian.Ext.mk₀ CategoryTheory.Limits.biprod.inl).comp β ⋯) (h₂ : (CategoryTheory.Abelian.Ext.mk₀ CategoryTheory.Limits.biprod.inr).comp α ⋯ = (CategoryTheory.Abelian.Ext.mk₀ CategoryTheory.Limits.biprod.inr).comp β ⋯) : α = β
• CategoryTheory.Abelian.Ext.biprodAddEquiv_apply_snd 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {X₁ X₂ Y : C} {n : ℕ} (e : CategoryTheory.Abelian.Ext (X₁ ⊞ X₂) Y n) : (CategoryTheory.Abelian.Ext.biprodAddEquiv e).2 = (CategoryTheory.Abelian.Ext.mk₀ CategoryTheory.Limits.biprod.inr).comp e ⋯
• CategoryTheory.Abelian.Ext.addEquivBiprod_symm_apply 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {X Y₁ Y₂ : C} {n : ℕ} (e : CategoryTheory.Abelian.Ext X Y₁ n × CategoryTheory.Abelian.Ext X Y₂ n) : CategoryTheory.Abelian.Ext.addEquivBiprod.symm e = e.1.comp (CategoryTheory.Abelian.Ext.mk₀ CategoryTheory.Limits.biprod.inl) ⋯ + e.2.comp (CategoryTheory.Abelian.Ext.mk₀ CategoryTheory.Limits.biprod.inr) ⋯
• CategoryTheory.kernelCokernelCompSequence.inr_π 📋 Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr (CategoryTheory.kernelCokernelCompSequence.π f g) = CategoryTheory.Limits.cokernel.π (CategoryTheory.CategoryStruct.comp f g)
• CategoryTheory.kernelCokernelCompSequence.inr_φ_fst 📋 Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr (CategoryTheory.CategoryStruct.comp (CategoryTheory.kernelCokernelCompSequence.φ f g) CategoryTheory.Limits.biprod.fst) = -CategoryTheory.CategoryStruct.id Y
• CategoryTheory.kernelCokernelCompSequence.inr_φ_fst_assoc 📋 Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) {Z✝ : C} (h : Y ⟶ Z✝) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr (CategoryTheory.CategoryStruct.comp (CategoryTheory.kernelCokernelCompSequence.φ f g) (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.fst h)) = CategoryTheory.CategoryStruct.comp (-CategoryTheory.CategoryStruct.id Y) h
• CategoryTheory.kernelCokernelCompSequence.inr_π_assoc 📋 Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) {Z✝ : C} (h : CategoryTheory.Limits.cokernel (CategoryTheory.CategoryStruct.comp f g) ⟶ Z✝) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.inr (CategoryTheory.CategoryStruct.comp (CategoryTheory.kernelCokernelCompSequence.π f g) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.cokernel.π (CategoryTheory.CategoryStruct.comp f g)) h
• CategoryTheory.Limits.pointwiseBinaryBicone_inr_app 📋 Mathlib.CategoryTheory.Limits.FunctorCategory.BinaryBiproducts
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasBinaryBiproducts C] {D : Type u_2} [CategoryTheory.Category.{v_2, u_2} D] (F G : CategoryTheory.Functor D C) (X : D) : (CategoryTheory.Limits.pointwiseBinaryBicone F G).inr.app X = CategoryTheory.Limits.biprod.inr
• CategoryTheory.IsPullback.hasPullback_biprod_fst_biprod_snd 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.BicartesianSq
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroMorphisms C] {X Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] : CategoryTheory.Limits.HasPullback CategoryTheory.Limits.biprod.inl CategoryTheory.Limits.biprod.inr
• CategoryTheory.IsPullback.pullbackBiprodInlBiprodInr 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.BicartesianSq
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroMorphisms C] {X Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] : CategoryTheory.Limits.pullback CategoryTheory.Limits.biprod.inl CategoryTheory.Limits.biprod.inr ≅ 0
• CategoryTheory.BicartesianSq.of_has_biproduct₂ 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.BicartesianSq
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroMorphisms C] {X Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] : CategoryTheory.BicartesianSq 0 0 CategoryTheory.Limits.biprod.inl CategoryTheory.Limits.biprod.inr
• CategoryTheory.IsPullback.inr_fst 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.BicartesianSq
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroMorphisms C] (X Y : C) [CategoryTheory.Limits.HasBinaryBiproduct X Y] : CategoryTheory.IsPullback CategoryTheory.Limits.biprod.inr 0 CategoryTheory.Limits.biprod.fst 0
• CategoryTheory.IsPullback.of_hasBinaryBiproduct 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.BicartesianSq
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroMorphisms C] (X Y : C) [CategoryTheory.Limits.HasBinaryBiproduct X Y] : CategoryTheory.IsPullback 0 0 CategoryTheory.Limits.biprod.inl CategoryTheory.Limits.biprod.inr
• CategoryTheory.IsPushout.inr_fst 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.BicartesianSq
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroMorphisms C] (X Y : C) [CategoryTheory.Limits.HasBinaryBiproduct X Y] : CategoryTheory.IsPushout CategoryTheory.Limits.biprod.inr 0 CategoryTheory.Limits.biprod.fst 0
• CategoryTheory.IsPushout.of_has_biproduct 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.BicartesianSq
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroMorphisms C] (X Y : C) [CategoryTheory.Limits.HasBinaryBiproduct X Y] : CategoryTheory.IsPushout 0 0 CategoryTheory.Limits.biprod.inl CategoryTheory.Limits.biprod.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