Loogle!

Result

Found 55 declarations mentioning CategoryTheory.Abelian.Ext.comp.

• CategoryTheory.Abelian.Ext.comp 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {X Y Z : C} {a b : ℕ} (α : CategoryTheory.Abelian.Ext X Y a) (β : CategoryTheory.Abelian.Ext Y Z b) {c : ℕ} (h : a + b = c) : CategoryTheory.Abelian.Ext X Z c
• CategoryTheory.Abelian.Ext.comp_mk₀_id 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {X Y : C} {n : ℕ} (α : CategoryTheory.Abelian.Ext X Y n) : α.comp (CategoryTheory.Abelian.Ext.mk₀ (CategoryTheory.CategoryStruct.id Y)) ⋯ = α
• CategoryTheory.Abelian.Ext.mk₀_id_comp 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {X Y : C} {n : ℕ} (α : CategoryTheory.Abelian.Ext X Y n) : (CategoryTheory.Abelian.Ext.mk₀ (CategoryTheory.CategoryStruct.id X)).comp α ⋯ = α
• CategoryTheory.Abelian.Ext.mk₀_comp_mk₀ 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) : (CategoryTheory.Abelian.Ext.mk₀ f).comp (CategoryTheory.Abelian.Ext.mk₀ g) ⋯ = CategoryTheory.Abelian.Ext.mk₀ (CategoryTheory.CategoryStruct.comp f g)
• CategoryTheory.Abelian.Ext.comp_assoc_of_second_deg_zero 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {X Y Z T : C} {a₁ a₃ a₁₃ : ℕ} (α : CategoryTheory.Abelian.Ext X Y a₁) (β : CategoryTheory.Abelian.Ext Y Z 0) (γ : CategoryTheory.Abelian.Ext Z T a₃) (h₁₃ : a₁ + a₃ = a₁₃) : (α.comp β ⋯).comp γ h₁₃ = α.comp (β.comp γ ⋯) h₁₃
• CategoryTheory.Abelian.Ext.comp_assoc_of_third_deg_zero 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {X Y Z T : C} {a₁ a₂ a₁₂ : ℕ} (α : CategoryTheory.Abelian.Ext X Y a₁) (β : CategoryTheory.Abelian.Ext Y Z a₂) (γ : CategoryTheory.Abelian.Ext Z T 0) (h₁₂ : a₁ + a₂ = a₁₂) : (α.comp β h₁₂).comp γ ⋯ = α.comp (β.comp γ ⋯) h₁₂
• CategoryTheory.Abelian.Ext.mk₀_comp_mk₀_assoc 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {X Y Z T : C} (f : X ⟶ Y) (g : Y ⟶ Z) {n : ℕ} (α : CategoryTheory.Abelian.Ext Z T n) : (CategoryTheory.Abelian.Ext.mk₀ f).comp ((CategoryTheory.Abelian.Ext.mk₀ g).comp α ⋯) ⋯ = (CategoryTheory.Abelian.Ext.mk₀ (CategoryTheory.CategoryStruct.comp f g)).comp α ⋯
• CategoryTheory.Abelian.Ext.comp_sum 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {X Y Z : C} {p : ℕ} (α : CategoryTheory.Abelian.Ext X Y p) {ι : Type u_1} [Fintype ι] {q : ℕ} (β : ι → CategoryTheory.Abelian.Ext Y Z q) {n : ℕ} (h : p + q = n) : α.comp (∑ i, β i) h = ∑ i, α.comp (β i) h
• CategoryTheory.Abelian.Ext.sum_comp 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {X Y Z : C} {p : ℕ} {ι : Type u_1} [Fintype ι] (α : ι → CategoryTheory.Abelian.Ext X Y p) {q : ℕ} (β : CategoryTheory.Abelian.Ext Y Z q) {n : ℕ} (h : p + q = n) : (∑ i, α i).comp β h = ∑ i, (α i).comp β h
• CategoryTheory.Abelian.Ext.comp_assoc 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {X Y Z T : C} {a₁ a₂ a₃ a₁₂ a₂₃ a : ℕ} (α : CategoryTheory.Abelian.Ext X Y a₁) (β : CategoryTheory.Abelian.Ext Y Z a₂) (γ : CategoryTheory.Abelian.Ext Z T a₃) (h₁₂ : a₁ + a₂ = a₁₂) (h₂₃ : a₂ + a₃ = a₂₃) (h : a₁ + a₂ + a₃ = a) : (α.comp β h₁₂).comp γ ⋯ = α.comp (β.comp γ h₂₃) ⋯
• CategoryTheory.Abelian.Ext.comp_zero 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {X Y : C} {n : ℕ} (α : CategoryTheory.Abelian.Ext X Y n) (Z : C) (m p : ℕ) (h : n + m = p) : α.comp 0 h = 0
• CategoryTheory.Abelian.Ext.zero_comp 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] (X : C) {Y Z : C} (n : ℕ) {m : ℕ} (β : CategoryTheory.Abelian.Ext Y Z m) (p : ℕ) (h : n + m = p) : CategoryTheory.Abelian.Ext.comp 0 β h = 0
• CategoryTheory.Abelian.Ext.comp_neg 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {X Y Z : C} {n : ℕ} (α : CategoryTheory.Abelian.Ext X Y n) {m : ℕ} (β : CategoryTheory.Abelian.Ext Y Z m) {p : ℕ} (h : n + m = p) : α.comp (-β) h = -α.comp β h
• CategoryTheory.Abelian.Ext.neg_comp 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {X Y Z : C} {n : ℕ} (α : CategoryTheory.Abelian.Ext X Y n) {m : ℕ} (β : CategoryTheory.Abelian.Ext Y Z m) {p : ℕ} (h : n + m = p) : (-α).comp β h = -α.comp β h
• CategoryTheory.Abelian.Ext.add_comp 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {X Y Z : C} {n : ℕ} (α₁ α₂ : CategoryTheory.Abelian.Ext X Y n) {m : ℕ} (β : CategoryTheory.Abelian.Ext Y Z m) {p : ℕ} (h : n + m = p) : (α₁ + α₂).comp β h = α₁.comp β h + α₂.comp β h
• CategoryTheory.Abelian.Ext.comp_add 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {X Y Z : C} {n : ℕ} (α : CategoryTheory.Abelian.Ext X Y n) {m : ℕ} (β₁ β₂ : CategoryTheory.Abelian.Ext Y Z m) {p : ℕ} (h : n + m = p) : α.comp (β₁ + β₂) h = α.comp β₁ h + α.comp β₂ h
• CategoryTheory.Abelian.Ext.comp_hom 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {X Y Z : C} [HasDerivedCategory C] {a b : ℕ} (α : CategoryTheory.Abelian.Ext X Y a) (β : CategoryTheory.Abelian.Ext Y Z b) {c : ℕ} (h : a + b = c) : (α.comp β h).hom = α.hom.comp β.hom ⋯
• CategoryTheory.Abelian.extFunctor_map_app 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] (n : ℕ) {X₁ X₂ : Cᵒᵖ} (f : X₁ ⟶ X₂) (Y : C) : ((CategoryTheory.Abelian.extFunctor n).map f).app Y = AddCommGrpCat.ofHom (AddMonoidHom.mk' (fun α => (CategoryTheory.Abelian.Ext.mk₀ f.unop).comp α ⋯) ⋯)
• 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.addEquivBiprod_apply_fst 📋 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 ⊞ Y₂) n) : (CategoryTheory.Abelian.Ext.addEquivBiprod e).1 = e.comp (CategoryTheory.Abelian.Ext.mk₀ CategoryTheory.Limits.biprod.fst) ⋯
• CategoryTheory.Abelian.Ext.addEquivBiprod_apply_snd 📋 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 ⊞ Y₂) n) : (CategoryTheory.Abelian.Ext.addEquivBiprod e).2 = e.comp (CategoryTheory.Abelian.Ext.mk₀ CategoryTheory.Limits.biprod.snd) ⋯
• CategoryTheory.Abelian.Ext.biprodAddEquiv_apply_fst 📋 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).1 = (CategoryTheory.Abelian.Ext.mk₀ CategoryTheory.Limits.biprod.inl).comp e ⋯
• 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.Abelian.Ext.biprodAddEquiv_symm_apply 📋 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 Y n × CategoryTheory.Abelian.Ext X₂ Y n) : CategoryTheory.Abelian.Ext.biprodAddEquiv.symm e = (CategoryTheory.Abelian.Ext.mk₀ CategoryTheory.Limits.biprod.fst).comp e.1 ⋯ + (CategoryTheory.Abelian.Ext.mk₀ CategoryTheory.Limits.biprod.snd).comp e.2 ⋯
• CategoryTheory.Abelian.Ext.bilinearComp_apply_apply 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] (X Y Z : C) (a b c : ℕ) (h : a + b = c) (α : CategoryTheory.Abelian.Ext X Y a) (β : CategoryTheory.Abelian.Ext Y Z b) : ((CategoryTheory.Abelian.Ext.bilinearComp X Y Z a b c h) α) β = α.comp β h
• CategoryTheory.ShortComplex.ShortExact.extClass_naturality 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.ExtClass
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {S₁ S₂ : CategoryTheory.ShortComplex C} (h₁ : S₁.ShortExact) (h₂ : S₂.ShortExact) (f : S₁ ⟶ S₂) : h₁.extClass.comp (CategoryTheory.Abelian.Ext.mk₀ f.τ₁) ⋯ = (CategoryTheory.Abelian.Ext.mk₀ f.τ₃).comp h₂.extClass ⋯
• CategoryTheory.ShortComplex.ShortExact.comp_extClass_assoc 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.ExtClass
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact) {Y : C} {n : ℕ} (γ : CategoryTheory.Abelian.Ext S.X₁ Y n) {n' : ℕ} (h : 1 + n = n') : (CategoryTheory.Abelian.Ext.mk₀ S.g).comp (hS.extClass.comp γ h) ⋯ = 0
• CategoryTheory.ShortComplex.ShortExact.extClass_comp_assoc 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.ExtClass
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact) {Y : C} {n : ℕ} (γ : CategoryTheory.Abelian.Ext S.X₂ Y n) {n' : ℕ} {h : 1 + n = n'} : hS.extClass.comp ((CategoryTheory.Abelian.Ext.mk₀ S.f).comp γ ⋯) h = 0
• CategoryTheory.ShortComplex.ShortExact.comp_extClass 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.ExtClass
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact) : (CategoryTheory.Abelian.Ext.mk₀ S.g).comp hS.extClass ⋯ = 0
• CategoryTheory.ShortComplex.ShortExact.extClass_comp 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.ExtClass
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact) : hS.extClass.comp (CategoryTheory.Abelian.Ext.mk₀ S.f) ⋯ = 0
• CategoryTheory.ShortComplex.ext_mk₀_f_comp_ext_mk₀_g 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.ExtClass
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] (S : CategoryTheory.ShortComplex C) : (CategoryTheory.Abelian.Ext.mk₀ S.f).comp (CategoryTheory.Abelian.Ext.mk₀ S.g) ⋯ = 0
• CategoryTheory.Abelian.Ext.singleFunctor_map_comp_hom 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] [HasDerivedCategory C] {X Y Z : C} (f : X ⟶ Y) {n : ℕ} (x : CategoryTheory.Abelian.Ext Y Z n) : CategoryTheory.CategoryStruct.comp ((DerivedCategory.singleFunctor C 0).map f) x.hom = ((CategoryTheory.Abelian.Ext.mk₀ f).comp x ⋯).hom
• CategoryTheory.Abelian.Ext.contravariant_sequence_exact₁ 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact) (Y : C) {n₀ : ℕ} (x₁ : CategoryTheory.Abelian.Ext S.X₁ Y n₀) {n₁ : ℕ} (hn₁ : 1 + n₀ = n₁) (hx₁ : hS.extClass.comp x₁ hn₁ = 0) : ∃ x₂, (CategoryTheory.Abelian.Ext.mk₀ S.f).comp x₂ ⋯ = x₁
• CategoryTheory.Abelian.Ext.contravariant_sequence_exact₃ 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact) (Y : C) {n₁ : ℕ} (x₃ : CategoryTheory.Abelian.Ext S.X₃ Y n₁) (hx₃ : (CategoryTheory.Abelian.Ext.mk₀ S.g).comp x₃ ⋯ = 0) {n₀ : ℕ} (hn₀ : 1 + n₀ = n₁) : ∃ x₁, hS.extClass.comp x₁ hn₀ = x₃
• CategoryTheory.Abelian.Ext.covariant_sequence_exact₁ 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] (X : C) {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact) {n₁ : ℕ} (x₁ : CategoryTheory.Abelian.Ext X S.X₁ n₁) (hx₁ : x₁.comp (CategoryTheory.Abelian.Ext.mk₀ S.f) ⋯ = 0) {n₀ : ℕ} (hn₀ : n₀ + 1 = n₁) : ∃ x₃, x₃.comp hS.extClass hn₀ = x₁
• CategoryTheory.Abelian.Ext.covariant_sequence_exact₃ 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] (X : C) {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact) {n₀ : ℕ} (x₃ : CategoryTheory.Abelian.Ext X S.X₃ n₀) {n₁ : ℕ} (hn₁ : n₀ + 1 = n₁) (hx₃ : x₃.comp hS.extClass hn₁ = 0) : ∃ x₂, x₂.comp (CategoryTheory.Abelian.Ext.mk₀ S.g) ⋯ = x₃
• CategoryTheory.Abelian.Ext.contravariant_sequence_exact₂ 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact) (Y : C) {n : ℕ} (x₂ : CategoryTheory.Abelian.Ext S.X₂ Y n) (hx₂ : (CategoryTheory.Abelian.Ext.mk₀ S.f).comp x₂ ⋯ = 0) : ∃ x₁, (CategoryTheory.Abelian.Ext.mk₀ S.g).comp x₁ ⋯ = x₂
• CategoryTheory.Abelian.Ext.covariant_sequence_exact₂ 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] (X : C) {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact) {n : ℕ} (x₂ : CategoryTheory.Abelian.Ext X S.X₂ n) (hx₂ : x₂.comp (CategoryTheory.Abelian.Ext.mk₀ S.g) ⋯ = 0) : ∃ x₁, x₁.comp (CategoryTheory.Abelian.Ext.mk₀ S.f) ⋯ = x₂
• CategoryTheory.Abelian.Ext.hom_comp_singleFunctor_map_shift 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] [HasDerivedCategory C] {X Y Z : C} {n : ℕ} (x : CategoryTheory.Abelian.Ext X Y n) (f : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp x.hom ((CategoryTheory.shiftFunctor (DerivedCategory C) ↑n).map ((DerivedCategory.singleFunctor C 0).map f)) = (x.comp (CategoryTheory.Abelian.Ext.mk₀ f) ⋯).hom
• CategoryTheory.Abelian.Ext.preadditiveCoyoneda_homologySequenceδ_singleTriangle_apply 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact) [HasDerivedCategory C] {X : C} {n₀ : ℕ} (x : CategoryTheory.Abelian.Ext X S.X₃ n₀) {n₁ : ℕ} (h : n₀ + 1 = n₁) : (CategoryTheory.ConcreteCategory.hom ((CategoryTheory.preadditiveCoyoneda.obj (Opposite.op ((DerivedCategory.singleFunctor C 0).obj X))).homologySequenceδ hS.singleTriangle ↑n₀ ↑n₁ ⋯)) x.hom = (x.comp hS.extClass h).hom
• CategoryTheory.Abelian.Ext.preadditiveYoneda_homologySequenceδ_singleTriangle_apply 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact) [HasDerivedCategory C] {Y : C} {n₀ : ℕ} (x : CategoryTheory.Abelian.Ext S.X₁ Y n₀) {n₁ : ℕ} (h : 1 + n₀ = n₁) : (CategoryTheory.ConcreteCategory.hom ((CategoryTheory.preadditiveYoneda.obj ((DerivedCategory.singleFunctor C 0).obj Y)).homologySequenceδ ((CategoryTheory.Pretriangulated.triangleOpEquivalence (DerivedCategory C)).functor.obj (Opposite.op hS.singleTriangle)) ↑n₀ ↑n₁ ⋯)) x.hom = (hS.extClass.comp x h).hom
• CategoryTheory.Abelian.Ext.comp_smul 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Linear
  {R : Type t} [Ring R] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.Linear R C] [CategoryTheory.HasExt C] {X Y Z : C} {a b : ℕ} (α : CategoryTheory.Abelian.Ext X Y a) (β : CategoryTheory.Abelian.Ext Y Z b) {c : ℕ} (h : a + b = c) (r : R) : α.comp (r • β) h = r • α.comp β h
• CategoryTheory.Abelian.Ext.smul_comp 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Linear
  {R : Type t} [Ring R] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.Linear R C] [CategoryTheory.HasExt C] {X Y Z : C} {a b : ℕ} (α : CategoryTheory.Abelian.Ext X Y a) (β : CategoryTheory.Abelian.Ext Y Z b) {c : ℕ} (h : a + b = c) (r : R) : (r • α).comp β h = r • α.comp β h
• CategoryTheory.Abelian.Ext.smul_eq_comp_mk₀ 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Linear
  {R : Type t} [Ring R] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.Linear R C] [CategoryTheory.HasExt C] {X Y : C} {n : ℕ} (x : CategoryTheory.Abelian.Ext X Y n) (r : R) : r • x = x.comp (CategoryTheory.Abelian.Ext.mk₀ (r • CategoryTheory.CategoryStruct.id Y)) ⋯
• CategoryTheory.Abelian.Ext.bilinearCompOfLinear_apply_apply 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Linear
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] (R : Type t) [CommRing R] [CategoryTheory.Linear R C] (X Y Z : C) (a b c : ℕ) (h : a + b = c) (α : CategoryTheory.Abelian.Ext X Y a) (β : CategoryTheory.Abelian.Ext Y Z b) : ((CategoryTheory.Abelian.Ext.bilinearCompOfLinear R X Y Z a b c h) α) β = α.comp β h
• CategoryTheory.Abelian.Ext.mapExactFunctor_comp 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Map
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] {D : Type u_2} [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Abelian D] (F : CategoryTheory.Functor C D) [F.Additive] [CategoryTheory.Limits.PreservesFiniteLimits F] [CategoryTheory.Limits.PreservesFiniteColimits F] [CategoryTheory.HasExt C] [CategoryTheory.HasExt D] {X Y Z : C} {a b : ℕ} (α : CategoryTheory.Abelian.Ext X Y a) (β : CategoryTheory.Abelian.Ext Y Z b) {c : ℕ} (h : a + b = c) : CategoryTheory.Abelian.Ext.mapExactFunctor F (α.comp β h) = (CategoryTheory.Abelian.Ext.mapExactFunctor F α).comp (CategoryTheory.Abelian.Ext.mapExactFunctor F β) h
• CategoryTheory.Abelian.Ext.mapExactFunctor_comp_mk₀_natTransApp 📋 Mathlib.Algebra.Homology.DerivedCategory.Ext.Map
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] {D : Type u_2} [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Abelian D] [CategoryTheory.HasExt C] [CategoryTheory.HasExt D] {X Y : C} {n : ℕ} (α : CategoryTheory.Abelian.Ext X Y n) {F G : CategoryTheory.Functor C D} [F.Additive] [G.Additive] [CategoryTheory.Limits.PreservesFiniteLimits F] [CategoryTheory.Limits.PreservesFiniteColimits F] [CategoryTheory.Limits.PreservesFiniteLimits G] [CategoryTheory.Limits.PreservesFiniteColimits G] (τ : F ⟶ G) : (CategoryTheory.Abelian.Ext.mapExactFunctor F α).comp (CategoryTheory.Abelian.Ext.mk₀ (τ.app Y)) ⋯ = (CategoryTheory.Abelian.Ext.mk₀ (τ.app X)).comp (CategoryTheory.Abelian.Ext.mapExactFunctor G α) ⋯
• CategoryTheory.Sheaf.H.map_apply 📋 Mathlib.CategoryTheory.Sites.SheafCohomology.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] {J : CategoryTheory.GrothendieckTopology C} [CategoryTheory.HasSheafify J AddCommGrpCat] [CategoryTheory.HasExt (CategoryTheory.Sheaf J AddCommGrpCat)] {F G : CategoryTheory.Sheaf J AddCommGrpCat} (f : F ⟶ G) {n : ℕ} (x : F.H n) : (CategoryTheory.Sheaf.H.map f n) x = CategoryTheory.Abelian.Ext.comp x (CategoryTheory.Abelian.Ext.mk₀ f) ⋯
• CategoryTheory.InjectiveResolution.mk₀_comp_extMk 📋 Mathlib.CategoryTheory.Abelian.Injective.Ext
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {X Y : C} (R : CategoryTheory.InjectiveResolution Y) {n : ℕ} (f : X ⟶ R.cocomplex.X n) (m : ℕ) (hm : n + 1 = m) (hf : CategoryTheory.CategoryStruct.comp f (R.cocomplex.d n m) = 0) {X' : C} (g : X' ⟶ X) : (CategoryTheory.Abelian.Ext.mk₀ g).comp (R.extMk f m hm hf) ⋯ = R.extMk (CategoryTheory.CategoryStruct.comp g f) m hm ⋯
• CategoryTheory.InjectiveResolution.extMk_comp_mk₀ 📋 Mathlib.CategoryTheory.Abelian.Injective.Ext
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {X Y : C} {R : CategoryTheory.InjectiveResolution Y} {n : ℕ} (f : X ⟶ R.cocomplex.X n) (m : ℕ) (hm : n + 1 = m) (hf : CategoryTheory.CategoryStruct.comp f (R.cocomplex.d n m) = 0) {Y' : C} {R' : CategoryTheory.InjectiveResolution Y'} {g : Y ⟶ Y'} (φ : R.Hom R' g) : (R.extMk f m hm hf).comp (CategoryTheory.Abelian.Ext.mk₀ g) ⋯ = R'.extMk (CategoryTheory.CategoryStruct.comp f (φ.hom.f n)) m hm ⋯
• CategoryTheory.ProjectiveResolution.extMk_comp_mk₀ 📋 Mathlib.CategoryTheory.Abelian.Projective.Ext
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {X Y : C} (R : CategoryTheory.ProjectiveResolution X) {n : ℕ} (f : R.complex.X n ⟶ Y) (m : ℕ) (hm : n + 1 = m) (hf : CategoryTheory.CategoryStruct.comp (R.complex.d m n) f = 0) {Y' : C} (g : Y ⟶ Y') : (R.extMk f m hm hf).comp (CategoryTheory.Abelian.Ext.mk₀ g) ⋯ = R.extMk (CategoryTheory.CategoryStruct.comp f g) m hm ⋯
• CategoryTheory.ProjectiveResolution.mk₀_comp_extMk 📋 Mathlib.CategoryTheory.Abelian.Projective.Ext
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasExt C] {X Y : C} {R : CategoryTheory.ProjectiveResolution X} {n : ℕ} (f : R.complex.X n ⟶ Y) (m : ℕ) (hm : n + 1 = m) (hf : CategoryTheory.CategoryStruct.comp (R.complex.d m n) f = 0) {X' : C} {R' : CategoryTheory.ProjectiveResolution X'} {g : X' ⟶ X} (φ : R'.Hom R g) : (CategoryTheory.Abelian.Ext.mk₀ g).comp (R.extMk f m hm hf) ⋯ = R'.extMk (CategoryTheory.CategoryStruct.comp (φ.hom.f n) f) m hm ⋯
• CategoryTheory.GrothendieckTopology.MayerVietorisSquare.mk₀_f_comp_biprodAddEquiv_symm_biprodIsoProd_hom 📋 Mathlib.CategoryTheory.Sites.SheafCohomology.MayerVietoris
  {C : Type u} [CategoryTheory.Category.{v, u} C] {J : CategoryTheory.GrothendieckTopology C} [CategoryTheory.HasWeakSheafify J (Type v)] [CategoryTheory.HasSheafify J AddCommGrpCat] [CategoryTheory.HasExt (CategoryTheory.Sheaf J AddCommGrpCat)] (S : J.MayerVietorisSquare) (F : CategoryTheory.Sheaf J AddCommGrpCat) {n : ℕ} (x : ↑(F.H' n S.X₂ ⊞ F.H' n S.X₃)) : (CategoryTheory.Abelian.Ext.mk₀ S.shortComplex.f).comp (CategoryTheory.Abelian.Ext.biprodAddEquiv.symm ((CategoryTheory.ConcreteCategory.hom ((F.H' n S.X₂).biprodIsoProd (F.H' n S.X₃)).hom) x)) ⋯ = (CategoryTheory.ConcreteCategory.hom (S.fromBiprod F n)) x
• CategoryTheory.GrothendieckTopology.MayerVietorisSquare.biprodAddEquiv_symm_biprodIsoProd_hom_toBiprod_apply 📋 Mathlib.CategoryTheory.Sites.SheafCohomology.MayerVietoris
  {C : Type u} [CategoryTheory.Category.{v, u} C] {J : CategoryTheory.GrothendieckTopology C} [CategoryTheory.HasWeakSheafify J (Type v)] [CategoryTheory.HasSheafify J AddCommGrpCat] [CategoryTheory.HasExt (CategoryTheory.Sheaf J AddCommGrpCat)] (S : J.MayerVietorisSquare) (F : CategoryTheory.Sheaf J AddCommGrpCat) {n : ℕ} (x : ↑(F.H' n S.X₄)) : CategoryTheory.Abelian.Ext.biprodAddEquiv.symm ((CategoryTheory.ConcreteCategory.hom ((F.H' n S.X₂).biprodIsoProd (F.H' n S.X₃)).hom) ((CategoryTheory.ConcreteCategory.hom (S.toBiprod F n)) x)) = (CategoryTheory.Abelian.Ext.mk₀ S.shortComplex.g).comp x ⋯

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

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 69fae59