Loogle!

Result

Found 61 declarations mentioning CategoryTheory.Functor.IsRightDerivedFunctor.

• CategoryTheory.Functor.IsRightDerivedFunctor 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H] (RF : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] : Prop
• CategoryTheory.Functor.instIsRightDerivedFunctorTotalRightDerivedTotalRightDerivedUnit 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H] (F : CategoryTheory.Functor C H) (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [F.HasRightDerivedFunctor W] : (F.totalRightDerived L W).IsRightDerivedFunctor (F.totalRightDerivedUnit L W) W
• CategoryTheory.Functor.HasRightDerivedFunctor.mk' 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H] (RF : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) {W : CategoryTheory.MorphismProperty C} [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] : F.HasRightDerivedFunctor W
• CategoryTheory.Functor.IsRightDerivedFunctor.isLeftKanExtension 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_1} {D : Type u_2} {H : Type u_3} {inst✝ : CategoryTheory.Category.{v_1, u_1} C} {inst✝¹ : CategoryTheory.Category.{v_2, u_2} D} {inst✝² : CategoryTheory.Category.{v_3, u_3} H} (RF : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) {inst✝³ : L.IsLocalization W} [self : RF.IsRightDerivedFunctor α W] : RF.IsLeftKanExtension α
• CategoryTheory.Functor.IsRightDerivedFunctor.mk 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H] {RF : CategoryTheory.Functor D H} {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} {α : F ⟶ L.comp RF} {W : CategoryTheory.MorphismProperty C} [L.IsLocalization W] (isLeftKanExtension : RF.IsLeftKanExtension α) : RF.IsRightDerivedFunctor α W
• CategoryTheory.Functor.isRightDerivedFunctor_iff_isLeftKanExtension 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H] (RF : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] : RF.IsRightDerivedFunctor α W ↔ RF.IsLeftKanExtension α
• CategoryTheory.Functor.rightDerivedDesc 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H] (RF : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (G : CategoryTheory.Functor D H) (β : F ⟶ L.comp G) : RF ⟶ G
• CategoryTheory.Functor.rightDerivedUnique 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H] (RF RF' : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (α'₂ : F ⟶ L.comp RF') (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] [RF'.IsRightDerivedFunctor α'₂ W] : RF ≅ RF'
• CategoryTheory.Functor.rightDerivedNatTrans_id 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_3} {D : Type u_1} {H : Type u_2} [CategoryTheory.Category.{v_1, u_3} C] [CategoryTheory.Category.{v_2, u_1} D] [CategoryTheory.Category.{v_3, u_2} H] (RF : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] : RF.rightDerivedNatTrans RF α α W (CategoryTheory.CategoryStruct.id F) = CategoryTheory.CategoryStruct.id RF
• CategoryTheory.Functor.isRightDerivedFunctor_iff_isIso_rightDerivedDesc 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_3} {D : Type u_1} {H : Type u_2} [CategoryTheory.Category.{v_1, u_3} C] [CategoryTheory.Category.{v_2, u_1} D] [CategoryTheory.Category.{v_3, u_2} H] (RF : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (G : CategoryTheory.Functor D H) (β : F ⟶ L.comp G) : G.IsRightDerivedFunctor β W ↔ CategoryTheory.IsIso (RF.rightDerivedDesc α W G β)
• CategoryTheory.Functor.rightDerivedNatIso 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H] (RF RF' : CategoryTheory.Functor D H) {F F' : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (α' : F' ⟶ L.comp RF') (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] [RF'.IsRightDerivedFunctor α' W] (τ : F ≅ F') : RF ≅ RF'
• CategoryTheory.Functor.rightDerivedNatTrans 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H] (RF RF' : CategoryTheory.Functor D H) {F F' : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (α' : F' ⟶ L.comp RF') (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (τ : F ⟶ F') : RF ⟶ RF'
• CategoryTheory.Functor.rightDerived_fac 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_3} {D : Type u_1} {H : Type u_2} [CategoryTheory.Category.{v_1, u_3} C] [CategoryTheory.Category.{v_2, u_1} D] [CategoryTheory.Category.{v_3, u_2} H] (RF : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (G : CategoryTheory.Functor D H) (β : F ⟶ L.comp G) : CategoryTheory.CategoryStruct.comp α (L.whiskerLeft (RF.rightDerivedDesc α W G β)) = β
• CategoryTheory.Functor.rightDerivedNatIso_hom 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H] (RF RF' : CategoryTheory.Functor D H) {F F' : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (α' : F' ⟶ L.comp RF') (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] [RF'.IsRightDerivedFunctor α' W] (τ : F ≅ F') : (RF.rightDerivedNatIso RF' α α' W τ).hom = RF.rightDerivedNatTrans RF' α α' W τ.hom
• CategoryTheory.Functor.rightDerivedNatIso_inv 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H] (RF RF' : CategoryTheory.Functor D H) {F F' : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (α' : F' ⟶ L.comp RF') (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] [RF'.IsRightDerivedFunctor α' W] (τ : F ≅ F') : (RF.rightDerivedNatIso RF' α α' W τ).inv = RF'.rightDerivedNatTrans RF α' α W τ.inv
• CategoryTheory.Functor.isRightDerivedFunctor_iff_of_iso 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_1} {D : Type u_3} {H : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_3} D] [CategoryTheory.Category.{v_3, u_2} H] {RF RF' : CategoryTheory.Functor D H} {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (α' : F ⟶ L.comp RF') (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] (e : RF ≅ RF') (comm : CategoryTheory.CategoryStruct.comp α (L.whiskerLeft e.hom) = α') : RF.IsRightDerivedFunctor α W ↔ RF'.IsRightDerivedFunctor α' W
• CategoryTheory.Functor.rightDerived_fac_app 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_3} {D : Type u_1} {H : Type u_2} [CategoryTheory.Category.{v_1, u_3} C] [CategoryTheory.Category.{v_2, u_1} D] [CategoryTheory.Category.{v_3, u_2} H] (RF : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (G : CategoryTheory.Functor D H) (β : F ⟶ L.comp G) (X : C) : CategoryTheory.CategoryStruct.comp (α.app X) ((RF.rightDerivedDesc α W G β).app (L.obj X)) = β.app X
• CategoryTheory.Functor.instIsRightDerivedFunctorCompCompWhiskerRightHomAssociatorOfIsLeftAdjoint 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_3} {D : Type u_4} {H : Type u_1} {H' : Type u_2} [CategoryTheory.Category.{v_1, u_3} C] [CategoryTheory.Category.{v_2, u_4} D] [CategoryTheory.Category.{v_3, u_1} H] [CategoryTheory.Category.{v_4, u_2} H'] (RF : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (G : CategoryTheory.Functor H H') [G.IsLeftAdjoint] : (RF.comp G).IsRightDerivedFunctor (CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight α G) (L.associator RF G).hom) W
• CategoryTheory.Functor.rightDerivedNatTrans_fac 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_1} {D : Type u_3} {H : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_3} D] [CategoryTheory.Category.{v_3, u_2} H] (RF RF' : CategoryTheory.Functor D H) {F F' : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (α' : F' ⟶ L.comp RF') (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (τ : F ⟶ F') : CategoryTheory.CategoryStruct.comp α (L.whiskerLeft (RF.rightDerivedNatTrans RF' α α' W τ)) = CategoryTheory.CategoryStruct.comp τ α'
• CategoryTheory.Functor.rightDerived_ext 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_3} {D : Type u_1} {H : Type u_2} [CategoryTheory.Category.{v_1, u_3} C] [CategoryTheory.Category.{v_2, u_1} D] [CategoryTheory.Category.{v_3, u_2} H] (RF : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (G : CategoryTheory.Functor D H) (γ₁ γ₂ : RF ⟶ G) (hγ : CategoryTheory.CategoryStruct.comp α (L.whiskerLeft γ₁) = CategoryTheory.CategoryStruct.comp α (L.whiskerLeft γ₂)) : γ₁ = γ₂
• CategoryTheory.Functor.rightDerived_fac_assoc 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_3} {D : Type u_1} {H : Type u_2} [CategoryTheory.Category.{v_1, u_3} C] [CategoryTheory.Category.{v_2, u_1} D] [CategoryTheory.Category.{v_3, u_2} H] (RF : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (G : CategoryTheory.Functor D H) (β : F ⟶ L.comp G) {Z : CategoryTheory.Functor C H} (h : L.comp G ⟶ Z) : CategoryTheory.CategoryStruct.comp α (CategoryTheory.CategoryStruct.comp (L.whiskerLeft (RF.rightDerivedDesc α W G β)) h) = CategoryTheory.CategoryStruct.comp β h
• CategoryTheory.Functor.rightDerived_fac_app_assoc 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_3} {D : Type u_1} {H : Type u_2} [CategoryTheory.Category.{v_1, u_3} C] [CategoryTheory.Category.{v_2, u_1} D] [CategoryTheory.Category.{v_3, u_2} H] (RF : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (G : CategoryTheory.Functor D H) (β : F ⟶ L.comp G) (X : C) {Z : H} (h : G.obj (L.obj X) ⟶ Z) : CategoryTheory.CategoryStruct.comp (α.app X) (CategoryTheory.CategoryStruct.comp ((RF.rightDerivedDesc α W G β).app (L.obj X)) h) = CategoryTheory.CategoryStruct.comp (β.app X) h
• CategoryTheory.Functor.rightDerivedNatTrans_app 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_1} {D : Type u_3} {H : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_3} D] [CategoryTheory.Category.{v_3, u_2} H] (RF RF' : CategoryTheory.Functor D H) {F F' : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (α' : F' ⟶ L.comp RF') (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (τ : F ⟶ F') (X : C) : CategoryTheory.CategoryStruct.comp (α.app X) ((RF.rightDerivedNatTrans RF' α α' W τ).app (L.obj X)) = CategoryTheory.CategoryStruct.comp (τ.app X) (α'.app X)
• CategoryTheory.Functor.rightDerivedNatTrans_comp 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_1} {D : Type u_3} {H : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_3} D] [CategoryTheory.Category.{v_3, u_2} H] (RF RF' RF'' : CategoryTheory.Functor D H) {F F' F'' : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (α' : F' ⟶ L.comp RF') (α'' : F'' ⟶ L.comp RF'') (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] [RF'.IsRightDerivedFunctor α' W] (τ : F ⟶ F') (τ' : F' ⟶ F'') : CategoryTheory.CategoryStruct.comp (RF.rightDerivedNatTrans RF' α α' W τ) (RF'.rightDerivedNatTrans RF'' α' α'' W τ') = RF.rightDerivedNatTrans RF'' α α'' W (CategoryTheory.CategoryStruct.comp τ τ')
• CategoryTheory.Functor.rightDerivedNatTrans_fac_assoc 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_1} {D : Type u_3} {H : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_3} D] [CategoryTheory.Category.{v_3, u_2} H] (RF RF' : CategoryTheory.Functor D H) {F F' : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (α' : F' ⟶ L.comp RF') (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (τ : F ⟶ F') {Z : CategoryTheory.Functor C H} (h : L.comp RF' ⟶ Z) : CategoryTheory.CategoryStruct.comp α (CategoryTheory.CategoryStruct.comp (L.whiskerLeft (RF.rightDerivedNatTrans RF' α α' W τ)) h) = CategoryTheory.CategoryStruct.comp τ (CategoryTheory.CategoryStruct.comp α' h)
• CategoryTheory.Functor.rightDerivedNatTrans_app_assoc 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_1} {D : Type u_3} {H : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_3} D] [CategoryTheory.Category.{v_3, u_2} H] (RF RF' : CategoryTheory.Functor D H) {F F' : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (α' : F' ⟶ L.comp RF') (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (τ : F ⟶ F') (X : C) {Z : H} (h : RF'.obj (L.obj X) ⟶ Z) : CategoryTheory.CategoryStruct.comp (α.app X) (CategoryTheory.CategoryStruct.comp ((RF.rightDerivedNatTrans RF' α α' W τ).app (L.obj X)) h) = CategoryTheory.CategoryStruct.comp (τ.app X) (CategoryTheory.CategoryStruct.comp (α'.app X) h)
• CategoryTheory.Functor.rightDerivedNatTrans_comp_assoc 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C : Type u_1} {D : Type u_3} {H : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_3} D] [CategoryTheory.Category.{v_3, u_2} H] (RF RF' RF'' : CategoryTheory.Functor D H) {F F' F'' : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (α' : F' ⟶ L.comp RF') (α'' : F'' ⟶ L.comp RF'') (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] [RF'.IsRightDerivedFunctor α' W] (τ : F ⟶ F') (τ' : F' ⟶ F'') {Z : CategoryTheory.Functor D H} (h : RF'' ⟶ Z) : CategoryTheory.CategoryStruct.comp (RF.rightDerivedNatTrans RF' α α' W τ) (CategoryTheory.CategoryStruct.comp (RF'.rightDerivedNatTrans RF'' α' α'' W τ') h) = CategoryTheory.CategoryStruct.comp (RF.rightDerivedNatTrans RF'' α α'' W (CategoryTheory.CategoryStruct.comp τ τ')) h
• CategoryTheory.LocalizerMorphism.isRightDerivedFunctor_iff_precomp 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerived
  {C₁ : Type u_1} {C₂ : Type u_2} {H₁ : Type u_3} {H₂ : Type u_4} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C₁] [CategoryTheory.Category.{v_2, u_2} C₂] [CategoryTheory.Category.{v_3, u_5} D] [CategoryTheory.Category.{v_4, u_3} H₁] [CategoryTheory.Category.{v_5, u_4} H₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) [Φ.IsLocalizedEquivalence] [Φ.functor.IsEquivalence] (L₁ : CategoryTheory.Functor C₁ H₁) (L₂ : CategoryTheory.Functor C₂ H₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] (G : CategoryTheory.Functor H₁ H₂) (iso : Φ.functor.comp L₂ ≅ L₁.comp G) {F₁ : CategoryTheory.Functor C₁ D} {RF₁ : CategoryTheory.Functor H₁ D} (α₁ : F₁ ⟶ L₁.comp RF₁) {F₂ : CategoryTheory.Functor C₂ D} {RF₂ : CategoryTheory.Functor H₂ D} (α₂ : F₂ ⟶ L₂.comp RF₂) (e₁ : Φ.functor.comp F₂ ≅ F₁) (e₂ : G.comp RF₂ ≅ RF₁) (h : α₁ = CategoryTheory.CategoryStruct.comp e₁.inv (CategoryTheory.CategoryStruct.comp (Φ.functor.whiskerLeft α₂) (CategoryTheory.CategoryStruct.comp (Φ.functor.associator L₂ RF₂).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight iso.hom RF₂) (CategoryTheory.CategoryStruct.comp (L₁.associator G RF₂).hom (L₁.whiskerLeft e₂.hom))))) := by cat_disch) : RF₂.IsRightDerivedFunctor α₂ W₂ ↔ RF₁.IsRightDerivedFunctor α₁ W₁
• CategoryTheory.Functor.isRightDerivedFunctor_of_inverts 📋 Mathlib.CategoryTheory.Functor.Derived.PointwiseRightDerived
  {C : Type u₁} {D : Type u₂} {H : Type u₃} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.Category.{v₃, u₃} H] {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] (F' : CategoryTheory.Functor D H) (e : L.comp F' ≅ F) : F'.IsRightDerivedFunctor e.inv W
• CategoryTheory.Functor.instIsRightDerivedFunctorLiftInvFac 📋 Mathlib.CategoryTheory.Functor.Derived.PointwiseRightDerived
  {C : Type u₁} {D : Type u₂} {H : Type u₃} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.Category.{v₃, u₃} H] {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] (hF : W.IsInvertedBy F) : (CategoryTheory.Localization.lift F hF L).IsRightDerivedFunctor (CategoryTheory.Localization.fac F hF L).inv W
• CategoryTheory.Functor.isPointwiseLeftKanExtensionOfHasPointwiseRightDerivedFunctor 📋 Mathlib.CategoryTheory.Functor.Derived.PointwiseRightDerived
  {C : Type u₁} {D : Type u₂} {H : Type u₃} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.Category.{v₃, u₃} H] (F' : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp F') (W : CategoryTheory.MorphismProperty C) [F.HasPointwiseRightDerivedFunctor W] [L.IsLocalization W] [F'.IsRightDerivedFunctor α W] : (CategoryTheory.Functor.LeftExtension.mk F' α).IsPointwiseLeftKanExtension
• CategoryTheory.Functor.isIso_of_isRightDerivedFunctor_of_inverts 📋 Mathlib.CategoryTheory.Functor.Derived.PointwiseRightDerived
  {C : Type u₁} {D : Type u₂} {H : Type u₃} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.Category.{v₃, u₃} H] {L : CategoryTheory.Functor C D} {W : CategoryTheory.MorphismProperty C} [L.IsLocalization W] {F : CategoryTheory.Functor C H} (RF : CategoryTheory.Functor D H) (α : F ⟶ L.comp RF) (hF : W.IsInvertedBy F) [RF.IsRightDerivedFunctor α W] : CategoryTheory.IsIso α
• CategoryTheory.Functor.isRightDerivedFunctor_iff_of_inverts 📋 Mathlib.CategoryTheory.Functor.Derived.PointwiseRightDerived
  {C : Type u₁} {D : Type u₂} {H : Type u₃} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.Category.{v₃, u₃} H] {L : CategoryTheory.Functor C D} {W : CategoryTheory.MorphismProperty C} [L.IsLocalization W] {F : CategoryTheory.Functor C H} (RF : CategoryTheory.Functor D H) (α : F ⟶ L.comp RF) (hF : W.IsInvertedBy F) : RF.IsRightDerivedFunctor α W ↔ CategoryTheory.IsIso α
• CategoryTheory.LocalizerMorphism.rightDerivedFunctorComparison 📋 Mathlib.CategoryTheory.Localization.DerivabilityStructure.PointwiseRightDerived
  {C₁ : Type u₁} {C₂ : Type u₂} {H : Type u₃} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] [CategoryTheory.Category.{v₃, u₃} H] {D₁ : Type u₄} {D₂ : Type u₅} [CategoryTheory.Category.{v₄, u₄} D₁] [CategoryTheory.Category.{v₅, u₅} D₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (L₁ : CategoryTheory.Functor C₁ D₁) (L₂ : CategoryTheory.Functor C₂ D₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] (F : CategoryTheory.Functor C₂ H) (F₁ : CategoryTheory.Functor D₁ H) (α₁ : Φ.functor.comp F ⟶ L₁.comp F₁) (F₂ : CategoryTheory.Functor D₂ H) (α₂ : F ⟶ L₂.comp F₂) [F₁.IsRightDerivedFunctor α₁ W₁] : F₁ ⟶ (Φ.localizedFunctor L₁ L₂).comp F₂
• CategoryTheory.LocalizerMorphism.instIsIsoFunctorRightDerivedFunctorComparison 📋 Mathlib.CategoryTheory.Localization.DerivabilityStructure.PointwiseRightDerived
  {C₁ : Type u₁} {C₂ : Type u₂} {H : Type u₃} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] [CategoryTheory.Category.{v₃, u₃} H] {D₁ : Type u₄} {D₂ : Type u₅} [CategoryTheory.Category.{v₄, u₄} D₁] [CategoryTheory.Category.{v₅, u₅} D₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (L₁ : CategoryTheory.Functor C₁ D₁) (L₂ : CategoryTheory.Functor C₂ D₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] (F : CategoryTheory.Functor C₂ H) (F₁ : CategoryTheory.Functor D₁ H) (α₁ : Φ.functor.comp F ⟶ L₁.comp F₁) (F₂ : CategoryTheory.Functor D₂ H) (α₂ : F ⟶ L₂.comp F₂) [F₁.IsRightDerivedFunctor α₁ W₁] [Φ.IsRightDerivabilityStructure] [(Φ.functor.comp F).HasPointwiseRightDerivedFunctor W₁] [F₂.IsRightDerivedFunctor α₂ W₂] : CategoryTheory.IsIso (Φ.rightDerivedFunctorComparison L₁ L₂ F F₁ α₁ F₂ α₂)
• CategoryTheory.LocalizerMorphism.isIso_iff_of_isRightDerivabilityStructure 📋 Mathlib.CategoryTheory.Localization.DerivabilityStructure.PointwiseRightDerived
  {C₁ : Type u₁} {C₂ : Type u₂} {H : Type u₃} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] [CategoryTheory.Category.{v₃, u₃} H] {D₁ : Type u₄} {D₂ : Type u₅} [CategoryTheory.Category.{v₄, u₄} D₁] [CategoryTheory.Category.{v₅, u₅} D₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (L₁ : CategoryTheory.Functor C₁ D₁) (L₂ : CategoryTheory.Functor C₂ D₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] (F : CategoryTheory.Functor C₂ H) (F₁ : CategoryTheory.Functor D₁ H) (α₁ : Φ.functor.comp F ⟶ L₁.comp F₁) (F₂ : CategoryTheory.Functor D₂ H) (α₂ : F ⟶ L₂.comp F₂) [F₁.IsRightDerivedFunctor α₁ W₁] [Φ.IsRightDerivabilityStructure] [(Φ.functor.comp F).HasPointwiseRightDerivedFunctor W₁] [F₂.IsRightDerivedFunctor α₂ W₂] (X : C₁) : CategoryTheory.IsIso (α₁.app X) ↔ CategoryTheory.IsIso (α₂.app (Φ.functor.obj X))
• CategoryTheory.LocalizerMorphism.rightDerivedFunctorComparison_fac_app 📋 Mathlib.CategoryTheory.Localization.DerivabilityStructure.PointwiseRightDerived
  {C₁ : Type u₁} {C₂ : Type u₂} {H : Type u₃} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] [CategoryTheory.Category.{v₃, u₃} H] {D₁ : Type u₄} {D₂ : Type u₅} [CategoryTheory.Category.{v₄, u₄} D₁] [CategoryTheory.Category.{v₅, u₅} D₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (L₁ : CategoryTheory.Functor C₁ D₁) (L₂ : CategoryTheory.Functor C₂ D₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] (F : CategoryTheory.Functor C₂ H) (F₁ : CategoryTheory.Functor D₁ H) (α₁ : Φ.functor.comp F ⟶ L₁.comp F₁) (F₂ : CategoryTheory.Functor D₂ H) (α₂ : F ⟶ L₂.comp F₂) [F₁.IsRightDerivedFunctor α₁ W₁] (X : C₁) : CategoryTheory.CategoryStruct.comp (α₁.app X) ((Φ.rightDerivedFunctorComparison L₁ L₂ F F₁ α₁ F₂ α₂).app (L₁.obj X)) = CategoryTheory.CategoryStruct.comp (α₂.app (Φ.functor.obj X)) (F₂.map ((CategoryTheory.CatCommSq.iso Φ.functor L₁ L₂ (Φ.localizedFunctor L₁ L₂)).hom.app X))
• CategoryTheory.LocalizerMorphism.rightDerivedFunctorComparison_fac_app_assoc 📋 Mathlib.CategoryTheory.Localization.DerivabilityStructure.PointwiseRightDerived
  {C₁ : Type u₁} {C₂ : Type u₂} {H : Type u₃} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] [CategoryTheory.Category.{v₃, u₃} H] {D₁ : Type u₄} {D₂ : Type u₅} [CategoryTheory.Category.{v₄, u₄} D₁] [CategoryTheory.Category.{v₅, u₅} D₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (L₁ : CategoryTheory.Functor C₁ D₁) (L₂ : CategoryTheory.Functor C₂ D₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] (F : CategoryTheory.Functor C₂ H) (F₁ : CategoryTheory.Functor D₁ H) (α₁ : Φ.functor.comp F ⟶ L₁.comp F₁) (F₂ : CategoryTheory.Functor D₂ H) (α₂ : F ⟶ L₂.comp F₂) [F₁.IsRightDerivedFunctor α₁ W₁] (X : C₁) {Z : H} (h : F₂.obj ((Φ.localizedFunctor L₁ L₂).obj (L₁.obj X)) ⟶ Z) : CategoryTheory.CategoryStruct.comp (α₁.app X) (CategoryTheory.CategoryStruct.comp ((Φ.rightDerivedFunctorComparison L₁ L₂ F F₁ α₁ F₂ α₂).app (L₁.obj X)) h) = CategoryTheory.CategoryStruct.comp (α₂.app (Φ.functor.obj X)) (CategoryTheory.CategoryStruct.comp (F₂.map ((CategoryTheory.CatCommSq.iso Φ.functor L₁ L₂ (Φ.localizedFunctor L₁ L₂)).hom.app X)) h)
• CategoryTheory.LocalizerMorphism.rightDerivedFunctorComparison_fac 📋 Mathlib.CategoryTheory.Localization.DerivabilityStructure.PointwiseRightDerived
  {C₁ : Type u₁} {C₂ : Type u₂} {H : Type u₃} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] [CategoryTheory.Category.{v₃, u₃} H] {D₁ : Type u₄} {D₂ : Type u₅} [CategoryTheory.Category.{v₄, u₄} D₁] [CategoryTheory.Category.{v₅, u₅} D₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (L₁ : CategoryTheory.Functor C₁ D₁) (L₂ : CategoryTheory.Functor C₂ D₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] (F : CategoryTheory.Functor C₂ H) (F₁ : CategoryTheory.Functor D₁ H) (α₁ : Φ.functor.comp F ⟶ L₁.comp F₁) (F₂ : CategoryTheory.Functor D₂ H) (α₂ : F ⟶ L₂.comp F₂) [F₁.IsRightDerivedFunctor α₁ W₁] : CategoryTheory.CategoryStruct.comp α₁ (L₁.whiskerLeft (Φ.rightDerivedFunctorComparison L₁ L₂ F F₁ α₁ F₂ α₂)) = CategoryTheory.CategoryStruct.comp (Φ.functor.whiskerLeft α₂) (CategoryTheory.CategoryStruct.comp (Φ.functor.associator L₂ F₂).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight (CategoryTheory.CatCommSq.iso Φ.functor L₁ L₂ (Φ.localizedFunctor L₁ L₂)).hom F₂) (L₁.associator (Φ.localizedFunctor L₁ L₂) F₂).hom))
• CategoryTheory.LocalizerMorphism.rightDerivedFunctorComparison_fac_assoc 📋 Mathlib.CategoryTheory.Localization.DerivabilityStructure.PointwiseRightDerived
  {C₁ : Type u₁} {C₂ : Type u₂} {H : Type u₃} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] [CategoryTheory.Category.{v₃, u₃} H] {D₁ : Type u₄} {D₂ : Type u₅} [CategoryTheory.Category.{v₄, u₄} D₁] [CategoryTheory.Category.{v₅, u₅} D₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (L₁ : CategoryTheory.Functor C₁ D₁) (L₂ : CategoryTheory.Functor C₂ D₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] (F : CategoryTheory.Functor C₂ H) (F₁ : CategoryTheory.Functor D₁ H) (α₁ : Φ.functor.comp F ⟶ L₁.comp F₁) (F₂ : CategoryTheory.Functor D₂ H) (α₂ : F ⟶ L₂.comp F₂) [F₁.IsRightDerivedFunctor α₁ W₁] {Z : CategoryTheory.Functor C₁ H} (h : L₁.comp ((Φ.localizedFunctor L₁ L₂).comp F₂) ⟶ Z) : CategoryTheory.CategoryStruct.comp α₁ (CategoryTheory.CategoryStruct.comp (L₁.whiskerLeft (Φ.rightDerivedFunctorComparison L₁ L₂ F F₁ α₁ F₂ α₂)) h) = CategoryTheory.CategoryStruct.comp (Φ.functor.whiskerLeft α₂) (CategoryTheory.CategoryStruct.comp (Φ.functor.associator L₂ F₂).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight (CategoryTheory.CatCommSq.iso Φ.functor L₁ L₂ (Φ.localizedFunctor L₁ L₂)).hom F₂) (CategoryTheory.CategoryStruct.comp (L₁.associator (Φ.localizedFunctor L₁ L₂) F₂).hom h)))
• CategoryTheory.LocalizerMorphism.Derives.isIso 📋 Mathlib.CategoryTheory.Localization.DerivabilityStructure.Derives
  {C₁ : Type u₁} {C₂ : Type u₂} {H : Type u₃} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] [CategoryTheory.Category.{v₃, u₃} H] {D₂ : Type u₄} [CategoryTheory.Category.{v₄, u₄} D₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} {Φ : CategoryTheory.LocalizerMorphism W₁ W₂} {F : CategoryTheory.Functor C₂ H} (h : Φ.Derives F) [Φ.IsRightDerivabilityStructure] {L₂ : CategoryTheory.Functor C₂ D₂} [L₂.IsLocalization W₂] {RF : CategoryTheory.Functor D₂ H} (α : F ⟶ L₂.comp RF) (X₁ : C₁) [RF.IsRightDerivedFunctor α W₂] : CategoryTheory.IsIso (α.app (Φ.functor.obj X₁))
• CategoryTheory.LocalizerMorphism.Derives.isIso_of_isRightDerivedFunctor 📋 Mathlib.CategoryTheory.Localization.DerivabilityStructure.Derives
  {C₁ : Type u₁} {C₂ : Type u₂} {H : Type u₃} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] [CategoryTheory.Category.{v₃, u₃} H] {D₂ : Type u₄} [CategoryTheory.Category.{v₄, u₄} D₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} {Φ : CategoryTheory.LocalizerMorphism W₁ W₂} {F : CategoryTheory.Functor C₂ H} (h : Φ.Derives F) [Φ.IsRightDerivabilityStructure] {L₂ : CategoryTheory.Functor C₂ D₂} [L₂.IsLocalization W₂] {RF : CategoryTheory.Functor D₂ H} (α : F ⟶ L₂.comp RF) (X₁ : C₁) [RF.IsRightDerivedFunctor α W₂] : CategoryTheory.IsIso (α.app (Φ.functor.obj X₁))
• CategoryTheory.LocalizerMorphism.Derives.isRightDerivedFunctor_of_isIso 📋 Mathlib.CategoryTheory.Localization.DerivabilityStructure.Derives
  {C₁ : Type u₁} {C₂ : Type u₂} {H : Type u₃} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] [CategoryTheory.Category.{v₃, u₃} H] {D₂ : Type u₄} [CategoryTheory.Category.{v₄, u₄} D₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} {Φ : CategoryTheory.LocalizerMorphism W₁ W₂} {F : CategoryTheory.Functor C₂ H} (h : Φ.Derives F) [Φ.IsRightDerivabilityStructure] {L₂ : CategoryTheory.Functor C₂ D₂} [L₂.IsLocalization W₂] {RF : CategoryTheory.Functor D₂ H} (α : F ⟶ L₂.comp RF) (hα : ∀ (X₁ : C₁), CategoryTheory.IsIso (α.app (Φ.functor.obj X₁))) : RF.IsRightDerivedFunctor α W₂
• CategoryTheory.LocalizerMorphism.Derives.isRightDerivedFunctor_iff_isIso 📋 Mathlib.CategoryTheory.Localization.DerivabilityStructure.Derives
  {C₁ : Type u₁} {C₂ : Type u₂} {H : Type u₃} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] [CategoryTheory.Category.{v₃, u₃} H] {D₂ : Type u₄} [CategoryTheory.Category.{v₄, u₄} D₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} {Φ : CategoryTheory.LocalizerMorphism W₁ W₂} {F : CategoryTheory.Functor C₂ H} (h : Φ.Derives F) [Φ.IsRightDerivabilityStructure] {L₂ : CategoryTheory.Functor C₂ D₂} [L₂.IsLocalization W₂] {RF : CategoryTheory.Functor D₂ H} (α : F ⟶ L₂.comp RF) : RF.IsRightDerivedFunctor α W₂ ↔ ∀ (X₁ : C₁), CategoryTheory.IsIso (α.app (Φ.functor.obj X₁))
• HomotopyCategory.Plus.instIsIsoAppOfInjectiveXIntAsHomologicalComplexUpHomotopicObjPlus 📋 Mathlib.Algebra.Homology.DerivedCategory.DerivabilityStructureInjectives
  {C : Type u_1} {H : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [CategoryTheory.Category.{v_2, u_2} H] [CategoryTheory.EnoughInjectives C] (F : CategoryTheory.Functor (HomotopyCategory.Plus C) H) [HasDerivedCategory C] (F' : CategoryTheory.Functor (DerivedCategory.Plus C) H) (α : F ⟶ DerivedCategory.Plus.Qh.comp F') [F'.IsRightDerivedFunctor α (HomotopyCategory.Plus.quasiIso C)] (K : HomotopyCategory.Plus C) [∀ (n : ℤ), CategoryTheory.Injective (K.obj.as.X n)] : CategoryTheory.IsIso (α.app K)
• CategoryTheory.Functor.instIsRightDerivedFunctorPlusPlusRightDerivedFunctorPlusRightDerivedFunctorPlusUnitQuasiIso 📋 Mathlib.Algebra.Homology.DerivedCategory.RightDerivedFunctorPlus
  {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Abelian C] [CategoryTheory.Abelian D] [HasDerivedCategory C] [HasDerivedCategory D] (F : CategoryTheory.Functor C D) [F.Additive] [CategoryTheory.EnoughInjectives C] : F.rightDerivedFunctorPlus.IsRightDerivedFunctor F.rightDerivedFunctorPlusUnit (HomotopyCategory.Plus.quasiIso C)
• CategoryTheory.Adjunction.derivedε 📋 Mathlib.CategoryTheory.Functor.Derived.Adjunction
  {C₁ : Type u_1} {C₂ : Type u_2} {D₁ : Type u_3} {D₂ : Type u_4} [CategoryTheory.Category.{v_1, u_1} C₁] [CategoryTheory.Category.{v_2, u_2} C₂] [CategoryTheory.Category.{v_3, u_3} D₁] [CategoryTheory.Category.{v_4, u_4} D₂] {G : CategoryTheory.Functor C₁ C₂} {F : CategoryTheory.Functor C₂ C₁} (adj : G ⊣ F) {L₁ : CategoryTheory.Functor C₁ D₁} {L₂ : CategoryTheory.Functor C₂ D₂} (W₂ : CategoryTheory.MorphismProperty C₂) [L₂.IsLocalization W₂] {G' : CategoryTheory.Functor D₁ D₂} {F' : CategoryTheory.Functor D₂ D₁} (α : L₁.comp G' ⟶ G.comp L₂) (β : F.comp L₁ ⟶ L₂.comp F') [(F'.comp G').IsRightDerivedFunctor (CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight β G') (L₂.associator F' G').hom) W₂] : F'.comp G' ⟶ CategoryTheory.Functor.id D₂
• CategoryTheory.Adjunction.derived 📋 Mathlib.CategoryTheory.Functor.Derived.Adjunction
  {C₁ : Type u_1} {C₂ : Type u_2} {D₁ : Type u_3} {D₂ : Type u_4} [CategoryTheory.Category.{v_1, u_1} C₁] [CategoryTheory.Category.{v_2, u_2} C₂] [CategoryTheory.Category.{v_3, u_3} D₁] [CategoryTheory.Category.{v_4, u_4} D₂] {G : CategoryTheory.Functor C₁ C₂} {F : CategoryTheory.Functor C₂ C₁} (adj : G ⊣ F) {L₁ : CategoryTheory.Functor C₁ D₁} {L₂ : CategoryTheory.Functor C₂ D₂} (W₁ : CategoryTheory.MorphismProperty C₁) (W₂ : CategoryTheory.MorphismProperty C₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] {G' : CategoryTheory.Functor D₁ D₂} {F' : CategoryTheory.Functor D₂ D₁} (α : L₁.comp G' ⟶ G.comp L₂) (β : F.comp L₁ ⟶ L₂.comp F') [G'.IsLeftDerivedFunctor α W₁] [F'.IsRightDerivedFunctor β W₂] [(G'.comp F').IsLeftDerivedFunctor (CategoryTheory.CategoryStruct.comp (L₁.associator G' F').inv (CategoryTheory.Functor.whiskerRight α F')) W₁] [(F'.comp G').IsRightDerivedFunctor (CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight β G') (L₂.associator F' G').hom) W₂] : G' ⊣ F'
• CategoryTheory.Adjunction.derived_counit 📋 Mathlib.CategoryTheory.Functor.Derived.Adjunction
  {C₁ : Type u_1} {C₂ : Type u_2} {D₁ : Type u_3} {D₂ : Type u_4} [CategoryTheory.Category.{v_1, u_1} C₁] [CategoryTheory.Category.{v_2, u_2} C₂] [CategoryTheory.Category.{v_3, u_3} D₁] [CategoryTheory.Category.{v_4, u_4} D₂] {G : CategoryTheory.Functor C₁ C₂} {F : CategoryTheory.Functor C₂ C₁} (adj : G ⊣ F) {L₁ : CategoryTheory.Functor C₁ D₁} {L₂ : CategoryTheory.Functor C₂ D₂} (W₁ : CategoryTheory.MorphismProperty C₁) (W₂ : CategoryTheory.MorphismProperty C₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] {G' : CategoryTheory.Functor D₁ D₂} {F' : CategoryTheory.Functor D₂ D₁} (α : L₁.comp G' ⟶ G.comp L₂) (β : F.comp L₁ ⟶ L₂.comp F') [G'.IsLeftDerivedFunctor α W₁] [F'.IsRightDerivedFunctor β W₂] [(G'.comp F').IsLeftDerivedFunctor (CategoryTheory.CategoryStruct.comp (L₁.associator G' F').inv (CategoryTheory.Functor.whiskerRight α F')) W₁] [(F'.comp G').IsRightDerivedFunctor (CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight β G') (L₂.associator F' G').hom) W₂] : (adj.derived W₁ W₂ α β).counit = adj.derivedε W₂ α β
• CategoryTheory.Adjunction.derived_unit 📋 Mathlib.CategoryTheory.Functor.Derived.Adjunction
  {C₁ : Type u_1} {C₂ : Type u_2} {D₁ : Type u_3} {D₂ : Type u_4} [CategoryTheory.Category.{v_1, u_1} C₁] [CategoryTheory.Category.{v_2, u_2} C₂] [CategoryTheory.Category.{v_3, u_3} D₁] [CategoryTheory.Category.{v_4, u_4} D₂] {G : CategoryTheory.Functor C₁ C₂} {F : CategoryTheory.Functor C₂ C₁} (adj : G ⊣ F) {L₁ : CategoryTheory.Functor C₁ D₁} {L₂ : CategoryTheory.Functor C₂ D₂} (W₁ : CategoryTheory.MorphismProperty C₁) (W₂ : CategoryTheory.MorphismProperty C₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] {G' : CategoryTheory.Functor D₁ D₂} {F' : CategoryTheory.Functor D₂ D₁} (α : L₁.comp G' ⟶ G.comp L₂) (β : F.comp L₁ ⟶ L₂.comp F') [G'.IsLeftDerivedFunctor α W₁] [F'.IsRightDerivedFunctor β W₂] [(G'.comp F').IsLeftDerivedFunctor (CategoryTheory.CategoryStruct.comp (L₁.associator G' F').inv (CategoryTheory.Functor.whiskerRight α F')) W₁] [(F'.comp G').IsRightDerivedFunctor (CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight β G') (L₂.associator F' G').hom) W₂] : (adj.derived W₁ W₂ α β).unit = adj.derivedη W₁ α β
• CategoryTheory.Adjunction.derivedε_fac_app 📋 Mathlib.CategoryTheory.Functor.Derived.Adjunction
  {C₁ : Type u_1} {C₂ : Type u_2} {D₁ : Type u_3} {D₂ : Type u_4} [CategoryTheory.Category.{v_1, u_1} C₁] [CategoryTheory.Category.{v_2, u_2} C₂] [CategoryTheory.Category.{v_3, u_3} D₁] [CategoryTheory.Category.{v_4, u_4} D₂] {G : CategoryTheory.Functor C₁ C₂} {F : CategoryTheory.Functor C₂ C₁} (adj : G ⊣ F) {L₁ : CategoryTheory.Functor C₁ D₁} {L₂ : CategoryTheory.Functor C₂ D₂} (W₂ : CategoryTheory.MorphismProperty C₂) [L₂.IsLocalization W₂] {G' : CategoryTheory.Functor D₁ D₂} {F' : CategoryTheory.Functor D₂ D₁} (α : L₁.comp G' ⟶ G.comp L₂) (β : F.comp L₁ ⟶ L₂.comp F') [(F'.comp G').IsRightDerivedFunctor (CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight β G') (L₂.associator F' G').hom) W₂] (X₂ : C₂) : CategoryTheory.CategoryStruct.comp (G'.map (β.app X₂)) ((adj.derivedε W₂ α β).app (L₂.obj X₂)) = CategoryTheory.CategoryStruct.comp (α.app (F.obj X₂)) (L₂.map (adj.counit.app X₂))
• CategoryTheory.Adjunction.derivedε_fac_app_assoc 📋 Mathlib.CategoryTheory.Functor.Derived.Adjunction
  {C₁ : Type u_1} {C₂ : Type u_2} {D₁ : Type u_3} {D₂ : Type u_4} [CategoryTheory.Category.{v_1, u_1} C₁] [CategoryTheory.Category.{v_2, u_2} C₂] [CategoryTheory.Category.{v_3, u_3} D₁] [CategoryTheory.Category.{v_4, u_4} D₂] {G : CategoryTheory.Functor C₁ C₂} {F : CategoryTheory.Functor C₂ C₁} (adj : G ⊣ F) {L₁ : CategoryTheory.Functor C₁ D₁} {L₂ : CategoryTheory.Functor C₂ D₂} (W₂ : CategoryTheory.MorphismProperty C₂) [L₂.IsLocalization W₂] {G' : CategoryTheory.Functor D₁ D₂} {F' : CategoryTheory.Functor D₂ D₁} (α : L₁.comp G' ⟶ G.comp L₂) (β : F.comp L₁ ⟶ L₂.comp F') [(F'.comp G').IsRightDerivedFunctor (CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight β G') (L₂.associator F' G').hom) W₂] (X₂ : C₂) {Z : D₂} (h : L₂.obj X₂ ⟶ Z) : CategoryTheory.CategoryStruct.comp (G'.map (β.app X₂)) (CategoryTheory.CategoryStruct.comp ((adj.derivedε W₂ α β).app (L₂.obj X₂)) h) = CategoryTheory.CategoryStruct.comp (α.app (F.obj X₂)) (CategoryTheory.CategoryStruct.comp (L₂.map (adj.counit.app X₂)) h)
• CategoryTheory.Adjunction.derived' 📋 Mathlib.CategoryTheory.Functor.Derived.Adjunction
  {C₁ : Type u_1} {C₂ : Type u_2} {D₁ : Type u_3} {D₂ : Type u_4} [CategoryTheory.Category.{v_1, u_1} C₁] [CategoryTheory.Category.{v_2, u_2} C₂] [CategoryTheory.Category.{v_3, u_3} D₁] [CategoryTheory.Category.{v_4, u_4} D₂] {G : CategoryTheory.Functor C₁ C₂} {F : CategoryTheory.Functor C₂ C₁} (adj : G ⊣ F) {L₁ : CategoryTheory.Functor C₁ D₁} {L₂ : CategoryTheory.Functor C₂ D₂} (W₁ : CategoryTheory.MorphismProperty C₁) (W₂ : CategoryTheory.MorphismProperty C₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] {G' : CategoryTheory.Functor D₁ D₂} {F' : CategoryTheory.Functor D₂ D₁} (α : L₁.comp G' ⟶ G.comp L₂) (β : F.comp L₁ ⟶ L₂.comp F') [G'.IsLeftDerivedFunctor α W₁] [F'.IsRightDerivedFunctor β W₂] (η : CategoryTheory.Functor.id D₁ ⟶ G'.comp F') (ε : F'.comp G' ⟶ CategoryTheory.Functor.id D₂) (hη : ∀ (X₁ : C₁), CategoryTheory.CategoryStruct.comp (η.app (L₁.obj X₁)) (F'.map (α.app X₁)) = CategoryTheory.CategoryStruct.comp (L₁.map (adj.unit.app X₁)) (β.app (G.obj X₁)) := by cat_disch) (hε : ∀ (X₂ : C₂), CategoryTheory.CategoryStruct.comp (G'.map (β.app X₂)) (ε.app (L₂.obj X₂)) = CategoryTheory.CategoryStruct.comp (α.app (F.obj X₂)) (L₂.map (adj.counit.app X₂)) := by cat_disch) : G' ⊣ F'
• CategoryTheory.Adjunction.derived'_counit 📋 Mathlib.CategoryTheory.Functor.Derived.Adjunction
  {C₁ : Type u_1} {C₂ : Type u_2} {D₁ : Type u_3} {D₂ : Type u_4} [CategoryTheory.Category.{v_1, u_1} C₁] [CategoryTheory.Category.{v_2, u_2} C₂] [CategoryTheory.Category.{v_3, u_3} D₁] [CategoryTheory.Category.{v_4, u_4} D₂] {G : CategoryTheory.Functor C₁ C₂} {F : CategoryTheory.Functor C₂ C₁} (adj : G ⊣ F) {L₁ : CategoryTheory.Functor C₁ D₁} {L₂ : CategoryTheory.Functor C₂ D₂} (W₁ : CategoryTheory.MorphismProperty C₁) (W₂ : CategoryTheory.MorphismProperty C₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] {G' : CategoryTheory.Functor D₁ D₂} {F' : CategoryTheory.Functor D₂ D₁} (α : L₁.comp G' ⟶ G.comp L₂) (β : F.comp L₁ ⟶ L₂.comp F') [G'.IsLeftDerivedFunctor α W₁] [F'.IsRightDerivedFunctor β W₂] (η : CategoryTheory.Functor.id D₁ ⟶ G'.comp F') (ε : F'.comp G' ⟶ CategoryTheory.Functor.id D₂) (hη : ∀ (X₁ : C₁), CategoryTheory.CategoryStruct.comp (η.app (L₁.obj X₁)) (F'.map (α.app X₁)) = CategoryTheory.CategoryStruct.comp (L₁.map (adj.unit.app X₁)) (β.app (G.obj X₁)) := by cat_disch) (hε : ∀ (X₂ : C₂), CategoryTheory.CategoryStruct.comp (G'.map (β.app X₂)) (ε.app (L₂.obj X₂)) = CategoryTheory.CategoryStruct.comp (α.app (F.obj X₂)) (L₂.map (adj.counit.app X₂)) := by cat_disch) : (adj.derived' W₁ W₂ α β η ε hη hε).counit = ε
• CategoryTheory.Adjunction.derived'_unit 📋 Mathlib.CategoryTheory.Functor.Derived.Adjunction
  {C₁ : Type u_1} {C₂ : Type u_2} {D₁ : Type u_3} {D₂ : Type u_4} [CategoryTheory.Category.{v_1, u_1} C₁] [CategoryTheory.Category.{v_2, u_2} C₂] [CategoryTheory.Category.{v_3, u_3} D₁] [CategoryTheory.Category.{v_4, u_4} D₂] {G : CategoryTheory.Functor C₁ C₂} {F : CategoryTheory.Functor C₂ C₁} (adj : G ⊣ F) {L₁ : CategoryTheory.Functor C₁ D₁} {L₂ : CategoryTheory.Functor C₂ D₂} (W₁ : CategoryTheory.MorphismProperty C₁) (W₂ : CategoryTheory.MorphismProperty C₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] {G' : CategoryTheory.Functor D₁ D₂} {F' : CategoryTheory.Functor D₂ D₁} (α : L₁.comp G' ⟶ G.comp L₂) (β : F.comp L₁ ⟶ L₂.comp F') [G'.IsLeftDerivedFunctor α W₁] [F'.IsRightDerivedFunctor β W₂] (η : CategoryTheory.Functor.id D₁ ⟶ G'.comp F') (ε : F'.comp G' ⟶ CategoryTheory.Functor.id D₂) (hη : ∀ (X₁ : C₁), CategoryTheory.CategoryStruct.comp (η.app (L₁.obj X₁)) (F'.map (α.app X₁)) = CategoryTheory.CategoryStruct.comp (L₁.map (adj.unit.app X₁)) (β.app (G.obj X₁)) := by cat_disch) (hε : ∀ (X₂ : C₂), CategoryTheory.CategoryStruct.comp (G'.map (β.app X₂)) (ε.app (L₂.obj X₂)) = CategoryTheory.CategoryStruct.comp (α.app (F.obj X₂)) (L₂.map (adj.counit.app X₂)) := by cat_disch) : (adj.derived' W₁ W₂ α β η ε hη hε).unit = η
• CategoryTheory.Functor.IsRightDerivedFunctor.instCompShiftFunctorPostcomposeShiftNatTrans 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerivedCommShift
  {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H] (RF : CategoryTheory.Functor H D) {F : CategoryTheory.Functor C D} {L : CategoryTheory.Functor C H} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] {A : Type u_4} [AddGroup A] [CategoryTheory.HasShift D A] (a : A) : (RF.comp (CategoryTheory.shiftFunctor D a)).IsRightDerivedFunctor (CategoryTheory.Functor.IsRightDerivedFunctor.postcomposeShiftNatTrans RF α a) W
• CategoryTheory.Functor.IsRightDerivedFunctor.commShift 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerivedCommShift
  {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H] (RF : CategoryTheory.Functor H D) {F : CategoryTheory.Functor C D} {L : CategoryTheory.Functor C H} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (A : Type u_4) [AddGroup A] [CategoryTheory.HasShift C A] [CategoryTheory.HasShift D A] [CategoryTheory.HasShift H A] [W.IsCompatibleWithShift A] [F.CommShift A] [L.CommShift A] : RF.CommShift A
• CategoryTheory.Functor.IsRightDerivedFunctor.instCompShiftFunctorPrecomposeShiftNatTrans 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerivedCommShift
  {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H] (RF : CategoryTheory.Functor H D) {F : CategoryTheory.Functor C D} {L : CategoryTheory.Functor C H} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] {A : Type u_4} [AddGroup A] [CategoryTheory.HasShift C A] [CategoryTheory.HasShift H A] [W.IsCompatibleWithShift A] [L.CommShift A] (a : A) : ((CategoryTheory.shiftFunctor H a).comp RF).IsRightDerivedFunctor (CategoryTheory.Functor.IsRightDerivedFunctor.precomposeShiftNatTrans RF α a) W
• CategoryTheory.Functor.IsRightDerivedFunctor.natTrans_commShift 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerivedCommShift
  {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H] (RF : CategoryTheory.Functor H D) {F : CategoryTheory.Functor C D} {L : CategoryTheory.Functor C H} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (A : Type u_4) [AddGroup A] [CategoryTheory.HasShift C A] [CategoryTheory.HasShift D A] [CategoryTheory.HasShift H A] [W.IsCompatibleWithShift A] [F.CommShift A] [L.CommShift A] : CategoryTheory.NatTrans.CommShift α A
• CategoryTheory.Functor.IsRightDerivedFunctor.comp_map_commShiftIso_hom_app 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerivedCommShift
  {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H] (RF : CategoryTheory.Functor H D) {F : CategoryTheory.Functor C D} {L : CategoryTheory.Functor C H} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (A : Type u_4) [AddGroup A] [CategoryTheory.HasShift C A] [CategoryTheory.HasShift D A] [CategoryTheory.HasShift H A] [W.IsCompatibleWithShift A] [F.CommShift A] [L.CommShift A] (a : A) (X : C) : CategoryTheory.CategoryStruct.comp (α.app ((CategoryTheory.shiftFunctor C a).obj X)) (CategoryTheory.CategoryStruct.comp (RF.map ((CategoryTheory.Functor.commShiftIso L a).hom.app X)) ((CategoryTheory.Functor.commShiftIso RF a).hom.app (L.obj X))) = CategoryTheory.CategoryStruct.comp ((CategoryTheory.Functor.commShiftIso F a).hom.app X) ((CategoryTheory.shiftFunctor D a).map (α.app X))
• CategoryTheory.Functor.IsRightDerivedFunctor.comp_map_commShiftIso_hom_app_assoc 📋 Mathlib.CategoryTheory.Functor.Derived.RightDerivedCommShift
  {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H] (RF : CategoryTheory.Functor H D) {F : CategoryTheory.Functor C D} {L : CategoryTheory.Functor C H} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (A : Type u_4) [AddGroup A] [CategoryTheory.HasShift C A] [CategoryTheory.HasShift D A] [CategoryTheory.HasShift H A] [W.IsCompatibleWithShift A] [F.CommShift A] [L.CommShift A] (a : A) (X : C) {Z : D} (h : (CategoryTheory.shiftFunctor D a).obj (RF.obj (L.obj X)) ⟶ Z) : CategoryTheory.CategoryStruct.comp (α.app ((CategoryTheory.shiftFunctor C a).obj X)) (CategoryTheory.CategoryStruct.comp (RF.map ((CategoryTheory.Functor.commShiftIso L a).hom.app X)) (CategoryTheory.CategoryStruct.comp ((CategoryTheory.Functor.commShiftIso RF a).hom.app (L.obj X)) h)) = CategoryTheory.CategoryStruct.comp ((CategoryTheory.Functor.commShiftIso F a).hom.app X) (CategoryTheory.CategoryStruct.comp ((CategoryTheory.shiftFunctor D a).map (α.app X)) h)

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