Loogle!
Result
Found 121 definitions mentioning CategoryTheory.Limits.IsZero. Of these, 39 match your pattern(s).
- CategoryTheory.Limits.IsZero.op Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X : C}, CategoryTheory.Limits.IsZero X → CategoryTheory.Limits.IsZero { unop := X } - CategoryTheory.Limits.isZero_zero Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects
∀ (C : Type u) [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroObject C], CategoryTheory.Limits.IsZero 0 - CategoryTheory.Limits.IsZero.of_iso Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C}, CategoryTheory.Limits.IsZero Y → (X ≅ Y) → CategoryTheory.Limits.IsZero X - CategoryTheory.Limits.IsZero.unop Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X : Cᵒᵖ}, CategoryTheory.Limits.IsZero X → CategoryTheory.Limits.IsZero X.unop - CategoryTheory.Limits.IsZero.mk Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X : C}, (∀ (Y : C), Nonempty (Unique (X ⟶ Y))) → (∀ (Y : C), Nonempty (Unique (Y ⟶ X))) → CategoryTheory.Limits.IsZero X - CategoryTheory.Functor.isZero Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {D : Type u'} [inst_1 : CategoryTheory.Category.{v', u'} D] (F : CategoryTheory.Functor C D), (∀ (X : C), CategoryTheory.Limits.IsZero (F.obj X)) → CategoryTheory.Limits.IsZero F - CategoryTheory.Limits.IsZero.obj Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {D : Type u'} [inst_1 : CategoryTheory.Category.{v', u'} D] [inst_2 : CategoryTheory.Limits.HasZeroObject D] {F : CategoryTheory.Functor C D}, CategoryTheory.Limits.IsZero F → ∀ (X : C), CategoryTheory.Limits.IsZero (F.obj X) - CategoryTheory.Limits.IsZero.of_epi Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] {X Y : C} (f : X ⟶ Y) [inst_2 : CategoryTheory.Epi f], CategoryTheory.Limits.IsZero X → CategoryTheory.Limits.IsZero Y - CategoryTheory.Limits.IsZero.of_mono Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] {X Y : C} (f : X ⟶ Y) [inst_2 : CategoryTheory.Mono f], CategoryTheory.Limits.IsZero Y → CategoryTheory.Limits.IsZero X - CategoryTheory.Limits.IsZero.of_epi_zero Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] (X Y : C) [inst_2 : CategoryTheory.Epi 0], CategoryTheory.Limits.IsZero Y - CategoryTheory.Limits.IsZero.of_mono_zero Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] (X Y : C) [inst_2 : CategoryTheory.Mono 0], CategoryTheory.Limits.IsZero X - CategoryTheory.Functor.zero_obj Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {D : Type u'} [inst_1 : CategoryTheory.Category.{v', u'} D] [inst_2 : CategoryTheory.Limits.HasZeroObject D] (X : C), CategoryTheory.Limits.IsZero (0.obj X) - CategoryTheory.Limits.IsZero.of_epi_eq_zero Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] {X Y : C} (f : X ⟶ Y) [inst_2 : CategoryTheory.Epi f], f = 0 → CategoryTheory.Limits.IsZero Y - CategoryTheory.Limits.IsZero.of_mono_eq_zero Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] {X Y : C} (f : X ⟶ Y) [inst_2 : CategoryTheory.Mono f], f = 0 → CategoryTheory.Limits.IsZero X - CategoryTheory.Functor.map_isZero Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D] [inst_2 : CategoryTheory.Limits.HasZeroMorphisms C] [inst_3 : CategoryTheory.Limits.HasZeroMorphisms D] (F : CategoryTheory.Functor C D) [inst_4 : F.PreservesZeroMorphisms] {X : C}, CategoryTheory.Limits.IsZero X → CategoryTheory.Limits.IsZero (F.obj X) - ModuleCat.isZero_of_subsingleton Mathlib.Algebra.Category.ModuleCat.Basic
∀ {R : Type u} [inst : Ring R] (M : ModuleCat R) [inst_1 : Subsingleton ↑M], CategoryTheory.Limits.IsZero M - AddCommGroupCat.isZero_of_subsingleton Mathlib.Algebra.Category.GroupCat.Zero
∀ (G : AddCommGroupCat) [inst : Subsingleton ↑G], CategoryTheory.Limits.IsZero G - AddGroupCat.isZero_of_subsingleton Mathlib.Algebra.Category.GroupCat.Zero
∀ (G : AddGroupCat) [inst : Subsingleton ↑G], CategoryTheory.Limits.IsZero G - CommGroupCat.isZero_of_subsingleton Mathlib.Algebra.Category.GroupCat.Zero
∀ (G : CommGroupCat) [inst : Subsingleton ↑G], CategoryTheory.Limits.IsZero G - GroupCat.isZero_of_subsingleton Mathlib.Algebra.Category.GroupCat.Zero
∀ (G : GroupCat) [inst : Subsingleton ↑G], CategoryTheory.Limits.IsZero G - CategoryTheory.ShortComplex.isZero_homology_of_isZero_X₂ Mathlib.Algebra.Homology.ShortComplex.Homology
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] (S : CategoryTheory.ShortComplex C), CategoryTheory.Limits.IsZero S.X₂ → ∀ [inst_2 : S.HasHomology], CategoryTheory.Limits.IsZero S.homology - CategoryTheory.ShortComplex.Exact.isZero_of_both_zeros Mathlib.Algebra.Homology.ShortComplex.Exact
∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_3, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] {S : CategoryTheory.ShortComplex C}, S.Exact → S.f = 0 → S.g = 0 → CategoryTheory.Limits.IsZero S.X₂ - CategoryTheory.ShortComplex.Exact.isZero_X₂ Mathlib.Algebra.Homology.ShortComplex.Exact
∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_3, u_1} C] [inst_1 : CategoryTheory.Preadditive C] {S : CategoryTheory.ShortComplex C}, S.Exact → S.f = 0 → S.g = 0 → CategoryTheory.Limits.IsZero S.X₂ - HomologicalComplex.isZero_zero Mathlib.Algebra.Homology.HomologicalComplex
∀ {ι : Type u_1} {V : Type u} [inst : CategoryTheory.Category.{v, u} V] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms V] {c : ComplexShape ι} [inst_2 : CategoryTheory.Limits.HasZeroObject V], CategoryTheory.Limits.IsZero HomologicalComplex.zero - HomologicalComplex.isZero_single_obj_X Mathlib.Algebra.Homology.Single
∀ {V : Type u} [inst : CategoryTheory.Category.{v, u} V] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms V] [inst_2 : CategoryTheory.Limits.HasZeroObject V] {ι : Type u_1} [inst_3 : DecidableEq ι] (c : ComplexShape ι) (j : ι) (A : V) (i : ι), i ≠ j → CategoryTheory.Limits.IsZero (((HomologicalComplex.single V c j).obj A).X i) - CategoryTheory.Pretriangulated.Triangle.isZero₁_of_isZero₂₃ Mathlib.CategoryTheory.Triangulated.Pretriangulated
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroObject C] [inst_2 : CategoryTheory.HasShift C ℤ] [inst_3 : CategoryTheory.Preadditive C] [inst_4 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C], ∀ T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles, CategoryTheory.Limits.IsZero T.obj₂ → CategoryTheory.Limits.IsZero T.obj₃ → CategoryTheory.Limits.IsZero T.obj₁ - CategoryTheory.Pretriangulated.Triangle.isZero₂_of_isZero₁₃ Mathlib.CategoryTheory.Triangulated.Pretriangulated
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroObject C] [inst_2 : CategoryTheory.HasShift C ℤ] [inst_3 : CategoryTheory.Preadditive C] [inst_4 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C], ∀ T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles, CategoryTheory.Limits.IsZero T.obj₁ → CategoryTheory.Limits.IsZero T.obj₃ → CategoryTheory.Limits.IsZero T.obj₂ - CategoryTheory.Pretriangulated.Triangle.isZero₃_of_isZero₁₂ Mathlib.CategoryTheory.Triangulated.Pretriangulated
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroObject C] [inst_2 : CategoryTheory.HasShift C ℤ] [inst_3 : CategoryTheory.Preadditive C] [inst_4 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C], ∀ T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles, CategoryTheory.Limits.IsZero T.obj₁ → CategoryTheory.Limits.IsZero T.obj₂ → CategoryTheory.Limits.IsZero T.obj₃ - CategoryTheory.Pretriangulated.Triangle.isZero₁_of_isIso₂ Mathlib.CategoryTheory.Triangulated.Pretriangulated
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroObject C] [inst_2 : CategoryTheory.HasShift C ℤ] [inst_3 : CategoryTheory.Preadditive C] [inst_4 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C], ∀ T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles, CategoryTheory.IsIso T.mor₂ → CategoryTheory.Limits.IsZero T.obj₁ - CategoryTheory.Pretriangulated.Triangle.isZero₃_of_isIso₁ Mathlib.CategoryTheory.Triangulated.Pretriangulated
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroObject C] [inst_2 : CategoryTheory.HasShift C ℤ] [inst_3 : CategoryTheory.Preadditive C] [inst_4 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C], ∀ T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles, CategoryTheory.IsIso T.mor₁ → CategoryTheory.Limits.IsZero T.obj₃ - CategoryTheory.Pretriangulated.Triangle.isZero₂_of_isIso₃ Mathlib.CategoryTheory.Triangulated.Pretriangulated
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroObject C] [inst_2 : CategoryTheory.HasShift C ℤ] [inst_3 : CategoryTheory.Preadditive C] [inst_4 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C], ∀ T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles, CategoryTheory.IsIso T.mor₃ → CategoryTheory.Limits.IsZero T.obj₂ - HomologicalComplex.isZero_single_obj_homology Mathlib.Algebra.Homology.SingleHomology
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] [inst_2 : CategoryTheory.Limits.HasZeroObject C] {ι : Type u_1} [inst_3 : DecidableEq ι] (c : ComplexShape ι) (j : ι) (A : C) (i : ι), i ≠ j → CategoryTheory.Limits.IsZero (((HomologicalComplex.single C c j).obj A).homology i) - CategoryTheory.Functor.isZero_leftDerived_obj_projective_succ Mathlib.CategoryTheory.Abelian.LeftDerived
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {D : Type u_1} [inst_1 : CategoryTheory.Category.{u_2, u_1} D] [inst_2 : CategoryTheory.Abelian C] [inst_3 : CategoryTheory.HasProjectiveResolutions C] [inst_4 : CategoryTheory.Abelian D] (F : CategoryTheory.Functor C D) [inst_5 : F.Additive] (n : ℕ) (X : C) [inst_6 : CategoryTheory.Projective X], CategoryTheory.Limits.IsZero ((F.leftDerived (n + 1)).obj X) - isZero_Ext_succ_of_projective Mathlib.CategoryTheory.Abelian.Ext
∀ {R : Type u_1} [inst : Ring R] {C : Type u_2} [inst_1 : CategoryTheory.Category.{u_3, u_2} C] [inst_2 : CategoryTheory.Abelian C] [inst_3 : CategoryTheory.Linear R C] [inst_4 : CategoryTheory.EnoughProjectives C] (X Y : C) [inst_5 : CategoryTheory.Projective X] (n : ℕ), CategoryTheory.Limits.IsZero (((Ext R C (n + 1)).obj { unop := X }).obj Y) - SemiNormedGroupCat.isZero_of_subsingleton Mathlib.Analysis.Normed.Group.SemiNormedGroupCat
∀ (V : SemiNormedGroupCat) [inst : Subsingleton ↑V], CategoryTheory.Limits.IsZero V - SemiNormedGroupCat₁.isZero_of_subsingleton Mathlib.Analysis.Normed.Group.SemiNormedGroupCat
∀ (V : SemiNormedGroupCat₁) [inst : Subsingleton ↑V], CategoryTheory.Limits.IsZero V - CategoryTheory.Functor.isZero_rightDerived_obj_injective_succ Mathlib.CategoryTheory.Abelian.RightDerived
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {D : Type u_1} [inst_1 : CategoryTheory.Category.{u_2, u_1} D] [inst_2 : CategoryTheory.Abelian C] [inst_3 : CategoryTheory.HasInjectiveResolutions C] [inst_4 : CategoryTheory.Abelian D] (F : CategoryTheory.Functor C D) [inst_5 : F.Additive] (n : ℕ) (X : C) [inst_6 : CategoryTheory.Injective X], CategoryTheory.Limits.IsZero ((F.rightDerived (n + 1)).obj X) - CategoryTheory.isZero_Tor'_succ_of_projective Mathlib.CategoryTheory.Monoidal.Tor
∀ (C : Type u_1) [inst : CategoryTheory.Category.{u_2, u_1} C] [inst_1 : CategoryTheory.MonoidalCategory C] [inst_2 : CategoryTheory.Abelian C] [inst_3 : CategoryTheory.MonoidalPreadditive C] [inst_4 : CategoryTheory.HasProjectiveResolutions C] (X Y : C) [inst_5 : CategoryTheory.Projective X] (n : ℕ), CategoryTheory.Limits.IsZero (((CategoryTheory.Tor' C (n + 1)).obj X).obj Y) - CategoryTheory.isZero_Tor_succ_of_projective Mathlib.CategoryTheory.Monoidal.Tor
∀ (C : Type u_1) [inst : CategoryTheory.Category.{u_2, u_1} C] [inst_1 : CategoryTheory.MonoidalCategory C] [inst_2 : CategoryTheory.Abelian C] [inst_3 : CategoryTheory.MonoidalPreadditive C] [inst_4 : CategoryTheory.HasProjectiveResolutions C] (X Y : C) [inst_5 : CategoryTheory.Projective Y] (n : ℕ), CategoryTheory.Limits.IsZero (((CategoryTheory.Tor C (n + 1)).obj X).obj Y)
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About
Loogle searches of Lean and Mathlib definitions and theorems.
You may also want to try the CLI version, the VS
Code extension, the lean.nvim
integration or the Zulip bot.
Usage
Loogle finds definitions and lemmas in various ways:
By constant:
🔍Real.sin
finds all lemmas whose statement somehow mentions the sine function.By lemma name substring:
🔍"differ"
finds all lemmas that have"differ"
somewhere in their lemma name.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
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 findtsum_lt_tsum
even though the hypothesisf i < g i
is not the last.
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, _ * _, _ ^ _, |- _ < _ → _
woould 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 currently provided by Joachim Breitner <mail@joachim-breitner.de>.
This is Loogle revision fa2ddf5
serving mathlib revision d874bdf