Loogle!
Result
Found 14 declarations mentioning CochainComplex.mappingCone.map.
- CochainComplex.mappingCone.triangleMap_homβ π Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated
{C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {Kβ Lβ Kβ Lβ : CochainComplex C β€} (Οβ : Kβ βΆ Lβ) (Οβ : Kβ βΆ Lβ) (a : Kβ βΆ Kβ) (b : Lβ βΆ Lβ) (comm : CategoryTheory.CategoryStruct.comp Οβ b = CategoryTheory.CategoryStruct.comp a Οβ) : (CochainComplex.mappingCone.triangleMap Οβ Οβ a b comm).homβ = CochainComplex.mappingCone.map Οβ Οβ a b comm - CochainComplex.mappingCone.map π Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated
{C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {Kβ Lβ Kβ Lβ : CochainComplex C β€} (Οβ : Kβ βΆ Lβ) (Οβ : Kβ βΆ Lβ) (a : Kβ βΆ Kβ) (b : Lβ βΆ Lβ) (comm : CategoryTheory.CategoryStruct.comp Οβ b = CategoryTheory.CategoryStruct.comp a Οβ) : CochainComplex.mappingCone Οβ βΆ CochainComplex.mappingCone Οβ - CochainComplex.mappingCone.map.eq_1 π Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated
{C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {Kβ Lβ Kβ Lβ : CochainComplex C β€} (Οβ : Kβ βΆ Lβ) (Οβ : Kβ βΆ Lβ) (a : Kβ βΆ Kβ) (b : Lβ βΆ Lβ) (comm : CategoryTheory.CategoryStruct.comp Οβ b = CategoryTheory.CategoryStruct.comp a Οβ) : CochainComplex.mappingCone.map Οβ Οβ a b comm = CochainComplex.mappingCone.desc Οβ ((CochainComplex.HomComplex.Cochain.ofHom a).comp (CochainComplex.mappingCone.inl Οβ) CochainComplex.mappingCone.homotopyToZeroOfId._proof_3) (CategoryTheory.CategoryStruct.comp b (CochainComplex.mappingCone.inr Οβ)) β― - CochainComplex.mappingCone.map_eq_mapOfHomotopy π Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated
{C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {Kβ Lβ Kβ Lβ : CochainComplex C β€} (Οβ : Kβ βΆ Lβ) (Οβ : Kβ βΆ Lβ) (a : Kβ βΆ Kβ) (b : Lβ βΆ Lβ) (comm : CategoryTheory.CategoryStruct.comp Οβ b = CategoryTheory.CategoryStruct.comp a Οβ) : CochainComplex.mappingCone.map Οβ Οβ a b comm = CochainComplex.mappingCone.mapOfHomotopy (Homotopy.ofEq comm) - CochainComplex.mappingCone.map.congr_simp π Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated
{C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {Kβ Lβ Kβ Lβ : CochainComplex C β€} (Οβ : Kβ βΆ Lβ) (Οβ : Kβ βΆ Lβ) (a aβ : Kβ βΆ Kβ) (e_a : a = aβ) (b bβ : Lβ βΆ Lβ) (e_b : b = bβ) (comm : CategoryTheory.CategoryStruct.comp Οβ b = CategoryTheory.CategoryStruct.comp a Οβ) : CochainComplex.mappingCone.map Οβ Οβ a b comm = CochainComplex.mappingCone.map Οβ Οβ aβ bβ β― - CochainComplex.mappingCone.map_id π Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated
{C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {K L : CochainComplex C β€} (Ο : K βΆ L) : CochainComplex.mappingCone.map Ο Ο (CategoryTheory.CategoryStruct.id K) (CategoryTheory.CategoryStruct.id L) β― = CategoryTheory.CategoryStruct.id (CochainComplex.mappingCone Ο) - CochainComplex.mappingCone.map_comp π Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated
{C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {Kβ Lβ Kβ Lβ Kβ Lβ : CochainComplex C β€} (Οβ : Kβ βΆ Lβ) (Οβ : Kβ βΆ Lβ) (Οβ : Kβ βΆ Lβ) (a : Kβ βΆ Kβ) (b : Lβ βΆ Lβ) (comm : CategoryTheory.CategoryStruct.comp Οβ b = CategoryTheory.CategoryStruct.comp a Οβ) (a' : Kβ βΆ Kβ) (b' : Lβ βΆ Lβ) (comm' : CategoryTheory.CategoryStruct.comp Οβ b' = CategoryTheory.CategoryStruct.comp a' Οβ) : CochainComplex.mappingCone.map Οβ Οβ (CategoryTheory.CategoryStruct.comp a a') (CategoryTheory.CategoryStruct.comp b b') β― = CategoryTheory.CategoryStruct.comp (CochainComplex.mappingCone.map Οβ Οβ a b comm) (CochainComplex.mappingCone.map Οβ Οβ a' b' comm') - CochainComplex.mappingCone.map_comp_assoc π Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated
{C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {Kβ Lβ Kβ Lβ Kβ Lβ : CochainComplex C β€} (Οβ : Kβ βΆ Lβ) (Οβ : Kβ βΆ Lβ) (Οβ : Kβ βΆ Lβ) (a : Kβ βΆ Kβ) (b : Lβ βΆ Lβ) (comm : CategoryTheory.CategoryStruct.comp Οβ b = CategoryTheory.CategoryStruct.comp a Οβ) (a' : Kβ βΆ Kβ) (b' : Lβ βΆ Lβ) (comm' : CategoryTheory.CategoryStruct.comp Οβ b' = CategoryTheory.CategoryStruct.comp a' Οβ) {Z : HomologicalComplex C (ComplexShape.up β€)} (h : CochainComplex.mappingCone Οβ βΆ Z) : CategoryTheory.CategoryStruct.comp (CochainComplex.mappingCone.map Οβ Οβ (CategoryTheory.CategoryStruct.comp a a') (CategoryTheory.CategoryStruct.comp b b') β―) h = CategoryTheory.CategoryStruct.comp (CochainComplex.mappingCone.map Οβ Οβ a b comm) (CategoryTheory.CategoryStruct.comp (CochainComplex.mappingCone.map Οβ Οβ a' b' comm') h) - CochainComplex.mappingConeCompTriangle_morβ π Mathlib.Algebra.Homology.HomotopyCategory.Triangulated
{C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {Xβ Xβ Xβ : CochainComplex C β€} (f : Xβ βΆ Xβ) (g : Xβ βΆ Xβ) : (CochainComplex.mappingConeCompTriangle f g).morβ = CochainComplex.mappingCone.map f (CategoryTheory.CategoryStruct.comp f g) (CategoryTheory.CategoryStruct.id Xβ) g β― - CochainComplex.mappingConeCompTriangle_morβ π Mathlib.Algebra.Homology.HomotopyCategory.Triangulated
{C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {Xβ Xβ Xβ : CochainComplex C β€} (f : Xβ βΆ Xβ) (g : Xβ βΆ Xβ) : (CochainComplex.mappingConeCompTriangle f g).morβ = CochainComplex.mappingCone.map (CategoryTheory.CategoryStruct.comp f g) g f (CategoryTheory.CategoryStruct.id Xβ) β― - CochainComplex.mappingConeCompTriangle_morβ_naturality π Mathlib.Algebra.Homology.HomotopyCategory.Triangulated
{C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {Xβ Xβ Xβ : CochainComplex C β€} (f : Xβ βΆ Xβ) (g : Xβ βΆ Xβ) {Yβ Yβ Yβ : CochainComplex C β€} (f' : Yβ βΆ Yβ) (g' : Yβ βΆ Yβ) (Ο : CategoryTheory.ComposableArrows.mkβ f g βΆ CategoryTheory.ComposableArrows.mkβ f' g') : CategoryTheory.CategoryStruct.comp (CochainComplex.mappingCone.map g g' (Ο.app 1) (Ο.app 2) β―) (CochainComplex.mappingConeCompTriangle f' g').morβ = CategoryTheory.CategoryStruct.comp (CochainComplex.mappingConeCompTriangle f g).morβ ((CategoryTheory.shiftFunctor (CochainComplex C β€) 1).map (CochainComplex.mappingCone.map f f' (Ο.app 0) (Ο.app 1) β―)) - CochainComplex.mappingConeCompTriangle_morβ_naturality_assoc π Mathlib.Algebra.Homology.HomotopyCategory.Triangulated
{C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {Xβ Xβ Xβ : CochainComplex C β€} (f : Xβ βΆ Xβ) (g : Xβ βΆ Xβ) {Yβ Yβ Yβ : CochainComplex C β€} (f' : Yβ βΆ Yβ) (g' : Yβ βΆ Yβ) (Ο : CategoryTheory.ComposableArrows.mkβ f g βΆ CategoryTheory.ComposableArrows.mkβ f' g') {Z : HomologicalComplex C (ComplexShape.up β€)} (h : (CategoryTheory.shiftFunctor (CochainComplex C β€) 1).obj (CochainComplex.mappingConeCompTriangle f' g').objβ βΆ Z) : CategoryTheory.CategoryStruct.comp (CochainComplex.mappingCone.map g g' (Ο.app 1) (Ο.app 2) β―) (CategoryTheory.CategoryStruct.comp (CochainComplex.mappingConeCompTriangle f' g').morβ h) = CategoryTheory.CategoryStruct.comp (CochainComplex.mappingConeCompTriangle f g).morβ (CategoryTheory.CategoryStruct.comp ((CategoryTheory.shiftFunctor (CochainComplex C β€) 1).map (CochainComplex.mappingCone.map f f' (Ο.app 0) (Ο.app 1) β―)) h) - CochainComplex.mappingConeCompHomotopyEquiv_commβ π Mathlib.Algebra.Homology.HomotopyCategory.Triangulated
{C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {Xβ Xβ Xβ : CochainComplex C β€} (f : Xβ βΆ Xβ) (g : Xβ βΆ Xβ) : CategoryTheory.CategoryStruct.comp (CochainComplex.mappingCone.inr (CochainComplex.mappingCone.map f (CategoryTheory.CategoryStruct.comp f g) (CategoryTheory.CategoryStruct.id Xβ) g β―)) (CochainComplex.mappingConeCompHomotopyEquiv f g).inv = (CochainComplex.mappingConeCompTriangle f g).morβ - CochainComplex.mappingConeCompHomotopyEquiv_commβ_assoc π Mathlib.Algebra.Homology.HomotopyCategory.Triangulated
{C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasBinaryBiproducts C] {Xβ Xβ Xβ : CochainComplex C β€} (f : Xβ βΆ Xβ) (g : Xβ βΆ Xβ) {Z : CochainComplex C β€} (h : CochainComplex.mappingCone g βΆ Z) : CategoryTheory.CategoryStruct.comp (CochainComplex.mappingCone.inr (CochainComplex.mappingCone.map f (CategoryTheory.CategoryStruct.comp f g) (CategoryTheory.CategoryStruct.id Xβ) g β―)) (CategoryTheory.CategoryStruct.comp (CochainComplex.mappingConeCompHomotopyEquiv f g).inv h) = CategoryTheory.CategoryStruct.comp (CochainComplex.mappingConeCompTriangle f g).morβ h
About
Loogle searches Lean and Mathlib definitions and theorems.
You can use Loogle from within the Lean4 VSCode language extension
using (by default) Ctrl-K Ctrl-S. 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:
By constant:
πReal.sin
finds all lemmas whose statement somehow mentions the sine function.By lemma name substring:
π"differ"
finds all lemmas that have"differ"
somewhere in their lemma name.By subexpression:
π_ * (_ ^ _)
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
πReal.sqrt ?a * Real.sqrt ?a
If the pattern has parameters, they are matched in any order. Both of these will find
List.map
:
π(?a -> ?b) -> List ?a -> List ?b
πList ?a -> (?a -> ?b) -> List ?b
By main conclusion:
π|- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of allβ
andβ
) 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,
π|- _ < _ β 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
π Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ β _
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.
This is Loogle revision ee8c038
serving mathlib revision 7a9e177