Loogle!

Result

Found 29 declarations mentioning groupHomology.map.

groupHomology.map_id 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G : Type u} [CommRing k] [Group G] {A : Rep k G} (n : ℕ) : groupHomology.map (MonoidHom.id G) (CategoryTheory.CategoryStruct.id A) n = CategoryTheory.CategoryStruct.id (groupHomology A n)
groupHomology.epi_map_0_of_epi 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G : Type u} [CommRing k] [Group G] {A B : Rep k G} (f : A ⟶ B) [CategoryTheory.Epi f] : CategoryTheory.Epi (groupHomology.map (MonoidHom.id G) f 0)
groupHomology.functor_map 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
(k G : Type u) [CommRing k] [Group G] (n : ℕ) {A B : Rep k G} (φ : A ⟶ B) : (groupHomology.functor k G n).map φ = groupHomology.map (MonoidHom.id G) φ n
groupHomology.map 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G H : Type u} [CommRing k] [Group G] [Group H] {A : Rep k G} {B : Rep k H} (f : G →* H) (φ : A ⟶ (Action.res (ModuleCat k) f).obj B) (n : ℕ) : groupHomology A n ⟶ groupHomology B n
groupHomology.map.eq_1 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G H : Type u} [CommRing k] [Group G] [Group H] {A : Rep k G} {B : Rep k H} (f : G →* H) (φ : A ⟶ (Action.res (ModuleCat k) f).obj B) (n : ℕ) : groupHomology.map f φ n = HomologicalComplex.homologyMap (groupHomology.chainsMap f φ) n
groupHomology.π_map 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G H : Type u} [CommRing k] [Group G] [Group H] {A : Rep k G} {B : Rep k H} (f : G →* H) (φ : A ⟶ (Action.res (ModuleCat k) f).obj B) (n : ℕ) : CategoryTheory.CategoryStruct.comp (groupHomology.π A n) (groupHomology.map f φ n) = CategoryTheory.CategoryStruct.comp (groupHomology.cyclesMap f φ n) (groupHomology.π B n)
groupHomology.map_id_comp 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G : Type u} [CommRing k] [Group G] {A B C : Rep k G} (φ : A ⟶ B) (ψ : B ⟶ C) (n : ℕ) : groupHomology.map (MonoidHom.id G) (CategoryTheory.CategoryStruct.comp φ ψ) n = CategoryTheory.CategoryStruct.comp (groupHomology.map (MonoidHom.id G) φ n) (groupHomology.map (MonoidHom.id G) ψ n)
groupHomology.π_map_assoc 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G H : Type u} [CommRing k] [Group G] [Group H] {A : Rep k G} {B : Rep k H} (f : G →* H) (φ : A ⟶ (Action.res (ModuleCat k) f).obj B) (n : ℕ) {Z : ModuleCat k} (h : groupHomology B n ⟶ Z) : CategoryTheory.CategoryStruct.comp (groupHomology.π A n) (CategoryTheory.CategoryStruct.comp (groupHomology.map f φ n) h) = CategoryTheory.CategoryStruct.comp (groupHomology.cyclesMap f φ n) (CategoryTheory.CategoryStruct.comp (groupHomology.π B n) h)
groupHomology.map₁_one 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G H : Type u} [CommRing k] [Group G] [Group H] {A : Rep k G} {B : Rep k H} (φ : A ⟶ (Action.res (ModuleCat k) 1).obj B) : groupHomology.map 1 φ 1 = 0
groupHomology.coresNatTrans_app 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
(k : Type u) {G H : Type u} [CommRing k] [Group G] [Group H] (f : G →* H) (n : ℕ) (X : Action (ModuleCat k) H) : (groupHomology.coresNatTrans k f n).app X = groupHomology.map f (CategoryTheory.CategoryStruct.id ((Action.res (ModuleCat k) f).obj X)) n
groupHomology.H0π_comp_map 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G H : Type u} [CommRing k] [Group G] [Group H] {A : Rep k G} {B : Rep k H} (f : G →* H) (φ : A ⟶ (Action.res (ModuleCat k) f).obj B) : CategoryTheory.CategoryStruct.comp (groupHomology.H0π A) (groupHomology.map f φ 0) = CategoryTheory.CategoryStruct.comp φ.hom (groupHomology.H0π B)
groupHomology.map_id_comp_H0Iso_hom 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G : Type u} [CommRing k] [Group G] {A B : Rep k G} (f : A ⟶ B) : CategoryTheory.CategoryStruct.comp (groupHomology.map (MonoidHom.id G) f 0) (groupHomology.H0Iso B).hom = CategoryTheory.CategoryStruct.comp (groupHomology.H0Iso A).hom ((Rep.coinvariantsFunctor k G).map f)
groupHomology.map_id_comp_H0Iso_hom_assoc 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G : Type u} [CommRing k] [Group G] {A B : Rep k G} (f : A ⟶ B) {Z : ModuleCat k} (h : (Rep.coinvariantsFunctor k G).obj B ⟶ Z) : CategoryTheory.CategoryStruct.comp (groupHomology.map (MonoidHom.id G) f 0) (CategoryTheory.CategoryStruct.comp (groupHomology.H0Iso B).hom h) = CategoryTheory.CategoryStruct.comp (groupHomology.H0Iso A).hom (CategoryTheory.CategoryStruct.comp ((Rep.coinvariantsFunctor k G).map f) h)
groupHomology.H0π_comp_map_assoc 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G H : Type u} [CommRing k] [Group G] [Group H] {A : Rep k G} {B : Rep k H} (f : G →* H) (φ : A ⟶ (Action.res (ModuleCat k) f).obj B) {Z : ModuleCat k} (h : groupHomology B 0 ⟶ Z) : CategoryTheory.CategoryStruct.comp (groupHomology.H0π A) (CategoryTheory.CategoryStruct.comp (groupHomology.map f φ 0) h) = CategoryTheory.CategoryStruct.comp φ.hom (CategoryTheory.CategoryStruct.comp (groupHomology.H0π B) h)
groupHomology.map_1_quotientGroupMk'_epi 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G : Type u} [CommRing k] [Group G] (A : Rep k G) (S : Subgroup G) [S.Normal] [Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)] : CategoryTheory.Epi (groupHomology.map (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv 1)
groupHomology.H1CoresCoinfOfTrivial_g 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G : Type u} [CommRing k] [Group G] (A : Rep k G) (S : Subgroup G) [S.Normal] [Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)] : (groupHomology.H1CoresCoinfOfTrivial A S).g = groupHomology.map (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv 1
groupHomology.map_comp 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k : Type u} [CommRing k] {G H K : Type u} [Group G] [Group H] [Group K] {A : Rep k G} {B : Rep k H} {C : Rep k K} (f : G →* H) (g : H →* K) (φ : A ⟶ (Action.res (ModuleCat k) f).obj B) (ψ : B ⟶ (Action.res (ModuleCat k) g).obj C) (n : ℕ) : groupHomology.map (g.comp f) (CategoryTheory.CategoryStruct.comp φ ((Action.res (ModuleCat k) f).map ψ)) n = CategoryTheory.CategoryStruct.comp (groupHomology.map f φ n) (groupHomology.map g ψ n)
groupHomology.coinfNatTrans_app 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
(k : Type u) {G : Type u} [CommRing k] [Group G] (S : Subgroup G) [S.Normal] (n : ℕ) (A : Rep k G) : (groupHomology.coinfNatTrans k S n).app A = groupHomology.map (QuotientGroup.mk' S) (A.mkQ (Representation.Coinvariants.ker (MonoidHom.comp A.ρ S.subtype)) ⋯) n
groupHomology.map_comp_assoc 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k : Type u} [CommRing k] {G H K : Type u} [Group G] [Group H] [Group K] {A : Rep k G} {B : Rep k H} {C : Rep k K} (f : G →* H) (g : H →* K) (φ : A ⟶ (Action.res (ModuleCat k) f).obj B) (ψ : B ⟶ (Action.res (ModuleCat k) g).obj C) (n : ℕ) {Z : ModuleCat k} (h : groupHomology C n ⟶ Z) : CategoryTheory.CategoryStruct.comp (groupHomology.map (g.comp f) (CategoryTheory.CategoryStruct.comp φ ((Action.res (ModuleCat k) f).map ψ)) n) h = CategoryTheory.CategoryStruct.comp (groupHomology.map f φ n) (CategoryTheory.CategoryStruct.comp (groupHomology.map g ψ n) h)
groupHomology.π_map_apply 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G H : Type u} [CommRing k] [Group G] [Group H] {A : Rep k G} {B : Rep k H} (f : G →* H) (φ : A ⟶ (Action.res (ModuleCat k) f).obj B) (n : ℕ) (x : CategoryTheory.ToType (groupHomology.cycles A n)) : (CategoryTheory.ConcreteCategory.hom (groupHomology.map f φ n)) ((CategoryTheory.ConcreteCategory.hom (groupHomology.π A n)) x) = (CategoryTheory.ConcreteCategory.hom (groupHomology.π B n)) ((CategoryTheory.ConcreteCategory.hom (groupHomology.cyclesMap f φ n)) x)
groupHomology.H1CoresCoinfOfTrivial_f 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G : Type u} [CommRing k] [Group G] (A : Rep k G) (S : Subgroup G) [S.Normal] [Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)] : (groupHomology.H1CoresCoinfOfTrivial A S).f = groupHomology.map S.subtype (CategoryTheory.CategoryStruct.id ((Action.res (ModuleCat k) S.subtype).obj A)) 1
groupHomology.H0π_comp_map_apply 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G H : Type u} [CommRing k] [Group G] [Group H] {A : Rep k G} {B : Rep k H} (f : G →* H) (φ : A ⟶ (Action.res (ModuleCat k) f).obj B) (x : CategoryTheory.ToType A.V) : (CategoryTheory.ConcreteCategory.hom (groupHomology.map f φ 0)) ((CategoryTheory.ConcreteCategory.hom (groupHomology.H0π A)) x) = (CategoryTheory.ConcreteCategory.hom (groupHomology.H0π B)) ((CategoryTheory.ConcreteCategory.hom φ.hom) x)
groupHomology.map_id_comp_H0Iso_hom_apply 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G : Type u} [CommRing k] [Group G] {A B : Rep k G} (f : A ⟶ B) (x : CategoryTheory.ToType (groupHomology A 0)) : (CategoryTheory.ConcreteCategory.hom (groupHomology.H0Iso B).hom) ((CategoryTheory.ConcreteCategory.hom (groupHomology.map (MonoidHom.id G) f 0)) x) = (Representation.Coinvariants.map A.ρ B.ρ (ModuleCat.Hom.hom f.hom) ⋯) ((CategoryTheory.ConcreteCategory.hom (groupHomology.H0Iso A).hom) x)
groupHomology.H1π_comp_map 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G H : Type u} [CommRing k] [Group G] [Group H] {A : Rep k G} {B : Rep k H} (f : G →* H) (φ : A ⟶ (Action.res (ModuleCat k) f).obj B) : CategoryTheory.CategoryStruct.comp (groupHomology.H1π A) (groupHomology.map f φ 1) = CategoryTheory.CategoryStruct.comp (groupHomology.mapCycles₁ f φ) (groupHomology.H1π B)
groupHomology.H2π_comp_map 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G H : Type u} [CommRing k] [Group G] [Group H] {A : Rep k G} {B : Rep k H} (f : G →* H) (φ : A ⟶ (Action.res (ModuleCat k) f).obj B) : CategoryTheory.CategoryStruct.comp (groupHomology.H2π A) (groupHomology.map f φ 2) = CategoryTheory.CategoryStruct.comp (groupHomology.mapCycles₂ f φ) (groupHomology.H2π B)
groupHomology.H1π_comp_map_assoc 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G H : Type u} [CommRing k] [Group G] [Group H] {A : Rep k G} {B : Rep k H} (f : G →* H) (φ : A ⟶ (Action.res (ModuleCat k) f).obj B) {Z : ModuleCat k} (h : groupHomology B 1 ⟶ Z) : CategoryTheory.CategoryStruct.comp (groupHomology.H1π A) (CategoryTheory.CategoryStruct.comp (groupHomology.map f φ 1) h) = CategoryTheory.CategoryStruct.comp (groupHomology.mapCycles₁ f φ) (CategoryTheory.CategoryStruct.comp (groupHomology.H1π B) h)
groupHomology.H2π_comp_map_assoc 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G H : Type u} [CommRing k] [Group G] [Group H] {A : Rep k G} {B : Rep k H} (f : G →* H) (φ : A ⟶ (Action.res (ModuleCat k) f).obj B) {Z : ModuleCat k} (h : groupHomology B 2 ⟶ Z) : CategoryTheory.CategoryStruct.comp (groupHomology.H2π A) (CategoryTheory.CategoryStruct.comp (groupHomology.map f φ 2) h) = CategoryTheory.CategoryStruct.comp (groupHomology.mapCycles₂ f φ) (CategoryTheory.CategoryStruct.comp (groupHomology.H2π B) h)
groupHomology.H1π_comp_map_apply 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G H : Type u} [CommRing k] [Group G] [Group H] {A : Rep k G} {B : Rep k H} (f : G →* H) (φ : A ⟶ (Action.res (ModuleCat k) f).obj B) (x : CategoryTheory.ToType (ModuleCat.of k ↥(groupHomology.cycles₁ A))) : (CategoryTheory.ConcreteCategory.hom (groupHomology.map f φ 1)) ((CategoryTheory.ConcreteCategory.hom (groupHomology.H1π A)) x) = (CategoryTheory.ConcreteCategory.hom (groupHomology.H1π B)) ((CategoryTheory.ConcreteCategory.hom (groupHomology.mapCycles₁ f φ)) x)
groupHomology.H2π_comp_map_apply 📋 Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
{k G H : Type u} [CommRing k] [Group G] [Group H] {A : Rep k G} {B : Rep k H} (f : G →* H) (φ : A ⟶ (Action.res (ModuleCat k) f).obj B) (x : CategoryTheory.ToType (ModuleCat.of k ↥(groupHomology.cycles₂ A))) : (CategoryTheory.ConcreteCategory.hom (groupHomology.map f φ 2)) ((CategoryTheory.ConcreteCategory.hom (groupHomology.H2π A)) x) = (CategoryTheory.ConcreteCategory.hom (groupHomology.H2π B)) ((CategoryTheory.ConcreteCategory.hom (groupHomology.mapCycles₂ f φ)) x)

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:
🔍 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 find tsum_lt_tsum even though the hypothesis f 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, _ * _, _ ^ _, |- _ < _ → _
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 ff04530 serving mathlib revision 8623f65