Loogle!

Result

Found 9 declarations mentioning CategoryTheory.GradedObject.mapBifunctorRightUnitor.

• CategoryTheory.GradedObject.mapBifunctorRightUnitor 📋 Mathlib.CategoryTheory.GradedObject.Unitor
  {C : Type u_1} {D : Type u_2} {I : Type u_3} {J : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [Zero I] [DecidableEq I] [CategoryTheory.Limits.HasInitial C] (F : CategoryTheory.Functor D (CategoryTheory.Functor C D)) (Y : C) (e : F.flip.obj Y ≅ CategoryTheory.Functor.id D) [∀ (X : D), CategoryTheory.Limits.PreservesColimit (CategoryTheory.Functor.empty C) (F.obj X)] (p : J × I → J) (hp : ∀ (j : J), p (j, 0) = j) (X : CategoryTheory.GradedObject J D) [(((CategoryTheory.GradedObject.mapBifunctor F J I).obj X).obj ((CategoryTheory.GradedObject.single₀ I).obj Y)).HasMap p] : CategoryTheory.GradedObject.mapBifunctorMapObj F p X ((CategoryTheory.GradedObject.single₀ I).obj Y) ≅ X
• CategoryTheory.GradedObject.ι_mapBifunctorRightUnitor_hom_apply 📋 Mathlib.CategoryTheory.GradedObject.Unitor
  {C : Type u_1} {D : Type u_2} {I : Type u_3} {J : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [Zero I] [DecidableEq I] [CategoryTheory.Limits.HasInitial C] (F : CategoryTheory.Functor D (CategoryTheory.Functor C D)) (Y : C) (e : F.flip.obj Y ≅ CategoryTheory.Functor.id D) [∀ (X : D), CategoryTheory.Limits.PreservesColimit (CategoryTheory.Functor.empty C) (F.obj X)] (p : J × I → J) (hp : ∀ (j : J), p (j, 0) = j) (X : CategoryTheory.GradedObject J D) [(((CategoryTheory.GradedObject.mapBifunctor F J I).obj X).obj ((CategoryTheory.GradedObject.single₀ I).obj Y)).HasMap p] (j : J) : CategoryTheory.CategoryStruct.comp (CategoryTheory.GradedObject.ιMapBifunctorMapObj F p X ((CategoryTheory.GradedObject.single₀ I).obj Y) j 0 j ⋯) ((CategoryTheory.GradedObject.mapBifunctorRightUnitor F Y e p hp X).hom j) = CategoryTheory.CategoryStruct.comp ((F.obj (X j)).map (CategoryTheory.GradedObject.singleObjApplyIso 0 Y).hom) (e.hom.app (X j))
• CategoryTheory.GradedObject.mapBifunctorRightUnitor_inv_naturality 📋 Mathlib.CategoryTheory.GradedObject.Unitor
  {C : Type u_1} {D : Type u_2} {I : Type u_3} {J : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [Zero I] [DecidableEq I] [CategoryTheory.Limits.HasInitial C] (F : CategoryTheory.Functor D (CategoryTheory.Functor C D)) {Y : C} (e : F.flip.obj Y ≅ CategoryTheory.Functor.id D) [∀ (X : D), CategoryTheory.Limits.PreservesColimit (CategoryTheory.Functor.empty C) (F.obj X)] (p : J × I → J) (hp : ∀ (j : J), p (j, 0) = j) (X X' : CategoryTheory.GradedObject J D) (φ : X ⟶ X') [(((CategoryTheory.GradedObject.mapBifunctor F J I).obj X).obj ((CategoryTheory.GradedObject.single₀ I).obj Y)).HasMap p] [(((CategoryTheory.GradedObject.mapBifunctor F J I).obj X').obj ((CategoryTheory.GradedObject.single₀ I).obj Y)).HasMap p] : CategoryTheory.CategoryStruct.comp φ (CategoryTheory.GradedObject.mapBifunctorRightUnitor F Y e p hp X').inv = CategoryTheory.CategoryStruct.comp (CategoryTheory.GradedObject.mapBifunctorRightUnitor F Y e p hp X).inv (CategoryTheory.GradedObject.mapBifunctorMapMap F p φ (CategoryTheory.CategoryStruct.id ((CategoryTheory.GradedObject.single₀ I).obj Y)))
• CategoryTheory.GradedObject.mapBifunctorRightUnitor_naturality 📋 Mathlib.CategoryTheory.GradedObject.Unitor
  {C : Type u_1} {D : Type u_2} {I : Type u_3} {J : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [Zero I] [DecidableEq I] [CategoryTheory.Limits.HasInitial C] (F : CategoryTheory.Functor D (CategoryTheory.Functor C D)) {Y : C} (e : F.flip.obj Y ≅ CategoryTheory.Functor.id D) [∀ (X : D), CategoryTheory.Limits.PreservesColimit (CategoryTheory.Functor.empty C) (F.obj X)] (p : J × I → J) (hp : ∀ (j : J), p (j, 0) = j) (X X' : CategoryTheory.GradedObject J D) (φ : X ⟶ X') [(((CategoryTheory.GradedObject.mapBifunctor F J I).obj X).obj ((CategoryTheory.GradedObject.single₀ I).obj Y)).HasMap p] [(((CategoryTheory.GradedObject.mapBifunctor F J I).obj X').obj ((CategoryTheory.GradedObject.single₀ I).obj Y)).HasMap p] : CategoryTheory.CategoryStruct.comp (CategoryTheory.GradedObject.mapBifunctorMapMap F p φ (CategoryTheory.CategoryStruct.id ((CategoryTheory.GradedObject.single₀ I).obj Y))) (CategoryTheory.GradedObject.mapBifunctorRightUnitor F Y e p hp X').hom = CategoryTheory.CategoryStruct.comp (CategoryTheory.GradedObject.mapBifunctorRightUnitor F Y e p hp X).hom φ
• CategoryTheory.GradedObject.mapBifunctorRightUnitor_inv_apply 📋 Mathlib.CategoryTheory.GradedObject.Unitor
  {C : Type u_1} {D : Type u_2} {I : Type u_3} {J : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [Zero I] [DecidableEq I] [CategoryTheory.Limits.HasInitial C] (F : CategoryTheory.Functor D (CategoryTheory.Functor C D)) (Y : C) (e : F.flip.obj Y ≅ CategoryTheory.Functor.id D) [∀ (X : D), CategoryTheory.Limits.PreservesColimit (CategoryTheory.Functor.empty C) (F.obj X)] (p : J × I → J) (hp : ∀ (j : J), p (j, 0) = j) (X : CategoryTheory.GradedObject J D) [(((CategoryTheory.GradedObject.mapBifunctor F J I).obj X).obj ((CategoryTheory.GradedObject.single₀ I).obj Y)).HasMap p] (j : J) : (CategoryTheory.GradedObject.mapBifunctorRightUnitor F Y e p hp X).inv j = CategoryTheory.CategoryStruct.comp (e.inv.app (X j)) (CategoryTheory.CategoryStruct.comp ((F.obj (X j)).map (CategoryTheory.GradedObject.singleObjApplyIso 0 Y).inv) (CategoryTheory.GradedObject.ιMapBifunctorMapObj F p X ((CategoryTheory.GradedObject.single₀ I).obj Y) j 0 j ⋯))
• CategoryTheory.GradedObject.ι_mapBifunctorRightUnitor_hom_apply_assoc 📋 Mathlib.CategoryTheory.GradedObject.Unitor
  {C : Type u_1} {D : Type u_2} {I : Type u_3} {J : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [Zero I] [DecidableEq I] [CategoryTheory.Limits.HasInitial C] (F : CategoryTheory.Functor D (CategoryTheory.Functor C D)) (Y : C) (e : F.flip.obj Y ≅ CategoryTheory.Functor.id D) [∀ (X : D), CategoryTheory.Limits.PreservesColimit (CategoryTheory.Functor.empty C) (F.obj X)] (p : J × I → J) (hp : ∀ (j : J), p (j, 0) = j) (X : CategoryTheory.GradedObject J D) [(((CategoryTheory.GradedObject.mapBifunctor F J I).obj X).obj ((CategoryTheory.GradedObject.single₀ I).obj Y)).HasMap p] (j : J) {Z : D} (h : X j ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.GradedObject.ιMapBifunctorMapObj F p X ((CategoryTheory.GradedObject.single₀ I).obj Y) j 0 j ⋯) (CategoryTheory.CategoryStruct.comp ((CategoryTheory.GradedObject.mapBifunctorRightUnitor F Y e p hp X).hom j) h) = CategoryTheory.CategoryStruct.comp ((F.obj (X j)).map (CategoryTheory.GradedObject.singleObjApplyIso 0 Y).hom) (CategoryTheory.CategoryStruct.comp (e.hom.app (X j)) h)
• CategoryTheory.GradedObject.mapBifunctorRightUnitor_inv_naturality_assoc 📋 Mathlib.CategoryTheory.GradedObject.Unitor
  {C : Type u_1} {D : Type u_2} {I : Type u_3} {J : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [Zero I] [DecidableEq I] [CategoryTheory.Limits.HasInitial C] (F : CategoryTheory.Functor D (CategoryTheory.Functor C D)) {Y : C} (e : F.flip.obj Y ≅ CategoryTheory.Functor.id D) [∀ (X : D), CategoryTheory.Limits.PreservesColimit (CategoryTheory.Functor.empty C) (F.obj X)] (p : J × I → J) (hp : ∀ (j : J), p (j, 0) = j) (X X' : CategoryTheory.GradedObject J D) (φ : X ⟶ X') [(((CategoryTheory.GradedObject.mapBifunctor F J I).obj X).obj ((CategoryTheory.GradedObject.single₀ I).obj Y)).HasMap p] [(((CategoryTheory.GradedObject.mapBifunctor F J I).obj X').obj ((CategoryTheory.GradedObject.single₀ I).obj Y)).HasMap p] {Z : CategoryTheory.GradedObject J D} (h : CategoryTheory.GradedObject.mapBifunctorMapObj F p X' ((CategoryTheory.GradedObject.single₀ I).obj Y) ⟶ Z) : CategoryTheory.CategoryStruct.comp φ (CategoryTheory.CategoryStruct.comp (CategoryTheory.GradedObject.mapBifunctorRightUnitor F Y e p hp X').inv h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.GradedObject.mapBifunctorRightUnitor F Y e p hp X).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.GradedObject.mapBifunctorMapMap F p φ (CategoryTheory.CategoryStruct.id ((CategoryTheory.GradedObject.single₀ I).obj Y))) h)
• CategoryTheory.GradedObject.mapBifunctorRightUnitor_naturality_assoc 📋 Mathlib.CategoryTheory.GradedObject.Unitor
  {C : Type u_1} {D : Type u_2} {I : Type u_3} {J : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [Zero I] [DecidableEq I] [CategoryTheory.Limits.HasInitial C] (F : CategoryTheory.Functor D (CategoryTheory.Functor C D)) {Y : C} (e : F.flip.obj Y ≅ CategoryTheory.Functor.id D) [∀ (X : D), CategoryTheory.Limits.PreservesColimit (CategoryTheory.Functor.empty C) (F.obj X)] (p : J × I → J) (hp : ∀ (j : J), p (j, 0) = j) (X X' : CategoryTheory.GradedObject J D) (φ : X ⟶ X') [(((CategoryTheory.GradedObject.mapBifunctor F J I).obj X).obj ((CategoryTheory.GradedObject.single₀ I).obj Y)).HasMap p] [(((CategoryTheory.GradedObject.mapBifunctor F J I).obj X').obj ((CategoryTheory.GradedObject.single₀ I).obj Y)).HasMap p] {Z : CategoryTheory.GradedObject J D} (h : X' ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.GradedObject.mapBifunctorMapMap F p φ (CategoryTheory.CategoryStruct.id ((CategoryTheory.GradedObject.single₀ I).obj Y))) (CategoryTheory.CategoryStruct.comp (CategoryTheory.GradedObject.mapBifunctorRightUnitor F Y e p hp X').hom h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.GradedObject.mapBifunctorRightUnitor F Y e p hp X).hom (CategoryTheory.CategoryStruct.comp φ h)
• CategoryTheory.GradedObject.mapBifunctor_triangle 📋 Mathlib.CategoryTheory.GradedObject.Unitor
  {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {D : Type u_4} {I₁ : Type u_5} {I₂ : Type u_6} {I₃ : Type u_7} {J : Type u_8} [CategoryTheory.Category.{v_1, u_1} C₁] [CategoryTheory.Category.{v_2, u_2} C₂] [CategoryTheory.Category.{v_3, u_3} C₃] [CategoryTheory.Category.{v_4, u_4} D] [Zero I₂] [DecidableEq I₂] [CategoryTheory.Limits.HasInitial C₂] {F₁ : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C₁)} {F₂ : CategoryTheory.Functor C₂ (CategoryTheory.Functor C₃ C₃)} {G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₃ D)} (associator : CategoryTheory.bifunctorComp₁₂ F₁ G ≅ CategoryTheory.bifunctorComp₂₃ G F₂) (X₂ : C₂) (e₁ : F₁.flip.obj X₂ ≅ CategoryTheory.Functor.id C₁) (e₂ : F₂.obj X₂ ≅ CategoryTheory.Functor.id C₃) [∀ (X₁ : C₁), CategoryTheory.Limits.PreservesColimit (CategoryTheory.Functor.empty C₂) (F₁.obj X₁)] [∀ (X₃ : C₃), CategoryTheory.Limits.PreservesColimit (CategoryTheory.Functor.empty C₂) (F₂.flip.obj X₃)] {r : I₁ × I₂ × I₃ → J} {π : I₁ × I₃ → J} (τ : CategoryTheory.GradedObject.TriangleIndexData r π) (X₁ : CategoryTheory.GradedObject I₁ C₁) (X₃ : CategoryTheory.GradedObject I₃ C₃) [(((CategoryTheory.GradedObject.mapBifunctor F₁ I₁ I₂).obj X₁).obj ((CategoryTheory.GradedObject.single₀ I₂).obj X₂)).HasMap τ.p₁₂] [(((CategoryTheory.GradedObject.mapBifunctor G I₁ I₃).obj (CategoryTheory.GradedObject.mapBifunctorMapObj F₁ τ.p₁₂ X₁ ((CategoryTheory.GradedObject.single₀ I₂).obj X₂))).obj X₃).HasMap π] [(((CategoryTheory.GradedObject.mapBifunctor F₂ I₂ I₃).obj ((CategoryTheory.GradedObject.single₀ I₂).obj X₂)).obj X₃).HasMap τ.p₂₃] [(((CategoryTheory.GradedObject.mapBifunctor G I₁ I₃).obj X₁).obj (CategoryTheory.GradedObject.mapBifunctorMapObj F₂ τ.p₂₃ ((CategoryTheory.GradedObject.single₀ I₂).obj X₂) X₃)).HasMap π] [CategoryTheory.GradedObject.HasGoodTrifunctor₁₂Obj F₁ G τ.ρ₁₂ X₁ ((CategoryTheory.GradedObject.single₀ I₂).obj X₂) X₃] [CategoryTheory.GradedObject.HasGoodTrifunctor₂₃Obj G F₂ τ.ρ₂₃ X₁ ((CategoryTheory.GradedObject.single₀ I₂).obj X₂) X₃] [(((CategoryTheory.GradedObject.mapBifunctor G I₁ I₃).obj X₁).obj X₃).HasMap π] (triangle : ∀ (X₁ : C₁) (X₃ : C₃), CategoryTheory.CategoryStruct.comp (((associator.hom.app X₁).app X₂).app X₃) ((G.obj X₁).map (e₂.hom.app X₃)) = (G.map (e₁.hom.app X₁)).app X₃) : CategoryTheory.CategoryStruct.comp (CategoryTheory.GradedObject.mapBifunctorAssociator associator τ.ρ₁₂ τ.ρ₂₃ X₁ ((CategoryTheory.GradedObject.single₀ I₂).obj X₂) X₃).hom (CategoryTheory.GradedObject.mapBifunctorMapMap G π (CategoryTheory.CategoryStruct.id X₁) (CategoryTheory.GradedObject.mapBifunctorLeftUnitor F₂ X₂ e₂ τ.p₂₃ ⋯ X₃).hom) = CategoryTheory.GradedObject.mapBifunctorMapMap G π (CategoryTheory.GradedObject.mapBifunctorRightUnitor F₁ X₂ e₁ τ.p₁₂ ⋯ X₁).hom (CategoryTheory.CategoryStruct.id X₃)

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