Theorems · Theorem · category theory

CategoryTheory.GradedObject.Monoidal.triangle

∀ {I : Type u} [inst : AddMonoid I] {C : Type u_1} [inst_1 : CategoryTheory.Category.{v_1, u_1} C]
  [inst_2 : CategoryTheory.MonoidalCategory C] [inst_3 : DecidableEq I] [inst_4 : CategoryTheory.Limits.HasInitial C]
  [inst_5 :
    ∀ (X₁ : C),
      CategoryTheory.Limits.PreservesColimit (CategoryTheory.Functor.empty C)
        ((CategoryTheory.MonoidalCategory.curriedTensor C).obj X₁)]
  [inst_6 :
    ∀ (X₂ : C),
      CategoryTheory.Limits.PreservesColimit (CategoryTheory.Functor.empty C)
        ((CategoryTheory.MonoidalCategory.curriedTensor C).flip.obj X₂)]
  (X₁ X₃ : CategoryTheory.GradedObject I C) [inst_7 : X₁.HasTensor X₃]
  [inst_8 :
    (CategoryTheory.GradedObject.Monoidal.tensorObj X₁ CategoryTheory.GradedObject.Monoidal.tensorUnit).HasTensor X₃]
  [inst_9 :
    X₁.HasTensor (CategoryTheory.GradedObject.Monoidal.tensorObj CategoryTheory.GradedObject.Monoidal.tensorUnit X₃)]
  [inst_10 : X₁.HasGoodTensor₁₂Tensor CategoryTheory.GradedObject.Monoidal.tensorUnit X₃]
  [inst_11 : X₁.HasGoodTensorTensor₂₃ CategoryTheory.GradedObject.Monoidal.tensorUnit X₃],
  CategoryTheory.CategoryStruct.comp
      (CategoryTheory.GradedObject.Monoidal.associator X₁ CategoryTheory.GradedObject.Monoidal.tensorUnit X₃).hom
      (CategoryTheory.GradedObject.Monoidal.tensorHom (CategoryTheory.CategoryStruct.id X₁)
        (CategoryTheory.GradedObject.Monoidal.leftUnitor X₃).hom) =
    CategoryTheory.GradedObject.Monoidal.tensorHom (CategoryTheory.GradedObject.Monoidal.rightUnitor X₁).hom
      (CategoryTheory.CategoryStruct.id X₃)

Defined in

Mathlib.CategoryTheory.GradedObject.Monoidal

Cited by

0 results in Mathlib

Foundations

Depth 44 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddMonoidCategoryTheory.CategoryCategoryTheory.MonoidalCategoryDecidableEqCategoryTheory.Limits.HasInitialCategoryTheory.Limits.PreservesColimitCategoryTheory.Limits.PreservesColimitCategoryTheory.GradedObject.HasTensorCategoryTheory.GradedObject.HasTensorCategoryTheory.GradedObject.HasTensorCategoryTheory.GradedObject.HasGoodTensor₁₂TensorCategoryTheory.GradedObject.HasGoodTensorTensor₂₃