Theorems · Theorem · category theory

CategoryTheory.Localization.Monoidal.tensor_comp_assoc

∀ {C : Type u_1} {D : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
  [inst_1 : CategoryTheory.Category.{v_2, u_2} D] (L : CategoryTheory.Functor C D)
  (W : CategoryTheory.MorphismProperty C) [inst_2 : CategoryTheory.MonoidalCategory C] [inst_3 : W.IsMonoidal]
  [inst_4 : L.IsLocalization W] {unit : D} (ε : L.obj (CategoryTheory.MonoidalCategoryStruct.tensorUnit C) ≅ unit)
  {X₁ Y₁ Z₁ X₂ Y₂ Z₂ : CategoryTheory.LocalizedMonoidal L W ε} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (g₁ : Y₁ ⟶ Z₁)
  (g₂ : Y₂ ⟶ Z₂) {Z : CategoryTheory.LocalizedMonoidal L W ε}
  (h : CategoryTheory.MonoidalCategoryStruct.tensorObj Z₁ Z₂ ⟶ Z),
  CategoryTheory.CategoryStruct.comp
      (CategoryTheory.MonoidalCategoryStruct.tensorHom (CategoryTheory.CategoryStruct.comp f₁ g₁)
        (CategoryTheory.CategoryStruct.comp f₂ g₂))
      h =
    CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.tensorHom f₁ f₂)
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.tensorHom g₁ g₂) h)

Defined in

Mathlib.CategoryTheory.Localization.Monoidal.Basic

Cited by

2 results in Mathlib

Foundations

Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.MonoidalCategoryCategoryTheory.MorphismProperty.IsMonoidalCategoryTheory.Functor.IsLocalization