Theorems · Definition · category theory

CategoryTheory.BraidedCategory.ofFaithful

{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] →
        [inst_2 : CategoryTheory.MonoidalCategory C] →
          [inst_3 : CategoryTheory.MonoidalCategory D] →
            (F : CategoryTheory.Functor C D) →
              [inst_4 : F.Monoidal] →
                [F.Faithful] →
                  [inst_6 : CategoryTheory.BraidedCategory D] →
                    (β :
                        (X Y : C) →
                          CategoryTheory.MonoidalCategoryStruct.tensorObj X Y ≅
                            CategoryTheory.MonoidalCategoryStruct.tensorObj Y X) →
                      autoParam
                          (∀ (X Y : C),
                            CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.LaxMonoidal.μ F X Y)
                                (F.map (β X Y).hom) =
                              CategoryTheory.CategoryStruct.comp (β_ (F.obj X) (F.obj Y)).hom
                                (CategoryTheory.Functor.LaxMonoidal.μ F Y X))
                          CategoryTheory.BraidedCategory.ofFaithful._auto_1 →
                        CategoryTheory.BraidedCategory C

Verifying the axioms for a braiding by checking that the candidate braiding is sent to a braiding by a faithful monoidal functor.

Defined in

Mathlib.CategoryTheory.Monoidal.Braided.Basic

Cited by

0 results in Mathlib

Foundations

Depth 20 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.MonoidalCategoryCategoryTheory.MonoidalCategoryCategoryTheory.Functor.MonoidalCategoryTheory.Functor.FaithfulCategoryTheory.BraidedCategory