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Mathlib.CategoryTheory.Monoidal.Center

Half braidings and the Drinfeld center of a monoidal category #

We define Center C to be pairs ⟨X, b⟩, where X : C and b is a half-braiding on X.

We show that Center C is braided monoidal, and provide the monoidal functor Center.forget from Center C back to C.

Implementation notes #

Verifying the various axioms directly requires tedious rewriting. Using the slice tactic may make the proofs marginally more readable.

More exciting, however, would be to make possible one of the following options:

  1. Integration with homotopy.io / globular to give "picture proofs".
  2. The monoidal coherence theorem, so we can ignore associators (after which most of these proofs are trivial).
  3. Automating these proofs using rewrite_search or some relative.

In this file, we take the second approach using the monoidal composition ⊗≫ and the coherence tactic.

A half-braiding on X : C is a family of isomorphisms X ⊗ U ≅ U ⊗ X, monoidally natural in U : C.

Thinking of C as a 2-category with a single 0-morphism, these are the same as natural transformations (in the pseudo- sense) of the identity 2-functor on C, which send the unique 0-morphism to X.

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    The Drinfeld center of a monoidal category C has as objects pairs ⟨X, b⟩, where X : C and b is a half-braiding on X.

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      A morphism in the Drinfeld center of C.

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        theorem CategoryTheory.Center.Hom.ext {C : Type u₁} :
        ∀ {inst : CategoryTheory.Category.{v₁, u₁} C} {inst_1 : CategoryTheory.MonoidalCategory C} {X Y : CategoryTheory.Center C} {x y : X.Hom Y}, x.f = y.fx = y
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        • CategoryTheory.Center.instQuiver = { Hom := CategoryTheory.Center.Hom }

        Construct an isomorphism in the Drinfeld center from a morphism whose underlying morphism is an isomorphism.

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          Auxiliary definition for the MonoidalCategory instance on Center C.

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            Auxiliary definition for the MonoidalCategory instance on Center C.

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              Auxiliary definition for the MonoidalCategory instance on Center C.

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                def CategoryTheory.Center.tensorHom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {X₁ : CategoryTheory.Center C} {Y₁ : CategoryTheory.Center C} {X₂ : CategoryTheory.Center C} {Y₂ : CategoryTheory.Center C} (f : X₁ Y₁) (g : X₂ Y₂) :
                X₁.tensorObj X₂ Y₁.tensorObj Y₂

                Auxiliary definition for the MonoidalCategory instance on Center C.

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                  Auxiliary definition for the MonoidalCategory instance on Center C.

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                    @[simp]
                    theorem CategoryTheory.Center.tensorUnit_fst {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :
                    CategoryTheory.Center.tensorUnit.fst = 𝟙_ C

                    Auxiliary definition for the MonoidalCategory instance on Center C.

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                      Auxiliary definition for the MonoidalCategory instance on Center C.

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                        Auxiliary definition for the MonoidalCategory instance on Center C.

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                          The forgetful monoidal functor from the Drinfeld center to the original category.

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                            Auxiliary definition for the BraidedCategory instance on Center C.

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                              Auxiliary construction for ofBraided.

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                                The functor lifting a braided category to its center, using the braiding as the half-braiding.

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                                  @[simp]
                                  theorem CategoryTheory.Center.ofBraided_toLaxMonoidalFunctor_μ_f (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : C) (Y : C) :
                                  ((CategoryTheory.Center.ofBraided C) X Y).f = CategoryTheory.CategoryStruct.id (CategoryTheory.MonoidalCategory.tensorObj ({ obj := CategoryTheory.Center.ofBraidedObj, map := fun {X Y : C} (f : X Y) => { f := f, comm := }, map_id := , map_comp := }.obj X) ({ obj := CategoryTheory.Center.ofBraidedObj, map := fun {X Y : C} (f : X Y) => { f := f, comm := }, map_id := , map_comp := }.obj Y)).fst