# Documentation

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.

## Future work #

Verifying the various axioms here is done by 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; I'm unsure if the monoidal coherence theorem is even usable in dependent type theory).
3. Automating these proofs using rewrite_search or some relative.
structure CategoryTheory.HalfBraiding {C : Type u₁} [] (X : C) :
Type (max u₁ v₁)

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.

Instances For
theorem CategoryTheory.HalfBraiding.naturality_assoc {C : Type u₁} [] {X : C} (self : ) {U : C} {U' : C} (f : U U') {Z : C} (h : ) :
def CategoryTheory.Center (C : Type u₁) [] :
Type (max u₁ u₁ v₁)

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.

Instances For
theorem CategoryTheory.Center.Hom.ext_iff {C : Type u₁} :
∀ {inst : } {inst_1 : } {X Y : } (x y : ), x = y x.f = y.f
theorem CategoryTheory.Center.Hom.ext {C : Type u₁} :
∀ {inst : } {inst_1 : } {X Y : } (x y : ), x.f = y.fx = y
structure CategoryTheory.Center.Hom {C : Type u₁} [] (X : ) (Y : ) :
Type v₁
• f : X.fst Y.fst
• comm : ∀ (U : C), =

A morphism in the Drinfeld center of C.

Instances For
@[simp]
theorem CategoryTheory.Center.Hom.comm_assoc {C : Type u₁} [] {X : } {Y : } (self : ) (U : C) {Z : C} (h : ) :
=
theorem CategoryTheory.Center.ext {C : Type u₁} [] {X : } {Y : } (f : X Y) (g : X Y) (w : f.f = g.f) :
f = g
@[simp]
theorem CategoryTheory.Center.id_f {C : Type u₁} [] (X : ) :
@[simp]
theorem CategoryTheory.Center.comp_f {C : Type u₁} [] {X : } {Y : } {Z : } (f : X Y) (g : Y Z) :
@[simp]
theorem CategoryTheory.Center.isoMk_hom {C : Type u₁} [] {X : } {Y : } (f : X Y) [] :
().hom = f
@[simp]
theorem CategoryTheory.Center.isoMk_inv_f {C : Type u₁} [] {X : } {Y : } (f : X Y) [] :
().inv.f =
def CategoryTheory.Center.isoMk {C : Type u₁} [] {X : } {Y : } (f : X Y) [] :
X Y

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

Instances For
instance CategoryTheory.Center.isIso_of_f_isIso {C : Type u₁} [] {X : } {Y : } (f : X Y) [] :
@[simp]
theorem CategoryTheory.Center.tensorObj_fst {C : Type u₁} [] (X : ) (Y : ) :
().fst =
def CategoryTheory.Center.tensorObj {C : Type u₁} [] (X : ) (Y : ) :

Auxiliary definition for the MonoidalCategory instance on Center C.

Instances For
@[simp]
theorem CategoryTheory.Center.tensorHom_f {C : Type u₁} [] {X₁ : } {Y₁ : } {X₂ : } {Y₂ : } (f : X₁ Y₁) (g : X₂ Y₂) :
def CategoryTheory.Center.tensorHom {C : Type u₁} [] {X₁ : } {Y₁ : } {X₂ : } {Y₂ : } (f : X₁ Y₁) (g : X₂ Y₂) :

Auxiliary definition for the MonoidalCategory instance on Center C.

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@[simp]
theorem CategoryTheory.Center.tensorUnit_snd_β {C : Type u₁} [] (U : C) :
CategoryTheory.HalfBraiding.β CategoryTheory.Center.tensorUnit.snd U =
@[simp]
theorem CategoryTheory.Center.tensorUnit_fst {C : Type u₁} [] :
CategoryTheory.Center.tensorUnit.fst =

Auxiliary definition for the MonoidalCategory instance on Center C.

Instances For

Auxiliary definition for the MonoidalCategory instance on Center C.

Instances For
def CategoryTheory.Center.leftUnitor {C : Type u₁} [] (X : ) :
CategoryTheory.Center.tensorObj CategoryTheory.Center.tensorUnit X X

Auxiliary definition for the MonoidalCategory instance on Center C.

Instances For
def CategoryTheory.Center.rightUnitor {C : Type u₁} [] (X : ) :
CategoryTheory.Center.tensorObj X CategoryTheory.Center.tensorUnit X

Auxiliary definition for the MonoidalCategory instance on Center C.

Instances For
@[simp]
theorem CategoryTheory.Center.tensor_fst {C : Type u₁} [] (X : ) (Y : ) :
=
@[simp]
theorem CategoryTheory.Center.tensor_f {C : Type u₁} [] {X₁ : } {Y₁ : } {X₂ : } {Y₂ : } (f : X₁ Y₁) (g : X₂ Y₂) :
@[simp]
@[simp]
theorem CategoryTheory.Center.associator_hom_f {C : Type u₁} [] (X : ) (Y : ) (Z : ) :
().hom.f = (CategoryTheory.MonoidalCategory.associator X.fst Y.fst Z.fst).hom
@[simp]
theorem CategoryTheory.Center.associator_inv_f {C : Type u₁} [] (X : ) (Y : ) (Z : ) :
().inv.f = (CategoryTheory.MonoidalCategory.associator X.fst Y.fst Z.fst).inv
@[simp]
theorem CategoryTheory.Center.leftUnitor_hom_f {C : Type u₁} [] (X : ) :
= ().hom
@[simp]
theorem CategoryTheory.Center.leftUnitor_inv_f {C : Type u₁} [] (X : ) :
= ().inv
@[simp]
theorem CategoryTheory.Center.rightUnitor_hom_f {C : Type u₁} [] (X : ) :
= ().hom
@[simp]
theorem CategoryTheory.Center.rightUnitor_inv_f {C : Type u₁} [] (X : ) :
= ().inv
@[simp]
theorem CategoryTheory.Center.forget_toLaxMonoidalFunctor_ε (C : Type u₁) [] :
().toLaxMonoidalFunctor =
@[simp]
theorem CategoryTheory.Center.forget_toLaxMonoidalFunctor_μ (C : Type u₁) [] (X : ) (Y : ) :
CategoryTheory.LaxMonoidalFunctor.μ ().toLaxMonoidalFunctor X Y =
@[simp]
theorem CategoryTheory.Center.forget_toLaxMonoidalFunctor_toFunctor_map (C : Type u₁) [] :
∀ {X Y : } (f : X Y), ().toLaxMonoidalFunctor.toFunctor.map f = f.f
@[simp]
theorem CategoryTheory.Center.forget_toLaxMonoidalFunctor_toFunctor_obj (C : Type u₁) [] (X : ) :
().toLaxMonoidalFunctor.toFunctor.obj X = X.fst

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

Instances For
@[simp]
theorem CategoryTheory.Center.braiding_hom_f {C : Type u₁} [] (X : ) (Y : ) :
().hom.f = (CategoryTheory.HalfBraiding.β X.snd Y.fst).hom
@[simp]
theorem CategoryTheory.Center.braiding_inv_f {C : Type u₁} [] (X : ) (Y : ) :
().inv.f = (CategoryTheory.HalfBraiding.β X.snd Y.fst).inv

Auxiliary definition for the BraidedCategory instance on Center C.

Instances For
@[simp]
theorem CategoryTheory.Center.ofBraidedObj_fst {C : Type u₁} [] (X : C) :
= X
@[simp]
theorem CategoryTheory.Center.ofBraidedObj_snd_β {C : Type u₁} [] (X : C) (Y : C) :
def CategoryTheory.Center.ofBraidedObj {C : Type u₁} [] (X : C) :

Auxiliary construction for ofBraided.

Instances For
@[simp]
theorem CategoryTheory.Center.ofBraided_toLaxMonoidalFunctor_μ_f (C : Type u₁) [] (X : C) (Y : C) :
(CategoryTheory.LaxMonoidalFunctor.μ ().toLaxMonoidalFunctor X Y).f = CategoryTheory.CategoryStruct.id (CategoryTheory.MonoidalCategory.tensorObj ((CategoryTheory.Functor.mk { obj := CategoryTheory.Center.ofBraidedObj, map := fun {X Y} f => }).obj X) ((CategoryTheory.Functor.mk { obj := CategoryTheory.Center.ofBraidedObj, map := fun {X Y} f => }).obj Y)).fst
@[simp]
theorem CategoryTheory.Center.ofBraided_toLaxMonoidalFunctor_toFunctor_map_f (C : Type u₁) [] :
∀ {X Y : C} (f : X Y), (().toLaxMonoidalFunctor.toFunctor.map f).f = f
@[simp]
theorem CategoryTheory.Center.ofBraided_toLaxMonoidalFunctor_toFunctor_obj (C : Type u₁) [] (X : C) :
().toLaxMonoidalFunctor.toFunctor.obj X =
@[simp]
theorem CategoryTheory.Center.ofBraided_toLaxMonoidalFunctor_ε_f (C : Type u₁) [] :
().toLaxMonoidalFunctor.f =

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

Instances For