The Center Construction
Let $\mathcal C$ be a monoidal category. The Drinfeld center of $\mathcal C$ is a category whose objects are tuples $(X,\gamma)$ such that $X \in \mathcal C$ and $\{\gamma_Z \colon X \otimes Z \to Z \otimes X \mid Z \in \mathcal C\}$ is a natural isomorphism such that
commutes for all $Y,Z \in \mathcal C$ and $\gamma_{\mathbb 1} = \mathrm{id}_X$.
Computing the Center
The Drinfeld center can be computed explicitely for reasonably small fusion categories. Any fusion category implementing the corresponding interface is supported.
Example
I = Ising()
C = center(I)
simples(C)5-element Vector{CenterObject}:
Central object: 𝟙
Central object: 𝟙
Central object: 𝟙 ⊕ χ
Central object: 2⋅χ
Central object: 4⋅XMethods
AbstractAlgebra.Generic.dim — Methoddim(X::CenterObject)Return the categorical dimension of X.
AbstractAlgebra.compose — Methodcompose(f::CenterMorphism, g::CenterMorphism)Return the composition g∘f.
AbstractAlgebra.direct_sum — Methoddirect_sum(f::CenterMorphism, g::CenterMorphism)Return the direct sum of f and g.
AbstractAlgebra.direct_sum — Methoddirect_sum(X::CenterObject, Y::CenterObject)Return the direct sum object of X and Y.
AbstractAlgebra.is_isomorphic — Methodis_isomorphic(X::CenterObject, Y::CenterObject)Check if X≃Y. Return (true, m) where mis an isomorphism if true, else return (false,nothing).
AbstractAlgebra.kernel — Methodkernel(f::CenterMoprhism)Return a tuple (K,k) where Kis the kernel object and kis the inclusion.
Base.inv — Methodinv(f::CenterMorphism)Return the inverse of fif possible.
Base.one — Methodone(C::CenterCategory)Return the one object of C.
Base.zero — Methodzero(C::CenterCategory)Return the zero object of C.
Hecke.center — Methodcenter(C::Category)Return the Drinfeld center of C.
Hecke.cokernel — Methodcokernel(f::CenterMorphism)Return a tuple (C,c) where Cis the cokernel object and cis the projection.
Hecke.dual — Methoddual(X::CenterObject)Return the (left) dual object of X.
Hecke.id — Methodid(X::CenterObject)Return the identity on X.
Hecke.is_central — Functionis_central(Z::Object)Return true if Z is in the categorical center, i.e. there exists a half-braiding on Z.
Hecke.tensor_product — Methodtensor_product(f::CenterMorphism,g::CenterMorphism)Return the tensor product of f and g.
Hecke.tensor_product — Methodtensor_product(X::CenterObject, Y::CenterObject)Return the tensor product of X and Y.
LinearAlgebra.tr — Methodtr(f:::CenterMorphism)Return the categorical trace of f.
TensorCategories.add_simple! — Methodadd_simple!(C::CenterCategory, S::CenterObject)Add the simple object S to the vector of simple objects.
TensorCategories.associator — Methodassociator(X::CenterObject, Y::CenterObject, Z::CenterObject)Return the associator isomorphism (X⊗Y)⊗Z → X⊗(Y⊗Z).
TensorCategories.braiding — Methodbraiding(X::CenterObject, Y::CenterObject)Return the braiding isomorphism γ_X(Y): X⊗Y → Y⊗X.
TensorCategories.central_projection — Functioncentral_projection(X::CenterObject, Y::CenterObject, f::Morphism)Compute the image under the projection Hom(F(X),F(Y)) → Hom(X,Y).
TensorCategories.coev — Methodcoev(X::CenterObject)Return the coevaluation morphism 1 → X⊗X∗.
TensorCategories.ev — Methodev(X::CenterObject)Return the evaluation morphism X⊗X → 1.
TensorCategories.half_braiding — Methodhalf_braiding(X::CenterObject, Y::Object)Return the half braiding isomorphism γ_X(Y): X⊗Y → Y⊗X.
TensorCategories.half_braiding — Methodhalf_braiding(Z::CenterObject)Return a vector with half braiding morphisms Z⊗S → S⊗Z for all simple objects S.
TensorCategories.half_braidings — Methodhalf_braidings(Z::Object)Return all objects in the center lying over Z.
TensorCategories.morphism — Methodmorphism(f::CenterMorphism)Return the image under the forgetful functor.
TensorCategories.object — Methodobject(X::CenterObject)Return the image under the forgetful functor.
TensorCategories.simples — Methodsimples(C::CenterCategory)Return a vector containing the simple objects of C.
TensorCategories.spherical — Methodspherical(X::CenterObject)Return the spherical structure X → X∗∗ of X.
TensorCategories.zero_morphism — Methodzero_morphism(X::CenterObject, Y::CenterObject)Return the zero morphism 0:X → Y.
TensorCategories.is_half_braiding — Functionis_half_braiding(X::Object, half_braiding::Vector{<:Morphism})TBW
TensorCategories.adjusted_dual_basis — Methodadjusted_dual_basis(V::AbstractHomSpace, U::AbstractHomSpace, S::Object, W::Object, T::Object)Compute a dual basis for the spaces Hom(S, W⊗T) and Hom(S̄⊗W, T̄)
TensorCategories.dual_basis — Methoddual_basis(V::AbstractHomSpace, W::AbstractHomSpace)Compute the dual basis for Hom(X,Y) and Hom(X̄,Ȳ)