• Fusion categories are the input to the Levin-Wen construction

  • Excitations (anyons) are described by the Drinfel'd center

  • If an invertible bimodule exists between two fusion categories, they lead to the same excitations

1. Finite set of simple objects

2. For each triple , a finite dimensional vector space

3. For each 4-tuple, isomorphisms

obeying the pentagon equation. After choosing bases for all , the category can be specified by a finite set of matrices. We will use the string-diagram notation:

Theorem 1:

Let be unitary, skeletal fusion categories, and an indecomposable, unitary, finitely semisimple, skeletal bimodule category.

Then is invertible if and only if and

Associated to any module category is an algebra , with basis

Associated to any module category , there is a fusion category called the Morita dual, denoted

• This is the unique fusion category such that is invertible

• Typically, is defined as the category of -module endofunctors on

We can associate a representation of to each object as follows:

Define be the vector space with the action of given by:

which remains in .

In the case that is equivalent to , the simple objects of correspond to the irreducible representations of

We can therefore verify invertibility by checking that:

1. Simple objects of label irreducible representations of
2. Distinct simples label to distinct irreducibles
3. All irreducibles of are labelled by simples of

We can verify the above conditions by utilizing a generalization of Schur orthogonality to weak Hopf algebras.

Given a finite group, we can always define two fusion categories:

• , the category of -graded vector spaces
• , the category of finite dimensional representations of

Recall from the introduction that we wanted to understand MPO-injectivity

The data used to define these tensors is exactly the skelatal data of a bimodule category

As an applicaiton of our main result, we can now understand MPO-injectivity as the requirement that the bimodule category is invertible

• Extending classical results to quantum symmetries

  • Schur orthogonality
  • Wigner-Eckart theorem: Constraints on symmetric tensors
  • Applications to tensor networks

• Open questions

  • Going beyond finite case?
  • Higher dimensions?
    • Fusion n-categories
    • Higher module module categories
    • Generalizing weak Hopf algebras?

Given bimodule categories , , the product is the category , with objects given by:

Objects , together with a collection of isomorphisms which must be compatible with all associators