The Motivation

This package began its journey asking the question "Can we play around with explicit categorical entities in the computer?".

By nature categorical operations and constructions are generic and abstract. The categorical language therefore provides a framework of construction that can be performed as long as the objects (or morphisms) play along. TensorCategories.jl aims to provide an interface for categories with additional structure like additive, linear, abelian, monoidal, tensor and fusion categories. The main focus though lies in fusion and finite tensor categories.

Realizing Categories in The Computer

Due to the nature of category theory the realization of certain categories is very dependent on themselves. Thus the internal workings are generally up to the user. As long as the interface for the desired additional structures is implemented.

Some kind of categories, i.e. fusion categories, are entirely described (up to equivalence) by discrete data known as $F$-symbols. Thus for such categories we can provide a datatype SixJCategory to quickly work with categories given by such data.

Mathematical Foundation

Throughout the package we will consider definitions and terminology as provided in [1].