Introduction
A tensor category is, roughly, a category equipped with a tensor product that lets you “multiply” objects and morphisms in a way that behaves like ordinary multiplication up to coherent isomorphism, together with a unit object playing the role of identity. It can be viewed as a categorical generalization of familiar algebraic settings such as vector spaces and representation categories, and it is often described as part of the broader idea of categorification, where ring-like structures are lifted to the level of categories. Tensor categories appear widely in modern mathematics and mathematical physics, especially in representation theory, Hopf algebras, topology, quantum field theory, and quantum computation, which is why they are a central language for studying symmetry and composition in a highly structured way.
Contents
Monoidal Categories
Finite Sets Category
Braided Monoidal Categories
Braided Monoidal Categories
Yang-Baxter Equation
The Braid Category
Tangles and Categories of Tangles