I read on the arXiv the following:

Let $\mathcal{\mathbf{C}}$ be a semisimple spherical tensor category with simple unit and let $\mathbf{\Gamma}$ be the set of isomorphism classes of simple objects.

Unfortunately I could not read further since I didn't understand the jargon:

- category: a collection of objects with morphisms satisfying certain rules.
- e.g. the category
**Set**of Sets and functions between them.

- e.g. the category
- tensor category has a notion of "tensor product" $\otimes$
- e.g.
**Vect**the category of vector spaces and linear maps between them

- e.g.
- spherical tensor category
- A spherical category is a monoidal category with duals that behaves as if its morphisms can be drawn and moved around on a sphere.
**Confused** - A spherical category is a pivotal category where the left and right trace operations coincide on all objects.
**Even more so since I don't know what "pivot" means or why there is left and right "trace"**

- A spherical category is a monoidal category with duals that behaves as if its morphisms can be drawn and moved around on a sphere.
- pivotal category
- A pivotal category is an autonomous category equipped with a monoidal natural isomorphism A→(A∗)∗. Pivotal categories have also been called “sovereign categories.” This is a kind of category with duals.
- E.g. (possibly?)
**Vect**with the duality operation $(V^\ast)^\ast = V$

- semicimple category
- A semisimple category is a category in which each object is a direct sum of finitely many simple objects, and all such direct sums exist.
- E.g. Category of representation of a finite group $G$ which have notions of tensor product $\otimes$ and direct sum $\oplus$, so they behave almost like a ring.

Another look through the paper suggests I am looking for a Frobenius algebra, which has zany rules like these:

Now I can read the second half the sentence:

If $\mathbf{\Gamma}$ is finite we can define $\dim \mathbf{\mathcal{C}} = \sum_{i \in \Gamma} d(X_i)^2$

Because the category is irreducible every object has a reduction to the direct sum of simple object. Presumably there is a way to compute these dimensions of these things? Continuing...

If $\mathcal{\mathcal{C}}$ is finite dimensional and braided then the Gauss sums of $\mathcal{\mathbf{C}}$ are defined by $$ \Delta_{\pm} \mathcal{\mathcal{C}} = \sum_{i \in \Gamma} \omega(X_i)^{\pm 1} d(X_i)^2$$ where $\theta(X) = \omega(X) id_X$ is the twist of the simple object $X$ defined by the spherical structure.

So I look up braided monoidal category the tensor product $\otimes$ has to satisfy "hexagon rules"...

What is an example of a **finite dimensional braided spherical tensor cateogry** ? And how do I compute each term in the sum above in such an instance?