I am interested in Deligne's Theorem regarding how all tensor categories satisfying some "nice" properties are equivalent to $\textrm{Rep}(G, \epsilon)$ where $G$ is a supergroup. I have two questions.
- What are precisely the "nice" properties of the tensor category that is assumed by Deligne's Theorem? In nLab and these lecture notes by Etingof and Kannan describe what these "nice" properties are and they seem to agree with each other. However, in this exposition by Ostrik on Deligne's original paper it states that the tensor category is further required to be finitely $\otimes$-generated (there exists some object $X$ such that every object in the category is a sub-summand of $X^{\otimes n}$ for some $n$). This assumption does not appear to be required based on the statement in nLab and notes by Etingof and Kannan. So I would like to confirm, is this additional assumption of being finitely $\otimes$-generated neccessary? I tried looking at the original paper by Deligne; however, it is in French and I couldn't find an English translation.
- Can anyone recommend a source which provides a proof of Deligne's Theorem that is accessible to someone that is not very familiar with category theory? In particular, I have a "nice" tensor category and I am interested in trying to the supergroup $G$ such that the tensor category is equivalent to $\textrm{Rep}(G,\epsilon)$.