This is an overview talk about Deligne categories. These categories interpolate (representation) categories to complex parameters (or parameters in any field). Initially Deligne considered the cases $\underline{Rep}(O_t)$, $\underline{Rep}(GL_t)$, $\underline{Rep}(S_t)$ for $t \in \mathbb{C}$ which interpolate representations of the orthogonal groups, the general linear and the symmetric groups (we refer to those as classical Deligne categories).

After that I will discuss briefly generalizations by Knop (complex representations of finite groups of Lie type) and Etingof (interpolating categories for Harish-Chandra modules, Rational Cherednik algebras, parabolic category $\mathcal{O}$, ...).

Deligne categories are typically not abelian (nor semisimple) for integer parameters. For the classical Deligne categories this leads to the complicated construction of abelian envelopes which should morally replace the original Deligne categories at such parameters.

I will mostly try to explain the connection of the classical Deligne categories to other tensor categories:

1) Deligne's original motivation (abelian tensor categories in characteristic 0 which are not super tannakian)

2) Connections to the stable representation category $Rep (GL(\infty))$ or the algebraic representations of the infinite symmetric group.

3) Classification of (tensor) ideals, applications to representations of supergroups

3) Deligne categories as limits in rank and characteristic of modular representation categories.

If this talk doesn't go completely astray, then there should be some time to talk about Deligne categories for quantum groups and the situation in characteristic p.

This talk should (hopefully) be the first of a couple of other talks about Deligne categories (by hopefully other people). I will make some suggestions at the end about possible topics.

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