Benjamin Andersson: Tensor Categories

Abstract.

Tensor categories, also known under the name of ”symmetric monoidal categories”, is an area of import for research in pure mathematics, and with links to computation, logic and physics. In this article, we go through their properties, introducing categorical concepts as needed as we go along. We prove some well-known theorems, such as maclanes coherence theorem, for monoidal categories (and in particular monoidal categories with a symmetry), and we zoom in on symmetric monoidal categories with some more restrictive structure (e.g. rigidity). The work tries to lay the ground for understanding the main theorem in P. Deligne & J.S. Milnes article ”Tannakian Categories”, wherein it is shown that an exact, faithful \(k\)-linear tensor functor yields an equivalence of categories between a more restricted type of tensor category, and the category of linear representations of a group (perphaps affine group scheme) \(G\).