2025/12/14 by Sebbag, Daniel
#Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2512.12556
Let C be a finite tensor category with a decomposition C≅ D⊕M into full abelian categories such that D is a tensor subcategory and M has a single simple projective object. We call C a simple extension of D. When C is braided and D is a braided subcategory, we will call C a braided simple extension. Applying the notion of a simple extension to a pointed non-semisimple category D gives a natural generalization to the family of near-group categories, which was first introduced by Siehler. Braided near-group categories were fully classified by Siehler and Thornton. In this paper, we study the structure of braided simple extensions of a braided finite tensor category D in general, and specifically when D is pointed. In particular, we classify non-degenerate braided non-semisimple near-group categories, and prove that any braided non-semisimple near-group category C is "an extension" of a non-degenerate braided non-semisimple near-group category by Rep(G), where G is the Picard group of a "canonical" symmetric subcategory of C.