Combinatorial criteria for the existence of abelian envelopes

Generally speaking, it is easy to construct linear monoidal categories, but hard to make them abelian. This subtle problem of finding abelian envelopes features prominently in the theory of interpolation categories, Verlinde categories, and tensor categories from pro-oligomorphic groups. I will explain how to use monoidal adjunctions and concrete combinatorial criteria to address it in certain cases. This is based on joint work with Robert Laugwitz and Sebastian Posur.