Algebra Seminar: Identity in the presence of adjunction

Welcome to join on site or via Zoom link (Meeting ID: 645 5572 6999, please contact the organizer for passcode)

Abstract: Given a possibly non-unital semigroup S, there is a canonical way to  adjoint a unit element to it: we add an element e, and declare that e*s=s=s*e for all s in S. The problem of  adjoining units is appreciably more subtle and difficult in the setting of semigroup categories and  monoidal categories.

Crucially, the known results in this direction assume that the semigroup  category/2-semicategory in question has duals/adjunctions - however, the ordinary definition of an adjunction refers to a unit object, so this approach requires developing a non-unital notion of an adjunction. In this talk, I will present my current work in progress on this problem, and  sketch potential generalizations of the previous results of Houston on promonoidality of rigid semigroup categories, of Ko-Mazorchuk-Zhang on lifting of module categories, and of Ko-Mazorchuk-Zhang, Coulembier and Benson-Etingof-Ostrik on the use of bar resolutions in this setting. Time permitting, I will also sketch how the obtained results may provide a characterization of finite tensor categories among finite abelian monoidal categories.

This is a seminar in our algebra seminar series.