Semiseparability of induction functors in a monoidal category

TL;DR

This paper investigates conditions under which induction and coinduction functors in monoidal categories are semiseparable, exploring their preservation under lax monoidal functors and applications to duoidal categories.

Contribution

It provides necessary and sufficient conditions for semiseparability of induction functors in monoidal categories, extending to coinduction and applications in duoidal categories.

Findings

Semiseparability characterized by algebra and coalgebra morphisms.

Semiseparability preserved under lax monoidal functors.

Applications to combinations of (co)induction functors in duoidal categories.

Abstract

For any algebra morphism in a monoidal category, we provide sufficient conditions (which are also necessary if the unit is a left tensor generator) for the attached induction functor being semiseparable. Under mild assumptions, we prove that the semiseparability of the induction functor is preserved if one applies a lax monoidal functor. Similar results are shown for the coinduction functors attached to coalgebra morphisms in a monoidal category. As an application, we study the semiseparability of combinations of (co)induction functors in the context of duoidal categories.