Semiseparability of induction functors in a monoidal category

TL;DR

This work develops a unified framework for semiseparability of (co)induction functors in monoidal categories, providing necessary and sufficient conditions via regularity of (co)algebra morphisms and revealing when semiseparability persists under lax/colax monoidal functors. A Rafael-type theorem translates semiseparability of left adjoints into regularity of units, yielding concrete criteria for induction and coinduction in a broad array of categories, including abelian, Set, bimodule, and bialgebra module settings. The paper then extends these ideas to duoidal categories, showing that semiseparability is preserved under combinations of two monoidal structures and providing natural corollaries for familiar structures such as cartesian products and pre-braided categories. Collectively, the results generalize and recover classical semiseparability notions (AB22, CGN97, ACMM06) while supplying new tools for analyzing (co)induction in advanced categorical frameworks. The findings have potential applications in areas that rely on controlled transfer functors between module/comodule categories and in the study of structured monoidal contexts like 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.

Theorems & Definitions (111)

• Proposition 2.1
• Lemma 2.2
• Proposition 2.3
• Corollary 2.4
• Theorem 2.5
• Proposition 2.6
• Proposition 2.7
• Proposition 2.8
• Definition 3.1
• Remark 3.2