Semiseparability of induction functors in a monoidal category

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.

💡 Research Summary

This paper investigates the semi‑separability of induction and co‑induction functors associated with algebra and coalgebra morphisms in a monoidal category (C, ⊗, 𝟙). After recalling the basic theory of semi‑separable functors and a Rafael‑type theorem characterizing semi‑separability via regularity of the (co)unit, the authors focus on the induction functor φ_* = –⊗R S attached to an algebra morphism φ : R → S. The main result (Theorem 3.5) shows that φ* is semi‑separable if and only if φ is regular as an R‑bimodule morphism, i.e. there exists an R‑bimodule map E : S → R with φ ∘ E ∘ φ = φ (equivalently φ E u_S = u_S). When the monoidal unit 𝟙 is a left‑⊗‑generator, this condition is also necessary. The proof constructs a natural transformation ν: φ_* φ_* ⇒ Id_{C_R} using the coequalizer description of the tensor product and shows η ∘ ν ∘ η = η, where η is the unit of the adjunction φ_* ⊣ φ^*. Conversely, assuming semi‑separability and the generator condition, the associated ν yields the required E, establishing the equivalence.

A parallel theory is developed for co‑induction. For a coalgebra morphism ψ : C → D, the co‑induction functor ψ^* = –□_D C is semi‑separable precisely when ψ is regular as a D‑bicomodule morphism, i.e. there exists χ : D → C with ψ ∘ χ ∘ ψ = ψ (Theorem 4.4). Again, the generator hypothesis (𝟙 left‑⊗‑cogenerator) makes the condition necessary. The authors also treat the stronger notions of separability (split‑epi) and natural fullness (split‑mono) for both induction and co‑induction.

Sections 3.4 and 4.4 examine how semi‑separability behaves under lax (or strong) monoidal functors. If F : C → D is a (strong) monoidal functor, then the induced algebra morphism F(φ) has an induction functor whose semi‑separability is equivalent to that of φ_*. This shows that the property is stable under change of monoidal context.

The final part of the paper moves to duoidal categories (C, ∘, ·). Given two algebra morphisms f₁, f₂ in the ∘‑monoidal structure, the authors prove that the induction functor attached to their ·‑product, (f₁·f₂)_*, inherits semi‑separability (as well as separability and natural fullness) from the individual functors (Proposition 5.2). Analogous results hold for co‑induction (Propositions 5.9, 5.10). These results apply to monoidal categories with finite products, pre‑braided monoidal categories, and concrete examples such as sets, matrix algebras, bimodules over a ring, and modules over a bialgebra.

Overall, the paper provides a comprehensive characterization of semi‑separability for (co)induction functors in monoidal and duoidal settings, establishes necessary and sufficient conditions under mild hypotheses, and demonstrates the robustness of the property under monoidal functors. The work extends classical separability results for ring extensions to a broad categorical framework, offering new tools for researchers working in categorical algebra, quantum groups, and related areas.