Monoidal adjunctions and abelian envelopes

We show how monoidal adjunctions can be used to prove the existence of monoidal abelian envelopes of pseudo-tensor categories, in particular, those admitting a combinatorial description with certain properties. We derive concrete general criteria, which we then demonstrate by giving relatively simple combinatorial proofs of the existence of new abelian envelopes for interpolation categories of the hyperoctahedral and of the modified symmetric groups.

💡 Research Summary

This paper develops a systematic method for constructing monoidal abelian envelopes of pseudo‑tensor categories by exploiting monoidal adjunctions. The authors introduce the notion of “pseudo‑diagrammatic” linear monoidal categories, i.e., categories whose Hom‑spaces are spanned by collections of morphisms closed under tensor product—a property typical of diagrammatic categories such as those built from cobordisms or partition diagrams.

First, they prove that every pseudo‑diagrammatic category satisfies the necessary exactness condition identified in recent work (CEOP23), which is a prerequisite for the existence of an abelian envelope (Theorem A, Proposition 3.10). The central technical tool is the concept of a splitting object: an object X such that tensoring with X turns any morphism into a split morphism. The presence of a splitting object guarantees that any abelian envelope will have the “quotient property” (every object in the envelope is a quotient of an object from the original category).

The key insight is that splitting objects can be transferred along suitable monoidal adjunctions. Specifically, if a pseudo‑tensor category C admits a linear monoidal functor F to an abelian tensor category T that has enough projectives and F possesses a left or right adjoint, then C inherits a splitting object and consequently an abelian envelope with enough projectives (Theorem B, Theorem 5.11, Corollary 6.2). The authors also treat variants where the adjunction exists only after passing to ind‑completions or on a family of full subcategories, yielding envelopes that may lack enough projectives but still enjoy the quotient property.

To make the adjunction hypothesis verifiable, they show that the existence of a right adjoint to F follows from the representability of the functor Hom(F(–),𝟙) (Corollary 5.5, Lemma 5.7). This provides a concrete, often combinatorial, criterion for checking the hypotheses in practice.

Next, they give a streamlined construction of abelian envelopes with enough projectives. They prove that for any pseudo‑tensor category C the following are equivalent: (a) C has an abelian envelope with enough projectives; (b) the unit object 𝟙 admits a presentation by global splitting objects; (c) C possesses a non‑zero global splitting object and satisfies the exactness condition of CEOP23 (Theorem C, Theorem 6.1). In the pseudo‑diagrammatic setting, condition (c) alone suffices, because the exactness condition is automatic.

The general theory is then applied to subcategories of Deligne’s interpolation categories Sₜ = Rep Sₜ. Theorem D states that if a subcategory C ⊂ Sₜ is generated by objects