The second Delannoy category

TL;DR

This paper introduces the second Delannoy category, a non-semi-simple tensor category associated with a specific measure on an oligomorphic group, and constructs its abelian envelope, advancing understanding of oligomorphic tensor categories.

Contribution

It constructs the abelian envelope of the second Delannoy category and fully characterizes its local abelian envelopes, a novel achievement in oligomorphic tensor category theory.

Findings

The second Delannoy category is non-semi-simple but uniform across fields.

The abelian envelope of the second Delannoy category is constructed and characterized.

The category admits exactly two local abelian envelopes, the functors $ ext{ extbackslash}Psi$ and $ ext{ extbackslash}Phi$.

Abstract

In recent work, Harman and Snowden constructed a symmetric tensor category associated to an oligomorphic group equipped with a measure. The oligomorphic group $\mathbb{G}$ of order preserving automorphisms of the real line admits exactly four measures. The category $\mathcal{C}$ associated to the first measure is called the (first) Delannoy category; it is semi-simple and pre-Tannakian, with numerous special properties. In this paper, we study the (non-abelian) category $\mathcal{A}$ associated to the second measure, which we call the second Delannoy category. We construct a new pre-Tannakian category $\mathcal{D}$ together with a fully faithful tensor functor $\Psi \colon \mathcal{A} \to \mathcal{D}$. The category $\mathcal{D}$ is the correct ``abelian version'' of the second Delannoy category. Like $\mathcal{C}$, it has remarkable properties: for instance, it is non-semi-simple, but behaves uniformly in the coefficient field (e.g., it has the same Grothendieck ring and $\mathrm{Ext}^1$ quiver over any field). Additionally, we completely solve the problem of understanding how $\mathcal{A}$ relates to general pre-Tannakian categories. We show that $\mathcal{A}$ admits exactly two local abelian envelopes: the functor $\Psi$, and a previously constructed functor $\Phi \colon \mathcal{A} \to \mathcal{C}$. This is the first case where the local envelopes of a category have been completely determined, outside of cases where there is at most one envelope. This work opens the door to constructing abelian versions of other oligomorphic tensor categories that do not admit a unique envelope.