Second Delannoy Category Overview

• Second Delannoy category is a rigid, additive symmetric tensor category constructed from oligomorphic groups and specifically chosen measures, encoding Schwartz spaces and invariant matrices.
• It features a unique tensor structure with indecomposable objects indexed by finite words and morphisms governed by a ruffle rule analogous to Delannoy paths.
• Its abelian envelope has dual incarnations—semi-simple and highest-weight—demonstrating universal mapping properties and characteristic-independent invariants.

The second Delannoy category is a rigid additive symmetric tensor category arising from the structure theory of permutation categories attached to oligomorphic groups equipped with specifically chosen measures. Its construction, universal properties, and rich abelian envelope theory make it central in the modern study of tensor categories defined via infinite permutation groups and their measures (Coulembier et al., 22 Jan 2026, Coulembier et al., 11 Oct 2025).

1. Construction from Oligomorphic Groups and Measures

Let $G = \mathrm{Aut}(\mathbb{R}, <)$ denote the oligomorphic group of order-preserving bijections of the real line. This group admits exactly four $k$-valued measures $\mu_1, \mu_2, \mu_3, \mu_4$, where $k$ is any field. The symmetric tensor categories associated to these measures are called the Delannoy categories. The objects in these permutation categories are the Schwartz spaces $S(X)$ of finitary smooth $G$-sets $X$, with morphisms defined as $G$-invariant matrices. For a measure $\mu$ and $G$-set $X$:

• $$S(X) = \{\varphi: X \to k \mid \varphi \text{ invariant under some open subgroup}\}$$
• Morphisms $S(X) \to S(Y)$ are $G$-invariant functions $K: Y \times X \to k$, with composition given by matrix multiplication using $\mu$ as integration.

The second measure $\mu_2$ satisfies $\mu_2(p_{2,1}) = -1$ and $\mu_2(p_{2,2}) = 0$ for natural coordinate projections, with $\mu_2(X)$ computed as the Euler characteristic of the closure of $X$ under a co-finite topology. The "second Delannoy category," denoted here as $\mathcal{A}$ or $C_2$, is defined as the Karoubian envelope of $\mathrm{Perm}(G, \mu_2)$ (Coulembier et al., 22 Jan 2026, Coulembier et al., 11 Oct 2025).

2. Structure of the Additive Tensor Category

Every object in the second Delannoy category $\mathcal{A}$ is a direct summand of a finite direct sum of the spaces $A(\mathbb{R}^{(n)})$, the Schwartz spaces on strictly increasing $n$-tuples. The category is Krull–Schmidt, and indecomposable objects are indexed by weights $\lambda$: finite words in $\{\bullet, \circ\}$.

Morphisms and Tensor Structure

• Nonzero morphisms between indecomposables $M_\lambda$, $M_\mu$ occur only for $\mu = \lambda, \lambda\circ, \lambda\bullet$, with unique (up to scalar) maps $d_\lambda$, their duals $u$, and identities (subject to relations like $u_\lambda \circ d_\lambda \neq 0$).
• The tensor product is governed by a ruffle or Delannoy-path rule:

$$M_\lambda \otimes M_\mu \cong \bigoplus_{\rho \in \Omega'_{\lambda, \mu}} M_{\omega(\rho)}$$

where $\Omega'_{\lambda, \mu}$ is a marked-ruffle set.

• The unit object is $M_{\emptyset} = A(\mathbb{R}^{(0)})$.

Characteristic Properties

• $\mathcal{A}$ is rigid, symmetric monoidal, Karoubian, and not abelian.
• The category admits exactly two tensor ideals: $\{0\}$ and the negligible ideal $n$, with $\mathcal{A}/n$ the functor to vector spaces.
• Simple objects are negligible; endomorphisms can have zero trace under $\mu_2$.

3. The Abelian Envelope and Functorial Structure

The second Delannoy category is not abelian, necessitating a construction of its "abelian envelope." This is achieved via a fully faithful symmetric monoidal functor $\Psi: \mathcal{A} \to \mathcal{D}$, where $\mathcal{D}$ is a highest-weight abelian category with an explicit combinatorial presentation.

Objects and Morphisms in $\mathcal{D}$

• Objects: indexed by weights $$\lambda \in \Lambda = \{\text{words in } \bullet, \circ\}$$.
• Morphism spaces: $$\dim_k \mathrm{Hom}_{\mathcal{D}}(\lambda, \mu) = 1$$ if $\lambda = \mu$ or $\mu = \lambda\nu$ for alternating $\nu$ ending in $\circ$, or $\lambda = \mu\nu$ with $\nu$ ending in $\bullet$; zero otherwise.
• Simple objects: $S_\lambda$; standard $\Delta_\lambda$, costandard $\nabla_\lambda$, indecomposable tiltings $T_\lambda$ of finite length.

Key Results

• $\mathcal{D}$ is a lower-finite, characteristic-independent highest-weight pre-Tannakian category; its structure does not depend on the base field.
• The classes $[T_\lambda]$ form a $\mathbb{Z}$-basis of $K_0(\mathcal{D})$, and the functor $M_\lambda \mapsto [T_\lambda]$ gives an isomorphism $K^{\oplus}(\mathcal{A}) \cong K_0(\mathcal{D})$.
• The Ext$^1$-quiver of $\mathcal{D}$ is the Cayley graph of the free monoid on two generators, with relations from basic morphisms.

4. Universal Properties and Local Abelian Envelopes

The second Delannoy category satisfies powerful universal mapping properties in the context of ordered étale algebras:

• For any Karoubian tensor category $T$, tensor functors $C_2 \to T$ correspond to ordered étale algebras $A$ in $T$ ("type 2 Delannic algebras") with unit and coordinate maps of prescribed "degree" matching $\mu_2$ on projections.
• The free type 2 Delannic algebra is $C_2(\mathbb{R})$.

Classification of Envelopes

For any rigid Karoubian $k$-linear tensor category $\mathcal{E}$, local abelian envelopes are initial exact, faithful tensor functors into pre-Tannakian categories. For the second Delannoy category $\mathcal{A}$, there are exactly two local abelian envelopes (Coulembier et al., 22 Jan 2026):

Envelope	Functor	Universal Property Condition
Semi-simple	$\Phi: \mathcal{A} \to \mathcal{C}$	2-Delannic algebra is bounded
Highest-weight	$\Psi: \mathcal{A} \to \mathcal{D}$	2-Delannic algebra is unbounded

No third envelope exists since the degenerate $\Theta: \mathcal{A} \to \text{Vec}$ collapses the essential structure.

5. Comparative Analysis with the First Delannoy Category

The first Delannoy category $\mathcal{C}$, corresponding to $\mu_1$, is semi-simple pre-Tannakian and is its own abelian envelope. Its simple objects $L_\lambda$ are indexed identically, and its tensor structure is governed by combinatorics of Delannoy paths.

By contrast, $\mathcal{A}$ (second Delannoy) is rigid but not abelian and requires construction of distinct abelian envelopes. $\mathcal{D}$, its main envelope, exhibits:

• Non-semi-simplicity
• Super-exponential, yet characteristic-independent, growth
• The same Grothendieck ring and Ext$^1$-quiver over arbitrary fields
• Both Grothendieck semirings and Adams operations are analogous, but semisimplicity fails

Table: Key Features of First vs. Second Delannoy Categories

Feature	First Delannoy ($\mathcal{C}$)	Second Delannoy ($\mathcal{A}, \mathcal{D}$)
Abelian	Yes	No ($\mathcal{A}$), Yes ($\mathcal{D}$)
Semi-simple	Yes	No
Grothendieck ring	Known, explicit	Same as $\mathcal{C}$
Envelopes	Unique	Two (semi-simple, highest-weight)

6. Delannic Algebras and Universal Generation

Second Delannoy category $\mathcal{A}$ universally encodes the tensor category generated by a single ordered étale algebra $A$ with $\dim A = 0$, and prescribed degrees for coordinate maps:

• $\gamma_1(A) = -1$
• $\gamma_2(A) = 0$

Any such algebra gives rise to a unique tensor functor from the second Delannoy category, confirming its role as a universal object in the ordered étale algebra context (Coulembier et al., 11 Oct 2025).

7. Structural Uniformity and Implications

$\mathcal{D}$'s key invariants (Grothendieck ring, Ext$^1$-quiver, tilting structure) are characteristic-independent, making it a robust framework for generalization to broader classes of symmetric tensor categories arising from oligomorphic groups. The determination of its two local abelian envelopes establishes the first explicit example of a pre-Tannakian category with more than one such envelope, indicating new directions for the construction and study of abelian versions of non-quasi-regular tensor categories (Coulembier et al., 22 Jan 2026, Coulembier et al., 11 Oct 2025).

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References:

• (Coulembier et al., 22 Jan 2026) Harman, Snowden, "The second Delannoy category," 2026.
• (Coulembier et al., 11 Oct 2025) Coulembier, Harman, Snowden, "Universal properties of Delannoy categories," 2025.