2-Segal sets from cuts of rooted trees

TL;DR

The paper constructs a concrete family of $2$-Segal sets $X^T$ from rooted trees $T$ using admissible cuts and layerings, and situates these within the broader $2$-Segal, Hall algebra, and operadic framework. It shows how $X^T$ yields a pointed stable double category $\\mathcal{P}X^T$ and analyzes the associated Hall algebra $\mathcal{H}^T$, including noncommutativity in the labelled case and commutativity in certain unlabelled path cases, while comparing to the underlying graph 2-Segal set via a simplicial map $U:X^T\to X^G$ that is not generally CULF or relatively Segal. The work further connects tree-based $2$-Segal structures to invertible operads/cooperads, giving explicit descriptions of $\mathcal{Q}_{X^T}$ and contrasting them with $\mathcal{Q}_{X^G}$, thereby enriching the interaction between combinatorial tree decompositions and operadic formalisms. Overall, the results place rooted-tree constructions squarely in the $2$-Segal/Hall algebra/operad landscape, providing computable, lattice-like examples with clear algebraic consequences.

Abstract

The theory of 2-Segal sets has connections to various important constructions such as the Waldhausen $S_\bullet$-construction in algebraic $K$-theory, Hall algebras, and (co)operads. In this paper, we construct 2-Segal sets from rooted trees and explore how these applications are illustrated by this example.

Figures (22)

• Figure 1: Example of a rooted tree.
• Figure 2: An example of a tree $T$ with two cuts (left), and its image under the map $d_0$ (right).
• Figure 3: Example of the 2-Segal property. To recover the trees with two cuts in the upper left corners, observe that the trees in the top right corners contain all the necessary data except one cut. The information of this missing cut is present in the bottom left corners. Their intersection, the tree in the bottom right corner, informs us how to glue these two trees, hence determines the missing cut.
• Figure 4: The failure of injectivity for the 1-Segal condition in the unlabelled case.
• Figure 5: The objects, morphisms and three squares of $\mathcal{P}X^T$.

Theorems & Definitions (60)

• Definition 2.1
• Definition 2.3
• Proposition 2.4
• Example 2.5
• Definition 2.6
• Example 3.2
• Definition 3.3
• Definition 3.4
• Example 3.5
• Definition 3.7