2-Segal sets and pseudomonoids in the bicategory of spans

TL;DR

This work establishes an elementary, set-theoretic bridge between $2$-Segal sets and pseudomonoids in the bicategory of spans, showing that $2$-Segal sets categorify associative algebraic structures present in spans. The authors develop a graphical calculus for $2$-Segal sets, prove the equivalence with pseudomonoids in $oldsymbol{Span}$, and translate $2$-Segal data into pseudomonoid coherences (and vice versa) via dual polygon diagrams. They then derive associative (co)algebra constructions—Hall algebras and incidence algebras—under finiteness hypotheses, with multiple illustrative examples including nerves of partial categories and an exponential-generating-function case. The approach emphasizes accessibility, avoiding heavy $\,\infty$-categorical machinery, and situates $2$-Segal sets as categorifications of algebraic structures arising from spans, with potential applications in combinatorics, representation theory, and categorification programs.

Abstract

In this survey article, we give an introduction to the notion of a 2-Segal set and prove that 2-Segal sets are equivalent to pseudomonoids in the bicategory of spans. The proof utilizes graphical techniques for 2-Segal sets and spans that should be useful in more general settings. There are procedures for obtaining an associative algebra from a 2-Segal set (satisfying finiteness conditions). We describe these procedures and give several examples of algebras arising from 2-Segal sets. Wherever possible, we avoid higher category theory so as to make the paper accessible to a wide audience.

Figures (18)

• Figure 1: The Hasse diagrams of the posets of subdivided length $2$ intervals (on the left) and subdivided length $3$ intervals (on the right).
• Figure 2: Visualization of the maps $P_{13}^5: [2] \to [5]$, given by $\{0,1,2\} \mapsto \{1,2,3\}$, and $Q_{13}^5: [4] \to [5]$, given by $\{0,1,2,3,4\} \mapsto \{0,1,3,4,5\}$.
• Figure 3: The poset of subdivided squares.
• Figure 4: The poset of subdivided pentagons.
• Figure 5: Graphical calculus for the face map $d_1^4: X_4 \to X_3$.

Theorems & Definitions (24)

• Proposition 2.1
• proof
• Definition 2.2
• Theorem 2.3: grothendieck:FGA3*Proposition 4.1
• Proposition 2.4
• proof : Proof of Theorem \ref{['thm:1segal']}
• Proposition 3.1
• Definition 3.2
• Remark 3.3
• Definition 3.4