2-Segal sets and pseudomonoids in the bicategory of spans

TL;DR

This survey introduces 2-Segal sets and demonstrates their equivalence to pseudomonoids in spans, providing graphical techniques and procedures to derive associative algebras, making the concepts accessible without heavy higher category theory.

Contribution

It establishes the equivalence between 2-Segal sets and pseudomonoids in spans, and provides accessible graphical methods and algebraic procedures derived from 2-Segal sets.

Findings

2-Segal sets are equivalent to pseudomonoids in spans

Procedures to obtain associative algebras from 2-Segal sets

Examples of algebras arising from 2-Segal sets

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.