The decomposition space perspective

TL;DR

This paper introduces and unifies the concepts of decomposition spaces and 2-Segal spaces, providing key criteria and examples within simplicial and category theory frameworks.

Contribution

It establishes the equivalence between decomposition spaces and 2-Segal spaces and introduces new criteria and examples for these structures.

Findings

Path space criterion characterizes decomposition spaces via upper and lower de9calages.

Edgewise subdivision criterion provides an alternative characterization.

Outer face complexes generate free decomposition spaces, enriching the class of examples.

Abstract

This paper provides an introduction to decomposition spaces and 2-Segal spaces, unifying the two perspectives. We begin by defining decomposition spaces using the active-inert factorization system on the simplicial category, and show their equivalence to 2-Segal spaces. Key results include the path space criterion, which characterizes decomposition spaces in terms of their upper and lower d\'ecalages, and the edgewise subdivision criterion. We also introduce free decomposition spaces arising from outer face complexes, providing a rich source of examples. Formal prerequisites are minimal -- readers should have a working knowledge of simplicial methods and basic category theory.