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The decomposition space perspective

Philip Hackney

TL;DR

This work provides a gentle yet comprehensive introduction to decomposition spaces and their equivalence with $2$-Segal spaces, anchored by the active–inert factorization on the simplicial category. It develops core criteria—the path-space (decalage) criterion and the edgewise subdivision criterion—that characterize decomposition spaces via upper/lower decalages and their Segal property, respectively. The text also highlights free decomposition spaces arising from outer face complexes and their rich supply of examples, linking to incidence algebras and Möbius inversion, and situates Segal spaces as a special case within this unified framework. Together, these results offer practical tools for constructing and recognizing decomposition spaces while clarifying their relations to classical Segal, $2$-Segal, and categorical nerves frameworks.

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 décalages, 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.

The decomposition space perspective

TL;DR

This work provides a gentle yet comprehensive introduction to decomposition spaces and their equivalence with -Segal spaces, anchored by the active–inert factorization on the simplicial category. It develops core criteria—the path-space (decalage) criterion and the edgewise subdivision criterion—that characterize decomposition spaces via upper/lower decalages and their Segal property, respectively. The text also highlights free decomposition spaces arising from outer face complexes and their rich supply of examples, linking to incidence algebras and Möbius inversion, and situates Segal spaces as a special case within this unified framework. Together, these results offer practical tools for constructing and recognizing decomposition spaces while clarifying their relations to classical Segal, -Segal, and categorical nerves frameworks.

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 décalages, 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.
Paper Structure (9 sections, 18 theorems, 17 equations)

This paper contains 9 sections, 18 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Ways to read this paper
  3. The active-inert factorization system on the simplicial category
  4. Decomposition spaces and 2-Segal spaces
  5. 2-Segal spaces are decomposition spaces
  6. Segal spaces are decomposition spaces
  7. The path space criterion
  8. The edgewise subdivision criterion
  9. Free decomposition spaces

Key Result

Lemma 2.2

The pair of subcategories $(\Delta_\textup{act}, \Delta_\textup{int})$ constitutes a factorization system on $\Delta$. ∎

Theorems & Definitions (41)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 3.1: Decomposition space
  • Definition 3.4
  • Example 3.5: Simplicial groupoid of rooted forests
  • Example 3.6: Partial monoids
  • Remark 3.9
  • Definition 3.10: Culf maps
  • Theorem 4.1
  • ...and 31 more