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An $\infty$-Category of 2-Segal Spaces

Jonte Gödicke

TL;DR

The work develops a precise bridge between 2-Segal spaces and algebra objects in span categories, establishing the equivalence $2\text{-Seg}_{\Delta}^{\leftrightarrow}(\mathcal{C}) \simeq \mathrm{Alg}(\mathrm{Span}(\mathcal{C}^{\times}))$ for ∞-categories with finite limits and extending it to birelative/bimodule contexts. It then treats module, left/right module, and bimodule structures through multicolored 2-Segal conditions, giving corresponding ∞-category equivalences with bimodule/spans in $\mathrm{Span}(\mathcal{C}^{\times})$. The paper also provides concrete K-theory inspired examples via Waldhausen’s $S_{\bullet}$-construction, its relative and hermitian variants, and demonstrates their use for homotopy-coherent representations of Hall algebras. Duality is integrated through twisted arrow constructions and $C_2$-actions, connecting to Carlier’s bicomodule configurations and enabling duality-preserving functorial constructions of relative 2-Segal objects and related module morphisms in span categories.

Abstract

Algebra objects in $\infty$-categories of spans admit a description in terms of $2$-Segal objects. We introduce a notion of span between $2$-Segal objects and extend this correspondence to an equivalence of $\infty$-categories. Additionally, for every $\infty$-category with finite limits $\mathcal{C}$, we introduce a notion of a birelative $2$-Segal object in $\mathcal{C}$ and establish a similar equivalence with the $\infty$-category of bimodule objects in spans. Examples of these concepts arise from algebraic and hermitian K-theory through the corresponding Waldhausen $S_{\bullet}$-construction. Apart from their categorical relevance, these concepts can be used to construct homotopy coherent representations of Hall algebras.

An $\infty$-Category of 2-Segal Spaces

TL;DR

The work develops a precise bridge between 2-Segal spaces and algebra objects in span categories, establishing the equivalence for ∞-categories with finite limits and extending it to birelative/bimodule contexts. It then treats module, left/right module, and bimodule structures through multicolored 2-Segal conditions, giving corresponding ∞-category equivalences with bimodule/spans in . The paper also provides concrete K-theory inspired examples via Waldhausen’s -construction, its relative and hermitian variants, and demonstrates their use for homotopy-coherent representations of Hall algebras. Duality is integrated through twisted arrow constructions and -actions, connecting to Carlier’s bicomodule configurations and enabling duality-preserving functorial constructions of relative 2-Segal objects and related module morphisms in span categories.

Abstract

Algebra objects in -categories of spans admit a description in terms of -Segal objects. We introduce a notion of span between -Segal objects and extend this correspondence to an equivalence of -categories. Additionally, for every -category with finite limits , we introduce a notion of a birelative -Segal object in and establish a similar equivalence with the -category of bimodule objects in spans. Examples of these concepts arise from algebraic and hermitian K-theory through the corresponding Waldhausen -construction. Apart from their categorical relevance, these concepts can be used to construct homotopy coherent representations of Hall algebras.
Paper Structure (18 sections, 48 theorems, 139 equations, 2 figures)

This paper contains 18 sections, 48 theorems, 139 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Acknowledgments
  4. 2-Segal Conditions
  5. 2-Segal Objects
  6. Birelative 2-Segal Objects
  7. Birelative Decomposition Spaces
  8. Relation to bicomodule configurations
  9. Segal objects from K-theory
  10. 2-Segal Spans from Exact inf-Functors
  11. Relative 2-Segal Spans from Duality preserving inf-Functors
  12. Modules in Span Categories
  13. Preliminaries
  14. Characterization of Bimodules
  15. Module Morphisms in Span Categories
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\mathcal{C}$ be an $\infty$-category with finite limits. There exists an equivalence of $\infty$-categories between the subcategory of the $\infty$-category $\mathop{\mathrm{\sf{Span}}}\nolimits(\sf{Fun}(\Delta^{\mathop{\mathrm{op}}\nolimits},\mathcal{C}))$ of spans of simplicial objects with objects $2$-Segal object and morphisms $2$-Segal spans and the $\infty$-category of algebra objects

Figures (2)

  • Figure 1: Proof of Lemma \ref{['lem:Diagram']}
  • Figure 2: Image under $F$ of the decomposition of the $2$-simplex $\sigma$ into a $4$-simplex. The original $2$-simplex is colored blue.

Theorems & Definitions (129)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Definition 2.1
  • Definition 2.2
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Definition 2.7
  • ...and 119 more