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

TL;DR

This paper develops an $$-category framework for 2-Segal spaces, establishing equivalences with algebra objects, bimodule objects, and applications to K-theory and Hall algebra representations.

Contribution

It introduces a notion of spans between 2-Segal objects and extends the correspondence to an equivalence of $$-categories, also defining birelative 2-Segal objects in $$ with similar equivalences.

Findings

Established an equivalence between algebra objects in spans and 2-Segal objects.

Defined a notion of span between 2-Segal objects and extended to an $$-category equivalence.

Applied concepts to algebraic and hermitian K-theory, enabling homotopy coherent Hall algebra representations.

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.