An $\mathrm{\infty }$-Category of 2-Segal Spaces

Algebra objects in $\mathrm{\infty }$-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 $\mathrm{\infty }$-categories. Additionally, for every $\mathrm{\infty }$-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 $\mathrm{\infty }$-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.