An -Category of 2-Segal Spaces

Algebra objects in -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 -categories. Additionally, for every -category with finite limits C, we introduce a notion of a birelative 2-Segal object in C 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 S_-construction. Apart from their categorical relevance, these concepts can be used to construct homotopy coherent representations of Hall algebras.