Towards real $2$-Segal spaces

The notion of $2$-Segal spaces, a generalization of that of $1$-Segal spaces, encodes structures in which composition may not always exist or be unique. These structures were independently defined by Dyckerhoff and Kapranov in [2], and (by the name of decomposition spaces) by Gálvez-Carrillo, Kock, and Tonks in [3].

Both sets of authors identified the $S$-construction—a fundamental piece in algebraic $K$-theory—as a prominent source of examples of such structures. This connection was further studied by Bergner, Osorno, Ozornova, Rovelli, and Scheimbauer in [1], where they show that (a version of) the $S$-construction actually provides an equivalence between $2$-Segal spaces and certain double Segal spaces.

In the realm of algebraic $K$-theory, we often care about extracting information in the presence of hermitian forms in our objects of study, which leads us to consider the so called real algebraic $K$-theory. In one of the existing approaches to this theory, we rely on a version of the $S$-construction introduced in [4]: the real $S$-construction.

Given the tight link in the classical case between the $S$-construction and $2$-Segal spaces, a natural question arises: Is there a variant of $2$-Segal spaces that is able to codify the extra structure on the real $S$-construction? In this talk, based on work in progress with Hadrian Heine, we discuss some obstacles for such generalization, and suggest a possible answer to this question.