The \(A_\infty\)-structure of the index map (Q1756156)

Let $F$ be a local field with residue field $k$. For any $F$-vector supace of finite dimension, the authors constructed an ``index map'' $BGL(V) \to BK_{k}$ in [\textit{O. Braunling} et al., Sel. Math., New Ser. 24, No. 2, 1039--1091 (2018; Zbl 1391.19005)]. \par In this paper under review, the authors construct a map of reduced Segal spaces $B_{\bullet}GL(V) \to K_{S_{\bullet}(\mathrm{Vect}_{f}(k))}$, whose geometric realization is the index map. Here, $\mathrm{Vect}_{f}(k)$ stands for the category of finite dimensional vector spaces. More generally, they construct this model in the framework of Tate objects in exact categories.