The $A_{\mathrm{\infty }}$-structure of the index map

Let $F$ be a local field with residue field $k$. The classifying space of $G{L}_n(F)$ comes canonically equipped with a map to the delooping of the $K$-theory space of $k$. Passing to loop spaces, such a map abstractly encodes a homotopy coherently associative map of A-infinity-spaces $G{L}_n(F)\to K_k$. Using a generalized Waldhausen construction, we construct an explicit model built for the $A_{\mathrm{\infty }}$-structure of this map, built from nested systems of lattices in $F^n$. More generally, we construct this model in the framework of Tate objects in exact categories, with finite dimensional vector spaces over local fields as a motivating example.