  Let $F$ be a local field with residue field $k$. The classifying space of
$GL_n(F)$ 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 $GL_n(F)\to K_k$.
Using a generalized Waldhausen construction, we construct an explicit model
built for the $A_\infty$-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.
