Let be a local field with residue field . The classifying space of comes canonically equipped with a map to the delooping of the -theory space of . Passing to loop spaces, such a map abstractly encodes a homotopy coherently associative map of -spaces . Using a generalized Waldhausen construction, we construct an explicit model built for the -structure of this map, built from nested systems of lattices in . 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.
"The $A_\infty$-structure of the index map." Ann. K-Theory 3 (4) 581 - 614, 2018. https://doi.org/10.2140/akt.2018.3.581