Multiplicative comparison of K-Theories and homology

K-Theory is a machine taking certain types of categories as input and producing spectra (or spaces) as its output. There are two notable versions, Segal and Waldhausen’s K-theory, depending on the specific kind of input, with a precise comparison between the two, due to A.-M. Bohmann and A. Osorno. We fully extend Bohmann and Osorno’s comparison to include a fourth object, spectra in chain complexes. This is joint work with Brandon Doherty and Arash Karimi.