  We show that the characteristic polynomial and the Lefschetz zeta function
are manifestations of the trace map from the $K$-theory of endomorphisms to
topological restriction homology (TR). Along the way we generalize
Lindenstrauss and McCarthy's map from $K$-theory of endomorphisms to
topological restriction homology, defining it for any Waldhausen category with
a compatible enrichment in orthogonal spectra. In particular, this extends
their construction from rings to ring spectra. We also give a revisionist
treatment of the original Dennis trace map from $K$-theory to topological
Hochschild homology (THH) and explain its connection to traces in bicategories
with shadow (also known as trace theories).
