  We develop a theory of generalized characters of local systems in
$\infty$-categories, which extends classical character theory for group
representations and, in particular, the induced character formula. A key aspect
of our approach is that we utilize the interaction between traces and their
categorifications. We apply this theory to reprove and refine various results
on the composability of Becker-Gottlieb transfers, the Hochschild homology of
Thom spectra, and the additivity of traces in stable $\infty$-categories.
