  Hochschild homology is a classical invariant of rings that plays an important
role because of its connection to algebraic $K$-theory via the Dennis trace. At
level zero, the Dennis trace is induced by the Hattori-Stallings trace. In this
paper, we introduce new algebraic $K$-theories of coalgebras and obtain
coalgebraic refinements of the Hattori-Stallings trace that connect these
algebraic $K$-theories to coHochschild homology (the invariant analogous to
Hochschild homology but for coalgebras). We employ bicategorical methods of
Ponto to show that coHochschild homology is a shadow. Consequently, we obtain
that coHochschild homology is Morita-Takeuchi invariant.
