  We flesh out the theory of "trace theories" and "trace functors" sketched in
arXiv:1308.3743, extend it to a homotopical setting, and prove a reconstruction
theorem claiming that a trace theory is completely determined by the associated
trace functor. As an application, we consider Topological Hoshschild Homology
$THH(A,M)$ of a algebra $A$ over a perfect field of positive characteristic,
with coefficients in a bimodule $M$, and prove two comparison results. Firstly,
we give a very simple algebraic model for THH in terms of Hochschild-Witt
Homology WHH of arXiv:1604.01588 (and we also identify $TP(A)$ with the
periodic version $WHP(A)$ of WHH). Secondly, we prove that $THH(A)$ is
identified with the zero term of the conjugate filtration on the co-periodic
cyclic homology $\overline{HP}(A)$ of arXiv:1509.08784, and the isomorphism
sends the Bokstedt periodicity generator to the Bott periodicity generator. We
also give an independent proof of Bokstedt periodicity that is somewhat shorter
than the usual ones.
