Trace theories, Bokstedt periodicity and Bott periodicity

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.