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We flesh out the theory of "trace theories" and "trace functors" sketched inarXiv:1308.3743, extend it to a homotopical setting, and prove a reconstructiontheorem claiming that a trace theory is completely determined by the associatedtrace 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-WittHomology WHH of arXiv:1604.01588 (and we also identify $TP(A)$ with theperiodic version $WHP(A)$ of WHH). Secondly, we prove that $THH(A)$ isidentified with the zero term of the conjugate filtration on the co-periodiccyclic homology $\overline{HP}(A)$ of arXiv:1509.08784, and the isomorphismsends the Bokstedt periodicity generator to the Bott periodicity generator. Wealso give an independent proof of Bokstedt periodicity that is somewhat shorterthan the usual ones.

Trace theories, Bokstedt periodicity and Bott periodicity