We introduce the notion of a polygonic spectrum which is designed to
axiomatize the structure on topological Hochschild homology THH(R,M)
of an E1​-ring R with coefficients in an R-bimodule M. For
every polygonic spectrum X, we define a spectrum TR(X) as the
mapping spectrum from the polygonic version of the sphere spectrum S
to X. In particular if applied to X=THH(R,M) this gives a
conceptual definition of TR(R,M).
Every cyclotomic spectrum gives rise to a polygonic spectrum and we prove
that TR agrees with the classical definition of TR in this case. We construct
Frobenius and Verschiebung maps on TR(X) by exhibiting
TR(X) as the Z-fixedpoints of a quasifinitely genuine
Z-spectrum. The notion of quasifinitely genuine Z-spectra
is a new notion that we introduce and discuss inspired by a similar notion over
Z introduced by Kaledin. Besides the usual coherences for genuine
spectra, this notion additionally encodes that TR(X) admits certain
infinite sums of Verschiebung maps.Comment: 61 pages, comments are welcom