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A cyclotomic spectrum is an $S^1$-equivariant spectrum $E$ with fixed points for all the finite cyclic groups $C_p = \mathbb{Z}/p\mathbb{Z} \hookrightarrow S^1$ inside the circle group, and equipped with $S^1$-equivariant identifications $E^{C_p} \stackrel{\simeq}{\to} E$ of the $C_P$-fixed points with the full object.
The topological Hochschild homology-spectra $E = THH(A)$ are naturally cyclotomic spectra, and this is where the concept originates: by the discussion at Hochschild cohomology $THH(A)$ is the E-infinity ring of functions on the free loop space of $Spec(A)$, and cyclotomic structure reflects the structure of free loop spaces: loops that repeat with period $p$ are equivalent to plain loops.
Cyclotomic structure is the origin of the cyclotomic trace map $THH \longrightarrow TC$ from topological Hochschild homology to topological cyclic homology.
Throughout, for $p$ a prime number write $C_p \subset S^1$ for the cyclic group $\mathbb{Z}/p\mathbb{Z}$ of order $p$, regarded as a subgroup of the circle group.
A definition says that a cyclotomic spectrum is an circle group-genuine equivariant spectrum $X$ (modeled on orthogonal spectra) equipped with equivalences to its naive point-set fixed point spectra $\Phi^{C_p} X$ for all the cyclic subgroups $C_p \subset S^1$.
A more abstract definition was given in (Nikolaus-Scholze 17):
A cyclotomic spectrum is
a spectrum $X$
a circle group ∞-action on $X$, i.e. an (∞,1)-functor $B S^1 \to Spectra$ which takes the unique point of $B S^1$ to $X$;
for each prime number $p$ a homomorphism of spectra with such circle group action
to the Tate spectrum (the homotopy cofiber $X^{t C_p} \coloneqq cofib( X_{C_p} \overset{norm_p}{\to} X^{C_p} )$ of the norm map), where the circle action on the Tate spectrum comes from the canonical identification $S^1/C_p \simeq C_p$.
(These morphisms $F_p$ are called the Frobenius morphisms of the cyclotomic structure, due to this def., this example).
(Nikolaus-Scholze 17, def. 1.3, def. II.1.1).
For $X$ a spectrum with stable homotopy groups bounded below, then def. is equivaent to the traditional:
There is an (∞,1)-functor
from traditional (“genuine”) cyclotomic spectra bounded below to bounded below cyclotomic spectra in the sense of def. , and this is an equivalence of (∞,1)-categories.
(Nikolaus-Scholze 17, prop. II.3.4, theorem II.6.9).
(topological Hochschild homology)
For every A-∞ ring $A$, the topological Hochschild homology spectrum $THH(A)$ naturally carries the structure of a cyclotomic spectrum (def. ).
(Nikolaus-Scholze 17, section II.2, def. III.2.3)
(trivial cyclotomic spectra)
Every spectrum $X$ becomes a cyclotomic spectrum $X^{triv}$ in the sense of def. by equipping it
with the trivial circle group ∞-action
for each prime $p$ with the composite morphism
(the first being the $( B C_p \times (-) \dashv (-)^{C_p} )$-unit into the homotopy fixed points, the second the defining morphism into the Tate spectrum )
the $S^1/C_p$-equivariant structure on these morphisms given under the adjunction between trivial action and homotopy fixed points by the adjunct morphisms
as the composite
This construction constitutes a left adjoint (infinity,1)-functor to taking topological cyclic homology
(Nikolaus-Scholze 17, example II.1.2 (ii) and middle of p. 126)
(cyclotomic sphere spectrum)
The sphere spectrum regarded as a cyclotomic spectrum via example is called the cyclotomic sphere spectrum.
As such it is equivalently its topological Hochschild homology according to example :
(Nikolaus-Scholze 17, example II.1.2 (ii))
The tensor unit in the symmetric monoidal (infinity,1)-category of cyclotomic spectra is the cyclotomic sphere spectrum from example (Blumberg-Mandell 13, example 4.9)
Andrew Blumberg, Michael Mandell, The homotopy theory of cyclotomic spectra (arXiv:1303.1694)
Clark Barwick, Saul Glasman, Cyclonic spectra, cyclotomic spectra, and a conjecture of Kaledin (arXiv:1602.02163)
Clark Barwick, Saul Glasman, Noncommutative syntomic realization (pdf)
Cary Malkiewich, A visual introduction to cyclic sets and cyclotomic spectra, 2015 (pdf)
Thomas Nikolaus, Peter Scholze, On topological cyclic homology (arXiv:1707.01799)
Last revised on April 6, 2018 at 18:00:43. See the history of this page for a list of all contributions to it.