# The homotopy theory of cyclotomic spectra

@article{Blumberg2013TheHT,
title={The homotopy theory of cyclotomic spectra},
author={Andrew J. Blumberg and Michael A. Mandell},
journal={arXiv: K-Theory and Homology},
year={2013}
}
• Published 2013
• Mathematics
• arXiv: K-Theory and Homology
We describe spectral model* categories of cyclotomic spectra and p-cyclotomic spectra (in orthogonal spectra) with triangulated homotopy categories. In the homotopy category of cyclotomic spectra, the spectrum of maps from the sphere spectrum corepresents the finite completion of TC, and in the homotopy category of p-cyclotomic spectra, the spectrum of maps from the sphere spectrum corepresents the p-completion of TC(-;p).
Cyclonic spectra, cyclotomic spectra, and a conjecture of Kaledin
• Mathematics
• 2016
With an explicit, algebraic indexing $(2,1)$-category, we develop an efficient homotopy theory of cyclonic objects: circle-equivariant objects relative to the family of finite subgroups. We constructExpand
Cartier modules and cyclotomic spectra
• Mathematics
• 2018
We construct and study a t-structure on p-typical cyclotomic spectra and explain how to recover crystalline cohomology of smooth schemes over perfect fields using this t-structure. Our main tool is aExpand
A naive approach to genuine $G$-spectra and cyclotomic spectra
• Mathematics
• 2017
For any compact Lie group $G$, we give a description of genuine $G$-spectra in terms of the naive equivariant spectra underlying their geometric fixedpoints. We use this to give an analogousExpand
Relative cyclotomic spectra and topological cyclic homology via the norm
• Mathematics
• 2014
We describe a construction of the cyclotomic structure on topological Hochschild homology ($THH$) of a ring spectrum using the Hill-Hopkins-Ravenel multiplicative norm. Our analysis takes placeExpand
Topological cyclic homology via the norm
• Mathematics
• 2014
We describe a construction of the cyclotomic structure on topological Hochschild homology ($THH$) of a ring spectrum using the Hill-Hopkins-Ravenel multiplicative norm. Our analysis takes placeExpand
On curves in K-theory and TR
We prove that TR is corepresentable by the reduced topological Hochschild homology of the flat affine line S[t] as a functor defined on the∞-category of cyclotomic spectra with values inExpand
Homotopy-theoretically enriched categories of noncommutative motives
AbstractWaldhausen’s K-theory of the sphere spectrum (closely related to the algebraic K-theory of the integers) is naturally augmented as an S0-algebra, and so has a Koszul dual. Classic work ofExpand
Cyclotomic structure in the topological Hochschild homology of $DX$
Let $X$ be a finite CW complex, and let $DX$ be its dual in the category of spectra. We demonstrate that the Poincar\'e/Koszul duality between $THH(DX)$ and the free loop space $\Sigma^\infty_+ LX$Expand
Comparing cyclotomic structures on different models for topological Hochschild homology
• Mathematics
• 2017
The topological Hochschild homology $THH(A)$ of an orthogonal ring spectrum $A$ can be defined by evaluating the cyclic bar construction on $A$ or by applying Bokstedt's original definition of $THH$Expand
Kaledin's degeneration theorem and topological Hochschild homology
We give a short proof of Kaledin's theorem on the degeneration of the noncommutative Hodge-to-de Rham spectral sequence. Our approach is based on topological Hochschild homology and the theory ofExpand

#### References

SHOWING 1-10 OF 25 REFERENCES
Localization theorems in topological Hochschild homology and topological cyclic homology
• Mathematics
• 2012
We construct localization cofibration sequences for the topological Hochschild homology (THH ) and topological cyclic homology (TC ) of small spectral categories. Using a global construction of theExpand
Cofibrations in Homotopy Theory
We define Anderson-Brown-Cisinski (ABC) cofibration categories, and construct homotopy colimits of diagrams of objects in ABC cofibration categories. Homotopy colimits for Quillen model categoriesExpand
Equivariant Orthogonal Spectra and S-Modules
• Mathematics
• 2002
Introduction Orthogonal spectra and $S$-modules Equivariant orthogonal spectra Model categories of orthogonal $G$-spectra Orthogonal $G$-spectra and $S_G$-modules ""Change"" functors for orthogonalExpand
On the K-theory of finite algebras over witt vectors of perfect fields
• Mathematics
• 1997
Introduction i 1. The topological Hochschild spectrum 1 2. Witt vectors 12 3. Topological cyclic homology 18 4. Topological cyclic homology of perfect fields 22 5. Topological cyclic homology ofExpand
Motivic structures in non-commutative geometry
We review some recent results and conjectures saying that, roughly speaking, periodic cyclic homology of a smooth non-commutative algebraic variety should carry all the additional "motivic"Expand
Homotopy Limits, Completions and Localizations
• Mathematics
• 1987
Completions and localizations.- The R-completion of a space.- Fibre lemmas.- Tower lemmas.- An R-completion of groups and its relation to the R-completion of spaces.- R-localizations of nilpotentExpand
Model categories and their localizations
Localization of model category structures: Summary of part 1 Local spaces and localization The localization model category for spaces Localization of model categories Existence of left BousfieldExpand
More Concise Algebraic Topology: Localization, Completion, and Model Categories
• Mathematics
• 2012
With firm foundations dating only from the 1950s, algebraic topology is a relatively young area of mathematics. There are very few textbooks that treat fundamental topics beyond a first course, andExpand
On the K-theory of local fields
In this paper we study the higher Quillen-K-groups of algebraically closed fields, local fields and real numbers. When k is an algebraically closed field we show that &(k) is divisible, its torsionExpand
MODEL CATEGORIES OF DIAGRAM SPECTRA
• Mathematics
• 2001
Working in the category $\mathcal{T}$ of based spaces, we give the basic theory of diagram spaces and diagram spectra. These are functors $\mathcal{D}\longrightarrow \mathcal{T}$ for a suitableExpand