Possible “way in”: http://chromotopy.org/?p=372
http://mathoverflow.net/questions/87174/absence-of-maps-between-p-local-and-q-local-spectra
Some slides of Ravenel
Talbot 2013 workshop on chromatic homotopy theory.
More slides of Ravenel
Here is something about uniqueness of triangulated cats.
Chromotopy write a lot about chromatic homotopy…
http://sbseminar.wordpress.com/2010/01/28/chromatic-stable-homotopy-theory-and-the-ahss
arXiv:0911.5238 Continuous homotopy fixed points for Lubin-Tate spectra from arXiv Front: math.AT by Gereon Quick We construct a stable model structure on profinite symmetric spectra with a continuous action of an arbitrary profinite group. This provides a natural framework for a new construction of homotopy fixed point spectra and of homotopy fixed point spectral sequences for the action of the extended Morava stabilizer group on Lubin-Tate spectra. These continuous homotopy fixed points are canonically equivalent to the homotopy fixed points of Devinatz and Hopkins but have a drastically simplified construction.
Toen: Note of Chern character, loop spaces and derived algebraic geometry. File Toen web publ Abel-2007.pdf. Notion of derived cat sheaves, a categorification of the notion of complexes of sheaves of O-modules on schemes (also quasi-coh and perfect versions). Chern character for these categorical sheaves, a categorified version of the Chern char for perfect complexes with values in cyclic homology. Using the derived loop space. “This work can be seen as an attempt to define algebraic analogues of of elliptic objects and char classes for them”. 1. Motivations: Elliptic cohomology, geometric interpretations, chromatic level and n-categorical level, 2-VBs. Maybe the typical generalized CT of chromatic level n should be related to n-cats, more precisely cohom classes should be rep by maps from X to a certain n-stack. Rognes red-shift conjecture: Intuitively saying that the K-th spectrum of a commutative ring spectrum of chrom level n is of chrom level (n+1). More on ell cohom and 2-cats. Idea of categorical sheaves: For X a scheme, should have a symmetric monoidal 2-cat Cat(X) which is a categorification of Mod(X), in the sense that Mod(X) should be the cat of endomorphisms of the unit objects in Cat(X). More details. Notions of secondary cohomology and secondary K-theory. Notion of derived categorical sheaves, more reasonable than nonderived version. Relation between -equivariant functions on LX and negative cyclic homology. 2. Categorification of homological algebra and dg-cats. 3. Loop spaces in DAG. More, including relations with variations of Hodge structures. Final remark on algebraic elliptic cohomology. “Algebraic K-theory determines complex topological K-theory”, ref to Walker 2002.
arXiv:1101.5201 Every K(n)-local spectrum is the homotopy fixed points of its Morava module from arXiv Front: math.AT by Daniel G. Davis, Takeshi Torii Let n \geq 1 and let p be any prime. Also, let E_n be the Lubin-Tate spectrum, G_n the extended Morava stabilizer group, and K(n) the nth Morava K-theory spectrum. Then work of Devinatz and Hopkins and some results due to Behrens and the first author of this note, show that if X is a finite spectrum, then the localization L_{K(n)}(X) is equivalent to the homotopy fixed point spectrum (L_{K(n)}(E_n \wedge X))^{hG_n}, which is formed with respect to the continuous action of G_n on L_{K(n)}(E_n \wedge X). In this note, we show that this equivalence holds for any (S-cofibrant) spectrum X. Also, we show that for all such X, the strongly convergent Adams-type spectral sequence abutting to \pi_\ast(L_{K(n)}(X)) is isomorphic to the descent spectral sequence that abuts to \pi_\ast((L_{K(n)}(E_n \wedge X))^{hG_n}).
nLab page on Chromatic picture