prompted by this MO question I saw that our entry *parameterized spectrum* was a bit thin on information. I have now at least briefly added a section *Yoga of six functors* with mentioning of and pointers to the Wirthmüller context property, the Beck-Chevalley condition and the interpretation as linear homotopy type theory.

added to homotopy groups of spheres the table

$k =$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | $\cdots$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\pi_k(\mathbb{S}) =$ | $\mathbb{Z}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | $\mathbb{Z}_{24}$ | $0$ | $0$ | $\mathbb{Z}_2$ | $\mathbb{Z}_{240}$ | $(\mathbb{Z}_2)^2$ | $(\mathbb{Z}_2)^3$ | $\mathbb{Z}_6$ | $\mathbb{Z}_{504}$ | $0$ | $\mathbb{Z}_3$ | $(\mathbb{Z}_2)^2$ | $\mathbb{Z}_{480} \oplus \mathbb{Z}_2$ |

I am slowly creating a bunch of entries on basic concepts of equivariant stable homotopy theory, such as

- equivariant suspension spectrum, equivariant sphere spectrum, equivariant homotopy groups, RO(G)-grading, fixed point spectrum, tom Dieck splitting

At the moment I am mostly just indexing Stefan Schwede’s

]]>added rough description and original citation to *Adams e-invariant*

I gave *chromatic homotopy theory* an Idea-section.

To be expanded eventually…

]]>I am touching various entries related to equivariant stable homotopy theory, adding basics from the literature. For instance I briefly added to *G-spectrum* the basic definition via indexing on a universe, and added the statement of the equivariant stable Whitehead theorem, cross-linked with the relevant bits at *equivariant homotopy theory*, etc. I have also been expanding a little more at *RO(G)-grading* and cross-linked more with old material at *equivariant cohomology*. Tried to make the link between RO(G)-grading and equivariant suspension isomorphism more explicit.

Just in case you are watching the logs and are wondering. I am not announcing every single edit, unless there is anything noteworthy.

]]>stubs for *chromatic tower* and *monochromatic layer*

(all these stubs just to chart the landscape and provide pointers, to be filled with more content…)

]]>Started a bare minimum at *cyclotomic spectrum*. So far it’s essentially just a pointer to the canonical reference by Blumberg-Mandell. (Thomas Nikolaus and Peter Scholze have a new foundation of the theory in preparation for which notes however are not public yet, also Clark Barwick has something in preparation, for which you may find notes by looking at his website and being clever in deducing hidden URLs, he says.)

For the moment the only fact that I have actually recorded in the entry is a fact that is trivial for anyone familiar with the theory,but which looks interesting from the point of view of the story at *Generalized cohomology of M2/M5-branes (schreiber)*: the global equivariant sphere spectrum for all the cyclic groups (all the A-type finite groups in the ADE classification…) carries canonical cyclotomic structure and as such is the tensor unit among cyclotomic spectra.

Apart from mentioning this, I have added brief cross-links with *topological cyclic homology*, *equivariant sphere spectrum*, *cyclic group* and maybe other entries.

started a minimum at *Anderson duality* just for compleness, see the other thread on *dualizing object in a closed category*.

gave *Lagrangian cobordism* an Idea-section added references related to the Fukaya category and cross-linked with relevant entries.

the keyword “cobordism theory” used to redirect to the entry *cobordism*. While often the latter term is used as shorthand for the former, the entry “cobordism” is really just about the basic notion of cobordisms between manifolds, so redirecting “cobordism theory” to there wasn’t satisfactory.

So I gave it an entry in its own right, added a little Idea-section briefly surveying the scope of cobordism theory proper, copied over the relevant references:

Then I included (in that entry directly and into related entry as a “floating table of contents”) a list of pointers to related entries:

]]>created a minimum at *connective cover*, just for completeness

added to *n-excisive functor* a section

added the definition to the old stub *t-structure*. Also redicrecting *heart*, of course. And also for stable $\infty$-categories.

We had material related to the “stabel Dold-Kan correspondence” (unbounded chain complexes mapping to spectra) scattered at the entries *Dold-Kan correspondence*, *module spectrum* and *algebra spectrum*. I have copy-and-pasted this material together into one dedicated entry *stable Dold-Kan correspondence*.

That deserves to be further expanded, for the moment it is just blindly copy-and-pasted from the existign material.

]]>stub for *conditional convergence* (of spectral sequences) for the moment just so as to record the references. (even *coctalos* says at some point “…I think this is what conditional convergence means…”)

After scanning a bunch of literature, my favorite survey of the Adams spectral sequence is now this gem here:

- Dylan Wilson
*Spectral Sequences from Sequences of Spectra: Towards the Spectrum of the Category of Spectra*, lecture at 2013 Pre-Talbot Seminar (pdf)

added an Idea-section to *Mackey functor* (which used to be just a list of references). Also added more references.

started a topic cluster table of contents *higher linear algebra - contents* and included it as a “floating table of contents” into relevant entries

started some minimum at *MR cohomology theory*

I am working on an entry *model structure on orthogonal spectra*. So far it contains a detailed construction of the symmetric monoidal strict model structure, and then a detailed proof of the stable model structure.

(I think its complete, but towards the end the expositional aspects need more polishing, i.e. more cross-links ect. But not today.)

This follows the writeup that I had started at *Model categories of diagram spectra*, but (besides being more complete and more polished by now) it works around the issue that I ran into there, by defining the weak equivalences to be the stable weak homotopy equivalences ($\pi_\ast$-isos) right away. This means that the proof still verbatim gives a proof also of the stable model structure on sequential sequential spectra and on excisive functors, but not on symmetric spectra.

brief entry *nilpotence theorem*

(in chromatic homotopy theory, maybe needs disambiguation later…)

]]>I have started *rational equivariant stable homotopy theory*, but so far there is nothing but references.

* rephrased the intro part, trying to make it more forcefully to the point (not claiming to have found the optimum, though)

* added a dedicated section <a href="http://ncatlab.org/nlab/show/stable+(infinity%2C1)-category#the_homotopy_category_of_a_stable_category_triangulated_categories_7">The homotopy cat of a stable (oo,1)-cat: traingulated categories</a> to highlight the important statement here, which was previously a bit hidden in the main text. ]]>

gave *Bousfield localization of spectra* a more informative Idea-section