Finally added to *fracture theorem* the basic statement of the “arithmetic fracture square”, hence the following discussion.

The number theoretic statement is the following:

+– {: .num_prop #ArithmeticFractureSquare}

The integers $\mathbb{Z}$ are the fiber product of all the p-adic integers $\underset{p\;prime}{\prod} \mathbb{Z}_p$ with the rational numbers $\mathbb{Q}$ over the rationalization of the former, hence there is a pullback diagram in CRing of the form

$\array{ && \mathbb{Q} \\ & \swarrow && \nwarrow \\ \mathbb{Q}\otimes_{\mathbb{Z}}\underset{p\;prime}{\prod} \mathbb{Z}_p && && \mathbb{Z} \\ & \nwarrow && \swarrow \\ && \underset{p\;prime}{\prod} \mathbb{Z}_p } \,.$Equivalently this is the fiber product of the rationals with the integral adeles $\mathbb{A}_{\mathbb{Z}}$ over the ring of adeles $\mathbb{A}_{\mathbb{Q}}$

$\array{ && \mathbb{Q} \\ & \swarrow && \nwarrow \\ \mathbb{A}_{\mathbb{Q}} && && \mathbb{Z} \\ & \nwarrow && \swarrow \\ && \mathbb{A}_{\mathbb{Z}} } \,.$=–

In the context of a modern account of categorical homotopy theory this appears for instance as (Riehl 14, lemma 14.4.2).

+– {: .num_remark}

Under the function field analogy we may think of

$Spec(\mathbb{Z})$ as an arithmetic curve over F1;

$\mathbb{A}_{\mathbb{Z}}$ as the ring of functions on the formal disks around all the points in this curve;

$\mathbb{Q}$ as the ring of functions on the complement of a finite number of points in the curve;

$\mathbb{A}_{\mathbb{Q}}$ is the ring of functions on punctured formal disks around all points, at most finitely many of which do not extend to the unpunctured disk.

Under this analogy the arithmetic fracture square of prop. \ref{ArithmeticFractureSquare} says that the curve $Spec(\mathbb{Z})$ has a cover whose patches are the complement of the curve by some points, and the formal disks around these points.

This kind of cover plays a central role in number theory, see for instance thr following discussions:

=–

]]>added the plain traditional definition to *J-homomorphism*

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

I have been indexing Kochman’s excellent book *Bordism, Stable Homotopy and Adams Spectral Sequences* and in the course I have touched all the relevant entries

created stub for *étale morphism of E-∞ rings* in order to record the theorem of essential uniqueness of lifts of étale morphism from underlying commutative rings to $E_\infty$-rings (which is crucial for the characterization of the moduli stack of derived elliptic curves, and I have cross-linked with that). But otherwise no content yet, due to lack of leisure.

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)

created some minimum at *Boardman homomorphism* (the thing generalising the Hurewicz homomorphism)

started an entry *global equivariant stable homotopy theory* with an Idea-section and some references.

I have also created a brief entry for the unstable version: *global equivariant homotopy theory*.

For anyone who wants to edit and wondering where to add what, let me just highlight that there is the following collection of existing entries (some of them with genuine content, some mostly stubby)

homotopy theory | stable homotopy theory |
---|---|

equivariant homotopy theory | equivariant stable homotopy theory |

global equivariant homotopy theory | global equivariant stable homotopy theory |

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.

]]>I have created a minimum at *global family* (a suitable family of groups in the sense of global equivariant homotopy theory).

Hm, the set of finite subgroups of $SO(3)$ or of $SU(2)$. Is that a global family? I.e. is it closed under quotient groups by normal subgroups?

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

To be expanded eventually…

]]>I have created an entry *spectral symmetric algebra* with some basics, and with pointers to Strickland-Turner’s Hopf ring spectra and Charles Rezk’s power operations.

In particular I have added amplification that even the case that comes out fairly trivial in ordinary algebra, namely $Sym_R R$ is interesting here in stable homotopy theory, and similarly $Sym_R (\Sigma^n R)$.

I am wondering about the following:

In view of the discussion at spectral super scheme, then for $R$ an even periodic ring spectrum, the superpoint over $R$ has to be

$R^{0 \vert 1} \;=\; Spec(Sym_R \Sigma R) \simeq Spec\left( R \wedge \left( \underset{n \in \mathbb{N}}{\coprod} B\Sigma(n)^{\mathbb{R}^n} \right)_+ \right) \,.$This of course is just the base change/extension of scalars under Spec of the “absolute superpoint”

$\mathbb{S}^{0\vert 1} \simeq Spec(Sym_{\mathbb{S}} (\Sigma \mathbb{S}))$(which might deserve this notation even though the sphere spectrum is of course not even periodic).

This looks like a plausible answer to the quest that David C. and myself were on in another thread, to find a plausible candidate in spectral geometry of the ordinary superpoint $\mathbb{R}^{0 \vert 1}$, regarded as the base of the brane bouquet.

]]>created an extemely stubby stub *Weiss topology*, just to record pointer to that cool fact which Dmitri Pavlov advertised on MO (here).

I have no time to expand on the entry right now. But maybe somebody else here does? Would be worthwhile.

]]>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.

Has anyone developed models for the homotopy theory of $H \mathbb{Q}$.module spectra over rational topological spaces a bit?

I expect there should be a model on the opposite category of dg-modules over rational dg-algebras. Restricted to the trivial modules it should reduce to the standard Sullivan/Quillen model of rational homotopy theory. Restricted to the dg-modules over $\mathbb{Q}$ it should reduce to the standard model for the homotopy theory of rational chain complexes, hence equivalently that of $H \mathbb{Q}$-module spectra.

Is there any work on this?

]]>added to *spectrum with G-action* brief paragraphs “Relation to genuine G-spectra”, and “relation to equivariant cohomology”.

Both would deserve to be expanded much more, but it’s a start.

]]>while I was adding more references and pointers to *KR-theory* I have created a brief stub for *real algebraic K-theory*, just to record the (still unpublished…?) references

I have given *infinity-group infinity-ring* its own entry (it used to be redirecting to *infinity-group of units*)

Then I added a section “H-group ring spectra” with some details on the simpler version $\Sigma^\infty(-)_+ \colon Ho(Top) \longrightarrow Ho(Spectra)$.

]]>(never mind)

]]>created a stub entry for *comodule spectrum*, for the moment just so as to briefly record the result by Hess-Shipley 14 that comodule spectra over suspension spectra of connected spaces $X$ are equivalently parameterized spectra over $X$. Added that reference also to *A-theory*. Needs to be expanded further.

(Thanks to Charles Rezk for the pointer.)

]]>for the purposes of having direct links to it, I gave a side-remark at *stable Dold-Kan correspondence* its own page: rational stable homotopy theory, recording the equivalence

I also added the claim that under this identification and that of classical rational homotopy theory then the derived version of the free-forgetful adjunction

$(dgcAlg^{\geq 2}_{\mathbb{Q}})_{/\mathbb{Q}[0]} \underoverset {\underset{U \circ ker(\epsilon_{(-)})}{\longrightarrow}} {\overset{Sym \circ cn}{\longleftarrow}} {\bot} Ch^{\bullet}(\mathbb{Q})$models the stabilization adjunction $(\Sigma^\infty \dashv \Omega^\infty)$. But I haven’t type the proof into the entry yet.

]]>Concerning our geologically slow discussion elsewhere, in various other threads, on $\infty$-toposes that contain non-trivial stable homotopy theory.

Here is a trivial thought:

while we grew fond of identifying the $\infty$-category and allegedly $\infty$-topos of parameterized spectra as the tangent (infinity,1)-category to $\infty Grpd$, maybe it’s after all more useful to think of it instead as the full sub-category of the slice $(\infty,2)$-category

$(\infty,1)Cat_{/Spectra}$on the $(\infty,0)$-truncated objects, which we are inclined to write

$\infty Grpd_{/Spectra} \hookrightarrow (\infty,1)Cat_{/Spectra} \,.$But suppose these two obviously plausible facts about $(\infty,2)$-toposes hold true:

slices of $(\infty,2)$-toposes are $(\infty,2)$-toposes;

full subcateories on $(\infty,0)$-truncated objects inside $(\infty,2)$-toposes are $(\infty,1)$-toposes

then it would follow immediately that $\infty Grpd_{/Spectra}$ is an $(\infty,1)$-topos.

Maybe somebody could remind me why this obvious (naive?) strategy for going about it is no good.

]]>I have expanded a little at *CW-spectrum*

started a minimum at *E-nilpotent completion* (the thing that an $E$-Adams tower converges to).