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- Discussion Type
- discussion topiccomparison map between algebraic and topological K-theory
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Apr 29th 2014

This used to be a super-brief paragraph at

*topological K-theory*; and now it is a slightly longer but still stubby entry*comparison map between algebraic and topological K-theory*

- Discussion Type
- discussion topicČech model structure on simplicial presheaves
- Category Latest Changes
- Started by Zhen Lin
- Comments 2
- Last comment by Urs
- Last Active Apr 26th 2014

There seem to be some misleading remarks at Čech model structure on simplicial presheaves.

Accordingly, the (∞,1)-topos presented by the Čech model structure has as its cohomology theory Čech cohomology.

Marc Hoyois seems to says the opposite: there is no deep relation between “Čech” in “Čech cohomology” and in “Čech model structure”.

[…] the corresponding Čech cover morphism .

Notice that by the discussion at model structure on simplicial presheaves - fibrant and cofibrant objects this is a morphism between cofibrant objects.

The Čech nerve is projective-cofibrant if we assume the site has pullbacks. I don’t know how to prove it otherwise. Of course, injective-cofibrancy is trivial.

this question is evidently also relevant to what the correct notion of internal ∞-groupoid may be

Based on the discussion here, it seems that the Čech model structure is

*not*site-independent, even though it can be defined on the category of simplicial*sheaves*. A very strange state of affairs…

- Discussion Type
- discussion topicsmooth spectrum
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Apr 26th 2014

am starting an entry

*smooth spectrum*(in the sense of*smooth infinity-groupoid*). But nothing much there yet.

- Discussion Type
- discussion topicSO orientation of elliptic cohomology
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Apr 24th 2014

minimum at

*spin orientation of Tate K-theory*, for the moment just as to record the reference and the proposition number in there (to go with this MO question)

- Discussion Type
- discussion topicreal-oriented cohomology theory
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Apr 23rd 2014

started some minimum at

*real-oriented cohomology theory*

- Discussion Type
- discussion topiccongruence subgroup
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Apr 22nd 2014

some basics at

*congruence subgroup*

- Discussion Type
- discussion topicquery at contravariant functor
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Zhen Lin
- Last Active Apr 21st 2014

Vladimir Sotirov has asked a question at contravariant functor.

- Discussion Type
- discussion topicFermat's little theorem
- Category Latest Changes
- Started by Colin Tan
- Comments 6
- Last comment by Colin Tan
- Last Active Apr 21st 2014

Stated Fermat’s little theorem.

- Discussion Type
- discussion topicbinomial theorem
- Category Latest Changes
- Started by Todd_Trimble
- Comments 3
- Last comment by Todd_Trimble
- Last Active Apr 20th 2014

Created binomial theorem, and added a relevant lemma to freshman’s dream.

- Discussion Type
- discussion topicvon Neumann algebras
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 37
- Last comment by Todd_Trimble
- Last Active Apr 20th 2014

I started rewriting von Neumann algebra from the nPOV. So far I rewrote the definition and added some remarks about Sakai's theorem and preduals, but you can already see a proposed list of sections to be written.

I also edited the remarks section to stress the nPOV.

- Discussion Type
- discussion topicClosed Category, Category of V-enriched Categories
- Category Latest Changes
- Started by Vladimir_Sotirov
- Comments 6
- Last comment by Vladimir_Sotirov
- Last Active Apr 19th 2014

- I have edited the articles on closed category and unit enriched category with a view toward a substantial revision of category of V-enriched categories. My primary purpose will be to document the extra structure necessary to realize a $2$-category as a $2$-category of $V$-enriched categories for the various contexts (at least for the monoidal and closed contexts anyway). In particular, I hope to motivate the structure of a closed category $C$ as the minimal amount of structure necessary to recover a $2$-category of categories enriched in self-enriched $C$.

- Discussion Type
- discussion topicString theory and the real world
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 18th 2014

created a category:Reference-entry

*String theory and the real world*(a set of lecture notes on string phenomenology)(This is to go along with this PhysicsOverflow reply)

- Discussion Type
- discussion topicSimpson's conjecture
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Apr 16th 2014

added to

*Simpson conjecture*a History section with a paragraph on how Carlos Simpson came up with the conjecture based on that claim by Kapranov-Voevodsky’s (the one whose delicacy Voevodsky now says made him formalize mathematics in HoTT…)

- Discussion Type
- discussion topicEilenberg-MacLane object
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Apr 16th 2014

I was unsatisfied with the entry Eilenberg-MacLane object. So I changed the wording at the beginning. Maybe it's an improvement, maybe something better needs to be done.

- Discussion Type
- discussion topicequivariant homotopy theory -- table
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Apr 14th 2014

created

*equivariant homotopy theory – table*displaying the various cohesive $\infty$-toposes and their bases $\infty$-toposes (for inclusion in “Related entries” at the relevant entries)

- Discussion Type
- discussion topicglobal orbit catgegory / global equivariant indexing category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 14th 2014

created

*global orbit category*and*global equivariant indexing category*.Both entries contain almost the same content at the moment. Both could use more editing, too.

- Discussion Type
- discussion topicG-space
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 13th 2014

created

*G-space*, a glorified disambiguation page.

- Discussion Type
- discussion topicj-invariant
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 11th 2014

I have been adding some stuff to

*j-invariant*, but it’s not really good yet (this here just in case you are watching the logs and are wondering what’s happening)

- Discussion Type
- discussion topicelliptic fibration
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 11th 2014

started something at

*elliptic fibration*

- Discussion Type
- discussion topicTmf(n)
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 11th 2014

created an entry for

*Tmf(n)*

- Discussion Type
- discussion topicM5-brane charge
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 11th 2014

created

*M5-brane charge*

- Discussion Type
- discussion topicGalois group
- Category Latest Changes
- Started by adeelkh
- Comments 3
- Last comment by Urs
- Last Active Apr 10th 2014

Here is a note to myself or anyone else to add the following new preprint to Galois group when the nLab is back online.

- Akhil Mathew, The Galois group of a stable homotopy theory, http://arxiv.org/abs/1404.2156.

I’ve only just read the introduction, but it looks pretty great…

- Discussion Type
- discussion topicmodel structure on operads
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Apr 10th 2014

started model structure on operads

by the way: I noticed that the page operad has not a single reference. Maybe somebody feels like filling in his favorite ones...

- Discussion Type
- discussion topicmodular curve
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 10th 2014

some basics at

*modular curve*

- Discussion Type
- discussion topicThe Goursat theorem.
- Category Latest Changes
- Started by TobyBartels
- Comments 5
- Last comment by Urs
- Last Active Apr 10th 2014

New stubs Édouard Goursat and Goursat theorem and some rearrangement of holomorphic function. I hope to put Goursat’s proof at Goursat theorem (but in the meantime you may see it PlanetMath) and consider its constructive content (probably assuming the fan theorem). But it might be a while before I get around to that.

- Discussion Type
- discussion topiccompact element
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by zskoda
- Last Active Apr 10th 2014

compact element in a lattice, defining also algebraic and coherent frames/locales and quantales.

- Discussion Type
- discussion topicFell bundles
- Category Latest Changes
- Started by davidoslive
- Comments 12
- Last comment by zskoda
- Last Active Apr 10th 2014

- I've created a page for Fell Bundles. It's only really a stub at the moment but I'll get around to expanding it eventually. The nLab POV of Fell bundles looks very different from the classical view but the two views can easily be reconciled (which I guess should form part of the expansion).

- Discussion Type
- discussion topiccomplex line
- Category Latest Changes
- Started by Urs
- Comments 21
- Last comment by Colin Tan
- Last Active Apr 10th 2014

- Discussion Type
- discussion topicnodal curve
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 10th 2014

added to

*nodal curve*a brief paragraph*over the complex numbers*

- Discussion Type
- discussion topiclevel structure on an elliptic curve
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 9th 2014

started

*level structure on an elliptic curve*with an Idea-section on what it means over the complex numbers.

- Discussion Type
- discussion topicinversion involution
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Apr 9th 2014

for some reason I created brief entry

*inversion involution*, but there is not really much of a point, I have to admit. But now it exists.

- Discussion Type
- discussion topicGoerss-Hopkins-Miller theorem
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Apr 8th 2014

I have expanded a bit the previous stub entry

*Goerss-Hopkins-Miller theorem*. It’s still stubby, but less so.I have added

more of the pertinent references;

an actual Idea-section

the statement of the Hopkins-Miller theorem in the version as it appears in Charles Rezk’s notes.

Maybe this feeble step forward inspires Aaron to add more… :-)

- Discussion Type
- discussion topicHochschild-Kostant-Rosenberg theorem
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Tim_Porter
- Last Active Apr 8th 2014

added details to Hochschild-Kostant-Rosenberg theorem

added the same to Hochschild cohomology

- Discussion Type
- discussion topiclocal-global principle
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 7th 2014

started a note at

*local-global principle*.Need to interrupt now. This clearly can be extended indefinitely…

- Discussion Type
- discussion topicMoebius transformation
- Category Latest Changes
- Started by Todd_Trimble
- Comments 10
- Last comment by Todd_Trimble
- Last Active Apr 7th 2014

Stubby beginning for Moebius transformation.

- Discussion Type
- discussion topicprime spectrum of a monoidal stable (infinity,1)-category
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Apr 6th 2014

Created prime spectrum of a monoidal stable (∞,1)-category and cross-linked vigorously with related entries.

this needs to be further expand, clearly. More references etc.

- Discussion Type
- discussion topicgenus of a surface
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Apr 5th 2014

I added the word ‘helps’ to the entry at genus of a surface since its genus does not fully classify the surface, you need orientability (and then is it a surface with boundary or not).

- Discussion Type
- discussion topiccubic curve
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 4th 2014

started an entry

*cubic curve*,For the moment I wanted to record (see the entry) a pointer to Akhil Mathew’s identification of that eight-fold cover of $\mathcal{M}_{cub}$ (hence of $\mathcal{M}_{ell}$) which is analogous to the 2-fold cover of the “moduli stack of formal tori” $B \mathbb{Z}_2$ that ends up being the reason for the $\mathbb{Z}_2$-action on $KU$.

So here is the question that I am after: that cover is classified by a map $\mathcal{M}_{ell} \to B \mathbb{Z}/8\mathbb{Z}$, hence we get a double cover of the moduli space of elliptic curves $d \colon \mathcal{M}_{ell} \to B\mathbb{Z}/2\mathbb{Z}$.

Accordingly there is a spectrum $Q \coloneqq d_\ast(\mathcal{O}^{top})$ equipped with a $\mathbb{Z}_2$-action whose homotopy fixed points is $tmf$, I suppose: $tmf \simeq Q^{\mathbb{Z}_2}$. (Hm, maybe I need to worry about the compactification…).

I’d like to say that $Q$ is to $tmf$ as $KU$ is to $KO$. This is either subject to some confusion (wich one?) or else is an old hat. In the second case: what would be a reference?

- Discussion Type
- discussion topicKSC-theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 4th 2014

started some minimum at

*KSC-theory*

- Discussion Type
- discussion topicstring theory and cohomology theory -- table
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 4th 2014

started a table-for-inclusion-in-relevant-entries:

*string theory and cohomology theory – table*Right now it reads like this:

**cohomology theories of string theory fields on orientifolds**string theory B-field $B$-field moduli RR-field bosonic string line 2-bundle ordinary cohomology $H\mathbb{Z}^3$ type II superstring super line 2-bundle $Pic(KU)//\mathbb{Z}_2$ KR-theory $KR^\bullet$ type IIA superstring super line 2-bundle $B GL_1(KU)$ KU-theory $KU^1$ | type IIB superstring super line 2-bundle $B GL_1(KU)$ KU-theory $KU^0$ type I superstring super line 2-bundle $Pic(KU)//\mathbb{Z}_2$ KO-theory $KO$ type $\tilde I$ superstring super line 2-bundle $Pic(KU)//\mathbb{Z}_2$ KSC-theory $KSC$

- Discussion Type
- discussion topicgeometric infinity-stack
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Apr 3rd 2014

have created geometric infinity-stack

gave Toën’s definition in detail (quotient of a groupoid object in an (infinity,1)-category in $T Alg_\infty^{op} \stackrel{Spec}{\hookrightarrow}Sh_\infty(C)$ ) and indicated the possibility of another definition, along the lines that we are discussing on the $n$Café

- Discussion Type
- discussion topictwisted forms
- Category Latest Changes
- Started by Jon Beardsley
- Comments 8
- Last comment by FinnLawler
- Last Active Apr 2nd 2014

I made a new page called twisted form. Unfortunately, this stole the redirect from a sub-heading on differential form. The page is still pretty much a stub. I hope to enlarge it soon.

- Discussion Type
- discussion topicinverse semigroup
- Category Latest Changes
- Started by zskoda
- Comments 5
- Last comment by zskoda
- Last Active Apr 2nd 2014

inverse semigroups behave very well, in some aspects almost like groups, and have close relation to etale groupoids and quantales. I added few references to its stubby entry.

- Discussion Type
- discussion topicBrauer infinity-group
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 1st 2014

briefly started

*Brauer infinity-group*with a quick remark on the relation to and cross-link to*Picard infinity-group*and*infinity-group of units*

- Discussion Type
- discussion topicelementary theory of the category of sets
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 1st 2014

I happened upon our entry

*ETCS*again (which is mostly a pointer to further entries and further resources) and found that it could do with a little bit more of an Idea-section, before it leaves the reader alone with the decision whether to follow any one of the many further links offered.I have expanded a bit, and now it reads as follows. Please feel invited to criticize and change. (And a question: didn’t we have an entry on ETCC, too? Where?)

The

**Elementary Theory of the Category of Sets**(Lawvere 65), or*ETCS*for short, is a formulation of set-theoretic foundations in a category-theoretic spirit. As such, it is the prototypical structural set theory.More in detail, ETCS is a first-order theory axiomatizing elementary toposes and specifically those which are well-pointed, have a natural numbers object and satisfy the axiom of choice. The idea is, first of all, that traditional mathematics naturally takes place “inside” such a topos, and second that by varying the axioms much of mathematics may be done inside more general toposes: for instance omitting the well-pointedness and the axiom of choice but adding the Kock-Lawvere axiom gives a smooth topos inside which synthetic differential geometry takes place.

Modern mathematics with emphasis on concepts of homotopy theory would more directly be founded in this spirit by an axiomatization not just of elementary toposes but by elementary (∞,1)-toposes. This is roughly what univalent homotopy type theory accomplishes, for more on this see at

*relation between type theory and category theory – Univalent HoTT and Elementary infinity-toposes*.Instead of increasing the higher categorical dimension (n,r) in the first argument, one may also, in this context of elementary foundations, consider raising the second argument. The case $(2,2)$ is the elementary theory of the 2-category of categories (ETCC).

- Discussion Type
- discussion topicsymmetric functions
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by Urs
- Last Active Mar 29th 2014

New references at symmetric function and new stub noncommutative symmetric function. An (unfinished?) discussion query from symmetric function moved here:

David Corfield: Why does Hazewinkel in his description of the construction of $\Lambda$ on p. 129 of this use a graded projective limit construction in terms of projections of polynomial rings?

John Baez: Hmm, it sounds like you’re telling me that there are ’projections’

$\Lambda_{n+1} \to \Lambda_n$given by setting the $(n+1)$st variable to zero, and that Hazewinkel defines $\Lambda$ to be the limit (= projective limit)

$\cdots \to \Lambda_2 \to \Lambda_1 \to \Lambda_0$rather than the colimit

$\Lambda_0 \to \Lambda_1 \to \Lambda_2 \to \cdots$Right now I don’t understand the difference between these two constructions well enough to tell which one is ’right’. Can someone explain the difference? Presumably there’s more stuff in the limit than the colimit.

Mike Shulman: I think the difference is that the limit contains “polynomials” with infinitely many terms, and the colimit doesn’t. That’s often the way of these things.

Actually, on second glance, I don’t understand the description of the maps in the colimit system; are you sure they actually exist? What exactly does it mean to “add in new terms with the new variable to make the result symmetric”?

David Corfield: The two constructions are explained very well in section 2.1 of the Wikipedia article.

Mike Shulman: Thanks! Here’s what I get from the Wikipedia article: the projections are easy to define. They are surjective and turn out to have sections (as ring homomorphisms). The ring of symmetric functions can be defined either as the colimit of the sections, or as the the limit of the projections

*in the category of graded rings*. The limit in the category of all rings would contain too much stuff.

- Discussion Type
- discussion topicequivariant complex oriented cohomology
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 27th 2014

recorded some references on equivariant complex oriented cohomology theory at

*equivariant cohomology – References – In complex oriented cohomology*.

- Discussion Type
- discussion topicgenera and partition functions - table
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 26th 2014

hm, it seems I never announced it: there is an old table-for-inclusion-in-relevant-entries called

*genera and partition functions - table*which I have been editing a bit more lately.

- Discussion Type
- discussion topicEisenstein series
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Mar 26th 2014

started

*Eisenstein series*with some formulas.

- Discussion Type
- discussion topicAtiyah-Bott-Shapiro orientation
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Mar 26th 2014

I have added to

*K-orientation*pointers to the articles by Atiyah-Bott-Shapiro and to Joachim (2004), together with a brief paragraph.

- Discussion Type
- discussion topicstructure in model theory
- Category Latest Changes
- Started by Todd_Trimble
- Comments 12
- Last comment by Urs
- Last Active Mar 26th 2014

I added some categorical POV on structure in model theory (which is being touched upon in another thread).

- Discussion Type
- discussion topicelliptic cohomology -- contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 25th 2014

Given the series of entries lately, I naturally came to the point that I started to want a “floating context” table of contents. So I started one and included it into relevant entries:

But this needs more work still, clearly.

- Discussion Type
- discussion topicJacobi form
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Mar 25th 2014

Created a minimum at

*Jacobi form*.

- Discussion Type
- discussion topicbraid group
- Category Latest Changes
- Started by Todd_Trimble
- Comments 1
- Last comment by Todd_Trimble
- Last Active Mar 25th 2014

Missing from braid group was the precise geometric definition, so I put that in.

- Discussion Type
- discussion topiclogarithmic cohomology operation
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Mar 24th 2014

am starting something at

*logarithmic cohomology operation*, but so far there are just some general statements and some references

- Discussion Type
- discussion topicthick subcategory theorem
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 24th 2014

Finally created

*thick subcategory theorem*with a quick statement of the theorem and a quick pointer to how this determines the prime spectrum of a monoidal stable (∞,1)-category of the (∞,1)-category of spectra.Cross-linked vigorously with related entries.

- Discussion Type
- discussion topicBousfield-Kuhn functor
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 24th 2014

Created some minimum at

*Bousfield-Kuhn functor*, for the moment just so as to record some references.

- Discussion Type
- discussion topicBICEP2
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by zskoda
- Last Active Mar 24th 2014

Created

*BICEP2*, currently with the following text:

*BICEP2*is the name of an astrophysical experiment which released its data in March 2014. The experiment claims to have detected a pattern called the “B-mode” in the polarization of the cosmic microwave background (CMB).This data, if confirmed, is widely thought to be due to a gravitational wave mode created during the period of cosmic inflation by a quantum fluctuation in the field of gravity which then at the era of decoupling left the characteristic B-mode imprint on the CMB. This fact alone is regarded as further strong evidence for the already excellent experimental evidence for cosmic inflation as such (competing models did not predict such gravitational waves to be strong enough to be detectable in this way).

What singles out the BICEP2 result over previous confirmations of cosmic inflation is that the data also gives a quantitative value for the energy scale at which cosmic inflation happened (the mass of the hypothetical inflaton), namely at around $10^{16}$GeV. This is ntoeworthy as being only two order of magnituded below the Planck scale, and hence 12 or so orders of magnitude above energies available in current accelerator experiments (the LHC). Also, it is at least a curious coincidence that this is precisely the hypothetical GUT scale.

It is thought that this value rules out a large number of variant models of cosmic inflation and favors the model known as

*chaotic inflation*.

- Discussion Type
- discussion topicmodalities, closure and reflection - contents
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by zskoda
- Last Active Mar 24th 2014

am creating a table

*modalities, closure and reflection - contents*and adding it as a floating table of contents to relevant entries

- Discussion Type
- discussion topicquantum fluctuation
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 24th 2014

just for completeness so that I don’t have gray links elsewhere, I have created some minimum (nothing exciting) at

*quantum fluctuation*.