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    • 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…

    • 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.

    • 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 $C$.

    • 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…)

    • 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.

    • 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)

    • created G-space, a glorified disambiguation page.

    • 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)

    • 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…

    • 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...

    • 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).

    • 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.

    • 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… :-)

    • started a note at local-global principle.

      Need to interrupt now. This clearly can be extended indefinitely…

    • 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).

    • 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 ℳ cub\mathcal{M}_{cub} (hence of ℳ ell\mathcal{M}_{ell}) which is analogous to the 2-fold cover of the “moduli stack of formal tori” Bℤ 2B \mathbb{Z}_2 that ends up being the reason for the ℤ 2\mathbb{Z}_2-action on KUKU.

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

      Accordingly there is a spectrum Q≔d *(𝒪 top)Q \coloneqq d_\ast(\mathcal{O}^{top}) equipped with a ℤ 2\mathbb{Z}_2-action whose homotopy fixed points is tmftmf, I suppose: tmf≃Q ℤ 2tmf \simeq Q^{\mathbb{Z}_2}. (Hm, maybe I need to worry about the compactification…).

      I’d like to say that QQ is to tmftmf as KUKU is to KOKO. This is either subject to some confusion (wich one?) or else is an old hat. In the second case: what would be a reference?

    • have created geometric infinity-stack

      gave Toën’s definition in detail (quotient of a groupoid object in an (infinity,1)-category in TAlg ∞ op↪SpecSh ∞(C)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 nnCafé

    • 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.

    • 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 Λ\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’

      Λ n+1→Λ n \Lambda_{n+1} \to \Lambda_n

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

      ⋯→Λ 2→Λ 1→Λ 0 \cdots \to \Lambda_2 \to \Lambda_1 \to \Lambda_0

      rather than the colimit

      Λ 0→Λ 1→Λ 2→⋯ \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.

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

    • 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.

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

    • 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 1610^{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.

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

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