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    • I keep drawing and re-drawing that Whitehead tower again and again. That needs to stop. So I created now an entry with a table, to be included where needed: higher spin structure - table

    • Added a fair bit of content to 7d Chern-Simons theory.

      Of the three examples discussed there, the first two are review. The third is inspired by something I have been talking about with D. Fiorenza, C. Rogers and H. Sati.

    • New entry IMU linked from ICM. Note that ICM page has the link to the archive of articles from old ICM-s. This is very precious as these are usually readable surveys of major contributions to mathematics covering over half a century.

    • I added a query box to the Holographic Principle page, referring to the work of Andersen and Ueno which I believe has now made rigorous that geometric quantization of Chern-Simons theory = quantum groups approach ala Reshetikhin-Turaev.

    • Before I forget, I uploaded a new version of my anafunctors paper to my page David Roberts. In particular, the finer points have been made a lot tighter. I even use technical phrases such as ’enough groupoids’ and ’admits cotensors’! :) It has also been submitted for publication.

    • Here it is: Lipschitz map. I don't know why I wrote it; I just felt like it. There really is much more to say, but I think that I've said enough for now!

    • stub for Noetherian poset

      (just needed to be able to point to it, no real content there yet)

    • added in an Examples-section to stable factorization system the statement that in an adhesive category, in particular in a topos, the (epi, mono)-factorization is stable.

    • started Hitchin functional but have to interrupt now in the middle of it. This entry is not in good shape yet.

    • expanded the Idea- and the Definition section at G2-manifold (also further at G2). (Still not really complete, though.) Highlighted the relation to 2-plectic geometry.

    • am starting line 2-bundle. I am headed for some discussion of the 2-stack of super line 2-bundles, its role as the twistings of K-theory in degrees 0,1, and 2/3 and so forth. But right now it’s still mostly a stub. More in a short while…

    • Someone anonymous has deleted a paragraph at red herring principle on non-associative algebra. This seems a bit strange. I am no expert on those beasties but although non-associative algebra includes the study of Lie algebras etc., amongst them are the modules and it seems to me that a module (with trivial multiplication) considered as a Lie algebra is an associative non-associative algebra! The query by Toby further down the entry is relevant but if we assume ‘non-unital’ as well (and that is sometimes done) there is no problem.

      There was no post to the Forum. The IP is 2.40.78.132. which is in Trieste it seems.

      Should the paragraph be reinstated?

    • The entry Dynkin diagram is a ‘My First Slide’ one. Sometimes this sort of attempt has resulted in a new entry with substance being generated later, but more often nothing happens. Does anyone want to create such an entry?

    • quick note semi-Segal space (just recording a reference of a speaker we had today)

    • I tried to start an entry called Fermat theory. Unfortunately I screwed up and forgot to capitalized "Fermat". Maybe a lab elf can clean up my mess?

      There are also lots of other obvious ways this page could be improved...

    • I have started making notes at differential cohesion on the axiomatic formulation of

      So far just the bare basics. To be expanded…

      The basic observation (easy in itself, but fundamental for the concept formation) is that for any differential cohesive homotopy type XX, the inclusion of the formally étale maps into XX into the full slice over XX is not only reflective but also co-reflective (since the formally étale maps are the Pi_inf-closed morphisms with the infinitesimal path groupoid functor / de Rham space functor Π inf\Pi_{inf} being a left adjoint).

      This means that for GG any differential cohesive ∞\infty-group with the corresponding de Rham coefficient object ♭ dRBG\flat_{dR}\mathbf{B}G (the universal moduli for flat 𝔤\mathfrak{g}-valued differential forms), the sheaf of flat 𝔤\mathfrak{g}-valued forms over any XX is given by the sections of the coreflection of the product projection X×♭ dRBG→XX \times \flat_{dR}\mathbf{B}G \to X into the formally étale morphisms into XX.

    • I created an entry realizability model. But I only got to put one single reference into it and now I am forced to go offline.

      I’ll try to add more later. But maybe somebody here feels inspired to add a brief explanation…

    • not sure why, but reading

      • Peter Dybjer, Thoughts on Martin-Löf’s Meaning Explanations (pdf)

      made me look at

      which seems to be about something deep and important that eventually I’d like to better grasp (but don’t yet), and that made me create computational consistency.

      But I admit that I don’t really know what I am doing, in this case. So I’ll stop.

    • Added some formulas and a manifestly relativistic version to action functional.

      I have also been reverting JA's changes to variant conventions of spelling and grammar.

    • I am starting an entry internal (infinity,1)-category about complete Segal-like things.

      This is prompted by me needing a place to state and prove the following assertion: a cohesive ∞\infty-topos is an “absolute distributor” in the sense of Lurie, hence a suitable context for internalizing (∞,1)(\infty,1)-categories.

      But first I want a better infrastructure. In the course of this I also created a “floating table of contents”

      and added it to the relevant entries.

    • Put a link to

      into weak omega-groupoid… only trouble being that this entry doesn’t exist yet but redirects to infinity-groupoid, which otherwise has no references currently ?!-o . Somebody should take care of editing this a bit. But it won’t be me right now.

    • I made some modifications to the definition section of root, and added the theorem that finite multiplicative subgroups of a field are cyclic. While I was at it, I added a bit to quaternion.

    • In email discussion with somebody I wanted to point to the nnLab entry A-infinity space only to notice that there is not much there. I have now spent a minute adding just a tiny little bit more…

    • It’s time to become serious about “higher order” aspects of applications of the the “sharp-modality” ♯\sharp in a cohesive (infinity,1)-topos H\mathbf{H} – I am thinking of the construction of moduli ∞\infty-stacks for differential cocycles.

      Consider, as usual, the running example H=Sh ∞(CartSp)=\mathbf{H} = Sh_\infty(CartSp) = Smooth∞Grpd.

      Simple motivating example: moduli of differential forms

      Here is the baby example, which below I discuss how to refine:

      there is an object called Ω 1∈H\Omega^1 \in \mathbf{H}, which is just the good old sheaf of differential 1-forms. Consider also a smooth manifold X∈HX \in \mathbf{H}. On first thought one might want to say that the internal hom object [X,Ω 1][X, \Omega^1] is the “moduli 0-stack of differential 1-forms on XX”. But that’s not quite right. For U∈U \in CartSp, the UU-plots of the latter should be smoothly UU-parameterized sets of differential 1-forms on XX, but the UU-plots of [X,Ω 1][X,\Omega^1] contain a bit more stuff. They are of course 1-forms on U×XU \times X and the actual families that we want to see are only those 1-forms on U×XU \times X which have “no leg along UU”. But one sees easily that the correct moduli stack of 1-forms on XX is

      Ω 1(X):=♯ 1[X,Ω 1]↪♯[X,Ω 1], \mathbf{\Omega}^1(X) := \sharp_1 [X,\Omega^1] \hookrightarrow \sharp [X, \Omega^1] \,,

      where ♯ 1[X,Ω 1]:=image([X,Ω 1]→♯[X,Ω 1])\sharp_1 [X,\Omega^1] := image( [X, \Omega^1] \to \sharp [X, \Omega^1] ) is the concretification of [X,Ω 1][X,\Omega^1].

      Next easy example: moduli of connections

      This above kind of issue persists as we refine differential 1-forms to circle-principal connections: write BU(1) conn∈H\mathbf{B}U(1)_{conn} \in \mathbf{H} for the stack of circle-principal connections. Then for XX a manifold, one might be inclined to say that the mapping stack [X,BU(1) conn][X, \mathbf{B}U(1)_{conn}] is the moduli stack of circle-principal connections on XX. But again it is not quite right: a UU-plot of [X,BU(1) conn][X,\mathbf{B}U(1)_{conn}] is a circle-principal connection on U×XU \times X, but it should be one with no form components along UU, so that we can interpret it as a smoothly UU-parameterized set of connections on XX.

      The previous example might make one think that this is again fixed by considering ♯ 1[X,BU(1) conn]\sharp_1 [X, \mathbf{B}U(1)_{conn}]. But now that we have a genuine 1-stack and not a 0-stack anymore, this is not good enough: the stack ♯ 1[X,BU(1) conn]\sharp_1 [X, \mathbf{B}U(1)_{conn}] has as UU-plots the groupoid whose objects are smoothly UU-parameterized sets of connections on XX – that’s as it should be – , but whose morphisms are Γ(U)\Gamma(U)-parameterized sets of gauge transformations between these, where Γ(U)\Gamma(U) is the underlying discrete set of the test manifold UU – and that’s of course not how it should be. The reflection ♯ 1\sharp_1 fixes the moduli in degree 0 correctly, but it “dustifies” their automorphisms in degree 1.

      We can correct this as follows: the correct moduli stack U(1)Conn(X)U(1)\mathbf{Conn}(X) of circle principal connections on some XX is the homotopy pullback in

      U(1)Conn(X) → [X,BU(1)] ↓ ↓ ♯ 1[X,BU(1) conn] → ♯ 1[X,BU(1)] \array{ U(1)\mathbf{Conn}(X) &\to& [X, \mathbf{B} U(1)] \\ \downarrow && \downarrow \\ \sharp_1 [X, \mathbf{B}U(1)_{conn}] &\to& \sharp_1 [X, \mathbf{B} U(1)] }

      where the bottom morphism is induced from the canonical map BU(1) conn→BU(1)\mathbf{B}U(1)_{conn} \to \mathbf{B}U(1) from circle-principal connections to their underlying circle-principal bundles.

      Here the ♯ 1\sharp_1 in the bottom takes care of making the 0-cells come out right, whereas the pullback restricts among those dustified Γ(U)\Gamma(U)-parameterized sets of gauge transformations to those that actually do have a smooth parameterization.

      More serious example: moduli of 2-connections

      The previous example is controlled by a hidden pattern, which we can bring out by noticing that

      [X,BU(1)]≃♯ 2[X,BU(1)] [X, \mathbf{B}U(1)] \simeq \sharp_2 [X, \mathbf{B}U(1)]

      where ♯ 2\sharp_2 is the 2-image of id→♯id \to \sharp, hence the factorization by a 0-connected morphism followed by a 0-truncated one. For the 1-truncated object [X,BU(1)][X, \mathbf{B}U(1)] the 2-image doesn’t change anything. Generally we have a tower

      id=♯ ∞→⋯→♯ 2→♯ 1→♯ 0=♯. id = \sharp_\infty \to \cdots \to \sharp_2 \to \sharp_1 \to \sharp_0 = \sharp \,.

      Moreover, if we write DKDK for the Dold-Kan map from sheaves of chain complexes to sheaves of groupoids (and let stackification be implicit), then

      BU(1) conn =DK(U(1)→Ω 1) BU(1) =DK(U(1)→0). \begin{aligned} \mathbf{B}U(1)_{conn} &= DK( U(1) \to \Omega^1 ) \\ \mathbf{B}U(1) &= DK( U(1) \to 0 ) \end{aligned} \,.

      If we pass to circle-principal 2-connections, this becomes

      B 2U(1) conn 1=B 2U(1) conn =DK(U(1)→Ω 1→Ω 2) BU(1) conn 2 =DK(U(1)→Ω 1→0) BU(1) conn 3=B 2U(1) =DK(U(1)→0→0) \begin{aligned} \mathbf{B}^2 U(1)_{conn^1} = \mathbf{B}^2U(1)_{conn} &= DK( U(1) \to \Omega^1 \to \Omega^2 ) \\ \mathbf{B}U(1)_{conn^2} & = DK( U(1) \to \Omega^1 \to 0 ) \\ \mathbf{B}U(1)_{conn^3} = \mathbf{B}^2 U(1) & = DK( U(1) \to 0 \to 0 ) \end{aligned}

      and so on.

      And a little reflection show that the correct moduli 2-stack (BU(1))Conn(X)(\mathbf{B}U(1))\mathbf{Conn}(X) of circle-principal 2-connections on some XX is the homotopy limit in

      (BU(1))Conn(X) → → [X,B 2U(1)] ↓ ♯ 2[X,B 2U(1) conn 2] → ♯ 2[X,B 2U(1)] ↓ ↓ ♯ 1[X,B 2U(1) conn] → ♯ 1[X,B 2U(1) conn 2]. \array{ (\mathbf{B}U(1))\mathbf{Conn}(X) &\to& &\to& [X, \mathbf{B}^2 U(1)] \\ && && \downarrow \\ && \sharp_2 [X, \mathbf{B}^2 U(1)_{conn^2}] &\to& \sharp_2 [X, \mathbf{B}^2 U(1)] \\ \downarrow && \downarrow \\ \sharp_1 [X, \mathbf{B}^2 U(1)_{conn}] &\to& \sharp_1 [X, \mathbf{B}^2 U(1)_{conn^2}] } \,.

      This is a “3-stage ♯\sharp-reflection” of sorts, which fixes the naive moduli 2-stack [X,B 2U(1)][X, \mathbf{B}^2 U(1)] first in degree 0 (thereby first completely messing it up in the higher degrees), then fixes it in degree 1, then in degree 2. Then we are done.

    • It should have its own announcement: Frankel model of ZFA added to the lab. I should say that in this model there is a map x→ℕx \to \mathbb{N} where every fibre has two elements, which has no section (which would be “choosing a sock out of a countable set of pairs”)

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