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    • 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”)

    • We had discussed here at some length the formalization of formally etale morphisms in a differential cohesive (infinity,1)-topos. But there is an immediate slight reformulation which I never made explicit before, but which is interesting to make explicit:

      namely I used to characterize formal étaleness in terms of the canonical morphism ϕ:i !→i *\phi : i_! \to i_* between the components of the geometric morphism i:H↪H thi : \mathbf{H} \hookrightarrow \mathbf{H}_{th} that defines the differential cohesion – because that formulation made close contact to the way Kontsevich and Rosenberg formulate formal étaleness.

      But there is a more suggestive/transparent but equivalent (in fact more general, since it works in all of H th\mathbf{H}_{th} not just in H\mathbf{H}) formulation in terms of the Π inf\mathbf{\Pi}_{inf}-modality, the “fundamental infinitesimal path ∞\infty-groupoid” operator:

      a morphism f:X→Yf : X \to Y in H th\mathbf{H}_{th} is formally étale precisely if the canonical diagram

      X → Π inf(X) ↓ f ↓ Π inf(f) Y → Π inf(Y) \array{ X &\to & \mathbf{\Pi}_{inf}(X) \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\mathbf{\Pi}_{inf}(f)}} \\ Y &\to& \mathbf{\Pi}_{inf}(Y) }

      is an ∞\infty-pullback.

      (It’s immediate that this is equivalent to the previous definition, using that i !i_! is fully faithful, by definition.)

      This is nice, because it makes the relation to general abstract Galois theory manifest: if we just replace in the above the infinitesimal modality Π inf\mathbf{\Pi}_{inf} with the finite path ∞\infty-groupoid modality Π\mathbf{\Pi}, then the above pullback characterizes the “Π\mathbf{\Pi}-closed morphisms” which precisely constitute the total space projections of locally constant ∞\infty-stacks over YY. Here we now characterize general ∞\infty-stacks over YY.

      And for instance in direct analogy with the corresponding proof for the Π\mathbf{\Pi}-modality, one finds for the Π inf\mathbf{\Pi}_{inf}-modality that, for instance, we have an orthogonal factorization system

      (Π inf−equivalences,formallyetalemorphisms). (\mathbf{\Pi}_{inf}-equivalences\;,\; formally\;etale\;morphisms) \,.

      I’ll spell out more on this at Differential cohesion – Structures a little later (that’s why this here is under “latest changes”), for the moment more details are in section 3.7.3 of differential cohomology in a cohesive topos (schreiber).

    • added to disk a brief pointer to Joyal’s combinatorial disks. Needs to be expanded, probably entry should be split and disambiguated. But no time right now.

    • I have started something in an entry

      layers of foundations

      which has grown out of the the desires expressed in the thread The Wiki history of the universe.

      This is tentative. I should have maybe created this instead on my personal web. I hope we can discuss this a bit. If it leads nowhere and/or if the feeling is that it is awkward for one reason or other, I promise to remove it again. But let’s give this a chance. I feel this is finally beginning to converge to something.

    • started adding list of references to the page Bill Lawvere

      not that I made it very far -- just three items so far :-)

      I was really looking for an online copy of "Categorical dynamics" as referenced at synthetic differential geometry and generalized smooth algebra, but haven't found it yet. I was thinking that the "Toposes of laws of motion" that I do reference must be something close. But I don't know.

    • New, mainly disambiguation, entry affine algebra. Note that affine algebra for most algebraists is not the same as affine Lie algebra. I have corrected a wrong link in Wess-Zumino model which links to affine algebra instead to affine Lie algebra; let us be careful when linking in future. Affine algebras are coordinate rings of affine varieties. I have split affine variety from algebraic variety which also got a redirect algebraic manifold (= smooth algebraic variety). New entry Igor Shafarevich.

    • I created Alex Heller at Jim’s suggestion. It is very stubby and could have a lot more added.

    • New entry (!) tangent Lie algebra. Significant changes at invariant differential form with redirect invariant vector field reflecting the vector fields and other tensor cases. Many more related entries listed at and the whole entry reworked extensively at Lie theory. Some changes at Lie’s three theorems and local Lie group. New stubs Chevalley group and Sigurdur Helgason.

      By the way, when writing tangent Lie algebra, I had the problem with finding the correct font for the standard symbol of Lie algebra of vector fields on a manifold. Usually one has varchi symbol which looks like Greek chi but with dash through middle. The varchi symbol is not recognized and I put mathcal X which is slanted and script, just alike, but without dash through middle.

      By the way, on a real Lie group GG of dimension nn, if one expresses the right invariant vector field in terms of left invariant vector fields then at each point there is a ℝ\mathbb{R}-linear operator which sends any frame of left invariant vector fields to the corresponding frame in right invariant vector fields; this gives a GL n(ℝ)GL_n(\mathbb{R})-valued real analytic function on GG (or, in local coordinates, on a neighborhood of the unit element). In other words, if I take a frame in a Lie algebra and interpret it in two ways, as a frame of left invariant vector fields and a frame in right invariant vector fields, then I can take a matrix of real analytic functions on a Lie group and multiply the frame of left invariant vector fields with this matrix to get the correspoding frame of right invariant vector fields. I use in my current research some computations involving this matrix function. Does anybody know of any reference in literature which does any computations involving this matrix valued function on GG ?

    • A stub Massey product and a longer Toda bracket (still plenty gaps of reference, many many unlinked words). No promises w.r.t. spellings.

      I now see I’ve missed the convention for capitalization. Will fix that now… done.

      Cheers

    • I am hereby moving the following old Discussion box from interval object to here

      Urs Schreiber: this is really old discussion by now. We might want to start putting dates on discussions. In principle it can be seen from the entry history, but readers glancing at this here hardly will. Maybe discussions like this here are better had at the forum after all.

      We had originally started discussing the notion of interval objects at homotopy but then moved it to this entry here. The above entry grew out of the following discussion we had, together with discussion at Trimble n-category.

      Urs: Let me chat a bit about what I am looking for here. It seems very useful to have a good notion of what it means in a context like a closed category of fibrant objects to say that path objects are compatibly corepresented.

      By this should be meant: there exists an object II such that

      • for BB any other object, [I,B][I,B] is a path object;

      • and such that II has some structure and property which makes it “nice”.

      In something I am thinking about the main point of II being nice is that it can model compositon: it must be possible to put two intervals end-to-end and get an interval of twice the length. In some private notes here I suggest that:

      a “category with interval object” should be

      I think there are a bunch of obvious examples: all familiar models of higher groupoids (Kan complexes, ω\omega-groupoids etc.) with the interval object being the obvious cellular interval {a→≃c}\{a \stackrel{\simeq}{\to} c\}.

      I also describe one class of applications which I think this is needed/useful for: recall how Kenneth Brown in section 4 of his article on category of fibrant objects (see theorems recalled there and reference given there) describes fiber bundles in the abstract homotopy theory of a pointed category of fibrant objects. This is pretty restrictive. In order to describe things like ∞\infty-vector bundles in an context of enriched homotopy theory one must drop this assumption of the ambient category being pointed. The structure of it being a category with an interval object is just the necessary extra structure to still allow to talk of (principal and associated) fiber bundles in abstract homotopy theory. It seems.

      Comments are very welcome.

      Todd: The original “Trimblean” definition for weak nn-categories (I called them “flabby” nn-categories) crucially used the fact that in a nice category TopTop, we have a highly nontrivial TopTop-operad where the components have the form hom Top(I,I ∨n)\hom_{Top}(I, I^{\vee n}), where X∨YX \vee Y here denotes the cospan composite of two bipointed spaces (each seen as a cospan from the one-point space to itself), and the hom here is the internal hom between cospans.

      My comment is that the only thing that stops one from generalizing this to general (monoidal closed) model categories is that “usually” II doesn’t seem to be “nice” in your sense here, and so one doesn’t get an interesting (nontrivial) operad when my machine is applied to the interval object. But I’m generally on the lookout for this sort of thing, and would be very interested in hearing from others if they have interesting examples of this.

      to be continued in the next comment

    • I noticed that the text at loop space didn’t point to smooth loop space and didn’t make clear that such a variant might even exist. So I have now expanded the Idea-section there a little to give a better picture.

    • I have received a question on the old entry directed object, so I am looking at that now. First of all I’ll clean it up a bit and move old discussion from there to here:

      [begin forwarded discussion]

      +–{.query}

      Eric: I don’t fully “grok” this constructive definition, but I like its flavor. Is it possible to formalize the procedure in a simple catchy phrase? In other words, when you begin with a “category CC with interval object II”, but whose objects are otherwise undirected (like Top), you construct the “supercategory d ICd_I C” with directed CC-objected (even though no objects in CC are directed). I used the term “directed internalization”, but is there a better term?

      I just think this concept is important and should have some really slick arrow theoretic description and I’m not having any luck coming up with one myself.

      =–

      [ continued in next post ]

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