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    • added missing cross-link with cyclic set.

      Some harmonizing might be necessary here, maybe entries should actually be merged.

      diff, v4, current

    • I made the definition of linear interval explicit at interval, also correspondingly streamlined slightly the corresponding text at classifying topos.

    • A stub on computer software Cadabra.

      v1, current

    • updated “counterxample” link from its ancient nForum location

      diff, v9, current

    • I noticed that Welcome to the nForum (nlabmeta) instructed users to make query boxes. I believe query boxes are now deprecated, so I removed that instruction.

    • For the discussion at Higgs field I created a category:people entry for Philip Gibbs, who was about the first to discuss the Higgs detection at LHC.

      diff, v1, current

    • There has been much attention in the nlab on groupoid cardinality of Leinster/Berger and Euler characteristics of a category of Leinster; and Baez's work with collaborators on groupoidification and his earlier talks and posts on cardinality. Urs noticed that it fits with Freed's ideas on Feynman path integrals and came up with Freed-Schreiber-Ulm "kantization" formulas. While this works well for some finite and TQFT situations, I would like to know what happens in general. Tom and Simon Willerton have been doing infinite extensions to metric spaces and heat-kernel like expressions were important there. This reminded me of the equivariant localization formulas of Atiyah-Bott, Duisterman-Heckmann, Witten and others which are used in a number of situations but also computing the first term in heat-like expansions for Feyman integrals. In nice examples, like WZNW model, TQFTs, Chern-Simons, the semiclassical expressions give exact result. That is why the "kantization" in its present version gievs a good result. But we should go beyond. Thus we should understand similar expansions from nPOV. So I started creating some elementary background entries (for a while) like semiclassical approximation and now something closer to topologically oriented people on the blog: Lefschetz trace point formula. Soon there will appear various related index formulas and equivariant index formulas.

      I should tell in advance: the usual Lefschetz formulas are for the traces for one mapping; the equivariant ones are for family index by elements of a group. So it is not a number but a numbered valued function on the group. Thus we are arriving to a character. How now about the case when the group is categorified and we have categorified traces ? In that case we should formulate an appropriate ellipticity notion for a complex of operators on 2-bundles, and come after a categorified index formula. And then to get the G-equivariant version for G a 2-group. Some good kantization formulas should come from index formulas of that kind. By transgression, of course, it should be related to ideas like index formula on loop space, like Witten's index theorem; and eventually also to elliptic cohomology. Right ?

      Edit: yet another thing are anomalies. We took some formulation of anomaly cancellation directly from geometric condition on equipping the space with a particular structure, which then boils down to lift and voila some (nonabelian) obstruction. But originally one looks at amplitudes in QFT, does various standard things to them like zeta function regularization and finds obstruction from there. This took some development in works of Alvarez-Gaume, Jackiw, Stora, Witten and so on, with the role of the geometry of determinant line bundle emphasised by Quillen, Atiyah-Singer, Freed...We did not really go to these origins, and we should I think.

    • mirror symmetry (needs more well chosen references, I am runnning out of time and will be busy next few days; there are hundereds of references available so we should choose important and/or well written ones)

    • Over in another thread, David Roberts asks for explanation of a bunch of terms in QFT (here).

      In reaction I have started a minimum of explanation for one more item in the list: hidden sector.

    • Edit to: Ángel Uranga by Urs Schreiber at 2018-04-01 01:00:59 UTC.

      Author comments:

      added pointer to string pheno textbook

    • Edit to: Luis Ibáñez by Urs Schreiber at 2018-04-01 01:00:18 UTC.

      Author comments:

      added pointer to string pheno textbook

    • Edit to: D6-brane by Urs Schreiber at 2018-04-01 00:53:41 UTC.

      Author comments:

      hyperlinked pointer to textbook by Ibaney-Uranga

    • Edit to: Taub-NUT space by Urs Schreiber at 2018-04-01 00:53:00 UTC.

      Author comments:

      hyperlinked pointer to textbook by Ibaney-Uranga

    • Page created: bullet cluster by Urs Schreiber at 2018-03-31 00:53:25 UTC.

      Author comments:

    • Page created: proof of the prime number theorem by Todd Trimble at 2018-03-31 00:45:16 UTC.

      Author comments:

      New page which gives the details of the proof of the prime number theorem, as extracted from Zagier (after Newman), but less compressed.

    • On the orthogonal group page, there is a table of π i(SO(n))\pi_i(SO(n)) for 1≤i≤121 \leq i \leq 12 and 2≤n≤122 \leq n \leq 12. The reference given is the Encyclopedic Dictionary of Mathematics which actually has a table which goes up to i=15i = 15 and n=17n = 17. In a comment to my last MO question, David Roberts suggested that I add these additional groups to the table. I am happy to do this, but as the person who created the table used the same resource, they made a decision to stop at i=12i = 12 and n=12n = 12; maybe there was a good reason for this.

      Should I expand the table or is it fine as is?

    • Added to simple group an example I was given on MO: the simple group of cardinality κ\kappa, given by taking the smallest normal subgroup of Aut(κ)Aut(\kappa) containing the 3-cycles. This is essentially the ’even’ permutations for an infinite set.

      This is nice, because I was trying to think of a simple group (or one with only small normal subgroups) with inaccessible cardinality, and some obvious tricks weren’t working.

    • I decided to make my first contributions to the nLab, so am following the request, and saying what I did here.

      I fixed a typo in the definition of composition of correspondences. It is now a little unclear, so I also started a new page, internal tensor product (the link was already there, just no page). I’ll try to add some details tomorrow when I have time.

    • Has anyone developed models for the homotopy theory of HℚH \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 ℚ\mathbb{Q} it should reduce to the standard model for the homotopy theory of rational chain complexes, hence equivalently that of HℚH \mathbb{Q}-module spectra.

      Is there any work on this?

    • I added to endomorphism the observation that in a cartesian monoidal category, if an internal endomorphism monoid End(c)End(c) exists and is commutative, then cc is subterminal.

    • Added the following to the page on the Gray tensor product:

      The Gray tensor product as the left Kan extension of a tensor product on the full subcategory Cu of 2Cat is on page 16 of

      Since I’m really new at this, feel very free to give advice or corrections, thanks, Keith

    • I have corrected and expanded my note (at 4-sphere: here) of the result of Roig-Saralegi 00, p. 2 on minimal rational dg-models of the following maps over S 3S^3

      S 4 ↓ S 3,AAS 4//S 1 ↓ S 3,AAS 0 ↓ S 3 \array{ S^4 \\ \downarrow \\ S^3 } \,,\phantom{AA} \array{ S^4//S^1 \\ \downarrow \\ S^3 } \,,\phantom{AA} \array{ S^0 \\ \downarrow \\ S^3 }

      induced from the “suspended Hopf action” of S 1S^1 on S 4S^4.

      My aim in extracting this is to rename the generators given in Roig-Saralegi 00, p. 2 such as to make their degrees and their pattern more manifest. I hope I got it right now:

      fibration vector space underlying minimal dg-model differential on minimal dg-model S 4 ↓ S 3 Sym •⟨h 3⟩⊗⟨ω 2p⏟deg=2p,f 2p+4⏟deg=2p+4|p∈ℕ⟩ d:{ω 0 ↦0 ω 2p+2 ↦h 3∧ω 2p f 4 ↦0 f 2p+6 ↦h 3∧f 2p+4 S 0 ↓ S 3 Sym •⟨h 3⟩⊗⟨ω 2p⏟deg=2p,f 2p⏟deg=2p|p∈ℕ⟩ d:{ω 0 ↦0 ω 2p+2 ↦h 3∧ω 2p f 0 ↦0 f 2p+2 ↦h 3∧f 2p S 4//S 1 ↓ S 3 Sym •⟨h 3,f 2⟩⊗⟨ω 2p⏟deg=2p,f 2p+4⏟deg=2p+4|p∈ℕ⟩ d:{ω 0 ↦0 ω 2p+2 ↦h 3∧ω 2p f 2 ↦0 f 2p+4 ↦h 3∧f 2p+2 \array{ \text{fibration} & \array{\text{vector space underlying} \\ \text{minimal dg-model}} & \array{ \text{differential on} \\ \text{minimal dg-model} } \\ \array{ S^4 \\ \downarrow \\ S^3 } & Sym^\bullet \langle h_3\rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \omega_{2p} }}, \underset{deg = 2p + 4}{ \underbrace{ f_{2p + 4} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \omega_0 & \mapsto 0 \\ \omega_{2p+2} &\mapsto h_3 \wedge \omega_{2p} \\ f_4 & \mapsto 0 \\ f_{2p+6} & \mapsto h_3 \wedge f_{2p + 4} \end{aligned} \right. \\ \array{ S^0 \\ \downarrow \\ S^3 } & Sym^\bullet \langle h_3\rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \omega_{2p} }}, \underset{ deg = 2p }{ \underbrace{ f_{2p} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \omega_0 & \mapsto 0 \\ \omega_{2p+2} &\mapsto h_3 \wedge \omega_{2p} \\ f_0 & \mapsto 0 \\ f_{2p+2} &\mapsto h_3 \wedge f_{2p} \end{aligned} \right. \\ \array{ S^4//S^1 \\ \downarrow \\ S^3 } & Sym^\bullet \langle h_3 , f_2 \rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \omega_{2p} }}, \underset{ deg =2p + 4 }{ \underbrace{ f_{2p + 4} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \omega_0 & \mapsto 0 \\ \omega_{2p+2} &\mapsto h_3 \wedge \omega_{2p} \\ f_2 & \mapsto 0 \\ f_{2p+4} & \mapsto h_3 \wedge f_{2p + 2} \end{aligned} \right. }

    • added to Massey product a paragraph on their relation to A ∞A_\infty-algebras and a bunch of references on that relation

    • created Cahiers topos.

      Do I understand correctly that this gadget is named after the journal that Dubuc’s original article appeared in? What a strange idea.

    • Created the bare minimum at limit spaces. I was surprised it was missing.

    • Currently, the page Tools for the advancement of objective logic says that the described paper discusses the “concrete particular”, but this paper does not discuss “concrete particular”.

      [For context, Lawvere regards “concrete particular” as a category error, as he has stated elsewhere—basically to Lawvere, the abstract and the concrete are two aspects of a universal/general concept, and not to be discussed at the level of particulars. The passage from the particular to the universal/general is marked by choosing a subclass of observables about the particular as definitive—this generates the abstract general, and thence the concrete generals. The connection between the particulars and the concrete general comes from the fact that each particular can be observed in the ways specified by the abstract general, and that the abstract generals embed as representables into the concrete generals.]

      To be precise, the 1994 paper talks about the particular, and then the abstract and concrete general.

      However, I haven’t edited the page yet because I believe that in order to keep the nlab internally consistent, one may need to also update the page abstract general, concrete general and concrete particular; however, I am aware of a significant discussion elsewhere in the nforum, and I see that maybe some consensus was already reached there in favor of the notion of the “concrete particular”.

      The latter page contains the nlab’s synthesis of these ideas which does not quite match Lawvere’s—which is generally fine, but I think we should be careful about attributing this interpretation to Lawvere himself (which we are in danger of doing in the page on the 1994 paper).

      Do you all have any thoughts on how to proceed?

    • at sober space the only class of examples mentioned are Hausdorff spaces. What’s a good class of non-Hausdorff sober spaces to add to the list?

    • Over in another thread, David Roberts asks for explanation of a bunch of terms in QFT (here).

      In further reaction I have started a minimum of explanation for one more item in the list: bottom-up and top-down model building.

    • I have tweaked the Idea-section at naturalness, and I added a pointer to the first decent discussion that I have seen: Clarke 17

    • I am running into the following simple question and am wondering if there is anything useful to be said.

      Let

      𝒜∈dgcAlg ℚ \mathcal{A} \in dgcAlg_\mathbb{Q}

      be a differential graded-commutative algebra in characteristic zero, whose underlying graded algebra is free graded-commutative on some graded vector space VV:

      𝒜=(Sym(V),d). \mathcal{A} = (Sym(V), d) \,.

      Consider an odd-graded element

      c∈𝒜 odd, c \in \mathcal{A}_{odd} \,,

      and write (c)(c) for the ideal it generates.

      In this situation I’d like to determine whether it is true that

      1. there is an inclusion 𝒜/(c)↪𝒜\mathcal{A}/(c) \hookrightarrow \mathcal{A};

      2. for every element ω∈𝒜\omega \in \mathcal{A} there is a decomposition

        ω=ω 0+cω 1 \omega = \omega_0 + c \omega_1

        for unique ω 0,ω 1∈𝒜/(c)↪𝒜\omega_0, \omega_1 \in \mathcal{A}/(c) \hookrightarrow \mathcal{A}.

      For example if c≠0∈V odd↪𝒜 odd↪𝒜c \neq 0 \in V_{odd} \hookrightarrow \mathcal{A}_{odd} \hookrightarrow \mathcal{A} is a generator, then these conditions are trivially true.

      On the other extreme, if cc is the product of an odd number >1\gt 1 of odd generators, then it is not true. For example if c=c 1c 2c 3c = c_1 c_2 c_3, with c 1,c 2,c 3∈V odd↪𝒜 oddc_1, c_2, c_3 \in V_{odd} \hookrightarrow \mathcal{A}_{odd}, then for instance c(1+c 1)=c(1+c 2)=cc (1 + c_1) = c (1 + c_2) = c and so the coefficient ω 1\omega_1 is not unique.

      Is there anything useful that one can say in general?

    • For purposes of linking, I had given an entry to decomposable differential form.

      In more general ℕ\mathbb{N}-graded-commutative algebras than just that of differential forms, is there any established terminology for

      0.\;\;\;\;\;0. elements that are sums of decomposables, i.e. sums of monomials in elements of degree 1?

      What I’d really need is terminology for:

      1. elements HH of degree n+1n+1 which split off at least one factor of degree 11, hence H=α⏟deg=1⋅βH = \underset{deg = 1}{\underbrace{ \alpha}} \cdot \beta;

      2. elements which are finite sums of these.

      Is there anything?

    • Here is another stub: Albert algebra.

      It would be nice to get a reference to clear up the number of (real) Albert algebras. John Baez's octonion paper, among other literature (including our Jordan algebra), takes it for granted that there is only one (which is true, over the complex numbers, but people are usually working over the real numbers). But John himself points out on a Wikipedia talk page that there are two (and that's what I followed).

    • We had a paragraph on split ocotnions buried in the entry composition algebra.

      In order to be able to link to it, I have given that paragraph its own entry, now split octonions. But this deserves to be expanded of course.

    • Over in another thread, David Roberts asks for explanation of a bunch of terms in QFT (here).

      Here I started a minimum of explanation for one item in the list: protection from quantum corrections.

    • at Bockstein homomorphism in the examples-section where it says

      B nU(1)≃B n+1ℤ \mathbf{B}^n U(1) \simeq \mathbf{B}^{n+1}\mathbb{Z}

      I have added the parenthetical remark

      (which is true in ambient contexts such as ETop∞GrpdETop\infty Grpd or Smooth∞GrpdSmooth \infty Grpd)

      Just to safe the reader from a common trap. Because it is not true in Top≃∞GrpdTop \simeq \infty Grpd. The problem is that in all traditional literature the crucial distinction between TopTop and ETop∞GrpdETop \infty Grpd (or similar) is often appealed to implicitly, but rarely explicitly. In Top≃∞GrpdTop \simeq \infty Grpd we have instead B nU(1)≃K(U(1),n)\mathbf{B}^n U(1) \simeq K(U(1), n).

    • if you have been looking at the logs you will have seen me work on this for a few days already, so I should say what I am doing:

      I am working on creating an entry twisted smooth cohomology in string theory . This is supposed to eventually serve as the set of notes for my lectures at the ESI Program K-Theory and Quantum Fields in the next weeks.

      This should probably sit on my personal web, and I can move it there eventually. But for the moment I am developing it as an nnLab entry because that saves me from prefixing every single wiki-link with

       nLab:

    • Took a stab at a general formulation of Poisson summation formula, although the class of functions to which it is supposed to apply wasn’t nailed down (yet).

      (Some of the ingredients of Tate’s thesis are currently on my mind.)

    • I have added the Fierz identities that give the S 2S^2-valued supercocycle in 5d here.

      Added this briefly also at Fierz identity: here

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