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    • 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

    • The observation that the conditional expectation enjoys a universal property inspired me to write some ''random text''.

    • At field (physics) I am beginning to write an actual introduction to the topic, now in a new section titled “A first idea of quantum fields”.

      This means to introduce the concept with precise detail, but in a simple context (trivial and bosonic field bundles over Minkowski spacetime, perturbatively quantized) that allows to get a quick idea of the idea of the concept of (quantum) fields as such, without being distracted by other details.

      So far I made it up to the derivation of the EOMs. Discussion of (deformation) quantization is to follow (maybe by tonight, depending on how much trouble I have with the trains) and I plan to sprinkle in the detailed example from scalar field in parallel with the abstract discussion.

    • At dichotomy between nice objects and nice categories I added a quote from Deligne about allowing awful schemes gave a nice category of schemes. I can’t find the page I was thinking of where this dichotomy is also mentioned along with attribution of the general idea to Grothendieck. I wanted to add it there as well.

    • I created a stub entry for Hörmander topology, just to record some references.

      The following seems to be waiting for somebody to answer it:

      Consider the deformed Minkowski metric

      η ε≔diag(−1+iε,1+iε,…,1+iε) \eta_\epsilon \coloneqq diag( -1 + i \epsilon, 1+ i \epsilon , \dots , 1 + i \epsilon )

      for ε>0∈ℝ\epsilon \gt 0 \in \mathbb{R}.Then consider the ε\epsilon-deformed Feynman propagator Δ F,ε,Λ\Delta_{F,\epsilon,\Lambda} with momentum cut off with scale Λ\Lambda.

      The question: does the limit satisfy

      Δ F=limε→0Λ→∞Δ F,ε,Λ \Delta_{F} \;=\; \underset{ {\epsilon \to 0} \atop {\Lambda \to \infty} }{\lim} \Delta_{F,\epsilon,\Lambda}

      in the Hörmander topology for tempered distributions with wave front set contained in that of the genuine Feynman propagator Δ F\Delta_F?

    • for ease of linking I have given antibracket its own little entry (it used to just redirect to BV-BRST complex).

      I had also given local antibracket an little entry of its own. Possibly these two should be merged…

    • have created extension of distributions with the statement of the characterization of the space of point-extensions of distributions of finite degree of divergence: here

      This space is what gets identified as the space of renormalization freedom (counter-terms) in the formalization of perturbative renormalization of QFT in the approach of “causal perturbation theory”. Accordingly, the references for the theorem, as far as I am aware, are from the mathematical physics literature, going back to Epstein-Glaser 73. But the statement as such stands independently of its application to QFT, is fairly elementary and clearly of interest in itself. If anyone knows reference in the pure mathematics literature (earlier or independent or with more general statements that easily reduce to this one), please let me know.

    • I created Bishop’s constructive mathematics by moving some material from Errett Bishop and adding some more discussion of what it is and isn’t. Comments and suggestions are very welcome; I’m still trying to figure out the best way to describe the relationship of this theory to other things like topos logic.

    • I added some references to convex space and began a new entry on homomorphism.

      It would be great to see the article on convex spaces continue... it sort of trails off now. I've tried to enlist Tobias Fritz.

    • This is probably a request for Todd!

      Over on colimits for categories of algebras there’s a corollary I really need right now, about Eilerberg-Moore categories being cocomplete, and the remark:

      The hypotheses of the preceding corollary hold when CC is a complete, cocomplete, cartesian closed category and CC is the monad corresponding to a finitary algebraic theory.

      That sounds like exactly what I want, but when I click on finitary algebraic theory I get taken to a page that doesn’t have the definition of “finitary algebraic theory”. I think I know what this means, so I could guess and stick it in, but I think I should let the expert do it.

      Oh, whoops! - as usual, I actually need a multi-sorted generalization. But still it would be nice to have this clarified.

    • I created locally compact groupoid with an attempt at a very general definition, that will be refined to connect with the notion in the literature. In particular, if one has a system of Haar measures on the source (or target) fibres, then this will most likely place further constraints on the topological structure.

    • I have given multigraph its own little entry, so that I can point to it (from discussion of Feynman diagrams).

    • I have given finite graph its own minimum entry, just so as to be able to point to it

    • we didn’t have face

    • I have created an entry-for-inclusion titled

      which is one “Summary box” that means to give a lightning but accurate summary of the origin and meaning of Feynman diagrams in the rigorous description via causal perturbation theory.

      This makes use of a set of nicely done slides in Brouder 10; a citation is contained.

      I am meaning to include this as a Summary-box into relevant entries, such as Feynman diagram and renormalization.

    • I created a new page distributivity of products and colimits, where I recorded what I learned after asking this question: http://nforum.mathforge.org/discussion/6255/commutativity-of-homotopy-sifted-colimits-and-products-in-categories-other-than-sets-or-spaces/

    • I gave product of distributions its own entry. For the moment it just points to the definition in Hörmander’s book.

      This should eventually supercede the section “Multiplication of distributions” at distributions, which I find suboptimal: that section starts very vaguely referring to physics as if the issue only appears there, and it keeps being very vague, with its three sub-subsections being little more than a pointer to one reference by Colombeau.

      I suggest to

      1. remove that whole subsection at distribution and leave just a pointer to product of distributions

      2. move the mentioning of Colombeau’s reference to product of distributions and say how it relates to Hörmander’s definition

      3. remove all vague mentioning of application in physics and instead add a pointer to Wick algebra and microcausal functional, which I will create shortly.

    • started an entry interacting vacuum with some pointers. For instance to

      • Johann Rafelski, Vacuum structure – An Essay, in pages 1-29 of H. Fried, Berndt Müller (eds.) Vacuum Structure in Intense Fields, Plenum Press 1990 (GBooks)

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