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    • 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 3ω 2pdeg=2p,f 2p+4deg=2p+4|p d:{ω 0 0 ω 2p+2 h 3ω 2p f 4 0 f 2p+6 h 3f 2p+4 S 0 S 3 Sym h 3ω 2pdeg=2p,f 2pdeg=2p|p d:{ω 0 0 ω 2p+2 h 3ω 2p f 0 0 f 2p+2 h 3f 2p S 4//S 1 S 3 Sym h 3,f 2ω 2pdeg=2p,f 2p+4deg=2p+4|p d:{ω 0 0 ω 2p+2 h 3ω 2p f 2 0 f 2p+4 h 3f 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. }
    • I have given Grothendieck construction for model categories its own entry, in order to have a place for recording references. In particular I added pointer to the original references (Roig 94, Stanculescu 12)

      (There used to be two places in the entry Grothendieck construction where an attempt was made to list the literature on the model category version, but they didn’t coincide and were both inclomplete. So I have replaced them with pointers to the new entry.)

    • I would like to include something on wheeled properads (or wheeled PROPs) in the nlab. It seems to me that a wheeled prop is something like a symmetric monoidal category with duals for every object generated by one object. Is this right? Is there a place in the litterature where i can find the relation between wheeled properads used by Merkulov and some kinds of symmetric monoidal categories with duality?

      Before changing the PROP entry to add this variant, i would like to have a nice reference on this.
    • added statement of and references for some of the homotopy groups of E 8E_8 to E8

    • At the old entry cohomotopy used to be a section on how it may be thought of as a special case of non-abelian cohomology. While I (still) think this is an excellent point to highlight, re-reading this old paragraph now made me feel that it was rather clumsily expressed. Therefore I have rewritten (and shortened) it, now the third paragraph of the Idea-section.

      (We had had long discussion about this entry back in the days, but it must have been before we switched to nForum discussion, because on the nForum there seems to be no trace of it.)

    • started a bare minimum at Poisson-Lie T-duality, for the moment just so as to have a place to record the two original references

    • 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.

    • Todd,

      when you see this here and have a minute, would you mind having a look at monoidal category to see if you can remove the query-box discussion there and maybe replace it by some crisp statement?


    • 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 principle of equivalence I have restructured the Examples-section: added new subsections in “In physics” on gauge transformations and on general covariance (just pointers so far, no text), and then I moved the section that used to be called “In quantum mechanics” to “Examples-In category theory” and renamed it to “In the definition of \dagger-categories” (for that is really what these paragraphs discuss, not any notion of equivalence in quantum mechanics, the application of \dagger-categories in that context notwithstanding)

    • pure morphism (much more to be said, and more references, but no time now)

    • an entry for mere proposition had been missing. Created a minimum, just so as to satisfy links.

    • Detection of the 21cm hydrogen absorption line expected in the CMB has been claimed now. Such a detection is thought to have implications for observational cosmology comparable in relevance to those of the recent gravitational wave detection. I have collected some original articles and reviews here in an otherwise empty entry hydrogen line

    • 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?

    • Added some remarks, mostly about extensivity and exactness, to quasitopos.

    • I started comma double category. Since I care about equipments more than double categories in general, and because it actually is an instance of a comma object, I made the article mostly about virtual double categories. I wrote down a couple of conjectures about when the comma has units and composites, but haven’t verified them yet and not sure when I will.