This is just a quick question which I hope will catch the attention of Noam in particular (but anyone is welcome to contribute).

For something I am working on in knot theory, I would like to appeal to a classical theorem of Whitney which says that any 3-connected planar graph has a unique embedding in the plane up to planar isotopy. It is not altogether easy to track down the original reference; indeed Noam asked about it here. I have seen at least two other papers cited (incorrectly) in the literature!

I agree with Noam that Theorem 11 in the paper he mentions seems the correct reference. However, what Whitney actually proves is that every 3-connected planar graph has a unique dual graph. Now, it seems to me that this is equivalent to having a unique embedding (again, up to planar isotopy). However, I have not seen this explicitly stated anywhere in the literature. Do people agree that this is the case, i.e. that unique dual implies unique embedding? If so, I will add some kind of entry to the nLab just to record this and the correct reference.

]]>I started finite ∞-group, and added that same reference to Sylow p-subgroup.

]]>This looks like an interesting program being rolled out by Clark Barwick and colleagues. So far there’s an introduction

- Parametrized higher category theory and higher algebra: A general introduction (with Emanuele Dotto, Saul Glasman, Denis Nardin, and Jay Shah). (pdf)

and the first Exposé

- Parametrized higher category theory and higher algebra: Exposé I – Elements of parametrized higher category theory (with Emanuele Dotto, Saul Glasman, Denis Nardin, and Jay Shah), (pdf)

It sounds like it’s worth a dedicated entry, so I’ll start one Parametrized Higher Category Theory and Higher Algebra. I guess we can then spin off uncapitalized entries for the concepts.

They want to “untether equivariant homotopy theory from dependence upon a group” by generalizing away from the orbit category of a group to ’atomic, orbital $\infty$-categories’.

]]>following a suggestion by Zoran, I have created a stub (nothing more) for Kuiper’s theorem

]]>At first Zoran's reply to my query at [[structured (infinity,1)-topos]] sounded as though he were saying "being idempotent-complete" were a structure on an (oo,1)-category rather than just a property of it. That had me worried for a while. It looks, though, like what he meant is that "being idempotent" is structure rather than a property, and that makes perfect sense. So I created [[idempotent complete (infinity,1)-category]].

]]>Stack entry says: "The notion of stack is the one-step vertical categorification of a sheaf." In Grothendieck's main works, like pursuing stacks and in the following works of French schools, stack is any-times categorification of a sheaf, and the one-step case is called more specifically 1-stack. We can talk thus about stack in narrow sense or 1-stacks and stacks in wider sense as n-stacks for all n. Topos literature mainly means that the stack is the same as internal 1-stack.

]]>I made some very minor changes to the introduction at descent. I hesitate to do more but at present the discussion does not seem that readable to me. Can someone look at it to see what they think? The intro seems to plunge in deep very quickly and so the ‘idea’ of descent as that of gluing local information together, does not come across to me. The article is lso quite long and perhaps needs splitting up a bit.

]]>Examples are sparse at closed monoidal (infinity,1)-category. What would be good examples to add? Perhaps the stable (infinity,1)-category of spectra, although that doesn’t mention closedness.

]]>created univalence axiom

]]>After sitting on this for days and hardly doing a thing, I added some applications to [[distribution]] and added a bit to the section on synthetic differential geometry. While I was dawdling, Andrew Stacey stepped in and added to some parts that needed expert attention -- thanks, Andrew.

]]>Older nLabians may or may not recall that the original URL of the nForum was nforum.mathforge.org, and that at times it was useful to have a back-up hostname for the nlab itself which was nlab.mathforge.org. As I still rent the mathforge.org domain, I’ve kept these pointing at the nforum and nlab to avoid dead links.

I’m in the process of doing a bit of housekeeping at mathforge, and in so doing I’m updating the nforum.mathforge.org and nlab.mathforge.org DNS entries so that they are *CNAME* not *A* entries. (If that means nothing to you, you probably should have stopped reading a while ago.) This is what they should have been originally: cname is like a symbolic link and means that if ncatlab.org moves then these will follow it.

For the record, once DNS hosts are updated, then they will alias as follows:

`nforum.mathforge.org`

to`nforum.ncatlab.org`

`nlab.mathforge.org`

to`ncatlab.org`

If they should be otherwise, let me know.

I’m pleased to see that the nLab/nForum are still going strong!

]]>I have started to add some of the basic definitions and facts to *Schwartz space*, *tempered distribution* and *Fourier transform of distributions*.

Notice that we had an entry titled “Schwartz space” already since May 2013 (rev 1 by Andrew Stacey) which considered not spaces of smooth functions with rapidly decreasing derivatives, but locally convex TVSs $E$ “with the property that whenever $U$ is an absolutely convex neighbourhood of $0$ then it contains another, say $V$, such that $U$ maps to a precompact set in the normed vector space $E_V$.”

I had not been aware of this use of “Schwartz space” before, and Andrew gave neither reference nor discussion of the evident question, whether “the” Schwartz space is “a” Schwartz space. In June 2015 somebody saw our entry and shared his confusion about this point on Maths.SE here, with no reply so far.

I see that this other use of “Schwartz space” appears in Terzioglu 69 (web) where it is attributed to Grothendieck.

]]>Created [[extranatural transformation]] by moving the relevant information from [[dinatural transformation]] and adding the definition. Disagreements are welcome, but I feel that since dinaturals that aren't extranatural are so rare and harder to deal with and understand, extranaturals merit their own page.

]]>New entry distribution of subspaces and a disambiguating remark at distribution.

]]>I have been adding some more (historical) references to the entry *quantum electrodynamics* (also at *quantum field theory*, *S-matrix* and *causal propagator*)

have been writing some Idea-section at *causal perturbation theory*

(currently this has much overlap with *locally covariant perturbative quantum field theory*, eventially the latter will contain more stuff)

Lavau extends it by a sequence of graded vector bundles `resolving' F

but was it written anywhere before Lavau ]]>

This used to be a side-remark at *wave front set*, I have given it its own entry: *Paley-Wiener-Schwartz theorem*.

Stephan Alexander Spahn has created descent object, with some definitions from Street’s *Categorical and combinatorial aspects of descent theory*.

If I get the opportunity this weekend I’ll add details from Street’s *Correction to ’Fibrations in bicategories’* and Lack’s *Codescent objects and coherence*. Anyone know of any other references?

Looking at Street’s paper again, what he describes as the ’$n=0$’ case of codescent objects looks to be just the notion of a coequalizer. I would have expected reflexive coequalizers, though, because the higher-$n$ case uses $n+2$-truncated simplicial objects. Is there a reason for this?

]]>[New thread because, although it existed since 2012, pasting scheme appears not to have had a LatestChanges thread]

Started to expand pasting schemes. Intend to do more on this soon, in an integrated fashion with digraph and planar graph.

PLEASE note: ACCIDENTALLY a page pasting schemes was created too, as a result of some arcane issues with pluralized names of pages-still-empty. Please delete pasting schemes.

]]>created *pullback of a distribution*, just for completeness

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.

]]>brief note on *Whitney extension theorem*

added an Idea-section to *Mackey functor* (which used to be just a list of references). Also added more references.

I’ve started running Firefox 57 in the Nightly distribution. The only difference I’ve noticed with nLab/nForum/nCafe is that the Stylish extension no longer works however Stylus seems fine as a replacement.

I’ve put the 2 styles I use on userstyles.org in case anybody else wants them:

If anybody has personal styles note that they should be exported before upgrading to Firefox 57. Here are Instructions on how to export from Stylish and import to Stylus.

(I’m not sure why userstyles is not showing the screenshot of small comment nesting but here is a direct link)

These user styles wouldn’t be necessary if the CSS of these sites were fixed to my preferences :}

]]>