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• I added a section on triangulable spaces and PL structures to simplicial complex, but this is the type of thing which gets beyond my ken pretty quickly. My real motivation is to convince myself that a space is homeomorphic to the realization of a simplicial complex (in short, is triangulable) if and only if it is homeomorphic to the realization of a simplicial set – perhaps this seems intuitively obvious, but it should be given a careful proof, and I want such a proof to have a home in the Lab. (Tim Porter said in a related discussion that there was a relevant article by Curtis in some early issue of Adv. Math., but I am not near a university library to investigate this.)

I’ll put down some preliminary discussion here. Let $P_{fin}(X)$ denote the poset of finite nonempty subsets of $X$. A simplicial complex consists of a set $V$ and a down-closed subset $\Sigma \subseteq P_{fin}(V)$ such that every singleton $\{v\}$ belongs to $\Sigma$. Thus $\Sigma$ is itself a poset, and we can take its nerve as a simplicial set. The first claim is that the realization of this nerve is homeomorphic to the realization of the simplicial complex. This I believe is or should be a basic result in the technique of subdivision. Hence realizations of simplicial sets subsume triangulable spaces.

For the other (harder) direction, showing that realizations of simplicial sets are triangulable, I want a lemma: that the realization of a nerve of a poset is triangulable. Basically the idea is that we use the simplicial complex whose vertices are elements of the poset and whose simplices are subsets $\{x_1, x_2, \ldots, x_n\}$ for which we have a strictly increasing chain $x_1 \lt x_2 \lt \ldots \lt x_n$. Then, the next step would use the following construction: given a simplicial set $X$, construct the poset whose elements are nondegenerate simplices (elements) of $X$, ordered $x \lt y$ if $x$ is some face of $y$. The claim would be that the realization of $X$ is homeomorphic to the realization of the nerve of this poset.

All of this could very well be completely standard, but it’s hard for me to find an account of this in one place. Alternatively, my intuitions might be wrong here. Or, perhaps I’m going about it in a clumsy way.

• So, I have some pending changes on operad that I made in the sandbox and am waiting for a go-ahead to post from the interested parties, but I was also wondering if someone would be willing to write up a follow-up to the very nice definition of an operad as a monoid in the blah blah monoidal category. That is, it seems like this should give us a very nice way to define an algebra, but I don't know how one would actually go about doing it.

• rearranged a bit and expanded category theory - contents. In particular I added a list with central theorems of category theory.

• added Eric’s illustrations to the Idea-section at representable presheaf. Also added a stub-section on Definition in higher category theory.

• Added complete topological vector space including various variants (quasi-complete, sequentially complete, and some others). Hopefully got all the redirects right!

I only have Schaefer’s book at home so couldn’t check “locally complete” - I know that Jarchow deals with this in his book. Kriegl and Michor naturally only consider it in the context of smootheology so I’m not sure what the “best” characterisation is. There’s also a notational conflict with “convenient” versus “locally complete”. As Greg Kuperberg pointed out, in some places “convenient” means “locally complete and bornological” whereas in others it means just “locally complete” (in the contexts where convenient is used the distinction is immaterial as the topology is not considered an integral part of the structure).

I added these whilst working on the expansion of the TVS relationships diagram. That brought up a question on terminology. In the diagram, we have entries “Banach space” and “Hilbert space” (and “normed space” and “inner product space”). These don’t quite work, though, as for a topological vector space the correct notion of a normed space should be normable space as the actual choice of norm is immaterial for the TVS properties. I’m wondering whether or not this is something to worry about. Here’s an example of where it may be an issue: a nuclear Banach space is automatically finite dimensional. That implies that its topology can be given by a Hilbert structure. However, the Hilbertian norm may not be the one that was first thought of. But that’s a subtlety that’s tricky to convey on a simple diagram. So I’d rather have “normable” than “normed”. Does anyone else have an opinion on this?

If “normable” is fine, then the important question is: what’s a better way of saying “Hilbertisable”, or “Banachable”? Length doesn’t matter here, as I’m putting the expanded names in tooltips and only using abbreviations in the diagram.

• With just slightly more it could also be called "Lie theory in an oo,1-topos" I suppose.

if you looked at this yesterday, as it was under construction, maybe have another look: I believe I could clarify the global story a bit better.

• Looking at the entry Banach spaces, I found the following in the introduction:

So every $n$-dimensional real Banach space may be described (up to linear isometry, the usual sort of isomorphism) as the Cartesian space $\mathbb{R}^n$ equipped with the $p$-norm for $1 \leq p \leq \infty$

which seems to imply that every norm on a finite dimensional Banach space is a $p$-norm for some $p$. That feels to me like a load of dingo’s kidneys. To define a norm on some $\mathbb{R}^n$ I just need a nice convex set, and there’s lots more of these than the balls of $p$-norms, surely.

Am I missing something?

• Moonshine, intentionally with capital M as most people do follow this convention for the Monster and (Monstrous) Moonshine VOA.

• somebody signing as “Anonymous Coward” had created special relativity and typed in a confused paragraph (the smallest confusion being that the paragraph concerned not special but general relativity).

I removed that paragraph and quickly wrote a brief “Idea”-section . But have no time to do this justice now.

• It got announced in another category, but here it is in Latest Changes:

Todd began (and then I edited) simple group.

• I wrote a quick entry conformal group, just from memory. Somebody could check and expand. In fact it would not be bad to have also a separate entry on conformal and on quasiconformal mappings.

• somehow I missed that there already is a page compact operator and created compact operators. The plural is another error :-) the unsatisfied link that I used to create the page was “compact operators”. When I tried to rename it to the singular term it failed, of course. Now the page compact operators is simply superfluous, but as a non-administrator I cannot delete it…

• Created sequential compactness, should probably link to all these compactness variations from compact space. Not sure if I got the “iff” bit right in the relationship with compactness itself; will check it myself if no-one fixes it in the meantime.

I decided that this was the key property in manifolds of mapping spaces and to stop trying to figure out a Froelicher version of sequentially compact for the time-being.

• the term “twist” or “twisted” is one of those insanely ambiguous terms in math. Trying to follow our recent agreement on how to deal with ambiguous page names, I tried to indicate this at twist .

• Created lax-idempotent 2-monad, with some definitions from Kelly–Lack. I think Kock has a couple of others. I’ll add more, like proofs of the equivalence of the definitions, and more on the cocompletion example, later (next week, probably).

• The final copy of my thesis is up on the lab. Available from Fundamental Bigroupoids and 2-Covering Spaces. I’ve fixed the typo in definition 5.1 that made it into the print copy ;)

Now to all the other projects that are on the back burner, time permitting…

• started a floating toc for topos theory. See at the right of topos.

Please feel encouraged to expand and improve the structure.

• finally noticed that (infinity,1)-sheaf was hardly even a stub. Have now filled some genuine content in there.

• Created free monad with a discussion of some of the subtleties and the notion of “algebraically-free”.

• I’ve started porting my notes “differential topology of loop spaces” over to the nlab, starting at differential topology of mapping spaces. As part of the transfer, I intend to map out the theory for general mapping spaces, not just loop spaces (that’ll give me a bit more motivation to do the transfer since just cut-and-paste is boring!). I’ve just copied over the contents and the introduction so far and haven’t edited them as yet. In particular, although I’ve wikilinked all the original section names, these will get changed as they currently focus on loop spaces.

The introduction to the original document ended as follows (not copied over to the new version):

This document began life as notes from talks given at NTNU and at Sheffield so I would like to thank the topologists at those institutions, and in particular Nils Baas, for letting me talk about my favourite mathematical subject. I would also like to thank Ralph Cohen and the “loop group” at Stanford.

This is by no means a ﬁnished document, as an example it is somewhat sparse on references. Any comments, suggestions, and constructive criticism will be welcomed.

The second paragraph is sort-of stating the obvious as it holds to some extent for any nLab page! And I would love to be able to add some more names to the list in the first paragraph. Again, I hope it goes without saying but I’ll say it anyway: although I anticipate being the main contributor to these pages, it is not my project! I would love it if people read it, add comments, add other stuff, write (constructive) graffiti, link it to other stuff.

• The entry cover was in a pitiful state. I tried to brush it up a bit. But I am afraid I am still not doing it justice. But also I don’t quite have the leisure for a good exposition right now. What I really want is to create an entry good cover in a moment…

• stub for Sullivan construction (I got annoyed that the entry didn’t exist, but also don’t feel like doing it justice right now)

• Because I want to point to it in a reply to the current discussion on the Category Theory Mailing list, I tried to brush up the entry k-tuply monoidal n-category a bit.

In particular I

• expanded the Idea section and added some statements that had been missing there;

• reacted to the old query box discussion there and moved the query box to the very bottom;

• added a section on k-tuply monoidal $\infty$-groupoids and $\infty$-stacks here.

• added a section on k-tuply monoidal $(n,1)$-categories here

• I had started an article on AT category (which I originally mis-titled as “AT categories” – thank you Toby for fixing this!), but getting a little stuck here and there. I’m using the exchange between Freyd and Pratt on the categories mailing list (what else is there?) as my reference, but as is so often the case, Freyd’s discussion is a little too snappy and terse for me to follow it down to all the nitty-gritty details.

There’s a minor point I’m having trouble verifying: that coproducts are disjoint (as a consequence of the AT axioms that Freyd had enunciated thus far where he made that claim, in his main post), particularly that the coprojections are monic. Presumably this isn’t too hard and I’m just being dense. A slightly less than minor point: I’m having trouble verifying Ab-enrichment of the category of type A objects. I believe Freyd as abelian-categories-guru implicitly – I don’t doubt him. Can anyone help?

• Todd is helping me understand opposite categories beginning with $FinSet^{op}$ here.

This discussion helped prompt some improvement of the page opposite category. When I look at that page now, I see the statement:

The idea of noncommutative geometry is essentially to define a category of spaces as the opposite category of a category of algebras.

This reminded me of a remark I made in the “Forward” to a paper I wrote back in 2002, so I’ve now itexified that “Foreward” here:

Noncommutative Geometry and Stochastic Calculus

By the way, this also suggests that the category $FinSet$ is the category of spaces opposite to the category of finite Boolean algebras in the sense of space and quantity.

• I noticed that recently Konrad Waldorf created a very nice article

I went through it and added definition/theorem/proof-environments and lots of hyperlinks. Some of them are unsaturated. Maybe somebody feels inspired to create corresponding entries.

• added the cosimplicial version of the statement to Eilenberg-Zilber theorem and included a reference that gives a proof

• Eric wanted to know about closed functors, so we started a page. Probably somebody has written about these before, so references would be nice, if anybody knows them. (Google gives some hits that look promising, but I can’t read them now.)

• I’ve done a bit of housekeeping at Froelicher space. I’ve split the page into pieces, putting each major section into its own section.

(This will necessitate a little reference chasing at manifolds of mapping spaces, and I need to put in some redirects)

I’ve put in a definition of curvaceous compactness at topological notions of Frölicher spaces. It works, but I’m not sure if it’s the right one yet.

• It seems to me that despite so lenghty discussions and entry related to the mapping space-hm adjunction, only the ideal situations are treated (convenient categories of spaces). For this reason, I have created a new entry exponential law for spaces containing the conditions usually used in the category of ALL topological spaces, as well as few remarks about the pointed spaces.

• Taking the advice that if I write something on the internet, it should be stuck on the n-Lab, I've converted my recent comments in the n-category cafe and some old blog posts into a new page on the relationship between categorification and groupoidification: categorification via groupoid schemes

• Split off the mapping spaces stuff from local addition into manifolds of mapping spaces. Still plenty to do and things to check (particularly on the linear stuff, and particularly figuring out what “compact” means). Haven’t actually deleted the relevant bit from local addition yet. Also, haven’t put a table of contents at manifolds of mapping spaces since I’ve learnt that the best way to get Urs to read something is to not put a toc in.

• Casson invariant count SU(2) local systems of integral homology spheres. Thomas considered its holomorphic counterpart which is ultimately related to counting BPS states on Calabi-Yau 3-folds.

P.S. Hmm. Tags. New option. Great. Is there a list of tags ?

• I am in a small wave of activity along one of my principal lonegr goals in nlab: the connection between the operator theory and geometry. This is of extreme importance for physics if we ever want to go beyond the TQFTs in quantization program. As Tom Leinster has in his work seen, there are heat-kernel like expansions involved all around the place even when one takes categorical approach and the first terms are of topological nature. This is exactly so in the WKB-expansions where the zeroth term is often the exact value for topological or more general integrable models. Witten's calculation of Witten's index (related to tmf) is an example where such WKB aprpoximations are evaluated and in presence of supersymmetry there are no other terms. Thus I believe that the kantization preferred in nlab is limited to work exactly in simiklar cases and that in general we will have more terms of WKB-like nature involved. We need to develop a categorified WKB method which will unify both.

On the other hand, the WKB method is not just expansion like in quantum mechanics books, it does involve cocycles right away in usual symplectic geometry. There is so-called Maslov index related to the multidimensional WKB method which has been pioneered by V. Maslov. The quantity which is slowly changing is an analogue of the eikonal in geometric optics, so the whole thing is a generalization of the geometric optics approximation. One can see some aspects of that on (free online, on the AMS web site, under books, here) book on symplectic geometry by Guillemin and Sternberg.

Harmonic side of the stationary phase approximation (which is just a variant of WKB in fact) is studied for long under the name oscillatory integrals. This is studied especially by Lars Hörmander and the Japanese school of microlocal analysis (btw, that one is the number 3000 entry in nlab!); the differential equations are often decribed via D-modules and in nonlinear case D-schemes which Gorthendieck described via crystals.

Strangely enough Kashiwara who worked much in microlocal analysis and D-modules has created a notion of crystal bases and crystals of quantum groups but these are NOT related to crystals. Thus I created crystal basis to fix the opinions in the nlab before they go astray...

I created entry hyperfunction, one of the tools of microanalysis, by Japanese school, a neat version of generalized functions, more flexible than distributions of Laurent Schwarz. They are obtained as boundary values of holomorphic functions (reminds me of BV formalism :)).

• I added a bit in the functionals section of locally convex space about coordinate projections being continuous for LCTVSs, and that there are counterexamples to this fact without local convexity. This was from memory, I hope I got it right.

I hope it’s not a fluke that I can edit from home tonight.

I also hyperlinked my front page of my web a bit, as Urs does (like it’s going out of fashion :), so I can present our model to my company, as I (and some others to whom I have explained it) would like to implement the ’open lab book’ research model we have here. I would loove to be able to do it in instiki (by which I mean the technically minded people), but we may be stuck with an awful free wiki platform, chosen for its ’minimal advertising’ (and I quote!).

Anyway, as a result, there are a bunch of new stubby pages there, that probably aren’t worth looking at yet.

• expanded the long-time stub entry (n,1)-topos a little more. Made Mike’s former query box an Example-section.

• Mike left a query box over at structured (infinity,1)-topos about admissibility structures. I am pretty sure that the admissibility structure is not, as the statement in the article says, a grothendieck topology. Rather, it is a class of morphisms that is in some way compatible with the grothendieck topology. At least looking at Toën's notes (which it seems are essentially a version of HAG II restricted to ordinary categories and ordinary stacks (I'm not positive that this is fully accurate, but I'm reasonably confident in the statement)), a geometric structure is a class of morphisms that is compatible with the grothendieck topology satisfying a number of conditions (that seem to match the axioms for an admissibility structure given here!). Correct me if I'm wrong, but it appears that an admissibility structure is precisely the class of morphisms P in the definition of a geometric context (or maybe even the pair (τ,P)).

Here's the link. Anyway, if this is true, it appears to answer Mike's question (once suitably generalized to (∞,1)-categories).

If I'm mistaken, please let me know.

I've put this in the (Latest Changes) category because at the moment, there is no nLab general category.