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    • Regarding that the nlabizens have discussed so much various generalizations of Grothendieck topology, maybe somebody knows which terminology is convenient for the setup of covers of abelian categories by finite conservative families of flat localizations functors, or more generally by finite conservative families of flat (additive) functors. Namely the localizations functors do not mutually commute so the descent data are more complicated but if you produce the comonad from a cover then the descent data are nothing but the comodules over the comonad on the product of the categories which cover. In noncommutative geometry we often deal with stacks in this generalization of topology and use ad hoc language, say for cocycles, but the thing is essentially very simple and the language barier should be overcome. There are more general and ore elaborate theories of nc stacks, but this picture is the simplest possible.

    • stub for crystalline cohomology

      There are notes by Jacob Lurie on crystals, but I forget where to find them. Does anyone have the link?

    • I got the book “Counterexamples in Topological Vector Spaces” out of our library, and just the sheer number of them made me realise that my goal of getting the poset of properties to be a lattice would produce a horrendous diagram. So I’ve gone for a more modest aim, that of trying to convey a little more information than the original diagram.

      Unfortunately, the nLab isn’t displaying the current diagram, though the original one displays just fine and on my own instiki installation then it also displays just fine so I’m not sure what’s going on there. Until I figure that out, you can see it here. The source code is in the nLab: second lctvs diagram dot source.

      A little explanation of the design:

      1. Abbreviate all the nodes to make the diagram more compact (with a key by the side, and tooltips to display the proper title).
      2. Added some properties: LF spaces, LB spaces, Ptak spaces, B rB_r spaces
      3. Taken out some properties: I took out those that seemed “merely” topological in flavour: paracompactness, separable, normal. I’m pondering taking out completeness and sequential completeness as well.
      4. Tried to classify the different properties. I picked three main categories: Size, Completeness, Duality. By “Size”, I mean “How close to a Banach space?”.

      (It seems that Instiki’s SVG support has … temporarily … broken. I’ll email Jacques.)

    • started at infinity-Lie groupoid a section The (oo,1)-topos on CartSp.

      Currently this gives statement and proof of the assertion that for a smooth manifold regarded as an object of sPSh(CartSp) proj,covsPSh(CartSp)_{proj,cov} the Cech nerve of a good open cover provides a cofibrant replacement.

    • 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)P_{fin}(X) denote the poset of finite nonempty subsets of XX. A simplicial complex consists of a set VV and a down-closed subset Σ⊆P fin(V)\Sigma \subseteq P_{fin}(V) such that every singleton {v}\{v\} belongs to Σ\Sigma. Thus Σ\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,…,x n}\{x_1, x_2, \ldots, x_n\} for which we have a strictly increasing chain x 1<x 2<…<x nx_1 \lt x_2 \lt \ldots \lt x_n. Then, the next step would use the following construction: given a simplicial set XX, construct the poset whose elements are nondegenerate simplices (elements) of XX, ordered x<yx \lt y if xx is some face of yy. The claim would be that the realization of XX 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.

    • started rational homotopy theory in an (infinity,1)-topos

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

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

    • 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 ;)

      Thus I’ve updated the links at David Roberts, the above linked page, and on my private web home page. If anyone knows of any other places it is linked, please let me know, or update the link to point to DMRthesis_final.pdf, instead of DMR_thesis.pdf.

      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)(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 opFinSet^{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 FinSetFinSet 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

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