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polished a bit and expanded a bit at interval category (nothing deep, just so that it looks better)
Todd is helping me understand opposite categories beginning with 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 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.
started essential geometric morphism
the topic-cluster “algebra” is stiil a little orphaned on the Lab, I find. I added a bit to
but these entries are still a bit pitiful. Maybe somebody feels challenged to turn them into good entries. Especially the latter.
Should we have pages algebra over a Lawvere theory, algebra over a PROP, etc? Some entries look like they want such links (for instance algebra itself).
expanded object
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.
I edited the entry enriched category theory a bit, in an attempt to eventually bring that into decent shape.
I think we should eventually expand the list of related entries there and make it a floating toc.
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.
created stub for circle bundle
proudly presenting the circle group ;-)
created concrete site
Created generalized kernel, with links from a lot of places.
split off smooth (infinity,1)-algebra from derived smooth manifold and (infinity,1)-algebraic theory
added more theorems to Cartesian fibration and polished the intro slightly
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.
created sub-quasi-category
I created cograph of a profunctor and added some references to it at cograph of a functor. All the cograph pages could probably use some unifying work.
split off inner fibration from fibrations of quasi-categories and added remarks on how it models the (oo,1)-version of the notion of cograph of a profunctor
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 :)).
added a section about how to compute limits and colimits in (infinity,1)Cat and in Infinity-Grpd in terms of coCartesian fibrations to these entries.
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.
Sharpened up some of the discussion at finitary monad (emphasizing equivalence with Lawvere theories), and added some technical applications to reflexive coequalizer. Both were used to support a detailed proof embedded at smooth algebra.
expanded the long-time stub entry (n,1)-topos a little more. Made Mike’s former query box an Example-section.