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    • I'd like to create an entry gauge fixing. the usual method I use to create new entries (typing their name in the search form) does not work, since search redirects me to examples for Lagrangian BV. now I'm trying to put a link here to see if this works (but I'm skeptic..). should it not work, how do I create the new entry?

      thanks

      edit: it works!!!! :-)

      I'll now start writing the entry

    • I am a bit unhappy with the present state of local system. This entry is lacking the good systematic nPOV story that would hold it together.

      (For instance at some point a local system is defined to be a locally constant sheaf with values in vector spaces . This is something a secret blogger would do, but not worthy of an nLab. Certainly that's in practice an important specia case, but still just a very special case).

      But I don*t just want to restructure the entry without getting some feedback first. So I added now at the very end a section

      A general picture

      To my mind this should become, with due comments not to scare the 0-category theorists away, the second section after a short and to-the-point Idea section. The current Idea section is too long (I guess I wrote it! :-)

      Give me some feedback please. If I see essential agreement, I will take care and polish the entry a bit, accordingly.

    • I would like to write an article at size issues about the various ways of dealing with them, and I've started linking to that page in anticipation.

      I still haven't written it, but I decided that there were enough links that I ought to put something. So now there are some links to other articles in the Lab on the subject.

    • I noticed the word codifferential was used in the page on chain complexes. I raised this sort of terminological problem before and cannot remember the result of the discussion! The boundary operator in a chain complex is classically called the `differential', and the extra co seems contrary to `tradition'. Perhaps cochain complexes should have a cofferential but even that seems unnecessary. I have not changed this in case someone else has a good reason for the terminology ... what is the concensus? The point is not important but is worth clearing up I think.

    • Somebody clicked some buttons to make an empty slideshow at essential supremum, so I wrote an extremely stubby article there with links to real articles on the rest of the WWW.

    • I started subframe; of course, a subframe corresponds to a quotient locale.

      But we really want regular subframes. Is there a convenient elementary description of those?

    • Found some exposition by Todd on the web to add to ultrafilter, and a quote by Michal Barr added at ultraproduct.

      This was prompted by a comment by Terry Tao

      >There are two main facts that makes ultralimit analysis powerful. The first is that one can take ultralimits of arbitrary sequences of objects, as opposed to more traditional tools such as metric completions, which only allow one to take limits of Cauchy sequences of objects.

      So are ultralimits in his sense a form of completion?

    • Definitions for topological spaces, locales, and toposes here: open map.

    • Does anybody know of a definition of CABA (which would not literally be complete atomic Boolean algebra) such that the theorem that the CABAs are (up to isomorphism of posets) precisely the power sets is constructively valid?

      If you do, you can put it here: CABA.

    • I further worked on the Idea-section at cohomology, expanding and polishing here and there.

      Hit "see changes" to see what I did, precisely: changes (additions, mostly) concern mainly the part on nonabelian cohomology, and then at the end the part about twisted and differential cohomology.

      by the way, what's your all opinion about that big inset by Jim Stasheff following the Idea section, ended by Toby's query box? I think we should remove this.

      This is really essentially something I once wrote on my private web. Jim added some sentences to the first paragraph. Possibly he even meant to add this to my private web and by accident put it on the main Lab. In any case, that part is not realy fitting well with the flow of the entry (which needs improvement in itself) and most of the information is repeated anyway. I am pretty sure Jim won't mind if we essentially remove this. We could keep a paragraph that amplifies the situation in Top a bit, in an Examples-section.

      What do you think?

    • This comment is invalid XML; displaying source. <p>Since Mike's thread <a href="http://www.math.ntnu.no/~stacey/Vanilla/nForum/comments.php?DiscussionID=679&page=1">questions on structured (oo,1)-topos</a> got a bit highjacked by general oo-stack questions, I thought I'd start this new thread to announce attempted answers:</p> <p>Mike had rightly complained in a query box that a "remark" of mine in which I had meant to indicate the intuitive meaning of the technical condition on an (oo,1)-structure sheaf had been "ridiculous".</p> <p>But that condition is important, and important to understand. I have now removed the nonsensical paragraph and Mike's query box complaining about it, inserted a new query box saying "second attempt" and then spelled out two archetypical toy examples in detail, that illustrate what's going on.</p> <p>The second of them can be found in StrucSp itself, as indicated. It serves mainly to show that an ordinary ringed space has a structure sheaf in the sense of structured oo-toposes precisely if it is a <a href="https://ncatlab.org/nlab/show/locally+ringed+space">locally ringed space</a>.</p> <p>But to try to bring out the very simple geometry behind this even better, I preceded this example now by one where a structure sheaf just of continuous functions is considered.</p> <p>Have a look.</p>

    • can anyone point me to some useful discussion of cosismplicial simplicial abelian groups

      \Delta \to [\Delta^{op}, Ab]

      and cosimplicial simplicial rings

      \Delta \to [\Delta^{op}, CRing]

      I guess there should be a Dold-Kan correspondence relating these to unbounded (co)chain complexes (that may be nontrivial both in positive as well as in negative degree). I suppose it's kind of straightforward how this should work, but I'd still ike to know of any literature that might discuss this. Anything?

    • I expanded the section Gradings at cohomology. Made three sub-paragraphs:

      • integer grading

      • bigrading

      • exoctic grading

      using the kind of insights that we were discussing recently in various places.

      I also put in a query box where I wonder about a nice way to define a Chern character construction for general oo-stack oo-toposes.

    • added an interesting reference by Kriegl and Michor to generalized smooth algebra, kindly pointed out by Thomas Nikolaus:

      C^\infty-algebras from the functional analyytic point of view

      Also added some other references.

    • I completely rewrote BrownAHT

      same message as before, but I think I could say it much better now

    • this MO question on relative cohomology made me create an entry with an attempt to give the fully general abstract definition in the spirit of the discussion at cohomology

      a very stubby note is now at relative cohomology

      this clearly needs more development, but I think the basic idea is obvious

    • I added to motivic cohomology the definition of motivic cohomology in terms of connected components of hom spaces in (the homotopy stabilization of) the (oo,1)-topos of oo-stacks on the Nisnevich site.

      This is now the section Homotopy localization of the (oo,1)-topos on Nis.

      Probably Zoran mentioned this before, without me getting it. Now I get it, and I like it a lot. This is the kind of nice formulation of motivic cohomology that one would expect from the point of view that is expressed at cohomology:

      in fact, really a priori one should consider nonabelian motivic cohomology, by considering genuine oo-stacks on Nis, without stabilizing. One could consider differential Motivic cohomology by not retricting attention to the homotopy localization .

    • I entered hypersimplex, which is a tip of an important iceberg in stable category business, I hope we can continue expanding the topic and do osme research on it.

    • at interval object we have a section that discusses how in a category with interval object for every object there are various incarnations of its "path groupoid".

      We had had two such incarnations there: the first one discusses the structure of a Trimble n-category on this "path groupoid", the second one the structure of a simplicial set.

      I want one more such incarnation: the structure of a planar dendroidal set.

      A proposal for how that should work I have now typed in the new section titled currently Fundamental little 1-cubes space induced from an interval.

      (This section title is bad, I need to think of something better...)

      Eventually I want to see if this can be pushed to constitute the necessary ingredients for a "May recognition principle" in a general oo-stack oo-topos: over a site C with interval object, I want for each k a dendroidal presheaf that encodes something like the C-parameterized little k-cubes operad, which should act on k-fold loop oo-stacks on C.

      That's the motivation, at least.

    • I started quasicoherent infinity-stack. Currently all this contains is a summary of some central definitions and propositions in Toen/Vezzosi's work. I tried to list lots of direct pointers to page and verse, as their two articles tend to be a bit baroque as far as notation and terminology is concerned.

      This goes parallel with the blog discussion here.

      In the process I also created stubs for SSet-site and model site. These are terms by Toen/Vezzosi, but I think these are obvious enough concepts that deserve an entry of their own. Eventually we should also have one titled "(oo,1)-site", probably, that points to these as special models.

    • created an entry smooth natural numbers

      I tried to extract there the fundamental mechanism that makes the "nonstandard natural numbers" in Moerdijk-Reyes Models for Smooth Infinitesimal Analysis tick. In their book the basic idea is a bit hidden, but in fact it seems that it is a very elementary mechanism at work. I try to describe that at the entry. Would be grateful for a sanity check from topos experts.

      I find it pretty neat how the sequences of numbers used to represent infinite numbers in nonstandard analysis appears (as far as I understand) as generalized elements of a sheaf in a sheaf topos here.

    • I noticed that we have an entry Fredholm operator. I added a very brief remark on the space of Fredholm operators as a classifying space for topological K-theory , and added there a very brief link back.

      eventually, of course, it would be nice to add some details.

      (also added sections and a toc).

    • I once again can't enter the edit pages. So this here is just to remind myself:

      I just discovered that the lecture notes for the Barcelona school a while ago are in fact online available, here:

      Advanced Course on Simplicial Methods in Higher Categories

      This should have been out as a book already, but keeps being delayed. It contains three important lectures, that we should link to from the respective entries:

      • Joyal's book on quasicategories

      • Moerdijk's book on dendroidal sets

      • Toen's lectures on simplicial presheaves

    • How can I upload a document with diagrams that people can view the problem.
    • in order to discuss weighted limits in my revision of limits, I introduced a stubby notion of weighted join of quasi-categories. The construction and the subsequent notion of weighted limit seem quite natural, but everithing now seems too simple, so I fear to have completely misunderstood the notion of weighted limit.. :(

      could anybody give a look?

    • started an entry cocycle to go along with the entry cohomology, motivated from my discussion with Mike on the blog here

      I mention the possible terminology suggeestion of "anamorphisms" for cocycles there, and added a link to it from anafunctor.

    • started Whitehead tower, plus some speculative comments on versions using higher categories.

      -David Roberts

    • I found the discusssion at internal infinity-groupoid was missing some perspectives

      I made the material originally there into one subsection called

      • Kan complexes in an ordinary category

      and added two more subsections

      • Kan complexes in an (oo,1)-category

      • Internal strict oo-groupoids .

      The first of the two currently just points to the other relevant entry, which is groupoid object in an (infinity,1)-category, the second one is currently empty.

      But I also added a few paragraphs in an Idea section preceeding everything, that is supposed to indicate how things fit together.

    • Comment at codomain fibration about the suggested categorification, Cat^2 --> Cat. I personally don't think we've got to the bottom of what a 2-fibration is, with the possible exception of Igor Bakovic.

      David Roberts

    • I've just discovered that, from back in the days before redirects, we have two versions of Eilenberg-Mac Lane space. I have now combined them, by brute force; I'll leave it to Urs to make it look nice.

    • I've modified over quasi-categories in my personal area, upgrading from Hom-Sets to Hom-Spaces (i.e. infinity-categories of morphisms). This seems to simplify a lot the definition, and to make the connection with limits clearer. I'll wait for your comments before moving (in case they are positive) the version from my area to the main lab.

      two technical questions:

      i) how do i remove a page from my area (that's what I'd do after moving its content on the main lab)
      ii) there's a link to over quasi-categories on the page Domenico Fiorenza, but it seems not to work, and I am missing the problem with it

    • I am pretty happy with what I just wrote at

      Modified Wedge Product (ericforgy)

      I proposed the idea years ago, but only now found a voice to express it in way that I think might resonate with others.

      Basically, we have differential forms \Omega(M) and cochains C^*(S) and maps:

      deRham (R): \Omega(M)\to C^*(S)

      and

      Whitney (W): C^*(S)\to\Omega(M)

      that satisfy

      R\circ W = 1,<br/>

      W\circ R \sim 1,<br/>

      d\circ W = W\circ d, and

      d\circ R = R\circ d.

      However, one thing that has always bugged me is that these maps do not behave well with products. The wedge product in \Omega(M) is graded commutative "on the nose" and the cup product in C^*(S) is not graded commutative "on the nose", but is graded commutative when you pass to cohomology.

      The image of W is called the space of "Whitney forms" and has been used for decades by engineers in computational physics due to the fact that Whitney forms provide a robust numerical approximation to smooth forms since the exterior derivative commutes with the Whitney map and we get exact conservation laws (cohomology is related to conserved quantities in physics).

      One thing that always bugged me about Whitney forms is that they are not closed as an algebra under the ordinary wedge product, i.e. the wedge product of two Whitney forms is not a Whitney form. Motivated by this I proposed a new "modified wedge product" that turned Whitney forms into a graded differential algebra.

      Now although in grade 0, Whitney forms commute, Whitney 0-forms and Whitney 1-forms do not commute except in the continuum limit where the modified wedge product converges to the ordinary wedge product and Whitney forms converge to smooth forms.

      I think this might be a basis for examining the "cochain problem" John talked about in TWFs Week 288.

      To the best of my knowledge, this is the first time a closed algebra of Whitney forms has been written down, although I would not be completely surprised if it is written down in some tome from 100 years ago (which I guess would be hard since it would predate Whitney).

      Another nice thing about the differential graded noncommutative algebra of Whitney forms is that they are known to converge to smooth forms with sufficiently nice simplicial refinements (a kind of nice continuum limit) and you have true morphisms from the category of Whitney forms to the category of cochains (or however you want to say it). In other words, I believe the arrow theoretic properties of Whitney forms will be nicer than those of smooth forms.

    • I expanded derivation a little:

      gave the full definition with values in bimodules and added to the examples a tiny little bit on examples for this case.

      I think I also corrected a mistake in the original version of the definition: the morphism d : A \to N is of course not required to be a module homomorphism (well, it is, but over the underlying ground ring, not over A).

      At Kähler differential I just polished slightly, adding a few words and links in the definition and adding sections. I don't really have time for this derivations/Kähler stuff at the moment. Am hoping that those actively talki9ng about this on the blog will find the time to archive their stable insights at this entry.

    • I started writing folk model structure on Cat with an explicit summary of the construction, and a description of how it can be modified to work if you assume only COSHEP. I feel like there should also be a "dual" model structure assuming some other weakening of choice, in which all categories are cofibrant and the fibrant objects are the "stacks", but I haven't yet been able to make it come out right.

    • Noticed that the entry topos was lacking an example-section, so I started one: Examples. Would be nice if eventually we'd have some discussion of non-Grothendieck topos examples.

      I won't do that now, off the top of my head. Maybe later.

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