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    • 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.

    • At Grothendieck fibration I wonder if we can make the definition less evil than the non-evil version there, with applications to Dold fibrations. Also the insertion of a necessary adjective at topological K-theory.

      -David Roberts

    • created infinity-limits - contents and added it as a toc to relevant entries

      (maybe I shoulod have titled the page differently, but it doesn't matter much for a toc)

    • created a section Contractible objects at lined topos.

      This introduces and discusses a bit a notion of objects being contractible with respect to a specified line object (maybe the section deserves to be at interval object instead, not sure).

      This notion is something I made up, so review critically. I am open for suggestions of different terminology. The concept itself, simple as it is (though not entirely trivial), I need for the discussion of path oo-groupoids of oo-stacks on my personal web:

      if a lined Grothendieck topos (\mathcal{T} = Sh(C),R) is such that all representable objects are contractible with respect to the line object R, then the path oo-groupoid functor

      \Pi : SSh(C) \to SSh(C)

      on simplicial sheaves, which a priori is only a Qulillen functor of oo-prestacks, enhances to a Quillen functor of oo-stacks (i.e. respects the local weak equivalences).

    • Added to the Idea section at space and quantity a short paragraph with pointers to the (oo,1)-categorical realizations. (Parallel to the blog discussion here)

    • no, I didn't create an entry with that title.

      but I added to n-fibration a brief link, though, to the concept that is currently described at Cartesian fibration, which models Grothendieck fibrations of (oo,1)-categories.

      This here is mainly to remind me that there is need to polish and reorganize the nLab entries on higher fibrations into something more coherent.

    • I fixed a bunch of broken links on the lab just now. In case anybody is wondering what all of those edits were.

    • I have just made links to all of the contentful orphaned paged on the main nLab web. However, they may still be walled gardens; Instiki doesn't find those automatically.

      In general, when you create a new page, it's a good idea to create a link to it from some existing page on a more general topic. (The links that I just made may not have been the best!) That way, it's more likely that people will actually find their way to your new page.

    • I wanted to start expanding on the big story at nonabelian Lie algebra cohomology, but then found myself wanting to polish first a bit further the background material.

      I came to think that it is about time to collect our stuff on "oo-Lie theory".

      So I created a floating table of contents

      and added it to most of the relevant entries.

      This toc is based on the one on my personal web here -- but much larger now -- and still contains some links to my web, where I am trying to develop the full story. If anyone feels ill-at-ease with these links to my personal web, let me know.

    • It looked to me like Urs hit Ctrl-V instead of Ctrl-C there, so I rolled back, but now Urs is editing again, so probably he's just doing something that I interrupted. Since I can't leave a note there now, I'll leave one here: I won't interfere again, Urs.

    • added to (infinity,1)-operad the definition/proposition of the model structure for the category of (oo,1)-categories of operations here

    • I added to vertical categorification the comments that I'd made at MathOverflow, as Urs has requested. I'm not sure that I'm happy with where I put them and how I labelled them, but maybe it's better if other people judge that.

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