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created rational homotopy equivalence
started model structure on cosimplicial rings and model structure on dg-algebras
want to eventually expand on this: does anyone happen have an electronic copy of the article by Jardine referenced at model structure on dg-algebras?
(in the process I also created a quick stub for differential graded ring)
edited homotopy coherent nerve a bit
I tried to bring out the structure more by adding more subsections. Have a look at the new table of contents. Then I did a bunch of trivial edits like indenting some equations etc. Have a look at "See changes" if you want to see it precisely.
I put a summary of the Chapman complement theorem at shape theory. I remember a discussion about duality on the blog some time ago and this may be relevant.
Some more discussion (Ian and myself) at quantum channel about the definition of QChan when taking into account classical information.
started something at Hamilton operator
I added a small subsection to the definition of an enriched category over which describes them as lax monoidal functors where the codomain is the monoidal category of endospans on in the bicategory of spans.
This is really just for Zoran although anyone else is welcome to help. I felt there needed to be a little more here, but you are also closely involved with this so please, check that what I have added is alright. Thanks. Tim
I wanted to add to rational homotopy theory a section that gives a summary overview of the two Lie theoretic approaches, Sullivan's and Quillen's, indicating the main ingredients and listing the relevant references, by collecting some of the information accumulated in the blog discussion.
But, due to my connection problem discused in another thread, even after trying repeatedly for about 45 minutes, the nLab software still regards me as a spammer and won't let me edit the entry.
I'll try again tomorrow. Meanwhile, in case a good soul here can help me out, I post the text that I wanted to add to the entry in the next message. It's supposed to go right after the section ""Rational homotopy type".
Wrote identity type and display map and dependent type.
Also, I have a question/correction at internal logic in a presheaf topos
I've created a stub article for equilogical spaces. I couldn't quite figure out how to make a link while preserving the subscripting; I guess I could rewrite that to avoid the formatting problem, but presumably someone else knows how to do it anyway
started category fibered in groupoids as I think this deserves a page of its own separated from Grothendieck fibration
I understand that there was some terminological opposition voiced at Grothendieck fibration concerning the term "category cofibered in groupoids", but am I right that this does not imply opposition against "category fibered in groupoids", only that the right term for the arrow-reversed situation should be "opfibration in groupoids"?
started expanding simplicial skeleton. But more needs to eventually go here.
I am expanding the entry homotopy group (of an infinity-stack) by putting in one previously missing aspect:
there are two different notions of homotopy groups of oo-stacks, or of objects in an (oo,1)-topos, in general
the "categorical" homotopy groups
the "geometric" homotopy groups.
See there for details. This can be seen by hand in same cases That this follows from very general nonsense was pointed out to me by Richard Williamson, a PhD student from Oxford (see credits given there). The basic idea for 1-sheaves is Grothendieck's, for oo-stacks on topological spaces it has been clarified by Toen.
While writing what I have so far (which I will probably rewrite now) I noticed that the whole story here is actually nothing but an incarnation of Tannak-Krein reconstruction! I think.
It boils down to this statement, I think:
IF we already know what the fundamental oo-groupoid of an object is, then we know that a "locally constant oo-stack" with finite fibers is nothing but a flat oo-bundle, namely a morphism (think about it for n=1, where it is a very familiar statement). The collectin of all these is nothing but the representation category (on finite o-groupoids)
For each point this comes with the evident forgetful funtor
that picks the object that we are representing on.
Now, Tannaka-Krein reconstruction suggests that we can reconstruct as the automorphisms of the functor.
And that's precisely what happens. This way we can find from just knowing "locally constant oo-stacks" on X, i.e. from known flat oo-bundles with finite fibers on X.
And this is exactly what is well known for the n=1 case, and what Toen shows for oo-stacks on Top.
Chenchang Zhu updated the reference to her work at Lie theory for stacky Lie groupoids
(need to rethink what I said here, sorry)
added to global section the statement that ooGrpd is indeed the terminal (oo,1)-topos.
started formal dg-algebra
I thought I'd amuse myself with creating a succinct list of all the useful structures that we have canonically in an (oo,1)-topos without any intervention by hand:
I started typing that at structures in a gros (oo,1)-topos on my personal web.
I think this gives a quite remarkable story of pure abstract nonsense. None of this is created "by man" in a way. It all just exists.
Certainly my list needs lots of improvements. But I am too tired now. I thought I'd share this anyway now. Comments are welcome.
Main point missing in the list currently is the free loop space object, Hochschild cohomology and Domenico's proposal to define the Chern character along that route. I am still puzzled by how exactly the derived loop space should interact with and .
I am re-reading Simpson/Teleman's "de Rham theorem for oo-stacks" and realize what I missed on first reading:
they have essentually the statement that I produced recently, that oo-stacks on a site with "geometrically contractible" objects form a localy contractible (infinity,1)-topos in that the terminal geometric morphism is essential.
They say this (slightly implicitly) at the level of homotopy cateories on page 5.
I briefly recorded this in a somewhat stubby way now at locally contracible (oo,1)-topos - Examples
I have created a brief entry for Phil Higgins. Clearly more could be added for instance his work on using groupoids to prove group theoretic results.
I have started an entry on the decalage functor.
I expanded the beginning of physicscontents and edited a bit to bring out more structure.
regular differential operator in noncommutative geometry (algebraic of course)
a handful further definitions and properties for bisimplicial sets
Jim Stasheff edited a bit at A-infinity category -- but did not yet work in the comments that he posted to the blog
as part of a reply in my blog discussion with Mike here I created a section
relation to the model structure for oo-groupoids at model structure for quasi-categories
created (oo,1)Cat
Added a couple of SVGs at Froelicher space using the new svg-editor.
New stub holonomic quantum field. Very deep relation between monodromy involving approach to integrable systems and a class of QFTs has been discovered in Japan in 1970s.
I worked on improving (hopefully) and further expanding (a bit) the discussion of geometric homotopy groups at
I give at
Idea now just the brief reminder of the simple situation for 1-toposes as described at locally connected topos;
and then at Defintion I state the very obvious and simple generalization of this to homotopy oo-groupoids of objects in a locally contractible (oo,1)-topos.
Then I say something like: while this definition is very obvious and simple, it seems it has not quite been stated in the literature (except possibly in the thesis by Richad Williamson), but that there are old well-known results in the literature that essentially, with only slight modification of language, already do say precisely this.
Then I go through this claim in detail. I list three subsections with three different methods of how to construct that left adjoint to the constant oo-stack functor, and then discuss in some detail how old and new references do already -- if slightly implicitly -- discuss precisely this. The three sections are
You'll notice that I also link to the discussioon of the absgtract oo-adjunction on my personal web. Currently I am thinking of the entry on my personal web as talking about the abstract notion of a path oo-groupoid, and of this page here on the nLab as providing all the "well-known" aspects of it (in that these are in the literature).
Please complain if somehow this doesn't look like the right thing to do. I am currently a bit undecided as to what bits of this discussion should be on my personal web and which on the nLab.
I expanded the text at (infinity,1)-category of (infinity,1)-categories a bit more
At category with weak equivalences we say that it is unclear whether every (oo,1)-category arises as the simplicial localization of a cat with weak equivalences, but that it seems plausible.
At (infinity,1)-category we say that indeed every (oo,1)-category arises as the simplicial localization of a homotopical category.
I had put in the paragraph that says this based on a message that Andre Joyal recently posted to the CatTheory mailing list.
It would be good to harmonize this with the discussion at category with weak equivalences and maybe to add some references.
I have started adding stuff to homotopy coherent diagram I have added new references into Rainer Vogt and have created Jean-Marc Cordier.
I think I understand locally contractible (oo,1)-toposes now, with their left-adjoint to the constant oo-sheaf functor.
The (simple) observation is here on my personal web.
stub for (n,1)-topos
(edit: typo in the headline: meant is "bare" path oo-groupoid)
I think I have the proof that when the structured path oo-groupoid of an oo-stack oo-topos exists, as I use on my pages for differential nonabelian cohomology, then its global sections/evaluation on the point yields the bare path oo-groupoid functor, left adjoint to the formation of constant oo-stacks.
A sketch of the proof is now here.
Recall that this goes along with the discussion at locally constant infinity-stack and homotopy group of an infinity-stack.
P.S.
Am in a rush, will get back to the other discussion here that are waiting for my replies a little later. Just wanted to et this here out of the way
For the record, all I did at geometric morphism#sheaftopoi was to add a paragraph at the beginning of the example, substitute ‘sober’ for ‘Hausdorff’ in appropriate places, and add to the query box there. I mention this because the diff thinks that I did much more than that, and I don't want anybody to waste time looking for such changes!
I still to make the proof apply directly to sober spaces; the part that used that the space was Hausdorff is still in those terms.
I fiddled a bit with direct image, but maybe didn't end up doing anything of real value...
I have just uploaded a new 10 chapter version of the menagerie notes. It can be got at via my n-lab page then to my personal n-lab page and follow the link.
I've started a section in the HowTo on the new SVG-editor.
I expanded the Examples-section at geometric morphism and created global section to go along with constant sheaf
I started
constant infinity-stack (existed before, edited it a bit)
still need locally constant function
I added a new section at curvature about the classical notion of curvature and renamed the idea section into Modern generalized ideas of curvature. The classical notion has to do with bending in a space, measured in some metrics. I wrote some story about it and moved the short mention of it in previous version into that first section on classical curvature. It should be beefed up with more details. I corrected the incorrect statement in the previous version that the curvature on fiber bundles generalized the Gaussian curvature. That is not true, the Gaussian curvature is the PRODUCT of the eigenvalues of the curvature operator, rather than a 2-form. Having said that, I wrote the entry from memory and I might have introduced new errors. Please check.
It's up high at model structure on simplicial presheaves. I think that I answered it, but Urs had probably better take a look.
Todd suggested an excellent rewording in the definition of horn, and I have made the necessary changes. Do check it out if you care about horns!
I noted the point made by, I think, Toby about there being stuff on profinite homotopy type in the wrong place (profinite group). I have started up a new entry on profinite homotopy types, but am feeling that it needs some more input of ideas, so help please.
added a paragraph on Lurie's HTT section 7.2.2 to References at cohomology
I edited fundamental group of a topos a bit
added more subsections, trying to make the structure of the entry clearer
added an Idea-in-words paragraph at the very beginning, before Tim Portert's idea-of-the-technical-construction part now following this.
where the discussion alluded to higher toposes I have now linked to the relevant entry fundamental group of an infinity-stack
added the reference to Johnstone's book
added at homotopy group of an infinity-stack links back to this entry here, as well as at Grothendieck's Galois theory.
These are used by Sridhar Ramesh to great expository effect at (n,r)-category.