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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.
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
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.
We've had a stub at representation for awhile; I rewrote it and let intertwiner redirect to it. But it's still a stub.
Created material-structural adjunction, along with a stub for Mostowski's collapsing lemma.
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.
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.
Expanded the References section at Coo-rings
(in the course of creating Seminar on Smooth Loci (schreiber))
came across one more Coo-ring reference by Peter Michor: on characterizong those Coo-rings that are algebras of smooth functions on some manifold. Interesting.
I want to eventually expand the stub entry relation between quasi-categories and simplicial categories. I just added a sentence only to give the entry something like an "Idea" section wher previously there had been just a lonely hyperlink.
I think the main theme here is that of "semi-strictification". So I added a remark to semi-strict infinity-category. But not really anything satisfactory yet.
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?
I felt that we needed an explicit entry
to collect the notions of category theory available in the context of (oo,1)-categories.
I had first started making this a subsection at (infinity,1)-category, but then I felt that this should parallel category and category theory.
In this context I then also felt like creating (infinity,1)-Grothendieck construction. This so far is just a collection of pointers, but eventually for instance some of the material currently hidden at limit in a quasi-category should be moved here.
Created idempotent adjunction. Does anyone know a published reference for this notion?
A little article: directed join
Several naive questions at structured (infinity,1)-topos.
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 created a page on this topic: preserved limit. (We also need reflected limit and created limit.)
I'm not sure about the stuff at the bottom.
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?
<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
and cosimplicial simplicial rings
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:
-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
added a quick mention of the relation between deRham space and derived free loop space object to derived stack and Hochschild cohomology, pointing to this reference here for all the details
B. Toen, G. Vezzosi, -Equivariant simplicial algebras and de Rham theory http://arxiv.org/abs/0904.3256
more later.
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
I started relative point of view based on material in the pages that link to it.
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.
I found the discusssion at internal infinity-groupoid was missing some perspectives
I made the material originally there into one subsection called
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.
added to (infinity,2)-category a section models for the (oo,1)-category of all (oo,2)-categories
I also added (infinity,2)-category and Theta-space to the floating TOC
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.
stubby stub for unitary group