Not signed in (Sign In)

A discussion forum about contributions to the nLab wiki and related areas of mathematics, physics, and philosophy.

Want to take part in these discussions? Sign in if you have an account, or apply for one below

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

- Discussion Type
- discussion topicNatural transformations
- Category Latest Changes
- Started by SridharRamesh
- Comments 9
- Last comment by SridharRamesh
- Last Active Feb 21st 2010

- Added a minor alternative rendition of the definition of natural transformation. I'm not sure that it's particularly useful for anything, but it is at least true, with some aesthetically pleasing qualities.

- Discussion Type
- discussion topicstructures in a gros (oo,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Feb 19th 2010

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:

- principal oo-bundles, covering oo-bundles, oo-vector-bundles, fundamental groupoid, flat cohomology, deRham cohomology, Chern character, differential cohomology.

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 .

- Discussion Type
- discussion topiclocally contractible (oo,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Feb 19th 2010

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

- Discussion Type
- discussion topicTim : New entries
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Feb 19th 2010

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.

- Discussion Type
- discussion topicdecalage
- Category Latest Changes
- Started by Tim_Porter
- Comments 14
- Last comment by Tim_Porter
- Last Active Feb 19th 2010

I have started an entry on the decalage functor.

- Discussion Type
- discussion topicphysicscontents
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by DavidRoberts
- Last Active Feb 18th 2010

I expanded the beginning of physicscontents and edited a bit to bring out more structure.

- Discussion Type
- discussion topicnc diff operators
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Feb 18th 2010

regular differential operator in noncommutative geometry (algebraic of course)

- Discussion Type
- discussion topicsolid functor
- Category Latest Changes
- Started by Mike Shulman
- Comments 7
- Last comment by Andrew Stacey
- Last Active Feb 18th 2010

Created solid functor with an SVG graphic. The SVG editor is awesome! Even if it is still a little buggy.

- Discussion Type
- discussion topicbisimplicial object
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 18th 2010

a handful further definitions and properties for bisimplicial sets

- Discussion Type
- discussion topicA-oo category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 17th 2010

Jim Stasheff edited a bit at A-infinity category -- but did not yet work in the comments that he posted to the blog

- Discussion Type
- discussion topicrelatin SSet-Joyal to SSet-Quillen
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 17th 2010

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

- Discussion Type
- discussion topic(oo,1)Cat
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 16th 2010

created (oo,1)Cat

- Discussion Type
- discussion topicSVGs at Frolicher spaces
- Category Latest Changes
- Started by Andrew Stacey
- Comments 7
- Last comment by Eric
- Last Active Feb 16th 2010

Added a couple of SVGs at Froelicher space using the new svg-editor.

- Discussion Type
- discussion topicholonomic quantum fields
- Category Latest Changes
- Started by zskoda
- Comments 5
- Last comment by zskoda
- Last Active Feb 15th 2010

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.

- Discussion Type
- discussion topichomotopy groups in an (oo,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 15th 2010

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.

- Discussion Type
- discussion topic(oo,1)-cat of (oo,1)-cats
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 15th 2010

I expanded the text at (infinity,1)-category of (infinity,1)-categories a bit more

- Discussion Type
- discussion topicall (oo,1)-cats from DW-localization
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Feb 15th 2010

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.

- Discussion Type
- discussion topichomotopy coherent diagram
- Category Latest Changes
- Started by Tim_Porter
- Comments 25
- Last comment by Tim_Porter
- Last Active Feb 15th 2010

I have started adding stuff to homotopy coherent diagram I have added new references into Rainer Vogt and have created Jean-Marc Cordier.

- Discussion Type
- discussion topiclocally contractible (oo,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 15th 2010

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.

- Discussion Type
- discussion topic(n,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Feb 12th 2010

stub for (n,1)-topos

- Discussion Type
- discussion topicGalois theory/bar path oo-groupoid of an oo-stack
- Category Latest Changes
- Started by Urs
- Comments 53
- Last comment by zskoda
- Last Active Feb 11th 2010

(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

- Discussion Type
- discussion topicGeometric morphisms of sheaf topoi
- Category Latest Changes
- Started by TobyBartels
- Comments 3
- Last comment by TobyBartels
- Last Active Feb 9th 2010

For the record, all I did at geometric morphism#sheaftopoi was to add a paragraph at the beginning of the example, substitute â€˜soberâ€™ for â€˜Hausdorffâ€™ 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.

- Discussion Type
- discussion topicdirect image
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 9th 2010

I fiddled a bit with direct image, but maybe didn't end up doing anything of real value...

- Discussion Type
- discussion topicMenagerie
- Category Latest Changes
- Started by Tim_Porter
- Comments 8
- Last comment by Andrew Stacey
- Last Active Feb 9th 2010

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.

- Discussion Type
- discussion topicSVG Editor
- Category Latest Changes
- Started by Andrew Stacey
- Comments 1
- Last comment by Andrew Stacey
- Last Active Feb 9th 2010

I've started a section in the HowTo on the new SVG-editor.

- Discussion Type
- discussion topicglobal sections
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Feb 9th 2010

I expanded the Examples-section at geometric morphism and created global section to go along with constant sheaf

- Discussion Type
- discussion topic(locally) constant n-stacks
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Feb 8th 2010

I started

constant infinity-stack (existed before, edited it a bit)

still need locally constant function

- Discussion Type
- discussion topiccurvature
- Category Latest Changes
- Started by zskoda
- Comments 4
- Last comment by Urs
- Last Active Feb 8th 2010

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.

- Discussion Type
- discussion topicQuestion from Todd.
- Category Latest Changes
- Started by TobyBartels
- Comments 4
- Last comment by Urs
- Last Active Feb 8th 2010

It's up high at model structure on simplicial presheaves. I think that I answered it, but Urs had probably better take a look.

- Discussion Type
- discussion topicHorns
- Category Latest Changes
- Started by Tim_Porter
- Comments 2
- Last comment by Urs
- Last Active Feb 8th 2010

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!

- Discussion Type
- discussion topicProfinite Homotopy Theory
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Tim_Porter
- Last Active Feb 8th 2010

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.

- Discussion Type
- discussion topiccohomology
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 5th 2010

added a paragraph on Lurie's HTT section 7.2.2 to References at cohomology

- Discussion Type
- discussion topicVocabulary words from Stone Spaces, II.1.3
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Feb 5th 2010

- Discussion Type
- discussion topicfundamental group of a topos
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 5th 2010

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.

- Discussion Type
- discussion topicn-Hom-categories
- Category Latest Changes
- Started by TobyBartels
- Comments 6
- Last comment by TobyBartels
- Last Active Feb 5th 2010

These are used by Sridhar Ramesh to great expository effect at (n,r)-category.

- Discussion Type
- discussion topicgauge fixing
- Category Latest Changes
- Started by domenico_fiorenza
- Comments 11
- Last comment by TobyBartels
- Last Active Feb 4th 2010

- 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

- Discussion Type
- discussion topiclocal system
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 4th 2010

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.

- Discussion Type
- discussion topicRepresentations and intertwiners
- Category Latest Changes
- Started by TobyBartels
- Comments 12
- Last comment by Urs
- Last Active Feb 4th 2010

We've had a stub at representation for awhile; I rewrote it and let intertwiner redirect to it. But it's still a stub.

- Discussion Type
- discussion topicmaterial-structural adjunction
- Category Latest Changes
- Started by Mike Shulman
- Comments 2
- Last comment by TobyBartels
- Last Active Feb 3rd 2010

Created material-structural adjunction, along with a stub for Mostowski's collapsing lemma.

- Discussion Type
- discussion topicSize issues
- Category Latest Changes
- Started by TobyBartels
- Comments 4
- Last comment by TobyBartels
- Last Active Feb 2nd 2010

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.

- Discussion Type
- discussion topicCodifferential
- Category Latest Changes
- Started by Tim_Porter
- Comments 4
- Last comment by zskoda
- Last Active Feb 2nd 2010

- 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.

- Discussion Type
- discussion topicEssential suprema
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Feb 2nd 2010

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.

- Discussion Type
- discussion topicCoo-ring
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 2nd 2010

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.

- Discussion Type
- discussion topicsSet-categories and quasi-categories
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Tim_Porter
- Last Active Feb 2nd 2010

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.

- Discussion Type
- discussion topic(Regular) subframes
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Feb 2nd 2010

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?

- Discussion Type
- discussion topic(oo,1)-category theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 2nd 2010

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.

- Discussion Type
- discussion topicidempotent adjunction
- Category Latest Changes
- Started by Mike Shulman
- Comments 9
- Last comment by zskoda
- Last Active Feb 1st 2010

Created idempotent adjunction. Does anyone know a published reference for this notion?

- Discussion Type
- discussion topicDirected joins
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by TobyBartels
- Last Active Feb 1st 2010

A little article: directed join

- Discussion Type
- discussion topicultrafilters
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active Feb 1st 2010

- 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?

- Discussion Type
- discussion topicquestions on structured (infty,1)-topoi
- Category Latest Changes
- Started by Mike Shulman
- Comments 57
- Last comment by Urs
- Last Active Feb 1st 2010

Several naive questions at structured (infinity,1)-topos.

- Discussion Type
- discussion topicOpen maps
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Jan 31st 2010

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

- Discussion Type
- discussion topicNew article: CABAs
- Category Latest Changes
- Started by TobyBartels
- Comments 6
- Last comment by TobyBartels
- Last Active Jan 31st 2010

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.

- Discussion Type
- discussion topicPreservation of limits
- Category Latest Changes
- Started by TobyBartels
- Comments 3
- Last comment by TobyBartels
- Last Active Jan 31st 2010

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.

- Discussion Type
- discussion topiccohomology
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Mike Shulman
- Last Active Jan 30th 2010

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?

- Discussion Type
- discussion topicon structured (oo,1)-topoi
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by zskoda
- Last Active Jan 29th 2010

`<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>`

- Discussion Type
- discussion topiccosimplicial simplicial abelian groups and rings
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jan 28th 2010

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?

- Discussion Type
- discussion topicbigrading on cohomology
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 28th 2010

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.

- Discussion Type
- discussion topicCoo-rings
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by zskoda
- Last Active Jan 27th 2010

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.

- Discussion Type
- discussion topicBrown's Abstract Homotopy Theory and Generalized Sheaf Cohomology
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 25th 2010

I completely rewrote BrownAHT

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

- Discussion Type
- discussion topicde Rham space as derived free loop space
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 24th 2010

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/0904.3256more later.