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-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty book bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex-geometry computable-mathematics computer-science constructive constructive-mathematics cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry goodwillie-calculus graph graphs gravity group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory history homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monoidal monoidal-category-theory morphism motives motivic-cohomology noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics 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 subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory 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 topicBott periodicity theorem
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Feb 22nd 2010

- Discussion Type
- discussion topicrational topological space
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 22nd 2010

expanded the list of examples at rational topological space

- Discussion Type
- discussion topiclocally contractible space
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 22nd 2010

At locally contractible space David Roberts had written a conjecture relating local contractibility to local triviality of constant n-stacks. I have added to that a converse proposition.

- 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 topicderivation
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 18th 2010

added to derivation at the very end in the exampls section a discussion of derivations on smooth functions (and how they are vector fields) and f derivations on continuous functions (and how they are trivial).

- 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 topichomotopy equivalence
- Category Latest Changes
- Started by Tim_Porter
- Comments 2
- Last comment by Tim_Porter
- Last Active Feb 16th 2010

I have been editing the entry homotopy equivalence to include a brief discussion of strong homotopy equivalences and Vogt's lemma. In so doing, I have followed my nose and found various other entries to edit. For instance that for Hans Baues, that for cylinder functor, etc. I am thinking that the general area of Henry Whitehead's idea of algebraic homotopy, may be a useful intermediate one between the infinity category ideas (which could be seen as just a 'souped up' version of Kan complexes), (I am not saying they are just that a cynic might make them out to be!) and the algebraic topologists desire to perform calculations. Note the quotes at algebraic homotopy. Of course, they d not say what 'compute' means in this context. (Note we do not have an entry on Whitehead as yet.)

- 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 topicrelation between quasi-categories and simplicial categories
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 11th 2010

one more remark at relation between quasi-categories and simplicial categories

(to be expanded...)

- 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 ‘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.

- 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.