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- Discussion Type
- discussion topicinterval object: dendroidal version of path groupoid
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 19th 2010

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.

- Discussion Type
- discussion topicquasicoherent oo-stack
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Jan 18th 2010

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.

- Discussion Type
- discussion topicsmooth natural numbers
- Category Latest Changes
- Started by Urs
- Comments 27
- Last comment by Urs
- Last Active Jan 18th 2010

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.

- Discussion Type
- discussion topicaxiom of foundation
- Category Latest Changes
- Started by Mike Shulman
- Comments 2
- Last comment by Mike Shulman
- Last Active Jan 18th 2010

Three questions at axiom of foundation.

- Discussion Type
- discussion topicpure set
- Category Latest Changes
- Started by Mike Shulman
- Comments 8
- Last comment by TobyBartels
- Last Active Jan 16th 2010

I'm in the process of reworking pure set to incorporate those of AN's points that were valid and remove as many of the query-boxes as possible, in order to make it readable.

Right now, the page describes trees, and then extensional graphs, and then briefly mentions the more general notion of which these are specializations. I would find it more intuitive, and more concise, to first describe the general notion, and then introduce trees and extensionality as two ways of ensuring that equivalent graphs (in the loose "bisimulation" sense appropriate for the general notion) are in fact isomorphic. Unless someone objects, I'll probably reorganize it that way.

- Discussion Type
- discussion topicFredholm operator and K-theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 15th 2010

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).

- Discussion Type
- discussion topiclecture notes
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Jan 15th 2010

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

- Discussion Type
- discussion topicCategory Theory Problem on Right Adjoints
- Category Latest Changes
- Started by yellsap
- Comments 3
- Last comment by TobyBartels
- Last Active Jan 15th 2010

- How can I upload a document with diagrams that people can view the problem.

- Discussion Type
- discussion topicNew page: [[relative point of view]]
- Category Latest Changes
- Started by TobyBartels
- Comments 4
- Last comment by zskoda
- Last Active Jan 14th 2010

I started relative point of view based on material in the pages that link to it.

- Discussion Type
- discussion topicweighted join of quasi-categories
- Category Latest Changes
- Started by domenico_fiorenza
- Comments 12
- Last comment by Mike Shulman
- Last Active Jan 14th 2010

- 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?

- Discussion Type
- discussion topicSullivan differential forms
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 13th 2010

- Discussion Type
- discussion topicinhabited object
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jan 12th 2010

I split off inhabited object from inhabited set.

(moved Mike's and Toby's old discussion query box to the new entry, too)

I added an Examples section with a remark about this issue in the context of Models for Smooth Infinitesimal Analysis, that I happen to be looking into.

personally, I feel I need more examples still at internal logic to follow this in its full scope. I guess I should read the Elephant one day, finally.

In the book Moerdijk-Reyes say in a somewhat pedestrian way that existential quantifiers in the internal logic of a sheaf topos are to be evaluated on covers, hence asking internally if a sheaf has a (internally global) element means asking if for any cover of the point, there is a morphism .

That's fine with me and I follow this in as far as the purpose of their book is concerned, but I need to get a better idea of how the logical quantifiers are formulate in internal logic in full generality.

- Discussion Type
- discussion topicModels for Smooth Infinitesimal Analysis
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 12th 2010

I am filling in more details (definitions, properties) about the various toposes at Models for Smooth Infinitesimal Analysis

- Discussion Type
- discussion topiccocycle
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jan 11th 2010

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.

- Discussion Type
- discussion topicWhitehead tower
- Category Latest Changes
- Started by Guest
- Comments 19
- Last comment by domenico_fiorenza
- Last Active Jan 10th 2010

- started Whitehead tower, plus some speculative comments on versions using higher categories.

-David Roberts

- Discussion Type
- discussion topicinternal oo-groupoid
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by domenico_fiorenza
- Last Active Jan 9th 2010

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.

- Discussion Type
- discussion topic(infinity,2)-category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 8th 2010

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

- Discussion Type
- discussion topic2-fibration at codomain fibration
- Category Latest Changes
- Started by Guest
- Comments 3
- Last comment by zskoda
- Last Active Jan 8th 2010

- 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

- Discussion Type
- discussion topicEilenbergâ€“Mac Lane shenanigans
- Category Latest Changes
- Started by TobyBartels
- Comments 3
- Last comment by Urs
- Last Active Jan 8th 2010

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.

- Discussion Type
- discussion topicuniray group
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by zskoda
- Last Active Jan 7th 2010

stubby stub for unitary group

- Discussion Type
- discussion topicdifferential cohomology - nonabelian case
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jan 7th 2010

started adding a detailed "Idea" section to the page differential cohomology - nonabelian case on my personal web

currently this consists of the section The classical case of U-principal bundles and revolves around an abstract-nonsense interpretation of the Chern character

so this is in parts to be read as one more contribution in my discussion, elsewhere, with Domenico.

- Discussion Type
- discussion topicQuadratic Functions in Geometry, Topology,and M-Theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 7th 2010

created reference entry on Quadratic Functions in Geometry, Topology,and M-Theory, the fundamental article on differential cohomology.

- Discussion Type
- discussion topicnon-canonical isomorphisms
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Jan 7th 2010

I added some remarks regarding Steve Lack's paper of that name to biproduct and distributive category.

- Discussion Type
- discussion topicover quasi-categories
- Category Latest Changes
- Started by domenico_fiorenza
- Comments 5
- Last comment by TobyBartels
- Last Active Jan 6th 2010

- 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

- Discussion Type
- discussion topicrn-Simons gerbe
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 6th 2010

stub for Chern-Simons gerbe

- Discussion Type
- discussion topicSimons-Sullivan structured bundle
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 6th 2010

created Simons-Sullivan structured bundle

eventually I want to move the discussion currently in a subsection at differential K-theory to this entry

- Discussion Type
- discussion topiclifting gerbe
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 6th 2010

quick entry for lifting gerbe

- Discussion Type
- discussion topicspace
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Jan 5th 2010

In search for a more independent existence of the general abstract notes "Notions of Space" that were still bundled in the "talk notes"-page A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves I copied the material to the entry space.

- Discussion Type
- discussion topichomotopy n-type
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 5th 2010

John Baez added a query box to homotopy n-type

- Discussion Type
- discussion topicDifferential Graded (Noncommutative) Algebra of Whitney Forms
- Category Latest Changes
- Started by Eric
- Comments 2
- Last comment by Urs
- Last Active Jan 4th 2010

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 and cochains and maps:

and

that satisfy

<br/>

<br/>

, and

However, one thing that has always bugged me is that these maps do not behave well with products. The wedge product in is graded commutative "on the nose" and the cup product in is not graded commutative "on the nose", but is graded commutative when you pass to cohomology.

The image of 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.