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• 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'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.

• 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

• How can I upload a document with diagrams that people can view the problem.
• 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?
• 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 $F$ has a (internally global) element means asking if for $U \to *$ any cover of the point, there is a morphism $U \to F$.

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.

• 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.

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

-David Roberts
• 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.

• 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
• 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.

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

• 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 $\Omega(M)$ and cochains $C^*(S)$ and maps:

$deRham (R): \Omega(M)\to C^*(S)$

and

$Whitney (W): C^*(S)\to\Omega(M)$

that satisfy

$R\circ W = 1,$<br/>

$W\circ R \sim 1,$<br/>

$d\circ W = W\circ d$, and

$d\circ R = R\circ d.$

However, one thing that has always bugged me is that these maps do not behave well with products. The wedge product in $\Omega(M)$ is graded commutative "on the nose" and the cup product in $C^*(S)$ is not graded commutative "on the nose", but is graded commutative when you pass to cohomology.

The image of $W$ 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.