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

• 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

• Tim Porter added references to microbundle and I edited the formatting of the entry a bit

• 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.

• I expanded derivation a little:

gave the full definition with values in bimodules and added to the examples a tiny little bit on examples for this case.

I think I also corrected a mistake in the original version of the definition: the morphism $d : A \to N$ is of course not required to be a module homomorphism (well, it is, but over the underlying ground ring, not over $A$).

At Kähler differential I just polished slightly, adding a few words and links in the definition and adding sections. I don't really have time for this derivations/Kähler stuff at the moment. Am hoping that those actively talki9ng about this on the blog will find the time to archive their stable insights at this entry.

• I started writing folk model structure on Cat with an explicit summary of the construction, and a description of how it can be modified to work if you assume only COSHEP. I feel like there should also be a "dual" model structure assuming some other weakening of choice, in which all categories are cofibrant and the fibrant objects are the "stacks", but I haven't yet been able to make it come out right.

• Noticed that the entry topos was lacking an example-section, so I started one: Examples. Would be nice if eventually we'd have some discussion of non-Grothendieck topos examples.

I won't do that now, off the top of my head. Maybe later.

• cellular set, mainly references for now

BTW, Does anybody have a file or scan of Joyal's original 1997 article ?

• At Grothendieck fibration I wonder if we can make the definition less evil than the non-evil version there, with applications to Dold fibrations. Also the insertion of a necessary adjective at topological K-theory.

-David Roberts
• created infinity-limits - contents and added it as a toc to relevant entries

(maybe I shoulod have titled the page differently, but it doesn't matter much for a toc)