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

• 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.

• 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)

• created a section Contractible objects at lined topos.

This introduces and discusses a bit a notion of objects being contractible with respect to a specified line object (maybe the section deserves to be at interval object instead, not sure).

This notion is something I made up, so review critically. I am open for suggestions of different terminology. The concept itself, simple as it is (though not entirely trivial), I need for the discussion of path oo-groupoids of oo-stacks on my personal web:

if a lined Grothendieck topos $(\mathcal{T} = Sh(C),R)$ is such that all representable objects are contractible with respect to the line object $R$, then the path oo-groupoid functor

$\Pi : SSh(C) \to SSh(C)$

on simplicial sheaves, which a priori is only a Qulillen functor of oo-prestacks, enhances to a Quillen functor of oo-stacks (i.e. respects the local weak equivalences).

• Added to the Idea section at space and quantity a short paragraph with pointers to the (oo,1)-categorical realizations. (Parallel to the blog discussion here)

• no, I didn't create an entry with that title.

but I added to n-fibration a brief link, though, to the concept that is currently described at Cartesian fibration, which models Grothendieck fibrations of (oo,1)-categories.

This here is mainly to remind me that there is need to polish and reorganize the nLab entries on higher fibrations into something more coherent.

• I fixed a bunch of broken links on the lab just now. In case anybody is wondering what all of those edits were.

• I have just made links to all of the contentful orphaned paged on the main nLab web. However, they may still be walled gardens; Instiki doesn't find those automatically.

In general, when you create a new page, it's a good idea to create a link to it from some existing page on a more general topic. (The links that I just made may not have been the best!) That way, it's more likely that people will actually find their way to your new page.

• I wanted to start expanding on the big story at nonabelian Lie algebra cohomology, but then found myself wanting to polish first a bit further the background material.

I came to think that it is about time to collect our stuff on "oo-Lie theory".

and added it to most of the relevant entries.

This toc is based on the one on my personal web here -- but much larger now -- and still contains some links to my web, where I am trying to develop the full story. If anyone feels ill-at-ease with these links to my personal web, let me know.

• created quick stub for framed bicategory

but my machine's battery will die any second now...

• It looked to me like Urs hit Ctrl-V instead of Ctrl-C there, so I rolled back, but now Urs is editing again, so probably he's just doing something that I interrupted. Since I can't leave a note there now, I'll leave one here: I won't interfere again, Urs.

• added to (infinity,1)-operad the definition/proposition of the model structure for the category of (oo,1)-categories of operations here

• I added to vertical categorification the comments that I'd made at MathOverflow, as Urs has requested. I'm not sure that I'm happy with where I put them and how I labelled them, but maybe it's better if other people judge that.

• Added some more to the ongoing discussion about composition at evil.

(fixed)
• I'd like to add the following "shape" to http://ncatlab.org/nlab/show/limit#types_of_shapes_of_limit_cones_17 :

The limit of the identity functor Id: C --> C is the initial object of C (it it exists).
• I've added the latest, almost complete, draft of my thesis to my personal web - go via David Roberts. Comments on introduction are welcome, if you feel so inclined. Just put them on David Roberts.

On a related note, is it quite legitimate to post updates on personal webs here? (Now that I've already done it)

David Roberts
• Edited the page category theory. Mostly about that certain presheaves are the same as categories and the long discussion at the end with an idea how to solve my problem about CW-complexes. Removed precursors link since there is nothing about them in nLab. This new logging is a bit confusing and harder to read.

-Rafael
• In the lab book metaphor, this page is some jottings of stuff that I'm pretty sure must be out there (as it's a fairly obvious thing to do) but have no idea of what it's called (hedgehogs, perhaps?). So I'd be grateful if someone strong in the ways of Lawvere theories could stop by and help me out.

(Plus I had to make up the notation and terminology as I went along so that's all horrible)

Hopefully the big box at the top of the page makes this clear!

• I apologize in case this discussion is already open and I have been unable to find it.

There is something I am unable to undrstand in the definition of extended TQFT as on the nLab page http://ncatlab.org/nlab/show/extended+topological+quantum+field+theory

Namely, it seems to me that the recursive definition should rather end with "smooth compact oriented (n-m+1)-manifolds to R-linear (m?2)-categories"
• One of these has started (or continued) a conversation at the bottom of graph.

• I'm guessing that ferrim is spam. If no-one says anything to the contrary within 24hrs then I'll add it to the spam category.

If it is spam, it's either a random spambot post or it's someone testing to see how vigilant we are. If the latter, as there's no content then they may simply test to see if the link stays active. In which case, our previous "policy" of blanking the content won't send the right signal here (especially as there's no content to blank). Is there any objection to renaming spam entries? Say, as 'spam (original title)' (or whatever the allowable punctuation characters are)?

• In entry groupoid object in an (infinity,1)-category there is a passage

"it is the generalization of Stasheff H-space from Top to more general ?-stack (?,1)-topoi: an object that comes equipped with an associative and invertible monoid structure, up to coherent homotopy"

I repeat what I documented in earlier discussion on H-space: H-spaces are widely used terminology since 1950, thus before Stasheff work which of course is an important work on coherencies for them. So it is likely improper to say Stasheff H-space...Stasheff has REFINEMENTS of H-spaces, namely $A_n$-spaces and the group-like case is A infty spaces.

• Somebody named ‘Harry’ has a comment at evil. Presumably it is of interest to Mike and me.

• Added topological cube to cube, and removed some JA-esque redirects from terms like succubi and so forth.

David Roberts
• I see Mike's 1-category equipment

May I vote for the following: we should "play Bourbaki" and correct the naming mistake made here. The obvious name one should use is "pro-morphism structure".

We equip a category with pro-morphisms.

We equip a category with a pro-morphism structure.

Or, if you insist,

We equip a category with pro-arrows.

We equip a category with a pro-arrow structure.

But the day will come when you want a pro-2-morphism structure. And then one will regret having used "arrow" instead of "morphism".

I mean, compared to issues like "presentable" versus "locally presentable", this idea of saying just "equipment" is a bit drastic, to my mind.

• I'd like to write something about a Quillen equivalence, if any, between model structures on

• n-connected pointed spaces

• grouplike E-n spaces .

With the equivalence given by forming n-fold look spaces.

But I need more input. I found a nice discussion of a model structure on n-connected pointed spaces in A closed model category on (n-1)-connected spaces. I suppose there is a standard model structure on E-k algebras in Top. Is a Quilen equivalence described anywhere?

• I added to directed colimit the $\kappa$-directed version, for some regular cardinal $\kappa$.

We should maybe also add to directed set the $\kappa$-directed version. What we currently descrribe there is just the $\kappa = \aleph_0$-directed version.

Accordingly then I also added to compact object the definition of the variant of $\kappa$-compact objects.

At small object previously it mentioned "$\kappa$-filtered colimits". I now made that read "$\kappa$-directed colimits".

I hope that's right. If not, do we need to beware of the differene?

• created entry for Dan Freed and added some links to articles by him here and there

• expanded the discussion of face maps at dendroidal set a little