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- 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 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.

- Discussion Type
- discussion topicregular and exact completion
- Category Latest Changes
- Started by Mike Shulman
- Comments 8
- Last comment by Todd_Trimble
- Last Active Jan 4th 2010

Started writing regular and exact completion.

- Discussion Type
- discussion topicderivation and Kähler differential
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by John Baez
- Last Active Jan 3rd 2010

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 is of course not required to be a module homomorphism (well, it is, but over the underlying ground ring, not over ).

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.

- Discussion Type
- discussion topicfolk model structure on Cat
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Jan 1st 2010

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.

- Discussion Type
- discussion topic[[weak factorization system on Set]]
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Jan 1st 2010

Mike Shulman wrote weak factorization system on Set, which is very nice.

- Discussion Type
- discussion topicanafunctors
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Jan 1st 2010

Motivated by this question on MO, created weak factorization system on Set and added some comments about size questions to anafunctor.

- Discussion Type
- discussion topictopos
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Mike Shulman
- Last Active Dec 30th 2009

- Discussion Type
- discussion topicoo-Lie differentiation and integration
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Dec 30th 2009

expanded the Idea-section at schreiber:oo-Lie differentiation and integration and polished the section of oo-Lie diffeentiation somewhat, following the blog discussion here

- Discussion Type
- discussion topicThose evil †-categories
- Category Latest Changes
- Started by TobyBartels
- Comments 5
- Last comment by Mike Shulman
- Last Active Dec 30th 2009

Discussion has spilt from the

`categories`

mailing list into evil.

- Discussion Type
- discussion topicQuestion on Grothendieck fibrations.
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by Mike Shulman
- Last Active Dec 30th 2009

Sridhar Ramesh has one at Grothendieck fibration.

- Discussion Type
- discussion topiccellular set
- Category Latest Changes
- Started by zskoda
- Comments 5
- Last comment by TobyBartels
- Last Active Dec 30th 2009

cellular set, mainly references for now

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

- Discussion Type
- discussion topicGrothendieck fibration+topological K-theory
- Category Latest Changes
- Started by Guest
- Comments 6
- Last comment by domenico_fiorenza
- Last Active Dec 28th 2009

- 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

- Discussion Type
- discussion topiccodomain stacks
- Category Latest Changes
- Started by Mike Shulman
- Comments 2
- Last comment by Urs
- Last Active Dec 28th 2009

Added comments to regular category, exact category, coherent category, and pretopos about under precisely what conditions the codomain fibration is a stack for the relevant Grothendieck topology. Also added some thoughts about "pre-lextensive categories" to extensive category.

- Discussion Type
- discussion topicAlgebraic categories
- Category Latest Changes
- Started by TobyBartels
- Comments 14
- Last comment by Mike Shulman
- Last Active Dec 24th 2009

I started algebraic category, with a note also at monadic adjunction.

- Discussion Type
- discussion topicoo-limits - contents
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by TobyBartels
- Last Active Dec 24th 2009

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)

- Discussion Type
- discussion topicbutterfly
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Dec 23rd 2009

started expanding butterfly

also butterfly diagrams at crossed profunctor

- Discussion Type
- discussion topicdeRham theorem for oo-Lie groupoids
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 22nd 2009

filled a gap in deRham theorem for oo-Lie groupoids on my personal web.

But I need to sleep over this...

- Discussion Type
- discussion topiccontractible objects in a lined topos
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Dec 22nd 2009

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 is such that all representable objects are contractible with respect to the line object , then the path oo-groupoid functor

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).

- Discussion Type
- discussion topicshape fibration
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by zskoda
- Last Active Dec 22nd 2009

An incomplete introduction to shape fibration and a related bio entry Sibe Mardesic.

- Discussion Type
- discussion topicBeck–Chevalley
- Category Latest Changes
- Started by TobyBartels
- Comments 7
- Last comment by zskoda
- Last Active Dec 22nd 2009

Somebody wrote quite the stub at Beck-Chevalley_Condition; I fixed the name but didn't mess with it otherwise. It seems to have appeared (properly named) on Mike's web too, all by the AnonymousCoward.

- Discussion Type
- discussion topicspace and quantity
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 21st 2009

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)

- Discussion Type
- discussion topicbar construction
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 18th 2009

homeopathic definition at bar construction

- Discussion Type
- discussion topic(n,r)-fibrations
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 18th 2009

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.

- Discussion Type
- discussion topicFixed broken links
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Dec 17th 2009

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

- Discussion Type
- discussion topicNew stubs
- Category Latest Changes
- Started by TobyBartels
- Comments 4
- Last comment by TobyBartels
- Last Active Dec 16th 2009

Just look at the list of all wanted pages, here are a few stubby articles: 1-topos, 2-Hilbert space, Alexandroff compactification, Banach manifold. (Yes, I went alphabetically.)

- Discussion Type
- discussion topicNo more orphaned pages!
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Dec 16th 2009

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.

- Discussion Type
- discussion topicoo-Lie theory - contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 16th 2009

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".

So I created a floating table of contents

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.

- Discussion Type
- discussion topicRiemann surfaces and Myers-Steenrod theorem
- Category Latest Changes
- Started by amathew
- Comments 3
- Last comment by amathew
- Last Active Dec 15th 2009

I created Riemann surface and Myers-Steenrod theorem.

- Discussion Type
- discussion topicNew page: [[classical mathematics]]
- Category Latest Changes
- Started by TobyBartels
- Comments 27
- Last comment by Urs
- Last Active Dec 15th 2009

I started classical mathematics to link to from internal logic.

- Discussion Type
- discussion topicframed bicategory
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by TobyBartels
- Last Active Dec 13th 2009

created quick stub for framed bicategory

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

- Discussion Type
- discussion topicEdit conflict at [[model structure on dendroidal sets]]
- Category Latest Changes
- Started by TobyBartels
- Comments 4
- Last comment by TobyBartels
- Last Active Dec 11th 2009

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.

- Discussion Type
- discussion topic(oo,1)-operads
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Dec 11th 2009

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

- Discussion Type
- discussion topicObjections by Maarten
- Category Latest Changes
- Started by TobyBartels
- Comments 3
- Last comment by Urs
- Last Active Dec 10th 2009

Or possibly not by Maarten. Anyway, there is some discussion at superdifferential form and Weil algebra that people into that sort of thing should look at.

- Discussion Type
- discussion topicCategorification vs laxification
- Category Latest Changes
- Started by TobyBartels
- Comments 20
- Last comment by TobyBartels
- Last Active Dec 10th 2009

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.

- Discussion Type
- discussion topichomotopy coinvariants
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Dec 10th 2009

New item homotopy coinvariants functor, after a paper of Kathryn Hess. For now just a definition.

- Discussion Type
- discussion topicPresenting categories with large simplicial sets
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by Harry Gindi
- Last Active Dec 10th 2009

More discussion about this at category theory.

- Discussion Type
- discussion topicGavin Wraith wrote [[geometric theory]].
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Dec 9th 2009

Gavin Wraith wrote geometric theory.

- Discussion Type
- discussion topicEvil composition
- Category Latest Changes
- Started by Harry Gindi
- Comments 2
- Last comment by Andrew Stacey
- Last Active Dec 9th 2009

- Discussion Type
- discussion topicTypes of shapes of limit cones
- Category Latest Changes
- Started by domenico_fiorenza
- Comments 5
- Last comment by TobyBartels
- Last Active Dec 9th 2009

- 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).

- Discussion Type
- discussion topicPersonal web updated
- Category Latest Changes
- Started by Guest
- Comments 7
- Last comment by Guest
- Last Active Dec 9th 2009

- 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

- Discussion Type
- discussion topicEckmann-Hilton duality
- Category Latest Changes
- Started by zskoda
- Comments 11
- Last comment by zskoda
- Last Active Dec 8th 2009

New entry Eckmann-Hilton duality. Discussion welcome.

- Discussion Type
- discussion topicCategory theory
- Category Latest Changes
- Started by Guest
- Comments 3
- Last comment by TobyBartels
- Last Active Dec 7th 2009

- 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

- Discussion Type
- discussion topicuniversal algebra in a monoidal category
- Category Latest Changes
- Started by Andrew Stacey
- Comments 35
- Last comment by Andrew Stacey
- Last Active Dec 7th 2009

Created universal algebra in a monoidal category

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!

- Discussion Type
- discussion topicDefinition of extended TQFT
- Category Latest Changes
- Started by domenico_fiorenza
- Comments 6
- Last comment by domenico_fiorenza
- Last Active Dec 5th 2009

- 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"

- Discussion Type
- discussion topicLong time lurker, first time writer
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Dec 5th 2009

One of these has started (or continued) a conversation at the bottom of graph.

- Discussion Type
- discussion topicferrim
- Category Latest Changes
- Started by Andrew Stacey
- Comments 9
- Last comment by Andrew Stacey
- Last Active Dec 4th 2009

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)?

- Discussion Type
- discussion topicNormal monomorphisms between dendroidal sets.
- Category Latest Changes
- Started by TobyBartels
- Comments 3
- Last comment by TobyBartels
- Last Active Dec 4th 2009

I asked a question at dendroidal set and wrote normal monomorphism to explain it.

- Discussion Type
- discussion topicsymmetric spectra
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 4th 2009

added to symmetric monoidal smash product of spectra a link to a pdf with seminar notes that nicely recall the definition of the symmetric monoidal category of spectra.

- Discussion Type
- discussion topicH-space
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Dec 4th 2009

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.

- Discussion Type
- discussion topicAnonymous question
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Dec 4th 2009

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

- Discussion Type
- discussion topiccube
- Category Latest Changes
- Started by Guest
- Comments 1
- Last comment by Guest
- Last Active Dec 4th 2009

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

David Roberts

- Discussion Type
- discussion topic1-category equipment
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by TobyBartels
- Last Active Dec 4th 2009

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.

- Discussion Type
- discussion topicn-connected spaces and E-n spaces
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Dec 3rd 2009

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?

- Discussion Type
- discussion topicequipment
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Dec 2nd 2009

Oh, and I copied over most of my exposition from the cafe post on equipments to 2-category equipped with proarrows.

- Discussion Type
- discussion topicsmall presheaves
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Dec 2nd 2009

Created small presheaf, and replaced the very old discussion at Grothendieck universe by a link to small presheaf.

Does anyone know any references that treat the case of small sheaves, in this sense?

- Discussion Type
- discussion topicdirected colimit
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by zskoda
- Last Active Nov 30th 2009

I added to directed colimit the -directed version, for some regular cardinal .

We should maybe also add to directed set the -directed version. What we currently descrribe there is just the -directed version.

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

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

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

- Discussion Type
- discussion topicthis and that
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 30th 2009

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