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

• I have just posed the question:

 If we want to weaken this even further to provide a
simplicial model of, for example, a ((?,2)-category?,
how would we do this?

Would we apply the lifting condition on all but three of
the indicies… and if so which three? (The first, last and ????)


Any and all thoughts would be appreciated.

• I created a page for Riemannian metric based on a "blog post": http://deltaepsilons.wordpress.com/2009/10/27/riemannian-metrics-and-connections/ and a suggestion of Urs Schreiber.

• I added an "idea" to loop space . Not claiming, though, that everybody will find this idea the most helpful one. But to some extent I think it is.

I had another look at delooping

Eric, you drew some nice-looking diagrams there in the discussion section. At some point in the discussion I say that I don't understand these diagram. I still don't! :-)

It would be nice if we could converge on this, because then we could move the diagrams out of the discussion into the text as a useful illustration.

Could you describe in words what you mean these diagrams are depicting? I am guessing that probably we are just thinking of what an arrow and a point means in such a diagram differently. Let's sort this out. If we agree that the diagrams make sense they should feature more prominently, if we come to the conclusion that there is some misunderstanding we should put a clearer warning to the reader.

• I have been polishing the entry Chevalley-Eilenberg algebra on my personal web a bit.

I thought it would be good to announce here what it is that I am currently thinking about. If nothing else, this will explain which entries you all see me working on here and thereby maybe facilitate interaction more.

So currently I am thinking about the sought-for proposition that is now stated in the section Properties at the above entry. It sure looks like something like this proposition ought to be right, but I am not there yet.

• I was kindly being alerted that the following long-awaited references are now available:

Paul Goerss's account of the Hopkins-Miller-Lurie theorem, now linked to at A Survey of Elliptic Cohomology

Lurie part VI on little cubes oo-operads, now linked to from Jacob Lurie

• Rather than ask whether it's worth it and have Urs say "do it, don't talk about it!", I started a page to compare different notions of completion. Fortunately, we are well supplied with experts on the subject. What would be great would be a comparison of different completion processes. How widely they are applicable, e.g., to the enriched case? In which situations two or more coincide, etc.
• A question at strict epimorphism for Mike Shulman. Or anyone else who has thought about bicategorical notions of epimorphisms.

-David Roberts
• Started a list at n-category of all the existing definitions of higher categories and comparisons between them. I'm sure I'm missing some, so please help!

• Discussion resumes at the bottom of graph.

• I made a UC Riverside Seminar on Cobordism and Topological Field Theories page to record all the notes from the seminar. Kind of like a seminar webpage, but in the nLab. Heh, I'm at home so I can't post comments to the n-category cafe (I'm trying to sort this out, my computer is not allowed to post comments currently).
• I got tired off looking at a question mark on my personal web and added a stub for Hilbert schemes.
• I weakened the definition of covering relation for directed graphs so that $(x,y)$ satisfy the covering relation if their is an edge $x\to y$ and no other path from $x$ to $y$. The condition that there is no $z$ with edges $x\to z$ and $z\to y$ is too strong.

• This comment is invalid XML; displaying source. Following discussion here <a href="http://www.math.ntnu.no/~stacey/Vanilla/nForum/comments.php?DiscussionID=244&page=1#Item_3" >here</a>, I decided it made sense to have an <a href="http://ncatlab.org/nlab/show/FAQ#how_can_i_get_a_personal_section_of_the_nlab_14" >FAQ entry on personal pages.</a> It doesn't say much at the moment, but I guess the only important piece of information is that you have to write Urs.
• created pages for Tony Pantev and Ludmil Katzarkov

(not much there yet, am just following the idea that it would be nice that most authors whose references are cited in some entry also have a page with at least a link to their website)

• I added the case of Set-enriched category theory to the example section of Cauchy complete category (thanks to David Corfield for fixing my LaTeX errors), and inserted the definition at Karoubi envelope. There is an issue of choosing how to split idempotents which someone like Toby might want to say something about.

• created page for Johan Louis Dupont, cited at simplicial deRham complex

(given that at that entry I am trying to merge some of Dupont's work with some of that of Anders Kock, it is curious that JL Dupont and Anders Kock are decade-long colleagues in Aarhus, as Anders Kock kindly reminds me a minute ago)

• To the entry on regular category I added a brief note describing an application of this idea and the calculus of relations to a paper of Knop. For the future I will try to flesh this note out as well as add a page on tensor categories.

By the way, does the definition of a tensor category have to include linearity? It seems that the definitions vary depending on where one looks (e.g. whether the monoidal structure is an additive functor). Thanks.

• I started an entry simplicial deRham complex

on differential forms on simplicial manifolds.

In parts this is for me to collect some standard references and definitions (still very incomplete on that aspect, help is appreciated -- is there a good reference by Dupont that is online available?)

and in parts this is to discuss the deeper abstract-nonsense origin of this concept.

I am thinking that

• with differential forms understood in the synthetic context as just the image under Dold-Kan of the cosimplicial algebra of functions on the simplicial object of infinitesimal simplices in some space

• it follows that the simplicial deRham complex of a simplicial object is just the image under Dold-Kan of the cosimplicial algebra of functions on the realization of the bisimplicial object of infinitesimal simplices in the given simplicial space.

This looks like it is prretty obvious, once one stares at the coend-formula, but precisely that makes me feel a bit nervous. Maybe i am being too sloppy here. Would appreciate you eyeballing this.

• Began entry with that name.
• I wrote Poincare group as an entree to the project of carrying on in nLab the blog discussion on unitary representations of the Poincare group. I'm not a specialist of course, so I ask the experts to please examine for accuracy.

• I expanded and polished the discussion of the abstract definition of of G-principal oo-bundles in an arbitrary (oo,1)-topos at principal infinity-bundle.

Parts of this could/should eventually be moved/copied to action and action groupoid, but I won't do that now.

I'd be interested in comments. One would expect that for the case that the ambient (oo,1)-topos is Top this style of definition should be well known in the literature, but I am not sure if it is. In fact, the examples listed further below in the entry, (the construction by Quillen and the Stasheff-Wirth construction) seems to indicate that this very simple very general nonsense picture has not been conceived as such before. Could that be true?

• I've removed the request for help link from the main contents. It didn't get used much (though I got answers to my questions there!). Since we have yet to actually delete a page, rather than just blank the request for help page I've put a pointer to where one can ask questions (pretty similar to that on the FAQ).

• I created a page for S-Sch as a notation for S-schemes to refer to in another post. Zoran pointed out that the notation is nonstandard (I do not know why I thought it was normal) and changed the title to Sch/S. I thus changed the first sentence to read Sch/S instead.

• I added a description of the degenerate affine Hecke algebra to the Hecke algebra page as one of the many variants.

I added the categorical generalization of Schur's lemma to that page.

I wrote a short stub on the additive envelope of a category, which Mike Shulman has expanded.

I mentioned the generalization of the Morse lemma to Hilbert manifolds.

I added the generalization of Hilbert's basis theorem to the case of where the ground ring is noetherian (not necessarily a field).

I wrote a short page on the Eilenberg swindle.
• I see that Akil Mathew has worked on a bunch of entries. Great! We should try to contact him and ask hom to record his changes here.

• I added Alex's recent lecture notes to cobordism hypothesis and in that process polished some typesetting there slightly.

Then I was pleased to note that Noah Snyder joined us and worked on fusion category. I created a page for him.

• I don't think that the (non-full) essential image of an arbitrary functor is well-defined.

• I added a fairly long (but still immensely incomplete) examples section to smooth topos.

I mention the "well adapted models" and say a few words about the point of it. Then I have a sectoin on how and in which sense algebraic geometry over a field takes place in a smooth topos. here the model is described easily, but I spend some lines on how to think of this. In the last example sections I have some remarks on non-preservation of limits in included subcategories of tame objects, but all that deserves further expansion of course.

• I continued working my way through the lower realms of the Whitehead tower of the orthogonal group by creating special orthogonal group and, yes, orthogonal group.

For the time being the material present there just keeps repeating the Whitehead-tower story.

But I want more there, eventually: I have a query box at orthogonal group. The most general sensible-nonsense context to talk about the orthogonal group should be any lined topos.

I am wondering if there is anything interesting to be said, from that perspective. Incidentally, I was prepared in this context to also have to create general linear group, only to find to my pleasant surprise that Zoran had already created that some time back. And in fact, Zoran discusses there an algebro-geometric perspective on GL(n) which, I think, is actually usefully thought of as the perspective of GL(n) in the lined topos of, at least, presheaves on $CRing^op$.

Presently I feel that I want eventually a discussion of all those seemingly boring old friends such as $\mathbb{Z}$ and $\mathbb{R} / \mathbb{Z}$ and $GL(n)$ etc. in lined toposes and smooth toposes. Inspired not the least by the wealth of cool structure that even just $\mathbb{Z}$ carries in cases such as the $\mathbb{B}$-topos in Models for Smooth Infinitesimal Analysis.

• created a page for Haynes Miller, since I just mentioned his name at string group as the one who coined that term.

not much on the page so far. Curiously, I found only a German Wikipedia page for him

• I've started listing differences between iTeX and LaTeX in the FAQ. That seemed the most logical place (I don't think we want a proliferation of places where users should look to find simple information) so either here or the HowTo seemed best. I chose the FAQ because the most likely time someone is going to look for this is when they notice something didn't look right.

The issue is that whilst iTeX is meant to be close to LaTeX they are never going to be the same so it's worth listing known differences with their work-arounds.

So far I've noted operator names, whitespace in \text, and some oddities on number handling.