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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!
Opened a page at higher order proposition.
expanded the previously pitiful monomorphism
Discussion resumes at the bottom of graph.
Added some references to Plebanski formulation of gravity
created stubs for gravity as a BF-theory and first-order formulation of gravity
On request by David Corfield, I wrote a bit about symplectic geometry and classical hamiltonian mechanics
I borrowed a nice description of Hasse diagrams from Toby.
I weakened the definition of covering relation for directed graphs so that satisfy the covering relation if their is an edge and no other path from to . The condition that there is no with edges and is too strong.
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)
created page for Lawrence Breen
John put in a definition at Grassmann algebra. Should these be distinguished from exterior algebra?
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.
Our new contributor Aaron F would like people to check this at induced representation.
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.
created bisimplicial object with two useful props.
Also linked to it from simplicial object
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.
Lars Kindler has joined to edit D-module.
I mentioned these at higher category theory and (n,n)-category, where it had been implied that the latter were as far as one could go in increasing the second parameter.
I think Ab-enriched category is a better name for the page than ringoid.
Jason Dusek joined and inspired me to start 2-rig.
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 wrote two-valued topos to help me tighten up Mike's latest edit to cocomplete well-pointed topos.
Reply to Roger Witte at foundations.
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 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.
Expanded the "Idea" section at A-infinity category.
Hugh Thomas joined to edit quiver
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.
Wrote a proof at cocomplete well-pointed topos that characterizes Grothendieck universes and Set.
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.
started filling in material into the planned database of smooth toposes at Models for Smooth Infinitesimal Analysis.
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 .
Presently I feel that I want eventually a discussion of all those seemingly boring old friends such as and and etc. in lined toposes and smooth toposes. Inspired not the least by the wealth of cool structure that even just carries in cases such as the -topos in Models for Smooth Infinitesimal Analysis.
created Fivebrane group but was being lazy:
essentially copy-and-pasted the intro from String group and then left a link to Fivebrane structure.
Then I went through String structure and Fivebrane structure and added links to String group and Fivebrane group.
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.
created homotopy group (of an infinity-stack)
a bit rough for the time being.
Also added a suitable link and short remark at homotopy group.
Vishal Lama joined the Lab!
on his page he promises to create Lab pages on some books on category theory and topos theory. Great, I am looking forward to it
started working on
schreiber:path-structured (infinity,1)-toposes
This is a kind of survey of some constructions I've recently been spamming the nLab with.
Roger Witte asks a question at foundations that looks interesting but which I haven't really thought about yet.
I added a paragraph on structural meaning to axiom of foundation.
I added the Lab itself to Online Resources, since that list is getting a lot of attention and may well be copied to other contexts.
I moved the redirect for de Rham cohomology from differential form to de Rham complex.
Todd just wrote Gram-Schmidt process; I added a bit.
As planned here
pairing — pretty simple, but not to be confused with the product
… we now have globular category.
created super smooth topos