nLab 2009 April changes

##Archive## * [current](http://www.math.ntnu.no/~stacey/Mathforge/nForum/?CategoryID=5) * 2009 September * 2009 August * 2009 July * 2009 June * 2009 May * 2009 April * 2009 March * 2009 February * 2009 January * 2008

Archive of changes made during April 2009. The substantive content of this page should not be altered. For past versions of this page beyond its own history, start here and work backwards.

2009-04-29

• Mike:

• Moved the discussion about comma categories from this page to the query box at comma category.
• Will look at descent when I get a chance.
• Created a couple of firefox search plugins for the nLab and uploaded them to the HowTo.
• Urs:

• Urs:

• filled the “details” section at descent for simplicial presheaves with the relevant material copy-and-pasted from descent.

• keep polishing, expanding and rearranging descent – when Mike comes back online I am hoping to discuss a bit more the relation between Street’s descent for $Str \omega Cat$-valued presheaves and the standard descent for their SSet-valued image under the $\omega$-nerve. As the new version indicates: the homotopy limit may be a red herring and the lack of monoidalness of the left adjoint of the nerve might be fixed by recourse to stratified simplicial sets using Verity’s results (?)

• added a pullback description to double comma object

• Mike:

• I’ve noticed that Urs (in particular) has been widely using the notation $(f,g)$ for comma categories that Toby tried to prevent me from denigrating. But I haven’t heard any convincing arguments in favor of that notation, and I have lots of reasons to dislike it; see the discussion at comma category. If general opinion is against me, I’ll shut up, but I want to hear from more people. If you use the notation $(f,g)$, do you have positive reasons to prefer it to $(f/g)$ or $(f\downarrow g)$?

2009-04-28

• Mike:

• Urs:

• created Bousfield-Kan map

• created category of simplices

• following Toby’s suggestion I moved descent and codescent to descent – then I entriely rewrote it! Now it starts with very general nonsense on localization of $(\infty,1)$-presheaves and then derives descent conditions as concrete realizations of that localization. Currently where it ends I am planning to add discussion about how to further get from descent to gluing conditions (i.e. to $\Delta$- and oriental-weighted limits) following discussion that I am having with Mike on the blog here

• added more details to ind-object, relating the two different definitions

• added standard examples of presheaves on open subsets to inverse image

• started adding a list “properties” to colimit analogous to the one at limit

• Since I'm making up a word at exponential object, I decided that both terms should be adjectives, leaving the verb simpler.
• I included some lower-dimensional cases at associahedron. I managed to get down to $K_1$.

2009-04-27

• Urs

• added a list of basic propeties to the end of limit

2009-04-23

• Mike: Since no one objected to my proposal on how to resolve the duplication between category of fractions and multiplicative system, I implemented it. The relevant material is now at calculus of fractions. I deleted a bit of the material about derived functors because it was not really specific to calculi of fractions, belonging more at derived functor.

• Toby Bartels: A question (not a dispute!, what do you know?) on terminology at exponential object.

• Mike:

• Urs:

• created path groupoid – other realizations of that idea should be stated there, too

• started creating a random list of some examples at limits and colimits by example – but not in the intended detailed form yet

• started reworking local system as we discussed there – still lots of room for improvement left, of course!, in particular many references could use more details and links

• notice that I moved the discussion box to the end of the entry; not to imply that the discussion is closed, but so that the box does not disturb the structure of the entry; I think this is reasonable for every discussion that concerns an entry as a whole more than a particular point in it
• From our first spam post (which I will not link to), I'd like to preserve this: ‘with a country like India, the limits are endless’ (links added).
• Wrote k-tuply connected n-category. I also made the numbering consistent on the other pages; connectedness should be $1$-tuple connectedness, not $0$-tuple connectedness.
• Tried my hand at a constructive version of cyclic order. Interestingly, the apartness relation doesn't seem to be recoverable from the cyclic order relation (which is related to the difficulty of doing antisymmetry and thus a reflexive version, which as well is not —even classically— equivalent to the negation of the irreflexive version).
• Urs:

• have a reply at local system

• found time to work a bit more on Lie infinity-algebroid representation

• request to Tim and others: there is naturally a kind of “twisting function” appearing there. One aim of my presentation is not to postulate it, but to derive that it arises from a (co)fibration of DGCAs. I am thinking that all the twisting functions and twisting cochains should have such a conceptual definition, from which one derives the usual component-wise definition by unravelling the component-wise mechanisms. I think here on the $n$Lab we should try to find and give conceptual categorical explanations as far as possible. My personal feelling is that all discussion of “twisting cochains” and related phenomena would become considerably clearer and less confusing if one had this.
• Toby Bartels: Please note that there are no actual links to Differential Nonabelian Cohomology (except this one just now). That there appear to be is actually a bug in Instiki (which I haven't bothered to report to Jacques yet).

• Tim:

• I have added a request at local system. Basically the current entry reads as if it related to a relatively recent idea. I suggest we look at the origins of the idea, at least as old as ‘Steenrod (1943)’ if not before. It is central to much of the nLab work. Probably we need to be much less restrictive in the motivation of this entry.

• I have added some historical and motivational perspective in twisting function and would suggest that a similar section is needed for twisting cochain. The two threads of twisting the fibre and deforming the local structure of a ‘product’ are at the origin of both concepts.

2009-04-20

• Urs:

• Toby Bartels: I think that the naming discussion at ind-object is still current until Eric is happy.

• Urs responds: sure, I just thought for readability the discussion would be better had at the bottom of the page – a big green box in the middle of the entry gives the reader the impression that he or she needs to beware of some urgent unsettled issue before reading on.
• Toby: Ah, whereas I see the big green box up top as simply indicating an unsettled issue, urgent or otherwise.
• Urs: okay, that makes sense, let’s leave it this way – maybe eventually we need some mechanism to decide when to move a disscussion box to the bottom of the page – probably the mechanism should be that the one who started the discussion is the one decide
• Finn Lawler: Replied to Toby at minimal logic, and slightly expanded paraconsistent logic to incorporate some of our discussion.

• Urs:

• added to Yoneda lemma the following: a word on the proof, two further corollaries, a word on the meaning of the first two corollaries

• Urs:

• started polishing the typesetting at bundle gerbe, but there is still plenty of room for further improvement

• added a summary list to the section “Example: universes in SET” at universe in a topos

• Mike:

• imported Urs’ material on bundle gerbe. nLab doesn’t seem to like latex within lists. How do we fix this?
• Urs:

• Yet more at pure set. See the pretty pictures! Or rather … make them pretty if you know how, for I do not. (;_;)
• Another question for Finn, now at minimal logic.
• More correct material at pure set.

2009-04-14

• Urs:

• added a discussion to universe in a topos with more details on how to get back the Grothendieck universe axioms in $SET$. Please check. I’d be grateful for improvement.

• to Andrew: I think we want here on the $n$Lab as much detail as we can get hold of – if an entry becomes too long, though, it might be an option to split off entries from it “more details on xyz” or the like and link to them

• Andrew Stacey: Added the basic definitions to Chen space and some of the other variants of generalised smooth space. How much detail do we want on these pages?

• The English Pedant: I’m not convinced about the use of the word “heuristic” on the n-Lab. I’ve started a discussion on the n-Forum rather than force my views on the English language on the n-Lab. I realise that this is a little against the Wiki-spirit but I figured that if I went through changing all occurrences of the word “heuristic” then someone would object and we’d have a discussion about it; so to save a bit of agro, I’m instigating the discussion first.

• Toby Bartels: I started a new category, foundational axiom, in which I put the pages that contain axioms that one might (or might not) want in one foundations of mathematics. That way, anyone with opinions on the matter can check them to see that one's views are represented. (A few don't really have much in the way of an axiom right now, but one could be added or noted there.) This does not include things like the axioms for a group, but rather axioms for set theory (or other foundational theory). (Although set theory itself I don't think should really be included, but ETCS is in there.)

2009-04-09

• Urs:

• I had been asked by students to say something about why they should care about learning about $(\infty,1)$-categories. I thought that would be a good thing to try to answer in an $n$Lab entry, so I started an entry why (infinity,1)-categories?. This is just a first attempt. Maybe somebody would enjoy adding his or her own points of views of correcting/improving mine.

• created Connes fusion, but filled in only pointers to further references

• Zoran Škoda: created von Neumann algebra emphasising on sources of relations to category theory and low dimensional topology (particularly G. Segal’s program on relations between CFT and elliptic cohomology). There is a good wikipedia entry on von Neumann algebras with lost of references and details, but neglecting the connection to the above topics which should be expanded on. Moreover somebody should mayve write entres on related topics as Connes fusion, modular functor etc. as those are relevant for some of us.

• Toby Bartels: I came to some sort of decision at direct sum.

2009-04-05

• Urs:

• added a few links to examples at space and quantity (we have a general problem that many entries created eraly on don’t currently point to entries created more recently which de facto they should point to)

• touched combinatorial spectrum: replied to Mike, added a list of examples and have further questions

2009-04-01

category: meta

Revised on September 4, 2010 21:42:36 by Toby Bartels (173.190.156.19)