nLab 2009 February changes

Archive of changes made during February 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.

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2009-02-28

• Tim Added new material to crossed square.

2009-02-27

• Urs

  • edited action a bit more – but that entry is still not nice

  • extended geometric function theory

  • continued replying to Mike at hyperstructure

2009-02-26

• Urs

  • created action of a category on a set in reply to a question raised over at Secret Blogging Seminar here

• Andrew

  • Created a request for help? page. There are query boxes scattered all over the lab and I think that it ought to be a priority to answer these. A query on a page might be answerable by someone not an expert in what’s on that page (or even not interested in what’s on that page) - in fact, that’s highly likely. Flagging questions on this page (i.e., Latest Changes) is also less than optimal as it’s not clear who might be able to answer them (and probably not everyone checks this page). I freely admit that I’m doing this because no-one seems bothered in my questions on Froelicher spaces which are, in fact, more general category theory than specifics to Frölicher spaces.

    I’ve added it to the sidebar (and taken off General Discussion).

2009-02-25

• Urs

  • created enriched functor category for completeness

  • added a bit more details and references concerning the SSet homotopy limit at weighted limit

  • after checking with Mike added details of the definition of homotopy limits in terms of weighted limits in the SSet-enriched context to weighted limit; also edited accordingly the section local definition at homotopy limit and created a section on weighted limits at representable functor

2009-02-24

• Urs

  • have a question at the end of weighted limit for the homotopy coherency experts

  • created weighted limit

  • added references to sheaf and simplicial set

2009-02-23

• Tim:

  • Created an entry on the Conduché decomposition of the group of $n$-simplices of a simplicial group: decomposition theorem for simplicial groups. (An idea for renaming this entry would be welcome!)
  • Edited group T-complex.

2009-02-19

• Mike: created n-fibration.

  Why do I think of the Beach Boys every time I read the word $n$-fibrations? - Eric

• Ronnie: over the last two days, created compositions in cubical sets which gives the basic definitions and axioms for cubical sets with compositions and connections, (and inverses), with the example of the cubical singular complex in mind.

2009-02-18

• Mike:

  • Created Euler characteristic in order to ask a question there.

2009-02-17

• Toby Bartels:

  • Separated equivalence of categories from equivalence.
  • Basically copied 2-category to 3-category, then added a bit to the end of each to make the difference nontrivial. Also created strict 3-category along similar lines, although it's still rather short.

• Mike:

  • Introduced the terminology strict homotopy limit by analogy with strict 2-limit.

  • Rearranged homotopy pullback to make it fit better with homotopy limit.

2009-02-16

• Toby Bartels:

  • Fixed links to equivalence that should really be to weak equivalence.
  • Separated Segal space from complete Segal space.

• Mike: asked a question about horizontal categorification.

2009-02-15

• Tim: Added the construction of $\overline{W}$, the classifying space construction for simplicial groups to the entry for simplicial group.

• Urs

  • created topological T-duality

  • created homotopy pullback

  • created weak equivalence

  • replied at dg-category

2009-02-14

• Ronnie Added entries on C. Ehresmann and on Grothendieck. The second probably should be expanded, but it has a link to a good account.

• Tim: Asked a question in dg-category as to the degree of the differentials. We may need a convention on this.

• Urs

  • added Toën-Vezzosi proposal $2Vect := Ch(Vect)-Mod$ to 2-vector space and created dg-category accompanying this

2009-02-13

• Mike:

  • Incorporated the reference Tim suggested at homotopy limit.

  • Replied at strict 2-limit and comma category.

• Tim:

  • Added something to homotopy limit in part to provide a possible ‘old’ reference in reply to Urs and Mike below.
  • Started adding some summaries of results to group T-complex.

• Toby Bartels:

  • Imposed the convention for a group $G$, that $\mathcal{B}G$ is a space while $\mathbf{B}G$ is a groupoid, on classifying space and homotopy 1-type (although the groupoid has disappeared from that one), to hopefully reduce confusion.
  • Linked FAQ and HowTo from one another.
  • Terminological notes at strict 2-limit and comma category.

• Mike: Urs, in my biased opinion, the reference for the homotopy limit comparison that I find easiest to understand is my own paper (reference added). Of course the fact has been known for a long time; I don’t even know whether there’s any “original” source one could cite.

2009-02-12

• Urs: at homotopy limit I am asking for further details and/or literature on the “central theorem” which identified local and global definition

• Mike Shulman:

  • Added what I think is a more standard viewpoint to classifying space, although there are so many different meanings of that term.

  • Created comma object and started messing around with using SVG for diagrams instead of \array. It’s more work but it sure does look nicer (in the output, anyway).

    Urs: looks good! We should have at FAQ a brief instruction for how to proceed to produce such diagrams (which software to download, how to get nicely typeset labels, etc.)

    Mike: I gave it a try, but I don’t have any great wisdom to impart; I just played around with the XML until it worked. If there is software that can do this for us that would be great (until someone manages to write an xypic-to-svg converter).

2009-02-11

• Toby Bartels:

  • Agreed with Mike at large site.
  • Replied to Andrew at Toby Bartels.

• Mike

  • I like the edits at 2-limit and strict 2-limit, so I removed the comments (but anyone else with opinions, feel free to restart the discussion).

  • Naming question for large sheaf.

  • Answered at model structure on simplicial presheaves

• Urs

  • have a question for Mike at model structure on simplicial presheaves

  • created Bousfield localization

• Toby Bartels:

  • Comments (with related edits) at 2-limit and strict 2-limit.
  • Expanded section and wrote retraction following it; similarly wrote split monomorphism following split epimorphism.
  • Separated k-tuply monoidal (n,r)-category from periodic table.
  • Linked to Pursuing Stacks from infinity-stack so it won't be an orphan. People more familiar with Pursuing Stacks than I am may want to link to it in other ways.

2009-02-10

• Urs

  • created sheaves on large sites mainly to ask the question which I ask there

  • created model structure on simplicial presheaves, model structure on simplicial sheaves

• Mike:

  • Wrote about effective-epimorphic sieves at subcanonical coverage, by way of answering questions about anafunctors.
  • Wrote comma category. We should probably be consistent between under category/coslice category and slice category/over category about which is the page and which is the redirect. I’m pulling for slice and coslice myself.

• Andrew

  • Started the FAQ. (The link on HomePage could possibly be removed - I put it there to start the FAQ but then also put a link in the sidebar.)
  • Okay, only a minor point: I corrected the syntax of the query block on the HowTo page. The syntax that was there works, but is not the “standard” syntax as defined by the maruku filter.

• Toby Bartels

  • Finally contributed to the discussion at anafunctor.
  • Asked a question at subcanonical coverage.
  • Wrote finite group, since they're mentioned on profinite group.
  • Clarified essentially algebraic theory a bit.

• David Roberts

  • Continued discussion with Mike at anafunctor, and moved the discussion to its own section. I disagreed about the use of ‘’regular epimorphism’‘ for covers in categories without all pullbacks.
  • Discussion at generalized smooth space, and created (a very sketchy!) page profinite group.

• Toby Bartels:

  • Fixed the theorem environments at cat-1-group (you need six hash marks).
  • Defined $(n,r)$-fold categories at n-fold category, just because I could.

• Urs

  • created Courant Lie algebroid

  • reacted to Toby‘s remark at n-fold category by incorporating it and removing the query box

2009-02-09

• Mike:

  • Replied at internal logic.
  • Merged the discussion at directed set into the entry and removed the discussion. Writing locally presentable category and accessible category is on my to-do list, if no one beats me to it.
  • Raised an objection at entire relation.

• Toby Bartels:

  • Removed the discussion at internal logic and entire relation (which I thought I'd already corrected!), since Mike is right and I'm satisfied.
  • Wrote entire relation and functional relation (but not yet relation!).
  • Asked a question about $n$-fold groupoids at n-fold category.

• Urs

  • fixed the mistake at interval object that Toby spotted and added a little bit of dicussion

  • added Berger-Moerdijk’s definition to interval object

• Toby Bartels:

  • Made an objection at interval object.
  • Talked to Mike at directed set, weak limit, and internal logic.

• Mike:

  • Replied to Toby at directed set, internal logic, and familial regularity and exactness.

  • Created weak limit.

2009-02-08

• Toby Bartels: Did some work on directed object, which I think should have half spun off to undirected object.

• Tim:

  • Created an entry on Dwyer-Kan loop groupoid.

• Toby Bartels:

  • A comment at directed space, which led me to write directed set just to clarify terminology.
  • Somewhat incorporated Ronnie Brown's remark on geometric shapes for higher structures.
  • Comments on pospace and cartesian monad.
  • Standardized the notation as $\Simp{C}$ for the category of simplicial objects in $C$, as on SimpSet, simplicial set (where I dropped an unrelated note to Tim), simplicial group, etc.
  • Explained where the name of Galois connection comes from.
  • Added some details to cobordism.
  • Wrote Practical Foundations about Paul Taylor's book and linked it from internal logic (where I also conversed with Mike).
  • I really like familial regularity and exactness, which confirms all my prejudices about regularity and exactness; I have a small, largely irrelevant, question.
  • I separated join and meet from union and intersection (which are terms that I'd use only for subobjects).

2009-02-07

• Ronnie Brown:

  I edited the discussion in fibration to give a reference to work of P.R. Heath on operations of the paths on the base on the fibres, a groupoid operation up to homotopy.

• Toby Bartels: I'm back! I'll be slowly catching up. So far, I've simply read latest changes and extended Andrew's navigation links for the archive. (So it looks like the sort of thing that daily updated websites create automatically using PHP, except that I did it all laboriously by hand!)

2009-02-06

• Jocelyn?: edited the product page to link to an interactive demonstration written to use the same notation as in product.

• Mike: responded to all of David’s edits.

• David Roberts:

  • Edited anafunctor to clean up conditions on covers, and some explanations of what each one is for.

  • Created the page Grothendieck pretopology

  • Added mention of Dold fibration on the page fibration.

2009-02-05

• Mike:

  • Continued discussion with Urs at hyperstructure.

  • Moved thin to thin element in line with the naming convention that page names be singular nouns. Also added some other contexts in which “thinness” appears.

• Urs

  • replied to Mike at hyperstructure

2009-02-04

• Ronnie Brown

  • added discussion of strict versions of higher homotopy groupoids to fundamental infinity-groupoid

• Urs:

  • added a “laboratory section” to hyperstructure in which I am proposing a formalization of the concept

  • created dagger category; started filling dagger compact category with content

2009-02-03

• Andrew

  • Implemented the archive of latest changes (scroll down to the bottom of this page to get the link)
  • Added a fair bit more on Isbell Duality in Froelicher spaces. A few more general queries for those a little more versed in the lore of categories than me.

2009-02-02

• Eric

  • created Position, Velocity, and Acceleration to help augment the discussion going on at the n-Cafe

• Andrew: continued with Froelicher spaces. In particular, laid out some of the details of the Isbell duality proof.

• Urs

  • added as a further example the computation of $holim(\mathbf{B}H \to \mathbf{B}G \leftarrow \mathbf{B}H) \simeq H \backslash\backslash G // H$ to homotopy limit

  • many thanks to Mike and Tim for all the further details at homotopy limit!

  • re Mike‘s suggestion: yes, I’d be in favor of archiving the latest changes list here more frequently

• Mike:

  • Answered Urs’s question in a very long-winded way by adding lots of stuff to homotopy limit.
  • What would people think about rotating (i.e. archiving) the latest changes? every month, rather than every year? It’s pretty big right now and it will probably be immense by next December.

2009-02-01

• Andrew: After crashing the nLab 3 times I’ve finally managed to upload something on Frolicher spaces. Something is seriously wrong with my syntax - if no one’s taken pity on me and cleaned it up I’ll do so myself tomorrow. More importantly, I have a few queries for the more experienced categorists so I’d appreciate someone taking a look at those for me.

• Urs: thanks, Mike, very useful. Is anything similar known for 3-categories?

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