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\section*{2009 June changes}

\hypertarget{archive}{}\subsubsection*{{Archive}}\label{archive}

[[!include all changes]]

Archive of changes made during June 2009. The substantive content of this page should \textbf{not} be altered.


\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{20090630}{}\section*{{2009-06-30}}\label{20090630}

\begin{itemize}%
\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item created [[Eilenberg-MacLane object]] prompted by the blog discussion \href{http://golem.ph.utexas.edu/category/2009/06/cohomology_and_homotopy.html#c024902}{here}


\item replied at [[homotopy]]



\end{itemize}

\item [[Toby Bartels]]:

\begin{itemize}%
\item Responded to [[Ronnie Brown]] at [[groupoid]].
\item Noted an incomplete thought at [[simplicial homotopy]].
\item Complained a bit at [[homotopy]].

\end{itemize}

\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item created [[Quillen adjunction]]


\item added definition to [[Quillen equivalence]]


\item reorganized [[model structure on simplicial presheaves]]:

\begin{itemize}%
\item expanded the idea/introduction-section


\item moved the original material to

\begin{itemize}%
\item [[global model structure on simplicial presheaves]] (and expanded that)


\item [[local model structure on simplicial presheaves]]



\end{itemize}

\item created [[local model structure on simplicial sheaves]]


\item [[rectified infinity-stack]]



\end{itemize}

\item fixed (hopefully) the nonsense paragraph (due to me) that [[Toby Bartels|Toby]] had pointed out at [[sieve]]



\end{itemize}

\item [[Tim Porter|Tim]]:

\begin{itemize}%
\item I have pointed out at [[simplicial homotopy]] that the combinatorial description has much greater validity than claimed there before. It works for simplicial objects in any finitely cocomplete category. (I think that is in Duskin's Memoir volume, but is not difficult to check directly.) It depends on having the [[copower]] with $\Delta[1]$, which brings me to


\item We have two notations for simplices in $SSet$. We use both $\Delta[n]$ and $\Delta^n$. I do not like the second one. I have started a query on this at [[simplicial set]].

\begin{itemize}%
\item [[Urs Schreiber|Urs]] that $\Delta^n$ is due to me -- I'll try to change it back to $\Delta[n]$ (same for horns) where I see it

\end{itemize}


\end{itemize}

\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item prompted by [[David Corfield|David]]`s remark at [[homotopy group]] I was led to all of the following:

I split [[homotopy]] into

\begin{itemize}%
\item [[homotopy (as a transformation)]]


\item [[homotopy (as an operation)]]



\end{itemize}
and moved the material originally found at [[homotopy]] to [[homotopy (as a transformation)]], kept only two commented links at [[homotopy]] and then wrote [[homotopy (as an operation)]] in the spirit of David's remark and in the light of the entry [[Eckmann-Hilton duality]] (but just some tentative abstract nonsense so far).

then finally I added to [[homotopy group]] a last section ``some abstract nonsense'' that indicates how the discussion of homtopy groups could be given analogous, just dual, to the discussion the way we have it (since recently) at [[cohomology group]]


\item expanded at [[cohomology group]] (standard abelian examples, standard nonabelian examples)

\begin{itemize}%
\item [[David Corfield]] has an interesting question there, I think he is right, will try to write something at [[homotopy group]]

\end{itemize}


\end{itemize}

\item [[David Corfield|David]]

\begin{itemize}%
\item added to [[Eckmann-Hilton duality]]

\end{itemize}

\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item continued polishing and adding statements and proofs to [[category of fibrant objects]]


\item created a stub for [[simplicial skeleton]] -- much more to be said here eventually



\end{itemize}


\end{itemize}
\hypertarget{20090629}{}\section*{{2009-06-29}}\label{20090629}

\begin{itemize}%
\item [[John Baez]]: I polished up the entry about the book [[John Baez:Towards Higher Categories|Towards Higher Categories]] on my personal web, and modified the [[Towards Higher Categories|nLab entry]] accordingly. I expect someday to have lots of my papers and books on the $n$Lab. Somehow I want them to be easy to find from the main $n$Lab, while discouraging other people from changing them (yet perhaps not making this impossible). The search feature on the main portion of the $n$Lab doesn't search people's personal webs, does it?

\begin{itemize}%
\item [[Urs Schreiber|Urs]]: I don't think it does

\end{itemize}

\item [[Toby Bartels]]: People keep linking to [[left adjoint]] and [[right adjoint]] when talking about [[adjoint functor]]s, even though these redirect to the general concept of [[adjunction]]. This is very natural, so I've now created those as pages in their own right. They consist of the definitions in a variety of contexts and then refer you to the other pages for more detail. While I was creating pages that collected information from other pages, I also wrote [[triangle identities]] and [[weak inverse]]. A picture in string diagrams would be very welcome at the former.


\item [[Urs Schreiber]] continued filling in propositions and detailed proofs at [[category of fibrant objects]]


\item [[Toby Bartels]] fleshed out [[discrete fibration]] a bit.


\item [[Eric Forgy]] created [[interval category]].


\item [[Zoran Škoda]]: created [[discrete fibration]], [[quasideterminant]] and made additions (please check for correctness) to [[orthogonal factorization system]], added Duskin's reference to [[gerbe (general idea)]] (btw it emphasises on internal point of view to gerbes -- as ``bouquets'', missing in division of gerbe entries in nlab).


\item [[Urs Schreiber]]

\begin{itemize}%
\item created reference entry [[Lectures on n-Categories and Cohomology]] in order to collect the existing $n$Lab entries on topics discussed there

\end{itemize}


\end{itemize}
\hypertarget{20090628}{}\section*{{2009-06-28}}\label{20090628}

\begin{itemize}%
\item [[Toby Bartels]] responded to Todd below.


\item [[Todd Trimble|Todd]]: piped in on a discussion between [[Bruce Bartlett]] and [[Urs Schreiber]] at [[sieve]]; put in a related two cents at [[subobject]].



\end{itemize}
\hypertarget{20090627}{}\section*{{2009-06-27}}\label{20090627}

\begin{itemize}%
\item [[Todd Trimble|Todd]]: added more material to [[Trimble n-category]], outlining Leinster and Cheng's extension from $n$ to $\infty$.


\item [[Eric Forgy]] has begun some pages related to administration of the Lab:

\begin{itemize}%
\item [[Organization of the nLab]]
\item [[On Scientific Contributions to the nLab]]
\item [[AnonymousCoward]] (which is not new but has a new discussion of privacy issues)
\item not so much administration but new suggestions for resources:\begin{itemize}%
\item [[Bibliography]]
\item [[Notation]]

\end{itemize}


\end{itemize}

\item [[Urs Schreiber]]

\begin{itemize}%
\item created a section ``Category theory for Trimble $n$-categories'' at [[Trimble n-category]] and put in a query box with a list of questions -- I'd be interested in whatever partial answers and comments


\item created [[Approaching Higher Category Theory]] (following blog discussion \href{http://golem.ph.utexas.edu/category/2009/06/this_book_needs_a_title.html#c024835}{here} and \href{http://golem.ph.utexas.edu/category/2009/06/this_weeks_finds_in_mathematic_36.html#c024834}{here})


\item added to [[point of a topos]] a section ``enough points'' with discussion about what it means for a topos to have ``enough points'' and two examples


\item created [[QFT with defects]]


\item created [[2-category of 2-dimensional cobordisms]]


\item added to [[cobordism hypothesis]] more details on formalization and proof by J. Lurie


\item reworked [[(infinity,n)-category of cobordisms]]


\item split off [[n-fold complete Segal space]] from [[(infinity,n)-category]]


\item created [[(infinity,2)-category]]



\end{itemize}


\end{itemize}
\hypertarget{20090626}{}\section*{{2009-06-26}}\label{20090626}

\begin{itemize}%
\item [[Toby Bartels]]:

\begin{itemize}%
\item Wrote [[upper set]] and [[lower set]] to satisfy more links (including the one from [[Chu construction]] that didn't mean what I thought that it meant!).
\item Started to move stuff from [[generalized tangle hypothesis]] to [[cobordism hypothesis]] but realised that I don't really know enough to disentangle the material ---at least not without reading the references, which I don't have time to do right now.

\end{itemize}

\item [[Urs Schreiber|Urs]]:

\begin{itemize}%
\item created [[gerbe (in nonabelian cohomology)]] based on the blog discussion \href{http://golem.ph.utexas.edu/category/2008/08/connections_on_nonabelian_gerb.html#c024774}{here}


\item removed the former section ``references'' at [[cohomology]] and replaced it with a new section ``History and references'' by using the material of my blog comment \href{http://golem.ph.utexas.edu/category/2009/06/cohomology_and_homotopy.html#c024778}{here}


\item added to [[Trimble n-category]] a section ``basic idea'' with a paragraph that [[John Baez|John]] suggested to include (\href{http://golem.ph.utexas.edu/category/2009/06/kan_lifts.html#c024776}{here})


\item added discussion that Kan complexes form a category of fibrant objects to [[category of fibrant objects]] that is slightly more direct than the argument using the full model category structure (crucial point being the theorem now at [[model structure on simplicial sets]] that acyclic fibrations of simplicial sets are characterized by a right lifting property).


\item added the example of a functor that is an equivalence of groupoids as inducing isomorphisms of simplicial homotopy groups under the nerve to [[simplicial homotopy group]]


\item created [[right proper model category]]


\item started filling in detailed proofs at [[category of fibrant objects]] -- am thinking that maybe eventually now the entry should be broken apart into a brief overview and one containing all the details



\end{itemize}

\item [[Toby Bartels]]:

\begin{itemize}%
\item Wrote [[up set]] and [[down set]] to satisfy some links.
\item Inspired [[David Roberts]] to write [[numerable open cover]].
\item Spun [[strict 2-group]] off of [[2-group]] and added a bit of material about weak and coherent $2$-groups.
\item Started [[(n,j)-transformation]] based on discussion at [[modification]]; also wrote [[j-morphism]] and [[n-functor]] to fill links from it.

\end{itemize}

\item [[Eric Forgy]] has a request at [[notation]].


\item [[David Roberts]]: created [[fibration theory]]. Also a comment at [[gerbe (general idea)]] about the fibration theory axioms.



\end{itemize}
\hypertarget{20090625}{}\section*{{2009-06-25}}\label{20090625}

\begin{itemize}%
\item [[Todd Trimble|Todd]]: Recently [[loop space]] was created. Also, I added some examples to [[Chu construction]] and to [[star-autonomous category]].


\item [[Urs Schreiber|Urs]]:

\begin{itemize}%
\item added the example of principal bundles to [[fibration sequence]]


\item added an example section to [[group cohomology]] with details of how the abstract-nonsense definition reproduces the familiar formulas


\item created [[twisted cohomology]]


\item created [[n-group]]


\item created [[group cohomology]]

\begin{itemize}%
\item the point of view adopted there is \emph{almost} fully explicit in Brown-Higgins-Sivera's book \emph{Nonabelian algebraic topology} , only that one needs to notice in addition that the morphisms out of free resolutions of crossed complexes discusses there are the [[anafunctor]]s that compute the morphisms in the corresponding homotopy category. Has this been made fully explicit anywhere in the existing literature?

\end{itemize}

\item fiddled a bit with the entry [[higher category theory]] (added one more introductory sentence, created a hyperlinked list of definitions of higher categories) but I still feel that we should put more energy in this particular entry. It is sort of the single central entry one would expect an ``$n$Lab'' to be built around, but currently it doesn't even come close to living up to playing such a pivotal role. I am imagining that it should carry some paragraphs that highlight the powerful recent developments in view of [[Pursuing Stacks]], of the kind that I filled in today in the entry [[Carlos Simpson]]. Does any higher category theory expert out there feel like writing an expositional piece for the $n$Lab here?


\item created [[simplicial localization]] but was then too lazy to draw the hammock. But main point here is the link to an article by [[Tim Porter]] that nicely collects all the relevant definitions and references

\begin{itemize}%
\item there is plenty of further material by [[Tim Porter]] that should eventually be referenced at various $(\infty,1)$-categorical entries. By chance I just came across the slides \href{http://www.ima.umn.edu/talks/workshops/SP6.7-18.04/may/PorterMay.pdf}{Weak categories} which I now linked to from [[weak omega-category]] and [[higher category theory]]

\end{itemize}

\item created [[principal infinity-bundle]], just for completeness


\item created [[principal 2-bundle]] -- this is just the result of what came to mind while typing, I am sure to have forgotten and misrepresented crucial aspects. [[Toby Bartels]] should please have a critical look and modify as necessary, as should [[Igor Bakovic]] and [[Christoph Wockel]] in case they are reading this.


\item based on a reaction I received concerning my comment below on the entry [[gerbe]] I have split the material into entries [[gerbe (as a stack)]], [[gerbe (general idea)]], [[bundle gerbe]] and kept at [[gerbe]] only pointers to these entries -- let me know what you think


\item created an entry for [[Carlos Simpson]] motivated by a link to a recent \href{http://math.unice.fr/~carlos/slides/ihesAGjan09.pdf}{pdf note} -- that [[Zoran Skoda]] kindly pointed me to -- where Simpson briefly sketches the topic of higher stacks, old and recent progress and putting his own contribution into context. I thought that was a nice short comprehensive collection of keywords and so I reproduce that text now at the entry [[Carlos Simpson]] with all the keywords hyperlinked


\item created [[fibration sequence]]


\item added an ``Idea''-section to [[gerbe]] supposed to be read as ``general idea'', where I try to describe the concept in a way independent of the notion of stack, relating it to princpal bundles, principal $\infty$-bundles, fibration sequences and cohomology . At the end I say ``in the following we spell out concrete realization of this idea''. Then I made [[Tim Porter]]`s material a section ``Realization of gerbes as stacks''. Eventually I'd like to add similarly ``Realization of gerbes as Stasheff-Wirth fibrations'', ``Realization of gerbes as bundle gerbes'' etc.

\begin{itemize}%
\item I hope that this is okay with you all, in particular I hope that [[Tim Porter|Tim]] doesn't mind me putting such a chunk of material in front of his work. If anyone has the impression that the chunk I added should rather be separated to a different entry, I'll have no objections. But please have a look first to see what I am trying to get at.

\end{itemize}


\end{itemize}

\item [[Tim Porter|Tim]]:

\begin{itemize}%
\item I have added material to [[gerbe]], although it is still a long way from explaining the link with cohomology with integer coefficients that was requested.


\item In the process of doing the above I have added a deconstruction section to [[torsor]], and created [[trivial torsor]].


\item I created a brief entry on [[Jean-Luc Brylinski]], but this is really a stub with a link to Wikipedia.



\end{itemize}


\end{itemize}
\hypertarget{20090624}{}\section*{{2009-06-24}}\label{20090624}

\begin{itemize}%
\item [[Finn Lawler]]: Added a proof (sketch) of the `lax Yoneda lemma' to [[lax natural transformation]]. Also replied re terminology at [[modification]] (thanks to Urs for a fantastic reply to my question there).


\item [[Zoran Škoda]]: created [[Hochschild-Serre spectral sequence]].


\item [[Eric Forgy|Eric]]: I've started the process of applying redirects for symbolic links. This maintains the original ascii titles and urls, but makes links look much better. I've also started adding redirects for plural nouns, e.g. you no longer have to type

. Now you can simply type

 and will be redirected automatically to [[category]]. To see the changes in action, have a look at [[higher category theory]]. Feedback welcome!


\item [[Zoran Škoda]]: Created [[Hopf envelope]], [[free Hopf algebra]] and [[matrix Hopf algebra]] simultaneously with posting a related comment to a discussion between Baez, Trimble and Vicary on free and cofree functors for bialgebras.


\item [[Tim Porter|Tim]]: I have created a stub on [[gerbe|gerbes]]. There was a request on the Caf\'e{} for en entry and I thought this was a good way to remind me to start one. \ldots{} but feel free to do it for me!


\item [[Zoran Škoda]]:

\begin{itemize}%
\item Created [[class of adapted objects]]. It would be better to say class of objects adapted to a functor (we talk half exact additive functors between abelian categories). Created [[Grothendieck spectral sequence]] and [[spectral sequence]]. Both need more input/work\ldots{}


\item I have added a general paragraph at the beginning of [[descent]] because far not all the cases of descent theory fit into the framework of sheaf/stack theory, Grothendieck topologies and homotopical methods. Namely something is a sheaf when it satisfies the descent for all covers, but there are cases when one considers just the descent problem for a single morphism, not all morphisms of anything like a Grothendieck topology. There is also a descent along families of noncommutative localizations, when one needs to resort to noncommutative generalizations of Grothendieck topologies. So descent theory is more general than the geometry of sheaves and stacks, though the latter is surely he most important part.



\end{itemize}

\item [[Urs Schreiber|Urs]]:

\begin{itemize}%
\item created [[K-theory spectrum]] with just a question


\item created a stub for [[tmf]] based on a recent Category Theory mailing list contribution -- am thinking that we should more generally try to move good stuff from the mailing list into $n$Lab entries


\item created [[Eilenberg-MacLane space]], [[Eilenberg-MacLane spectrum]]


\item am offering a reply to [[Finn Lawler]] at [[modification]]



\end{itemize}

\item [[Toby Bartels]]: Just more questions, instead of answers, for Finn and Gavin.



\end{itemize}
\hypertarget{20090623}{}\section*{{2009-06-23}}\label{20090623}

\begin{itemize}%
\item [[Gavin Wraith]] wrote [[matrix theory]], [[tensor product theory]], and [[bimodel]], with a question at the last.


\item [[Finn Lawler]]: created [[modification]] (and asked a question there) and [[lax natural transformation]], including a statement of what I think is the Yoneda lemma for them. I have a proof (I think), which I'll add a sketch of later, if no-one finds the statement obviously wrong in the meantime.


\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item added to [[weak factorization system]] the full details of the (elementary) proof that morphisms defined by a right lifting property are stable under pullback --


\item added the theorem to [[Kan fibration]] that Kan fibrations between Kan complexes that are also weak equivalences are precisely those with the right lifting property with respect to simplex boundary inclusions


\item added example of fundamental $\infty$-groupoid = singular simplicial complex to [[Kan complex]]


\item edited [[Kan complex]] slightly and added two little propositions characterizing groupoids in terms of their Kan complex nerves


\item added further illustrations to [[Kan fibration]]



\end{itemize}


\end{itemize}
\hypertarget{20090622}{}\section*{{2009-06-22}}\label{20090622}

\begin{itemize}%
\item [[Toby Bartels]]: A section on [[measurable space]] about the constructive theory due to Cheng. (This is \emph{not} a straightforward variation on the classical theory but distinctly a generalisation, so may be of interest even to classical mathematicians.)


\item [[Urs Schreiber|Urs]] linked to [[normal complex of groups]] by a little lemma now added to [[Moore complex]]


\item [[Tim Porter]] added [[normal complex of groups]]


\item [[Urs Schreiber|Urs]]:

\begin{itemize}%
\item worked on [[simplicial group]]:

\begin{itemize}%
\item added section about the adjunction between simplicial groups and simplicial sets, in particular mentioning the free simplicial abelian group functor (since that plays a role in the proof at [[abelian sheaf cohomology]], which now links to it)


\item moved statement and proof that every simplicial group is a Kan complex into a formal theorem-proof environment


\item added references to Goerss-Jardine, to [[Tim Porter]]`s [[Tim Porter:crossed menagerie|Crossed Menagerie]] and to Peter May's book ``Simplicial objects in algebraic topology''


\item added an ``Idea'' section



\end{itemize}

\item added to [[Moore complex]]

\begin{itemize}%
\item the theorem of how it relates to the other two complexes in the game (it is quasi-isomorphic to the alternating sum complex and isomorphic to that divided by degeneracies)


\item illustrations of cells in low degree


\item the reference to [[Tim Porter]]`s [[Tim Porter:crossed menagerie|The Crossed Menagerie]] notes


\item a brief comment on use of terminology (after checking with [[Tim Porter]])



\end{itemize}


\end{itemize}

\item [[Toby Bartels]]: Started more articles that I'd linked earlier: [[exclusive disjunction]] (with a philosophical claim that I need to back up), [[countable choice]] (very much a stub), [[sigma-ideal]], [[Hartog's number]].


\item [[Urs Schreiber|Urs]]:

\begin{itemize}%
\item added the detailed formal proof to [[abelian sheaf cohomology]] that shows how it is sitting as a special case inside the more conceptual [[nonabelian cohomology]] (= hom-sets of [[infinity-stack]]s, really)

\begin{itemize}%
\item in this context I also rewrote the ``Idea'' section in the desire to drive home the main point better, but without removing the previous material. As a result the text of the entry is now somewhat repetitive, I am afraid. Maybe a little later, when I have more distance to it, I'll try to trim it down again. Or else, maybe one of you feels like polishing it.

\end{itemize}


\end{itemize}

\item [[Toby Bartels]]: Wrote [[Moore closure]], since I linked it and it's a neat idea. As I was writing it, I realised that it was more abstract than what I'd been writing lately and must correspond to something very nice categorially; by the time I was done, I realised what it was: a special case of a [[monad]]. Then I saw that there were really no examples of monads on our page!, so I added a few, but the majority of examples of monads on the wiki now are Moore closures. (And believe me, Moore closures are everywhere.) Anyway, if you're trying to understand what monads are, why not try [[Moore closure]]s first?


\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item added an ``Idea'' section to [[Moore complex]], edited the section headers a bit -- and have a question onm terminology: isn't this really the ``normalized chain complex'' whereas the Moore complex is the one on all cells with differential the alternating sum of face maps?


\item created [[organization of the nLab]] -- please see there for what this is about (or in fact, see the corresponding \href{http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=36&page=1#Item_2}{nForum thread} that is being linked to there)


\item added references to Goerss-Jardine's \emph{Simplicial homotopy theory} to [[Dold-Kan correspondence]] and [[Moore complex]]


\item reacted a bit at [[An Exercise in Kantization]] -- behind the scenes this is being developed further, I'd be happy to provide more detailed replies in a while, when things have stabilized a bit more


\item added a mention of and a link to [[Kan lift]] to the ``Idea'' section of [[Kan extension]]


\item replied to the discussion about pullback notation at [[Kan extension]]: I originally had ``$p^*$'' there. After somebody changed that to ``$p_*$'' I wrote the section ``note on terminology''. I'd be happy to have the $p^*$ reinstalled.



\end{itemize}


\end{itemize}
\hypertarget{20090621}{}\section*{{2009-06-21}}\label{20090621}

\begin{itemize}%
\item [[Toby Bartels]]:

\begin{itemize}%
\item Over at [[An Exercise in Kantization]], [[Eric Forgy]] and [[Daniel de França MTd2]] would like confirmation or correction from somebody that understands the paper (Alm's \emph{[[Kantization09May27.pdf:file]]}) better than I do.
\item Some additions to [[Vect]], [[Hilb]], and [[measurable space]]. (I plan to do more on the last, then move to [[measure space]].)

\end{itemize}

\item [[Todd Trimble|Todd]]:

\begin{itemize}%
\item I have a notational comment at [[Kan extension]]. Spurred by [[David Corfield]]`s \href{http://golem.ph.utexas.edu/category/2009/06/kan_lifts.html}{post} at the Caf\'e{}, I hope to get started on [[Kan lift]] soon. (Done.)


\item There's a running discussion between [[Toby Bartels]] and me at [[cyclic order]], centering on whether Connes' cycle category $\Lambda$ has been correctly characterized, or if not how to fix it. We agree now that a notion of ``total cyclic order'' is classically equivalent to the ``linear cyclic order'' notion used in the article, and equally feasible for purposes of trying to characterize $\Lambda$, but I'm currently perplexed by the apparent presence of a terminal object.



\end{itemize}


\end{itemize}
\hypertarget{20090620}{}\section*{{2009-06-20}}\label{20090620}

\begin{itemize}%
\item [[Tim Porter|Tim]]: Finally I have created [[2-crossed complex]], and in the process needed to create [[normal complex of groups]]. What would be nice is to work out exactly what $\infty$/$\omega$-groupoids correspond to 2-crossed complexes. Any ideas? (It probably is obvious viewed from the right perspective but I fear I do not yet have the right perspective to say `aha!') I can characterise these objects in homotopy theoretic language, but really would like some neat way of describing them in $\infty$-cat terms.


\item [[Todd Trimble]]: asked a question of [[Mike Shulman|Mike]] and [[Toby Bartels|Toby]] over at [[cyclic order]].


\item [[Toby Bartels]]:

\begin{itemize}%
\item Wrote [[countable set]], [[infinite set]], [[relative complement]], and [[preimage]].
\item Has the `Change page name.' feature broken for anybody else?

\end{itemize}


\end{itemize}
\hypertarget{20090619}{}\section*{{2009-06-19}}\label{20090619}

\begin{itemize}%
\item [[Tim Porter|Tim]]:

\begin{itemize}%
\item I have added more material to [[2-crossed module]] including some exercises (at the foot of the `page'! (Have fun!) I will not get around to doing an entry on 2-crossed complexes today.


\item I have adjusted [[homotopy coherent nerve]] in an attempt to answer some of the points made there by [[Todd Trimble|Todd]],



\end{itemize}

\item [[Urs Schreiber|Urs]]:

\begin{itemize}%
\item almost missed [[Tim Porter]]`s addition about the [[Dwyer-Kan loop groupoid]] to [[simplicial homotopy group]] -- that sounds very good, I'd be happy if we make this the default point of view at that entry and derive the more traditional description only as a special case from that


\item added the example of the bar construction of a group $G$ as the nerve of $\mathbf{B} G$ to [[nerve]]


\item added illustration to [[Kan fibration]]


\item added illustrative diagrams to [[boundary of a simplex]] and [[horn]]


\item added a section with details on ordinary nerves of ordinary categories to [[nerve]]



\end{itemize}

\item [[Toby Bartels]]:

\begin{itemize}%
\item Expanded [[measurable space]] quite a bit, but I need to go, so it hasn't really been proofread.
\item Split [[measure space]] off from [[measurable space]].

\end{itemize}


\end{itemize}
\hypertarget{20090618}{}\section*{{2009-06-18}}\label{20090618}

\begin{itemize}%
\item [[Todd Trimble]] keeps creating analysis pages that [[Toby Bartels|I]] link to; today's were [[measurable space]] and [[topological vector space]].


\item [[Tim Porter|Tim]]:

\begin{itemize}%
\item I finally added some stuff into [[crossed complex]] giving more information on the link with [[simplicial group]] and [[Moore complex]]. I hope to get around to creating 2-crossed complex in a day or two!


\item The discussion on [[simplicial homotopy group]] possibly needs more opinions! I just added a bit more of my viewpoint to Urs's thoughts box, but I do find that I am not sure what the idea / context / viewpoint (or whatever) of this entry should `optimally' be!



\end{itemize}

\item [[Eric Forgy|Eric]]: In an attempt to understand Kan extension, I cooked up an example of a functor and added it to [[functor]]. Let me know if I made any mistake.


\item [[Toby Bartels]]: Instead of changing links to [[path-connected space]] (which doesn't yet exist) to [[connected space]], I instead added {\colorbox[rgb]{1.00,0.93,1.00}{\tt \char91\char91\char33redirects\char32path\char45connected\char32space\char93\char93}} to [[connected space]]. That way, if we ever decide to separate out a new page [[path-connected space]], we don't need to go through all of the links and fix them!


\item [[Urs Schreiber|Urs]]: added the defintion of the product to [[simplicial homotopy group]]


\item [[Todd Trimble|Todd]]: wrote the beginnings of an apparently long-awaited article on [[weighted colimit]]s.


\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item filled in the definition at [[simplicial homotopy]] together with an intentionally pedestrian proof that simplicial homotopy in a Kan complex is an equivalence relation


\item slightly polished the ``remarks'' section at [[simplicial set]]



\end{itemize}

\item [[Todd Trimble|Todd]]:

\begin{itemize}%
\item wrote [[measurable space]], including material on measures and basic integration theory.


\item responded to queries of [[David Roberts]] at [[covering space]] and [[locally path-connected space]]. David wondered whether some material at [[covering space]] ought to be moved to [[universal covering space]]. I think we're covering similar material but with slightly different emphases; I'd like to see David put his imprimatur on the article on universal covering spaces, but leave [[covering space]] essentially intact while the material continues to settle into place; I'm thinking of adding more to it anyway (so that ``to be continued'' might still be appropriate).



\end{itemize}

\item [[Urs Schreiber|Urs]]:

\begin{itemize}%
\item added a reference (and minor comments on references) to [[model structure on simplicial presheaves]]


\item added an opinion to [[simplicial homotopy group]]

\begin{itemize}%
\item will add the definition of the group structure later today

\end{itemize}


\end{itemize}

\item [[Andrew Stacey]]: Changed the example in [[topological concrete category]] for [[generalized smooth space]] since I think that all the examples listed there are topological over set.


\item [[David Roberts]]:

\begin{itemize}%
\item Edited [[covering space]] as requested by [[Todd Trimble|Todd]]. I still wonder whether some of the discussion would be better off at [[universal covering space]].


\item Created stubs at [[semi-locally simply-connected space]] and [[locally path-connected space]].



\end{itemize}


\end{itemize}
\hypertarget{20090617}{}\section*{{2009-06-17}}\label{20090617}

\begin{itemize}%
\item [[Toby Bartels]]: Fixed mistakes at [[topological concrete category]], thanks to having a good online reference.


\item [[Todd Trimble|Todd]]:

\begin{itemize}%
\item Wrote [[approximation of the identity]], linked to by one thing I wrote at [[Hilbert space]].

\end{itemize}

\item [[Tim Porter|Tim]]:

\begin{itemize}%
\item Did my idea of a partial fix for the dilemma at [[simplicial homotopy group]]. Please edit! The entry still needs the definition of the group structure on the homotopy groups. I have included the link with the [[Moore complex]] of the [[Dwyer-Kan loop groupoid]]. I have also added a request for discussion here.

\end{itemize}

\item [[Todd Trimble|Todd]]:

\begin{itemize}%
\item Added a bit to the examples section at [[Hilbert space]], mentioning in particular the orthonormal basis theorem.

\end{itemize}

\item [[Tim Porter|Tim]]:

\begin{itemize}%
\item Added a useful reference to [[topological category]], plus a comment on terminology. It should be noted that the authors (Adamek, Herrlich and Strecker) use `topological concrete category' in the index, which they shorten to `topological category' in the text. Does that provide a solution to the terminological problem?

\end{itemize}

\item [[Toby Bartels]]:

\begin{itemize}%
\item Wrote [[Hilbert space]]. I finished the definitions and then got tired, so there's not much else. But we need this if we're ever going to discuss $2$-[[2-Hilbert space|Hilbert spaces]]!
\item [[David Roberts]] and I would appreciate any terminological suggestions at the unfortunately-named [[topological category]].

\end{itemize}


\end{itemize}
\hypertarget{20090616}{}\section*{{2009-06-16}}\label{20090616}

\begin{itemize}%
\item [[Toby Bartels]]:

\begin{itemize}%
\item Wrote [[topological category]], including a lot of vague stuff there, since I don't have good references here now. But I'm pretty sure that everything that I said is at least true.
\item Wrote [[core]], using a term that I learned from [[Mike Shulman]] for an [[underlying groupoid]].
\item Wrote [[semigroup]] and [[magma]], since they're such basic topics, but I didn't say much.
\item Added stuff to [[reflective subcategory]] about how obects of the ambient category can be seen as objects of the subcategory equipped with [[extra structure]]. Besides the examples there, see also [[core]].
\item In reply to the last point in what Urs has said immediately below, if you want to put in something that you \emph{expect} to be controversial, and hesitate for that reason, then put it in and just \emph{label} it as controversial. Put in a [[HowTo|query box]] asking `Is this right?' or saying `This is my new definition.' or whatever. Or put it in a new section at the bottom of the page, marked `\#\# Uncertain material' or whatever. The point is, get it down one way or another, and we will be very happy!

\end{itemize}

\item [[Urs Schreiber|Urs]]: I have a comment and appeal at [[nInsights]]

\begin{itemize}%
\item in this context I also want to ask again everybody:

\begin{itemize}%
\item please don't forget to drop latest changes logs here alerting us of which changes you made where, if it's anything beyond fixing typos


\item if you feel something needs to have a place in the nLab, don't feel hesitant to add it. I am getting the impression there is in parts some hesitation to insert material without checking with everybody else first. I'd say more efficient would be: add the material you would like to see, and if it should really turn out that there is serious disagreement with some other contributor, we can still roll back things and look for compromises.



\end{itemize}


\end{itemize}

\item [[Urs Schreiber|Urs]]:

\begin{itemize}%
\item expanded [[inverse image]]: added an ``Idea'' section, restructured discussion into presheaf and sheaf and sheaf on top-space bits, added the crucial lemmas and theorems and provided detailed proofs of some of them.


\item replied again at [[simplicial homotopy group]]: [[Tim Porter|Tim]] or [[Toby Bartels|Toby]] should please feel free to implement the remaining terminology adjustment


\item reacted and replied at [[simplicial homotopy group]] (also have a question) and added a few further bits



\end{itemize}

\item [[Tim Porter|Tim]]:

\begin{itemize}%
\item Discussion at [[simplicial homotopy group]].

\end{itemize}


\end{itemize}
\hypertarget{20090615}{}\section*{{2009-06-15}}\label{20090615}

\begin{itemize}%
\item [[Toby Bartels]]:

\begin{itemize}%
\item Discussions at [[connected space]] and [[simplicial homotopy group]].
\item New page at [[neighborhood]].

\end{itemize}

\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item created [[simplicial homotopy group]]


\item added to [[model structure on simplicial sets]] a puny beginning of a discussion of what is supposed to eventually become a discussion of the relation to model structures on strict $\infty$-groupoids


\item created [[omega-nerve]] and [[model structure on strict omega-groupoids]]


\item expanded a bit on the discussion at the beginning of [[model structure on simplicial sets]]


\item added detailed proof to [[geometric morphism]] of the fact that geometric morphisms $Sh(X) \to Sh(Y)$ are bijectively induced from continuous functions $X \to Y$



\end{itemize}

\item [[Tim Porter|Tim]]: I have continued at [[homotopy coherent nerve]]. This ended up with several new links to not-yet-existing pages!


\item [[Urs Schreiber]]:

\begin{itemize}%
\item created [[homeomorphism]] and [[local homeomorphism]] just to satisfy links


\item added theorem about stalkwise characterization of epi/mono/isos of sheaves to [[stalk]] and included an example


\item added a bit more to [[string structure]] (discussion of description in terms of class on total space, also some references, but very incomplete) - but still not done


\item split off remaining Friday session [[Oberwolfach Workshop, June 2009 -- Friday, June 12]] for [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]] following [[Toby Bartels|Toby]]



\end{itemize}

\item [[Todd Trimble]] created [[connected space]].


\item [[Toby Bartels]]: [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]] was getting too long for me to handle, so I removed the duplication of abstracts and broke the rest up by day.



\end{itemize}
\hypertarget{20090614}{}\section*{{2009-06-14}}\label{20090614}

\begin{itemize}%
\item [[Tim Porter|Tim]]: I have had a go at [[homotopy coherent nerve]]. Still more to add though.


\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item added more details on the sheaf-case to [[point of a topos]] and created [[skyscraper sheaf]]

\end{itemize}


\end{itemize}
\hypertarget{20090613}{}\section*{{2009-06-13}}\label{20090613}

\begin{itemize}%
\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item started creating [[string structure]] but had to quit before fully finished -- will continue tomorrow


\item created [[point of a topos]] and [[stalk]]


\item created [[Oberwolfach]]


\item pasted notes on Thomas Schick's talk on differential cohomology at [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]] into [[differential cohomology]]


\item created

\begin{itemize}%
\item [[local net]]


\item [[TQFT]]


\item [[FFRS-formalism]]


\item [[modular tensor category]]



\end{itemize}
but more like stubs to get some material in place, didn't find the leisure to really polish this


\item started adding links to keywords in the list of abstracts that [[Bruce Bartlett|Bruce]] added to [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]] -- am hoping that eventually we'll fill these links with content, partly just by copy-and-pasting the relevant bits from the lecture notes (as a start)



\end{itemize}

\item [[Tim Porter|Tim]]:

\begin{itemize}%
\item Mike, ditto from me!


\item Made some changes to [[simplex category]] and [[join of simplicial sets]]. A more explicit mention of ordinal sum has been made and some minor corrections done.



\end{itemize}

\item [[Bruce Bartlett|Bruce]]:

\begin{itemize}%
\item Congrats Mike!


\item Added lots of talk summaries and notes to [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]]. Lots of editing help needed\ldots{} what would be nice would be to have the links to the notes for the talks next to the talk titles --- both my notes as well as those of Gabriel Drummond-Cole (by linking to his \href{http://www.math.sunysb.edu/~blafard/notes/}{webpage}).



\end{itemize}

\item [[Urs Schreiber|Urs]]: I gather it is in order to express congratulations to [[Mike Shulman]] given \href{http://ncatlab.org/nlab/show/diff/Mike+Shulman}{the latest changes} to his page


\item [[Toby Bartels]]: I think that the (unordered) notion of \emph{category with inverses} at [[partially ordered category]] should be the same as that of \emph{[[dagger category]]}.



\end{itemize}
\hypertarget{20090612}{}\section*{{2009-06-12}}\label{20090612}

\begin{itemize}%
\item [[Victor Porton]]: Some other newer terms added (notably [[partially ordered category]]) to [[Categories and Sheaves]].


\item [[Todd Trimble]]: I finally found some time to write up more of what I had planned for [[covering space]]s, and tried to link up with the comments made by David Roberts. David and Urs, please take a look.



\end{itemize}
\hypertarget{20090611}{}\section*{{2009-06-11}}\label{20090611}

\begin{itemize}%
\item [[Urs Schreiber|Urs]]:

\begin{itemize}%
\item created [[tetracategory]]


\item added Thursday talks to [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology



\end{itemize}


\end{itemize}
]]

\begin{itemize}%
\item replied to [[Tim Porter]] at [[join of simplicial sets]]


\item [[Bruce Bartlett|Bruce]]: Added some comments abotu query boxes in personal nLab pages in the [[HowTo]] page.


\item [[Tim Porter|Tim]]: I have added a query to [[join of simplicial sets]]. Am I missing something, but I find the initial definition confusing as it does not mention the ordinal structure nor ordinal sum. (There may be a subtlety that I am missing so \ldots{})


\item New pages: [[doctrine]], [[pretopological space]].



\end{itemize}
\hypertarget{20090610}{}\section*{{2009-06-10}}\label{20090610}

\begin{itemize}%
\item New pages: [[Green-Schwarz mechanism]], [[quasigroup]].


\item [[Urs Schreiber]] added Wednesday talk notes to [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]]


\item [[David Roberts]]:

\begin{itemize}%
\item Put next, very drafty, chapter of my thesis on my personal web. Just click on my name and follow the white rabbit.
\item Added another comment to [[covering space]]. This time about the (total) tangent groupoid as applied to the non-basepoint version of the universal covering space.

\end{itemize}


\end{itemize}
\hypertarget{20090609}{}\section*{{2009-06-09}}\label{20090609}

\begin{itemize}%
\item More new pages: [[Freudenthal suspension theorem]], [[replete subcategory]], [[strictly full subcategory]].


\item [[Zoran Škoda]]: much expanded [[orbifold]], created [[Lie group]], [[compact Lie group]], [[orbispace]].


\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item created [[orbifold]] and [[orientifold]]


\item added further talk notes to [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]]


\item added notes on first part of lecture by Thomas Schick on \emph{differential cohomology} to [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]] -- am hoping to eventually move a polished version of that to [[differential cohomology]]



\end{itemize}

\item [[David Roberts]]

\begin{itemize}%
\item Created [[universal covering space]], but it's still pretty much a stub.
\item Added some remarks to [[covering space]], but they will probably need to move to [[universal covering space]].

\end{itemize}

\item [[Toby Bartels]]

\begin{itemize}%
\item Started [[discrete space]].
\item There seems to be a link missing from the [[HowTo]] on CSS in personal webs.

\end{itemize}


\end{itemize}
\hypertarget{20090608}{}\section*{{2009-06-08}}\label{20090608}

\begin{itemize}%
\item New pages:

\begin{itemize}%
\item [[Mike Shulman]] wrote [[2-monad]].
\item Besides asking people questions, [[Todd Trimble]] also started [[covering space]].

\end{itemize}

\item [[Urs Schreiber]]: created [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]], linked to from \href{http://golem.ph.utexas.edu/category/2009/06/strings_fields_topology_in_obe.html}{this} blog entry (am hoping to get around to add lots of further keyword links to existing $n$Lab entries, but maybe won't)


\item [[Todd Trimble]]: asked Alex a question at [[Alex Hoffnung]].


\item [[Bruce Bartlett|Bruce]]: Added link to notes to the strings, fields and topology workshop at Oberwolfach. Also added a ``How To'' section for creating CSS gismos like query boxes on your personal nLab wiki space.



\end{itemize}
\hypertarget{20090607}{}\section*{{2009-06-07}}\label{20090607}

\begin{itemize}%
\item [[John Baez|John]]: I created myself a [[John Baez:HomePage|new personal web]] since I lost the password for my old one --- and besides, the name of my old one was nonstandard. So far the only thrilling feature of this new web is the introduction to a paper I'm writing with [[James Dolan]], tentatively titled `Doctrines of Algebraic Geometry'.


\item [[Mike Shulman|Mike]]: Tried to distill a bit of the \href{http://golem.ph.utexas.edu/category/2009/06/mathematical_principles.html#c024277}{cafe discussion} about the [[empty space]].



\end{itemize}
\hypertarget{20090604}{}\section*{{2009-06-04}}\label{20090604}

\begin{itemize}%
\item [[Mike Shulman|Mike]]: Added some exposition to [[Hopf algebra]].

\end{itemize}
\hypertarget{20090603}{}\section*{{2009-06-03}}\label{20090603}

\begin{itemize}%
\item [[Toby Bartels]]:

\begin{itemize}%
\item Thanks for [[prometric space]], Mike. I didn't know about those, but they seem quite reasonable. I was thinking that I might want to figure out the categorial meaning of the definition of [[gauge space]], but maybe it's just that it's a halfway attempt at prometric space! (But do you have any good example of a nongaugeable prometric space?)
\item I seem to have gotten into an edit conflict with you at [[Zorn's lemma]], Todd. I didn't tell it to override your edit, but something happened regardless. Anyway, I think that I've fixed it.
\item Also, our page is at [[Hausdorff maximal principle]], which seems to be more common, but maybe your name is better. But look!, now it redirects! And you can move it too!

\end{itemize}

\item [[Todd Trimble]]: gave a proof of [[Zorn's lemma]]. Wouldn't mind expanding that entry to include the mutual equivalence between AC, Zorn, and well-ordering principle (assuming excluded middle). May get around to putting in something at [[Hausdorff maximality principle]].


\item [[Mike Shulman|Mike]]: Inspired by [[gauge space]], created [[prometric space]].


\item [[Toby Bartels]]:

\begin{itemize}%
\item [[Eric Forgy|Eric]] and I have been testing the new move and redirect features at the \emph{top} (for some reason) of the [[Sandbox]]. It seems possible to move and to create new redirects (make sure that you see how, it's backwards from MediaWiki), but not (yet) possible to regularise all of the current redirect pages.
\item Created a few requested pages. The interesting ones are [[Bill Lawvere]], [[Boolean ring]], and [[ideal]].

\end{itemize}


\end{itemize}
\hypertarget{20090602}{}\section*{{2009-06-02}}\label{20090602}

\begin{itemize}%
\item [[Toby Bartels]]:

\begin{itemize}%
\item An [[AnonymousCoward|anonymous coward]] created a blank version of [[Hausdorff maximal principle]], so I started that \ldots{} but it needs proofs! (at least).
\item A new reference, [[HAF]], since I've now cited it thrice.
\item Created [[gauge space]]. A lot of this is original research; years ago, I wanted to work out the general theory (even constructively) of quasigauge spaces, and perhaps now I will. (Or perhaps I will stop here, with only results that are known or easy to prove from what is known.)

\end{itemize}

\item [[Todd Trimble]]: Added some details on categorical operations on the category of [[Banach space]]s.


\item [[Toby Bartels]]:

\begin{itemize}%
\item Added the standard examples to [[uniform space]].
\item Corrected Andrew's reply to Mike at [[Froelicher space]]. (I'm pretty sure that this is what you meant, Andrew!)
\item Replied to [[David Roberts]] at [[Atiyah Lie groupoid]].

\end{itemize}

\item [[Andrew Stacey]]: replied to [[Mike Shulman]] at [[Froelicher space]]


\item [[Toby Bartels]]:

\begin{itemize}%
\item Archived [[2009 May changes]].
\item Fixed some mistakes at [[uniform space]].

\end{itemize}


\end{itemize}
\hypertarget{20090601}{}\section*{{2009-06-01}}\label{20090601}

\begin{itemize}%
\item [[Andrew Stacey|Andrew]] proved a theorem about [[Froelicher space|Hausdorff Frölicher spaces]] and the relationship to limits and colimits of manifolds.


\item [[Bruce Bartlett|Bruce]] added some stuff to Section 4 [[geometric infinity-function theory]] (with about query $QC(X)$ when $X$ is an $\omega$-groupoid internal to dg-manifolds) and ticked some things.



\end{itemize}

\vspace{.5em} \hrule \vspace{.5em}
[[2008 changes|First list]] --- [[2009 May changes|Previous list]] --- [[2009 July changes|Next list]] --- \href{http://www.math.ntnu.no/~stacey/Mathforge/nForum/?CategoryID=5}{Current list}


\vspace{.5em} \hrule \vspace{.5em}
category: meta



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