2009 June changes

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Archive of changes made during June 2009.
The substantive content of this page should **not** be altered.

***

# 2009-06-30 #

* [[Urs Schreiber|Urs]]

  * created [[Eilenberg-MacLane object]] prompted by the blog discussion [here](http://golem.ph.utexas.edu/category/2009/06/cohomology_and_homotopy.html#c024902)

  * replied at [[homotopy]]

*  [[Toby Bartels]]:
   *  Responded to [[Ronnie Brown]] at [[groupoid]].
   *  Noted an incomplete thought at [[simplicial homotopy]].
   *  Complained a bit at [[homotopy]].

* [[Urs Schreiber|Urs]]

  * created [[Quillen adjunction]]

  * added definition to [[Quillen equivalence]]

  * reorganized [[model structure on simplicial presheaves]]:

    * expanded the idea/introduction-section

    * moved the original material to

      * [[global model structure on simplicial presheaves]] (and expanded that)

      * [[local model structure on simplicial presheaves]]

    * created [[local model structure on simplicial sheaves]]

    * [[rectified infinity-stack]]

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

* [[Tim Porter|Tim]]:

  * 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

   * 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]].

     * [[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

* [[Urs Schreiber|Urs]]

  * prompted by [[David Corfield|David]]'s remark at [[homotopy group]] I was led to all of the following:

    I split [[homotopy]] into

    * [[homotopy (as a transformation)]]

    * [[homotopy (as an operation)]]

    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]]

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

    * [[David Corfield]] has an interesting question there, I think he is right, will try to write something at [[homotopy group]]

* [[David Corfield|David]]

  * added to [[Eckmann-Hilton duality]]

* [[Urs Schreiber|Urs]]

  * continued polishing and adding statements and proofs to [[category of fibrant objects]]

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

# 2009-06-29 #

* [[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?

  * [[Urs Schreiber|Urs]]: I don't think it does

*  [[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.

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

*  [[Toby Bartels]] fleshed out [[discrete fibration]] a bit.

*  [[Eric Forgy]] created [[interval category]].

* [[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).

* [[Urs Schreiber]]

  * created reference entry [[Lectures on n-Categories and Cohomology]] in order to collect the existing $n$Lab entries on topics discussed there

# 2009-06-28 #

*  [[Toby Bartels]] responded to Todd below.

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

# 2009-06-27 #

* [[Todd Trimble|Todd]]: added more material to [[Trimble n-category]], outlining Leinster and Cheng's extension from $n$ to $\infty$.

*  [[Eric Forgy]] has begun some pages related to administration of the Lab:
   *  [[Organization of the nLab]]
   *  [[On Scientific Contributions to the nLab]]
   *  [[AnonymousCoward]] (which is not new but has a new discussion of privacy issues)
   *  not so much administration but new suggestions for resources:
      *  [[Bibliography]]
      *  [[Notation]]

* [[Urs Schreiber]]

  * 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

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

  * 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

  * created [[QFT with defects]]

  * created [[2-category of 2-dimensional cobordisms]]

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

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

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

  * created [[(infinity,2)-category]]

# 2009-06-26 #

*  [[Toby Bartels]]:
   *  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!).
   *  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.

* [[Urs Schreiber|Urs]]:

  * created [[gerbe (in nonabelian cohomology)]] based on the blog discussion [here](http://golem.ph.utexas.edu/category/2008/08/connections_on_nonabelian_gerb.html#c024774)

  * 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 [here](http://golem.ph.utexas.edu/category/2009/06/cohomology_and_homotopy.html#c024778)

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

  * 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).

  * 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]]

  * created [[right proper model category]]

  * 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

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

*  [[Eric Forgy]] has a request at [[notation]].

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

# 2009-06-25 #

* [[Todd Trimble|Todd]]: Recently [[loop space]] was created. Also, I added some examples to [[Chu construction]] and to [[star-autonomous category]].

* [[Urs Schreiber|Urs]]:

  * added the example of principal bundles to [[fibration sequence]]

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

  * created [[twisted cohomology]]

  * created [[n-group]]

  * created [[group cohomology]]

    * the point of view adopted there is _almost_ fully explicit in Brown-Higgins-Sivera's book _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?

  * 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?

  * 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

    * 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 [Weak categories](http://www.ima.umn.edu/talks/workshops/SP6.7-18.04/may/PorterMay.pdf) which I now linked to from [[weak omega-category]] and [[higher category theory]]

  * created [[principal infinity-bundle]], just for completeness

  * 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.

  * 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

  * created an entry for [[Carlos Simpson]] motivated by a link to a recent [pdf note](http://math.unice.fr/~carlos/slides/ihesAGjan09.pdf) -- 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

  * created [[fibration sequence]]

  * 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.

    * 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.
* [[Tim Porter|Tim]]:

   * 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.

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

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

# 2009-06-24 #

* [[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).

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

* [[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 <nowiki>[[category|categories]]</nowiki>. Now you can simply type <nowiki>[[categories]]</nowiki> and will be redirected automatically to [[category]]. To see the changes in action, have a look at [[higher category theory]]. Feedback welcome!

* [[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.

* [[Tim Porter|Tim]]:  I have created a stub on [[gerbe|gerbes]].  There was a request on the Café for en entry and I thought this was a good way to remind me to start one.  ... but feel free to do it for me!

* [[Zoran Škoda]]:

  * 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...

  * 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.

* [[Urs Schreiber|Urs]]:

  * created [[K-theory spectrum]] with just a question

  * 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

  * created [[Eilenberg-MacLane space]], [[Eilenberg-MacLane spectrum]]

  * am offering a reply to [[Finn Lawler]] at [[modification]]

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

# 2009-06-23 #

*  [[Gavin Wraith]] wrote [[matrix theory]], [[tensor product theory]], and [[bimodel]], with a question at the last.

* [[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.

* [[Urs Schreiber|Urs]]

  * added to [[weak factorization system]] the full details of the (elementary) proof that morphisms defined by a right lifting property are stable under pullback --

  * 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

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

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

  * added further illustrations to [[Kan fibration]]

# 2009-06-22 #

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

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

* [[Tim Porter]] added [[normal complex of groups]]

* [[Urs Schreiber|Urs]]:

  * worked on [[simplicial group]]:

    * 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)

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

    * 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"

    * added an "Idea" section

  * added to [[Moore complex]]

    * 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)

    * illustrations of cells in low degree

    * the reference to [[Tim Porter]]'s [[Tim Porter:crossed menagerie|The Crossed Menagerie]] notes

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

*  [[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]].

* [[Urs Schreiber|Urs]]:

  * 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)

    * 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.

*  [[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?

* [[Urs Schreiber|Urs]]

  * 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?

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

  * added references to Goerss-Jardine's _Simplicial homotopy theory_ to [[Dold-Kan correspondence]] and [[Moore complex]]

  * 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

  * added a mention of and a link to [[Kan lift]] to the "Idea" section of [[Kan extension]]

  * 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.

# 2009-06-21 #

*  [[Toby Bartels]]:
   *  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 _[[Kantization09May27.pdf:file]]_) better than I do.
   *  Some additions to [[Vect]], [[Hilb]], and [[measurable space]].  (I plan to do more on the last, then move to [[measure space]].)

* [[Todd Trimble|Todd]]:

   * I have a notational comment at [[Kan extension]]. Spurred by [[David Corfield]]'s [post](http://golem.ph.utexas.edu/category/2009/06/kan_lifts.html) at the Café, I hope to get started on [[Kan lift]] soon. (Done.)

   * 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.

# 2009-06-20 #

*  [[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.

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

*  [[Toby Bartels]]:
   *  Wrote [[countable set]], [[infinite set]], [[relative complement]], and [[preimage]].
   *  Has the 'Change page name.' feature broken for anybody else?

# 2009-06-19 #

* [[Tim Porter|Tim]]:

  *  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.

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

* [[Urs Schreiber|Urs]]:

  * 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

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

  * added illustration to [[Kan fibration]]

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

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

*  [[Toby Bartels]]:
   *  Expanded [[measurable space]] quite a bit, but I need to go, so it hasn\'t really been proofread.
   *  Split [[measure space]] off from [[measurable space]].

# 2009-06-18 #

*  [[Todd Trimble]] keeps creating analysis pages that [[Toby Bartels|I]] link to; today\'s were [[measurable space]] and [[topological vector space]].

* [[Tim Porter|Tim]]:

   * 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!

   * 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!

* [[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.

*  [[Toby Bartels]]:  Instead of changing links to [[path-connected space]] (which doesn\'t yet exist) to [[connected space]], I instead added `[[!redirects path-connected space]]` 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!

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

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

* [[Urs Schreiber|Urs]]

  * filled in the definition at [[simplicial homotopy]] together with an intentionally pedestrian proof that simplicial homotopy in a Kan complex is an equivalence relation

  * slightly polished the "remarks" section at [[simplicial set]]

* [[Todd Trimble|Todd]]:

  * wrote [[measurable space]], including material on measures and basic integration theory.

  * 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).

* [[Urs Schreiber|Urs]]:

  * added a reference (and minor comments on references) to [[model structure on simplicial presheaves]]

  * added an opinion to [[simplicial homotopy group]]

    * will add the definition of the group structure later today

* [[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.

*  [[David Roberts]]:

   * 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]].

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

# 2009-06-17 #

*  [[Toby Bartels]]:  Fixed mistakes at [[topological concrete category]], thanks to having a good online reference.

* [[Todd Trimble|Todd]]:

   * Wrote [[approximation of the identity]], linked to by one thing I wrote at [[Hilbert space]].

*  [[Tim Porter|Tim]]:

   * 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.

* [[Todd Trimble|Todd]]:

   * Added a bit to the examples section at [[Hilbert space]], mentioning in particular the orthonormal basis theorem.

*  [[Tim Porter|Tim]]:

   * 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?

*  [[Toby Bartels]]:
   *  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]]!
   *  [[David Roberts]] and I would appreciate any terminological suggestions at the unfortunately-named [[topological category]].

# 2009-06-16 #

*  [[Toby Bartels]]:
   *  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.
   *  Wrote [[core]], using a term that I learned from [[Mike Shulman]] for an [[underlying groupoid]].
   *  Wrote [[semigroup]] and [[magma]], since they\'re such basic topics, but I didn\'t say much.
   *  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]].
   *  In reply to the last point in what Urs has said immediately below, if you want to put in something that you *expect* to be controversial, and hesitate for that reason, then put it in and just *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!

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

  * in this context I also want to ask again everybody:

    * 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

    * 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.

* [[Urs Schreiber|Urs]]:

  * 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.

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

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

*  [[Tim Porter|Tim]]:

   *  Discussion at [[simplicial homotopy group]].

# 2009-06-15 #

*  [[Toby Bartels]]:
   *  Discussions at [[connected space]] and [[simplicial homotopy group]].
   *  New page at [[neighborhood]].

* [[Urs Schreiber|Urs]]

  * created [[simplicial homotopy group]]

  * 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

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

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

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

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

* [[Urs Schreiber]]:

  * created [[homeomorphism]] and [[local homeomorphism]] just to satisfy links

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

  * 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

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

*  [[Todd Trimble]] created [[connected space]].

*  [[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.

# 2009-06-14 #

*  [[Tim Porter|Tim]]: I have had a go at [[homotopy coherent nerve]].  Still more to add though.

* [[Urs Schreiber|Urs]]

  * added more details on the sheaf-case to [[point of a topos]] and created [[skyscraper sheaf]]

# 2009-06-13 #

* [[Urs Schreiber|Urs]]

  * started creating [[string structure]] but had to quit before fully finished -- will continue tomorrow

  * created [[point of a topos]] and [[stalk]]

  * created [[Oberwolfach]]

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

  * created

    * [[local net]]

    * [[TQFT]]

    * [[FFRS-formalism]]

    * [[modular tensor category]]

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

  * 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)

* [[Tim Porter|Tim]]:

  * Mike, ditto from me!

  * 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.

* [[Bruce Bartlett|Bruce]]:

  * Congrats Mike!

  * Added lots of talk summaries and notes to [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]]. Lots of editing help needed... 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 [webpage](http://www.math.sunysb.edu/~blafard/notes/)).

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

*  [[Toby Bartels]]:  I think that the (unordered) notion of _category with inverses_ at [[partially ordered category]] should be the same as that of _[[dagger category]]_.

# 2009-06-12 #

*  [[Victor Porton]]:  Some other newer terms added (notably [[partially ordered category]]) to [[Categories and Sheaves]].

* [[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.

# 2009-06-11 #

* [[Urs Schreiber|Urs]]:

  * created [[tetracategory]]

  * added Thursday talks to [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology
]]

  * replied to [[Tim Porter]] at [[join of simplicial sets]]

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

*  [[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 ...)

*  New pages: [[doctrine]], [[pretopological space]].

# 2009-06-10 #

*  New pages: [[Green-Schwarz mechanism]], [[quasigroup]].

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

* [[David Roberts]]:

  * Put next, very drafty, chapter of my thesis on my personal web. Just click on my name and follow the white rabbit.
  * 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.

# 2009-06-09 #

*  More new pages:  [[Freudenthal suspension theorem]], [[replete subcategory]], [[strictly full subcategory]].

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

* [[Urs Schreiber|Urs]]

  * created [[orbifold]] and [[orientifold]]

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

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

* [[David Roberts]]

  * Created [[universal covering space]], but it's still pretty much a stub.
  * Added some remarks to [[covering space]], but they will probably need to move to [[universal covering space]].

*  [[Toby Bartels]]
   *  Started [[discrete space]].
   *  There seems to be a link missing from the [[HowTo]] on CSS in personal webs.

# 2009-06-08 #

*  New pages:
   *  [[Mike Shulman]] wrote [[2-monad]].
   *  Besides asking people questions, [[Todd Trimble]] also started [[covering space]].

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

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

* [[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.

# 2009-06-07 #

* [[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'.

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

# 2009-06-04 #

* [[Mike Shulman|Mike]]: Added some exposition to [[Hopf algebra]].

# 2009-06-03 #

*  [[Toby Bartels]]:
   *  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?)
   *  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.
   *  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!

* [[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]].

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

*  [[Toby Bartels]]:
   *  [[Eric Forgy|Eric]] and I have been testing the new move and redirect features at the *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.
   *  Created a few requested pages.  The interesting ones are [[Bill Lawvere]], [[Boolean ring]], and [[ideal]].

# 2009-06-02 #

*  [[Toby Bartels]]:
   *  An [[AnonymousCoward|anonymous coward]] created a blank version of [[Hausdorff maximal principle]], so I started that ... but it needs proofs! (at least).
   *  A new reference, [[HAF]], since I\'ve now cited it thrice.
   *  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.)

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

*  [[Toby Bartels]]:
   *  Added the standard examples to [[uniform space]].
   *  Corrected Andrew\'s reply to Mike at [[Froelicher space]].  (I\'m pretty sure that this is what you meant, Andrew!)
   *  Replied to [[David Roberts]] at [[Atiyah Lie groupoid]].

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

*  [[Toby Bartels]]:
   *  Archived [[2009 May changes]].
   *  Fixed some mistakes at [[uniform space]].

# 2009-06-01 #

* [[Andrew Stacey|Andrew]] proved a theorem about [[Froelicher space|Hausdorff Frölicher spaces]] and the relationship to limits and colimits of manifolds.

* [[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.

***

[[2008 changes|First list]] --- [[2009 May changes|Previous list]] --- [[2009 July changes|Next list]] --- [Current list](http://www.math.ntnu.no/~stacey/Mathforge/nForum/?CategoryID=5)

***

category: meta


2009 March changes

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Archive
+-- {: .hide}
[[!include all changes]]
=--
=--
=--

Archive of changes made during March 2009.
The substantive content of this page should **not** be altered.
For past versions of this page beyond its own history, start [here](http://ncatlab.org/nlab/revision/2009+June+changes/459) and work backwards.

***

# 2009-03-31

* [[Urs Schreiber|Urs]]:

  * started adding comments on embeddings at [[geometric morphism]], but ran out of time

  * following [[Mike Shulman|Mike]]'s suggestion I moved the material that was at [[sheafification in a Lawvere-Tierney topos]] over to [[Lawvere-Tierney topology]]

  * have (started) a discussion with [[David Roberts]] on the relation between internal [[anafunctor]]s and localization in higher sheaf categories over at his private area [[davidroberts:comments on chapter 2]]

  * incorporated in part [[Zoran Skoda|Zoran]]'s comment at [[sheafification]] of non-set-valued presheaves and created an entry on the [[IPC-property]] in the course of that

  * started [[Sheaves in Geometry and Logic]], on the MacLane-Moerdijk book

  * created [[dense monomorphism]] (thanks to [[Mike Shulman|Mike]] for the relation to [[local isomorphism]]) and then [[sheafification in a Lawvere-Tierney topos]]

  * created [[(infinity,1)-essentially surjective functor]] and [[(infinity,1)-fully faithful functor]]

* [[Mike Shulman|Mike]]:

  * Mentioned at [[triangulated category]] that the definition is   redundant.  If I had time I would fix it myself.

  * We are having an interesting discussion at [[derived functor]].  Unfortunately I am leaving town for a couple of weeks (starting with  [PSSL88](http://www.cheng.staff.shef.ac.uk/pssl88/)) so will be less active for a while.

* [[Urs Schreiber|Urs]]:

  * created [[category of open subsets]] and [[presheaf on open subsets]] to satisfy links at [[Categories and Sheaves]]

  * fixed and expanded [[local isomorphism]] and used that to create [[category of sheaves]], [[(infinity,1)-category of (infinity,1)-sheaves]] and [[sheafification]]

  * created [[base change]] and [[cobase change]]

  * started listing [[Zoran Skoda|Zoran]]'s latest entries and previous ones on examples/refinements of additive and abelian categories at [[additive and abelian categories]].

*  [[Tim Porter|Tim]]:

   *  Last night I continued my 'lexicon' with an entry on [[differential graded vector space]]s (yes I know these are just chain complexes, but my thought is to put the lexicon that I have on then prune and adjust!) (My connection crashed at that point as I was editing this page!)

* [[Zoran Škoda]]: Created [[suspended category]] and [[Quillen exact category]]. This is a continuation of our efforts to enter various classes of additive categories useful in homological algebra and K-theory.

# 2009-03-30

* [[David Roberts]]:

  * Asked question about Grothendieck's tame topology at [[nice topological space]].

* [[Urs Schreiber|Urs]]:

  * QUESTION: came across MacLane's
    _Foundations for Categories and Sets_ where it it argued that neither standared set/class set theory nor [[Grothendieck universe]]s provide decent foundations for categories and a formalism of _schools_ is introduced instead -- can anyone comment on that in the light of our discussion at [[Grothendieck universe]]?

  * added an "Idea" section to [[derived functor]] and split off [[derived functor on a derived category]] from that in order to discuss the special homological algebra aspects of derived functors separately -- but incomplete for the moment

  * created [[null system]]

  * created [[category of chain complexes]], but then didn't quite know where to go with this...

  * agreed at [[homological algebra]] and [[filtrant category]]

  * created [[homology]]

  * expanded [[differential object]] and created [[differential]] just to satisfy links

  * created [[matrix calculus]] and [[mapping cone]]

*  [[Tim Porter|Tim]]:
   *  I have tried to edit [[homological algebra]] somewhat along the lines that my query suggested but taking in ideas from [[Zoran Škoda|Zoran's]] query at that entry.
   * I have added some background material on [[dérivateur]]s on the [[triangulated category]] entry.
   * I have a **lexicon** for the concepts needed for rational homotopy theory that may be useful.  I have started adding some of this to [[graded vector space]].  My intention is to put quite a lot of routine stuff on this there and elsewhere and then to sort out links, conflicts of notation afterwards. Feedback, queries etc welcome. (They will encourage me to put more of the lexicon on there!)

*  [[Toby Bartels]]:
   *  Wrote [[biproduct]], [[direct sum]], and [[direct product]].
   *  Added more versions to [[additive and abelian categories]].
   *  Questioned the purpose of [[filtrant category]].
   *  I hope that people saw Zoran\'s addition to a query box below.

* [[Urs Schreiber|Urs]]:

  * concerning the entry [[homological algebra]]: [[Tim Porter]] and [[Zoran Skoda]] (or anyone else): please feel free to improve/revise the exposition

  * created [[triangulated category]] -- have a question about _dérivateurs_ there

  * created [[local epimorphism]]

  * created [[multiplicative system]]

* [[David Roberts]]

  * Uploaded notes on anafunctors to my [web](http://ncatlab.org/davidroberts/show/HomePage). Comments welcome and wanted.

# 2009-03-29

* [[Mike Shulman|Mike]]:

  * A grammatical suggestion at [[stuff, structure, property]].

  * Did some work on [[chain complex]] and all the [[additive and abelian categories]] pages.

* [[Urs Schreiber|Urs]]:

  * began [[derived category]] before running out of time again...

  * did some layout-editing for [[Mike Shulman|Mike]]'s additions to [[homotopy category of an (infinity,1)-category]]

* [[Finn Lawler]]: Continued experimenting with graphics for diagrams.  Have a look and see what you think:

  * Added some PNGs at [[Kan extension]]

  * Replaced PNGs at [[adjunction]] with larger ones.

* [[David Corfield|David]]

  * asked a question at [[differential object]]

* [[Urs Schreiber|Urs]]

  * started preparing the ground for [[derived category]] and [[triangulated category]] by creating [[category with translation]], [[chain complex]] and [[differential object]] before running out of time...

  * created [[homotopy category of an (infinity,1)-category]]

  * created [[homological algebra]], mainly as a collection of links to the keywords listed there

  * thanks to [[Mike Shulman|Mike]] for his polishing of my original [[filtrant category]] at [[filtered category]]

  * in a first attempt to clean up the entries surrounding [[abelian category]] I created the overview entry [[additive and abelian categories]] and branched off [[Ab-enriched category]], made [[pre-additive category]] a commented redirect to that and "commented out" the respective discussion still to be found at [[additive category]]; also made [[pre-abelian category]] a separate entry, so that now there is in order of increasing structure/property

    * [[pre-additive category]]

    * [[additive category]]

    * [[pre-abelian category]]

    * [[abelian category]]

  * am all in favor of [[Finn Lawler|Finn]]'s graphics! The only reason I don't include nice graphics myself a lot is that currently these take me longer to create than the MathML hacks

* [[Finn Lawler]]: Uploaded PNG images of the zig-zag identities and added them to [[adjunction]].  They're probably a bit too small, but what do people think of this approach as a work-around until there's an easy way to convert TeX to SVG?  Any other suggestions?
(Note: I tried converting these diagrams to SVG as described [here](http://meta.wikimedia.org/wiki/Help:Displaying_a_formula#Commutative_diagrams) but the resulting files were huge and didn't display anyway when inserted into the markdown source.  Instead I used `pdfcrop` and `convert` on the `pdflatex` output.)

# 2009-03-28

* [[Finn Lawler]]:

  * Edited [[linear logic]] in response to Mike's question.

  * Created [[star-autonomous category]].

* [[Mike Shulman|Mike]]: Prompted by discussion with Zoran, created [[strict epimorphism]] and added a lengthier discussion of types of epimorphism to [[epimorphism]].

* [[Zoran Škoda]]: Created [[etale space]]. The order of exposition is important, particularly in view of anticipated additional details. In [[Kan extension]] added a detailed paragraph on an example how left Kan extension pointwise formula has intuitive meaning in the case of constructing pullback for (pre)sheaves on topological spaces. Created [[torsor with structure category]] following the version in Moerdijk's book.

* [[Finn Lawler]]: Created [[linear logic]] -- just a short stub with
  basic ideas on motivation and models, plus a couple of references.
  Comments effusively welcomed.  (Edit: also removed Thursday's query
  box from [[context]]).

# 2009-03-27

* [[Mike Shulman|Mike]]:

  * Zoran and I are having a discussion about definitions of [[abelian category]].

  * Redirected [[filtrant category]] to [[filtered category]].

  * Moved the discussion about the word "bimorphism" from [[balanced category]] to [[bimorphism]].

  * Redirected [[parallel morphism]] to [[parallel morphisms]] (which _is_ in line with the naming conventions).

* [[Urs Schreiber|Urs]]:

  * created [[exact functor]]

  * created [[filtrant category]]

  * added to [[Higher Topos Theory]] more introductory/overview remarks which are supposed to be helpful for the newbie

  * created [[Yoneda extension]]

  * added section to [[Kan extension]] on formulas in terms of limits and colimits over comma categories;

  * added a section on the "local" computation of adjoint functors at [[adjoint functor]] and point out how this induces the local/global dichotomy at [[limit]], [[homotopy limit]] and [[Kan extension]] (see my previous modification below)

  * if I noticed correctly, [[Mike Shulman|Mike]] had changed my original notation $p^* := - \circ p : [C',D] \to [C,D]$ for precomposition with a functor $p : C \to C'$ (pullback notation) at [[Kan extension]] to $p_*$ (pushforward notation). I have now added a section _Remark on terminology: pushforward vs. pullback_ which is supposed to clarify this terminology issue.

    +--{: .query}
    [[Mike Shulman|Mike]]: That wasn't me.  I'm not sure that such a discussion belongs at [[Kan extension]]; it might belong somewhere but I would rather than the page [[Kan extension]] just pick one notation and possibly link to a discussion.

    [[Zoran Škoda]]: It was me who changed, though I better did not. I am happy with the original notation as well. For as your discussion on pushfowards I am less happy. Namely, if one is not happy with the direction of maps between open sets, one just redefines what is a morphism of sites (opposite to the functor direction), so that the morphism of sites is always correct direction. So, unless one does not have strong feeling on the choice of pushfoward pullback meaning, what is not in this case, mayeb original notation just caring about covariant vs contravariant was better.
    =--

  * addressed [[Zoran Skoda|Zoran]]'s and [[Tim Porter|Tim]]'s remarks at [[Kan extension]]: I have added now to [[Kan extension]] as well as to [[limit]] -- in analogy to what we already had at [[homotopy limit]] -- an explicit discussion of the difference between _local_ and _global_ definitions of the universal constructions

  * created [[universal construction]] -- but filled in just a question/query

* [[Tim Porter|Tim]]: I have raised a query at [[Kan extension]].

*  [[Toby Bartels]]:
   *  Wrote [[hereditarily finite set]], which is more pretty than useful.
   *  Threatened to rewrite [[Grothendieck universe]] once again.
   *  Wrote about the $3$-way factorisation system at [[stuff, structure, property]].
   *  Accepted Mike\'s terminology ('moderate') at [[Grothendieck universe]].

* [[Zoran Škoda]]: Created [[abelian category]] with multiple equivalent definitions.

*  [[Toby Bartels]]:
   *  If people don\'t like having several entries in one day per Mike\'s request, another option (hopefully good enough for Mike) is to move your entire list up to the top when you add to it (being sure to add to the top of your list too).
   *  Finn has nothing to apologise for at [[context]].
   *  Zoran and I are discussing terminology at [[projective limit]].
   *  Zoran and Mike are discussing terminology at [[representable functor]] (I only made a more philosophical comment).
   *  Compare [[nice category of spaces]] with [[convenient category of topological spaces]].
   *  I accept Mike\'s terminology at [[set theory]].
   *  I refactored [[kernel]] to use primarily the equaliser definition in any pointed-enriched category.

# 2009-03-26

* [[Mike Shulman|Mike]]:

  * I have a question about the meaning of "large" at [[Grothendieck universe]].

  * I'd like to request that people not add new sub-bullets under their own names on a given day if other people have since listed more changes above; rather, add a new bullet point at the top with your name.  If that didn't make sense, it's what I'm doing now, rather than (what I could have done) adding a new bullet point below the other copy of my name today.

  * Added some comments on syntactic categories to [[internal logic]], since Toby kindly saved me the work of defining them at [[context]].

* [[Finn Lawler]]: (Hello all -- long-time lurker, first-time editor.)  For my first edit, I asked a silly question at [[context]] and then answered it myself a little later.  I'll delete the query box if nobody has any comments.  Apologies for noise.

* [[Mike Shulman|Mike]]:

  * Split [[nice category of spaces]] from [[nice topological space]] and fixed all the links I could find.

  * Wrote [[universe in a topos]] by way of responding to the query at [[Grothendieck universe]].

  * Focused [[adjunction]] on the internal version in a 2-category, to distinguish it from [[adjoint functor]], which I reorganized and added a definition to.  (The zig-zag identities are crying out for SVG!)

  * Corrected [[generalized element]] to distinguish it from [[global element]].

  * Made a terminological suggestion at [[set theory]].

  * Commented about property-like structure at [[stuff, structure, property]].  It would be nice to move the examples earlier on this page.

* [[David Corfield|David]]

  * created [[algebraic set theory]]

* [[Urs Schreiber|Urs]]

  * started a stub-entry on [[Stable Infinity-Categories]] (Lurie's PhD part I) and advertized this little program of textbook $n$labification [here](http://golem.ph.utexas.edu/category/2009/01/nlab_general_discussion.html#c022834) at the blog

  * started an entry [[Higher Topos Theory]] (on Lurie's book) in a style analogous to [[Categories and Sheaves]] -- I included a link to [[Mike Shulman|Mike]]'s personal page [[michaelshulman:n-topos for large n|n-topos for large n]]; eventually it would be nice if we had an entry on the general idea and purpose of [[higher topos theory]]

  * started expanding [[Kan extension]]

  * created [[fiber product]], [[parallel morphism]], [[zero morphism]], [[kernel]], added example to [[pointed object]]

  * good to see that [[Mike Shulman|Mike]] is back! Mike, there is a request for you at [[Grothendieck universe]]: can you say something about rephrasing that as a "topos object internal to the [[ETCS]]-version of $Set$"?

  * added the globular zig-zag diagrams to [[adjunction]]

  * started [[continuous functor]]

  * added a link to [[David Corfield|David]]'s new entry [[algebraic set theory]] at [[set theory]]. It would be nice to put it into context there, eventually.

  * started expanding [[limit]] (and also a bit [[colimit]]): more motivation, more details on definition, more examples

  * created [[generalized element]]

  * more or less completed the hyperlinked keyword list of chapter one of [[Categories and Sheaves]]

  * filled in three equivalent definitions at [[adjoint functor]]

# 2009-03-25

* [[Urs Schreiber|Urs]]

  * expanded a bit at [[natural transformation]]

  * added simple remarks to [[contravariant functor]]

  * added the illustrative diagram to [[over category]] and added a remark on over categories to [[subobject]];

  * added an illustrative diagram to [[comma category]] and added a section there on how a comma category is a pullback;

  * added a little bit of discussion that every presheaf is a colimit of representables to [[presheaf]];

  * expanded a bit more at [[stuff, structure, property]]

# 2009-03-24

* [[Urs Schreiber|Urs]]

  * have more stupid questions (for [[Toby Bartels|Toby]], probably) at [[Grothendieck universe]] and incorporated more of Toby's replies

*  [[Toby Bartels]]:
   *  Gave a structural definition at [[Grothendieck universe]], but somebody should check it (Mike, this means you).
   *  Some more foundational material, including [[set theory]], [[pure set]], and [[point]] (which is a bit purple).
   *  Note that Urs also started [[stuff, structure, property]].

# 2009-03-23

* [[Urs Schreiber|Urs]]: created [[Grothendieck universe]] -- but have questions. Mentioned Grothendieck universes at [[small category]] and [[locally small category]].

# 2009-03-22

*  [[Toby Bartels]]:  I added a section on morphisms between [[context]]s (the substitutions, or interpretations), including (as an example) a complete description of the category of contexts of the theory of a group.  There is an exercise (to describe that category in group-theoretic terms) whose formatting all authors might want to look at.

# 2009-03-21

* [[Tim Porter|Tim]]:

   * I added a link to p-adic solenoid in [[shape theory]] as that example gives insights on the links between this area and dynamical systems.

# 2009-03-20

* [[Andrew Stacey]]: lifted the tangent/cotangent section from "Comparative Smootheology" to [[Froelicher space]].  I intend to remove this section from that paper and this seems like a good place to put and develop it.

* [[Zoran Škoda]]: created [[algebraic monad]], [[generalized ring]], [[compact object]], [[noncommutative algebraic geometry]], [[spectrum (geometry)]], [[Pierce spectrum]], [[filter]] (thanks Mike for an essential typographic correction), [[generator]], [[cogenerator]] (the latter were prompted by editing [[Morita equivalence]], paragraph on classical Morita). Zoran and Toby distributed the paragraph on ultrafilters from long growing entry [[filter]] partly to the new entry [[ultrafilter]]. Uploaded [[warsaw.gif:file]] with link within [[shape theory]].

# 2009-03-19

* [[Urs Schreiber|Urs]]

  * followed [[Mike Shulman|Mike]]'s remark and moved the previous content in [[stable infinity-category]] to [[stable (infinity,1)-category]], keeping just a general nonsense statement at [[stable infinity-category]]

  * added a remark on this and a link at [[spectrum]]

* [[Zoran Škoda]]: moved the earlier material from entry algebra to new entry [[associative unital algebra]], and put new material into [[algebra]]; one should have separate entry for any framework for algebras, and general entry [[algebra]] should have pointers to the major classes (like [[algebra over operad]]). Thanks Toby, we continue together on that: now there is an entry [[nonassociative algebra]] and so on. I have also addressed concerns of Mike in [[Connes' cyclic category]] which now has I hope correct definitions, plus more foundational issues and relevant literature and link to just uploaded file  [[krasauskas.pdf:file]]

* [[Andrew Stacey|Andrew]]: Added some more to [[Froelicher space]].  I've started on a new project on this: adapting topological notions to Froelicher spaces.

* [[Urs Schreiber|Urs]]:

  * created [[stable infinity-category]]

  * implemented [[Zoran Skoda|Zoran]]'s remark below by rephrasing a bit at [[Categories and Sheaves]]

  * added a bit more to [[large category]]

*  [[Toby Bartels]]:  Let [[specialization topology]] lead me to [[specialization order]].

* [[David Roberts]]:

  * Created [[Grothendieck's Galois theory]]

* [[Mike Shulman|Mike]]:

  * Removed the discussion about accented characters from   [[fundamental group of a topos]] and put it in the [[FAQ]].

  * Displayed my happiness at [[quotient object]] by removing the discussion.

# 2009-03-18

* [[Zoran Škoda]] has created [[dense subcategory]] (intentionally organized different than the entry for the entry for slight generalization, [[dense functor]]); created [[shape theory]] but needs much more work; I copied here references from [[fundamental group of a topos]] (plus to a Batanin's article) and in [[fundamental group of a topos]] I added the reference and link to Pataraia's article important for the abstract notion of fundamental groupoid in internal contexts.

* [[Urs Schreiber|Urs]]:

  * created an entry on the book [[Categories and Sheaves]] (comment you say that "sheaf condition is localization" there; well the category of sheaves is a localization of the category of presheaves, but the sheaf condition...you really meant what you say? -- Zoran)

  * in reaction to [[Toby Bartels|Toby]]'s discussion at [[large category]] I created entries for [[accessible category]], [[locally presentable category]] and [[sketch]]. But very incomplete.

*  [[Toby Bartels]]:  Started a discussion about [[large category]].

* [[Andrew Stacey|Andrew]]:

  * More done on [[Froelicher space]]s.  I think that I have finally figured out the relationship between Frölicher spaces and Isbell duality so if anyone else is interested in taking a look I'd appreciate your comments.

    I also found the standard layout of the page a little hard to work with, in particular with regard to delimiting proofs and definitions (both of which could get quite long) so I've been experimenting with alternative ways of demarking them (on [[Froelicher space]]).  Let me know if you like or dislike what you see.

* [[David Corfield|David]]:

  * Added a reference in response to one of Mike's questions at [[fundamental group of a topos]].

* [[Mike Shulman|Mike]]:

  * Created [[quantale]], [[adjoint functor theorem]], and [[total category]].
  * Asked some questions at [[fundamental group of a topos]].

* [[Tim Porter|Tim]]:

   * Created [[fundamental group of a topos]].  At present this is lifted / adapted from an article I wrote some time ago so needs some attention.  (This necessary attention includes fixing some diacriticals on some names.)

# 2009-03-17

* [[Mike Shulman|Mike]]:

  * Yes, homotopy theorists call a [[k-tuply groupal n-groupoid]] a _grouplike $E_k$-space_.

  * Continued discussion at [[Crans-Gray tensor product]].

*  [[Toby Bartels]]:
   *  Wrote [[congruence]] to mean an internal equivalence relation.
   *  Wrote [[k-tuply groupal n-groupoid]] based on [[k-tuply monoidal n-category]].  But I strongly suspect that homotopy theorists have something to say here in a completely different language.  (Perhaps [[Mike Shulman]] knows.)
   *  Created [[wedge sum]] to link from the following, and [[smash product]] to go with it.
   *  Put in a general definition at [[homotopy group]], trying to show how $\pi_n$ has $n$ products that are all the same.

* [[Tim Porter|Tim]]: I have been trying to give an adequate categorical treatment of [[profinite completion of a group]].

* [[Urs Schreiber|Urs]]:

  * made a little remark on [[Mike Shulman|Mike]]'s question at [[Crans-Gray tensor product]] and ask another question myself

  * added a remark to [[Waldhausen category]] on its relation to [[category of fibrant objects]] and ask a [question](http://golem.ph.utexas.edu/category/2009/01/nlab_general_discussion.html#c022572) about the precise statement to be made here over at the $n$Café

*  [[Toby Bartels]]:  Combined [[proset]] into [[preorder]], etc, as planned there.

* [[Andrew Stacey|Andrew]]:

  * Shifted [[generalized object|generalised object]] to [[Isbell envelope]] in line with the comments from the categories mailing list.  Mainly just altered the terminology in line with this nomenclature, but added a small section on how it relates to profunctors as well.
  * Modified [[Froelicher space]] a little as well in line with the Isbell envelope nomenclature.

* [[Zoran Škoda]] created [[Waldhausen category]] and made a remark into entry [[cofibration category]].

* [[Toby Bartels]]:  I created [[symmetric set]] out of material that [[Zoran Škoda]] added to [[FinSet]].

# 2009-03-16

* [[Toby Bartels]]:  Tired of writing `[[constructivism|constructive mathematics]]`, I moved [[constructivism]] to [[constructive mathematics]] and fixed links.  Similarly, I moved [[predicativism]] to [[predicative mathematics]].  After some thought, I also moved [[finitism]] to [[finite mathematics]] and expanded it a bit to fit the new name better.  To go with this, I finally created [[FinSet]].

* [[Zoran Škoda|Zoran]]: Created [[Loday-Pirashvili category]], [[dense functor]] and [[equivariant object]]. There are two different notions of dense subcategory, first of which has two different definitions and is related to colimits and nerve functor, and second which is related to pro-objects. In the entry [[Bousfield localization]] I added a paragraph on Bousfield localization for triangulated categories; made changes to [[nerve]] (more to be done: one needs to clarify the example of geometric realization etc.).

* [[Tim Porter|Tim]]:
  * I have put a request on [[Loday-Pirashvili category]] for some discussion of the links between the ideas here and those in [[2-vector space]], as there is overlap.
  * I have started a discussion on the exposition of [[dg-algebra]]. My preferred approach is via graded object,  differential algebra and  chain complex. The present approach I find a bit confusing.
  * In [[profinite completion of a group]], I have pointed out a difference in the use of the term [[profinite group]], that may need discussion.

# 2009-03-15

* [[Zoran Škoda|Zoran]]: Created (in last two days) several entries mainly related to co- Hopf- algebras and algebras in categories of chain complexes: [[Frechet-Uryson space]], [[Hopf module]], [[Hopf-Galois extension]], [[Maurer-Cartan equation]], [[category of elements]], [[compactly generated space]], [[coring]], [[dg-algebra]],[[distributive law]], [[torsor]] (very unfinished!), [[twisted module of homomorphisms]], [[twisted tensor product]], [[twisting cochain]] and made changes to few other entries including many changes in entry [[Hopf algebra]] and some in [[A-infinity-algebra]]. With the (Fukaya) convention used there $D_0$ should not exist.

* [[Mike Shulman|Mike]]: Created [[homotopy equivalence]] and [[weak homotopy equivalence]].

* [[Tim Porter|Tim]]:

  * created [[I-category]] which includes an alternative axiomatisation of [[cylinder functor]].  This is needed for Baues' version of abstract homotopy theory. Some results and examples will need to be added later.

# 2009-03-14

* [[Bruce Bartlett]] has created [[nInsights]].

* [[Ronnie Brown]]: I've rewritten and expanded [[homotopy group]]s partly to clarify the operation of dimension 1 on higher dimensions  and to emphasise the groupoid aspects.

*  [[Toby Bartels]]:  I\'ve written [[sequence]], [[net]], [[multi-valued function]], [[partial function]], and the long-delayed [[surjection]] and [[injection]].  Those interested in foundations may be particularly interested in my proposed alternative definition of [[sequence]].

*  [[Tim Porter|Tim]]: I have included a discussion of the nerve of an [[internal category]] at that entry.

# 2009-03-13

*  [[Tim Porter|Tim]]: I have changed the initial sentence of [[homotopy n-type]].  I think this is converging well thanks to the efforts of Mike and Toby.

*  [[Toby Bartels]]:  Since we already have [[fundamental groupoid]] and even [[fundamental infinity-groupoid]], I started [[fundamental group]] and [[homotopy group]].  But I only wrote #Idea# sections.

# 2009-03-12

* [[Mike Shulman|Mike]]:

  * Responded at [[homotopy n-type]] and [[proset]].

  * Did a little more bettering of [[profinite group]], and was inspired to create [[filtered category]], [[pro-object]], [[ind-object]], and (as stubs) [[free completion]] and [[topological group]].

*  [[Tim Porter|Tim]]:
    * I have edited [[homotopy type]] as recent changes seem to have altered the idea away from the usual one making it perhaps too narrow.  A similar problem seems to occur with [[homotopy n-type]] and I have raised a terminological query there as well.
   * In [[homotopy n-type]] I have also queried the use of 'nice space' and its relation with [[nice topological space]].  This latter entry is perhaps misnamed as it is really the category of these spaces that is nice rather than the spaces themselves.
   *  I have raised a question on *Disambiguation* at [[proset]].  Have we a policy as to how to handle  terms with perfectly acceptable multiple meanings?
   *  I have tried to 'better' the previous entry on [[profinite group]]! (see the old version to see why I say it this way.)

* [[Mike Shulman]]: A question about the [[Crans-Gray tensor product]].

*  [[Toby Bartels]]:
   *  Broke [[undirected object]] off of [[directed object]] as planned.  Discussion stayed at [[directed object]], even though it discusses both concepts (and that of a [[directed space]]).
   *  Finally wrote [[full subcategory]] and directed links to it.
   *  I\'ve continued the conversation that [[Eric Forgy]] started at [[preorder]].
   *  I have my own terminological question at [[linear relation]].

# 2009-03-11

* [[Mike Shulman|Mike]]: I have a suggestion for a terminological change at [[k-tuply monoidal n-category]].

# 2009-03-10

* [[Tim Porter]]:

  * Eventually we probably need a summary of some of the theory of algebraic homotopy that Baues has developed as if impinges on the [[homotopy hypothesis]] and on [[homotopical cohomology theory]].  To this end I have created a sort of historical entry on [[algebraic homotopy]].

  *  Created [[cofibration category]] as the first of the 'Bauesian' detailed entries.

* [[Toby Bartels]]:  I\'ve written several more articles on very basic topics, such as those that used to be '?'-links below.  You can see them on Recently Revised; I don\'t think that anything merits great attention.

# 2009-03-09

*  [[Toby Bartels]]:  I finally wrote [[relation]], which makes me realise that there is no [[subset]] yet ....  Also [[order]], but that\'s just a list of links to more specific pages.

# 2009-03-08

*  [[Toby Bartels]]:  I tried to clarify the difference between a [[preorder]] (a structure on a given set that satisfies certain properties) and a [[proset]] (a set equipped with such a structure).  I need to finish that for [[partial order]]/[[poset]] and [[total order]]/[[toset]], although I would also entertain the idea that these should all be redirected one way or the other.  But I got sidetracked writing [[linear order]] and [[loset]] instead.  (And then there\'s [[quasiorder]]; I don\'t think that [[quoset]] is necessary for reasons that I don\'t want to get into here.)

* [[John Baez]]:

  * Attempted to answer Eric's plea for a category-theoretic definition of 'Hasse diagram', in the discussion at the bottom of [[preorder]].  Unfortunately I don't know the official definition of 'Hasse diagram' --- though I know one when I see one.

  * Made a short page on [[proset]], since Toby seems to be using this as a synonym for [[preorder]].

# 2009-03-05

* [[Andrew Stacey]]:

  * Continued debate on what to call "[[generalized object|generalised object]]" since there seem to be some generalised objections (sorry).

# 2009-03-03

* [[Andrew Stacey]]:
  * Archived February
  * Started [[generalized object]]

* [[Toby Bartels]]:  Since Urs is using both 'over category' and 'over-category' (and not 'slice category'), I tried to standardise things as 'over category' to diminish the temptation to slip further into 'over-category'.  Principally this means that I moved [[slice category]] to [[over category]] and [[over-category in quasi-categories]] to [[over quasi-category]].

# 2009-03-02

* [[Urs Schreiber|Urs]]

  * created [[limit in quasi-categories]] and in the course of saturating links from there also created [[join of simplicial sets]], [[join of categories]], [[join of quasi-categories]], [[over-category in quasi-categories]], [[terminal object in a quasi-category]] and edited [[quasi-category]] itself a bit

# 2009-03-01

*  [[Toby Bartels]]:
   *  Slightly revised [[Tim Porter]]\'s new introduction to [[homotopy 2-type]] and copied it (with appropriate changes) to [[homotopy 1-type]] and [[homotopy 3-type]].  Also combined it with what was already at [[homotopy n-type]], so that has the most complete discussion of the issues.
   *  Created a bunch of articles on specific notions of $n$-category, especially for $n \leq 3$: [[3-poset]], [[3-groupoid]], [[n-poset]], [[2-poset]], [[locally posetal 2-category]], [[2-groupoid]], [[1-category]], [[1-poset]], [[1-groupoid]], [[(-2)-groupoid]], [[(-1)-poset]].  Similarly, I moved some material from [[(-1)-groupoid]] to [[truth value]] to fit the purposes of the former page (as explained in old discussion that was also moved).
   *  Spun [[simplicial groupoid]] off of [[simplicially enriched category]].  Experts please check that the definitions there (based somewhat on a remark in [[Dwyer-Kan loop groupoid]]) are correct.

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