2009 January changes

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Archive of changes made during January 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/217) and work backwards.

***

# 2009-01-31

* [[Mike Shulman|Mike]]

  * Created [[retract]] and [[idempotent]].

  * Clarified homotopy limits at [[model 2-category]] and [[strict 2-limit]].

* [[Urs Schreiber|Urs]]

  * created [[model 2-category]] and tried to subsume our discussion about it at [[strict 2-limit]] -- would like ask [[Mike Shulman|Mike]] to check

* [[Ronnie Brown]]

  * created [[higher dimensional algebra]]

# 2009-01-30

* [[Urs Schreiber|Urs]]

  * added more examples to [[homotopy limit]] and [[span trace]] (see [blog comment](http://golem.ph.utexas.edu/category/2009/01/benzvi_on_geometric_function_t.html#c021787))

  * added _based loop object_ to _free loop object_ at [[loop space object]] and linked that with [[generalized universal bundle]]

* [[Tim Porter|Tim]]:

  * Continued work on [[crossed module]], and [[crossed n-cube]]. I have added in the fibration example into [[crossed module]], and that may be of interest to others.  It says, effectively, that a fibration induces a weak crossed module structure on the loop spaces.

# 2009-01-29

* [[Mike Shulman|Mike]]:

  * Finally decided I need to get over my Australian training and make weak things the default, at least when writing on the nLab.  So I split up [[2-categorical limit]] into [[2-limit]] for the weak notion and [[strict 2-limit]] for the strict one.  The exposition here could probably use help, and I have a question about terminology for [[strict 2-limit]]s.

  * Redirected [[Gray category]] to the already-existing [[Gray-category]].  Should we have an official policy on the use or non-use of hyphens?  (-:

  * I haven't done much here recently because I've been working in my [[michaelshulman:HomePage|own web]] on a project developing [[michaelshulman:2-categorical logic|2-categorical logic]].

* [[Eric Forgy|Eric]]:

  * changed yet [[generalized universal bundle|another definition]] from _italic_ to **bold**. Recall that we agreed to follow that convention.

* [[David Corfield|David]]:

  * Added a remark after Urs' remark on [[philosophy]].

* [[Tim Porter|Tim]]:

  * Continued work on [[crossed module]], [[crossed n-cube]] and related entries.

  * Linked the [[Dold-Kan correspondence]] to the entry on the [[Moore complex]] where there was already some discussion of this topic.

* [[Urs Schreiber|Urs]]

  * have a question/request at [[2-limit]] about "homotopical enrichment". Can anyone say more about this? [[Timothy Porter|Tim]]?

  * created [[loop space object]] and included at [[span trace]] as an example the statement that the [[loop space object]] is the [[homotopy limit|homotopy trace]] of the identity span on the object

  * have a request at the end of [[2-categorical limit]] that we eventually give a formulation in terms of [[homotopy limit]]s.

  * added the defintiion in terms of adjoints to the constant diagram functor to [[limit]]

  * replied to [[Eric Forgy|Eric]] at [[generalized universal bundle]]

  * created [[Dold-Kan correspondence]] and also [[Dold-Kan theorem]] as a redirect

  * created [[simplicial presheaf]], [[SSet]]

  * have a remark/question at one point in [[David Corfield|David]]'s text on [[philosophy]]

  * started giving a discussion of the point of (higher) category theory in [[physics]]

# 2009-01-28

* [[Ronnie Brown]]

  * created/worked on [[cat-n-group]], [[cat-2-group]], [[crossed square]], [[simplicial T-complex]], [[homotopy 3-type]], [[homotopy hypothesis]]

* [[Urs Schreiber|Urs]]

  * added examples and a bit more to [[distributor]] and implemented [[Todd Trimble|Todd]]'s suggestion there

  * have a question at [[homotopy theory]] on Loday's result

  * created [[Gray category]]

  * created [[2-vector space]], [[bimodule]], [[distributor]] (and [[profunctor]] as a redirect)

* [[Tim Porter|Tim]]:

  * Continued adding stuff to various entries on $cat^n$-groups and references for Loday's theorem at [[cat-n-group]].

  * Created [[2-crossed module]].

# 2009-01-27

* [[Urs Schreiber|Urs]]:

  * have a question for [[Andrew Stacey|Andrew]] at [[Froelicher space]]

  * added a reference to [[dualizing object]]

  * created [[cat-n-group]]

  * added a bunch of references to [[directed homotopy theory]] which [[Tim Porter]] kindly provided

  * started adding/searching for references on $n$-fold groupoids to [[homotopy hypothesis]], added a respective remark to [[n-fold category]]

* [[Tim Porter|Tim]]:

  * Continued [[Moore complex]], created [[homotopy 3-type]], included a  list of some of the algebraic and categorical models for 3-types (with the intention of starting pages on the various types). (Request help on other models, please.)

# 2009-01-26

* [[Andrew Stacey|Andrew]]:

  * Joined the $n$-Community.  Hello.

  * Modified [[HomePage]] to point to the $n$-Forum.

  * Made list of generalised smooth spaces on the [[generalized smooth space|generalised smooth spaces]] page (with the intention of starting pages on the various types).

* [[Tim Porter|Tim]]:

  * Created [[Moore complex]] which contains the definition of the homotopy groups of a simplicial group.

  * Added query to [[homotopy theory]]:  should a summary of the Baues approach to abstract homotopy theory be included somewhere?

* [[Urs Schreiber|Urs]]

  * incorporated [[Todd Trimble|Todd]]'s remark into [[span trace]] and created a general entry on [[trace]] in monoidal categories; tried to add some clarification that the point of [[span trace]] and [[co-span co-trace]] was not to describe the concept of trace in general, but to describe how for spans it harmonizes with the interpretation of spans as linear maps in [[groupoidification]] and nicely matches with the fact that on co-spans regarded as [[cobordism]]s it realizes the idea that one glues the two ends of a cobordism together to get the trace -- I found that simple observation [noteworthy](http://golem.ph.utexas.edu/category/2008/05/hopkinslurie_on_baezdolan.html#c021486) in the context of the cobordisms hypothesis, in that all [[extended cobordism]]s seem to be generated by the interval just under cartesian product and co-span co-trace, so any map from extended cobordisms to [[multispan]]s which sends pushouts to pullbacks and regards the interval as weakly equivalent to the point should be fixed by its value on the point (that last statement should be scrutinized, but it is what made me want to make the (obvious) notion of co-span co-trace explicit)

* [[Mike Shulman|Mike]]:

  * Started trying to bring some order to the treatment of coverages by creating [[coverage]] and working on [[Grothendieck topology]] and [[Lawvere-Tierney topology]].  I wasn't quite bold enough to try to excise the word "topology" from this subject entirely yet.

  * Added relevant comments to [[anafunctor]] and [[folk model structure]].

# 2009-01-25

* [[Todd Trimble|Todd]]:

  * added more to [[geometric shapes for higher structures]];

  * gave an answer to Eric's question at [[globe]]

  * took some issue with Urs about whether span traces (and implicitly, cospan traces) have or haven't appeared in the literature

* [[Tim Porter|Tim]]:

  * Created [[pospace]] to help the entry on [[directed space]], at which I gave a link through to the new page.

  * Created [[directed homotopy theory]]. This is at present a stub plus an inadequate list of references that do not do justice to the area ... as yet. I have built in some links but feel there should be others.

  * Created [[group T - complex|group T - complex]], but more needs adding here.

# 2009-01-24

* [[Urs Schreiber|Urs]]

  * somewhat hastily created [[fundamental infinity-groupoid]] (have the vague recollection that I wrote something like that before... but can't see it anymore...)

  * created [[cobordism]] and [[extended cobordism]] and included a bunch of literature related to that and to [[multispan]], notably to [[Marco Grandis]]' work on [[Cospans in Algebraic Topology]] and to [[Jeffrey Morton]]'s work on [[multispan]]s ans [[extended cobordism]]s

  * further details at [[multispan]] (or in the file linke to there)

  * created an entry on Nils Baas' concept of [[hyperstructure]]s, mainly to point out how close it is at least in spirit to [[span]]s appearing in [[groupoidification]] -- and in fact to [[multispan]]s

  * created [[span trace]] and [[co-span co-trace]]
   (I am not sure about the (best) spelling convention!)
   after I had checked with [[John Baez|John]] that this has not previously been said explicitly in the literature

* [[Tim Porter|Tim]]:

  * Created [[simplicial group]]s which was needed by several entries.
  * Commented in [[simplicial set]] about a notational problem that needs attention. The notation for face and degeneracies in [[simplicial set]] is at odds with the standard one in _the literature_. \(*Now fixed*\)
  * Created [[simplicial identities]].
  * Created [[simplicial object in Cat]] referring to [[simplicial object]] as it was a 'hanging link'in [[simplicial category]].

# 2009-01-23

* [[Mike Shulman|Mike]]:

  * Replied at [[cartesian monad]], [[2-categorical limit]], [[internal logic]], and [[regular monomorphism]].
  * Responded to David's question at [[internal logic]] by adding mention of [[sketch]]es and of disjunctive and geometric logic.
  * Created [[familial regularity and exactness]].
  * Added the internal version to [[anafunctor]].

* [[Urs Schreiber|Urs]]

  * worked on [[directed object]]:

    * added discussion of degenerate examples of interval objects

    * suggested (and inserted) a formalization of the reparametrization axiom that [[Toby Bartels|Toby]] had left open

    * replied to the discussion at [[directed space]] asking Toby to suggest how to deal with the slight terminological subtlety we're running into

    * added references to Fahrenberg and Raussen at [[directed space]] (there must be more literature on _directed homotopy theory_! somebody should look it up)

    * replied to [[Toby Bartels|Toby]]'s remark on the need for directed homotopies at [[directed space]].

* [[David Corfield|David]]:

  * Asked a question at [[internal logic]] and another at [[regular monomorphism]]. Thanks for starting this page, Mike. There's still so much I don't understand about this topic.

* [[John Baez]]:

  * Wrote [[pushout]] in the same gentle style as the previous article [[pullback]].  I want nice easy introductions to all our favorite limits and colimits!

  * Wrote brief stubs for [[colimit]] and [[totally ordered set]].

* [[Toby Bartels]]:

  * Content:
    * Wrote [[context]], an idea that should be better appreciated.
    * Tried a 'constructive' definition at [[directed object]], with relevant comments at [[directed space]].
    * Wrote about the [[axiom of choice]] in superextensive sites.
    * Discussed Banach spaces at [[concrete category]].

  * Discussion:
    * Asked a terminological question at [[cartesian monad]].
    * Comments on comments at [[monoidal category]].
    * Made a terminological suggestion at [[2-categorical limit]].

  * Wiki structure:
    * Set up [[ambimorphic object]] as a redirect.
    * Separated [[finite object]] from [[finite set]].
    * Moved [[Trimble's notion of weak n-category]] to [[Trimble n-category]].

  * Other, more minor, edits.

# 2009-01-22

* [[Urs Schreiber|Urs]]

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

# 2009-01-21

* [[Mike Shulman|Mike]]:

  * Finally got around to creating [[internal logic]], which coincidentally probably answers David's questions at [[regular category]].

* [[David Corfield|David]]

  * Asked some questions at [[regular category]].

* [[Mike Shulman|Mike]]:

  * Changed [[enriched homotopy theory]], to reflect my feeling that it is more general than [[homotopy coherent category theory]].

  * Created [[enriched factorization system]], [[orthogonality]], and [[Galois connection]].

  * Created [[split epimorphism]], [[strong epimorphism]], and [[extremal epimorphism]] just to satisfy links.

* [[Eric Forgy|Eric]]

  * asked a question about "directed internalization" on [[directed space]]
  * continues to have a brain aneurysm at [[directed space]] but is making progress

* [[Urs Schreiber|Urs]]

  * added a bit of [[John Baez|John]]'s material to [[span]]

  * created [[complete Segal space]] and [[(infinity,n)-category]]

  * added diagrams to [[weak factorization system]] and created a stub entry [[monomorphism]] to saturate all the wanted links to it

# 2009-01-20

* [[John Baez|John]]:

  * Since Urs and Eric didn't respond to my question about deleting the discussion in [[monoidal category]] (see 2009-01-19), I went ahead and deleted it --- restore it if you like, after reading the comments at the end of that entry!

  * The picture of the pentagon identity in [[monoidal category]] has mysteriously disappeared, though the source code is still present.  Help!

  * I added [[endofunctor]] and [[strict monoidal category]].

* [[Mike Shulman|Mike]]:

  * Replied at [[fibration]], [[directed object]], and [[homotopy hypothesis]].

* [[Urs Schreiber|Urs]]

  * commented in the discussion at [[fibration]] on the use of the word _transport_

* [[Tim Porter|Tim]]

  *  I have added a question to [[homotopy hypothesis]] asking what criteria should be added to give 'good' categorical or groupoidal models for homotopy types.

  * For  [[directed object]] I have, similarly, tried to pose question about the criteria that should be 'directing' our search for good concepts in this case.

  *  I have also added a question about the 'optimal' definition of [[fibration]], as it seems to me that the lifting property is nearer the idea of fibration than the [[transport]] one that Mike has put forward.

* [[Mike Shulman|Mike]]:

  * Created [[fibration]], [[Grothendieck fibration]], and [[pseudofunctor]].

# 2009-01-19

* [[Urs Schreiber|Urs]]

  * separated [[directed object]] from [[directed space]] and included the definition of directed topological space by Grandis

  * further reacted at [[directed space]] and created [[homotopy hypothesis]]

  * following discussion by [[Eric Forgy|Eric]] at [[directed space]] I propose in the discussion section a formalization of the notion "an object $X$ is _directed_" and "an object $X$ is _undirected_" for the case that $X$ is an object in a category with [[interval object]].

* [[John Baez|John]]:

   * expanded the entry on [[monoid]], giving lots of examples of monoid objects in monoidal categories.  I think lists of examples like can be very useful and fun, and I want more!  I would like a list of [[PROP|PROPs]], for example, saying that $FinSet$ is the PROP for commutative monoids, and so on.

   * If Urs is happy with how the discussion at the end of [[monoidal category]] has been incorporated into the body of the article, maybe we can remove that discussion.

   * slightly expanded the entry on [[braided monoidal category]] - but it really needs some diagrams!

# 2009-01-18

* [[Mike Shulman|Mike]]:

  * Added detail, examples, and terminological comments to [[bicategory]].

  * Created [[2-categorical limit]].

  * Replied at [[finite set]].

* [[Todd Trimble]]

  * added a bit of material on terminal coalgebras to [[coalgebra]].

  * had an organizational thought at the end of [[coalgebra]].

* [[Toby Bartels]]
  * Adjustments to [[source]], [[target]], and [[identity assigning morphism]].
  * Comments and questions at [[finite set]], [[internal category]], and [[bicategory]].

* [[Urs Schreiber|Urs]]

  * created [[bi-pointed object]] and started to describe what should be its closed monoidal structure to be used at [[interval object]], but are being interrupted now and have to run -- hope this is about right so far...

* [[Eric Forgy|Eric]]:

  * Plagiarized the definition of [[internal category]] from [Baez and Crans](http://arxiv.org/abs/math/0307263) since the previous definition stated things should work in "the obvious way", which was not obvious to me
  * Asked if [[directed space|directed spaces]] could be defined using [[interval object]] on the directed spaces page.

* [[Toby Bartels]]:
  * Added other terms and meanings to [[decidable object]]; some of this should probably go to other (new) pages.
  * Clarified my question at [[interval object]].
  * Asked an organisational question at [[Trimble's notion of weak n-category]].
  * Other minor edits.

* [[Mike Shulman|Mike]]:

  * Added another argument in favor of using $B$ at [[category algebra]].

  * Expanded the topos-theoretic discussion at [[finite set]], including some examples.

# 2009-01-17

* [[Todd Trimble|Todd]]

  * responded to a plea of Eric at [[internal category]] to discuss an example; cf. Tim's latest change below.

* [[Tim Porter]]

   * I created [[2-group]].  I have only as yet discussed the strict form. I also added in some discussion of the Brown-Spencer theorem and took the definition of 2-group apart a little partially in response to [[Eric]]'s comment on [[internal category]].
   *  I fixed the definition of [[internal category]]. Somehow the condition that the composites $s\cdot i$ and $t\cdot i$ had been omitted, as far as I could see at least.

* [[Emily Riehl]]

  * created [[small object argument]]

* [[Urs Schreiber|Urs]]

  * continued fiddling with [[interval object]]

  * added references to [[Trimble's notion of weak n-category]]

# 2009-01-16

* [[Mike Shulman|Mike]]

  * Added the internal version to [[group]].

  * Created [[well-powered category]].

* [[Jim Stasheff]] added a word of caution at [[Hopf algebra]]

* [[Urs Schreiber|Urs]]

  * added references to [[interval object]] which should be moved to [[Trimble's notion of weak n-category]] (which is however locked at the moment and I have to run now)

  * meant to further develop [[interval object]] but got distracted -- the little I could get done is on [pages 5,6,7](http://ncatlab.org/schreiber/files/nacqJan15.pdf#page=5) [[schreiber:Nonabelian cocycles and their sigma model QFTs|here]]

  * replied to discussion at [[interval object]]

# 2009-01-15

* [[Toby Bartels]]: Wrote [[algebraic theory]], mostly to distinguish it from [[Lawvere theory]].

* [[Mike Shulman|Mike]]:

  * Toby and I appear to have finally reached a consensus on terminology at [[extensive category]].  I did some work on this page and [[coherent category]], [[pretopos]], and [[Grothendieck topos]] making them more consistent, and also created [[disjoint coproduct]] and [[coproduct]].

  * Responded to Toby at [[Lawvere-Tierney topology]].

* [[Tim Porter]] I have added in a new entry on the notion of a [[ simplicial T-complex]], which should help provide background for the still needed entry on complicial sets.

* [[Todd Trimble]]:

  * constructed an entry for "my" notion of weak $n$-category. Needs more revision and more work, but in part it was written in response to discussion between Urs and me (see e.g., [[interval object]]).

* [[Urs Schreiber]]

  * to saturate links from [[Trimble's notion of weak n-category]] and [[interval object]] I created [[A-infinity algebra]] and [[A-infinity category]] and filled in a little bit of information

  * started proposing a full definition at [[interval object]] following the discussion at that entry and at [[Trimble's notion of weak n-category]] -- please check!

  * the entry [[smooth Lorentzian space]] is currently oscillating a bit between showing a desire to state a new nice definition for (possibly generalized) Lorentzian spaces  and the aim to first establish some facts about the usual definition

  * filled in stuff at [[causet]]

  * added **questions** to [[Trimble's notion of weak n-category]]

  * expanded on the examples in [[generalized universal bundle]]

* [[Toby Bartels]]: I added stuff that I\'m trying to understand to [[Lawvere-Tierney topology]]. It is probably correct, but I had some questions. (See also the relevant terminological discussion at [[Grothendieck topology]].)

# 2009-01-14

* [[Mike Shulman|Mike]]:

  * Added detail about internal-homs and smash products to [[pointed object]], in response to Todd's comment.

  * Created [[adjunct]].

* [[Urs Schreiber|Urs]]:

  * started [[generalized universal bundle]] after all -- now I really need to run...

  * created [[pointed set]] and [[under category]], am planning to expand on the bigger story indicated at [[pointed set]], but not today

  * expanded the entry [[bi-brane]] in order to accompany some discussion about [[geometric function theory]] at [Ben-Zvi on geometric function theory](http://golem.ph.utexas.edu/category/2009/01/benzvi_on_geometric_function_t.html)

  * said more at the beginning of [[interval object]] -- and are having a discussion with [[Todd Trimble|Todd]] about examples and the necessity to relax the structure, and his Trimblean definition of $\infty$-category -- which we ought to have an entry about

  * in the entry [[category of fibrant objects]]

    * added comments on relation to [[pointed object]] and [[pointed category]]

    * made the _factorization lemma_ more explicit and related it to the notion of [[Morita equivalence]]

  * created a `categories: reference`-entry for [[Brown -- Abstract Homotopy Theory and Generalized Sheaf Cohomology]]

  * added a remark on the definition of kernel and cokernel of morphisms to [[pointed object]] -- some expert please have a look at that

  * added [[pointed category]] and also a handful of trivial entries which are actually superfluous, since I hadn't had my first coffee yet and wasn't paying proper attention...

* [[Mike Shulman|Mike]]:

  * Incorporated the discussion into [[k-tuply monoidal n-category]] and added more examples.  Perhaps some of this page would better go at [[delooping hypothesis]]?

  * Created [[pointed object]] and [[spectrum]].

# 2009-01-13

* [[Toby Bartels]]:
  * Conversations, including a new one with [[Todd Trimble|Todd]] at [[balanced category]].
  * Revised [[k-tuply monoidal n-category]]. Later I need to revise [[periodic table]] and write [[k-connected n-category]].
  * Wrote [[Elephant]] and [[Categories Work]], creating `category: reference` for them.

  +--{.query}
  [[Urs Schreiber|Urs]]: concerning [[Toby Bartels|Toby]]'s
  `category: reference`: I like that.
    I was thinking about including separate entries on references, too. We have a couple of further references already  which we refer to often enough to justify a separate entry for them eventually
  =--

* [[Urs Schreiber|Urs]]

  * created [[derived infinity-stack]], so far containing a sketch of the rough idea, a few references and pointers to current discussion on the blog

* [[Mike Shulman|Mike]]:

  * Created [[bijection]] and [[subsingleton]].

  * Answered Toby's three questions, and continued discussion at [[inhabited set]].

* [[Toby Bartels]]:
  * I took part in conversations that are already listed here.
  * I have questions for Mike at [[k-tuply monoidal n-category]], [[choice object]], and [[finite set]]. (Listed in order of decreasing importance.)

# 2009-01-12

* [[Mike Shulman|Mike]]:

  * Joined the discussion at [[category algebra]].

  * Removed the comments at [[constructivism]] that have now been incorporated into the main text (thanks for writing that detailed section, Toby).

  * Created [[finite set]] and [[choice object]].

* [[Tim Porter|Tim]]

  * I have asked on the page relating to [[action]] if the definition needs to be slightly wider for optimal later use.

* [[Todd Trimble|Todd]]

  * added some discussion to [[classifying topos]], linking up to the entry [[Lawvere theory]]

  * created entry on [[Cauchy complete category]]

* [[Toby Bartels]]:
  * I\'m having a conversation with myself at [[local ring]].
  * I\'m otherwise caught up and will rewrite [[k-tuply monoidal n-category]] before I do anything else.

* [[Mike Shulman|Mike]]:

  * Removed the discussion at [[field]], since it had converged and been incorporated into the entry.

  * Incorporated the apparent conclusion of the discussion at [[extensive category]] into the entry.

  * Added some details to [[excluded middle]].

# 2009-01-11

* [[Todd Trimble|Todd]]

  * began, at Urs's request, an entry for [[Yoneda reduction]]

  * finally submitted draft of regular category, after having it locked for a few days and doing very little with it in the end.

  * would like to point out that material relevant to entries like [[universal property]], [[universal construction]] and [[Yoneda lemma]] exists at [[representable functor]]; this could use further expansion

* [[Urs Schreiber]]:

  * created a separate entry [[Yoneda embedding]], even though the information there is also recalled in [[Yoneda lemma]]

* [[Toby Bartels]]:
  * I created [[local ring]] and [[excluded middle]].
  * Discussions seem to be converging at [[constructivism]], [[extensive category]], [[field]], [[inhabited set]], [[k-tuply monoidal n-category]], [[predicativism]], and [[Grothendieck topology]] (whew!).
  * However, the discussion at [[(-1)-groupoid]] has become [a Café post](http://golem.ph.utexas.edu/category/2009/01/nlab_general_discussion.html#c021235).
  * I don\'t consider myself caught up yet, so if you\'re waiting for me to do something ... you\'ll have to wait another day.

# 2009-01-10

* [[Urs Schreiber|Urs]]:
  * created [[Kan fibration]] and edited [[horn]], [[hypercover]] and [[Kan complex]]
  * added remark on relation to [[FQFT]] at [[generalized tangle hypothesis]]
  * have some questions for [[Toby Bartels|Toby]] (or other topos experts) at [[(-1)-groupoid]] -- see also my general discussion question at the blog: [enriched sheaf and topos theory?](http://golem.ph.utexas.edu/category/2009/01/nlab_general_discussion.html#c021224)

* [[Toby Bartels]]:
  * I wrote [[Boolean topos]], [[COSHEP]], and [[finitism]].
  * I expanded [[equivalence relation]] and [[kernel pair]].
  * I wrote [[(-1)-groupoid]]; I\'ll try to do one of these a day, but they\'re dully repetitive.
  * I talked with Mike at [[k-tuply monoidal n-category]] and [[inhabited set]].

# 2009-01-09

* [[Toby Bartels|Toby]]:

  * I talked with Mike (and occasionally made substantive edits, too) at [[field]], [[Grothendieck topology]], [[extensive category]], [[predicativism]], and [[constructivism]].

* [[Mike Shulman|Mike]]:

  * Created [[coherent category]], [[Boolean category]], [[Heyting category]], [[equivalence relation]], [[kernel pair]], [[exact category]], [[balanced category]], and [[pretopos]].

  * Finally got fed up with the nonexistent links to it everywhere and created [[product]].  Also created [[generalized the]].

  * Did some rephrasing at [[infinity-category]].

* [[David Corfield|David]]

  * started a page on [[generalized tangle hypothesis]].

* [[Urs Schreiber|Urs]]:

  * created [[directed space]]

  * added a bit to [[David Corfield|David]]'s entry on the [[generalized tangle hypothesis]]

  * reacted to [[Mike Shulman|Mike]]'s comments at [[infinity-stack homotopically]]

* [[Mike Shulman]]:

  * Continued my discussions with Toby at [[Grothendieck topology]], [[extensive category]], and [[constructivism]], including one that should be moved to [[predicativism]].

  * Started new discussions at [[infinity-stack homotopically]], [[k-tuply monoidal n-category]], and [[inhabited set]].

  * Added the unit axiom to [[monoidal model category]].

# 2009-01-08

* [[Toby Bartels]]
  * Mike and I are discussing terminology at [[Grothendieck topology]] and [[extensive category]] (possibly also elsewhere).
  * I wrote [[regular epimorphism]] and [[epimorphism]].
  * I prepared the wiki for an article [[closed category]] (as such) by moving most of the material to [[closed monoidal category]] and fixing the links. (But I didn\'t actually write it yet.)

* [[John Baez]]

  * worked pertinent discussion into body of [[monoidal category]]

* [[Urs Schreiber]]

  * reacted to the discussion at [[path object]]

  * created [[cylinder functor]]

  * added my 2cent and my questions to the discussion at [[homotopy]] on [[interval object]]s -- and added the entry [[interval object]] but filling it not with a statement but with discussion: I say what I came to think a well behaved interval object should satisfy and am asking for information on what is known in this regard

  * created [[pushout-product axiom]] and linked to it from [[monoidal model category]]

  * tried to provide a more unified perspective on [[infinity-stack]] and [[infinity-category]] and [[nonabelian cohomology]] at [[infinity-stack homotopically]]

  * had the honor of creating [[Yoneda lemma]]

  * created [[geometric function theory]] to accompany a [guest post](http://golem.ph.utexas.edu/category/2009/01/benzvi_on_geometric_function_t.html) by [[David Ben-Zvi]] -- maybe somebody of the [[groupoidification]] team can eventually add an entry on [[Hecke algebra]] and its relation to groupoidification

* [[Toby Bartels]]: I wrote about [[constructivism]] and the [[empty set]].

* [[Mike Shulman|Mike]]

  * Created [[geometric morphism]], [[locale]], and [[sober space]].

  * Added a more classical version to [[homotopy]].

# 2009-01-07

* [[Todd Trimble|Todd]]

  * created [[regular category]]

  * created [[image]]

  * created [[Rel]]

  * created [[Lawvere theory]]

  * started some discussion at [[simplex category]]

* [[Timothy Porter]] created

  * [[simplicial complex]]

  * [[simplicially enriched category]]

  * [[horn]]

  * [[nerve]] (but see below)

* [[Urs Schreiber|Urs]]

  * gave the bare definition of [[crossed module]]

  * finally created [[algebraic definition of higher category]] and [[n-fold category]]. Just stubs so far, this deserves much (much) more discussion, clearly.

  * did [[geometric realization]]

  * started filling in [[nerve]] after I found an empty entry of that name -- now [[Timothy Porter]] sends me by email his version. Either he or I  should merge the material...

  * added details (definition and theorems) to [[category of fibrant objects]]

  * restructured the index page [[mathematics]] slightly and tried to complete the lists of keywords at [[foundations and logic]]. Can I suggest that everybody who creates a new entry considers adding a link to that entry, if appropriate, to the lists of "related entries" at one of the big index pages? I am thinking it would be good to give readers a chance to browse our material by topic and get an impression for what is there and what not (yet).

  * added references to [[Timothy Porter]] and Cordier at [[homotopy coherent category theory]]

  * Expanded slightly on [[Timothy Porter]]'s entry on [[simplicially enriched category]].

  * added a few remarks at [[higher category theory]]. I am not happy with that entry. Clearly we need a more comprehensive discussion there eventually.

* [[Mike Shulman|Mike]]: Created [[constructivism]] and imported the relevant discussion from [[apartness relation]].

# 2009-01-06

* [[Toby Bartels]]: [[Mike Shulman]] and I are having terminological discussions. Also, he fixed my theorem at [[extensive category]] (which will go in our paper, John).

* [[Urs Schreiber|Urs]]

  * further expanded on a bit and harmonized a bit more the circle of entries [[globe category]], [[simplex category]], [[cube category]] and [[globe]], [[simplex]], [[cube]] linked to and summarized in   [[geometric shapes for higher structures]]. At [[globe]] I give a reply to a question by Eric on how to think of globes as "pointed spheres" by stating a claim that the $(n+1)$-globe is the double cone over the $n$-globe in a precise sense. I believe this is true, but am not entirely happy with the proof I have. Would be great if somebody could check this.

* [[Mike Shulman|Mike]]

  * Created [[power]] and [[copower]].  Possibly these should be just one page?
    +--{.query}
     I think it is good to have many separate entries for sub-concepts if they all link to each other, maybe with a brief comment. --[[Urs Schreiber|Urs]]

    I agree. —Toby
    =--

# 2009-01-05

* [[Toby Bartels]]: I reacted to everything that everybody did this year, and I interacted with everything that everybody did today. More specifically:
  * I created [[parallel morphisms]], although perhaps I should have called it [[parallel pair]].
  * I added material, some possible irrelevant, to [[global element]].
  * I fixed typos in the stuff that I posted this year and copied some relevant bits to other pages, like [[2-category]].
  * I added some false material to [[extensive category]].
  * I added my opinion to [[Grothendieck topology]] and [[subcategory]].
  * I followed Mike\'s changes to [[simplicial category]] and applied them to [[cubical category]], creating [[cube category]] and (for good measure) [[globe category]].
  * I disambiguated links to [[simplicial category]], [[omega-category]], and [[internalization]].
  * I gave my favourite example of the [[red herring principle]].
  * I probably did some other stuff too, which I can no longer recall.

* [[Mike Shulman|Mike]]

  * Split off [[internal category]] from [[internalization]].  Probably a lot of links need to be updated.

  * Summarized the discussion at [[subcategory]] into a section called "Non-evil variants."

  * Created the entry [[red herring principle]].

* [[Todd Trimble|Todd]]

  * created the entry [[CW complex]]

  * added an alternative definition of [[cube category]], linking it to $\Delta$ and string diagrams

* [[Urs Schreiber|Urs]]

  * am reading [[Mike Shulman|Mike]]'s [Homotopy limits and colimits and enriched category theory](http://arxiv.org/abs/math.AT/0610194v1) and started adding some central definitions as entries here, such as [[closed monoidal homotopical category]] and [[enriched homotopical category]] and some related entries such as [[homotopical category]] and [[homotopy coherent category theory]]. Eventually I would like to see if some of my favorite monoidal homotopical categories are _examples_. See my request [here](closed monoidal homotopical category).

  * added [[Crans-Gray tensor product]]

  * started replying to [[Todd Trimble|Todd]]'s remarks at [[stack]] by addding a section _Descent in terms of pseudo-functors_ to [[descent and codescent]].

  * following the discussion which I now moved to [[discussion on terminology -- omega-category]] I moved the original material on that to [[strict omega-category]] and kept at [[omega-category]] just a pointer to that entry. This means we should go through a bunch of entries and rename links to [[omega-category]] into links to [[strict omega-category]].

* [[David Corfield]]

  * created [[philosophy]], [[Klein 2-geometry]]

* [[Emily Riehl]]

  * created [[model structure on simplicial sets]]

* [[Mike Shulman]]:

  * Added [[nice topological space]].

# 2009-01-04

* [[Mike Shulman]]:

  * Added disambiguation comments to [[simplicial category]].

  * Created [[Gray tensor product]] and [[Gray-category]].

* [[Toby Bartels]]:
  * I posted [[k-tuply monoidal n-category]]. I\'ll correct typos and look at everything else tonight (when it\'s technically -05).
  * Urs is right about $\infty$ vs $\omega$, and I\'ve changed the other appearances in [[(n,r)-category]] and [[periodic table]] (as well as fixing the new one!).

* [[Mike Shulman]]:

  * Added a terminological suggestion to [[omega-category]].

  * Refactored [[tensor product]], removing the discussion which prompted the refactoring.  Created [[fc-multicategory]].

* [[Urs Schreiber|Urs]]:

   * meant to follow up on [[Todd Trimble|Todd]]'s discussion of [[sheaf]] and [[stack]] in terms of [[sieve]]s and relate it to the entry on [[descent and codescent]] and  [[infinity-stack]] but was distracted by other things. But I did write up a quick rough note on this into a LaTeX file, which you can see here for the time being: [[if.pdf:file]].

   * expanded on the [[sheaf]]-condition, offering in parallel to the definition in terms of [[sieve]]s the definition in terms of [[oriental]]s with a a few remarks on how both are related and what the issue is with generalizing to  [[stack]] and [[infinity-stack]].

   * added, following email discussion with [[Todd Trimble|Todd]], a discussion to [[sieve]] of which sieves are the presheaf incarnations of which [[cover|covers]], and why

   * adapted [[n-category]] and [[n-groupoid]] to [[Toby Bartels|Toby]]'s [[(n,r)-category]] (which is a great entry -- we should flesh that out further).

   * took the liberty of changing in [[Toby Bartels|Toby]]'s entry [[(n,r)-category]] the "$\omega$-categories" appearing there to "$\infty$-categories", since I thought we did agree that "$\infty$-category" is the generic term while [[omega-category]] specifically means _strict globular $\infty$-category_.

   * separated [[infinity-category]] from [[higher category theory]] and added some brief stub entries such as [[n-category]], [[n-groupoid]] and [[Kan complex]].

   * created a _blog_ entry [nLab -- General Discussion](http://golem.ph.utexas.edu/category/2009/01/nlab_general_discussion.html) and added a remark at the beginning of the original _wiki_ entry [[General Discussion]] asking to post further general discussion not to the wiki, but to the blog.

   * rearranged [[tensor product]] following the comments and corrections by [[Toby Bartels|Toby]] and [[Mike Shulman|Mike]] (thanks!).

# 2009-01-03

* [[Mike Shulman]]:

  * there is a discussion going on at [[subcategory]] over whether any faithful functor should be called a "subcategory."  Input welcomed.

  * added a comment on terminological objections to [[Grothendieck topology]].

  * corrected the example of abelian groups at [[tensor product]].

  * added examples and comment on non-monoidal closure to [[closed category]]

* [[Todd Trimble]]:

  * added to (and edited) examples under [[closed category]]

  * created [[dinatural transformation]]

  * added examples to [[extensive category]]

  * added to the discussion on [[Day convolution]]

  * added some detail to the definition of [[Lawvere-Tierney topology]]

* [[Urs Schreiber]]:

  * did [[closed category]] and linked to it from Mike's [[cartesian monoidal category]]

  * expanded on [[Day convolution]]

  * split the information previously contained in [[site]] into several separate entries: [[cover]], [[sieve]], [[Grothendieck topology]], [[Lawvere-Tierney topology]] and, of course, [[site]].

* [[Mike Shulman]] created [[extensive category]], [[cartesian monoidal category]]

* [[Todd Trimble]] expanded on [[stack]]
and connected it to the discussion of covering sieves at
[[site]]

* [[Toby Bartels]]: I posted [[periodic table]] and [[(n,r)-category]], which I wrote today offline. There are probably lots of typos right now; I don\'t have a lot of time online today. Also, the former one really only contains an appendix that I wanted to write, not so much the article itself. But at least you can read what\'s there now.

# 2009-01-02

* [[Todd Trimble]] created [[operad]]

* [[Urs Schreiber|Urs]]: started adding remarks to [[opposite category]] about the big story hidden here related to [[duality]].

* [[Todd Trimble]] has expanded on [[site]] and [[sheaf]]. I have started adding a bit more motivational detail to [[site]] and links to [[descent and codescent|descent]], [[stack]] and [[infinity-stack]]. There is still plenty of room here for saying more about this general story.

* [[Urs Schreiber|Urs]]: motivated by some [discussion](http://golem.ph.utexas.edu/category/2008/12/organizing_the_pages_at_nlab.html) we had on the blog, in the entry [[About]] I started giving some indications on what the $n$Lab is and what it is not.

* [[Owen Biesel]]: added [[diagram]], [[global element]], [[terminal object]], [[limit]].

# 2009-01-01

Apparently, we all took a break from the $n$Lab for the New Year!

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