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\section*{2009 January changes}

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

[[!include all changes]]

Archive of changes made during January 2009. The substantive content of this page should \textbf{not} be altered. For past versions of this page beyond its own history, start \href{http://ncatlab.org/nlab/revision/2009+June+changes/217}{here} and work backwards.


\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{20090131}{}\section*{{2009-01-31}}\label{20090131}

\begin{itemize}%
\item [[Mike Shulman|Mike]]

\begin{itemize}%
\item Created [[retract]] and [[idempotent]].


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



\end{itemize}

\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item created [[model 2-category]] and tried to subsume our discussion about it at [[strict 2-limit]] -- would like ask [[Mike Shulman|Mike]] to check

\end{itemize}

\item [[Ronnie Brown]]

\begin{itemize}%
\item created [[higher dimensional algebra]]

\end{itemize}


\end{itemize}
\hypertarget{20090130}{}\section*{{2009-01-30}}\label{20090130}

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

\begin{itemize}%
\item added more examples to [[homotopy limit]] and [[span trace]] (see \href{http://golem.ph.utexas.edu/category/2009/01/benzvi_on_geometric_function_t.html#c021787}{blog comment})


\item added \emph{based loop object} to \emph{free loop object} at [[loop space object]] and linked that with [[generalized universal bundle]]



\end{itemize}

\item [[Tim Porter|Tim]]:

\begin{itemize}%
\item 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.

\end{itemize}


\end{itemize}
\hypertarget{20090129}{}\section*{{2009-01-29}}\label{20090129}

\begin{itemize}%
\item [[Mike Shulman|Mike]]:

\begin{itemize}%
\item 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.


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


\item 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]].



\end{itemize}

\item [[Eric Forgy|Eric]]:

\begin{itemize}%
\item changed yet [[generalized universal bundle|another definition]] from \emph{italic} to \textbf{bold}. Recall that we agreed to follow that convention.

\end{itemize}

\item [[David Corfield|David]]:

\begin{itemize}%
\item Added a remark after Urs' remark on [[philosophy]].

\end{itemize}

\item [[Tim Porter|Tim]]:

\begin{itemize}%
\item Continued work on [[crossed module]], [[crossed n-cube]] and related entries.


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



\end{itemize}

\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item have a question/request at [[2-limit]] about ``homotopical enrichment''. Can anyone say more about this? [[Timothy Porter|Tim]]?


\item 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


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


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


\item replied to [[Eric Forgy|Eric]] at [[generalized universal bundle]]


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


\item created [[simplicial presheaf]], [[SSet]]


\item have a remark/question at one point in [[David Corfield|David]]`s text on [[philosophy]]


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



\end{itemize}


\end{itemize}
\hypertarget{20090128}{}\section*{{2009-01-28}}\label{20090128}

\begin{itemize}%
\item [[Ronnie Brown]]

\begin{itemize}%
\item created/worked on [[cat-n-group]], [[cat-2-group]], [[crossed square]], [[simplicial T-complex]], [[homotopy 3-type]], [[homotopy hypothesis]]

\end{itemize}

\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item added examples and a bit more to [[distributor]] and implemented [[Todd Trimble|Todd]]`s suggestion there


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


\item created [[Gray category]]


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



\end{itemize}

\item [[Tim Porter|Tim]]:

\begin{itemize}%
\item Continued adding stuff to various entries on $cat^n$-groups and references for Loday's theorem at [[cat-n-group]].


\item Created [[2-crossed module]].



\end{itemize}


\end{itemize}
\hypertarget{20090127}{}\section*{{2009-01-27}}\label{20090127}

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

\begin{itemize}%
\item have a question for [[Andrew Stacey|Andrew]] at [[Froelicher space]]


\item added a reference to [[dualizing object]]


\item created [[cat-n-group]]


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


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



\end{itemize}

\item [[Tim Porter|Tim]]:

\begin{itemize}%
\item 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.)

\end{itemize}


\end{itemize}
\hypertarget{20090126}{}\section*{{2009-01-26}}\label{20090126}

\begin{itemize}%
\item [[Andrew Stacey|Andrew]]:

\begin{itemize}%
\item Joined the $n$-Community. Hello.


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


\item 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).



\end{itemize}

\item [[Tim Porter|Tim]]:

\begin{itemize}%
\item Created [[Moore complex]] which contains the definition of the homotopy groups of a simplicial group.


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



\end{itemize}

\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item 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 \href{http://golem.ph.utexas.edu/category/2008/05/hopkinslurie_on_baezdolan.html#c021486}{noteworthy} 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)

\end{itemize}

\item [[Mike Shulman|Mike]]:

\begin{itemize}%
\item 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.


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



\end{itemize}


\end{itemize}
\hypertarget{20090125}{}\section*{{2009-01-25}}\label{20090125}

\begin{itemize}%
\item [[Todd Trimble|Todd]]:

\begin{itemize}%
\item added more to [[geometric shapes for higher structures]];


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


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



\end{itemize}

\item [[Tim Porter|Tim]]:

\begin{itemize}%
\item Created [[pospace]] to help the entry on [[directed space]], at which I gave a link through to the new page.


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


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



\end{itemize}


\end{itemize}
\hypertarget{20090124}{}\section*{{2009-01-24}}\label{20090124}

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

\begin{itemize}%
\item somewhat hastily created [[fundamental infinity-groupoid]] (have the vague recollection that I wrote something like that before\ldots{} but can't see it anymore\ldots{})


\item 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


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


\item 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


\item 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



\end{itemize}

\item [[Tim Porter|Tim]]:

\begin{itemize}%
\item Created [[simplicial group]]s which was needed by several entries.
\item 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 \emph{the literature}. (\emph{Now fixed})
\item Created [[simplicial identities]].
\item Created [[simplicial object in Cat]] referring to [[simplicial object]] as it was a `hanging link'in [[simplicial category]].

\end{itemize}


\end{itemize}
\hypertarget{20090123}{}\section*{{2009-01-23}}\label{20090123}

\begin{itemize}%
\item [[Mike Shulman|Mike]]:

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

\end{itemize}

\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item worked on [[directed object]]:

\begin{itemize}%
\item added discussion of degenerate examples of interval objects


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


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


\item added references to Fahrenberg and Raussen at [[directed space]] (there must be more literature on \emph{directed homotopy theory}! somebody should look it up)


\item replied to [[Toby Bartels|Toby]]`s remark on the need for directed homotopies at [[directed space]].



\end{itemize}


\end{itemize}

\item [[David Corfield|David]]:

\begin{itemize}%
\item 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.

\end{itemize}

\item [[John Baez]]:

\begin{itemize}%
\item Wrote [[pushout]] in the same gentle style as the previous article [[pullback]]. I want nice easy introductions to all our favorite limits and colimits!


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



\end{itemize}

\item [[Toby Bartels]]:

\begin{itemize}%
\item Content:

\begin{itemize}%
\item Wrote [[context]], an idea that should be better appreciated.
\item Tried a `constructive' definition at [[directed object]], with relevant comments at [[directed space]].
\item Wrote about the [[axiom of choice]] in superextensive sites.
\item Discussed Banach spaces at [[concrete category]].

\end{itemize}

\item Discussion:

\begin{itemize}%
\item Asked a terminological question at [[cartesian monad]].
\item Comments on comments at [[monoidal category]].
\item Made a terminological suggestion at [[2-categorical limit]].

\end{itemize}

\item Wiki structure:

\begin{itemize}%
\item Set up [[ambimorphic object]] as a redirect.
\item Separated [[finite object]] from [[finite set]].
\item Moved [[Trimble's notion of weak n-category]] to [[Trimble n-category]].

\end{itemize}

\item Other, more minor, edits.



\end{itemize}


\end{itemize}
\hypertarget{20090122}{}\section*{{2009-01-22}}\label{20090122}

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

\begin{itemize}%
\item created [[(infinity,n)-category of cobordisms]]

\end{itemize}


\end{itemize}
\hypertarget{20090121}{}\section*{{2009-01-21}}\label{20090121}

\begin{itemize}%
\item [[Mike Shulman|Mike]]:

\begin{itemize}%
\item Finally got around to creating [[internal logic]], which coincidentally probably answers David's questions at [[regular category]].

\end{itemize}

\item [[David Corfield|David]]

\begin{itemize}%
\item Asked some questions at [[regular category]].

\end{itemize}

\item [[Mike Shulman|Mike]]:

\begin{itemize}%
\item Changed [[enriched homotopy theory]], to reflect my feeling that it is more general than [[homotopy coherent category theory]].


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


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



\end{itemize}

\item [[Eric Forgy|Eric]]

\begin{itemize}%
\item asked a question about ``directed internalization'' on [[directed space]]
\item continues to have a brain aneurysm at [[directed space]] but is making progress

\end{itemize}

\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item added a bit of [[John Baez|John]]`s material to [[span]]


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


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



\end{itemize}


\end{itemize}
\hypertarget{20090120}{}\section*{{2009-01-20}}\label{20090120}

\begin{itemize}%
\item [[John Baez|John]]:

\begin{itemize}%
\item 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!


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


\item I added [[endofunctor]] and [[strict monoidal category]].



\end{itemize}

\item [[Mike Shulman|Mike]]:

\begin{itemize}%
\item Replied at [[fibration]], [[directed object]], and [[homotopy hypothesis]].

\end{itemize}

\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item commented in the discussion at [[fibration]] on the use of the word \emph{transport}

\end{itemize}

\item [[Tim Porter|Tim]]

\begin{itemize}%
\item I have added a question to [[homotopy hypothesis]] asking what criteria should be added to give `good' categorical or groupoidal models for homotopy types.


\item 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.


\item 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.



\end{itemize}

\item [[Mike Shulman|Mike]]:

\begin{itemize}%
\item Created [[fibration]], [[Grothendieck fibration]], and [[pseudofunctor]].

\end{itemize}


\end{itemize}
\hypertarget{20090119}{}\section*{{2009-01-19}}\label{20090119}

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

\begin{itemize}%
\item separated [[directed object]] from [[directed space]] and included the definition of directed topological space by Grandis


\item further reacted at [[directed space]] and created [[homotopy hypothesis]]


\item following discussion by [[Eric Forgy|Eric]] at [[directed space]] I propose in the discussion section a formalization of the notion ``an object $X$ is \emph{directed}'' and ``an object $X$ is \emph{undirected}'' for the case that $X$ is an object in a category with [[interval object]].



\end{itemize}

\item [[John Baez|John]]:

\begin{itemize}%
\item 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.


\item 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.


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



\end{itemize}


\end{itemize}
\hypertarget{20090118}{}\section*{{2009-01-18}}\label{20090118}

\begin{itemize}%
\item [[Mike Shulman|Mike]]:

\begin{itemize}%
\item Added detail, examples, and terminological comments to [[bicategory]].


\item Created [[2-categorical limit]].


\item Replied at [[finite set]].



\end{itemize}

\item [[Todd Trimble]]

\begin{itemize}%
\item added a bit of material on terminal coalgebras to [[coalgebra]].


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



\end{itemize}

\item [[Toby Bartels]]

\begin{itemize}%
\item Adjustments to [[source]], [[target]], and [[identity assigning morphism]].
\item Comments and questions at [[finite set]], [[internal category]], and [[bicategory]].

\end{itemize}

\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item 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\ldots{}

\end{itemize}

\item [[Eric Forgy|Eric]]:

\begin{itemize}%
\item Plagiarized the definition of [[internal category]] from \href{http://arxiv.org/abs/math/0307263}{Baez and Crans} since the previous definition stated things should work in ``the obvious way'', which was not obvious to me
\item Asked if [[directed space|directed spaces]] could be defined using [[interval object]] on the directed spaces page.

\end{itemize}

\item [[Toby Bartels]]:

\begin{itemize}%
\item Added other terms and meanings to [[decidable object]]; some of this should probably go to other (new) pages.
\item Clarified my question at [[interval object]].
\item Asked an organisational question at [[Trimble's notion of weak n-category]].
\item Other minor edits.

\end{itemize}

\item [[Mike Shulman|Mike]]:

\begin{itemize}%
\item Added another argument in favor of using $B$ at [[category algebra]].


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



\end{itemize}


\end{itemize}
\hypertarget{20090117}{}\section*{{2009-01-17}}\label{20090117}

\begin{itemize}%
\item [[Todd Trimble|Todd]]

\begin{itemize}%
\item responded to a plea of Eric at [[internal category]] to discuss an example; cf. Tim's latest change below.

\end{itemize}

\item [[Tim Porter]]

\begin{itemize}%
\item 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]].
\item 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.

\end{itemize}

\item [[Emily Riehl]]

\begin{itemize}%
\item created [[small object argument]]

\end{itemize}

\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item continued fiddling with [[interval object]]


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



\end{itemize}


\end{itemize}
\hypertarget{20090116}{}\section*{{2009-01-16}}\label{20090116}

\begin{itemize}%
\item [[Mike Shulman|Mike]]

\begin{itemize}%
\item Added the internal version to [[group]].


\item Created [[well-powered category]].



\end{itemize}

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


\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item 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)


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


\item replied to discussion at [[interval object]]



\end{itemize}


\end{itemize}
\hypertarget{20090115}{}\section*{{2009-01-15}}\label{20090115}

\begin{itemize}%
\item [[Toby Bartels]]: Wrote [[algebraic theory]], mostly to distinguish it from [[Lawvere theory]].


\item [[Mike Shulman|Mike]]:

\begin{itemize}%
\item 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]].


\item Responded to Toby at [[Lawvere-Tierney topology]].



\end{itemize}

\item [[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.


\item [[Todd Trimble]]:

\begin{itemize}%
\item 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]]).

\end{itemize}

\item [[Urs Schreiber]]

\begin{itemize}%
\item 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


\item 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!


\item 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


\item filled in stuff at [[causet]]


\item added \textbf{questions} to [[Trimble's notion of weak n-category]]


\item expanded on the examples in [[generalized universal bundle]]



\end{itemize}

\item [[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]].)



\end{itemize}
\hypertarget{20090114}{}\section*{{2009-01-14}}\label{20090114}

\begin{itemize}%
\item [[Mike Shulman|Mike]]:

\begin{itemize}%
\item Added detail about internal-homs and smash products to [[pointed object]], in response to Todd's comment.


\item Created [[adjunct]].



\end{itemize}

\item [[Urs Schreiber|Urs]]:

\begin{itemize}%
\item started [[generalized universal bundle]] after all -- now I really need to run\ldots{}


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


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


\item 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


\item in the entry [[category of fibrant objects]]

\begin{itemize}%
\item added comments on relation to [[pointed object]] and [[pointed category]]


\item made the \emph{factorization lemma} more explicit and related it to the notion of [[Morita equivalence]]



\end{itemize}

\item created a {\colorbox[rgb]{1.00,0.93,1.00}{\tt categories\char58\char32reference}}-entry for [[Brown -- Abstract Homotopy Theory and Generalized Sheaf Cohomology]]


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


\item 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\ldots{}



\end{itemize}

\item [[Mike Shulman|Mike]]:

\begin{itemize}%
\item Incorporated the discussion into [[k-tuply monoidal n-category]] and added more examples. Perhaps some of this page would better go at [[delooping hypothesis]]?


\item Created [[pointed object]] and [[spectrum]].



\end{itemize}


\end{itemize}
\hypertarget{20090113}{}\section*{{2009-01-13}}\label{20090113}

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

\begin{itemize}%
\item Conversations, including a new one with [[Todd Trimble|Todd]] at [[balanced category]].
\item Revised [[k-tuply monoidal n-category]]. Later I need to revise [[periodic table]] and write [[k-connected n-category]].
\item Wrote [[Elephant]] and [[Categories Work]], creating {\colorbox[rgb]{1.00,0.93,1.00}{\tt category\char58\char32reference}} for them.

\end{itemize}
[[Urs Schreiber|Urs]]: concerning [[Toby Bartels|Toby]]`s {\colorbox[rgb]{1.00,0.93,1.00}{\tt category\char58\char32reference}}: 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


\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item created [[derived infinity-stack]], so far containing a sketch of the rough idea, a few references and pointers to current discussion on the blog

\end{itemize}

\item [[Mike Shulman|Mike]]:

\begin{itemize}%
\item Created [[bijection]] and [[subsingleton]].


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



\end{itemize}

\item [[Toby Bartels]]:

\begin{itemize}%
\item I took part in conversations that are already listed here.
\item I have questions for Mike at [[k-tuply monoidal n-category]], [[choice object]], and [[finite set]]. (Listed in order of decreasing importance.)

\end{itemize}


\end{itemize}
\hypertarget{20090112}{}\section*{{2009-01-12}}\label{20090112}

\begin{itemize}%
\item [[Mike Shulman|Mike]]:

\begin{itemize}%
\item Joined the discussion at [[category algebra]].


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


\item Created [[finite set]] and [[choice object]].



\end{itemize}

\item [[Tim Porter|Tim]]

\begin{itemize}%
\item I have asked on the page relating to [[action]] if the definition needs to be slightly wider for optimal later use.

\end{itemize}

\item [[Todd Trimble|Todd]]

\begin{itemize}%
\item added some discussion to [[classifying topos]], linking up to the entry [[Lawvere theory]]


\item created entry on [[Cauchy complete category]]



\end{itemize}

\item [[Toby Bartels]]:

\begin{itemize}%
\item I'm having a conversation with myself at [[local ring]].
\item I'm otherwise caught up and will rewrite [[k-tuply monoidal n-category]] before I do anything else.

\end{itemize}

\item [[Mike Shulman|Mike]]:

\begin{itemize}%
\item Removed the discussion at [[field]], since it had converged and been incorporated into the entry.


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


\item Added some details to [[excluded middle]].



\end{itemize}


\end{itemize}
\hypertarget{20090111}{}\section*{{2009-01-11}}\label{20090111}

\begin{itemize}%
\item [[Todd Trimble|Todd]]

\begin{itemize}%
\item began, at Urs's request, an entry for [[Yoneda reduction]]


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


\item 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



\end{itemize}

\item [[Urs Schreiber]]:

\begin{itemize}%
\item created a separate entry [[Yoneda embedding]], even though the information there is also recalled in [[Yoneda lemma]]

\end{itemize}

\item [[Toby Bartels]]:

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

\end{itemize}


\end{itemize}
\hypertarget{20090110}{}\section*{{2009-01-10}}\label{20090110}

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

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

\end{itemize}

\item [[Toby Bartels]]:

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

\end{itemize}


\end{itemize}
\hypertarget{20090109}{}\section*{{2009-01-09}}\label{20090109}

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

\begin{itemize}%
\item I talked with Mike (and occasionally made substantive edits, too) at [[field]], [[Grothendieck topology]], [[extensive category]], [[predicativism]], and [[constructivism]].

\end{itemize}

\item [[Mike Shulman|Mike]]:

\begin{itemize}%
\item Created [[coherent category]], [[Boolean category]], [[Heyting category]], [[equivalence relation]], [[kernel pair]], [[exact category]], [[balanced category]], and [[pretopos]].


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


\item Did some rephrasing at [[infinity-category]].



\end{itemize}

\item [[David Corfield|David]]

\begin{itemize}%
\item started a page on [[generalized tangle hypothesis]].

\end{itemize}

\item [[Urs Schreiber|Urs]]:

\begin{itemize}%
\item created [[directed space]]


\item added a bit to [[David Corfield|David]]`s entry on the [[generalized tangle hypothesis]]


\item reacted to [[Mike Shulman|Mike]]`s comments at [[infinity-stack homotopically]]



\end{itemize}

\item [[Mike Shulman]]:

\begin{itemize}%
\item Continued my discussions with Toby at [[Grothendieck topology]], [[extensive category]], and [[constructivism]], including one that should be moved to [[predicativism]].


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


\item Added the unit axiom to [[monoidal model category]].



\end{itemize}


\end{itemize}
\hypertarget{20090108}{}\section*{{2009-01-08}}\label{20090108}

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

\begin{itemize}%
\item Mike and I are discussing terminology at [[Grothendieck topology]] and [[extensive category]] (possibly also elsewhere).
\item I wrote [[regular epimorphism]] and [[epimorphism]].
\item 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.)

\end{itemize}

\item [[John Baez]]

\begin{itemize}%
\item worked pertinent discussion into body of [[monoidal category]]

\end{itemize}

\item [[Urs Schreiber]]

\begin{itemize}%
\item reacted to the discussion at [[path object]]


\item created [[cylinder functor]]


\item 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


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


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


\item had the honor of creating [[Yoneda lemma]]


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



\end{itemize}

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


\item [[Mike Shulman|Mike]]

\begin{itemize}%
\item Created [[geometric morphism]], [[locale]], and [[sober space]].


\item Added a more classical version to [[homotopy]].



\end{itemize}


\end{itemize}
\hypertarget{20090107}{}\section*{{2009-01-07}}\label{20090107}

\begin{itemize}%
\item [[Todd Trimble|Todd]]

\begin{itemize}%
\item created [[regular category]]


\item created [[image]]


\item created [[Rel]]


\item created [[Lawvere theory]]


\item started some discussion at [[simplex category]]



\end{itemize}

\item [[Timothy Porter]] created

\begin{itemize}%
\item [[simplicial complex]]


\item [[simplicially enriched category]]


\item [[horn]]


\item [[nerve]] (but see below)



\end{itemize}

\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item gave the bare definition of [[crossed module]]


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


\item did [[geometric realization]]


\item 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\ldots{}


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


\item 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).


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


\item Expanded slightly on [[Timothy Porter]]`s entry on [[simplicially enriched category]].


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



\end{itemize}

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



\end{itemize}
\hypertarget{20090106}{}\section*{{2009-01-06}}\label{20090106}

\begin{itemize}%
\item [[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).


\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item 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.

\end{itemize}

\item [[Mike Shulman|Mike]]

\begin{itemize}%
\item Created [[power]] and [[copower]]. Possibly these should be just one page?

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



\end{itemize}


\end{itemize}
\hypertarget{20090105}{}\section*{{2009-01-05}}\label{20090105}

\begin{itemize}%
\item [[Toby Bartels]]: I reacted to everything that everybody did this year, and I interacted with everything that everybody did today. More specifically:

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

\end{itemize}

\item [[Mike Shulman|Mike]]

\begin{itemize}%
\item Split off [[internal category]] from [[internalization]]. Probably a lot of links need to be updated.


\item Summarized the discussion at [[subcategory]] into a section called ``Non-evil variants.''


\item Created the entry [[red herring principle]].



\end{itemize}

\item [[Todd Trimble|Todd]]

\begin{itemize}%
\item created the entry [[CW complex]]


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



\end{itemize}

\item [[Urs Schreiber|Urs]]

\begin{itemize}%
\item am reading [[Mike Shulman|Mike]]`s \href{http://arxiv.org/abs/math.AT/0610194v1}{Homotopy limits and colimits and enriched category theory} 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 \emph{examples}. See my request hereonoidal homotopical category).


\item added [[Crans-Gray tensor product]]


\item started replying to [[Todd Trimble|Todd]]`s remarks at [[stack]] by addding a section \emph{Descent in terms of pseudo-functors} to [[descent and codescent]].


\item 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]].



\end{itemize}

\item [[David Corfield]]

\begin{itemize}%
\item created [[philosophy]], [[Klein 2-geometry]]

\end{itemize}

\item [[Emily Riehl]]

\begin{itemize}%
\item created [[model structure on simplicial sets]]

\end{itemize}

\item [[Mike Shulman]]:

\begin{itemize}%
\item Added [[nice topological space]].

\end{itemize}


\end{itemize}
\hypertarget{20090104}{}\section*{{2009-01-04}}\label{20090104}

\begin{itemize}%
\item [[Mike Shulman]]:

\begin{itemize}%
\item Added disambiguation comments to [[simplicial category]].


\item Created [[Gray tensor product]] and [[Gray-category]].



\end{itemize}

\item [[Toby Bartels]]:

\begin{itemize}%
\item I posted [[k-tuply monoidal n-category]]. I'll correct typos and look at everything else tonight (when it's technically -05).
\item 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!).

\end{itemize}

\item [[Mike Shulman]]:

\begin{itemize}%
\item Added a terminological suggestion to [[omega-category]].


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



\end{itemize}

\item [[Urs Schreiber|Urs]]:

\begin{itemize}%
\item 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\xspace  file, which you can see here for the time being: [[if.pdf:file]].


\item 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]].


\item 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


\item 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).


\item 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 \emph{strict globular $\infty$-category}.


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


\item created a \emph{blog} entry \href{http://golem.ph.utexas.edu/category/2009/01/nlab_general_discussion.html}{nLab -- General Discussion} and added a remark at the beginning of the original \emph{wiki} entry [[General Discussion]] asking to post further general discussion not to the wiki, but to the blog.


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



\end{itemize}


\end{itemize}
\hypertarget{20090103}{}\section*{{2009-01-03}}\label{20090103}

\begin{itemize}%
\item [[Mike Shulman]]:

\begin{itemize}%
\item there is a discussion going on at [[subcategory]] over whether any faithful functor should be called a ``subcategory.'' Input welcomed.


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


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


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



\end{itemize}

\item [[Todd Trimble]]:

\begin{itemize}%
\item added to (and edited) examples under [[closed category]]


\item created [[dinatural transformation]]


\item added examples to [[extensive category]]


\item added to the discussion on [[Day convolution]]


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



\end{itemize}

\item [[Urs Schreiber]]:

\begin{itemize}%
\item did [[closed category]] and linked to it from Mike's [[cartesian monoidal category]]


\item expanded on [[Day convolution]]


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



\end{itemize}

\item [[Mike Shulman]] created [[extensive category]], [[cartesian monoidal category]]


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


\item [[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.



\end{itemize}
\hypertarget{20090102}{}\section*{{2009-01-02}}\label{20090102}

\begin{itemize}%
\item [[Todd Trimble]] created [[operad]]


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


\item [[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.


\item [[Urs Schreiber|Urs]]: motivated by some \href{http://golem.ph.utexas.edu/category/2008/12/organizing_the_pages_at_nlab.html}{discussion} 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.


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



\end{itemize}
\hypertarget{20090101}{}\section*{{2009-01-01}}\label{20090101}

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


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


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category: meta



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