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\section*{2009 March changes}

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

[[!include all changes]]

Archive of changes made during March 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/459}{here} and work backwards.


\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{20090331}{}\section*{{2009-03-31}}\label{20090331}

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

\begin{itemize}%
\item started adding comments on embeddings at [[geometric morphism]], but ran out of time


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


\item 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]]


\item 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


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


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


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



\end{itemize}

\item [[Mike Shulman|Mike]]:

\begin{itemize}%
\item Mentioned at [[triangulated category]] that the definition is redundant. If I had time I would fix it myself.


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



\end{itemize}

\item [[Urs Schreiber|Urs]]:

\begin{itemize}%
\item created [[category of open subsets]] and [[presheaf on open subsets]] to satisfy links at [[Categories and Sheaves]]


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


\item created [[base change]] and [[cobase change]]


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



\end{itemize}

\item [[Tim Porter|Tim]]:

\begin{itemize}%
\item 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!)

\end{itemize}

\item [[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.



\end{itemize}
\hypertarget{20090330}{}\section*{{2009-03-30}}\label{20090330}

\begin{itemize}%
\item [[David Roberts]]:

\begin{itemize}%
\item Asked question about Grothendieck's tame topology at [[nice topological space]].

\end{itemize}

\item [[Urs Schreiber|Urs]]:

\begin{itemize}%
\item QUESTION: came across MacLane's \emph{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 \emph{schools} is introduced instead -- can anyone comment on that in the light of our discussion at [[Grothendieck universe]]?


\item 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


\item created [[null system]]


\item created [[category of chain complexes]], but then didn't quite know where to go with this\ldots{}


\item agreed at [[homological algebra]] and [[filtrant category]]


\item created [[homology]]


\item expanded [[differential object]] and created [[differential]] just to satisfy links


\item created [[matrix calculus]] and [[mapping cone]]



\end{itemize}

\item [[Tim Porter|Tim]]:

\begin{itemize}%
\item 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.
\item I have added some background material on [[dérivateur]]s on the [[triangulated category]] entry.
\item I have a \textbf{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!)

\end{itemize}

\item [[Toby Bartels]]:

\begin{itemize}%
\item Wrote [[biproduct]], [[direct sum]], and [[direct product]].
\item Added more versions to [[additive and abelian categories]].
\item Questioned the purpose of [[filtrant category]].
\item I hope that people saw Zoran's addition to a query box below.

\end{itemize}

\item [[Urs Schreiber|Urs]]:

\begin{itemize}%
\item concerning the entry [[homological algebra]]: [[Tim Porter]] and [[Zoran Skoda]] (or anyone else): please feel free to improve/revise the exposition


\item created [[triangulated category]] -- have a question about \emph{d\'e{}rivateurs} there


\item created [[local epimorphism]]


\item created [[multiplicative system]]



\end{itemize}

\item [[David Roberts]]

\begin{itemize}%
\item Uploaded notes on anafunctors to my \href{http://ncatlab.org/davidroberts/show/HomePage}{web}. Comments welcome and wanted.

\end{itemize}


\end{itemize}
\hypertarget{20090329}{}\section*{{2009-03-29}}\label{20090329}

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

\begin{itemize}%
\item A grammatical suggestion at [[stuff, structure, property]].


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



\end{itemize}

\item [[Urs Schreiber|Urs]]:

\begin{itemize}%
\item began [[derived category]] before running out of time again\ldots{}


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



\end{itemize}

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

\begin{itemize}%
\item Added some PNGs at [[Kan extension]]


\item Replaced PNGs at [[adjunction]] with larger ones.



\end{itemize}

\item [[David Corfield|David]]

\begin{itemize}%
\item asked a question at [[differential object]]

\end{itemize}

\item [[Urs Schreiber|Urs]]

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


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


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


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


\item 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

\begin{itemize}%
\item [[pre-additive category]]


\item [[additive category]]


\item [[pre-abelian category]]


\item [[abelian category]]



\end{itemize}

\item 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



\end{itemize}

\item [[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 \href{http://meta.wikimedia.org/wiki/Help:Displaying_a_formula#Commutative_diagrams}{here} but the resulting files were huge and didn't display anyway when inserted into the markdown source. Instead I used {\colorbox[rgb]{1.00,0.93,1.00}{\tt pdfcrop}} and {\colorbox[rgb]{1.00,0.93,1.00}{\tt convert}} on the {\colorbox[rgb]{1.00,0.93,1.00}{\tt pdflatex}} output.)



\end{itemize}
\hypertarget{20090328}{}\section*{{2009-03-28}}\label{20090328}

\begin{itemize}%
\item [[Finn Lawler]]:

\begin{itemize}%
\item Edited [[linear logic]] in response to Mike's question.


\item Created [[star-autonomous category]].



\end{itemize}

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


\item [[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.


\item [[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]]).



\end{itemize}
\hypertarget{20090327}{}\section*{{2009-03-27}}\label{20090327}

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

\begin{itemize}%
\item Zoran and I are having a discussion about definitions of [[abelian category]].


\item Redirected [[filtrant category]] to [[filtered category]].


\item Moved the discussion about the word ``bimorphism'' from [[balanced category]] to [[bimorphism]].


\item Redirected [[parallel morphism]] to [[parallel morphisms]] (which \emph{is} in line with the naming conventions).



\end{itemize}

\item [[Urs Schreiber|Urs]]:

\begin{itemize}%
\item created [[exact functor]]


\item created [[filtrant category]]


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


\item created [[Yoneda extension]]


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


\item 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)


\item 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 \emph{Remark on terminology: pushforward vs. pullback} which is supposed to clarify this terminology issue.

[[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.


\item 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 \emph{local} and \emph{global} definitions of the universal constructions


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



\end{itemize}

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


\item [[Toby Bartels]]:

\begin{itemize}%
\item Wrote [[hereditarily finite set]], which is more pretty than useful.
\item Threatened to rewrite [[Grothendieck universe]] once again.
\item Wrote about the $3$-way factorisation system at [[stuff, structure, property]].
\item Accepted Mike's terminology (`moderate') at [[Grothendieck universe]].

\end{itemize}

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


\item [[Toby Bartels]]:

\begin{itemize}%
\item 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).
\item Finn has nothing to apologise for at [[context]].
\item Zoran and I are discussing terminology at [[projective limit]].
\item Zoran and Mike are discussing terminology at [[representable functor]] (I only made a more philosophical comment).
\item Compare [[nice category of spaces]] with [[convenient category of topological spaces]].
\item I accept Mike's terminology at [[set theory]].
\item I refactored [[kernel]] to use primarily the equaliser definition in any pointed-enriched category.

\end{itemize}


\end{itemize}
\hypertarget{20090326}{}\section*{{2009-03-26}}\label{20090326}

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

\begin{itemize}%
\item I have a question about the meaning of ``large'' at [[Grothendieck universe]].


\item 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.


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



\end{itemize}

\item [[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.


\item [[Mike Shulman|Mike]]:

\begin{itemize}%
\item Split [[nice category of spaces]] from [[nice topological space]] and fixed all the links I could find.


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


\item 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!)


\item Corrected [[generalized element]] to distinguish it from [[global element]].


\item Made a terminological suggestion at [[set theory]].


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



\end{itemize}

\item [[David Corfield|David]]

\begin{itemize}%
\item created [[algebraic set theory]]

\end{itemize}

\item [[Urs Schreiber|Urs]]

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


\item 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]]


\item started expanding [[Kan extension]]


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


\item 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 $Set$''?


\item added the globular zig-zag diagrams to [[adjunction]]


\item started [[continuous functor]]


\item 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.


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


\item created [[generalized element]]


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


\item filled in three equivalent definitions at [[adjoint functor]]



\end{itemize}


\end{itemize}
\hypertarget{20090325}{}\section*{{2009-03-25}}\label{20090325}

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

\begin{itemize}%
\item expanded a bit at [[natural transformation]]


\item added simple remarks to [[contravariant functor]]


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


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


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


\item expanded a bit more at [[stuff, structure, property]]



\end{itemize}


\end{itemize}
\hypertarget{20090324}{}\section*{{2009-03-24}}\label{20090324}

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

\begin{itemize}%
\item have more stupid questions (for [[Toby Bartels|Toby]], probably) at [[Grothendieck universe]] and incorporated more of Toby's replies

\end{itemize}

\item [[Toby Bartels]]:

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

\end{itemize}


\end{itemize}
\hypertarget{20090323}{}\section*{{2009-03-23}}\label{20090323}

\begin{itemize}%
\item [[Urs Schreiber|Urs]]: created [[Grothendieck universe]] -- but have questions. Mentioned Grothendieck universes at [[small category]] and [[locally small category]].

\end{itemize}
\hypertarget{20090322}{}\section*{{2009-03-22}}\label{20090322}

\begin{itemize}%
\item [[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.

\end{itemize}
\hypertarget{20090321}{}\section*{{2009-03-21}}\label{20090321}

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

\end{itemize}


\end{itemize}
\hypertarget{20090320}{}\section*{{2009-03-20}}\label{20090320}

\begin{itemize}%
\item [[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.


\item [[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]].



\end{itemize}
\hypertarget{20090319}{}\section*{{2009-03-19}}\label{20090319}

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

\begin{itemize}%
\item 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]]


\item added a remark on this and a link at [[spectrum]]



\end{itemize}

\item [[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]]


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


\item [[Urs Schreiber|Urs]]:

\begin{itemize}%
\item created [[stable infinity-category]]


\item implemented [[Zoran Skoda|Zoran]]`s remark below by rephrasing a bit at [[Categories and Sheaves]]


\item added a bit more to [[large category]]



\end{itemize}

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


\item [[David Roberts]]:

\begin{itemize}%
\item Created [[Grothendieck's Galois theory]]

\end{itemize}

\item [[Mike Shulman|Mike]]:

\begin{itemize}%
\item Removed the discussion about accented characters from [[fundamental group of a topos]] and put it in the [[FAQ]].


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



\end{itemize}


\end{itemize}
\hypertarget{20090318}{}\section*{{2009-03-18}}\label{20090318}

\begin{itemize}%
\item [[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.


\item [[Urs Schreiber|Urs]]:

\begin{itemize}%
\item 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\ldots{}you really meant what you say? -- Zoran)


\item 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.



\end{itemize}

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


\item [[Andrew Stacey|Andrew]]:

\begin{itemize}%
\item More done on [[Froelicher space]]s. I think that I have finally figured out the relationship between Fr\"o{}licher 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.



\end{itemize}

\item [[David Corfield|David]]:

\begin{itemize}%
\item Added a reference in response to one of Mike's questions at [[fundamental group of a topos]].

\end{itemize}

\item [[Mike Shulman|Mike]]:

\begin{itemize}%
\item Created [[quantale]], [[adjoint functor theorem]], and [[total category]].
\item Asked some questions at [[fundamental group of a topos]].

\end{itemize}

\item [[Tim Porter|Tim]]:

\begin{itemize}%
\item 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.)

\end{itemize}


\end{itemize}
\hypertarget{20090317}{}\section*{{2009-03-17}}\label{20090317}

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

\begin{itemize}%
\item Yes, homotopy theorists call a [[k-tuply groupal n-groupoid]] a \emph{grouplike $E_k$-space}.


\item Continued discussion at [[Crans-Gray tensor product]].



\end{itemize}

\item [[Toby Bartels]]:

\begin{itemize}%
\item Wrote [[congruence]] to mean an internal equivalence relation.
\item 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.)
\item Created [[wedge sum]] to link from the following, and [[smash product]] to go with it.
\item Put in a general definition at [[homotopy group]], trying to show how $\pi_n$ has $n$ products that are all the same.

\end{itemize}

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


\item [[Urs Schreiber|Urs]]:

\begin{itemize}%
\item made a little remark on [[Mike Shulman|Mike]]`s question at [[Crans-Gray tensor product]] and ask another question myself


\item added a remark to [[Waldhausen category]] on its relation to [[category of fibrant objects]] and ask a \href{http://golem.ph.utexas.edu/category/2009/01/nlab_general_discussion.html#c022572}{question} about the precise statement to be made here over at the $n$Caf\'e{}



\end{itemize}

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


\item [[Andrew Stacey|Andrew]]:

\begin{itemize}%
\item 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.
\item Modified [[Froelicher space]] a little as well in line with the Isbell envelope nomenclature.

\end{itemize}

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


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



\end{itemize}
\hypertarget{20090316}{}\section*{{2009-03-16}}\label{20090316}

\begin{itemize}%
\item [[Toby Bartels]]: Tired of writing {\colorbox[rgb]{1.00,0.93,1.00}{\tt \char91\char91constructivism\char124constructive\char32mathematics\char93\char93}}, 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]].


\item [[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.).


\item [[Tim Porter|Tim]]:

\begin{itemize}%
\item 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.
\item 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.
\item In [[profinite completion of a group]], I have pointed out a difference in the use of the term [[profinite group]], that may need discussion.

\end{itemize}


\end{itemize}
\hypertarget{20090315}{}\section*{{2009-03-15}}\label{20090315}

\begin{itemize}%
\item [[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.


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


\item [[Tim Porter|Tim]]:

\begin{itemize}%
\item 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.

\end{itemize}


\end{itemize}
\hypertarget{20090314}{}\section*{{2009-03-14}}\label{20090314}

\begin{itemize}%
\item [[Bruce Bartlett]] has created [[nInsights]].


\item [[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.


\item [[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]].


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



\end{itemize}
\hypertarget{20090313}{}\section*{{2009-03-13}}\label{20090313}

\begin{itemize}%
\item [[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.


\item [[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.



\end{itemize}
\hypertarget{20090312}{}\section*{{2009-03-12}}\label{20090312}

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

\begin{itemize}%
\item Responded at [[homotopy n-type]] and [[proset]].


\item 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]].



\end{itemize}

\item [[Tim Porter|Tim]]:

\begin{itemize}%
\item 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.
\item 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.
\item I have raised a question on \emph{Disambiguation} at [[proset]]. Have we a policy as to how to handle terms with perfectly acceptable multiple meanings?
\item I have tried to `better' the previous entry on [[profinite group]]! (see the old version to see why I say it this way.)

\end{itemize}

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


\item [[Toby Bartels]]:

\begin{itemize}%
\item 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]]).
\item Finally wrote [[full subcategory]] and directed links to it.
\item I've continued the conversation that [[Eric Forgy]] started at [[preorder]].
\item I have my own terminological question at [[linear relation]].

\end{itemize}


\end{itemize}
\hypertarget{20090311}{}\section*{{2009-03-11}}\label{20090311}

\begin{itemize}%
\item [[Mike Shulman|Mike]]: I have a suggestion for a terminological change at [[k-tuply monoidal n-category]].

\end{itemize}
\hypertarget{20090310}{}\section*{{2009-03-10}}\label{20090310}

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

\begin{itemize}%
\item 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]].


\item Created [[cofibration category]] as the first of the `Bauesian' detailed entries.



\end{itemize}

\item [[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.



\end{itemize}
\hypertarget{20090309}{}\section*{{2009-03-09}}\label{20090309}

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

\end{itemize}
\hypertarget{20090308}{}\section*{{2009-03-08}}\label{20090308}

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


\item [[John Baez]]:

\begin{itemize}%
\item 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.


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



\end{itemize}


\end{itemize}
\hypertarget{20090305}{}\section*{{2009-03-05}}\label{20090305}

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

\begin{itemize}%
\item Continued debate on what to call ``[[generalized object|generalised object]]'' since there seem to be some generalised objections (sorry).

\end{itemize}


\end{itemize}
\hypertarget{20090303}{}\section*{{2009-03-03}}\label{20090303}

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

\begin{itemize}%
\item Archived February
\item Started [[generalized object]]

\end{itemize}

\item [[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]].



\end{itemize}
\hypertarget{20090302}{}\section*{{2009-03-02}}\label{20090302}

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

\begin{itemize}%
\item 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

\end{itemize}


\end{itemize}
\hypertarget{20090301}{}\section*{{2009-03-01}}\label{20090301}

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

\end{itemize}


\end{itemize}

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


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



\end{document}