Archive of changes made during July 2009. The substantive content of this page should not be altered.
Mike Shulman fixed a mistake at axiom of foundation.
Ronnie Brown: Added in connection on cubical sets a reference to a recent preprint of Maltsiniotis showing that cubical sets with connections form a strict test category in the sense of Grothendieck, thus correcting a well known disadvantage of cubical sets in comparison with simplicial sets.
Zoran ?koda?: added a version (in my understanding) of main definitions in almost scheme and made a longer heuristical quote from Gabber-Romero Introduction chapter. Created stub Catégories Tannakiennes about the Deligne’s seminal paper and Des catégories abéliennes about Pierre Gabriel’s thesis.
wrote a long “Idea” section at twisted cohomology and polished/expanded the rest, following comments by Jim Stasheff
added to the “Idea” section at differential form a paragraph that gives some categorical or otherwise abstract nonsense description of how to make precise the statement that “a differential form is something that may be integrated”.
created an entry Models for Smooth Infinitesimal Analysis (so far containing just a summary and an incomplete link list) and linked to it from synthetic differential geometry and Ehresmann connection
I am thinking that section IV there, Cohomology and Integration would be a good candidate for the high-brow abstract nonsense aspect section to be written at differential form
added expositional text to Kan fibrant replacement prodded by the blog discussion here
edited the intro to algebraic K-theory a bit more
added to Waldhausen category the Weibel reference
linked to Grothendieck group from K-theory and algebraic K-theory and Waldhausen category
added to Waldhausen category an example section with the examples needed at Grothendieck group
added various things to Grothendieck group
Toby Bartels: More discussion at Grothendieck group and (infinity,1)-categorical hom-space.
Zoran ?koda?: It looks correctly to me (I do not recall what wording I wrote and it looks like reading my mind). Bt it is late night and I should reread the whle entry rested at the day time. I created a stub version of schematic algebras (with references) and linked it at noncommutative algebraic geometry. It would be nice to have for comparison a more thorough entry on Gabber’s notion of almost schemes (just created unfinished entry) in commutative geometry, but the reflective localizations are used to define ‘exotic’ affines to start with. Gabber-Lorenzo’s book is an egregious sample of how a modern algebraic geometer of Grothendieck’s school develops the theory of schemes and in a way it is a build ground-up with requiring sofistication level, but not many concrete details from the usual theory of schemes.
Toby Bartels: Please check that I rephrased the definition correctly at noncommutative scheme. The original formulation did not make sense to me, but I think that I understood what was meant.
replied at (infinity,1)-categorical hom-space
have to think about the question at category with weak equivalences (which in any case shows that I phrased the sentence too carelessly)
Zoran ?koda?: created first draft of noncommutative scheme (after Rosenberg), and plan in few minutes to start schematic algebra (after van Oystaeyen). Somebody should also add noncommutative projective geometry (after Artin and Zhang). It is a pity that only few of the most stubborn contributors use summer to add new material…
Urs Schreiber added some clauses to Grothendieck group and have some comments in a query box: I am thinking what the entry tries to define should be discussed at algebraic K-theory while “Grothendieck group” should be the definition of a group structure on $A \times A$ for a monoid $A$
Zoran ?koda?: Created cop (it should have been maybe created by the police department). I thank people from Indiana math dept who discussed with me in Fall 2002 when I was searching for the appropriate name and discuraged me from using “mud” for that amorphous structure. BTW, does anybody know what to do on W XP when you loose permanently language bar ?
John Baez: created representation ring, Grothendieck group, Grothendieck ring, and symmetric function. Tidied up lambda-ring a little, but not enough.
Zoran ?koda?: Created coderivation and a new paragraph in derivation.
created (∞,1)-categorical hom-space with the Dwyer-Kan theorem, linked to it from simplicial localization and model category and category with weak equivalences
in the course of this I alsow expanded/rewrote the introductions to
Tim: I changed the e to a é (easy on a Mac!) in Les Dérivateurs and created an entry for Georges Maltsiniotis, note his first name has an ‘s’ on the end.
replied to David at category theory – am waiting for Rafael Borowiecki to get back to us before taking action outside of the query box
created Sjoerd Crans
in that context I also added reference to his work to model structure on simplicial sheaves and model structure on presheaves of simplicial groupoids
also wrote a quick “Idea” section for the latter entry
Toby Bartels: Moved the homotopy theory of Grothendieck to homotopy theory of Grothendieck. Normally, I don't mention that sort of thing here, but this time there's nothing in the naming conventions about this; it just feels right. Complaints are solicitied.
David Roberts: added comment to discussion at category theory - possibly Rafael means homotopy types when he says spaces. Added point about Grothendieck’s view on Cat as a category of models. Also a stub: the homotopy theory of Grothendieck.
Urs Schreiber: more reactions in that discussion at category theory on that one paragraph by Rafael Borowiecki – I suggest that this needs to be rewritten somehow. I make one suggestion, but chances are that I am still missing Rafael Borowiecki’s true intention.
Toby Bartels: Noticed the many wanted links at differential form and started filling them out; a lot of basic stuff in differential topology. Many will be stubs, and many will be capable of unreported generalisation, internalisation, and categorification. So far:
Rafael Borowiecki has figured out how categories are spaces at category theory. (I don't think that Rafael reads this page, so copy comments there.)
Zoran ?koda?: I have made changes to noncommutative algebraic geometry. Le Bruyn has kindly added a lot of material with ring-theoretic flavour (mainly references) and placed my unfinished text below his comment part. I have integrated his and my part, more chronologically and balancing categorical and ring-theoretic aspects; role of cyclic homology and many other directions (e.g. 30 years old subject of D-schemes of Beilinson) are missing. I would kindly invite Le Bruyn to write an adiditional separate entry on noncommutative projective geometry of Artin/Zhang flavour (I am not competent) as well on geometry at n, he is expert on. Quantum group aspects are planned to appear in entry equivariant noncommutative algebraic geometry which I just started.
Zoran ?koda?: created formally smooth morphism, quasi-free algebra, universal differential envelope, Kähler differential.
added a diagram and a sentence in the section “The homotopy category” at category of fibrant objects that makes the statement given there more explicit: that and how every cocycle out of a weak equivalence can be refined by a cocycle out of an acyclic fibration (i.e. by an “$\infty$-anafunctor”).
slightly expanded and polished the examples at Reedy model structure further
David Roberts: cleaning up the links is clearly the job of the golf department
John Baez: warning: I believe “special lambda-ring” is an old-fashioned name for what almost everyone now calls a lambda-ring. This is explained by Hazewinkel in his article cited on lambda-ring. So, I do not believe we should have a separate article on special lambda-ring.
Zoran: created Dennis trace (experts please expand!)
Incorporated results of discussion into effective epimorphism and regular epimorphism.
Everyone who uses ‘$\sqcup$’ (\sqcup
) should be aware of ‘$\amalg$’ (\amalg
) and ‘$\coprod$’ (\coprod
), as in ‘$A \amalg B$’ and ‘$\coprod_i A_i$’.
Interaction with Rafael Borowiecki at category theory.
Golf department? Golf department??
Urs Schreiber created Kan fibrant replacement
Zoran: created Grothendieck Festschrift, and quoted it as an addition to the timeline entry.
spelled out the simplest nontrivial example at Reedy model structure
created global model structure on functors that was requested long ago as projective model structure and injective model structure which I made redirects to it
updated accordingly the list of examples at model category and interlinked with Reedy model structure
I tracked down a reference that discusses enriched Reedy model structures for enriched functor categories from enriched Reedy categories to enriched model categories – I made corresponding additions to Reedy model structure and also created
thanks, Lab Elf. I just donated something to the Society for the Promotion of Elfish Welfare for that.
Lab Elf (service department): The problem was that there was a reference to a theorem in the section that you wanted to remove. By removing that section, you were trying to create a non-existent link. By reformatting that sentence, I was able to remove the section. Another Lab Elf (golf department) may be along later to clean up the links since I just did what was necessary to remove the section.
split off Reedy model structure from Reedy category – added an “Idea” section and started expanding on some of the technical details
by the way: it’s funny I can’t remove the old section “Model structures” from Reedy category which is now reproduced and expanded at Reedy model structure: I always get an “Internal server error” when I try to do that. I am familiar with the occurence of this error when one adds certain things (such as double-dollar included displyed math without line breaks before and after) but I couldn’t figure out which problem the removal of that paragraph causes. So that section is still sitting there, duplicated now.
added clarifying remarks to the references at K-theory
in a similar vein to the below comment i added to the beginning of delooping a remark how the one-object groupoid $\mathbf{B}G$ and the classifying space $B G$ are the same object under the homotopy hypothesis
from private discussion with somebody it became clear to me that the entry homotopy hypothesis failed to get across one of the main points with the required emphasis. I now added the central theorem about the Quillen equivalence between Top and SSet right at the beginning. The disucssion of all the subtleties and generalizations should come after that.
Andrew Stacey: Mathematical and non-mathematical stuff going on at Tall-Wraith monoids. Folded up the mathematical bit of the middle discussion (on what’s so special about $AbGrp$) into the main text, but probably not in the nicest and clearest way. Also continued the discussion on fonts and the like further down.
replied a bit at effective epimorphism – but don’t trust me, it’s way beyond my bed time – see you tomorrow :-)
moved Jim Stasheff’s insertion at cohomology one paragraph further down not to have it tear apart the main definition – this should eventually be merged more with the rest of the entry, it overlaps in parts.
Toby Bartels: More discussion at Tall-Wraith monoid (not actually about math) and effective epimorphism (actually about math!).
Urs Schreiber: added to effective epimorphism a remark on how the 1-categorical case is a special case of the $(\infty,1)$-categorical case and replied to the discussion
Lab Elf (service department): ‘Recently Revised’ now gets redirected to this page. This redirection happens at the server level (i.e. before anything gets to instiki) so shouldn’t affect performance. This also means that the ‘Recently Revised’ pages for labs other than then n-lab work as they ought to. If hitting those slows up the system then Other Steps Will Be Taken.
Please remember that this is hopefully a temporary problem and once we migrate to warmer climes, Normal Service can be resumed.
created petit topos and made gros topos redirect to that
added the links to both versions of David Spivak’s work at derived smooth manifold, structured generalized space and geometry (for structured (infinity,1)-toposes)
Zoran ?koda?: created heap, quantum heap.
created geometry (for structured (infinity,1)-toposes)
in that context I also
renamed structured generalized space to structured (infinity,1)-topos
created a stub for effective epimorphism
created simplicial resolution
Andrew Stacey parried the latest Baezian riposte, and whinged about the lack of a
<sarcasm> tag in XHTML.</sarcasm>PS Welcome back, Bruce.
John Baez: answered Andrew Stacey’s latest comments over at Tall-Wraith monoid.
Bruce Bartlett: Corrected a faulty link to the nLab Stylish theme for FireFox at HowTo. Works now.
thanks to Mike Shulman for the comments at small object argument – we should give both statements, for the non-locally presentable category, then with that other extra assumption, as well as for the locally presentable case – I’ll work that in later (my notation was following Lurie, by the way, but I agree that it is a bit weird)
edited the references section at structured generalized space
added a section with links to higher dimensional and homotopical generalizations to group
started creating symmetric monoidal functor but then noticed that monoidal functor didn’t even exist yet and postponed this to another time
hyperlinked some more keywords at HQFT and sigma-model.
Andrew Stacey: spotted a sneaky paragraph at Tall-Wraith monoid and put a query for its originator (John?).
Tim:
Toby Bartels: Lots of changes (mostly additions) to measure space. Please see if the notation is comprehensible. I have to check on a couple of things, but I left query boxes. There are several variations, but I only included things that people can actually get tenure by studying. No centipede mathematics just for the sake of it! (well, except for one comment, appropriately linked).
Andrew Stacey: continued the sparring at Tall-Wraith monoid (and answered the serious query). I wish I’d known the fascination with centipedes earlier, we caught one today and I could have gotten a good picture of it.
John Baez: inserted centipedes in quasigroup, magma, and the section on weakened definitions in group. Made a few other small changes in these.
John Baez: meddled a bit with centipede mathematics. Accidentally created a page called ‘semigroups’ — sorry, Toby; it looks like you’re merging it with semigroup, which said a lot of the same stuff in a more sophisticated lingo.
Mike: A couple of comments at small object argument.
asked a question about notation near the top of Tall-Wraith monoid, and tried to polish the proof that a Tall-Wraith monoid in abelian groups is just a ring, and enjoyed bickering a bit more with Andrew Stacey in the big green box near the bottom.
had some fun with centipede mathematics - see also my reply to Toby below.
deleted query by Rafael over on category, which had been answered by me and untouched for a while. He’d asked about ‘categories as 1d CW complexes’, but I think the item on categories as ‘directed graphs with composition law’ now answers that — even for people who don’t know what a CW complex is.
reduced the number of appearances of the word ‘isic’ over on isomorphisms; while it’s fun to make up new jargon, I don’t think we should actually use ‘isic’ when explaining concepts when ‘invertible’ will do. We don’t want to convey an impression of quirkiness, and we don’t want to require the reader to look through the whole page to understand new jargon when well-known jargon already exists.
deleted discussion by Toby and Tom over at regular monomorphism, since Tom said it was okay to do so, and some time has passed.
Todd Trimble wrote generalized multicategory, and added a reference at Crans-Gray tensor product to Sjoerd Crans’s papers. Guessed that his papers on teisi might be relevant to an inquiry Mike Shulman made there.
Toby Bartels wrote separation axioms (and a stub at disjoint sets).
John Baez wrote comments on isomorphism, bicategory, measure space, and under a July 24 comment of Eric's here. Toby has responded to all of them except the one at bicategory.
Eric: Responded to Toby at measure space and Densitized Pseudo Twisted Forms.
Tim: I have started an entry on HQFTs. Initially this will summarise Turaev’s theory, but I hope to get a bit more daring later on. I hope someone will tell me (then) if I am talking through my hat. (I rarely wear one.)
Toby Bartels: Comments for Eric at measure space and on his web.
Eric:
Remembered there IS a beautiful arrow theoretic way to think of measures, i.e. Leinster measure. Added a comment about it at measure space.
Created Leinster measure with, for now, just a link to the n-Cafe.
created K-theory spectrum
created Waldhausen S-construction and edited Waldhausen category a bit, but needs still more work – help is appreciated, I am not sure yet if I found the best literature
Eric:
Added more to questions on measure space. Whenever I see a long convoluted definition, e.g. measurable space, I tend to think there should be some short, concise, arrow theoretic description that incorporates all the little factoids into one pretty picture. A wild guess (that I know is wrong, but hopefully inspires someone to write down what is right): a measurable space is some kind of presheaf or maybe a representation on ????.
John Baez: the definition of measurable space is pretty darn simple and quick: it’s a set with a collection of subsets that’s closed under complements and countable unions. Such a collection is usually called a $\sigma$-algebra, and all this is explained pretty early on in the page measurable space. Whoever wrote the longer discussion below was just having fun analyzing the definition into little bite-sized pieces (I have my guess as to who this might be.)
Toby Bartels: Guilty as charged.
John Baez: The guilty conscience need not be accused by name. I think we should warn the reader when we go off on an excursion like this. Perhaps just a warning like: The following passage might be considered centipede mathematics, together with a small version of the following picture. I wish I knew how to center a picture!
Tom Ellis? created extremal monomorphism
edited algebraic K-theory a bit
created K-theory – as opposed to my previous take on this which was then moved to the “Idea” section at topological K-theory this time this is aiming for the fully general bird’s eye picture with indications how that produces all the special realizations in special cases
created decategorification – evidently much more can be said here, but it’s a start
expanded the “Idea” section at spectrum and effectively rewrote it – added a link to combinatorial spectrum at the end, which probably should be thought of as a concrete realization of the idea of $\mathbb{Z}$-category – accordingly I changed the title of the last section from “Conjectures” to “Combinatorial models”.
added a reference to Weibel’s online book to algebraic K-theory
Andrew Stacey: Tried answering John’s questions over at Tall-Wraith monoid. Probably lots for the lab elves to work on there.
(I confess that I did have the Hogwartian house elves uppermost in mind, but the shoemaker elves were not far off either. Being now in the Nordic realm I probably should have said ‘lab troll’ but trolls already have a place on the internet and it is Not Here)
John Baez: I answered some remarks by Mike Stay and Eric over on free cocompletion. I also had an hour-long chat with Mike that should eventually push this exposition forward quite a bit: I explained coends to him, which is a lot easier in words than on paper. But I hope we get the explanation into the $n$Lab eventually!
Toby Bartels: Answered Eric's first question; I'm not ready to think about the second one yet.
Eric: Asked some questions on measure space.
Toby Bartels: Remarks on notation at measure space.
Ben Webster created Hecke algebra
Zoran ?koda?: created Dunkl operator, double derivation; it is a start of a series which should include entries on Cherednik algebras, Knizhnik-Zamolodchikov connection, Calogero-Moser system, Gauss-Manin connection, Calogero-Moser space, deformed preprojective algebras and so on…
effectively rewrote infinity-stack – expanding it considerably, adding the relevant pointers to all the new material that has come together since I first wrote this
removed the discussion of costacks entirely. I’ll turn that into a separate entry in its own right eventually
John’s question there had been about my notation $\to\gt$ – that was a hack for the symbol for a fibration, an arrow with a double tip. In my new version this no longer appear, though it may still appear at hypercover, which is linked to, and elsewhere.
Toby knows how to typeset such arrows correcty. Maybe he could add a section to HowTo with the relevant information and links to special symbol lists
added a reference at (infinity,1)-functor to (infinity,1)-category of (infinity,1)-functors – the discussion of this issue that I like most is currently at models for infinity-stack (infinity,1)-toposes. Eventually that should be discussed better at the relevant entries.
Put in some gunk about Tall-Wraith monoid, which Andrew Stacey improved. Later I put in two queries!
Put in a query about D-modules.
Put in a query under infinity-stack.
created Dominic Verity
created Verity on descent for strict omega-groupoid valued presheaves
reworked the How to get started according to my opinions and our disucssions here
there are now two sections, one on how to paste source code of a comment one is about to submit to the blog, the other about how to paste non-source code material
in the course of this I have removed lots of the previous discussion on these points – the goal is to keep that particular page clean of auxiliary discussion and as brief and to the point as possiblle, because that’s the point of this page
if anyone feels I removed too much, please use Rollback to grab the deleted material and then cancel the rollback and insert the missing material in a suitable section at the main HowTo entry
Andrew Stacey: I’ve banned ‘Recently Revised’ for the time being. My method of banning has probably blocked it for all the private webs as well. If that’s really annoying then let me know and I’ll try to find a more specific method of banning just the nlab one.
Andrew Stacey was pleasantly pleased to stumble across Tall-Wraith monoids and made a few minor alterations (mainly style, and added a couple of references). I’ll shove this question over on the forum as well, but should we have a lab convention on fonts for categories, functors, objects, and the like?
renamed the new easy-basic-HowTo page to How to get started
then I reworked the formatting and edited pieces here and there
to Bruce Bartlett: I think on that particular page we don’t want query boxes, as that page is supposed to provide quick unambiguous information that tries to deconfuse people instead of to confuse them – please see my reply and check if you can work something into the paragraph right before the query box that allows to remove that query box
Andrew Stacey I concur, but couldn’t delete the query box as I made a remark in it and so if I delete the box now then that would permanently remove that remark. Someone else could do it (or I could in half an hour’s time).
David Corfield: Started Lambda-ring with some Baezian exposition and an abstract of James Borger. Hmm, is there a difference between $\lambda$-ring and $\Lambda$-ring? This paper uses both.
Toby Bartels welcomed Sebastian Thomas? at (n,k)-transformation.
Tim Silverman?: Answered a request from the n-Cafe by creating How to Copy and Paste Material from the n-Cafe and Include Links Back and Forth
slightly polsihed further at strong monad and removed the tentative-alert, now that Todd also approved of the statement
filled in a bit of text and some references at conformal field theory
added the notion of Frobenius lax-and-oplax functors to lax functor and provided pointers to their use in CFT
added a remark by Todd Trimble to associahedron on their relation to orientals that I asked him about by private email
Added more information to tensorial strength. Some of this should be checked.
Added more examples to lax functor. I’m in a lax mood these days, and I really enjoyed it when Paul-André Melliès told me a definition of ‘enriched category’ in terms of lax functors. This works for categories enriched over a bicategory, not just a monoidal category. Do we have any entry on enrichment over bicategories? If so, maybe someone could add a link.
Urs Schreiber: we had some old discussion on the blog on this description of enriched categories – I used to be interested in that in the context of A Note on RCFT and Quiver Reps – I’ll maybe add something about this to the entry
created strong monad
created lax functor
replied at (n,k)-transformation – I think that in principle this gives all the required information, but I am aware that eventually someone should describe that all explicitly in detail at that entry
Toby Bartels: Copied to (n,k)-transformation a question that was sent to me by email, and partially answered it. (Urs could probably answer the rest.)
added references and links to vertex operator algebra
created tensorial strength with material that Todd Trimble provided on the blog
+-- {: .un_remark}
…=--
at category theory.added links and references to tricategory
worked on the Idea section at category theory: I reformatted a bit the existing material, included lots of hyperlinks and filled in various further bits, such as a paragraph that lists the fundamental classes of examples and the quote from Barry Mitchell that Todd just mentioned on the blog
I was surprised to find the entry in a much more developed and pleasant state than I remembered it – maybe I missed the announcement here, or could it be that there was a major edit to the entry that wasn’t logged here at Latest Changes? Please remember to alert us here.
I am now hopeful that eventually we’ll be able to turn what should be the pivotal $n$Lab entry into something decent, too: that on higher category theory. At the moment that one is not a good advertisement of the $n$Lab project.
replied and reacted at locally presentable category
Eric: Asked, “What is a ‘component of a cocone’?” on An Exercise in Kantization.
Urs Schreiber where did you see that term used? Maybe the question (or its answer) belongs at colimit. Do you have an idea what a cocone itself is? It consists of lots of morphisms from the objects of a diagram to the cocone tip. If we regard the cocone as a natural transformation to a constant functor, then the components of that natural transformation are these single morphism from objects to the tip of the cocone. These I would call “components of the cocone”.
Zoran ?koda?: Thank you Toby, your new clarifications in essential image and replete subcategory are pretty helpful and clear, and I agree with them. Still I would like to think of more clean scheme of thinking of various kinds of images internally, in connection to various kinds of factorization systems and even multistep factorizations like Postnikov systems. There is probably a framework where, despite the differences all the kinds of images including homotopy image belong. The crucial is choice of a sort of factorization system: a variety of an image is basically the second morphism in the factorization (or less precisely its domain). In higher categories sometimes multistep factorizations systems are interesting, like Postnikov towers in topology. This way it may satisfy the point of view of Urs, who was IMHO not precise at the beginning but eventually pointed in the right general direction, and the reference of Barwick which he found seems to be really useful.
after discussion with Zoran Skoda I split off homotopy image from essential image, reserving the latter for the essential image of a functor of categories – I haven’t touched the content of essential image otherwise
updated link list at Higher Topos Theory (mostly under Appendix/Category Theory). For what that’s worth, the appendix is now getting pretty close to being fully indexed.
added reference to Richard Garner’s Understanding the small object argument to small object argument
created transfinite composition
made at small object argument the theorem a formal theorem (with theorem environment and all), added a list of references and – in the paragraph that is now right before the theorem – tweaked the former assumptions a bit, which I guess were taken by Mike Shulman from Hovey’s book. My impression is that in the “modern” literature the ambient category is assumed to be locally presentable – but it would be great if an expert checked my modifications (see also the further literature that I list)
Andrew Stacey took the hint and started incorporating the discussion into the main thread at paracompact space.
added Jeff Smith’s theorem to combinatorial model category and made Smith's theorem redirect to that
added the Barwick reference also to Bousfield localization, to combinatorial model category and to small object argument
added an “Idea” section to essential image, created subsections for different definitions and created one subsection with the definition of homotopy image as found in Clark Barwick’s work and as kindly pointed out by John Baez on the blog here
did some editing and have a discussion with David Corfield at group homotopy
added to folk model structure a sentence that these model structures present $(\infty,1)$-categories of the collection of the given $n$-categories, as part of a reply to Rafael Borowiecki’s question to the Cat-theory mailing list that I just posted
added to (infinity,n)-category the reference to Lurie’s “Goodwillie”-article and a few remarks on some pertinent definitions there
Zoran ?koda?: I actually do not think that Toby’s correction to essential image is correct. I mean that essential image is removing evil from image. No, image SUBcategory is just a specific and unique internal (subcategory in narrow sense) CHOICE of the (external) image of the functor within Cat as a category. Essential image subcategory is just a specific and unique choice of the bicategorical image of the functor considered as a 2-functor within Cat as a bicategory. The same with higher version. The homotopy image which Urs looks is just about image in external sense and not about the internal choice of which subset of k-cells for every k is chosen. Making a replete choice of subcategory is like taking a maximal atlas of a manifold to remove nonuniquness in the class of all atlases - so in a sense it is a maximal choice with respect to the target; the usual image of a functor is more calculated with respect to the domain of the functor. In bicategory Cat the two are equivalent; in category Cat they are not isomorphic.
Urs Schreiber: I am not sure I know what you mean by external vs internal. But I supppose one point you are making is that an essential image is/should be defined only up to the relevant notion of equivalence. Do you mean by “external” a characterization of essential image by a universal property, whereas by “internal” you mean a concrete representative of that (unique only up to equivalence)?
Do we agree on what the “external” definition should be? Is it the one I suggested it should be? In that case we might reorganize the entry by startiing it with the abstract nonsense definition and then taking the replete version as one concrete realization in Cat.
Zoran Skoda No (I have the feeling that you are not reading what I wrote), we disagree on what external definition should be because the essential image is not a notion which is external. It is a CHOICE of literally a subcategory, not a choice of embedding of categories in abstract sense, it is a choice of a SUBSET of n-cells which is a n-subcategory which is replete. On the other hand there are two (three) DIFFERENT notions of image of a functor. One is the image in external sense, that is image in Cat taken as a category or as a 2-category. Another is image as a subcategory in literal sense. Image in literal sense is of course a very specific representative of an image in Cat 1-categorical sense and essential image is a very specifical choice of an image in 2-categorical sense; actually it is a specifical choice of such 2-categorical image that the embedding of essential image into the codomain is also literally surjective on objects. This is a bit strange from external point of view: you have something what is just equivalent to 1-categorical image, while it additional property is again of 1-categorical type. Thus it mixes the two. Hence it is by no means superimposable to homotopy limits in any case.
Andrew Stacey: Responded at paracompact space and Froelicher space. Incidentally, if the start of a query box is indented for some reason (as on paracompact space) then it seems that all its contained paragraphs should be indented by at least the same amount.
thinking about it, I followed Zoran’s suggestion and moved the entire “Idea” part that I had t-yped into K-theory to topological K-theory – also the query box with the discussion is now there, and K-theory is once again just a link list…
brief reply and question at K-theory: what is the big global picture on K-theory that deserves to be put in the first sentences of the “Idea” section and really captures the full topic? Is there even any?
have a question at essential image: we should consider the weak/homotopy version of the definition of limit as the equalizer of the cokernel pair of a morphism, is there any literature/knowledge about that?
quick reply to Toby at locally presentable category: I didn’t mean to leave out the “locally”, but now that we are at it: what’s the point of saying “locally” here in the first place?
Toby Bartels: Added quite a bit to free monoid.
created presentable category for questionable reasons
added to locally presentable category the explicit charascterization
started adding an “Application” section to models for infinity-stack (infinity,1)-toposes
David Corfield: imported Patrick Schultz’s cafe comment on cartesian monads to cartesian monad, but now have doubts as to whether it ought to appear there first under ‘Idea’. Are there other uses for cartesian monads? And anyway similar material appears at multicategory.
Toby Bartels: Answered an anonymous question at regular monomorphism.
Eric: Installed Cygwin so that I could convert Dugger’s Sheaves and Homotopy Theory from dvi to pdf. I uploaded the pdf to the nLab and added links to all references to the paper.
Zoran ?koda?: created comonad, added more on connection for coring and semifree dga. I think David’d confusion might be genuine: not to call with dash or not, that is easy question of exact synonyms, but rather how to cleanly separate DIFFERENT but similar notions of say biadjunction and pseudoadjunction; setups in which they appear: strict and nonstrict 2-categories and Gray categories; and kinds of (pseudo/2)-monads they induce…to mention a few. The Memoirs booklet by Tom Fiore and some papers by Lack, Marmolejo, Vitale, Kelly…may be useful to compare and decide in this regard.
David Corfield: Added a comment at free cocompletion, which got me looking for “pseudoadjunction”. I would trigger a new page for it, but don’t know optimal naming conventions.
Eric: Hi David. Now that we have redirects, you can feel less concerned about naming conventions. For example, if you start a page pseudoadjunction and people come out with pitchforks saying it should be pseudo-adjunction, we now have the capability to simply change the page name. Better than that, we can add redirects so that both pseudoadjunction and pseudo-adjunction point to the same page and then it doesn’t matter. People can use either one when linking to your page.
David: The worry was more about the name itself. I recall John Baez in TWF wishing to avoid the term pseudomonad, and I see 2-monad covers various levels of weaknesses. Oh, I see we have lax 2-adjunction.
Eric: Would it make sense to add redirects for pseudoadjunction and pseudo-adjunction to lax 2-adjunction?
Tim: I have started an entry on dg-quiver. I have paused because I cannot decide whether this is the right version to put there or whether to use Peter May’s discussion in the talk that is linked to from that page.
Urs Schreiber made a remark at free cocompletion in between the exchange between John Baez and Eric Forgy: the Yoneda extension discussed there is at least a special case of a left Kan extension
Goncalo Marques replied at field
Added some words on ? to reflect Toby’s preference for the format “$L_\infty$-algebroid”. Also added a section with comments on specific pages and moved Toby’s comment (and my response) from Lie infinity-algebroid representation to this section.
Requested a PDF copy of Daniel Dugger, Sheaves and Homotopy Theory (web) at free cocompletion. This reference appears several places and those without the ability to read DVIs could use a PDF copy. Instructions on how to upload files to the nLab are given here. Once we have a PDF copy, I can go around and update links to this reference to make the PDF available.
Added some very pedestrian stuff (to help me understand it) to the Decategorified Theorem section of free cocompletion.
wrote a lot more on free cocompletion. I hope other students of category theory, not just Mike Stay and David Corfield, ask questions or do some of the exercises!
created refinement of a cover
started list of gauge fields at gauge theory, so far I have
puny start with Yang-Mills theory and Yang-Mills field
created twisted K-theory presenting it as a special case of the discussion at twisted cohomology – feeling slightly uneasy about making this public, though, maybe later I get scared and remove that content again, or move it to my private web. Or else, you tell me how obvious and well-known this is, then I can leave it there without further worries.
created twisted bundle and bundle gerbe module
added at homology an “Idea” section that introduces the concept as the image of homotopy under the Dold-Kan correspondence. Also added as an example explicitly the ordinary case of homology in chain complexes of abelian groups
yes, the duality mentioned at cohomotopy is the one called Eckmann-Hilton duality, at leat when the $(\infty,1)$-topos in question is Top. I have made that explicit now at cohomotopy.
yes, thanks for the improvement at RR field
added links to gauge theory fields in the example section of differential cohomology
created field strength
created RR-field
created differential K-theory
added an “Idea” section to free cocompletion, made John and Mike’s central statement a standout box, gave the theorem a theorem environment and added various links. Noticed to my surprise that the entry decategorification is, as yet, missing.
created concordance
created vectorial bundle (notice the difference to vector bundle)
Eric:
created cohomotopy and linked to it from cohomology
added “Idea”-section to K-theory
expanded further the entry on Deligne cohomology: gave maps to underlying classes and characteristic forms and made chain maps explicit – also reorganized slightly, making the perspective of the Deligne complex as the image under Dold-Kan of functors from the path $n$-groupoid the primary one.
in order to link to the new article by Martins and Picken I created path n-groupoid
added a pointer to some notes by Daniel Dugger to the discussion of free cocompletion at presheaf – Dugger gives a nice pedagogical description
added material about infinite-dimensional manifolds to paracompact space, taken from private email discussion that David Roberts kindly provided – but I notice that I still have a question, see there
started a pedagogical discussion of free cocompletion at presheaf, then followed Urs’ suggestion and moved it to the entry free cocompletion
added remark and question at field
Eric, why don’t you make the material on electromagnetism in media that you added into a proper section at electromagnetic field? Then we could move what is proper discussion between us into a query box, after all, while having the genuine material visible in the netry and not hidden in the section Discussion.
Eric:
Started tinkering with a draft Discrete Causal Spaces. Help is more than welcome.
Made a few comments on electromagnetic field and electric charge.
Asked a question on connection for a differential graded algebra (which Urs replied to).
added to twisted cohomology the May-Sigurdsson reference, mentioned their definition of twisted cohomology in terms of associated bundles of spectra and added a discussion on how that relates to the rest of the entry
it seems to me that linking to a page via a redirect has as a consequence that the linking page is not listed at the bottom of the linked page under Linked from . That’s too bad.
added the reference to Abad and Crainic at Lie infinity-algebroid representation
realized only now that there is an entry semifree dga, so I added to that entry a remark on Lie infinity-algebroid and conversely added there a pointer to the former
replied at connection for a differential graded algebra, remarking that this seems to be essentially the structure discussed at Lie infinity-algebroid representation – this concept seems to be reinvented many times, just recently it seems that what Abad and Crainic describe in 0901.0319 is the same idea
if indeed calling recently revised should be avoided for the time being, it is all the more important that you indicate even small changes/additions here at latest changes
yes, I am relieved to see Mike back, if only temporarily, I was worried that the $n$Lab had lost one of its most valuable contributors
Mike:
Replied to discussions at replete subcategory and pseudofunctor.
Sorry for suddenly disappearing; after graduating (thanks for the thoughts everyone!) I had immediate obligations to three coauthors that took priority, so I lost track of the nCommunity for a bit. (One of those papers, which people here might be interested in, should be appearing on the arXiv shortly.) Unfortunately I’ll now be traveling and out of touch for the next month, but then I’ll be back.
added three basic examples to metric space
filled more information provided by Todd Trimble into the entry paracompact space
created cup product
checked by private email with Todd Trimble and probably see my confusion at paracompact space now – replied there and added explicitly the example of second countable fin-dim manifolds
created magnetic current and electric current
I am getting the impression that the server runs much more smoothly when one avoids to call the “recently revised” page. This is a pity, because I used to go there all the time to see what’s happening, but it would be helpful to figure out if maybe the cause of the performance problems we see can be narrowed down further. Maybe calling “recently revised” causes the software to go through the entire database in an inefficient way.
have a question at paracompact space concerning what it says there about the “long line” compared to what it says at locally compact space – this seems to be inconsistent to me
added theorems about relation with abelian sheaf cohomology to Cech cohomology
Zoran ?koda?: created Euler number (including Euler polynomial(s)) and expanded Legendre polynomial. Wasted part of the day browsing programming manuals about Ruby…interesting. Maybe something prompts me to be doing something about it :)
Urs Schreiber: created Karoubi K-theory
Andrew Stacey: Stumbled across the discussion at Timeline of category theory and related mathematics on bibliographies and realised that more people were keen for this to be sorted out than I’d thought. A few possibilities are laid out in the corresponding discussion on the forum. Please stop by and let us know what you want from a bibliography system so that we can design it according to what everyone wants rather than just what a few of us want.
On that note, seeing as my mathematical skills are not in the mainstream of the current focus of the n-lab, I’m concentrating a bit more on technical support (stuff like the forum, bibliography, diagrams, useful little scripts like how to download the entire lab for offline browsing). There are lots of things that I (and the others who do a little hacking like this) could do but only so much time in which to do it. So if there’s something you’d like done, say a bibliography, that you think I could help with then the fact that someone actually wants it done pushes it up my priority list. However, unless you tell me about it or mention it somewhere that I will actually see it then I’m not going to do anything about it because I won’t know about it!
created Kalb-Ramond field
expanded the list of examples at model category and added at the beginning a sentence on combinatorial simplicial model categories
after a little reflection I moved the previous content at electromagnetism to electromagnetic field and kept just a brief note at the former, for later expansion
then I worked on electromagnetic field
I renamed the section I was working on into Mathematical model from physical input . This now starts with quick and concise derivation of the fact that the EM field is modeled by a Cech-Deligne cocycle based on a quick definition of Maxwell’s equations and the quantization condition.
the following sections “the local picture” and “the global picture” are supposed to provide the remaining details and background. Still needs polishing.
started working on electromagnetism, but no nice entry yet – will have to call it quits now – won’t mind if anyone feels like improving on the current situation, otherwise I’ll continue tomorrow
created gauge theory but only in order to create electromagnetism
Zoran ?koda?: created Otto Schreier and made some corrections and additions to timeline. Many attributions give too late dates, e.g. Vladimir Voevodsky’s motives dated to 2000, while 1st versions emerged already around 1995-1996. When I saw his 1996 paper Homology of schemes in 1996 I immediately after reading about a page said to myself this is a Fields medal (and it was only in 2002 to my surprise); his preprint on K-theory arXiv on the solution of Milnor conjecture which was in that circle of methods is 1995 or 1996 as well. Note the usage of some concepts of homological algebra by Cayley before Hilbert.
added a remark about the general nonsense at nerve and realization to homotopy coherent nerve
added some links to new entries to the link list at Higher Topos Theory
Zoran ?koda?: created scheme, Nikolai Durov, model stack, pseudomodel stack. For timeline enthusiasts, I noticed that there is a big overlap (and some disagreements) with knowledgeable 40-page article
added references to John’s lectures and TWFs to generalized (Eilenberg-Steenrod) cohomology and to Postnikov system
brought models for infinity-stack (infinity,1)-toposes to a reasonable completion
added homotopy coherent nerve as a further example at nerve and realization
Andrew Stacey: started fleshing out an example over at Frolicher space.
added to limit the $(\infty,1)$-categorical definition with a pointer to the entry on limits in quasi-categories
created Cartesian fibration
added to adjoint functor the definition for $(\infty,1)$-categories
created local equivalence
expanded Bousfield localization by discussion of localization of model categories
replied at chain homology and cohomology and have myself a comment – but no time at the moment to work on this entry, will try to come back to it later, unless some helpful soul takes care of it in the meantime
created models for infinity-stack (infinity,1)-toposes – but still working
David Roberts: Comment at chain homology and cohomology re: type of space represented by a positive-degrees chain complex. Also comment at sphere regarding topology on infinite sphere for the purposes of contractibility.
David Roberts: Posted kernel of an idea at microbundle that has been sitting at the back of mind for a while, in the hope someone might be able to use it or do something interesting with it.
Toby Bartels: Wrote subset collection (a foundational axiom of set theory intermediate between power sets and function sets, justified by type theory and strong enough to define the located Dedekind real numbers).
Toby Bartels: Wrote sphere and pointed space to fill some gaps. The former has a reference to (yet unwritten) Whitehead's theorem with the provacative claim that this shows that generalised (Eilenberg?Steenrod) homotopy theory? is unnecessary; I don't really intend to defend that, but maybe it will interest the people working on that subject?
Andrew Stacey: Started a discussion on the n-forum about how to get a snapshot of the n-lab (since this is really an announcement page rather than a discussion page). Discussion is here.
Toby Bartels: Added some illustrations to simplicial set, based on those at cubical set, as requested by an InterestedAnonymousCoward.
Zoran ?koda? created microbundle. Note that classical references do not mention morphisms, just isomorphisms or equivalences of microbundles. Did anybody notice my update downstairs on the issue of export_html (answer to Urs/Toby answers) ? I suggested that once a week an export_html be posted as a file to be downloaded which is not up-to-date with a warning, as I think (maybe I should be corrected) that Jacques stopped serving export_html because of long generation/compilation time, while static file and new cimpilation once a week will do less harm. And leave generation of export_markup as it is, up-to-date.
Urs:
Urs:
added model category version to local object
added remark about relation of Quillen equivalences to the corresponding presented $(\infty,1)$-categories to presentable (infinity,1)-category
created combinatorial model category
Tim: I have started to reorganise some of the entries on Cech methods since David has started on homotopy (as an operation) and had an idea about Cech homotopy. I have encorporated a point made by Zoran? about the history of Cech methods.
Zoran ?koda?: created a version (to be expanded) of Legendre polynomial and M M Postnikov and added references to Postnikov system. I think historically tower and system differed by inclusion of universal cohomological class representing the fibration into the notion of system (cf. Whitehead’s big book, ch.9). This should be still noted: if one does not specify the cohomological class this is I think like missing the choice of isomorphism when the isomorphism exists. Technical issues: I encountered a problem that sqrt{fraction} puts sqrt only such that the numerator is under the root. I do not know how one should write correctly. Toby thank you for the tip for getting the SOURCE of old versions. I sometimes write some items partly motivated by need to have them for my students, and plan to incorporate something close to my version into student scripta. You are very knowledgable about wiki world. :) I was also trying to take the export of the whole nlab and succeeded to get the markupML version but not html: when asking nlab/export_html i get 403 FORBIDDEN message in my browser.
Urs Schreiber: I also get this error message when trying to export the $n$Lab as html – I remember the html export was particularly heavy on the server and maybe it was truned off in view of the server being a bit weak – we are trying to move to a better host eventually
Toby: Yes, Jacques turned that off because it was such a load on the server; I expect that we can turn it on again when we get a better host. In principle, you can get the HTML by exporting the source and compiling it on a local copy of Instiki, but of course you have to install Instiki to do that. (Also note that you'll need the CSS files if you want the HTML to look the same, including fonts, query boxes, etc.) And neither of these includes old versions; I think that Urs(?) is backing those up periodically in case the server crashes.
Zoran: could then somebody make one copy of html zip file 2-3 times a month ? It would not be updated but still it would be useful for browsing math when offline. If I get the zip-file I can put it online on my homepage. Or simply could Jacques put one zip file of export_html weekly with link and warning that it is not up-to-date; and for editing we can anywy use markup version. Then the server does not get heavy with generation, I think that probably generating, compiling all takes time, the shear downloading from time to time would maybe not burden the server ?
David: began an experiment on homotopy (as an operation) of dualizing the cohomology page. Began generalized (Eilenberg-Steenrod) homotopy.
rewrote the intro to Cech cohomology (see there) and started adding a list of examples for the abelian case: the standard series line bundles, line bundle gerbes, with connection, etc. – but not well written at the moment and no effort to get signs straight
created nerve and realization in order to host Kan’s general idea of how a functor into a category with colimits induces an adjoint pair of functors
I think I know what I am doing, but I’d like to ask people like Tim Porter and Todd Trimble to have a critical look at it (where is Mike Shulman??) – in particular at the moment I allowed myself to assume that we are copowered over the enriching category in order to get nice formulas, I wouldn’t object if somebody finds the time to give the more general discussion
then of course I adjusted links and made some comments accordingly at nerve, geometric realization and Dold-Kan correspondence
following Eric’s question I typed a quick reply into cohomology on how the ordinary notion of cohomology in cochain complexes is reproduced. In principle this gives all the necessary information, but I’ll try to find the time later to give a long detailed exposition of how this basic important special case arises from the very general perspective
added an “Idea” section to the beginning of Postnikov system in triangulated category
Toby Bartels: A quick note: I also have been changing [[apple]]s
to [[apples]]
but I will not do it now unless I have reason to think that it was written by someone who prefers [[apples]]
. In general, I do not edit matters of taste; I didn't know that this was a matter of taste, but now I do know that. (Sometimes if I'm changing something else, then I will change matters of taste at the same time, but only if I have substantially rewritten the sentence or if helps to standardise the notation and terminology. And this does not qualify as notation and terminology.) I'm sorry if I caused offence, but please understand that I did not know that there was a difference of style.
[[!redirects apples]]
to the bottom of [[apple]]
, then both http://ncatlab.org/nlab/show/apple
and http://ncatlab.org/nlab/show/apples
should work. I will still add such redirects myself, for the benefit of the style preferred by Urs, Eric, and me; but I will no longer change styles in what Zoran has written.[[apple]]s
to get [[apple]]s
if you find it convenient to do so.Zoran ?k?oda: but the plurals are NOT there – if I write [[apple]]s I did not use the code for plural. Let me clear the issue (I will write round brackets): Eric is doing TWO things 1) he is taking entry ((apple)) and adding the redirection instruction inside to allow for ((apples)). This creates one new version, not too bad, you consider this robust, I can tolerate it. 2) he is changing every occurence of my reference ((apple))s which used to be correct usage from within entries ((banana)), ((pear)), ((ananas)) and ((strawberry)) to new format ((apples)). This amounts to not allowing me to use legitimate ((apple))s from within ((bananas)). This second thing, unlike the first, I can not tolerate, as it has no rational explanation. I do not know if it is good :)).
Eric: The fact that many items appear as [[apple]]s on the nLab is an artifact of the period prior to having redirects. Prior to having redirects, we’d have to write that as [[apple|apples]] to get it to render correctly which gets old after a while, so people naturally gravitated to the easier [[apple]]s. If we’d had redirects from the beginning, there would be a redirect at [[apple]] for [[apples]] and no one in their right mind would ever write [[apple]]s again (which is distracting to look at) if they could just write [[apples]] instead. I’m at a total loss as to why you oppose this. Currently, I am trying to reverse the damage so that we can make things cleaner from here. Whenever, I see “]]s”, I instinctively change this to “s]]” and add a redirect if it doesn’t exist. In time, this should work itself out and we should have plural redirects for most links that are commonly used. It would work itself even faster if people stopped writing “]]s” and used the plural link form instead.
Zoran: I disagree that for example I write [[apple]]s rather than [[apple|apples]] beause it is easier. I write it because the appearance of [[apple]]s I like more: it tells me by the color which part of the name is real URL (as I often type URL), plus I have no dislike for multicolored words. I will continue writing like that. If you like to write your way write, but leave my links the way they are. Otherwise I can not ENJOY writing. It takes sometimes the whole minute to reload the page and I often reload the page is somebody has changes it, and it disappoints if the change is a matter of taste, and is legitimate. Second there may be day when you will have no time to write redirects etc. and one will not memorize which redirects exist and which do not. If I write the way I do, I will never have problem with this. If I want a single-color appearance of the link for some reason, I do not mind writing it long way like [[apple|apples]], it is about 2 seconds more, rather than spending far more time to check if there is a created redirect and wait half a minute to load the page, specially if in addition to slowness of the server I have my own internet connection problems what is very often.
I started going a bit through the Timeline of category theory and related mathematics and added links to $n$Lab entries wherever I saw a possibility
I STRONGLY disagree with creation of spurios PLURAL items when they are of NO use. Namely like Eric just created new version of bialgebra cocycle to say that it redirects bialgebra cocycles, while it is simpler and better from memory point of view to write bialgebra cocycles. Why having whole page archived and one more page to browse in history just to distiguish if s is before or after the brackets ?? Dear Eric we have thousands of items to enter and there will be thousands of new pages in future and why to increase entropy and spend yuor valuable time on this – please take some book and help us entering something NEW and not messing with plurals and creating new versions for nothing. I have hard time browsing history when something is messed up and if I am going to spend it on such a nonsense than I will leave the idea to enter new items myself.
Zoran: look at my explanation few paragraphs above: I accept your creating redirects, but do not accept not allowing me having my own format of calling links within my own text. I changed your redirect back because I want to assert my right to have the link called any possible way I like it. It functions, it correctly displays and it si even more informative: ((apple))s tells you even the information that s is not the name of the page, and I often TYPE the URL name of the page rather than clicking on the link, because often because of slowness of the server working on laptop and desktop simultaneously. Of course this kind of strange usage is useful just for the author, but if I am in process of improving the page which i largely created i think I have the right for the convenience.
created bialgebra cocycle, Drinfel'd twist
by the way how does one download the source of a previous version ? I sometimes create a page and then there are changes after and I want to have my file for other purposes with what I wrote and I do not know how to access it.
added to nerve an “Idea” section and a further “examples”-section on nerves for chain complexes and the relation to the Dold-Kan correspondence
after a bit of work and with a bit of luck, I found the old reference by Kan where the description of the Dold-Kan correspondence is given in terms of nerves – this is the nice way to do it – added that reference to the Dold-Kan correspondence
by coincidence I came across the old entry crossed module and noticed that there were meanwhile plenty of links to add to it – so I did
added two further references, by Birgit Richter, on the ($\infty$-)monoidal structure of the Dold-Kan correspondence to Dold-Kan correspondence
added a small section on and a link to matching family at sheafification
reorganized Dold-Kan correspondence in an attempt to make the material more systematic – then I started adding a detailed def/lemma/theorem/proof list of the classical statement, but didn’t get very far yet
made a remark at Timeline of category theory and related mathematics
pasted a blog comment by David Ben-Zvi into n-categorical physics
Zoran ?koda?: created essential image, added n-category generalization to replete subcategory, additions to image, model structure on chain complexes, remark due Leinster to Gray-category; created matching family following mainly conventions in MacLane-Moerdijk. This last item surely overlaps with sheafification but the approach and exposition is rather different; created micro-entry maximal sieve.
Gave Tim a link at category theory.
Started suspension to fill links. (We already have reduced suspension. There's also a suspension defined at delooping which doesn't seem to be quite the same; what's the connection?)
created Cech cover and Cech model structure on simplicial presheaves
I notice that Timeline of category theory and related mathematics is a renamed version of the original Sandbox(!). (As you can see by clicking on its “history” link. But also the current Sandbox’s very first link claims to lead to the historical Sandbox, but takes on to the Sandbox poage with title “Timeline of category theory-…”) I guess that’s not intended.
tried to improve the exposition at groupoid object in an (infinity,1)-category. More to be done eventually, though.
Reminds Urs that he no longer needs to type [[Urs Schreiber|Urs]] because there is now a redirect from Urs to Urs Schreiber., i.e. the link [[Urs]] points to Urs. Try it! :)
Along similar lines, after seeing [[internalization|internal to]] about a million times, I just added a redirect so that typing [[internal to]] gets redirected automatically to internal to. Try it! :)
I would like to suggest that the appearance of links to nonexistent pages is a feature that we should not break. Thus we should not create blank pages (or pages that are blank except for redirects) but instead create pages only when we have something to put there. Conversely, we shouldn't change links to go to redirected forms (as at geometric infinity-function theory currently) unless the redirects have actually been created. If this means that we have things like [[∞-foo|infinity-foo]]
(when nobody has written about $\infty$-foos yet), then we're no worse off than before we had redirects, and the appearance of links to nonexistent pages still tells us something. (Note: there is some related discussion now hidden under July 3.)
A question at cogroup about what we really want there. (Surely more than just the empty set?)
added to principal bundle a long detailed section called “the $G$-action from the homotopy pullback” – this may look like weird overkill, but the point is that this serves as a warmup for an analogous discussion at principal infinity-bundle
rediscovered that we had an entry Cech methods and added lots of links to that
provided explicit details at Cech cohomology for the general (nonabelian) case in low degree
Zoran ?koda?: created small fibration, added more general discussion on endomorphism monoids.
Urs created Cech cohomology
more suggestions than changes, but it would be good to have entries for cogroup and co-H-space. Could homotopy group and cohomology group be made to resemble each other more? I.e., must the former be restricted to $n$-spheres as domain? Hmm, is something suboptimal about H-group and H-cogroup, whereas H-space and co-H-space? Perhaps ‘co’ and ‘H’ commute.
created groupoid object in an (infinity,1)-category
added a tiny bit of discussion to principal infinity-bundle about how the homotopy pullback definition gives the $G$-action and vice versa. But need to say more here.
Toby Bartels: Mentioned automorphism groups.
Tim: I have asked a silly question at pseudofunctor, but would appreciate an answer. Can I make a plea to someone to provide a more detailed treatment of the Grothendieck construction as well. (I mean the one which is related to the pseudocolimit. At least a general reference to that should be in the entry.)
Tim: I have added new references to distributor and Grothendieck fibration.
Toby Bartels: Added a bit to the scandalously short page isomorphism.
Eric Forgy created exchange law blank (perhaps by accident?) and Toby Bartels wrote a very brief stub there.
edited and expanded the link list at generalized smooth space
created smooth infinity-stack – took the liberty of declaring this to be the term for “$\infty$-stacks on CartesianSpaces”
created smooth space – took the liberty of declaring this to be the term for “sheaves on Diff” = “sheaves on CartesianSpaces”.
this way a “diffeological space” is precisely a “concrete smooth space” and I
edited diffeological space
and created concrete sheaf (but concrete site still missing, and also maybe not the best definition at concrete sheaf yet)
to make this statement.
Toby Bartels: Created Cauchy sequence and complete space, including brief references at the end of each to the relationship with Cauchy complete categories.
started a section “concrete realizations” at principal infinity-bundle. So far I recall the old result by Quillen on certain “1-categorica topological bundles” and their $\infty$-action groupoids. Then I start making some remarks on Jardine’s approach using what he calls “diagrams” and have a remark on how that compares to the Bartels-Baković-etc-style. This requires eventually much more discussion, but I have to call it quits for today.
But the point is that Jardine works in the “petit topos” perspective where all bundles live over the fixed site. So the terminal object in his setup is not the point, but the underlying space over which one works. This means that the simple picture of a principal $\infty$-bundle as the homotopy pullback of the point no longer works and is the reason why he introduces the yoga of what he calls “diagrams”.
On the other hand, when one works with simplicial sheaves in the context of a “gros topos” such as sheaves on $Diff$ or on open subsets of $\mathbb{R}^n$ or equivalent, then the simple conceptual picture remains valid.
Notice that placing oneself into the context of $n$-groupoids internal to diffeological spaces or the like is doing precissely this: working with simplicial sheaves on $Diff$.
Evidently i shouldn’t be discussing this here but in some entry. But it’s time for me to go to bed now.
Zoran ?koda? created grouplike element. It contains few words on Amitsur complex for a coring with a (semi-)grouplike. The aim is to soon (using the setup) introduce entries for connections for corings; and then correpsondence between falt connections and descent data in the comonadic and coring setups (after Menini et al; all coming back to the example which is the correspondence between 1-order costratifications and flat connections in the crystalline setup due Grothendieck).
started adding the discussion of the model given by (pre)sheaves with values in simplicial groupoids to model structure on simplicial presheaves (contained in a new big diagram containing all Quillen equivalent models for $\infty$-sheaves)
Zoran ?koda? created coseparable coring,Sweedler coring, two dimensional sheaf theory; expanded stratifold (which was empty, but existing!), added a reference to fibration in a 2-category and somewhere else. I think that in K-theory delooping has a bit different multiple meanings which are related but are more procedures making from something what can not be delooped strictly in the sense of delooping to the delooping of something what is the best approximation of deloopable; there are procedures due to Quillen, Waldhausen, Karoubi etc.
Eric: Added a redirect for Urs so that he no longer has to type [[Urs Schreiber|Urs]] and can simply type [[Urs]] and will be redirected to the correct page.
proudly presenting what is now the widest (horizontally speaking) $n$Lab entry as yet: at model structure on simplicial presheaves I added a section “Map” where I draw a big diagram that indicates at least part of the collection of model structures, their interrelation, definition and authors
Yeah, I was thinking of reworking that map to run vertically …. —Toby
Would that really be better, though? Optimally, eventually we’d produce a LaTeXed pdf diagram. Here and elsewhere. That, however, inhibits joint editing a bit. —Urs
moved the key statement of Toby’s remark to the very beginning of group
expanded at group on the statement that “a group is a groupoid with a single object”
added to loop space object the true definition, added also an “idea” section, links to related entries and an introductory blurb to the original material
created delooping for the matter discussed by Eric and Toby at Dijkgraaf-Witten theory
added Jardine-Luo reference to principal 2-bundle and principal infinity-bundle
added to Froelicher space an “Idea” and a “Reference” section, prompoted by the blog entry here
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