Archive of changes made during August 2009. The substantive content of this page should not be altered.
made infinitesimal space and infinitesimal quantity redirect to infinitesimal object – this way we can use the links and still wait with deciding whether to split the latter entry or not
created cosimplicial algebra (redirecting also cosimplicial ring)
created Chevalley-Eilenberg algebra in synthetic differential geometry where I aim to derive the Chevalley-Eilenberg algebra of a Lie group $G$ in terms of functions on the infinitesimal neighbourhood of the identity in a manner entirely analogous to how the deRham dgca is obtained at differential forms in synthetic differential geometry
created infinitesimal singular simplicial complex – it reiterates things said at differential forms in synthetic differential geometry but the motivation for creating it was to accomodate a reference that Zoran Skoda provided which gives a version of this notion in nonstandard analysis
Jon Awbrey added a stub and a link for future development at Peirce's logic of information.
created an Idea section at space and quantity, mainly to list some links.
Given the pivotal role that this entry is playing I am unhappy with its rough appearance. Hope we can eventually improve on that.
replied to Toby at infinitesimal object by agreeing that, yes, now that you mention it, it sounds like a good idea to rename that entry into infinitesimal space and create a parallel entry infinitesimal quantity.
thanks to Toby for the rephrasing, it’s much better now, yes
Urs Schreiber: various very minor edits:
added a first sentence at twisted bundle that links the concept with twisted cohomology (but more details could be given on that)
slightly edited the beginning of semifree dga (added the statement for the graded-commutative case)
Urs Schreiber: there was some funny behaviour by the software ater I had restarted the server a few minutes ago, which resulted in the Lab-Elf announcement below to disappear for some mysterious reason. I have rolled back now, but that, too, involved some funny effects. It should all be restored now, but if anyone wonders where his or her comment disappeared to, it may have been eaten by the software in this process (but I don’t think anything is lost, but let me know).
added to differential forms in synthetic differential geometry references by Breen-Messing and by Kock with a few lines of remarks. Am hoping to eventually expand and polish this entry.
slightly edited the first paragraphs of infinitesimal object and nonstandard analysis and inserted links back and forth
Lab Elf (public relations)?: the migration will happen this week. I had hoped to get everything done before the semester started at the Cobblers School, but was not able to do so, which means it has to fit in around my schedule. What I don’t want to do is do the migration, then go off for another violin lesson, and come back a few hours later to find that you’ve all crashed the site again.
Once that has been done, then the technical department can get on with figuring out how to improve things around here. However, whinging on this page is pretty pointless. If there’s some feature you’d like implemented then you could bug the Chief Lab Elf himself (though rumour has it that he’s quite busy at the moment) or one of the minions, but a much better idea would be to start a discussion about it over on the n-Forum. Treat it as a lab elf notice board. If there’s something you want all denizens of the lab to see, put it here, but if it’s primarily for the lab elves, put it on the forum.
(Note: the high honour of being a Lab Elf is one that anyone can have. There are various sub-departments, all with different roles and skills.)
Urs Schreiber: replied to Rafael Borowiecki at category theory
following various suggestions I didn’t quite delete the big discussion box there yet, but instead moved it to the very end of the entry, so that it doesn’t interrupt the reading flow anymore where it used to be (since we reached agreement there).
Urs Schreiber: replied to Rafael Borowiecki at Bousfield localization and asked for further details
I second John’s and Toby’s comments: the reason why it is important to log changes here is because this is the only reasonable way that we alert each other of changes and thereby give us a chance to interact with each other. If nobody would log here, everybody would be bound to blindly edit the Lab on his or her own, whithout any interaction. I have added a remark to this extent now also to the blurb that appears on the top of this page.
And yes, we are all suffering from the slowness of the Lab. But the moment that Andrew Stacey decides he is ready, we will switch to a better server. He has already set up everything and is just waiting for a good moment to migrate.
Jon Awbrey added content to relation theory. Incidentally, does anyone know how to get an @ symbol in a math context?
I have been busy writing on the timeline of category theory and reading a prehistory of n-categorical physics. But i managed to write something in nLab.
To log everything just takes more time, and i am already tired of waiting while doing the normal editing. Neither is there a direct link to this page. And is not recently revised? enough? I always used it. Since i don’t see the wonderful thing with this page i won’t have the energy and time to log everything here.
John Baez: if you don’t log your changes here, we probably won’t read your work, since we won’t know what you’ve done! That also means we’re not likely to work on further developing your articles.
The first thing I do when visiting the $n$Lab is go to this page… and I’m probably not the only one! That’s why this page is important: it’s where we meet and point each other to what we’ve just done. The rest of us log our entries on this page for this reason… why not you?
Rafael Borowiecki: OK, this is a good reason. But nLab is still too slow! Is nobody using recently revised?? Except that it is broken now.
Often if you get an error when saving, the save did go through, and you should check that before going back. (Maybe you know this, and just didn't bother to mention it.)
If you use the back button and submit again, then this can mess up the locking mechanism if somebody else starts to edit the page in the meantime. It is by starting a fresh ‘edit’ command that you tell people that you are still working on it. (If it helps, you can type in the URI directly —http://ncatlab.org/nlab/edit/whatever+the+page+title+is
— without showing the page.)
It's always a good idea to copy your material to the clipboard (or even a text file) before submitting it, just in case.
Zoran: The error was not about the entry but the content. When I cut and paste some paragraphs to a sandbox it did not work either. Only after I changed some sentences, characters and formating in the text the text was allowed. The problem seemed to me with the two formulas in double dollar signs as they were the last which I changed before in about 40th attempt the thing worked.
Zoran ?koda? added a section on classical topological version in deformation retract. Corrected big chunk which Jon Awbrey has erased from latest changes by an editing error. Added Pareigis classical reference to actegory.
Jon Awbrey is adding boolean domain, boolean function, and boolean-valued function — according to his custom starting with a middling level of abstraction that comes up a lot in pedagogy and practice and is calculated to avoid scaring too many children and horses.
Zoran ?koda? created locally convex space and additions to topological vector space; the latter wasted an additional one hour or so because the server was not accepting the input but sending internal application error messages for some reason which according to many tries seemed to have originated in some characters in formulas in previous (Toby’s text) it seems as only after changing his formulas it worked and removing my own paragraphs did not help, but who knows which combination of mine and earlier text is actually bugging the system. The repeated trials or removing and adding text took more time than necessary because of the old problem with nlab default that nlab rejects the submition if it is 4th subsequent trial without refreshing edit view from within show view (hence back button not allowed more than 3 times in a row). I think the default should be put to 10 or so, as I often hit over 3 submits from the text which I edit and do not want to wait (and also fight and pay slow internet when on wireless) to go back to show view and then again to ask for another edit: my mozilla keeps the old edit and I can go back to the text immeditaly if the submit was unsucessful or if I notice I do not like the outcome. Thus when server is slow I can reedit what I was editing 5 seconds ago instanteneously without repressing the edit and waiting for the server which may fail again. I think 10 resubmits within the frame could be a better deafult than 3. Created entries Fourier transform and Pontrjagin dual. Yesterday wrote few lines in functional analysis but on the submit the loss of connection lost it; I’ll try again today.
Jon Awbrey added multigrade operator and parametric operator. Added the missing Figures to minimal negation operator but didn’t know how to scale them here — 80% or 500px would probably be good.
Toby Bartels: Edited nonstandard analysis a bit. Grammatically, I changed ‘infinitesimally small’ to ‘infinitesimal’, which already means ‘infinitely small’ (in absolute value).
Zoran ?koda? created entries actegory (category with an action of a monoidal category), yet another sandbox SVGsandbox, (mathematical) analysis and Weierstrass preparation theorem (the latter is in a basis of connections between analytic and algebraic geometry; I want eventually to say about algebraic approaches to analytic geometry like analytic local algebras and rigid analytic geometry; the theory is rather parallel to some aspects of modern algebraic geometry, e.g. in the way differential forms and regular differential operators are introduced); created subfunctor (needing your checking). Earlier this week much extended noncommutative geometry and did not have time to report it here. I do not see nonstandard analysis in usual sense in Moerdijk-Reyes (also at page 385 bottom they themselves also include that the relation of their axiomatics SIA with nonstanard analysis is not clear). The discussion in chapter 7 is in generalized sense kind of nonstandard analysis, in the general sense that it entails a version of the transfer principle. However the discussion relies on a very general setup of complicated axiomatics based on commutative rings the transfer refers to certain language using coherent formulas. Thus it is not about extending language $L(\mathbb{R})$; or about ultrafilter model of the nonstandard extension of $\mathbb{R}$. Of course, at the topos level, one can express everything in terms of internal language of the topos and there is a general transfer principle in that setup. This is much more abstract sense of nonstandard analysis than in the rather concrete and conservative article I started. Of course further discussions and contributions in both directions are necessary for us.
Urs Schreiber: thanks, Zoran. By the way: in the later chapters of Models for Smooth Infinitesimal Analysis there is some detailed discussion of the relation between synthetic differential geometry and nonstandard analysis
Zoran ?koda? created nonstandard analysis containing basically an (incomplete) introduction to basic terms in the ultrafilter model of the nonstandard extension of real numbers. One of the reasons to create this right now is that Urs is talking about infinitesimal object in synthetic differential geometry, and it could be a moment to (hopefully usefully) compare with the notions of nonstandard analysis. Created Rouqier's cocovering (in subject of triangulated category).
Jon Awbrey noticed that graph was unsaturated, so he whetted it. There are many definitions of graph and many dialects of graph theory. I added one of my first and favorite. Read my …
Tim Porter: I forgot.. I created braid group yesterday, as I needed them for examples and Toby already has improved the presentation! Thanks.
Jon Awbrey: more almost ready for prime time $\nu$‘s
Andrew Stacey solicits input about making the database of categories a real database; see the Forum.
Welcome, Jon! It would be ideal if you actually wrote about this material instead of just linking to discussion elsewhere; hopefully, you're already planning to do this. Assuming that you wrote the material linked elsewhere (or have permission from those who did write it), you could mostly copy and paste it here; we regulars would be happy to help with formatting and the like. Some more technical notes: we have a convention of lowercase page titles, so that people can make links to your pages from the middle of sentences; and speaking of making links, you should probably link this stuff from the main logic page. (I'll do some of this right now.) —Toby Bartels
Jon Awbrey: almost ready for prime time $\nu$‘s
Tim Porter: I have created derivation on a group to provide some back up for Fox derivatives. This is also needed for the linearisation functor going from crossed complexes to chain complexes.
David Roberts: further tinkering at history of cohomology with local coefficients - mentioned link of Reidemeister’s original approach to crossed complexes. Forgive my ignorance Zoran, but I’m not sure what to do about your comment there. Is it meant to extend cohomological history generally or provide details on related topics?
Toby Bartels: I'll never be done with affine space. Never! Mwa-ha-ha-ha-ha!!!
Tim Porter wrote a comment down at the bottom of August 26, so look there for it.
Mike Shulman: I think I’m done with affine space for now.
Alex Nelson? commented at crossed complex.
Toby Bartels: Yes, always more at affine space, such as morphisms.
Mike Shulman: A bunch more at affine space, including an “unbiased” definition. There’s still a lot more to say.
Urs Schreiber: created infinitesimal object, with a bit of material, but haven’t found yet the time to round this up and polish
Zoran ?koda?: created cobordism category; created Fox derivative (cute construction in combinatorial group theory). I think, it is the same Fox of the famous article Fox-Neuwirth on the topology of configuration spaces.
David Corfield: And the Ralph Hartzler Fox of ‘Crowell and Fox’ about whom Ronnie Brown wrote
Crowell and Fox in [43, p. 153] took the view that a few definitions ‘like that of a group, or a topological space, have a fundamental importance for the whole of mathematics that can hardly be exaggerated. Others are more in the nature of convenient, and often highly specialised, labels which serve principally to pigeonhole ideas. As far as this book is concerned, the notions of category and groupoid belong in this latter class. It is an interesting curiosity that they provide a convenient systematisation of the ideas involved in developing the fundamental group.’
Tim: The wikipedia article is of interest on Ralph Fox. He was the doctoral supervisor of Milnor, Stallings and Barry Mazur!
Did some editing of heap, including adding two alternate definitions of the automorphism group, considering its functoriality, and incorporating the discussion into the main text.
Claimed that Toby’s fix of affine space contains superfluous data.
Zoran ?koda?: added links to heap and affine space at affine space; and the original (B and PS) references to BPS-state.
Zoran ?koda?: created dilogarithm.
Urs Schreiber: in case anyone is waiting for reactions from me: I am currently on a small vacation with little internet access. Will be back at full speed next Sunday or else next Monday
Zoran ?koda?: added synonym flasque to flabby sheaf; query at history of cohomology with local coefficients. Created quantum dilogarithm, but for now it consists only of references and links.
Question/correction at affine space.
Comment at symmetric function.
Discovered, and restarted, the terminological discussion at lax natural transformation from back in June.
Zoran ?koda?: $A$ and $B$ in fine sheaf are already closed, so I removed Toby’s correction taking closure.
David Roberts: some dates at twisted cohomology on the earliest references, and added title, date and small clarification on Reidemeister’s 1938 article at history of cohomology with local coefficients.
Zoran ?koda? created soft sheaf, fine sheaf, family of supports, analytic geometry.
David Corfield: Mike’s right at symmetric function isn’t he? So the definition needs redoing. Would do it myself, but how does one put the bit about grading properly?
Andrew Stacey pondered the format of database of categories.
By the way, it would still be really useful if people could take a look at the migrated n-lab. The main question I want to know is whether the pages look right. The migration involved a step or two that were pretty much guesswork and I want to know whether I guessed right or not. If a page looks horribly wrong, or something doesn’t work how it ought, please let me know. The best ways to let me know are either by email or by commenting over at the forum.
David Roberts: trivial comment at synthetic differential geometry in response to Mike.
Gonçalo Marques? has a report at symmetric function.
Toby Bartels: Tried to clean up the formatting at Jim Stasheff's new history of cohomology with local coefficients. In particular, this involved removing some stuff that seemed to refer to a bibliography that wasn't there, so please complain (or just put it back) if that was wrong.
added material to database of categories — more on Rel and SimpSet — and replied to Toby’s and Rafael’s comments there. What we want — I think — is not a massive list of all the categories we know (I could make up 5 a minute for the rest of my life), but a list of categories including their basic categorical properties. I don’t want to worry too much about the format of this list until it’s gotten a lot longer.
tried to get Rafael to visit this ‘latest changes’ page (and the corresponding pages for many previous months), log his changes here, and interact with us a bit more here.
replied to David’s and Toby’s comments on symmetric function.
wondered why each of these sub-entries is starting with an asterisk intead of a little circle, and tried in vain to correct it. Help!
*__<span class='newWikiWord'>John_Baez<a href='/nlab/new/John_Baez'>?</a></span>:
(spaces added), see how the *__
takes up $3$ characters? Then every line that comes under it should begin with $3$ spaces. (Sometimes you can get away with fewer.)Toby Bartels: Slight edits to space.
Rafael Borowiecki has a question at Bousfield localization.
expanded simplicial object a little, added Idea, Examples and mentioned cosimplicial objects
corrected at singular cohomology the alleged simplicial ring of functions to a cosimplicial ring
replied to Ronnie Brown at singular cohomology
thanks to Toby for space – I have just one request for a change of wording there
Toby Bartels: Thanks to Rafael Borowiecki, I added a link at Higher Topos Theory to the published version on Lurie's MIT website.
Vaughan Pratt: changed “category: categories” to “category: category” for Chu construction. (Rationale: The plural form was just my misremembering the name of that category, I didn’t mean to start a new category. Chu(V,k) is a category and therefore should be findable as such, but it would take a while to write a separate page for every pair (V,k).) (I also edited “Editing latest changes” but presumably that goes without saying. :) -vp)
category: category
, then you'd see how silly it is), so I do not think that you should actually change anything (other than plural to singular, which you did).Vaughan Pratt has started category: categories and put Chu construction in it. I'm not sure what this category is supposed to be for; the entry wouldn't fit into category: category either, since it's not a specific category (although maybe that is not the best criterion).
Ronnie Brown: Query (badly formatted) added to singular cohomology.
Urs Schreiber: further polished and edited ∞-quantity, gave more details on how the characterization of the deRham complex as the normalized Moore complex of functions on simplices follows from Anders Kock’s results and provided a reference that provides the statements about the relation between functions on $\mathbf{B}G$ and the Chevalley-Eilenberg algebra in the algebraic context.
Zoran ?koda?: created Schur's lemma.
Toby Bartels: Some breaking up into paragraphs and things at hyperplane line bundle, symplectic manifold, and Kähler manifold.
Zoran ?koda?: created Killing form with some words on Casimir operators. Eventually Casimir operators should be treated separately, but it is beneficial to develop the unique entry first to accumulate common facts, conventions and notation, because the two are closely related.
I am preparing some ground for a comprehensive discussion of the theorem – which I think we have now, using the latest addition at monoidal Dold-Kan correspondence – that Anders Kock’s chacraterization of the deRahm DGCA is precisely nothing but the characterization of the image of the cosimplicial algebra of functionas on infinitesimal simplicies under the normalized Moore cochain complex.
I wanted to do that at the entry on $(\infty,1)$-quantity. It could also go at differential forms in synthetic differential geometry, but as the $(\infty,1)$-quantity pages is globally marked as “research material” I thought it might be good to put it there and then just point to it from elsewhere.
But in the course of this I noticed first of all that my use “$(\infty,1)$-quantity” was a misnomer. It should be “$\infty$-quantity”. To explain (to myself) why, I created the dual entry ∞-space. Then I moved the material from the former (infinity,1)-quantity to the new ∞-quantity and started editing a bit and put redirects.
Zoran ?koda?: created symplectic vector space, symplectic manifold, Kähler manifold, hyperplane line bundle having not only $\mathcal{O}(1)$ but also the basics for $\mathcal{O}(-1)$, $\mathcal{O}(n)$.
moved the new discussion of references by David Roberts at topological T-duality to the References section, edited slightly and inserted some links
expanded symmetric monoidal category which was very stubby (and still is): added an “Idea” section with pointers to the general context – and started adding a list of examples
David Corfield: asked a question at symmetric function.
David Roberts: links at topological T-duality and mention of early work on the topic.
Omar Antolín-Camarena?: added a definition to accessible category, could use checking by someone who knows.
Omar Antolín-Camarena?: created small object, whose definition was missing from locally presentable category.
John Baez: started a database of categories. This is very preliminary, but it could be very useful if we keep working on it. The idea is to list lots of categories and their categorical properties. If this list becomes long we can try to organize it somehow.
David Roberts: question at topological T-duality. No mention has been made of the Adelaide school’s treatement there!
John Baez: so, please tell us what that school has done, or at least add some links to papers on the arXiv.
David Roberts: I did provide a couple of references in my comment. But to quote Galois, 'I have not time!' (but I’m not about to duel anyone). I’ll do a little bit now.
Started singular cohomology by copying the definition from cup product.
An important reference at database of categories. We're already using Instiki's category system here!
Moved some stuff about the structure from topology to topological structure (which is really topological space now).
Moved stuff from connection to connection on a bundle. It's possible that much of the latter could be put on a more specific page, something like parallel transport (which currently redirects).
A bit more at continuous map.
A suggestion for Zoran ?koda? at differential form.
Added links and such to nonabelian algebraic topology. Normally I don't log this sort of thing, but as much of this is a personal essay, I want Ronnie Brown to make sure that I didn't warp anything.
Roger Witte? has joined us with an edit to category theory.
Zoran ?koda?: added several (very carefully chosen, though I am not competent enough) references into BPS-state.
Urs Schreiber: split off monoidal Dold-Kan correspondence from Dold-Kan correspondence – moved the material in the original section at the latter to the former and linked back and forth – moreover I added a section Lax monoidalness of the Moore cochain complex functor where I claim to prove the statement asserted by this headline. CHECK.
Zoran ?koda?: created an outline for the BPS-state, expanded group theory. I side with Mike with long-surpressed urge to have nice conceptual explanation of the terminology semantics-structure adjunction.
Mike Shulman: Corrected a handedness error I made in my initial reply at monadic adjunction, and made a request: can anyone give a nice conceptual explanation of the terminology “semantics-structure adjunction”?
Zoran ?koda?: expanded slightly geometry, diffeomorphism and much topology (with groupings of similar items, few hours of work), created Hurewicz connection, continuous map and a stub for diffeity with few references. Additions to connection.
Urs Schreiber: added the standard singular cohomology version to cup product
Zoran ?koda?: created projection measure. I agree with Toby that, in common convention in published literature, if one uses a German phrase involving word Satz (like in his Theorem example) one will still write it with capital letter. As a physicist, I witness and referree papers on daily basis which are using the word Ansatz in English physics texts, as rule capitalized. Mathematicians do not like the term and logic of using something as a means to solve the problem if it is not justified a priori (one mathematician was telling me words of disgust: “what is physics ? Nothing! Ansatz!”).
I posted a query under differential form on interpretations as inner hom, i.e. functions on $\Pi TM$.
Ronnie Brown: I have rewritten nonabelian algebraic topology to incorporate historical comments I made made on discussion lists, and so to show my experience of the relation with nonabelian cohomology.
Urs Schreiber: reacted to David Roberts‘ latest comment at category theory by expaning the item on “categories = 1d spaces” with a discussion of one way how this statement can be thought of as being made precise
Mike Shulman: continued discussion at monotone function.
David Roberts: comment at category theory re category as a sort of directed space.
Lab Elf (wildebeest department)?: We will be migrating the entire $n$-Category Lab to a new server soon. Please see the announcement on the Café and report there all of the many problems!
Toby Bartels: I did as David urged below; see category theory and universe.
David Roberts: Re: category theory - Toby, go ahead. In particular, I found last night the foundations and philosophy page about which some comments were directed (the people involved, including me, didn’t know it already existed). There is a link to a paper on category theory about structuralism which might go there, as well as the (short) discussion surrounding it.
David Corfield: re foundations and philosophy, I can’t do more than log its creation here (see 2009-08-17). Now to find time to write it up.
Andrew Stacey: I’m hesitant to weigh in on this as I’m as guilty as everyone else, but merely flagging something here is not really enough. We should all think about how to organise the material here to make it easily findable. Of course, linking from related page to related page is good, but there should also be some hierarchical organisation. For example, there should be a philosophy index page and foundations and philosophy should be on it. Perhaps, appropriately enough, we should make more use of the category features in Instiki. At the moment, we have the following categories: biography, category, delete, drafts, foundational axiom, lexicon, meta, people, place, redirect, reference, spam.
This comment is getting a little long, so I’ve started a forum discussion with more details on how to implement this, so please reply there.
Toby Bartels: Reading Zoran's work reminds me that there's another way to interpretation notation for an integral, which I've now recorded at measure space. (I think that capitalising the ‘N’ in ‘Nullstellensatz’ in English is like capitalising the ‘T’ in ‘Cantor's Theorem’, but I'm not going to worry about it in a world with redirects.)
Zoran ?koda?: added standard references to commutative algebra and links to Murfet’s online notes. Nullstellensatz as a German noun compound starts with a capital letter, and it is usually (though not always) quoted so in English references.
Toby Bartels: Created algebraically closed field.
Zoran ?koda?: created commutative algebra both for the notion of a commutative $k$-algebra and the subject of the commutative algebra, which is one of the foundations of algebraic geometry. I have removed the redirect commutative algebra from associative unital algebra. Toby: a variety is an affine, quasiaffine, projective or quasiprojective variety. Hence if one talks about the category of all varieties, then all of those are objects simultaneously, and not by convention of choice. True enough quasiprojective includes all others as the affine space is a Zariski open subset of the projective space. I corrected nonlogical usage of the maximal compact to the maximal torus. The thing is that I usually use K for maximal compact in G which is complex, and now I used G for compact and $G^C$ for complexification, so in my normal notation it would be $K/T=G/B$ where $T$ is the maximal torus. Once I lost $K$ in the complexification notation I put it automatically at the place of the torus. Of course $SL(n,C)/B = SU(n)/T$, what is called Gram-Schmidt orthogonalization procedure :) Thanks for catching the inconsistency in language. I will use $T$ now and remove $K$ everywhere (better I used my notation).
Zoran ?koda?: created algebraic variety, additions to integral scheme and lists mathematics,geometry
Toby Bartels: I'm going to mess with category theory again, if somebody doesn't stop me.
Zoran ?koda?: after lots of work expanded the entry list for geometry (the first try was eaten up by internet explorer: nlab server did not accept my first submission and IE does not allow going back to the data (while never had this problem with firefox – just can go back to the form with data still in)).
Urs Schreiber: added a bit more on the monoidalness and the shuffle map at Dold-Kan correspondence
Zoran ?koda?: created flag variety, Borel-Weil theorem and expanded coherent state
Urs Schreiber: following David Roberts‘s reaction below I took the following action at category theory
I split the item on “categories as spaces” and “categories as mathematical universes” in two
I edited a bit the statement about them being spaces
then in the item on categories as mathematical universes I pasted in the paragraphs that me and David Roberts had suggested in the quesry box discussion below.
David Roberts: responded to Urs’ suggestion at category theory with one edited from his. We need to wrap up some of the discussion there, in particular, and, in my opinion, remove some points that would probably not be considered core to an introductory page on category theory - I hesitate to say 'points that are not generally supported by the category theory community' because I know that there are some new and/or minority points of view that are enlightening. Certainly a couple of the extended discussions might be moved to a discussion section. I blame the lateness of the hour in Adelaide for not doing that particular task myself.
Zoran ?koda?: created coherent state (it will be much longer later)
Urs Schreiber: replied at category theory by suggesting as an alternative an expanded version of the sentence on “mathematical universes” under discussion.
Zoran ?koda?: additions and changes to symplectic geometry
Lab Elf (golf department)?: added an entry at the How To on how to put parentheses (and other “unsafe” characters) in links.
David Roberts: comment at category theory - nothing substantial, just adding my voice to Toby’s comment.
Omar Antolín-Camarena?: added a section to adjoint functor theorem about the version for presentable (∞,1)-categories.
Omar Antolín-Camarena?: created solution set condition (has link from adjoint functor theorem).
Toby Bartels: Comments at equipment and category theory.
Urs Schreiber much as I regret it, I still have complaints – but also constructive suggestions – in the discussion at category theory
and i notice that a long discussion in a query box is getting cumbersome. Maybe we should move that to the blog.
commented on one discussion at category theory
finally created equipment
replied at monadic adjunction
added “Idea” section to generalized complex geometry
created geometric quantization collecting some links and importing John’s old material from here
created symplectic geometry with just some things to come back to later
Zoran ?koda?: created Leibniz algebra
added to groupoid cardinality the general definition for infinity-groupoids as well as a handful of further examples
started working on the entry exercise in groupoidification - the path integral
Toby Bartels: Wrote variety of algebras.
created a stub for quantization
renamed Jim Stasheff‘s entry Larmore into Larmore twisted cohomology and edited it; linked to it from the reference section at twisted cohomology
made Chu space a redirect to Chu construction
created entry for Michael Barr
added more references to Batanin omega-category, especially references that discuss modelling homotopy types using Batanin omega-groupoids.
added a reference to Simona Paoli’s paper at homotopy hypothesis. This contains an answer to Urs’ question about the functor that turns spaces into $cat^n$ groups. I also moved his question down to the ‘Discussion’ section near the end of the page.
posed a question on monadic adjunction. The end of my question contains a formatting error of the sort Toby knows how to fix! I forget how!
Jim Stasheff wrote Larmore; I'm not sure if it's supposed to be about the person or about a cohomology theory.
found time to expand the discussion at examples for geometric function objects – now this contains details on the simplest but somewhat archetypical example: that of over-categories. This is really the notion of geometric $\infty$-functions that is implicit in John Baez‘ notion of groupoidification. I am thinking that making its structure in the context of geometric $\infty$-function theory explicit is useful for putting this together with the discussion at geometric infinity-function theory into perspective. Will try to say more about the other examples indicated tomorrow.
slightly edited and updated links at geometric function theory – one of them now points to the new examples for geometric function objects, which however is still to be written, but I have to interrupt for a moment
created entry for Michael Batanin
created stub for Batanin omega-category
Cleaned up twisted cohomology; there's still a link to Urs's web that's broken.
Noted the compact nature of the Gelfand spectrum (also at maximal spectrum).
Mentioned $B^*$-algebras at operator algebra; but I really need to write C-star-algebra.
filled T-duality with a bit of content (motivation being to link to the Cavalcanti-Gualtieri article now linked to there)
created stub entry for our esteemed new contributor professor Andrew Ranicki
activity at twisted cohomology:
Jim Stasheff added a long list with chronology of references on twisted cohomology
Andrew Ranicki added a hyperlink to a reference (as far as I, Urs Schreiber, can tell from the logs)
edited Jim Staheff’s chronology slightly
and replaced the bulk of the entry with a newer version as it has evolved meanwhile in interaction between Urs and Jim at Urs’ private web – the new version has a longer motivational piece and includes details on the proof that and how the fibration-sequence definition of twisted cohomology that is used here reproduces the traditional one in terms of sections of bundles of spectra as a special case
created transversal maps
expanded generalized smooth algebra, adding more theorems, links and remarks (also corrected some minor mistakes in the previous version) – but I also suffered a stupid data loss due to the system having an “internal error” and myself srewing up the backup copy of my edit and had to type everything twice. Hopefully I didn’t delete unintentionally old content this way…
created a stub entry for our new esteemed contributor, Prof. Charles Wells
Zoran ?koda?: created very incomplete entries operator algebra, maximal ideal, Gelfand spectrum. I have mentioned mostly just the unital case, as the nonunital case will be more tricky when completed.
David Corfield: Started ideal completion and foundations and philosophy.
Charles Wells Lightly edited category theory, notably introducing posets as another example of a category.
Toby Bartels: More talk on category theory. Also note the existence of the article foundations; there's not a lot of philosophy there, but there could be and probably should be.
Zoran ?koda?: created coquasitriangular bialgebra, cosemisimple coalgebra; added the redirect corepresentation and few words on this terminology to comodule
David Roberts: More discussion at category theory. There are some statements that need unraveling, and I don’t quite feel up to it. In particular, Rafael Borowiecki pointed out the need for a philosophical page on foundations, as he referenced a paper on structuralism, sets and categories I didn’t feel fitted on category theory (or at least in the section where it was referenced). So maybe this is a plea to David Corfield, who is more qualified to talk about such things than I.
Toby Bartels: Started prime ideal theorem and maximal ideal theorem. Eventually I'd like to have precise equivalences of these, constructively valid (preferably in any pretopos), to various forms of choice, with proofs. Now is just a list of very basic results, possibly valid only in a model of ETCS.
Zoran ?koda? created homotopical algebra as a rather terminologically-historical entry (as opposed to more descriptive and concrete homotopy theory and compared to homological algebra); created quasitriangular bialgebra with redirects quasitriangular Hopf algebra and universal R-element. Added links to mathematics, algebra and maybe to some more entry(s).
John Baez: Why does the main front page look so weird? Did the site get moved to a new host, or is it the result of an alien invasion?
Hmm, now it’s back to normal.
Urs Schreiber: I can’t tell what happened. Andrew is indeed preparing the migration and has set up the nLab on another server by now. That shouldn’t affect what’s going on here. Or might it be that the phenomenon was something just on your side? What was it actually?
Toby Bartels: I was thinking of writing articles on $\Omega$-groups and ??????? varieties to talk about ideals in them. I agree that three articles, one on that subject, one on ideals in rings (and rigs, maybe even monoid objects in general) and one on lattices (and other partially ordered or preordered sets), would be a good idea.
Zoran ?koda? created quantum group, additions to fiber bundle, Hopf-Galois extension and Timeline of category theory and related mathematics including a query discussion and 1970 Timeline entry for Benabou-Roubaud theorem. Some thoughts on ideal entry. Toby, I think we should eventually have an entry which would have a general and lattice notions of ideal separated from the entry for rings/algebras. Otherwise half of the entry is incomprehensible for ring/algebra theorists. For example for noncommutative rings commutative notion of prime ideal splits into nonequivalent notions of prime and completely prime ideals (just to start, compare also primitive etc.), which are now difficult even to list as the rest of the list is lattice-worded. So I would opt to have a general entry and specialized entries for lattices, rings/algebras. Another interesting context are $\Omega$-groups (additively written not necessarily commutative groups with a family of operations, not necessarily unary ones which distribute over group “addition”; I am going to add a stub now) where ideals correspond to quotient $\Omega$-groups; interesting is to compare those version of ideals to the categorical notions of normal subobject. Thus we have ideals for sheaves of rings (e.g. defining ideal of a subvariety) etc.
Toby Bartels: Added principal ideals to ideal.
Zoran ?koda? created flabby sheaf
David Roberts: More comments at category theory, this time to statements in the section about the contrast with set theory due to Rafael Borowiecki.
David Roberts added to the discussion at category theory.
Zoran ?koda? created Benabou-Roubaud theorem
created equivalence of quasi-categories – at the moment just to record a useful lemma
added various new links to the list at Higher Topos Theory – also edited the paragraphs at the beginning a little
removed the old and meanwhile highly incomplete link list at sheaf and topos theory and instead included links to our main link list pages on these topics – would be nice if eventually we’d find the time to write a nice overview and exposition here on par with that at category theory
created model structure on marked simplicial over-sets – this is used to model the $(\infty,1)$-categorical Grothendieck construction, so I edited the latter accordingly
renamed right proper model category to proper model category and added the missing cases.
Rafael Borowiecki is back at category theory.
added an “Idea” section to cartesian morphism
started adding details on the relation between $(\infty,1)$Cartesian fibrations and (infinity,1)-presheaves to universal fibration of (infinity,1)-categories – not sure yet where this material should ultimately go: this is really the $(\infty,1)$-Grothendieck construction. Maybe it should be at Cartesian fibration.
created marked simplicial set
Toby Bartels: Went through Section 1.1 of Stone Spaces; see the links from there.
added to Grothendieck construction the definition in terms of pullback of the “universal Cat-bundle” – and added pointers to Grothendieck fibration and to category of elements at generalized universal bundle
created fibrations of simplicial sets
added to Kan fibration a sentence on right/left Kan fibrations and made inner Kan fibration, left Kan fibration, right Kan fibration and weak Kan fibration? redirect to it
made lifting property a redirect to weak factorization system
created anodyne morphism
started cartesian morphism in order to host, among the standard stuff, the $(\infty,1)$-categorical version
corrected some minor points at limit in a quasi-category a bit
created Karoubi envelope in order to collect references to the $\infty$-version of it
added a section “Idea” and a section “Generalization” to Grothendieck construction
added an “Idea” section and started a section “Properties” at Cartesian fibration
added some links to K-theoretic issues that have meanwhile come into existence at Grothendieck construction.
Concerning the comment there: my feeling is that people pretty consistently use the term “Grothendieck construction” for the reconstruction of a fibration from a pseudofunctor, whereas for the K-theoretic aspect they say “Grothendieck group”.
added pointers to the entry Cartesian fibration at fibered category and Grothendieck fibration
added to the discussion on universal fibrations at stuff, structure, property (in the section on logic) a comment about and pointer to the universal fibration of (∞,1)-categories.
Andreas Holmstrom has created a user page, including a link to a blog.
Urs Schreiber: added to pullback stability that of right lifting property and hence that of fibrations – restructured a bit.
Urs Schreiber: created pullback stability to satisfy links, but included so far just a pointer to commutativity of limits and colimits. Maybe we want to split that latter entry into the relevant subentries.
Urs Schreiber: created universal fibration of (∞,1)-categories and linked to it from generalized universal bundle and limit in a quasi-category.
Zoran ?koda?: created normal variety
Urs Schreiber added universal colimits as the last remaining entry on the four Giraud's axioms that was still missing at (infinity,1)-topos. The entry currently points to the relevant discussion at commutativity of limits and colimits for the content, but is supposed to serve for providing the particular terminology used here.
Urs Schreiber: created Thomason model category
Eric: Based on a discussion at the nCafe, I created Online Resources.
Andrew Stacey marked his return with a small but significant observation about extensions of the category of smooth manifolds: if the inclusion of manifolds into an extension category preserves limits and colimits then the extension category cannot be locally cartesian closed. At the moment, this is contained in a remark at the end of the third example of a Froelicher space - please check my reasoning!
(I’d like to be able to say that I figured this out whilst flying over Damascus, but I think it was actually Novosibirsk.)
Zoran ?koda?: created combinatorial group theory, free product of groups, Nielsen-Schreier theorem, Hopfian group; quoted references for algebraic proofs of Nielsen-Schreier, somebody should add a reference with explanation of the topological proof (I think Massey’s book would do but I do not have it at the moment).
Urs Schreiber: added to category algebra the description of (at least groupoid algebras) in terms of the weak colimit over the constand 2-functor to $Vect-Mod$. That’s kind of remarkable. I have to admit to my shame that I wasn’t aware of this fact before. It’s extracted from Free-Hopkins-Lurie-Teleman’s latest, where it is the starting point for a huge story.
Zoran ?koda?: I will most likely be on vacation for next about 10 days, what means mainly offline, though I hope to contribute with an item here and there within that period. To live safely (=with nlab) when offline I downloaded the whole html version of the site which I backed up online at my institute’s server (which is pretty well working). Here is the today’s file nlab.tar.gz, only about 10Mb but with 77 Mb after gunzipping back to tar and about the same after untarring. Of course I will not update this file at least till full return back.
Zoran ?koda? surely Toby, this is true for any fixed object $M$ (I wrote subobject?? hmm last night I entered a wrong building instead of the one in which my flat is). Just created coherent sheaf; because of severe time constraints it is rather short for a significant entry which will be large in future.
Toby Bartels: Rewrote the definition at property sup slightly; it seems to me that $\Omega$ should be an ascending chain of subobjects of a fixed object $M$, rather than an ascending chain of a fixed subobject $M$ (which I can't even parse). That also fits in with a noetherian category's having the property, but I mention it here in case I'm wrong.
Aleks Kissinger has joined us, adding examples to dagger category.
Zoran ?koda? created short entries simple object, semisimple object, socle, nilpotent ideal; noticed that if I create a redirect ((apples)) for ((apple)) and used it in ((pear)), the entry ((apple)) is NOT listed in the list of “linked from” entries at the bottom of the page. Thus if I link by a redirect name, I will miss the backpointer. This happened with simple objects listed in semisimple object, but simple object does not say that it is linked from semisimple object. Added more to artinian ring. Toby, I agree that $n Cat$ and $n$-$Cat$ are accepted synonyms, that hyphen looks better than minus, and that there is no a priori rational reason for $R$-$Mod$ as opposed to $R Mod$, however the tradition in math community and in professional typerighting (say in numerous journals of AMS) do not use $R Mod$, at least not noticably often, in favour of other versions, and if nlab has strange (even if abstractly correct) conventions in conventional part of math, it may be less attractive to students and professionals. Additional confusion may arise in confusion the name $R$ or so before $Mod$ with a modifier like $gr$, $dg$, $co$ or alike, which are more often (and with stronger arguments) in tradition written without hyphen.
Urs Schreiber: expanded and rearranged germ a bit
Toby Bartels: Added to germ the example that causes me to keep linking to it.
Urs Schreiber: created germ, added references to group cohomology on continuous/smooth case
$R$-$Mod$
(which won't work inside a displayed equation), $R\text{-}Mod$
, or $R‐Mod$
(which is a little funky but arguably the most proper). I will change minus signs to one of those from now on, if that is what you prefer. (Of course, $_R Mod$
also works.) Compare: ‘$R-Mod$’, ‘$R Mod$’, ‘$R$-$Mod$’, ‘$R\text{-}Mod$’, ‘$R‐Mod$’, ‘$_R Mod$’.Urs Schreiber: added a section “Details” to the end of (infinity,1)-quantity to go with the blog discussion here
Zoran ?koda?: started Bredon cohomology and a stub for Mackey functor with few references.
Urs Schreiber: created superconnection
Zoran ?koda?: created double of algebra with involution,alternative algebra,nilpotent element, Azumaya algebra. We should have things much more detailed and explicit here, e.g. showing the correspondence between sheaves of Azumaya algebras and abelian gerbes…Though Duskin’s construction of Azumaya complex deserves a separate entry of course and the Brower group does as well. Toby why do you use nonstandard $R Mod$ instead of $R-Mod$ or ${}_R Mod$ – the standard algebraic literature places either dash or places $R$ in subscript…and it is even more weird for right modules with $Mod R$.
David Roberts: added comment to fundamental groupoid about topology thereon and relation to local connectedness. In other news, I have a (non-academic) job, so as of next week my contributions will slow down a bit.
Zoran ?koda?: created several shorter entries open subscheme, open immersion of schemes, reduced scheme, integral scheme, integral domain, noetherian scheme (warning: the listed proposition is not that obvious to prove) and made some changes to noetherian ring.
Eric: Created Hasse diagram. It may need polishing to make it technically (and morally) correct.
Sridhar Ramesh: The article on Kleisli category erroneously made the remark that the free functor associated with an Eilenberg-Moore category would necessarily be faithful (as a simple counterexample, consider the monad on Set which sends everything to 1); I’ve reworded the line which stated this, as well as one other
Zoran ?koda?: created coherent module, noetherian category (containing also some material on locally noetherian abelian categories), Hilbert's basis theorem, noetherian ring, noetherian object, irreducible topological space, noetherian topological space, integrable system, differential topology, general topology, algebraic approaches to differential calculus, skewfield (with redirect ‘division ring’; I chose 1-word synonym as a main version however). Some users (outside of the contributor community) asked for more browsing access to nlab, namely to be able to descend to entries in specific subfields differently than via the long all pages list, or guessing and google search for notions; thus I gave small contribution to this by expanding algebra, topology and geometry and mathematics; I do not think that we should care to ever have something like a complete tree, but having overview entries with lots of links in subfields will create more awareness of non-obvious entries. I also think that the descent from anything like top entries to concrete entries should not be unique in general and link overlap whenever logical and intelligent cross-referencing would not hurt. I have moved much of the material from reconstruction theorem to Lawvere's reconstruction theorem and added a link to latter and few more references, mainly for Tannakian and scheme cases.
Zoran ?koda?: Created exact sequence of Hopf algebras (well, for now short exact).
Oh John, you gave me an opportunity for recalling some sweet memories: I clumsily used word lazy for a lack of interaction (actually a barrier of shyness/fear between experts and non-experts) at a summer school on geometry and strings in 1999 and got excommunicated there for about a week :)
John Baez: What, I’m the first one working on the $n$Lab and it’s already past noon (in Paris)? The rest of you must be getting lazy! :-) I added a little bit about cosimplicial objects vs cochain complexes to Dold-Kan correspondence.
Zoran ?koda?: To assist John Baez‘s links, created Eilenberg-Moore category, Kleisli category and added details and redirects to monadic functor. To John’s question: I think most basic facts in Tannaka reconstruction including sort of those on coalgebra level are in chapter 3 of Bodo Pareigis’ online notes; his emphasis is on ends and coends. However I hope you find Kazhdan’s notes, they must be great. I created entry Tomasz Maszczyk with very interesting past seminar abstract at the bottom relating a new reconstruction theorem coming from nc geometry.
Urs Schreiber: expanded the example list at twisted cohomology – in particular added “cohomology with local coefficients” as a special case. Added to the very beginning of local system a paragraph on how strictly speaking local systems were meant to be such “local systems of coefficients”.
Zoran ?koda?: created smooth scheme, flat morphism, smooth morphism of schemes, EGA IV, locally affine space, relativization in algebraic geometry. Notice that this is extending a series on regularity/smoothness/differential calculus in algebraic framework which included our earlier entries regular differential operator, quasi-free dga, formally smooth morphism and so on.
John and Ramesh thank you for the nice detailed material on Lawvere reconstruction theorem, I’d suggest to create a separate lab entry Lawvere reconstruction theorem on it, moving the most of the material there (regarding the size and number of other reconstruction theorems waiting to be covered in detail the same way) with a link and short comment on the main reconstruction page.
John Baez: that’s fine — go ahead and do it! We can talk about all the reconstruction theorems on the reconstruction page, but put details on separate pages. I’d been wanting to write a book about reconstruction theorems, but this will either help me write it or help eliminate the need for doing so.
Zoran ?koda?: Done, now we have separate Lawvere's reconstruction theorem and a sentence on its content with the link at reconstruction theorem.
John Baez expanded reconstruction theorem by adding the example Lawvere theories.
Zoran ?koda?: posted jibladzeCoeffLargeCats.djvu and linked it to crossed profunctor. Urs, how the integration approach to diff. forms fits with existance of classes of smooth, $C^1$-only, $L^1$-integrable etc. differential forms and the currents (“differential form-valued distributions”), and it seems it puts n-forms on n-manifolds in special position, than say k-forms on n-manifolds. There is a subject of geometric integration theory where integrability is related to geometric properties like rectifiability (Federer); how this fits with that. And finally with differential forms on singular varieties. It seems to me that this approach has advantages and applicability in some cases, while the easy approach via dualizing vector fields to get 1-forms and then proceeding algebraically in others. One should maybe also compare to Lurie’s usage of cotangent bundle in expressing an alternative approach to higher descent.
created (infinity,1)-quantity – some comments:
this accompanies a blog question here
among other things this aims to provide the full general nonsense $\infty$-version of the statement at the beginning of differential form
I could move this to my private web. Let me know if you feel that would be better.
Zoran ?koda?: created crossed profunctor (of crossed modules) and butterfly (papillon). First after M. Jibladze (1990) and second after B. Noohi (2005). I did not draw the diagram but wrote equations explicitly.
Eric:
Zoran ?koda?: created reconstruction theorem. There should eventually be separate entries for each of the main classes and examples, which are listed in the main entry.
added something on the dual Dold-Kan correspondence relating co -chain complexes and co-simplicial abelian groups to Dold-Kan correspondence.
renamed differentials at chain complex from “$d$” to “$\partial$”
expanded cochain complex
created differential forms in synthetic differential geometry with two purposes: it reviews the definition found in the literature and then proposes a – supposedly nicer – reformulation
Zoran ?koda?: added a query in compact object: the stated characterization of categories of $R$-modules is known at least about 3 decades before Ginzburg’s lectures. Maybe we should look into classical sources.
Added theorem characterizing categories of $R$-modules to compact object.
Added same theorem and also the Mitchell embedding theorem to abelian category.
Gleefully shot down an idea of Eric’s over at directed graph — but this idea can be resuscitated using the concept of ‘resolution’.
Sridhar Ramesh added a small note that Lawvere-Tierney topologies are the same as (internal) closure operators on truth values (Being new here, I’m not exactly sure what the protocols are; in “Please log all your non-trivial changes”, does “non-trivial” mean everything above, say, typo-correction?). I also observed, in the article on presheaves, that the representable presheaves on a category of presheaves are precisely those which turn colimits into limits (i.e., a functor of type $[C, Set]^{op} \to Set$ is representable just in case it is limit-preserving).
Urs Schreiber: I added a remark to the text above on what “non-trivial” means. Generally, I think you can’t log too much of your activity here, just too little. So if in doubt whether anyone else might be interested in your changes or not, drop a note here.
John Baez: Welcome, Sridhar Ramesh! I’d like to add: we are not trying to force you to do lots of bureaucratic work, so please don’t feel we are demanding you to log changes here. If you make a change and feel too tired to log it here, we won’t be mad at you. We just like to see changes here, even smallish ones.
Eric: Asked a question of Mike (or anyone else interested in SDG) at synthetic differential geometry.
Sridhar Ramesh has joined, editing module and Mitchell-Benabou language so far.
Urs Schreiber: replied and reacted at differential form
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Last revised on August 23, 2015 at 22:30:34. See the history of this page for a list of all contributions to it.