Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
Thanks, Zoran! I have added to these, especially multiset.
<div>
<blockquote>
Additions to <a href="https://ncatlab.org/nlab/show/Brown+representability+theory">Brown representability theory</a>
</blockquote>
<p>Thanks for that, Zoran! That was certainly missing.</p>
</div>
I added the example of ringed space and ringed site to ringed topos, and linked back and forth with lined topos.
Comment on ring, and a correction to what I think you meant about symmetric monoidal categories. If that's not what you meant, then I don't know what you meant; maybe you can clarify.
I understood that we were replacing by an arbitrary monoidal category , rather than replacing . So has its own notion of the additive abelian group.
However, I think that it's very strange to refer to a monoid in an arbitrary such as a ring in . I would accept this if were a monoidal additive category (with compatibility between the tensor product and biproduct), maybe even a monoidal -enriched category. Similarly, I would accept calling a monoid in a rig in if were a monoidal category with compatible biproducts (making it enriched over but not necessarily over ).
Note that this includes the case of an -ring for a possibly noncommutative ring; an -ring is a monoid in , which is a monoidal additive category.
Just for the record, I just added a section with pointers to oo-rings ro ring, just for completeness.
I was going to say also something leading over to ringed topos, but now Zoran has the page locked.
okay, I added to ring a pointer to rings in an arbitrary topos and tried to structure the various generalizations a bit. But probably still suboptimal.
Okay, so you are actually thinking of an object of a category equipped with both an "addition" morphism and a "multiplication" morphism ? Unfortunately you can't define a ring object that way unless your monoidal category is not just symmetric but cartesian; you need diagonal maps in order to write down the diagram saying that multiplication distributes over addition, and you also need augmentation maps in order to write down the condition that the addition have inverses.
@ Zoran
I never said that.
I know. But it seemed that Mike was saying that.
However, I think that it's very strange to refer to a monoid in an arbitrary such C as a ring in C
I didn't mean to imply that, that was sloppy editing on my part. I agree that it only makes sense to call a monoid in C a "ring" if C is Ab-like in some way. I wouldn't require it to be literally Ab-enriched, though; it also makes sense to call a monoid object in the category of spectra a "ring spectrum."
I think that I've rewritten it clearly and correctly; please check.
I removed this line about -rings:
Unlike for the -algebras, the multiplication is not -linear in the second factor, but only -linear.
Remember these are bimodules; the multiplication in the first factor is -linear on the left, while the multiplication in the second factor is -linear on the right.
Toby, nobody conventionally says for k-algebras algebras under k, nor for A-rings, rings under A.
No, certainly not!
Why say ‘only -linear’ when it is -linear? There is no asymmetry in the concept, only in a language that assumes that left is the default. A better language is to specify ‘left’ and ‘right’, as I did. In any case, ‘only’ is quite misleading.
You asked me to answer, but I don't know. Ronnie Brown wrote that, and my best guess as to what he meant matches Mike's guess. But I haven't thought about whether that makes the statement true.
Hi Zoran,
I don't know how these notions of deformation retracts are related, precisely, but it sure looks like they deserve separate entries.
Concerning the query at point of a topos: I suppose that, yes, what really matters are just the isomorphism classes of topos points.
Zoran, I think I didn't respond to your question about deformation retracts because I didn't understand what you were asking. Now that I do understand, I agree.
I split off deformation retract of a homotopical category from deformation retract, and also replied at point of a topos.
Hm, that broke all the links to this entry: most of them were intended to go to the def-retract of the homotopical version.
I guess I'll have to go through the source entries by hand...
I created an entry gebras. I put it intentionally with pural as it is more dedicated to the subject than to the linguistic notion of 'a gebra', but I am open to opposite veiwpoint.
If a ‘gebra’ is an object, then the subject would be ‘gebra theory’ in English. Am I correct that the point is to not use the word analogously to how ‘algebra’ is used?
After having read the article, I moved gebras to gebra theory.
@Urs: sorry about breaking the links, I didn't think of that. I do think that it's the topological notion which deserves to be at the page deformation retract, though.
Right. I have fixed the links meanwhile.
Maybe we should think about the links deformation retract for the enrichement and closed monoidal deformation retract. These of course all refer to the deformation retract of a homotopical category, so maybe their titles should be expanded.
After birational geometry, continuing in the mood of algebraic geometry, I wrote etale morphism. But then a query on Hurewitz theorem on fibrations, at Milnor slide trick.
Additions at Legendre transform.
Added reference Norman E. Steenrod, A convenient category of topological spaces, Michigan Math. J. 14 (1967) 133–152 to convenient category of topological spaces (and to compactly generated space), together with a link to the project euclid where it can be downloaded. The paper is not only a classical exposition of the compactly generated Hausdorff spacesm but also of the basic homotopy theory in that setup (like fibrations, NDRs etc.) with proofs. George Whitehead's book "Elements of homotopy theory" takes much from this paper but omits the proofs (in chapter I).
I have included the proof of the main statement in Hurewicz connection: a map is a fibration iff there is a Hurewicz connection for that map.
Urs has changed the layout, now I do not see word Theorem and word Proof in the central section. Is my browser or syntax ?
Hm. I tried to be helpful and added the formal theorem and proof environments.
They do display on my machine.
I thought that using this should be preferred, since only when we use these formal environments can we eventually program the CSS style such as to format theorems etc appropriately. At the moment one advantage is that the proof environment does make an automatically generated end-of-proof box appear.
Or at least it does on my machine! Now, I wasn't aware that this depends on the installation.
I get the end/of/proof box well, but do not get word Theorem and do not get word Proof in that entry (neither in regular nor in print view). On IE on a new desktop which fairly well displays the rest of nlab.
If you have JavaScript turned off, these environments don't appear correctly, but even then you should still get the words. Did my latest edit also remove the word ‘Definition’? Can you make a screen capture?
Yes, the definition disappeared. I will see later what is going on, now need to hurry.
The things work on firefox on the same computer, but the IE on that computer is quite well equipped and works for other nlab purposes and surprisingly not for this.
Created stub universal enveloping algebroid.
I complained again at reflective subcategory. Requirement to have the localization functor left exact to use localization terminology for the unit of adjunction is not required outside of topos community. New entry simple ring.
Adding details to simple ring led me to write zero ideal just for completness's sake.
New entry Whitehead product, now not having a complete definition yet (even that may need corrections). I see that the join of spaces has been defined only at suspension. Eventually it would be good to have better and separate entry join of spaces different from suspension and join of simplicial sets. I am too tired to look into Fuks' paper who has some point in finding the Eckmann-Hilton dual of that bifunctor (for compactly generated Hausdorff spaces in his case). After Whitehead product it would be nice to have entries on Massey products and Toda brackets. These play crucial role in "secondary" constructions of Baues which are categorification of a sort which should be compared to the stuff the nlab is centered about.
I added internal category in a monoidal category (aka noncartesian internal category) but did not spell out the main definition yet, I have to hurry to a bus right now...Bio entry George Janelidze (needs partial bibliography) and Hvedri Inassaridze.
I suppose it should be in Gabriel-Demazure book (exists djvu) though I am not sure how much is said in that generality. It is also quoted in intro parts of some papers of Toen, Vezzosi.
There might be something in
but I do not recall if the original book is treating the subject in quite that language, but to some extent it should be there.
I have created cofibration.
Yes, sheaves of sets.
I have to run as well now.
New entry arithmetic geometry and more links and references, mainly to Mikhail Kapranov and Alexander Rosenberg. Corrections to enhanced triangulated category; the three variants should not be equated in general. New details at higher monadic descent.
Élie Cartan, , Henri Cartan, Claudio Hermida, Sophus Lie, local Lie group and some crosslinks in Timeline (I can not spell so long title without effort). Stub Verdier's abelianization functor.
New bio entries Raoul Bott, Paul Bressler. In wikipedia Bressler is not listed among Bott's students, but I know he was.
The entry left adjoint has antiLeibniz ordering for the composition of natural transformations; as this is used much less widely, I think as a basic item, it is more audience friendly if we rephrase it in the more standard compoisition notation and Leibniz ordering. I will not change it unless I get confirmation from others. By the way, adjoint functor is for half na hour locked by an anonymous user. Though it is implied by other definition, I think we could have a paragraph or two for adjoint pair in terms of one of the two transformations but with stating its universal property (e.g. by a counit and its universal property).
New entries: compact operator, relatively compact subset, Fredholm operator. The purpose is of course, to set some background for the discussion of index theorems, which are closer to the central focus of the nlab. I'd like also to have entries on some relations to symplectic topology (e.g. Maslov index, Floer homology...) where the Fredholm theory is of central importance. New entries Morse theory, perfect Morse function, heat kernel. More crosslinks at geometry.
New entries fundamental solution, Sobolev space, Lebesgue space (in the latter the title is slightly ambiguous, see the first paragraph). For the motivating context see the post above. More crosslinks at functional analysis and at analysis.
Some additions to Chern character and another paragraph at Fredholm operator dealing with Fredholm complexes (I do not want to separate the latter unless the entry becomes huge).
new: Janez Mr?un (well the character gets crippled in nForum, look at the redirect Janez Mrcun), foliation, integrable connection; additions to Ieke Moerdijk
You mean integrable distribution, not integrable connection, yes?
right, integrable distribution -- many thanks to your sharp eye
New entry Jacobian conjecture. In CW complex, I added condition locally finite and the link to Milnor's article on geometric realization: a geometric realization of a locally finite simplicial complex is a CW complex. So the same condition for simplicial sets. Am I wrong ? (It seems so: not written in the Milnor's article; where the locally finite is useufl for other assertions; but isn't it that only locally finite simplicial complexes are Hausdorff while all CW-complexes are Hausdorff ??).
I added few more references and crosslinks, and made small formatting changes at homological algebra and at some related items.
I extended largely the list of related entries inalgebraic topology and created CW approximation (for now only the definition).
In addition to the entries I anounced earlier today in other entries (wall crossing, geometric representation theory, Alexander Beilinson, BBDG decomposition theorem, Ofer Gabber) I created Victor Ginzburg and am just going to create Ivan Mirkovic. Added Alexandre Kirillov.
I wish I had a better idea of "wall crossing". If you feel like adding a brief "Idea"-section describing how you think about it, I'd very much appreciate it
Unfirtunately, I have feeling only about some special cases, and not yet the general idea. Maybe in a bit time I can get some progress...though I am at the moment more excited about trying to prove that the homotopy stable quasicategory has a Maltsiniotis strong triangulated structure, not only usual triangulated structure, as in Lurie's paper.
I added more references to hypersimplex hoping it will make me closer to bite the problem. New stub barycenter.
Urs: there are 3 general areas in which I saw wall crossing: Stokes phenomenon (connecting the local WKB-solutions of (nonlinear) wave/eikonal equations across Stokes lines), representation theory (wall crossing functors) and Bridgeland stability conditions (relevant in string theory). Presumably one should understand the relation between the three.
I constructed a first version of overview section in wall crossing. We should have entries on isomonodromic deformation, Stokes line, Gauss-Manin connection, regular connection, Riemann-Hilbert correspondence, constructible sheaf etc. to support this more substantially. New entry Valery Lunts.
Yesterday i was not allowed to post comments to nlab, from home address which is dynamic address, so I can understand that. But today I can not post comments to cafe (while I can to the nlab) from my official institute's address 161.53.130.104:
Your comment submission failed for the following reasons:
You are not allowed to post comments
Please correct the error in the form below, then press PREVIEW to preview your comment.
Adjective complicial has two quite distinct meanings in higher category theory.
Wow, I've never heard that second meaning. Why don't they just say "dg-"?
Zoran,
about the Cafe not letting you post:
as I mentioned once, this is some annyoing bug or overreaction of the spam filter, that I can't do anything about. It happened to Bruce Bartlett a while ago, and he was stuck with it for some time. Unfortunately.
I have no idea what causes this, and it shouldn't happen. In principle Jacques Distler is the only one who can do anything about this. But a quick workaround should be to use any other machine or otherwise try to get a different IP address. After a while the problem tends to go away.
myself, I am making good experience now following Andrew's advice. When I connect now, I first log into a "VPN client" at Hamburg university. Don't ask me exactly what that is, I can only guess. But the result is that I am being proxied through the Hamburg university server and appear to any website as if I came from there. Hamburg University is not yet regarded as polluting the world with spam (though some people there are working on it ;-), so that helps me stay clear of spam traps.
Maybe you can ask somebody at Ruder Boskovich about this. Probably they have this, too.
No, this is the IP address of the whole internal network of our institute, so I can not change it.
Wow, I've never heard that second meaning. Why don't they just say "dg-"?
I guess it is reasonably good english to make an adjective complicial from complex, isn't it ? The terminology may be older than Verity's, though I don't know. But dg pertains only to a very specific kind of category, while complicial has much wider meaning pertaining to the whole case of derived geometry. So there is a complicial algebraic geometry which is for example not the same what some people call dg-schemes (dg schemes are a very old version of derived algebraic geometry which is not good in general). Plus I guess one wants to have a special term when emphasising that one does NOT mean bounded complexes from any side. I just added a quote to a new version of the entry complicial. I added the redirect complicial algebraic geometry.
New entry Higgs bundle. References at cyclic cohomology.
thanks for finding the Higgs field, Zoran, should we call Geneva?
I have once heard lectures on this, but I seem to have forgotten:
so there is no condition that satisfies a Leibnitz rule?
And how exactly do you form ? You wedge the 1-form part and do what to the sections of ?
Thanks for the joke. "Higgs field" is a widespread and actually pretty standard terminology in the business of monodromy and related vector bundles, connections and differential equations; I am not much familiar with this area but I should be (some of my long-term research objectives much intersect with this); papers of e.g. Tony Pantev are the state of the art in the field.
Thank you for your notational question: one starts with , sends it to , then another acts on the factor and the result wedge with the previously existing factor, and we get something in id est to . Thus wedge refers to the differential forms part only while on part you compose.
I have had some argument with Toby at representation. Toby is right in emphasising clean general perspective which can easily cover some of my objections. I have some doubts that not all classical notions of representation fit into this, while of course everything is OK with groups and vector spaces and their higher categorical and enriched analogues. It would be nice to say something about horizontal categorification there. I like to emphasis a concept of a symmetry which is not necessarily internal categorical notion (but surely is or should be in vast majority of cases), or whose internal origin is not yet known for some algebraic structures known to encode symmetries.
I created from memory the book entry Gabriel-Zisman. Please check and correct, those who remember the essential content and its historical novelty and importance better. I wrote more carefully path category and added a redirect free category.
New entries Como, separable algebras, separable field extension (should relate it to etale morphism at some point and define separable functors!), additional remark at Catégories Tannakiennes on Magid's theory of differential Galois theory. Magid's insight into separable algebras lead Janelidze to breakthrough in the subject of categorical Galois theory, into whose entry I added additional references.
Thanks for all this! Will try to look at it later. A bit short of time right now...
Marta Bunge, externalization (the latter vague from memory, somebody from topos theory or having more time to rethink and look into literature could write a more precise entry I hope)
Improvements at formal spectrum some stylistic, some mathematical (previous statements were true just in special cases). In particular the idea section had it wrong that the infintesimal neighborhoods are introduced with formal schemes, rather nilpotents in usual schemes can express infinitesimals: the point is that with formal schemes we can have functions depending on infinitely many neighborhoods at once. Thus not only first or second infinitesimal neighborhood but all at once. Second it is not true that the underlying space of Specf R for I-adic topology is having the same underlying space as Spec R, but the same as Spec (R/I). Once you make elements in I nilpotent the power of nilpotency does not count nilpotents do not affect the underlying space.
New entry FGA explained and related links in related entries.
Two new stubs: intersection cohomology, perverse sheaf.
Noticed we do not have Poincaré duality !
More at heat kernel: reference to Spin geometry and a concrete definition extracted from it (e.g. around page 208). More references at Fredholm operator.
I wrote five lemma in main nlab, as well as a shortened version at my personal lab, not to mess my students with the semi-abelian version section.