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New entry spectral cookbook with sketch of some very nice constructions of A. Rosenberg. New stub sheaf on a noncommutative space, pretty contentless so far, and a redirect page noncommutative sheaf, where the latter may have a different meaning (that is why a separate page).
I have removed this sentence from AnonymousCoward:
(Well, usually. Urs Schreiber —or for all we know, possibly somebody impersonating him (^_^)— has managed to keep his IP address out sometimes.)
This makes it sound as if I did something intentionally to hide my IP, which is not the case. Rather there must be a problem with the software, if something that should not have happened did happen.
I have also removed the following old discussion, which is better had here on the forum:
Eric: Can we change this? I am not anonymous, but I also do not want my IP listed (since it resolves to my employer, which I think should be private.) I guess I can always just not post from work, but small distractions now and then are nice.
Toby: IP addresses are almost always logged by web software, even for readers; in the past, these logs were usually deleted after a while, but now storage space is so cheap that this may no longer be true. People like to have the IP address available in case of problems —spam, DoS attacks, etc—. I like having that sort of information publicly available, rather than tucked into logs that are hidden behind passwords, to prevent the devlopment of hierarchies.
But if you want to be anonymous on the web, try searching for ’web proxy’ or the like. However, Jacques's software makes a fair attempt to defeat these, since they are often used to spam. (Even in general, I don't know how well they work, and ultimately they become the people with the secret information.)
Toby: I see that Urs managed to post from ’from bogus address’ today (June 27). Maybe we should ask him what he did differently!
Eric: I don’t mind if administrators can see my IP for security reasons, but it is not clear what purpose it serves to actually display it publicly for all to see. For example, I can see the IP addresses of people who comment on my blog, but it is not displayed for everyone to see.
Toby: That creates a hierarchy (of information if not power, but one leads to the other) where administrators are above everybody else. The wiki way gives the same information to everybody.
New stub copyright both about copyright attitude of the Community and as a place to collect links to interesting analysis of copyright, free literature, protection from plagiarism and similar issues. It also links to citations (zoranskoda).
I created a page emptypage. It would belong to meta category of pages but I do not want to attach even that label to it. I want it empty, I want it orphane, non-aliased and non-classified, truly minimal content and minimal sourcecode page.
With one click of the mouse I call the label of nlab:HomePage in my bookmarks bar, and then I change the URL by hand or go from HomePage to one of the links or use the search from there. If I am on slow connection, sometimes even HomePage loads longer. I think that some other users can smartly use the initial page like that. HomePage has information for newcomers, experienced users can sometimes prefer emptypage as their cleaner and leaner Lab homepage.
So emptypage is a quick way to see that the lab is up with a minimal length page and to get the basis for Lab search window or to change the URL without the cost of the HomePage load and HomePage html display time. Now with HomePage having also an additional Terms of usage section it grown today another bit more, so a reason more to create emptypage and to hopefully leave it empty.
I use emptypage to have it easier to type than empty page.
I hope other people won’t find it offending that I created a lean-expert-user depart point without consulting others, but I think it has obvious usages for some and it is not on the way to others, I hope.
I’ve been thinking a lot about degeneration of Hodge to de Rham spectral sequence lately. I checked out the page on the nlab about it. I saw that there was a link to Cartier operator but no page, so I created it.
This actually got me thinking. In some sense degeneration at is “intrinsic” to the derived category (I just made that up based on what I wrote in the article). There is a naive way to try to prove that if and are derived equivalent and if the SS degenerates for one, the other should too. I couldn’t see a way to make it work. Is there an obvious reason this should be true, or an obvious counterexample?
New stubs noncommutative projective geometry and Michael Artin.
An anonymous correspondent has put a question on lax functor, or rather has edited a previous query.
have split off geometric surjection/embedding factorization from the relevant entries. Maybe I find the time to spell out the proof there.
stub for B-bordism – just to record the Manifold Atlas-reference for the moment
stub for differential orientation
stub for fundamental class
As supplementary entries for sigme model I have created
New entry defining ideal of topologizing subcategory (of an abelian category), wanted at conormal bundle. It is in fact a subfunctor of the identity functor and if we evaluate it on projective generator in the case of a module category then we get the usual ideal in the corresponding ring.
created geometric transformation, just for completeness.
split of differential function complex from differential cohomology and starzed expanding it. But not done yet.
created reference-entry Three Roles of Quantum Field Theory
since I needed the link to exist, I have created a stub for Morita morphism (of Lie groupoid)s. Made Hilsum-Skandalis morphism redirect to it, for the moment.
Affinity in the context of D-modules, as defined by Alexander Beilinson is the subject of a new stub D-affinity. There is a categorical generalization in the MPI1996-53 preprint (pdf) of Lunts and Rosenberg in terms of differential monads. Many generalizations of Beilinson-Bernstein localization theorem have their intuitive explanation in a two-step reasoning. First the noncommutative algebra in question is understood as a noncommutative (or maybe categorical) resolution of singularities of a commutative object. Then the latter satisfies D-affinity and one can localize.
New entry domain globalization of functors (zoranskoda) under development. The codomain globalization is more trivial. This are questions of extending the constructions related to Beck’s comonadicity from categories to functors. Our interest with Gabi Bohm are mainly for covers by localizations with some equivariance/compatibility with respect to additional (co)monad, which are a matter of ongoing work. This compatibility is like, or some dual of the one in the definition of morphisms of Q-categories and also the compatibility of differential monads and localization, studied by Lunts and Rosenberg. The latter is related to the classical fact that the assignment of ring of regular differential operators to a commutative ring is compatible with exact localizations, in the sense that ; and also to Beilinson’s notion of D-affinity.
created Noether’s theorem – no-nonsense version
I have created stubs for the missing entries to complete this table:
The main actual content I added are, (at 2-type theory and 2-logic): pointers to Dan Licata’s thesis and to Mike’s personal wiki pages.
I’d hope that one outcome of the present Café discussion is that eventually some of these entries get equipped with some useful content.
(P.S. I would have linked to material by Mike Stay, too, but I don’t know what to link to.)
I am about to create an entry called locally algebra-ed topos in the spirit of the section for local algebras at classifying topos.
I tend to think this terminology is better than the undescriptive “structured topos”, but please let me know what you think.
I would like to amplify the following fact:
if we agree to say (which is reasonable) that
an algebra is a model of some essentially algebraic theory, hence a lex functor out of a finite-limite category;
a local algebra with respect to a coverage on the category is such a lex functor that preserves covers.
then the statement is:
I have started a new subject complex variational calculus - contents and have included it as a floating TOC into relevant entries
stub for charge
I started an important entry differential monad. According to Lunts-Rosenberg MPI 1996-53 pdf differential calculus on schemes and noncommutative schemes can be derived from the yoga of coreflective topologizing subcategories in the abelian category of quasicoherent sheaves on the scheme, like the -filtration, and -part, in the case when the topologizing subcategory is the diagonal in the sense of the smallest subcategory of the category of additive endofunctors having right adjoint which contains the identity functor – in that case we say differential filtration and differential part. The regular differential operators are the elements of the differential part of the bimodule of endomorphisms. Similarly, one can define the conormal bundle etc.
I’ve disambiguated links to cartesian category. I suggest that we avoid this term.
stub for coherent (infinity,1)-operad
Created a stubby entry for Gereon Quick, and added some more into (or fixed typos on) Daniel Isaksen, Cech homotopy,profinite homotopy theory and pro-homotopy theory.
I created an entry for Jonathan Pridham.
have added to monoidal (infinity,1)-category the definition of -monoidal -category, for an -operad
(though maybe this definition either deserves its own entry or ought to be included instead at symmetric monoidal (infnity,1)-category)
A point of information. These constructions are due to Charles Wells in this particular setting and to Jonathan Leech, (H-coextensions of monoids, vol. 1, Mem. Amer. Math. Soc, no. 157, American Mathematical Society, 1975) in the single object case, and McLane introduces the category of factorisations I think. Charlie Wells even pushes things a bit further than Baues. Hans does not seem to have known of that work. (Charles Wells, Extension theories for categories (preliminary report), (available from http://www.cwru.edu/artsci/math/wells/pub/pdf/catext.pdf), 1979. ) I have been meaning to have a go at this entry as I have written up a modern version of Wells especially in the non-Abelian case. There is a very nice interpretation of Natural System as a lax functor. (I will do this some time…. but I can make the notes available to anyone interested.)
New entry characteristic class of a structure to complement characteristic class and historical note on characteristic classes. I did not link to it from outside so far.
universally closed morphism and improvements at proper morphism
New entry proadjoint.
New stub nuclear topological vector space with redirect nuclear space. Grothendieck’s reference also at Fredholm operator.
Urs created Frechet manifold, so I created Frechet space. (We violated the naming conventions too, but I guess it's OK since we have the redirects in.)
I am trying to begin to coherently add some of the topics of part D of the Elephant into the Lab.
Currently I am creating lots of stub entries, splitting them off from existing entries if necessary, cross-link them appropriately, and then eventually add content to them.
so far I have for instance created new (mostly stub) entries for things like
I have created
and made it a disambiguation page.
I have edited the linked table of contents at Elephant, etc.
(or rather I will have in a few minutes. All my save-windows are currently stalled. Will have to restart the server.)
New entry representable fibered category.
created standard site (maybe not a great term, but since I am Labifying the Elephant). Added the theorem that every sheaf topos has a standard site of definition to site
have created Zariski site
added to overcategory the statement about lifs to adjunctions to slices, here
a bit of content at syntactic site
started indexed topos
I have touched indexed category and then started filling some first content into base topos.
I have created the following web of entries
cartesian category, cartesian functor, cartesian logic, cartesian theory
regular category, regular functor, regular logic, regular theory, regular coverage, regular topos
coherent category, coherent functor, coherent logic, coherent theory, coherent coverage, coherent topos
geometric category, geometric functor, geometric logic, geometric theory
Most of them stubs. Partly just material split off from other entries. But all with the relavent pointers to the Elephant or other literature. To be expanded.
I have created degeneration conjecture required at Dmitri Kaledin. In my memory, I never heard ofthis degeneration conjecture by precisely that name and I do not like it (there are so many degeneration conjectures in other fields, some of which I heard under that name). It is usually said the degeneration of Hodge to de Rham spectral sequence (conjecture). It has a classical analogue. I put redirect degeneration of Hodge to de Rham spectral sequence.
In differential cohomology in an (∞,1)-topos – survey, I can’t guess what ’nothing’ should be here:
The curvature characteristic forms / Chern characters in the traditional formulation of differential cohomology take values in abelian -Lie algebras and are therefore effectively nothing differential forms with values in a complex of vector spaces
This is an excerpt I wrote at logical functor:
As far as cartesian morphism there are two different universal properties in the literature, which are equivalent for Grothendieck fibered categories but not in general. In what Urs calls the “traditional definition” (but is in fact a later one) one has for every , for every , for every such that … there exist a unique da da da. This way it is spelled in Vistoli’s lectures. This is in fact the strongly cartesian property, stronger than one in Gabriel-Grothendieck SGA I.6. The usual, Grothendieck, or weak property takes for the identity, and the unique lift is of the identity at . Then a Grothendieck fibered category is the one which has cartesian lifts for all morphisms below and all targets, and cartesian morphisms are closed under composition. With the strong cartesian property one does not need to require the closedness under composition. Now a theorem says that in a Grothendieck fibered category, a morphism is strongly cartesian iff it is cartesian.
Now I have made some changes to cartesian morphism, so that the entry is aware of the two variants of the universal property, which are not equivalent in general but are equivalent for Grothendieck fibered categories.
There was also a statement there
In words: for all commuting triangles in Y and all lifts through p of its 2-horn to X, there is a unique refinement to a lift of the entire commuting triangle.
which is too vague and I am not happy with, as it does not involve the essential parameter: the morphism for which we test cartesianess. I made a hack to it, and still it is not something I happy with (I like the idea of horn mentioned, however not the lack of appropriate quantifiers/conditions etc.). It is cumbersome to talk horn. (Maybe we could skip the whole statement in this imprecise form, and just mention please note the filling of the horn in with prescribed projection in or alike). Here is the temporary hack:
In imprecise words: for all commuting triangles in (involving as above) and all lifts through of its 2-horn to (involving as above), there is a unique refinement to a lift of the entire commuting triangle.
started a reference-entry The Cauchy Problem in Classical Supergravity, but please don’t look at it before I have polished it tomorrow, when I am more awake.
Also created a stub for super-groupoid in the process. Same comment applies to that.
I have started a puny disambiguation page projection
stub for gauge boson
stub for photon
stub for gluon
stub for gaugino
Stubby little page for connection on a double category.
Zoran has created adjoint triple, I have added adjoint quadruple
I have split off formally unramified morphism from unramified morphism. Then I added the general-abstract topos theoretic characterization, by essentially copy-and-pasting the discussion from formally smooth morphism (and replacing epimorphisms by monomorphisms)
stub for formal groupoid. Just collecting references so far.
After Zoran had emphasized it for years without me ever really looking into it, now I have finally read the beginning of Kontsevich-Rosenberg’s article on “Q-categories” in more details… and was struck:
their notion of “generalized sheaves” is essentially nothing but the kind of condition that Lawvere considered in cohesive toposes . More precisely, Lawvere considered the objects for which the canonical morphism is an isomorphism. What Kontsevich-Rosenberg call generalized sheaves are those objects for which the other canonical morphism is an isomorphism: .
There are mainly two kinds of applications in Kontsevich-Rosenberg:
the original one was to find the right notion of sheaves over formal duals of non-commutative algebras. Apparently Rosenberg is fond of the insight that for a suitable cohesive presheaf topos (my words of course) the right condition is that is an iso.
Apparently (if I remember correctly what Zoran told me) Kontsevich added the observation that formal smoothness and hence infinitesimal thickening is naturally described in this context. Now that I looked through it, I realize that what they talk about in this context is really pretty much exactly what I axiomatized as infinitesimal cohesion.
So I am happy: at once now the entire 79 page article by Kontsevich-Rosenberg turns out to be a great resource of examples and applications of cohesive topos technology! Notably they shed more light on the role of those infamous extra axioms that involve the two canoical natural transformations that come with any cohesive topos.
For that reason I have now begun expanding the Lab entry Q-category that Zoran once started