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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• I moving the following old discussion from dg-algebra to here:

## Discussion

A previous version of this entry gave rise to the following discussion

+–{.query} Zoran, why would you not say that this is ’following the product rule from ordinary calculus’, as I wrote? Not that this can be proved like the product rule can, but it's an easy mnemonic (and a similar one works for direct sums too). —Toby

I find it very confusing for me at least. The Leibniz rule is about the coproduct in a single algebra; here one has several algebras with different differentials, not a single derivative operators, and not acting on a tensor square of a single algebra, so it is a bit far. If $A=B$ then I would be happy, but otherwise it is too general. —Zoran

You mean that if $A = B$, then the Leibniz rule is a special case of this? Then surely it is also a special case of the more general case without $A = B$? Anyway, I think that it's more an example of categorification than generalisation. —Toby

For some special algebras this is true. For example, the dual of symmetric algebra as a Hopf algebra can be identified with the infinite order formal differential operators with constant coefficients (the isomorphism is given by evaluation at zero). Thus the Leibniz rule for derivatives is indeed the dual coproduct to the product on the symmetric algebras. There are braided etc. generalizations to this, and a version for computing the coproduct on a dual of enveloping algebras. In physics the addition of momenta and angular momenta for multiparticle systems is exactly coming from this kind of coproduct. But in all these cases the operators whose product you are taking live in a representation of a single algebra. — Zoran

=–

• A coupld additions to measurable space that I've been sitting on for a while, and which I've realised that I'm not going to write more clearly anytime soon.

But someday I would like to move a lot of the discussion about various approaches to measure theory and make measurable space itself simpler, with pointers to variations.

• 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 $n$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 $n$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 $n$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 $E_1$ is “intrinsic” to the derived category $D(X)$ (I just made that up based on what I wrote in the article). There is a naive way to try to prove that if $X$ and $Y$ 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?

• An anonymous correspondent has put a question on lax functor, or rather has edited a previous query.

• 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.

• 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 $R\mapsto Diff(R)$ is compatible with exact localizations, in the sense that $S^{-1}R \mapsto S^{-1}Diff(R)$; and also to Beilinson’s notion of D-affinity.

• 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 $n$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:

• geometric theories are equivalently theories of local algebras.
• 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 $\mathbb{T}$-filtration, and $\mathbb{T}$-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.

• have added to monoidal (infinity,1)-category the definition of $\mathcal{O}$-monoidal $(\infty,1)$-category, for $\mathcal{O}$ an $\infty$-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.)

• 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.

(or rather I will have in a few minutes. All my save-windows are currently stalled. Will have to restart the server.)

• created standard site (maybe not a great term, but since I am $n$Labifying the Elephant). Added the theorem that every sheaf topos has a standard site of definition to site

• I have created the following web of entries

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 $\infty$-Lie algebras and are therefore effectively nothing differential forms with values in a complex of vector spaces

• touched string structure. Added some formal discussion, also polsihed layout and added references. But didn’t change the previous informal discussion.

• 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 $x'$, for every $h$, for every $g$ 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 $g$ the identity, and the unique lift is of the identity at $p(x_1)$. 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 $X$ with prescribed projection in $Y$ or alike). Here is the temporary hack:

In imprecise words: for all commuting triangles in $Y$ (involving $p(f)$ as above) and all lifts through $p$ of its 2-horn to $X$ (involving $f$ as above), there is a unique refinement to a lift of the entire commuting triangle.