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

• At principal ideal domain, I stated and proved the theorem that for modules over a pid, submodules of free modules are free (assuming the axiom of choice), and gave a couple of corollaries. This is at the head of a section on the structure theory of modules, which obviously could be expanded to treat the structure theory of finitely generated modules over a pid.

• Copying old query box here from pseudofunctor (having incorporated its content into the entry):

Tim: in specifying a pseudo functor $F$ you have to decide whether the isomorphism goes from $F(g f)$ to $F(g) F(f)$ or in the other direction. Of course they are equivalent as each will be inverse to the other. You might say that one is lax and pseudo the other op-lax and pseudo. When specifying the Grothendieck construction for such a functor, which is to be preferred?

Both are about equally represented in the literature that I have seen which gets confusing. (In other words, I’m confused!)

Toby: As you suggest, the two versions are equivalent, so in a way it doesn't make a difference. But it might be nice to settle a convention in case we need it.

Tim: I have been using (for the Menagerie) the idea that there are pseudofunctors presented in two equivalent flavours lax pseudofunctor and oplax ones.

Mike: Well, the natural comparison maps that you get in a Grothendieck fibration go in the “lax” direction $F(g) F(f) \to F(g f)$, since they are induced by the universal property of cartesian arrows. In particular, if you have a functor with “weakly cartesian” liftings that don’t compose, then it is a lax functor. Not a very strong argument, but if we just want some convention it might be a reason to pick lax. I think that making too big a deal out of the difference would be misleading, though.

• D-geometry and Riemann-Hilbert problem. In order to make more visible one of the principal directions, where the series of entries which I am writing these days is heading to.

• David Corfield and I came to start something at EPR paradox

• I created adequate subcategory. However, once I’d done so then I found it linked from dense functor and after reading that I wasn’t sure I ought to have created the original page. I did so because I wanted to record Isbell’s idea as it’s fairly relevant to categories of generalised smooth spaces - the test spaces form an adequate subcategory (or sort of do, I need to work out the details).

It seems to be old terminology (reading dense functor) so maybe a page devoted to it isn’t right. I could shift it to dense functor?

• Maybe I am looking at the wrong places: is there somewhere a discussion of examples for classes of toposes that satisfy COSHEP?

What is known about which sites induce toposes that validate COSHEP?

• Extended the entry Cohn localization now starting with the ring viewpoint. Urs: I hope you will now agree that it is justified to call it a localization of a ring $R\to \Sigma^{-1} R$.

• On some pages it is desireable to have cardinalities “$\aleph$” be provided with a link to their explanation. I have cerated a redirect-page for that purpose.

• I started creating the following tables for the entry geometry of physics. After having created them there I found that these deserve to be put into the related entries, too. So therefore I put them into their own pages now and included them in related entries via

  [[!include .... - table]]


These are the tables that I have so far:

These need a bit mor attention. But I have to quit now for the time being. Also, I am afraid I may be running here again against Mike’s preference for notation here and there.

But I am not dogmatic about this, I just created these tables as they happened to occur to me. I try to polish them later.

• at variational calculus I have started a section In terms of smooth spaces where I discuss a bit how for

$S \colon [\Sigma, X]_{\partial \Sigma} \to \mathbb{R}$

a smooth “functional”, namely a smooth map of smooth spaces, its “functional derivative” is simply the plain de Rham differential of smooth functions on smooth spaces

$\mathbf{d}S \colon [\Sigma, X]_{\partial \Sigma} \stackrel{S}{\to} \mathbb{R} \stackrel{\mathbf{d}}{\to} \Omega^1 \,.$

The notation can still be optimized. But I am running out of energy now. Has been a long day.

• (Edited.) An anonymous poster has created a page with Vesselin’s comments on MO simply copied and pasted. I don’t know what others think of this, but whether this is an appropriate use of the nLab seems open to debate. What do others think?

• I created a page Lax equation having no content so far but soon there will be some content.

• I just have met Jamie Vicary in Brussels, at QPL 2012. In his nice talk he pointed to an $n$Lab page which I didn’t know existed:

It’s about a computer algebra software that can handle KV-2-vector spaces. I have just now added some cross links.

• I have worked on the general structure of the entry locally presentable category. The previous structure was a bit erratic at times, due to the way it had grown. I have tried to collect paragraphs by topic, give them numbered environments, move theorems from the Examples-section to the Properties-section and so forth.

• Something odd has been happening at a new entry entitled exchange structure. Someone signing in as Carol entered in quite a lot of material relating to J. Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. This has just been deleted from the same IP address. This probably means nothing important but it is worth noting.

• I wanted to be able to point to practical foundations more directly than pointing to foundations and hoping that the reader would spot the paragraph on practical foundations there. So I split off an entry practical foundations. For the moment it contains nothing but the relevant material from foundations copy-and-pasted

• since I needed to point to it explicitly, not just via the entry BRST complex, I have created an entry ghost field with a paragraph of text.

I promise that one day I fill in more detailed discussion, but not right now.

• created metric jet after a mention at the Café.

• there already was a bit of case distinction at functional between the notion in functional analysis and the nonlinear notion in mapping space theory. I have edited a bit more, trying to polish a bit.

• Ross Tate has pointed out a mismatch in terminology: Kleisli objects and the Grothendieck construction (of a covariant Cat-valued functor) are both asserted to be “lax colimits”, but they are not the same kind of colimit (the 2-cells go in different directions). Thinking about this more, I have concluded that Kleisli objects are lax colimits and the Grothendieck construction is an oplax colimit. I wrote a bit about my reasoning here. But before I go changing all references to the Grothendieck construction to say “oplax colimit”, I thought I should do a sanity check — does this make sense to everyone else?

• I added to category of elements an argument for why $El$ preserves colimits.

• at additive functor there was a typo in the diagram that shows the preservation of biproducts. I have fixed it.

Also formatted a bit more.

• started complex analytic space

but I really have some basic questions on this topic, at the time of posting this I am really a layperson:

is it right that every complex analytic space is locally isomorphic to a polydisk?

So then they are all locally contractible as topological spaces. Are they also locally contractible as seen by étale homotopy? (So: do they admit covers by polydsisks such that if in the Cech-nerves of these covers all disks are sent to points, the resulting simplicial set is contractible?)

• I have added some information on the work of Henry Whitehead which is related to this topic, and referred to work of Graham Ellis, and of Higgins and I, which is relevant.

I expect I have not given the best code for all of this so someone may want to improve it in that respect.

Graham, also writes in his paper:

In view of the ease with which Whitehead's methods handle the
classifications of Olum and Jajodia, it is surprising that the
papers \cite{olum:1953} and \cite{jaj:1980} (both of which were
written after the publication of \cite{whjhc:1949}) make
respectively no use, and so little use, of \cite{whjhc:1949}.

We note here that B. Schellenberg, who was a student of Olum, has
rediscovered in \cite{sch:1973} the main classification theorems
of \cite{whjhc:1949}. The paper \cite{sch:1973} relies heavily on
earlier work of Olum.
• I came to wonder about the words “empty context” in type theory, for what is really the context of the unit type. For there is also the context of the empty type.‘ That that might also seem to be called the “empty context”.

I suppose nobody probably bothers to call the context of the empty type anything, because type theory over the empty type is the empty theory. :-)

But still, it feel terminologogically unsatisfactory. Any suggestions?

Would it not be better to speak of the unit context instead of the empty context for the context of the unit type?

Also, I keep thinking that type theory in the context of the empty type is not entirely without use. For instance it appears in the type-theoretic version of what topos-theoretically is the base change maps over

$\emptyset \to Type \to *$

and that is the codomain fibration

$\mathbf{H}_{/Type} \to \mathbf{H}$

with its strutcure as a pointed map remembered, since the point is

$* \simeq \mathbf{H}_{/\emptyset} \,.$

I don’t know yet if this is super-relevant for anything, but it seems non-irrlelevant enough not to preclude it from being speakable.

• Back in the early ’80s, Kriegl and Michor came up with a variant on the notion of “smooth manifold” that produced a cartesian closed category. Their remarks on this in A Convenient Setting … are interesting reading for putting this in context, but nonetheless I’ve been meaning to take a look at their definition for a while to see what the bones of the proposal are.

I’ve put up a basic page with just the definition at Kriegl and Michor’s cartesian closed category of manifolds. There’s more detail at A convenient setting for differential geometry and global analysis (lspace). I wasn’t sure how to split the pages; at the moment there’s not enough detail on the nlab page but I think that the nlab page shouldn’t have details on the actual paper.

• Created Dedekind completion. Probably not very satisfactory, but I lifted the main definition from Paul Taylor’s page on Dedekind cuts, so should be ok with a little tweaking.

• needed matter to point somewhere

• Edited the definition at the article axiom.