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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.
I discovered that there was a well-hidden entry Nielsen-Schreier theorem. I have now cross-linked it with free group, subgroup and in particular with free module, where the generalization is stated.
Copying old query box here from pseudofunctor (having incorporated its content into the entry):
Tim: in specifying a pseudo functor you have to decide whether the isomorphism goes from to 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 , 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.
I separated Cauchy filter from Cauchy space.
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.
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?
New quantum groups-related stubs: classical r-matrix, Yang-Baxter equation, quantum Yang-Baxter equation and small additions to quantum group, Vladimir Drinfel’d etc.
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 .
created localization of a module
created finite abelian group and added statements of some basic facts
On some pages it is desireable to have cardinalities “” be provided with a link to their explanation. I have cerated a redirect-page ℵ 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
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
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?
the entry idempotent complete (infinity,1)-category was missing the actual definition. I have now added it.
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 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?
created BPTS instanton
added a basic remark on the Relation to torsion groups to Tor.
created locally free module
added some basic paragraphs on The closed monoidal structure on RMod to Mod.
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 copied (not: moved) the last material that Todd had added to projective resolution to create an entry Hilbert’s theorem 90
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
and that is the codomain fibration
with its strutcure as a pointed map remembered, since the point is
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.
QFT on non-commutative spacetime, for the moment just to record a review 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
Since it appeared as a prominent grey link in integration of differential forms (and is a grey link in many other places), I wrote absolutely continuous measure.
as mentioned in another thread, I have created an entry minimal coupling
stub for divisible group
I have been adding basic propositons and their (farily) detailed proofs at injective object in the section Existence of enough injectives.
This expands on statements and proofs mentioned in other entries, notably at injective object, also at coextension of scalars (stuff added by Todd, I think).
Generally, it is often hard to decide in which entry exactly to put a theorem. Often there are several choices. Best of course to copy stuff to each relevant point or at least link to it from there.
But I am quite a bit time pressured now (and I hope that does not already show too much in what I just typed). So I won’t do any further such organization right now. But if anyone feels like looking into this, please don’t hesitate.
Created the page telescope conjecture since I noticed it was linked to by Morava K-theory but didn’t exist. Might add more later, specifically about how this is generalized to the setting of axiomatic stable homotopy categories and how it is true after localizing at , and some other spectra, but believed to be false in general.
Since I was being asked I briefly expanded automorphism infinity-group by adding the internal version and the HoTT syntax.
Mike, what’s the best type theory syntax for the definition of via -image factorization of the name of ?
I made hypothesis and conclusion entries or redirects to make deduction and induction - contents look nicer
(Gee, and I was really just editing injective module when the detour through Zorn’s lemma made me get distracted again by all this foundational business…)
added to composition a new section with trivial remarks on composition in enriched category theory.
created geometric fibre. Can someone lease check these algebraic geometry entries as that area is quite far from my safety zone! so I will get some things wrong.
Created Lévy hierarchy.
added to free module and to submodule a remark on the characterization of submodules of free modules.
thought we’d need an entry homotopy category of chain complexes
In stratified space, many of the references had page numbers given as if 123 { 234, rather than 123 - 234. This is probably a paste from somewhere else, but I was wondering how it happened so as to avoid it myself. I changed it. (Might it be a strange font?)
I have touched quasi-isomorphism, expanded the Idea-section and polished the Definition-section, added References
Recorded some facts from http://arxiv.org/abs/1101.2792 at supercompact cardinal, Vopenka’s principle, and a newly created page C(n)-extendible cardinal (with bonus stub for [[extendible cardinal]).
Urs had a framework at deduction and I put in something very brief. Also disambiguation at derivation.
I have given the section Existence of enough injectives at injective object a bit of structure. Then I started adding some similar basics to Existence of enough projectives at projective object.
Created Burali-Forti’s paradox.