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- Discussion Type
- discussion topicfree modules over a pid
- Category Latest Changes
- Started by Todd_Trimble
- Comments 2
- Last comment by Urs
- Last Active Oct 22nd 2012

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.

- Discussion Type
- discussion topicNielsen-Schreier theorem
- Category Latest Changes
- Started by Urs
- Comments 42
- Last comment by Urs
- Last Active Oct 22nd 2012

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.

- Discussion Type
- discussion topicpseudofunctor
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Oct 22nd 2012

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.

- Discussion Type
- discussion topicCauchy filter
- Category Latest Changes
- Started by TobyBartels
- Comments 10
- Last comment by David_Corfield
- Last Active Oct 21st 2012

I separated Cauchy filter from Cauchy space.

- Discussion Type
- discussion topicD-geometry
- Category Latest Changes
- Started by zskoda
- Comments 15
- Last comment by zskoda
- Last Active Oct 20th 2012

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.

- Discussion Type
- discussion topicEPR paradox
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Oct 20th 2012

David Corfield and I came to start something at

*EPR paradox*

- Discussion Type
- discussion topicdifferential homotopy type theory
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Mike Shulman
- Last Active Oct 19th 2012

While working at

*geometry of physics*on the next chapter*Differentiation*I am naturally led back to think again about how to best expose/introduce infinitesimal cohesion. To the reader but also, eventually, to Coq.First the trivial bit, concerning terminology: I am now tending to want to call it

*differential cohesion*, and*differential cohesive homotopy type theory*. What do you think?Secondly, I have come to think that the extra right adjoint in an infinitesimally cohesive neighbourhood need not be part of the axioms (although it happens to be there for $Sh_\infty(CartSp) \hookrightarrow Sh_\infty(CartSp_{th})$ ).

So I am now tending to say

**Definition.**A*differential structure*on a cohesive topos is an ∞-connected and locally ∞-connected geometric embedding into another cohesive topos.And that’s it. This induces a homotopy cofiber sequence

$\array{ CohesiveType &\hookrightarrow& InfThickenedCohesiveType &\to& InfinitesimalType \\ & \searrow & \downarrow & \swarrow \\ && DiscreteType }$Certainly that alone is enough axioms to say in the model of smooth cohesion all of the following:

- reduced type, infinitesimal path ∞-groupoid, de Rham space, jet bundle, D-geometry, ∞-Lie algebra (synthetically), Lie differentiation, hence “Formal Moduli Problems and DG-Lie Algebras” , formally etale morphism, formally smooth morphism, formally unramified morphism, smooth etale ∞-groupoid, hence ∞-orbifold etc.

So that seems to be plenty of justification for these axioms.

We should, I think, decide which name is best (“differential cohesion”?, “infinitesimal cohesion”?) and then get serious about the “differential cohesive homotopy type theory” or “infinitesimal cohesive homotopy type theory” or maybe just “differential homotopy type theory” respectively.

- Discussion Type
- discussion topicelectromagnetic potential
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 19th 2012

- Discussion Type
- discussion topicadequate subcategory
- Category Latest Changes
- Started by Andrew Stacey
- Comments 9
- Last comment by Andrew Stacey
- Last Active Oct 19th 2012

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?

- Discussion Type
- discussion topicYang-Baxter, r-matrices
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Oct 18th 2012

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.

- Discussion Type
- discussion topicLazard's criterion
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Oct 18th 2012

I gave this its own page, in order to have a convenient way to point to it:

*Lazard’s criterion*.

- Discussion Type
- discussion topicCOSHEP
- Category Latest Changes
- Started by Urs
- Comments 34
- Last comment by TobyBartels
- Last Active Oct 18th 2012

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?

- Discussion Type
- discussion topicCohn localization
- Category Latest Changes
- Started by zskoda
- Comments 5
- Last comment by zskoda
- Last Active Oct 17th 2012

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 $R\to \Sigma^{-1} R$.

- Discussion Type
- discussion topiclocalization of a module
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by zskoda
- Last Active Oct 17th 2012

created

*localization of a module*

- Discussion Type
- discussion topicfinite abelian group
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 17th 2012

created

*finite abelian group*and added statements of some basic facts

- Discussion Type
- discussion topicℵ
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by TobyBartels
- Last Active Oct 17th 2012

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

- Discussion Type
- discussion topicnatural deduction tables
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by TobyBartels
- Last Active Oct 17th 2012

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.

- Discussion Type
- discussion topicvariational calculus
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Oct 17th 2012

at

$S \colon [\Sigma, X]_{\partial \Sigma} \to \mathbb{R}$*variational calculus*I have started a section*In terms of smooth spaces*where I discuss a bit how fora 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.

- Discussion Type
- discussion topic"Mochizuki's proof of abc"
- Category Latest Changes
- Started by Todd_Trimble
- Comments 16
- Last comment by Mike Shulman
- Last Active Oct 16th 2012

(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?

- Discussion Type
- discussion topicidempotent complete (infinity,1)-category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 16th 2012

the entry

*idempotent complete (infinity,1)-category*was missing the actual definition. I have now added it.

- Discussion Type
- discussion topicLax equation
- Category Latest Changes
- Started by zskoda
- Comments 5
- Last comment by Mike Shulman
- Last Active Oct 16th 2012

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

- Discussion Type
- discussion topicTwoVect
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by jamievicary
- Last Active Oct 15th 2012

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.

- Discussion Type
- discussion topicLiouville integrability
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Oct 15th 2012

- Discussion Type
- discussion topicmodel topos
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 15th 2012

I needed an entry for Charles Rezk’s term

*model topos*in order to complete*locally presentable categories - table*, and so I created it. But we should have had that anyway. I have also cross-linked it with relevant entries

- Discussion Type
- discussion topiclocally presentable category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 15th 2012

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.

- Discussion Type
- discussion topicexchange structure
- Category Latest Changes
- Started by Tim_Porter
- Comments 4
- Last comment by TobyBartels
- Last Active Oct 14th 2012

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.

- Discussion Type
- discussion topicadjoint triangle theorem
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Oct 12th 2012

created adjoint triangle theorem.

- Discussion Type
- discussion topicpractical foundations
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Oct 12th 2012

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

- Discussion Type
- discussion topicghost field
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 12th 2012

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.

- Discussion Type
- discussion topicmetric jet
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Oct 12th 2012

created metric jet after a mention at the Café.

- Discussion Type
- discussion topicfunctional
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 11th 2012

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.

- Discussion Type
- discussion topiclax vs oplax colimits
- Category Latest Changes
- Started by Mike Shulman
- Comments 4
- Last comment by Todd_Trimble
- Last Active Oct 11th 2012

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?

- Discussion Type
- discussion topicBPTS instanton
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 10th 2012

created

*BPTS instanton*

- Discussion Type
- discussion topichafnian
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Oct 9th 2012

- Discussion Type
- discussion topicTor
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Oct 8th 2012

added a basic remark on the

*Relation to torsion groups*to*Tor*.

- Discussion Type
- discussion topiclocally free module
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Todd_Trimble
- Last Active Oct 8th 2012

created

*locally free module*

- Discussion Type
- discussion topiccategory of modules
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 8th 2012

added some basic paragraphs on

*The closed monoidal structure on RMod*to*Mod*.

- Discussion Type
- discussion topictensor product of modules
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Oct 8th 2012

split off

*tensor product of modules*from*tensor product*

- Discussion Type
- discussion topiccoherence theorem for monoidal bicategories
- Category Latest Changes
- Started by Mike Shulman
- Comments 2
- Last comment by Todd_Trimble
- Last Active Oct 7th 2012

- Discussion Type
- discussion topiccategory of elements
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Oct 6th 2012

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

- Discussion Type
- discussion topiccalculus of constructions
- Category Latest Changes
- Started by Urs
- Comments 21
- Last comment by Mike Shulman
- Last Active Oct 6th 2012

I have added the following paragraph to

*calculus of constructions*, I’d be grateful if experts could briefly give me a sanity check that this is an accurate characterization:More in detail, the

*Calculus of (co)Inductive Constructions*isa system of natural deduction with dependent types;

with the natural-deduction rules for dependent product types specified;

and with a rule for how to introduce new such natural-deduction rules for arbitrary (co)inductive types.

and with a type of types (hierarchy).

- Discussion Type
- discussion topiccoherence for bicategories with finite bilimits
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Oct 5th 2012

- Discussion Type
- discussion topicadditive functor
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Oct 4th 2012

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.

- Discussion Type
- discussion topiccomplex analytic space
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Oct 3rd 2012

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?)

- Discussion Type
- discussion topicpullback of differential forms
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Oct 3rd 2012

quick entry for

*pullback of differential forms*, to be further expanded

- Discussion Type
- discussion topicHilbert's theorem 90
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 2nd 2012

I copied (not: moved) the last material that Todd had added to

*projective resolution*to create an entry*Hilbert’s theorem 90*

- Discussion Type
- discussion topicexistential quantifier
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 2nd 2012

I have tried to brush-up

*existential quantifier*a little more. But not really happy with it yet.

- Discussion Type
- discussion topicjudgment
- Category Latest Changes
- Started by Mike Shulman
- Comments 11
- Last comment by Mike Shulman
- Last Active Oct 1st 2012

Created judgment.

- Discussion Type
- discussion topiclocal systems and coefficients
- Category Latest Changes
- Started by ronniegpd
- Comments 4
- Last comment by ronniegpd
- Last Active Oct 1st 2012

- 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.

- Discussion Type
- discussion topicempty context
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by TobyBartels
- Last Active Oct 1st 2012

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

$* \simeq \mathbf{H}_{/\emptyset} \,.$*pointed*map remembered, since the point isI don’t know yet if this is super-relevant for anything, but it seems non-irrlelevant enough not to preclude it from being speakable.

- Discussion Type
- discussion topicKriegl and Michor's cartesian closed category of manifolds
- Category Latest Changes
- Started by Andrew Stacey
- Comments 3
- Last comment by Andrew Stacey
- Last Active Sep 28th 2012

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.

- Discussion Type
- discussion topictopological ring
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Tim_Porter
- Last Active Sep 28th 2012

- Discussion Type
- discussion topicQFT on non-commutative spacetime
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 28th 2012

*QFT on non-commutative spacetime*, for the moment just to record a review paper

- Discussion Type
- discussion topic[Dedekind completion]
- Category Latest Changes
- Started by DavidRoberts
- Comments 9
- Last comment by Urs
- Last Active Sep 28th 2012

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.

- Discussion Type
- discussion topicmatter
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by TobyBartels
- Last Active Sep 28th 2012

needed

*matter*to point somewhere

- Discussion Type
- discussion topicsheaf of meromorphic functions
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 28th 2012

Adeel Khan created

*sheaf of meromorphic functions*.(He currently has problems logging into here, that’s why I am posting this for the moment.)

- Discussion Type
- discussion topicAbsolutely continuous measures
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Sep 27th 2012

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.

- Discussion Type
- discussion topicaxiom
- Category Latest Changes
- Started by Todd_Trimble
- Comments 7
- Last comment by Todd_Trimble
- Last Active Sep 27th 2012

Edited the definition at the article axiom.

- Discussion Type
- discussion topicupper bound
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by TobyBartels
- Last Active Sep 27th 2012

- Discussion Type
- discussion topicminimal coupling
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Sep 27th 2012

as mentioned in another thread, I have created an entry

*minimal coupling*