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- Discussion Type
- discussion topicFermat number
- Category Latest Changes
- Started by DavidRoberts
- Comments 1
- Last comment by DavidRoberts
- Last Active Apr 14th 2015

I created Fermat number and linked at prime number, making a few spots for other famous sorts of primes.

- Discussion Type
- discussion topicgeometry of physics -- principal bundles
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 9th 2015

started working on

*geometry of physics – principal bundles*. So far there is discussion of smooth principal 1-bundles in terms of homotopy pullbacks of smooth groupoids.In the course of this I fixed some formatting issues at

*category of fibrant objects*.

- Discussion Type
- discussion topicInjective hull
- Category Latest Changes
- Started by David_Corfield
- Comments 7
- Last comment by Mike Shulman
- Last Active Apr 8th 2015

- Started injective hull.

- Discussion Type
- discussion topicBorel's theorem
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by zskoda
- Last Active Apr 2nd 2015

- Discussion Type
- discussion topicsupergeometry via Aufhebung
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Urs
- Last Active Apr 2nd 2015

I keep trying to see to which degree one may nail down supergeometry via the yoga of adjoint modalities.

My the starting point (maybe there is a better one, but for the time being that’s the best I have come up with) is that the inclusion of commutative algebras into supercommutative algebras is reflective and coreflective, the reflector quotients out the ideal generated by the odd part, the coreflector picks the even subalgebra.

Passing to sheaves over the sites of formal duals to algebras, this gives an adjoint modality which (and you may not like that now, but keep in mind that it’s just notation which doesn’t reall matter) I decided to denote

$\rightrightarrows \; \dashv \; \rightsquigarrow$(The mnemonic is this: in Feynman diagrams $\rightsquigarrow$ is the symbol for the bosonic particles, so that denotes taking the bosonic subspace. Similarly a single fermion in a Feynman diagram appears as $\to$, so fermion bilinears look like $\rightrightarrows$ and that has to suffice to remind you of general even numbers of fermions.)

The details (there are not many, it’s straightforward) are at

*super formal smooth infinity-groupoid*.Now, while nice, this falls short of

*characterizing*supercommutative algebras.For instance the inclusion of commutative algebras into commutative algebras equipped with $\mathbb{Z}/n\mathbb{Z}$-grading (for any $2 \leq n \lt \infty$, with no extra signs introduced when swapping factors) works just as well. (For the inclusion into $\mathbb{Z}$-graded algebras there are also left and right adjoints but they coincide, and so I declare that this case is excluded by demanding faithfulness/non-degeneracy of the model.)

So the question is, which natural-looking further conditions could we impose on the above adjoints such as to narrow in a bit more on supercommutative algebras?

Here is one observation:

with the setup as described at

$\Re \; \dashv \; \Im \,.$*super formal smooth infinity-groupoid*we also have the reduction modality $\Re$ which, on function rings, takes away the nilpotent ideal in an algebra, and its right adjoint, the de Rham stack functor $\Im$:For both supercommutative algebras and for commutative algebras with grading we have inclusions of images of these functors

$\array{ \rightrightarrows &\dashv& \rightsquigarrow \\ \bot && \bot \\ \rightsquigarrow &\dashv& \R \\ \vee && \vee \\ \Re &\dashv& \Im }$On the other hand, for supercommutative algebras but not for commutative algebras with grading, there is also an inclusion of images diagonally

$\rightsquigarrow \Im \; \simeq \; \Im \,.$Because evaluating this via Yoneda on representables, then by adjunction this means that on these

$\Re \rightrightarrows \; \simeq \; \Re$hence that the reduced part of the even part is the reduced part of the full algebra.

But this says that the odd part of the algebra is nilpotent! This is something that is true for supercommutative algebras, but which need not be true for any old commutative algebras with grading.

It is tempting to say that this is a kind of

*Aufhebung*, though it differs from what is currently discussed in that entry in that the diagonal inclusion is along nw-se instead of ne-sw. But I think that just means the entry should be generalized.In summary, while the Aufhebung-condition here gets closer on narrowing in the axioms on supercommutative algebras, it still does not quite characterize them. So far these axioms still allow in particular also (sheaves on formal duals to) cyclically-graded algebras whose non-0 graded parts are nilpotent.

The first question seems to be: do we have further natural-looking axioms on the modalities that would narrow in further on genuine supergeometry?

But possibly a second question to consider is: might there be room to declare that the further generality allowed by the axioms is something to consider instead of to discard. If there is a nice axiomatics that characterizes something a tad more general than supergeometry, maybe that’s indication that this generalization is of interest?

Not sure yet.

- Discussion Type
- discussion topicderived algebraic geometry over E-n algebras
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by zskoda
- Last Active Apr 1st 2015

I have added the pair of references

John Francis,

*Derived algebraic geometry over $\mathcal{E}_n$-Rings*(pdf)John Francis,

*The tangent complex and Hochschild cohomology of $\mathcal{E}_n$-rings*(pdf)

to the References-section at various related entries, such as at

*derived noncommutative geometry*.(Thanks to Adeel in the MO-comments here. He watched me ask the question there on three different forums, before then giving a reply on the fourth ;-)

- Discussion Type
- discussion topicGabriel and Zisman
- Category Latest Changes
- Started by Tim_Porter
- Comments 7
- Last comment by zskoda
- Last Active Mar 31st 2015

Someone created a page Gabriel-Zisman. I have changed the name to Gabriel and Zisman and made a start on a (stub) entry.

- Discussion Type
- discussion topicMorita equivalence
- Category Latest Changes
- Started by Urs
- Comments 27
- Last comment by David_Corfield
- Last Active Mar 30th 2015

Just added a references to the entry

*Morita equivalence*. Noticed that the entry is in a woeful state.Can’t edit right now, but hereby I move some ancient query boxes that were sitting there from there to here:

— forwarded query boxes:

+–{: .query} David Roberts: More precisely, a Morita morphism is a span of Lie groupoids such that the 'source leg' has an anafunctor pseudoinverse. Anafunctors are only examples of Morita morphisms, in the sense that open covers $\coprod U_i \to M$ are examples of surjective submersions.

I’m also not sure that this should be called the folk model structure, as I don’t think it exists for groupoids internal to $Diff$. Details of the model structure are in a paper by Everaert, Kieboom and van der Linden, but seem to be tailored towards groupoids internal to categories of algebraic things (e.g semiabelian categories). I think the best one can do is a category of fibrant objects, but that is not something I’ve looked at much.

*Toby*: For me, an anafunctor involves a surjective submersion rather than an open cover, which is how that got in there. The important thing is to have equivalent hom-categories.David Roberts: I’m not what I was thinking at the time, but you are pretty much right: the definition of anafunctors depends on a choice of a subcanonical singleton Grothendieck pretopology, so it was remiss of me to demand the use of open covers :) As for the definition of Morita morphism, I now can’t remember if that referred to the span which is an arrow in the localised 2-category or the arrow in the unlocalised 2-category that is sent to an equivalence under localisation. At least for me, the terminology Morita morphism evokes the generalisation from a Morita equivalence (a span of weak equivalences) to a more general morphism in that setting.

*Toby*: I think that a Morita morphism should be a span, although now I'm not sure that this is what the text says, is it? I should check a reference and then change it.David Roberts: I’m fairly sure I’ve heard Lie groupoid people (Ping Xu springs immediately to mind) speak about a Morita morphism as being a fully faithful, essentially surjective (in the appropriate sense) internal functor, but I disagree with their usage. If this is indeed the case, we could note the terminological discrepancies =–

+– {: .query}

So is it true that there is a model category structure on algebras such that Morita equivalences of algebras are spans of acyclic fibrations with respect to that structure?

Zoran Škoda: Associative (nonunital) algebras make a semi-abelian category, ins’t it ? So one could then apply the general results of van den Linden published in TAC to get such a result, using regular epimorphism pretopology, it seems to me. It is probably known to the experts in this or another form.

=–

- Discussion Type
- discussion topicconformal structure
- Category Latest Changes
- Started by Urs
- Comments 33
- Last comment by Urs
- Last Active Mar 30th 2015

stub for conformal structure

- Discussion Type
- discussion topicCombinatorial Homotopy
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Mar 28th 2015

I fixed two grey links by creating pages for Whitehead’s two papers on Combinatorial Homotopy. If I get around to it I may add in a table of section headings with some comments. The entries are stubs at the moment.

- Discussion Type
- discussion topicdouble negation topology
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Mar 28th 2015

added to double negation topology some of the basic statements (in the section on topos theory)

- Discussion Type
- discussion topicadjoint cylinder / unity of opposites
- Category Latest Changes
- Started by Urs
- Comments 20
- Last comment by Urs
- Last Active Mar 27th 2015

I gave “adjoint cylinder” its own entry.

This is the term that Lawvere in

*Cohesive Toposes and Cantor’s “lauter Einsen”*(see the entry for links) proposes for adjoint triples that induce idempotet (co-)monads, and which he proposes to be a formalization of Hegel’s “unity of opposites”.In the entry I expand slightly on this. I hope the terminology does not come across as overblown. If it does, please give it a thought. I believe it is fun to see how this indeed formalizes quite well several of the examples from the informal literature.

I am not sure if “adjoint cylinder” is such a great term. I like “adjoint modality” better. Made that a redirect.

- Discussion Type
- discussion topicSnigdhayan Mahanta
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Mar 27th 2015

I added two links to interesting looking ArXiv papers at Snigdhayan Mahanta. I also updated the link to his webpage which was no longer valid.

- Discussion Type
- discussion topiccombinatorial map
- Category Latest Changes
- Started by Noam_Zeilberger
- Comments 6
- Last comment by Noam_Zeilberger
- Last Active Mar 26th 2015

I created a page for combinatorial map, as well as a stub for topological map.

- Discussion Type
- discussion topicOperads & Grothendieck-Teichmüller groups
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Mar 26th 2015

I have added links at Malcev completion and Benoit Fresse to his draft book: Operads & Grothendieck-Teichmüller groups. This looks interesting!

- Discussion Type
- discussion topicde Sitter gravity
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by DavidRoberts
- Last Active Mar 26th 2015

added more references to

*de Sitter gravity*. Cross-linked more with*Chern-Simons gravity*and*3d quantum gravity*.

- Discussion Type
- discussion topiccohomology of superconformal super Lie algebras
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 25th 2015

I have been adding stuff related to Green-Schwarz sigma models on super-AdS target spaces – added a list of references here – , and in the course of this touched a bunch of related entries and created a few more minimal entries.

These references show (more or less) that the relevant super Lie algebra cocycles all lift through the Inönü-Wigner contraction from the super anti-de-Sitter/superconformal group to the super Poincaré-group. What I am really after is seeing which of the cocycles lift to super Möbius space, i.e. not to the coset of the super adS/superconrormal group by a super-Lorentz subgroup, but by a parabolic conformal subgroup. This I don’t understand yet at all, but it might as well be trivial, and I wouldn’t see it at this stage.

Anyway, I have touched/created

*local and global geometry - table*,*anti de Sitter group*,*superconformal group*,*super anti de Sitter spacetime*and maybe others, too. At

*conformal group*I have added alightning statement of the key isomorphism with a pointer to the literature.

- Discussion Type
- discussion topicweakly distributive category
- Category Latest Changes
- Started by David_Corfield
- Comments 3
- Last comment by Todd_Trimble
- Last Active Mar 25th 2015

Robin Cockett started weakly distributive category. We already have linearly distributive category, so there needs to be a merger.

- Discussion Type
- discussion topicLaurent Lafforgue
- Category Latest Changes
- Started by Tim_Porter
- Comments 23
- Last comment by Urs
- Last Active Mar 24th 2015

I created a new entry on Laurent Lafforgue so as to point out his views on the importance of Olivia Caramello’s work on topos theory. I also updated her site.

- Discussion Type
- discussion topicreal irreducible spin representations -- table
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 24th 2015

I have moved the table of real spinor reps that was sitting at

*spin representation*into a entry-for-inclusion of its ownand then !included it back into the entry, and now also !included into at

*supersymmetry – Classification – Super Poincaré symmetry*.

- Discussion Type
- discussion topiclocal and global geometry - table
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Mar 20th 2015

made the following table, which had been copy-and-pasted into the relevant entries, a standalong table for automatic inclusion:

*local and global geometry - table*

- Discussion Type
- discussion topicterminology for modal types etc.
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by David_Corfield
- Last Active Mar 20th 2015

I am still after getting a more fine-grained idea of how to best and systematically attach words (notions) to the pluratily of structures encoded by adjoint pairs of modal operators.

Over in the GoogleGroup “nLab talk” Mike and I were discussing this in the thread called “modal and co-modal”, I am now moving this to here.

This issue is to a large extent just a linguistic one, and so should maybe better be ignored by readers with no tolerance for that.

Myself, I may be very slow in discussing this, as I come back to the question every now and then when something strikes me. For the moment I am just looking more closely at our examples to explore further what in their known situation is natural terminology.

Here is one random thought:

given a sharp modality, then we look for the “concrete” objects $X$ for which the unit $X \hookrightarrow \sharp X$ is a monomorphism. In a sense the anti-modal objects for which $\sharp X\simeq \ast$ are at the “opposite extreme” of these where a concrete object is all “supported on points”, a sharp-anti-modal object is maximally not supported on points.

By the way, regarding just the terminology for this special case: I have come to think that

*this*is what should be referred to by “intensive quantity” and “extensive quantity”. In modern language an extensive quantity in thermodynamics is a differential form in positive degree (e.g. the mass density 3-form on Euclidean 3-space) while an intensive quantity is a 0-form, hence a function (for instance a temperature function on Euclidean 3-space).Now the sheaf of functions $\mathbb{R}$ is indeed such that $\mathbb{R}\hookrightarrow \sharp \mathbb{R}$, while the sheaves $\mathbf{\Omega}^p$ of differential forms in positive degree are indeed such that $\sharp \mathbf{\Omega}^p \simeq \ast$. The objects $X$ with $\sharp X\simeq \ast$ are “extensive” in the literal sense that they “extend further than a single point”. This is much the “extension” by which also Grassmann’s Ausdehnungslehre refers to his forms.

A related random thought: so given any modal operator $\bigcirc$ we should probably be looking at objects $X$ such that $X \to \bigcirc X$ is a mono, or is $n$-truncated for general $n$.

For instance the “separated objects” for the constant $\ast$-modality are the mere propositions. Generally, the $n$-truncated objects are maybe more naturally thought of as being part of what the $\ast$-modality encodes. From this perspective the $n$-truncation monads on the one hand and the monads appearing in cohesion on the other play somewhat different roles in the whole system, which maybe explains some of the terminological mismatch (if that’s what it was) that Mike and I were struggling with in the earlier thread.

- Discussion Type
- discussion topicDold-Kan correspondence
- Category Latest Changes
- Started by John Baez
- Comments 8
- Last comment by Urs
- Last Active Mar 19th 2015

- I tried to
*slightly*polish the article Dold--Kan correspondence (section: Details) to reduce my confusion concerning the concept of "normalized Moore complex" for a simplicial abelian group. Before I did this work, two different definitions of this concept were given, and it wasn't made vividly clear that these two definitions are naturally isomorphic. In fact that's made clear on the page Moore complex, but I didn't think to look there at first. So, I've tried to make this page a bit more clear and self-contained.

This page could still use lots of work. There's a half-completed proof, and I suspect the notation and terminology regarding "normalized Moore complex", "alternating face complex", etc. is not completely consistent throughout the $n$Lab.

I'm also quite sure that in the*abelian*case, about 100 times as many people have heard of the alternating face complex --- except they call it the "normalized chain complex" of a simplicial abelian group. It may be okay to force people to learn about the normalized Moore complex, which has the advantage of applying to nonabelian simplicial groups. But, it's good to explain what's going on here.

- Discussion Type
- discussion topicsuper 1-brane in 3d, super 2-brane in 4d
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Mar 18th 2015

am adding entries for the Green-Schwarz super p-brane sigma models in the lowest possible dimension, for the moment mainly to record references.

At

*super 1-brane in 3d*there is now some genuine content, albeit bare minimum. At*super 2-brane in 4d*I have as yet just a stub with a single (and not very satisfactory) reference.

- Discussion Type
- discussion topicparabolic geometry
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Mar 18th 2015

stub for

*parabolic geometry*to record some references

- Discussion Type
- discussion topicsubtoposes
- Category Latest Changes
- Started by Mike Shulman
- Comments 2
- Last comment by Urs
- Last Active Mar 12th 2015

Stubs for dense subtopos, closed subtopos, open subtopos.

- Discussion Type
- discussion topicconnectology
- Category Latest Changes
- Started by Mike Shulman
- Comments 10
- Last comment by Mike Shulman
- Last Active Mar 11th 2015

Created connectology.

- Discussion Type
- discussion topicelasticity
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Mar 10th 2015

inspection of the original sources shows that since we have an entry

*cohesion*we should also have an entry*elasticity*. Created it with some minimum of pointers.

- Discussion Type
- discussion topicStatman thesis: Structural Complexity Of Proofs
- Category Latest Changes
- Started by Noam_Zeilberger
- Comments 2
- Last comment by Urs
- Last Active Mar 10th 2015

I recently obtained a copy of Richard Statman’s thesis, “Structural Complexity of Proofs”, so I created a little entry (Structural Complexity of Proofs) to describe some of its contents. For now there is just his definition of the “genus of a proof” and some references.

- Discussion Type
- discussion topicmixed Tate motive
- Category Latest Changes
- Started by adeelkh
- Comments 3
- Last comment by adeelkh
- Last Active Mar 10th 2015

new page mixed Tate motive, mostly to record some references for now

- Discussion Type
- discussion topichopfian object
- Category Latest Changes
- Started by Todd_Trimble
- Comments 1
- Last comment by Todd_Trimble
- Last Active Mar 9th 2015

Wonderful to have the nForum back. Testing with the following message.

I can’t remember everything I did at the nLab over the weekend (I guess I could look up Recently Revised), but I did add to Hopfian group, and linked to my current favored proof that epis in $Grp$ are surjections.

- Discussion Type
- discussion topicschanuel's conjecture
- Category Latest Changes
- Started by Todd_Trimble
- Comments 1
- Last comment by Todd_Trimble
- Last Active Mar 2nd 2015

Wrote a beginning for Schanuel’s conjecture.

- Discussion Type
- discussion topicSpinoza
- Category Latest Changes
- Started by trent
- Comments 6
- Last comment by trent
- Last Active Mar 1st 2015

added the following to the Baruch Spinoza and Spinoza’s System pages:

## Spinoza and Motifs

Spinoza seeks, in the vein of deep unification programs in mathematics and natural science, to find structural uniformities behind the segregative diversities analytic philosophy so prizes. Spinoza is “musical” in this search for unity and unification of all being(s), prizing a motif (as [a tragic soul mate of Spinoza’s], Grothendieck, so wisely dubbed it three hundred years later) - a single structural archetype idea, recurring throughout the theory of cognition, the theory human (and not “intentional’) action, the study of the mathematics of space-time, the nature of God and the interlocked natures of power and right - in short a motif unifying all beings and all actions in Nature.

*Everything in Its Right Place: Spinoza and Life by the Light of Nature*, Joseph Almog, pg xi

- Discussion Type
- discussion topicZeno's paradox of motion
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 27th 2015

created as stub for

*Zeno’s paradox of motion*, for the moment mainly in order to record pointers to texts that expand on its relation to the modern concept of convegence and differentiation, for which I found- Carl Benjamin Boyer,
*The history of the Calculus and its conceptual development*, Dover 1949

Added cross-pointer to this from the References-section of

*differential calculus*,*convergence*,*differentiation*and*analysis*- Carl Benjamin Boyer,

- Discussion Type
- discussion topicind-pro-object
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Feb 26th 2015

I started a new page dedicated to ind-pro-objects, more general pro-ind-objects and their iterations.

- Discussion Type
- discussion topiclax (infinity,1)-limit
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 25th 2015

created stub for

*lax (infinity,1)-limit*, so far just pointing to- David Gepner, Rune Haugseng, Thomas Nikolaus, “Lax colimits and free fibrations in ∞-categories” (arXiv:1501.02161)

Added the statement that the $\infty$-Grothendieck construction is an example briefly to (infinity,1)-Grothendieck construction. Cross-linked the corresponding piece at

*Grothendieck construction*.

- Discussion Type
- discussion topicCech cohomology
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Feb 25th 2015

I finally gave

*Cech cohomology*what should be a decent*General idea section*(I forget the history: there used to be an inadequate Idea section written long ago and then forgotten, then at some point in the past somebody rightly complained about it and I seem to remember having fixed it then, but maybe I didn’t, or not thoroughly enough, or something. Anyway.)

The whole entry still needs more love, of course. If anyone has time and energy, please feel invited.

(This here triggered by David Roberts asking about it here: https://plus.google.com/u/0/photos/117663015413546257905/albums/6013553739687182081/6013553742667883026?cfem=1&pid=6013553742667883026&oid=117663015413546257905)

- Discussion Type
- discussion topiccobordism ring
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Feb 25th 2015

stub for cobordism ring

- Discussion Type
- discussion topicTaub-NUT
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Feb 25th 2015

Someone started a page, Taub-NUT spacetime, on this with no content. I have added a link to a wikipedia page.

- Discussion Type
- discussion topiccurrent algebra
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Feb 23rd 2015

stub for current algebra

- Discussion Type
- discussion topicPoisson bracket Lie n-algebra
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Feb 23rd 2015

added to

*Poisson bracket Lie n-algebra*the two definitions we have and the statement of their equivalence.(I am about to edit at

*conserved current*and need to point to these ingredients from there)

- Discussion Type
- discussion topicdefinite globalization of WZW terms
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 19th 2015

Am starting a minimum at

*definite globalization of WZW terms*, to go along with*definite form*and*definite parameterization of WZW terms*.

- Discussion Type
- discussion topicMichael Spivak
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 17th 2015

I have added a stubby category:reference entry for Spivak’s

*Calculus on Manifolds*. Then I have allowed myself to put the quote from the preface thereThere are good reasons why the theorems should all be easy and the definitions hard.

into the entries

*theorem*and*definition*.

- Discussion Type
- discussion topiccoercion
- Category Latest Changes
- Started by David_Corfield
- Comments 14
- Last comment by Mike Shulman
- Last Active Feb 13th 2015

Thought I should put some content into coercion. I’m sure it could be much better phrased. What should be said about the kind of $c:(A \to B)$ that can be used to coerce?

- Discussion Type
- discussion topicmanifolds in differential cohesion
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Feb 12th 2015

I have fine-tuned the definition of manifolds in differential cohesion a bit more (here).

I think now a good axiomatization is like this:

Let $V$ be a differentially cohesive homotopy type equipped with a framing. Then a

$\array{ && U \\ & \swarrow && \searrow \\ V && && X }$*$V$-manifold*is an object $X$ such that there exists a*$V$-cover*, namely a correspondencesuch that both morphisms are formally étale morphisms and such that $U \to X$ is in addition an effective epimorphism.

This style of definition very naturally leads to a good concept of integrable G-structures (in differential cohesion).

What I find particularly charming is that if we take such a correspondence and “prequantize” it in the sense of prequantized Lagrangian correspondences, i.e. if we pick a differential coefficient object $\mathbf{B}\mathbb{G}_{conn}$ and complete to a correspondence in the slice

$\array{ && U \\ & \swarrow && \searrow \\ V && \swArrow_{\simeq} && X \\ & {}_{\mathllap{\mathbf{L}_{WZW}}}\searrow && \swarrow_{\mathrlap{\mathbf{L}^X_{WZW}}} \\ && \mathbf{B}\mathbb{G}_{conn} }$then this captures precisely the globalization problem of WZW terms that we have been discussing elsewhere: on the left we pick a WZW term on the model space, and completing the diagram to the right means finding a globalization of this term to $X$ that locally restricts to the canonical term, up to equivalence.

I think I have now full proof of one direction of the corresponding obstruction (details in this pdf):

**Theorem**Given $V$ a differentially cohesive $\infty$-group, $X$ a $V$-manifold, and $\mathbf{L}_{WZW}$ an equivariant WZW-term on $V$, then an obstruction to $\mathbf{L}_{WZW}^X$ to exist as above is the existence of an integrable $QuantMorph(\mathbf{L}_{WZW}^{\mathbb{D}^V})$-structure on $X$,(i.e. a lift of the structure group of the frame bundle to the quantomorphism n-group of the restriction $\mathbf{L}_{WZW}^{\mathbb{D}^V_e}$ of the WZW term to the infinitesimal neighbourhood of the neutral element in $V$, such that this lift restricts over a $V$-cover $U$ to the canonical $QuantMorph(\mathbf{L}_{WZW}^{\mathbb{D}^V_e})$-structure on $V$).

I still need to prove that this is not just a necessary but also a sufficient condition. This is harder…

- Discussion Type
- discussion topicétale theta function
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active Feb 12th 2015

Started étale theta function to record Mihnyong Kim’s MO question on it. Presumably it bears a relation to theta function.

- Discussion Type
- discussion topiccosimplicial object
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Feb 10th 2015

When somebody asked me about the sSet enrichment of cosimplicial objects, I noticed that the nLab didn’t have the pointers. So I have now split off a stub

*cosimplicial object*from*simplicial object*and added a bare minimum of pointers. No time for more at the moment.

- Discussion Type
- discussion topicmodule over a derived stack
- Category Latest Changes
- Started by adeelkh
- Comments 4
- Last comment by Urs
- Last Active Feb 5th 2015

New entry module over a derived stack (maybe this is already on some other page?), and some edits to perfect complex.

- Discussion Type
- discussion topic"Cohesive Toposes and Cantor's 'lauter Einsen'"
- Category Latest Changes
- Started by Urs
- Comments 31
- Last comment by Thomas Holder
- Last Active Feb 4th 2015

I gave Lawvere’s

*Cohesive Toposes and Cantor’s lauter Einsen*a category:reference-entry.The article makes an interesting claim: that Cantor’s original use of terminology was distorted by its editors to become what we now take to be the standard meaning in set theory. But that instead Cantor really meant “cohesive types” when saying “Menge” and used “Kardinale” really for what we call the underlying set of a cohesive type.

The article also contains the proposal that adjoint modalities capture Hegel’s “unity of opposites”.

Notice that where the English translations of

*Science of Logic*say “the One”, the original has “Das Eins”, which might just as well be translated with “The Unit”. In view of this and looking through Hegel’s piece on discreteness and repulsion, I think it is clear that Hegel’s “Einsen” is precisely Cantor’s “Einsen” as recalled by Lawvere. Namely: copies of the unit type.

- Discussion Type
- discussion topicheterotic string
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Feb 4th 2015

I have added a paragraph on the relevance of parameterized WZW terms also to the entry

*heterotic string theory*,now*Properties – General gauge backgrounds and parameterized WZW models*

- Discussion Type
- discussion topicterm model
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 3rd 2015

Just for completeness I should say that in the course of a recent discussion I had created a bare minimum at

*term model*.

- Discussion Type
- discussion topicNatural deduction
- Category Latest Changes
- Started by TobyBartels
- Comments 15
- Last comment by Thomas Holder
- Last Active Jan 29th 2015

Since the Idea section was so big, I put most of it in a Summary section instead. I also a few remarks about it along the way.

- Discussion Type
- discussion topicCrans-Gray tensor product
- Category Latest Changes
- Started by Harry Gindi
- Comments 39
- Last comment by Richard Williamson
- Last Active Jan 28th 2015

Steiner gave a beautiful description of the so-called Crans-Gray tensor product in the paper that I linked which gives a simple description of the tensor product of strict ω-categories that admit a basis with certain loop-freeness assumptions.

Among these objects are the objects of Θ, the orientals, the qbicals (defined by Crans), and a ton of other families of primitive shapes.

I’m surprised that this hasn’t gotten more coverage, especially since it’s about a trillion times easier to make sense of than Crans’s original paper, which is made obsolete by this paper, which is better in every way. It should be the standard reference.

I have added this link and a short description to Crans-Gray tensor product.

Also, not to disrespect Crans, but other mathematicians gave constructions of this tensor product before him (I believe there is an earlier construction using orientals by Street), and the nLab is the only place that calls it the Crans tensor product, so therefore, I recommend a name change to something like that “Lax Gray Tensor Product”.

It might also be instructive to show that the 2-categorical version is obtained by taking the 2-category whose 2-cells are connected 2-components and similarly that the same holds for the cartesian product of 1-categories.

- Discussion Type
- discussion topicBohr compactification
- Category Latest Changes
- Started by Todd_Trimble
- Comments 1
- Last comment by Todd_Trimble
- Last Active Jan 23rd 2015

I have been adding material to Bohr compactification.

- Discussion Type
- discussion topicn-image
- Category Latest Changes
- Started by Urs
- Comments 30
- Last comment by Mike Shulman
- Last Active Jan 23rd 2015

(…)

- Discussion Type
- discussion topicintegrability of G-structures
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jan 22nd 2015

have added a few more pointers to

*integrability of G-structures*, in particular to the classical textbook Sternberg 64 and to the Encyclopedia-of-Mathematics entry by Alekseeviskii.Then I wrote a section

*Formalization in differential cohesion*.

- Discussion Type
- discussion topicrelation between type theory and topology
- Category Latest Changes
- Started by David_Corfield
- Comments 3
- Last comment by spitters
- Last Active Jan 19th 2015

Someone just started relation between type theory and topology, referring to a talk by Martin Escardo.

- Discussion Type
- discussion topicdescent spectral sequence
- Category Latest Changes
- Started by Jon Beardsley
- Comments 1
- Last comment by Jon Beardsley
- Last Active Jan 16th 2015

I couldn’t find an open discussion on the page descent spectral sequence so I figured I’d start one and just everyone know that I added a little bit there to the introduction to try to put it into context. I’m not sure how well it fits with the general nlab POV, but thought it might be good just to have a little more there than what was there before.

- Discussion Type
- discussion topicHarmonic Analysis
- Category Latest Changes
- Started by trent
- Comments 3
- Last comment by trent
- Last Active Jan 15th 2015

added a link to the (video recorded) Nov 2014 MSRI Categorical Structures in Harmonic Analysis workshop to the harmonic analysis page.

- Discussion Type
- discussion topic[[atom]]
- Category Latest Changes
- Started by RodMcGuire
- Comments 3
- Last comment by RodMcGuire
- Last Active Jan 15th 2015

I noticed that Todd had just edited atom. I had been meaning to add something explaining the difference between “atomic” and “atomistic” so I added the following at atom#remarks_on_terminology.

“Atomic” and “Atomistic” differ for the simple example of the divisor lattice for some number $n$. The atoms in this lattice are prime numbers while it may also contain semi-atoms which are powers of primes. This lattice is atomic because any object not the bottom ($1$) is divisible by a prime. However it is not atomistic but instead uniquely semi-atomistic (every non-bottom object is the product of a unique set of semi-atoms), which is one way of stating the fundamental theorem of arithmetic, also known as the

*unique factorization theorem*.EDIT: I improved my nLab edit and reflected it above.