Not signed in (Sign In)

A discussion forum about contributions to the nLab wiki and related areas of mathematics, physics, and philosophy.

Want to take part in these discussions? Sign in if you have an account, or apply for one below

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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

If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

- Discussion Type
- discussion topicMore knot theory stuff
- Category Latest Changes
- Started by Andrew Stacey
- Comments 4
- Last comment by Andrew Stacey
- Last Active Apr 11th 2011

Expanded Vassiliev invariant, started Kontsevich integral, did a bit of reorganisation on knot theory (in particular, linking to more pages).

In case anyone’s wondering, there was a book put on the arXiv a couple of days ago touting itself as an introduction to Vassiliev invariants. I’m reading through it and taking notes as I go. I left in a bit of a rush today so the formatting of the Kontsevich integral went a bit haywire, and I made a statement on the Vassiliev invariant page that I know I didn’t say quite right.

In the arXiv book, Vassiliev invariants are introduced first using the Vassiliev skein relations, not their “proper” way (which I haven’t gotten to yet so I don’t know it). The formula looked very like a boundary map on a complex, but I think it has to be a cubical complex rather than a simplicial one. Only it isn’t the full boundary map, rather a partial boundary map (going to opposite faces), but I didn’t get it straight in my head until later. But now I think I’m going to wait until I read the bit about the

*true*definition - which I guess will be something like this - before correcting it (unless anyone gets there before me, of course).Drew a few more SVGs relevant for knots as well. The code for producing the trefoil knot is very nice now, though I say so myself!

- Discussion Type
- discussion topicPaolo Salvatore
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Apr 10th 2011

Stub for Paolo Salvatore.

- Discussion Type
- discussion topicframed little disk operad
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Apr 10th 2011

created framed little disk operad

- Discussion Type
- discussion topicGiansiracusa
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Apr 9th 2011

I created a stub for Jeff Giansiracusa.

- Discussion Type
- discussion topicconstructive Riesz representation theorem
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 8th 2011

reference for constructive proof of Riesz representation theorem

- Discussion Type
- discussion topicconstructive Gelfand duality theorem
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 8th 2011

Bas Spitters has kindly pointed out to me that the proof by Banaschewski and Mulvey of Gelfand duality is not actually constructive, as it invokes Barr’s theorem, and that he has a genuine constructive and also simpler proof with Coquand. I have added that to the refrences at constructive Gelfand duality theorem

- Discussion Type
- discussion topichomotopy colimit
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active Apr 8th 2011

I added to Quillen bifunctor as a further "application" the discussion of Bousfield-Kan type homotopy colimits.

At some point I want to collect the material on homotopy (co)limits currently scattered at Bousfield-Kan map at weighted limit and now at Quillen bifunctor into one coherent entry.

- Discussion Type
- discussion topicTuraev
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Urs
- Last Active Apr 6th 2011

I have created a (stubby) entry for Turaev. It needs more links.

- Discussion Type
- discussion topiccompact Lie algebra
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Apr 6th 2011

stub for compact Lie algebra

- Discussion Type
- discussion topicCrossed G-algebras
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Apr 5th 2011

I have started a stub on crossed G-algebras

- Discussion Type
- discussion topicendomorphism infinity-Lie algebra
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Apr 5th 2011

- Discussion Type
- discussion topiccohomology operation
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 31st 2011

created cohomology operation, just to record the two references that they are discussing curently on the ALG-TOP list

- Discussion Type
- discussion topicWrithe
- Category Latest Changes
- Started by Andrew Stacey
- Comments 1
- Last comment by Andrew Stacey
- Last Active Mar 31st 2011

Couple of minor knot changes: writhe is new, and I added the missing diagram (and some redirects) to framed link.

- Discussion Type
- discussion topicEquivalence classes
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Mar 29th 2011

Both equivalence class and partition used to redirect to equivalence relation, but neither term even appeared there. I removed the redirects and wrote equivalence class (but not partition).

By the way, I wrote this article entirely on my new phone (Android); I’m kind of getting the hang of this!

- Discussion Type
- discussion topicsuper L-oo algebra
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Mar 29th 2011

have created super L-infinity algebra

- Discussion Type
- discussion topicnoncommutative Gelfand-Naimark theorem
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active Mar 29th 2011

I need to be looking again into the subject of the Gelfand-Naimark theorem for noncommutative $C^*$-algebras $A$ regarded as commutative $C^*$-algebras in the copresheaf topos on the poset of commutative subalgebras of $A$, as described in

Heunen, Landsman, Spitters, A topos for algebraic quantum theory.

While it seems clear that something relevant is going on in these constructions, I am still trying to connect all this better to other topos-theoretic descriptions of physics that I know of.

Here is just one little observation in this direction. Not sure how far it carries.

If I understand correctly, we have in particular the following construction: for $\mathcal{H}$ a Hilbert space and $B(\mathcal{H})$ its algebra of bounded operators, let $A : \mathcal{O}(X) \to CStar$ be a local net of algebras on some Minkowski space $X$. landing (without restriction of generality) in subalgebras of $B(\mathcal{H})$.

By the internal/noncommutative Gelfand-Naimark theorem we have that each noncommutative $C^*$-algebra that $A$ assigns to an open subset corresponds bijectively to a locale internal to the topos $\mathcal{T}_{B(\mathcal{H})}$ of copresheaves on the commutative subalgebras of $B(\mathcal{H})$.

So using this, our Haag-Kastler local net becomes an internal-locale-valued presheaf

$A : \mathcal{O}(X)^{op} \to Loc(\mathcal{T}_{B(\mathcal{H})}) \,.$So over the base topos $B(\mathcal{H})$ this is a “space-valued presheaf”. we could think about generalizing this to $\infty$-presheaves, probably (though I’d need to think about if we really get there given that the locales need not come from actual spaces). The we could think about if this generalization dually corresponds indeed to the “higher order local nets” such as factorization algebras.

Just a very vague thought. Have to run now.

- Discussion Type
- discussion topicThe long exact sequence of a pair
- Category Latest Changes
- Started by domenico_fiorenza
- Comments 3
- Last comment by zskoda
- Last Active Mar 28th 2011

do we already have this in nLab? it seems that the long exact sequence in cohomology

$\cdots \to H^n(X,Y;A)\to H^n(X,A)\to H^n(Y,A) \to H^{n+1}(X,Y;A)\to \cdots$for an inclusion $Y\hookrightarrow X$ should have the following very simple and natural interpretation: for a morphism $f:Y\to X$ in a (oo,1)-topos $\mathbf{H}$ and a coefficient object $A$ together with a fixed morphism $\varphi:Y\to A$, consider the induced morphism $f^*:\mathbf{H}(X,A)\to \mathbf{H}(Y,A)$ and take its (homotopy) fiber over the point $*\stackrel{\varphi}{\to}\mathbf{H}(Y,A)$. In particular, when the coefficient object $A$ is pointed, we can consider the case where $\varphi:Y\to A$ is the distinguished point of $\mathbf{H}(Y,A)$. In this case the homotopy fiber one is considering should be denoted $\mathbf{H}(X,Y;A)$ and is the hom-space for the cohomology of the pair $(X,Y)$ with coefficients in $A$ (here one should actually make an explicit reference to the morphism $f:Y\to X$ in the notation, unless it is “clear” as in the case of the inclusion of the classical cohomology of a pair). then, for a deloopable coefficients object $A$, the long exact sequence in cohomology should immediately follow from the fiber sequence

$\array{ \mathbf{H}(X,Y;A) &\to& \mathbf{H}(X,A) \\ \downarrow && \downarrow \\ * &\to& \mathbf{H}(Y,A) }$

- Discussion Type
- discussion topicsupergravity Lie 3-algebra
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Mar 24th 2011

created supergravity Lie 3-algebra

- Discussion Type
- discussion topicsuperpoint
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 23rd 2011

created superpoint

- Discussion Type
- discussion topicGerstenhaber-Schack cohomology
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by zskoda
- Last Active Mar 23rd 2011

Tim has touched a bit entries on Drinfel’d twist and the more general bialgebra cocycle a la Shahn Majid and I have added another kind of bialgebra cocycles, namely those defining the Gerstenhaber-Schack cohomology. I added a tag gebra to this post (cogebras, bigebras etc.).

- Discussion Type
- discussion topicPhyics
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Mar 19th 2011

Can someone with more access than I have do a search and replace for Phyics. I have changed two entries to Physics (which I assume is correct :-)) but as it is not an important typo and there are five or six other occurrences a block replace is probably easy to do.

- Discussion Type
- discussion topicalgebra over a monad
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 17th 2011

I wanted to archive a pointer to Isbells

*Generic algebras*somewhere on the nLab, and now did so in algebra over a monad. But it is sitting a bit lonesomely there now by itself in the References-section…

- Discussion Type
- discussion topicvanishing cycle
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by zskoda
- Last Active Mar 17th 2011

Big stub (lots of references and links) for vanishing cycle and related microstub for Milnor fiber. Some related changes at intersection cohomology, perverse sheaf.

- Discussion Type
- discussion topicnormal operator
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Mar 16th 2011

created a stub for normal operator and noticed/remembered that Tim van Beek had once created the beginning of an entry spectral theorem that he ended with an empty section on the version for normal operators. If we are lucky he will come back some day and complete this, but it looks like he won’t. Maybe somebody else feels inspired to work on this entry.

- Discussion Type
- discussion topicaffine morphism
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Mar 16th 2011

New entry affine morphism (redirecting also affine morphism of schemes) and a related post at MathOverflow, linked there. Expansion of the material at affine scheme including few words on relative affine schemes and on fundamental theorem on morphisms of schemes.

- Discussion Type
- discussion topicDeligne conjecture
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Mar 11th 2011

have split off Deligne conjecture from Hochschild cohomology and expanded it. Various references only indicated by author name, still need to fill in publication informaiton

- Discussion Type
- discussion topicmodel structure on dg-modules
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Mar 11th 2011

I have created an entry model structure on dg-modules in order to record some references and facts.

I think using this I now have one version of the statement at derived critical locus (schreiber) that is fully precise. But I am still trying to see a better way. This is fiddly, because

contrary to what one might expect, thre is not much at all in the literature on general properties homotopy limits/colimits in dg-geometry;

and large parts of the standard toolset of homotopy theory of oo-algebras does not apply:

the fact that we are dealing with

*commutative*dg-algebras makes all Schwede-Shipley theory not applicable,the fact that we are dealing with oo-algebras in chain complexes makes all Berger-Moerdijk theory not apply;

and finally the fact that we are dealing with dg-algebras under another dg-algebra makes Hinich’s theory not apply!

That doesn’t leave many tools to fall back to.

- Discussion Type
- discussion topicEhlers
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Mar 10th 2011

I have created an entry on Phil Ehlers since Stephen Gaito has kindly scanned Phil’s MSc Thesis from 1991. (Phil’s PhD thesis was already on the Lab. The MSc is also there now.)

- Discussion Type
- discussion topicNotes of Utrecht talk
- Category Latest Changes
- Started by domenico_fiorenza
- Comments 25
- Last comment by Tim_Porter
- Last Active Mar 8th 2011

I’ve started writing the notes of the talk I’ll be giving in Utrecht next week. They are here

- Discussion Type
- discussion topicseminar on derived critical loci
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by zskoda
- Last Active Mar 7th 2011

with only about a month delay, I am starting now on my personal web a page

*Seminar on derived critical loci (schreiber)*

- Discussion Type
- discussion topicKoszul-Tate resolution
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 3rd 2011

stub for Koszul-Tate resolution

- Discussion Type
- discussion topicvariational calculus on diffeological spaces
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active Mar 3rd 2011

I have added to variational calculus a definition of critical loci of functionals, hence a definition of Euler-Lagrange equations, in terms of diffeological spaces. It’s a very natural definition which is

*almost*explicit in Patrick Iglesias-Zemmour’s book, only that he cannot make it fully explicit since the natural formulation involves the sheaf of forms $\Omega^1_{cl}(-)$ which is not concrete and hence not considered in that book.I was hoping I would find in his book the proof that the critical locus of a function on a diffeological space defined this was coincides with the “EL-locus” – it certainly contains it, but maybe there is some discussion necessary to show that it is not any larger – but on second reading it seems to me that the book also only observes the inclusion.

- Discussion Type
- discussion topicTarski’s Two Approaches to Modal Logic
- Category Latest Changes
- Started by Tim_Porter
- Comments 7
- Last comment by Tim_Porter
- Last Active Feb 28th 2011

This is a ’latest changes’, but for the Café rather than the backroom! Can David C (or someone) fix the link that does not work to Steve Awodey’s paper (It should be http://www.andrew.cmu.edu/user/awodey/preprints/FoS4.phil.pdf).

- Discussion Type
- discussion topicStructures de Dérivabilité
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 28th 2011

created Reference-entry Structures de Dérivabilité

- Discussion Type
- discussion topicpresymplectic structure
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 26th 2011

stub for presymplectic structure (just to record a reference on presymplectic reduction)

- Discussion Type
- discussion topicquotient space
- Category Latest Changes
- Started by Mike Shulman
- Comments 2
- Last comment by TobyBartels
- Last Active Feb 24th 2011

Stub for quotient space.

- Discussion Type
- discussion topicMilnor mu-bar invariants
- Category Latest Changes
- Started by Andrew Stacey
- Comments 4
- Last comment by Tim_Porter
- Last Active Feb 22nd 2011

I wanted to understand Milnor’s paper on Link Groups, so I basically rewrote the main bits in to Milnor mu-bar invariants. (I don’t understand the difference between $\mu$-invariants and $\bar{\mu}$-invariants, but I was only working on the original paper so presumably haven’t gotten that far yet.)

I even put a TOC in so Urs will be happy!

- Discussion Type
- discussion topicLisbon meeting notes
- Category Latest Changes
- Started by Tim_Porter
- Comments 10
- Last comment by Tim_Porter
- Last Active Feb 22nd 2011

I have just added a link to the notes that I prepared for the Lisbon meeting on my personal page. I would love to have some feedback, and in particular suggestions for incorporating some more of this in the nLab. The new material also forms part of the extended version of the Menagerie (which is now topping 800 pages.)

- Discussion Type
- discussion topicString 2-group
- Category Latest Changes
- Started by Guest
- Comments 7
- Last comment by Urs
- Last Active Feb 21st 2011

- At string 2-group it is claimed that the sequence of classifying spaces ending --> BSO(n) --> BO(n) is the Whitehead tower of O(n). Also mentioned is the version for smooth infinity groupoids (so I assume it is Urs who put that there). It is certainly not true that the sequence of classifying spaces so stated is the Whitehead tower for O(n), but the details for groups considered as one-object infinity groupoids are open to interpretation, so I haven't changed anything. Just a heads up.

-David Roberts

- Discussion Type
- discussion topiclocally equi-connected space
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 21st 2011

stub for locally equi-connected space

- Discussion Type
- discussion topicimpredicativity
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Feb 19th 2011

I have added a brief note about type-theoretic polymorphism to the list of impredicative axioms at predicative mathematics.

- Discussion Type
- discussion topicstructured ring spectrum
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 18th 2011

I see there is quite a bit of room for improvement of the $n$Lab material on ring spectra.

I noticed that smash product of ring spectra still pointed to a stub entry, while in parallel we have a fairly good beginning of a genuine entry at symmetric monoidal smash product of spectra. So I blanked the former and made it redirect to the latter.

I also made structured ring spectrum a redirect to this for the moment.

- Discussion Type
- discussion topiceffects of foundations
- Category Latest Changes
- Started by DavidRoberts
- Comments 4
- Last comment by DavidRoberts
- Last Active Feb 16th 2011

At effects of foundations on “real” mathematics I’ve put in the example of Fermat’s last theorem as being potentially derivable from PA, and pointed to two articles by McLarty on this topic.

(Edit: the naive wikilink to the given page breaks, due to the ” ” pair)

- Discussion Type
- discussion topicQuiver: discussion; Directed graph: discussion
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Feb 15th 2011

I’ve removed old discussion from quiver and directed graph. They can be found at revision #20 and revision #24, respectively.

- Discussion Type
- discussion topicSegal refined Lie group cohomology as intrinsic cohomology in SmoothooGrpd
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 13th 2011

You may or may not recall the observation, recorded at Lie group cohomology, that there is a natural map from the Segal-Blanc-Brylinski refinement of Lie group cohomology to the intrinsic cohomology of Lie groups when regarded as smooth infinity-groupoids.

For a while i did not know how to see whether this natural map is an equivalence, as one would hope it is. The subtlety is that the Cech-formula that Brylinski gives for refined Lie group cohomology corresponds to making a degreewise cofibrant replacement of $\mathbf{B}G$ in $Smooth \infty Grpd$ and then taking the diagonal, and there is no reason that this diagonal is itself still cofibrant (and I don’t think it is). So while Segal-Brylinski Lie group cohomology is finer and less naive than naive Lie group cohomology, it wasn’t clear (to me) that it is fine enough and reproduces the “correct” cohomology.

So one had to argue that for certain coefficients the degreewise cofibrant resolution in $[CartSp^{op}, sSet]_{proj,loc}$ is already sufficient for computing the derived hom space. It was only yesterday that I realized that this is a corollary of the general result at function algebras on infinity-stacks once we embed smooth infinity-groupoid into synthetic differential infinity-groupoids.

So I believe I have a proof now. I have written it out in synthetic differential infinity-groupoid in the section Cohomology and principal $\infty$-bundles.

- Discussion Type
- discussion topic(infinity,n)-category with duals
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Feb 13th 2011

stub for (infinity,n)-category with duals

- Discussion Type
- discussion topichigher and derived geometry
- Category Latest Changes
- Started by Urs
- Comments 24
- Last comment by zskoda
- Last Active Feb 12th 2011

in reply to Jim's question over on the blog, I was looking for a free spot on the nLab where I could write some general motivating remarks on the point of "derived geometry".

I then noticed that the entry higher geometry had been effectively empty. So I wrote there now an "Idea"-section and then another section specifically devoted to the idea of derived geometry.

(@Zoran: in similar previous cases we used to have a quarrel afterwards on to which extent Lurie's perspective incorporates or not other people's approaches. I tied to comment on that and make it clear as far as I understand it, but please feel free to add more of a different point of view.)

- Discussion Type
- discussion topiccategory of cobordisms
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 12th 2011

I hadd added a little bit of this and that to category of cobordisms earlier today in a prolonged coffee break.

This was in reaction to learning about the work by Ayala, now referenced there, whou considers categories of cobordisms equipped with

*geometric structure*given by morphisms into an $\infty$-stack $\mathcal{F}$.

- Discussion Type
- discussion topicBord(X) as free symmetric monoidal on Pi(X)
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by domenico_fiorenza
- Last Active Feb 11th 2011

A while back I had a discussion here with Domenico on how the framed cobordim $(\infty,n)$-category $Bord^{fr}_n(X)$ of cobordisns

*in*a topological space $X$ should be essentially the free symmetric monoidal $(\infty,n)$-category on the fundamental $\infty$-groupoid of $X$.This can be read as saying

Every flat $\infty$-parallel transport of fully dualizable objects has a unique $\infty$-holonomy.

(!)

Some helpful discussion with Chris Schommer-Pries tonight revealed that this is (unsurprisingly) already a special case of what Jacob Lurie proves. He proves it in more generality, which makes the statement easy to miss on casual reading. So I made it explicit now at cobordism hypothesis in the new section For cobordisms in a manifold.

- Discussion Type
- discussion topicmonoidal Dold-Kan correspondence
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by Urs
- Last Active Feb 8th 2011

the invaluable Denis-Charles Cisinski provided a useful reference with a bit on cosimplicial algebras at MO (here). I added that reference to monoidal Dold-Kan correspondence and to cosimplicial algebra.

- Discussion Type
- discussion topicde Rham theorem for smooth oo-groupoids
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by zskoda
- Last Active Feb 8th 2011

At synthetic differential infinity-groupoid I have entered statement and detailed proof that flat and infinitesimally flat real coefficients are equivalent in $SynthDiff\infty Grpd$

$\mathbf{\flat}_{inf} \mathbf{B}^n \mathbb{R} \simeq \mathbf{\flat} \mathbf{B}^n \mathbb{R} \,.$The proof proceeds by presentation of $\mathbf{\flat}_{inf} \mathbf{B}^n \mathbb{R}$ by essentially (a cofibrant resolution of) Anders Kocks’ s infinitesimal singular simplicial complex. In this presentation cohomology with coefficients in this object is manifestly computed as in de Rham space/Grothendieck descent-technology for oo-stacks.

But we also have an intrinsic notion of de Rham cohomology in cohesive $\infty$-toposes, and the above implies that in degree $n \geq 2$ this coincides with the de Rham space presentation as well as the intrinsic real cohomoloy.

All in all, this proves what Simpson-Teleman called the “de Rham theorem for $\infty$-stacks” in a note that is linked in the above entry. They consider a slightly different site of which I don’t know if it is cohesive, but apart from that their model category theoretic setup is pretty much exactly that which goes into the above proof. They don’t actually give a proof in this unpublished and sketchy note and they work (or at least speak) only in homotopy categories. But it’s all “morally the same”. For some value of “morally”.

- Discussion Type
- discussion topicformal smooth manifold
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 8th 2011

have created an entry formal smooth manifold, but without much beyond references for the moment.

- Discussion Type
- discussion topiccohomology localization
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Feb 8th 2011

quick entry for cohomology localization, but have to interrupt now

- Discussion Type
- discussion topicgroupoid of spin structures
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Feb 7th 2011

A manifold has

a set of orientations;

an xyz of topological spin structures

a 3-groupoid of topological string structures;

a 7-groupoid of topological fivebrane stuctures, etc.

and for some reason it is common in the literature (which of course is small in the last cases) to speak of these $n$-groupoids, but not so common to speak of the xyz here:

- A manifold has a
*groupoid*of spin structures.

Namely the homotopy fiber of the second Stiefel-Whitney class

$Spin(X) \to Top(X,B SO) \stackrel{(w_2)_*}{\to} Top(X, B^2 \mathbb{Z}_2) \,.$I have added one reference that explicitly discusses the groupoid of spin structures to spin structure.

Do you have further references?

- Discussion Type
- discussion topic2-out-of-3
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 7th 2011

I have split off an entry 2-out-of-3 property

- Discussion Type
- discussion topicdifferential fivebrane structures
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active Feb 6th 2011

created stub for differential fivebrane structure

sounds easy, but due to lots of software trouble that took me a good bit of the afternoon! :-(

- Discussion Type
- discussion topicchain homopy
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 6th 2011

wrote something at chain homotopy

- Discussion Type
- discussion topicline Lie n-algebra
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 4th 2011

I had created line Lie n-algebra, just for the sake of completeness and so that I know where to link to when I mention it

- Discussion Type
- discussion topicdifferential characteristic class
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 4th 2011

I have created an entry differential characteristic class.

I felt need for this as the traditional term secondary characteristic class first of all has (as discussed there) quite a bit of variance in convention of meaning in the established literature, and secondly it is unfortunately undescriptive (which is probably the reason for the first problem, I guess!).

Moreover, I felt the need for a place to discuss the concept “differential characteristic class” in the fully general abstract way in the spirit of our entry on cohomology, whereas “secondary characteristic class” has a certain association with concrete models. Some people use it almost synonymously with “Cheeger-Simons differential character”.

Anyway, so I created a new entry. So far it contains just the “unrefined” definition. I’ll try to expand on it later,

- Discussion Type
- discussion topicproblem at "essential image"
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Feb 4th 2011

I notice that the entry essential image is in a bad state:

it starts out making two statements, the first of which is then doubted by Mike in a query box, the second doubted by Zoran in a query box.

If there is really no agreement on what should go there, we should maybe better clear the entry, and discuss the matter here until we have a minimum of consensus.

But I guess the problems can easily be dealt with and somebody should try to polish this entry right away.