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- Discussion Type
- discussion topicseparable functor, S-category
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Apr 13th 2011

New microstubs S-category, separable coring and finally some substantial material at separable functor at last. The monograph by Caenapeel, Militaru and Zhu listed at separable functor studies Frobenius functors and separable functors in parallel; there are relations in a number of interesting situations. Frobenius functors are those where left and right adjoint are the same (hence in particular we have adjoint n-tuple for every $n$). Separable is a notion which is about certain spliting condition. This spliting is of the kind as spliting in Galois theory, I mean the Grothendieck’s version of classical Galois theory involves separable algebras at one side of Galois equivalence.

S-category due Tomasz Brzeziński is a formalism something similar to Q-categories of Alexander Rosenberg. Tomasz studies formal smoothness and separability in the setup of abelian categories, motivated by corings, Hopf algebras and similar applications. I would guess that understanding those could be useful into better understanding the Galois theory in cohesive topos, but I do not know.

I also created Maschke’s theorem which is one of the motivations for separable functors.

- Discussion Type
- discussion topicformally smooth morphism
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 12th 2011

I have expanded the entry formally smooth morphism:

I have first of all added the general-abstract formalization by Kontsevich-Rosenberg, taking the liberty of polishing it a bit from Q-category language to genuine (cohesive) topos-theoretic language and making contact with the notion of infinitesimal cohesion .

Then I added their theorems about how the general abstract topos-theoretic definitions do reproduce the traditional explicit notions.

Except for one clause : in prop. 5.8.1 of

*Noncommutative spaces*they show that the correct notion of formal smoothness for morphisms is reproduced in the non-commutative case (via the relative Cuntz-Quillen condition). But for the commutative case I see the corresponding statement only for objects (in section 4.1) not for morphisms.Zoran, do you know if they also discuss the relative version in the commutative case? Maybe it’s trivial, I haven’t thought it through yet.

- 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 topicsupergravity Lie 6-algebra
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 11th 2011

added to supergravity Lie 6-algebra a brief discussion of how the equations of motion of 11-d supergravity encode precisely the “rheonomic” $\infty$-connections with values in the supergravity Lie 6-algebra.

- 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 topicbilinear form
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by TobyBartels
- Last Active Apr 7th 2011

it has annoyed me for a long time that bilinear form did not exist. Now it does. But not much there yet.

- 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 topicenriched bicategory
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Mike Shulman
- Last Active Apr 2nd 2011

have created enriched bicategory in order to help Alex find the appropriate page for his notes.

- Discussion Type
- discussion topicinternal category in a monoidal category
- Category Latest Changes
- Started by FinnLawler
- Comments 1
- Last comment by FinnLawler
- Last Active Apr 1st 2011

I came across the page internal category in a monoidal category, which was lacking even a definition, so I put one in.

- 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 topicnilradical
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 29th 2011

created nilradical

- 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.