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

• 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.

• 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.

• 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!

• 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.

• 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

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

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

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

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

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

• 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.

• 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

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

• 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…