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• I have started a puny disambiguation page projection

• After Zoran had emphasized it for years without me ever really looking into it, now I have finally read the beginning of Kontsevich-Rosenberg’s article on “Q-categories” in more details… and was struck:

their notion of “generalized sheaves” is essentially nothing but the kind of condition that Lawvere considered in cohesive toposes $(u_! \dashv u^* \dashv u_* \dashv u^!) : T \to S$. More precisely, Lawvere considered the objects $x$ for which the canonical morphism $u_* x \to u_! x$ is an isomorphism. What Kontsevich-Rosenberg call generalized sheaves are those objects for which the other canonical morphism is an isomorphism: $u^* x \to u^! x$.

There are mainly two kinds of applications in Kontsevich-Rosenberg:

1. the original one was to find the right notion of sheaves over formal duals of non-commutative algebras. Apparently Rosenberg is fond of the insight that for a suitable cohesive presheaf topos (my words of course) the right condition is that $u^* x \to u^! x$ is an iso.

2. Apparently (if I remember correctly what Zoran told me) Kontsevich added the observation that formal smoothness and hence infinitesimal thickening is naturally described in this context. Now that I looked through it, I realize that what they talk about in this context is really pretty much exactly what I axiomatized as infinitesimal cohesion.

So I am happy: at once now the entire 79 page article by Kontsevich-Rosenberg turns out to be a great resource of examples and applications of cohesive topos technology! Notably they shed more light on the role of those infamous extra axioms that involve the two canoical natural transformations that come with any cohesive topos.

For that reason I have now begun expanding the $n$Lab entry Q-category that Zoran once started

• This was the query in topologizing subcategory which I summarized shortly:

Mike: Where does the word ’topologizing’ come from?

Zoran Skoda: I am not completely sure anymore, but I think it is from ring theory, where people looked at the localizations at topologizing categories. There exist some topologies on various sets of ideals like Jacobson topology, so it is something of that sort in the language of subcategories instead of the language of filters of ideals. I’ll consult old references like Popescu, maybe I recall better. In any case it is pretty standard and has long history in usage: both classical and modern. No, it is not in Popescu…old related term is in fact talking about topologizing filters of ideals in a ring, so that must be the source…for example, the classical algebra by Faith, vol I, page 520 defines when the set of right ideals is topologizing. I am not good with that notion, but I can make an entry with quotation to be improved later.

• 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.

• 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…

• 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.

• 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

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

2. 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.

• 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.)

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

• added to multivector field further references on how the divergenc/BV operator is the dual of the de Rham differential.

Domenico, could you tell me if you think that the following statement is correct?

in full abstractness, the content of Lagrangian BV is this:

we

• and assume we have a fixed isomorphism between its Hochschild cohomology and Hochshcild homology, which we think of as an iso between its differential forms and its multivector fields induced by a volume form (which it is for finite dimensional spaces);

• then we think of an action functional times a volume form on our configuration space as a closed differential form $\exp(i S) vol$, hence as an element in the Hochschild homology that is also in the cyclic homology

• and then use the above isomorphism to think of this equivalently an element in Hochschild cohomology, being a cocycle in cyclic cohomology.

• the cyclic differential is the BV-operator and the closure condition is the “master equation” $\Delta \exp(i S) = 0$;

• the fact that Lagrangian BV is controled by BV-algebra and hence, by Getzler’s theorem, by algebra over the homology of the little framed disk operad now follows from the fact that Hochschild homology of our space is given by the derived loop space.

Is that right? Is that the NiceStoryAboutLagrangianBV™? If so, is this written out in this fashion explicitly somewhere?

• 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.

• wrote a few lines at differential calculus, just so that the link does point somewhere. Clearly just a stub, to be expanded.

• 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).