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

• I have only now discovered that Gonzalo Reyes is (or has been) running a blog where he has posted lots of useful-looking notes.

For instance in the Physics-section he has a long series of expositions on basics of differential geometry with an eye towards general relativity in terms of synthetic differential geometry. I have added pointers to this to various related entries now.

The factorizing morphism $c \to im(f)$ is sometimes called the corestriction of $f$:

• The entry quantum state had been a bad mess with much dubious material. Where it was not dubious, it was superceded by the parallel state in AQFT and operator algebra.

For the time being I have mostly cleared this entry and added a pointer to state in AQFT and operator algebra. I think the best would be to delete the content of this entry entirely and merge the material from “state in AQFT and operator algebra” into here. But I am not energetic enough at this time of night to do so yet.

• I have split off classical state as a separate entry, which was implicit in some other entries.

• started the trinity of entries

But not done yet. So far: the basic idea in words and a pointer in each entry to the corresponding section in Zeidler’s textbook.

• Another meaning at operator, and the connection between them.

• At homomorphism, an incorrect definition was given (at least for monoids, and this was falsely claimed to generalise to the definition of functor). So I fixed this, and in the process expanded it (spelling out the inadequacy of the traditional definition for monoids) and made several examples (made explicit in the text) into redirects.

• I have created a stub type II string theory, because I needed the link. Hopefully at some point I find the time to write something substantial about the classification of critical 2d SCFTs. But not right now.

• I have added to perturbation theory and to AQFT a list of literature on perturbative constructions of local nets of observables.

This is in reply to a question Todd was asking: while the rigorous construction of non-perturbative interacting QFTs in dimension $\gt 2$ is still open, there has at least been considerable progress in grasping the perturbation theory and renormalization theory known from standard QFT textbooks in the precise context of AQFT.

This is a noteworthy step: for decades AQFT had been suffering from the lack of examples and lack of connection to the standard (albeit non-rigorous) literature.

• Urs, while it is good that spectral theorem is included into functional analysis table of contents, and it has functional analysis toc bar, I do not like that spectral theory is also included and also has this toc bar. My understanding is that spectral theory is much wider subject on the relation between the possibly categorified and possibly noncommutative function spaces (sheaf categories, noncommutative analogues) and the specifical “singular” features of those like prime ideals, like certain special objects in abelian categories, points of spectra in operator framework etc. In any case, in $n$POV, it is NOT a part of functional analysis, though some manifestations are. Like the concept of a space is not a subject of functional analysis, though some spaces are defined in the language of operator algebras. I find spectral theory on equal footing like space, “quantity” etc. Of course, the entry currently does not reflect this much (though it has a section on spectra in algebraic geometry), but it eventually will! Thus I will remove it from functional analysis contents.

One should also point out that using generators in the proof of Giraud’s reconstruction theorem of a site out of a topos is a variant of spectral idea: like points form certain spaces, so the generators of various kind generate or form a category. This is behind many spectral constructions (including recent Orlov’s spectrum which is very laconic but stems from that) and reconstruction theorems and if the category corresponds to coherent sheaves over a variety than often the geometric features of the variety give certain contributions to the spectrum.

• Can someone look at Three Roles of Quantum Field Theory. There was an unsigned change there and a box that does not work. I do not know what was intended so will not try to fix it.

• New entry Dmytro Shklyarov; he seems to be now in Augusburg. Lots of interesting recent work in several subfields of our interest. I did not know where to put his 2-representations paper into 2-vector space as the bibliography is scattered there with some classification of subtopics.

• At Urs’ urging, I have created functional analysis - contents. It needs considerable extending; and I’ve yet to include it anywhere.

As hinted by the contents, I plan to move the diagram from TVS to its own page (but still include it on TVS).

• I wrote about these at measurable space, following to reference to M.O answers by Dmitri Pavlov that were already being cited.

• have a look into (the) future

• I have added some new material to Boolean algebra and to ultrafilter. In the former, I coined the term ’unbiased Boolean algebra’ for the notion which describes Boolean algebras as equivalent to finite-product-preserving functors $Fin_+ \to Set$ from the category of finite nonempty sets, and the term $k$-biased Boolean algebra to refer to the multiplicity of ways in which Boolean algebras could be considered monadic over $Set$.

In ultrafilter, I added some material which gives a number of universal descriptions of the ultrafilter monad. This is in part inspired by some discussions I’m having with Tom Leinster, who remarked recently at the categories list that the ultrafilter monad could be described as a codensity monad. All this is related to the unbiased Boolean algebras and to the remarks due to Lawvere, which were described on an earlier revision; this material has been reworked.

• created an entry (infinity,1)Toposes on the $(\infty,1)$-catgeory (or $(\infty,2)$-category) of all $(\infty,1)$-toposes.

Also split off an entry (infinity,1)-geometric morphism

• I have tried to make the page torsion look more like a disambiguation page and less like a mess. But only partially successful.

• added to locally ringed topos the characterization as algebras over the geometric theory of local rings.

I give pointers to two references that I know which say this more or less explicitly: Johnstone and Lurie. But I lost the page where Johnstone says this. I had it a minute ago, but then somebody distracted me, and now it is as if the paragraph has disappeared…

• I have split off from smooth infinity-groupoid – structures the section on concrete objects, creating a new entry concrete smooth infinity-groupoid.

Right now there is

• a proof that 0-truncated concrete smooth $\infty$-groupoids are equivalent to diffeological spaces;

• and an argument that 1-truncated concrete smooth $\infty$-groupoids are equivalent to “diffeological groupoids”: groupoids internal to diffeological spaces.

That last one may require some polishing.

I am still not exactly sure where this is headed, in that: what the deep theorems about these objects should be. For the moment the statement just is: there is a way to say “diffeological groupoid” using just very ygeneral nonsense.

But I am experimenting on this subject with Dave Carchedi and I’ll play around in the entry to see what happens.

• I have introduced a new section in nlab intitled functorial analysis.

It talks about the functor of point approach to functional analysis, using partially defined functionals.
• I thought about starting a floating toc for classifying objects and related, but then decided to subsume it into Yoneda lemma - contents. There I have now added the list of entries

and, conversely, included that toc into all these entries.

• since the link was requested somewhere, I have created a stub for n-topos

• In convenient category of topological spaces, I rewrote a little under the section on counterexamples, and I added a number of examples and references. Some of this came about through a useful exchange with Alex Simpson at MO, here.

• I got a question by email about the equivariant tubular neighbourhoods in loop spaces (as opposed to those defined using propagating flows so I figured it was time to nLabify that section of differential topology of mapping spaces. Of course, in so doing I figured out a generalisation: given a fibre bundle $E \to B$, everything compact, we consider smooth maps $E \to M$ which are constant on fibres. This is a submanifold of the space of all smooth maps $E \to M$. Assuming we can put a suitable measure on the fibres of $E$, then we can define a tubular neighbourhood of this submanifold.

Details at equivariant tubular neighbourhoods. Title may be a bit off now, but it’s that because the original case was for the fibre bundle $S^1 \to S^1$ with fibre $\mathbb{Z}_n$.

This entry is also notable because I produced it using a whole new LaTeX-to-iTeX converter. Details on the relevant thread.

• I added a reference to a paper of Connes and Rovelli (1994) and a link (in modular theory) to

where André Henriques asks about some Connes philosophy. But André quotes in explaining the background to his question, that in full generality there is a homomorphism from imaginary line into the 2-group of invertible bimodules of the given von Neumann algebra $M$, which in the presence of state lifts to the homomorphism into $Aut(M)$. I learned just the case when there is a state, and am delighted to hear that this is just a strengthening of some categorical structure which exists even more generally. If somebody is familiar or can dig more on that general case, it would be nice to have such categorical picture in the $n$Lab entry modular theory.

• you may recall (okay, probably not ;-) what I once wrote in the entry on exterior differential systems: while in the classical literature these are thought of as dg-ideals in a de Rham complex, we should think of them as sub-Lie algebroids of tangent Lie algebroids.

Since exterior differential systems over X encode and are encoded by partial differential equations on functions on X, this means that such sub-Lie algebroids are partial differential equations.

This perspective is amplified much more in the literature on D-modules: I think we can think of a D-scheme as an infinite-order analog of a Lie algebroid, which is the corresponding first-order notion. The Jet-bundle with its D-scheme structure is the infinite-order analog of the tangent Lie algebroid.

And sub-D-schemes of Jet-D-schemes are partial differential equations, this is what everyone on D-geometry tells you first.

So I think there is a nice story here.

• I have updated the reference section on BV formalism by the following:

i think the Beilinson-Drinfeld book does not treat the classical BV formalism in full generality, even if
they give a natural language to formalize this (pseudo-tensor, i.e., local operations).

I changed the corresponding references by saying they give a formalism for quantum BV on algebraic curves.
The general quantum BV formalism is being studied by Costello-Gwilliam and the formalism of chiral algebras
in higher dimension that has to be used to generalize Beilinson-Drinfeld to higher dimension is being studied
by Gaitsgory-Francis in their Chiral Koszul duality article (using infinity categorical localizations to replace model category
tools for homotopy theory, that are not directly available).

I also precised the reference to my article about this that uses the language of Beilinson-Drinfeld book and particularly
local operations, to deal with classical BV formalism for general gauge theories. Beilinson-Drinfeld only treat the
classical BRST formalism and not classical BV i think (at least not for general base manifold, only for curves).
• New entry affiliated operator of a $C^\ast$-algebra aka affiliated element. This is important for the circle of entries on algebraic QFT, as the operator algebras are formed by bounded operators, while we typically need unbounded operators like derivative operator to do quantum mechanics.

I sent a version of that entry but the $n$Lab stuck in the middle of the operation so I am not sure if I succeeded. So here is the copy:

## Motivation

Most of the applications of operator algebras stuck in the problem that (hermitean or not) unbounded operators do not form an algebra under composition (or under Jordan multiplication); while the algebras of bounded operators are insufficient as most of applications involve also unbounded operators like the partial derivative operator on $L^2(\mathbb{R}^n)$ which is proportional to the momentum operator in quantum mechanics.

## Idea

The motivational problem is typically resolved by considering an operator algebra which contains operators which properly approximate the unbounded operators as close as one wishes, and formalize this by defining the larger class of “approximable” operators by means of operator algebra itself. One way to do this is to define the affiliated elements of $C^\ast$-algebra, or the operators affiliated with the $C^\ast$-algebra. The idea is that if there is an unbounded self-adjoint operator then we can consider its spectral projections; they are bounded and if we include them into the algebra, the convergence of the spectral decomposition will supply the approximation.

## Literature

• S. L. Woronowicz, K. Napiórkowski, Operator theory in $C^\ast$-framework, Reports on Mathematical Physics 31, Issue 3 (1992), 353-371, doi, pdf
• S. L. Woronowicz, $C^\ast$-algebras generated by unbounded elements, pdf
• wikipedia affiliated operator
• I was forced to split off the section on infinitesimal cohesion from the entry cohesive (infinity,1)-topos – because after I had expanded it a little more, the nLab server was completely refusing to safe the entry (instead of just being absurdly slow with doing so). I guessed that it is was its length that caused the software to choke on it, and it seems I was right. The split-off subsection is now here:

cohesive (infinity,1)-topos – infinitesimal cohesion

Things I have edited:

• added a bried Idea-paragraph at the beginning;

• changed the terminology from “$\infty$-Lie algebroid” to “formally cohesive infinity-groupoid” , making the former a special case (first order) of the latter;

• expanded the definition of formal smoothness, added remarks on formal unramifiedness in the $\infty$-context.

• I’ll try to start add some actual content to the entries classical mechanics, quantum mechanics, etc. For the time being I added a simple but good definition to classical mechanics. Of course this must eventually go with more discussion to show any value. I hope to be able to use some nice lecture notes from Igir Khavkine for this eventually.

For the time being, notice there was this old discussion box, which I am herby mving to the forum here:

+–{.query} Edit: I changed the above text, incorporating a part of the discussion (Zoran).

Zoran: I disagree. Classical mechanics is classical mechanics of anything: point particles, rigid bodies (the latter I already included), infinite systems (mechanics of strings, membranes, springs, elastic media, classical fields). It includes statics, not only dynamics. The standard textbooks like Goldstein take it exactly in that generality.

One could even count the simplified beginning part of the specialized branches like aerodynamics and hydrodynamics (ideal liquids for example), which are usually studied in separate courses and which in full formulation are not just mechanical systems, as the thermodynamics also affects the dynamics. There are also mechanical models of dissipative systems, where the dissipative part is taken only phenomenologically, e.g. as friction terms. Hydrodynamics can also be considered as a part of rheology.

Toby: I take your point that ’dynamics’ was not the right word. But do you draw any distinction between ’classical mechanics’ and ’classical physics’? Conversely, what word would you use to restrict attention to particles instead of fields, if not ’mechanics’? (Incidentally, I would take point particles as possibly spinning, although I agree that I should not assume that the particle are points anyway.)

Zoran: you see, in classical mechanics you express all you have by attaching mass, position, velocity etc. to the parfts of mechanical systems. Not all classical physics belongs to this kind of description. The thermodynamical quantities may influence the motion of the systemm, but their description is out of the frame of classical mechanics. If you study liquids you have to take into account both the classical mechanics of the liquid continuum but also variations of its temperature, entropy and so on, which are not expressable within the variables of mechanics. Formally speaking of course, the thermodynamics has very similar formal structure as mechanics, for example Gibbs and Helmholtz free energies and enthalpy are like Lagrangean, the quantities which are extremized when certain theremodynamical quantities are kept constant. To answer the terminological question, there is a classical mechanics of point particles and it is called classical mechanics of point particles, there is also cm of fields and cm of rigid bodies.

Toby: So ’mechanics’ for you means ‹not taking into account thermal physics›? That's not the way that I learned it! But I admit that I do not have a slick phrase for that (any more than you have a slick phrase for ‹mechanics of point particles›), so I will try to ascertain how the term is usually used and defer to that. =–

• I wanted to test something in the Sandbox (for this question of David Roberts on the TeX Stackexchange) and it was looking a bit cluttered so I gave it a clean-out.

• I am about to create D-scheme, but currently the Lab is down and the server does not react to my login attempts…

• I am about to write something at jet bundle and elsewhere about the general abstract perspective.

In chapter 2 of Beilinson-Drinfeld’s Chiral Algebras they have the nice characterization of the Jet bundle functor as the right adjoint to the forgetful functor $F : Scheme_{\mathcal{D}}(X) \to Scheme(X)$ from D-schemes over $X$ to just schemes over $X$.

Now, since D-modules on $X$ are quasicoherent modules on the de Rham space $\Pi_{inf}(X)$, I guess we can identify

$Scheme_{\mathcal{D}}(X)$

with

$Schemes/\Pi_{inf}(X)$

and hence the forgetful functor above is the pullback functor

$\array{ F(E) &\to& E \\ \downarrow && \downarrow \\ X &\to& \Pi_{inf}(X) }$

aling the lower canonical morphism (“constant infinitesimal path inclusion”).

This would mean that we have the following nice general abstract characterization of jet bundles:

let $\mathbf{H}$ be a cohesive (infinity,1)-topos equipped with infinitesimal cohesion $\mathbf{H} \hookrightarrow \mathbf{H}_{th}$. For any $X \in \mathbf{X}$ we then have the canonical morphism $i : X \to \mathbf{\Pi}_{inf}(X)$.

The Jet bundle functor is then simply the corresponding base change geometric morphism

$Jet := (i^* \dashv i_*) : \mathbf{H}/X \to \mathbf{H}/\mathbf{\Pi}(X)$

or rather, if we forget the $\mathcal{D}$-module structure on the coherent sheaves on the jet bundle, it is the comonad $i^* i_*$ induced by that.

Does that way of saying it ring a bell with anyone?

• I am starting an entry twisted differential c-structures. This is supposed to eventually contain the general statements of which statements in the following entries are special cases:

• started a Reference entry FHT theorem with a brief rough statement of what the theorem says. For the moment mainly in order to include pointers to where in the three articles the theorem is actually hidden (I think it is hidden quite well… ;-)