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starting a page on Poisson/commutator brackets of flux observables in (higher) gauge theory.
The title of the entry follows the title of Freed, Moore & Segal 2007a because that’s a good succinct description of the subject matter, but I don’t mean the entry to be restricted to their particular perspective (in fact, is their uncertainty relation not ultimately a definition – their Def. 1.29 – rather than a derivation from first principles?)
The most insightful discussion of the matter that I have seen so far is that in Cattaneo & Perez 2017, which is motivated by application to first-order formulation of gravity (where this has found a lot of attention), but I think the arguments apply verbatim to Yang-Mills theory, too (where however I haven’t seen it find any attention yet(?)).
Mike Stay kindly added the standard QM story to path integral.
I changed the section titles a bit and added the reference to the Baer-Pfaeffle article on the QM path integral. Probably the best reference there is on this matter.
I created effect algebra.
I have added to string theory a new section Critical strings and quantum anomalies.
Really I was beginning to work on a new entry twisted spin^c structure (not done yet) and then I found that a summary discussion along the above lines had been missing.
Added to Lagrangian correspondence after the Definition a remark on how Lagrangian correspondence are correspondences in the slice topos of smooth spaces over the moduli space of closed differential 2-forms.
I pasted in something Mike wrote on sketches and accessible models to sketch. But now it needs tidying up, and I’m wondering if it might have been better placed at accessible category. Alternatively we start a new page on sketch-theoretic model theory. Ideas?
Create a page for this theorem (mostly copied the text from initial algebra of an endofunctor).
added these two pointers:
Karen Uhlenbeck, notes by Laura Fredrickson, Equations of Gauge Theory, lecture at Temple University, 2012 (pdf)
Simon Donaldson, Mathematical uses of gauge theory (pdf)
(if anyone has the date or other data for the second one, let’s add it)
An old query removed from universal enveloping algebra and archived here:
Eric: Is this a special case of universal enveloping algebra as it pertains to Lie algebras? I thought the concept of a universal enveloping algebra was more general than this. I scribbled some notes here. They are far from rigorous, but the references at the bottom of the page are certainly rigorous. I don’t remember them being confined to Lie algebras. I’m likely confused.
[Edit: Oh! I see now. From enveloping algebra you link to this page and call it enveloping algebra of a Lie algebra. Would that be a better name for this page? Or maybe universal enveloping algebra of a Lie algebra? Something to make it clear this page is specific to Lie algebras?]
Zoran: if you read the above article than you see that it distingusihes the enveloping algebra of a Lie algebra and universal enveloping algebra of a Lie algebra which is a universal one among all such. There is also an enveloping algebra of an associative algebra what is a different notion.
Also added to universal enveloping algebra, a link to a MathOverflow question What is the universal enveloping algebra which is looking for a rather general construction in a class of symmetric monoidal pseudoabelian categories. I also created a minimal literature section.
I have expanded the Idea-section at deformation quantization a little, and moved parts of the previous material there to the Properties-section.
under “Relation to the Weyl algebra” (here) it used to say (I wrote this, some time ago) without qualification that the universal envelope of the Heisenberg algeba becomes the Weyl algebra only after identifying the extra central generator with the unit in the ground field.
But this depends on convention: If the Weyl algebra is regarded as the formal deformation quantization of the given symplectic vector space, then the central element is the formal parameter of the deformation and not identified with the unit.
I have adjusted the wording and added a couple of (so far somewhat random) references whose authors regard the situation in this second sense.
The same discussion should be had at Weyl algebra and under “Examples” at universal enveloping algebra. Maybe it is worth splitting this example off as a separate page and re-!include it in these entries.
As far as I can tell, Ehrhard’s definition of comprehension requires not just that the fibers have terminal objects but that these are preserved by the reindexing functors. This is automatic if the fibration is a bifibration, as in Lawvere’s version; it’s fairly explicit in Ehrhard’s formulation, and somewhat implicit in Jacobs’ but I believe still present (his “terminal object functor” must, I think, be a fibered terminal object).
while adding to representable functor a pointer to representable morphism of stacks I noticed a leftover discussion box that had still be sitting there. So hereby I am moving that from there to here:
[ begin forwarded discussion ]
+–{+ .query} I am pretty unhappy that all entries related to limits, colimits and representable things at nlab say that the limit, colimit and representing functors are what normally in strict treatment are just the vertices of the corresponding universal construction. A representable functor is not a functor which is naturally isomorphic to Hom(-,c) but a pair of an object and such isomorphism! Similarly limit is the synonym for limiting cone (= universal cone), not just its vertex. Because if it were most of usages and theorems would not be true. For example, the notion and usage of creating limits under a functor, includes the words about the behaviour of the arrow under the functor, not only of the vertex. Definitions should be the collections of the data and one has to distinguish if the existence is really existence or in fact a part of the structure.–Zoran
Mike: I disagree (partly). First of all, a functor equipped with an isomorphism is not a representable functor, it is a represented functor, or a functor equipped with a representation. A representable functor is one that is “able” to be represented, or admits a representation.
Second, the page limit says “a limit of a diagram … is an object of equipped with morphisms to the objects for all …” (emphasis added). It doesn’t say “such that there exist” morphisms. (Prior to today, it defined a limit to be a universal cone.) It is true that one frequently speaks of “the limit” as being the vertex, but this is an abuse of language no worse than other abuses that are common and convenient throughout mathematics (e.g. “let be a group” rather than “let be a group”). If there are any definitions you find that are wrong (e.g. that say “such that there exists” rather than “equipped with”), please correct them! (Thanks to your post, I just discovered that Kan extension was wrong, and corrected it.)
Zoran Skoda I fully agree, Mike that “equipped with” is just a synonym of a “pair”. But look at entry for limit for example, and it is clear there that the limiting cone/universal cone and limit are clearly distinguished there and the term limit is used just for the vertex there. Unlike for limits where up to economy nobody doubt that it is a pair, you are right that many including the very MacLane representable take as existence, but then they really use term “representation” for the whole pair. Practical mathematicians are either sloppy in writing or really mean a pair for representable. Australians and MacLane use indeed word representation for the whole thing, but practical mathematicians (example: algebraic geometers) are not even aware of term “representation” in that sense, and I would side with them. Let us leave as it is for representable, but I do not believe I will ever use term “representation” in such a sense. For limit, colimit let us talk about pairs: I am perfectly happy with word “equipped” as you suggest.
Mike: I’m not sure what your point is about limits. The definition at the beginning very clearly uses the words “equipped with.” Later on in the page, the word “limit” is used to refer to the vertex, but this is just the common abuse of language.
Regarding representable functors, since representations are unique up to unique isomorphism when they exist, it really doesn’t matter whether “representable functor” means “functor such that there exists an isomorphism ” or “functor equipped with an isomorphism .” (As long as it doesn’t mean something stupid like “functor equipped with an object such that there exists an isomorphism .”) In the language of stuff, structure, property, we can say that the Yoneda embedding is fully faithful, so that “being representable” is really a property, rather than structure, on a functor.
[ continued in next comment ]
Recording the result from Triantafillou 82, characterizing injective/projective objects in diagrams of vector spaces over (the opposite of) the orbit category.
(The degreewise ingredients in the rational model for topological G-spaces)
I added some first statements about projective resolutions also to projective object.
stub for jet bundle
As there had been a change to the entry for Ross Street I gave it a glance. Is there a reason that the second reference is to a paper without Ross as an author?I hesitate to delete it as there may be a hidden reason. (I have edited this discussion entry to remedy the point that Todd and Urs have made below. I also edited the title of this discussion!)
created stub for symplectic groupoid, effectively just regording my blog entries on Eli Hawkins' program of geometric quantization of Poisson manifolds
Spurred by an MO discussion, I added the observation that coproduct inclusions are monic in a distributive category.
it has annoyed me for a long time that bilinear form did not exist. Now it does. But not much there yet.
I have touched the Idea-section at first-order formulation of gravity, trying to improve a little.
added a chunk of some standard basics to elliptic curve – Definition over a general ring.
Also touched/briefly created various related entries, such as Weierstrass equation, Weierstrass elliptic function, cubic curve, j-invariant etc.
added to modular form a brief paragraph with a minimum of information on modular forms As automorphic forms. Needs to be expanded.
created a brief entry IKKT matrix model to record some references. Cross-linked with string field theory, and with BFSS matrix model
notes for fermionic path integral
in the course of this also created Pfaffian, added a line to Berezinian integral and linked to everything from various places.
added brief mentioning of the equivalence with a pointer to
This is a brief description of the construction that started appearing in category-theoretic accounts of deep learning and game theory. It appeared first in Backprop As Functor (https://arxiv.org/abs/1711.10455) in a specialised form, but has slowly been generalised and became a cornerstone of approaches unifying deep learning and game theory (Towards Foundations of categorical Cybernetics, https://arxiv.org/abs/2105.06332), (Categorical Foundations of Gradient-based Learning, https://arxiv.org/abs/2103.01931).
Our group here in Glasgow is using this quite heavily, so since I couldn’t find any related constructions on the nLab I decided to add it. This is also my first submission. I’ve read the “HowTo” page, followed the instructions, and I hope everything looks okay.
There’s quite a few interesting properties of Para, and eventually I hope to add them (most notably, it’s an Para is an oplax colimit of a functor BM -> Cat, where B is the delooping of a monoidal category M).
A notable thing to mention is that I’ve added some animated GIF’s of this construction. Animating categorical concepts is something I’ve been using as a pedagogical tool quite a bit (more here https://www.brunogavranovic.com/posts/2021-03-03-Towards-Categorical-Foundations-Of-Neural-Networks.html) and it seems to be a useful tool getting the idea across with less friction. If it renders well (it seems to) and is okay with you, I might add more to the Optics section, and to the neural networks section (I’m hoping to get some time to add our results there).
Bruno Gavranović