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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(?)).

      v1, current

    • Started page on generalized symmetries, with brief description of main Idea.

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • Create a stub. I will expand the list shortly.

      v1, current

    • 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.

    • added also the complementary cartoon for D-branes in string perturbation theory (the usual picture)

      diff, v48, current

    • 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?

    • Updated the linkref weak initial algebras' toweak inital’ (a.k.a. ‘weakly initial’)

      diff, v25, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • 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 \hbar 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.

      diff, v12, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • I added few words about the derived background behind the Duflo map (all comes from a deep insight of Kontsevich, later detailed by many authors).

      diff, v5, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • Redirect and few words about Wiener integral.

      diff, v2, current

    • 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).

      diff, v14, current

    • 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 FF equipped with an isomorphism Fhom C(,c)F\cong hom_C(-,c) 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 F:DCF : D \to C … is an object limFlim F of CC equipped with morphisms to the objects F(d)F(d) for all dDd \in D…” (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 GG be a group” rather than “let (G,,e)(G,\cdot,e) 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 Fhom C(,c)F\cong hom_C(-,c)” or “functor equipped with an isomorphism Fhom C(,c)F\cong hom_C(-,c).” (As long as it doesn’t mean something stupid like “functor equipped with an object cc such that there exists an isomorphism Fhom C(,c)F\cong hom_C(-,c).”) 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 ]


    • brief category:people-entry for hyperlinking references

      v1, current

    • brief cateory:people-entry for hyperlinking references

      v1, current

    • 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)

      v1, current

    • I added some first statements about projective resolutions also to projective object.

    • Added a reference to

      • John W. Gray, Coherence for the Tensor Product of 2-Categories, and Braid Groups , pp.62-76 in Heller, Tierney (eds.), Algebra, Topology, and Category Theory , Academic Press New York 1976.

      diff, v19, current

    • Create a new page to keep record of PhD theses in category theory (with links to the documents where possible), particularly older ones that are harder to discover independently. At the moment, this is just a stub, but I plan to fill it out more when I have the chance.

      v1, current

    • I corrected the ’historical’ comment. The Cartan seminar is December 1956 and E. H. Brown’s paper was submitted in 1958 so essentially the two theories were developed in tandem.

      diff, v13, current

    • starting page on hierarchy of universes in type theory

      Anonymouse

      v1, current

    • 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!)

    • starting article on Coquand universes, aka universes à la Coquand

      Anonymouse

      v1, current

    • created stub for symplectic groupoid, effectively just regording my blog entries on Eli Hawkins' program of geometric quantization of Poisson manifolds

    • gave this entry some formatting and added links

      diff, v4, current

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

    • started adding something (the example of the Hopf fibration and some references).

      What’s a canonical reference on the Whitehead products corresponding to the Hopf fibrations? Like what is an original reference and what is a textbook account?

      diff, v11, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • Add page structure, idea section.

      I preserved most of the original page under a definition section.

      diff, v6, current

    • a bare list of references, to be !include-ed into relevant entries (as announced in the thread on “higher gauge theory” here)

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • added brief mentioning of the equivalence H grp(G)H (BG) H^{grp}_\bullet(G) \;\simeq\; H_\bullet\big( B G \big) with a pointer to

      diff, v2, current

    • 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

      v1, current

    • Asked a question there.