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

• table for inclusion as floating TOCs

• I have tried to brush up the entry dense subcategory a little

(moved the references to the References, moved the part that alluded to the application with nerves to its own section and expanded slightly, added the relevant back-links).

• I am at the Croatian black hole school organized by Jarah Evslin, and I am partially taking care of Croatia related issues (visa, trasnportation advice. communication to the owners of the housing). Lots of interesting things here about star formation, black hole formation, making massive black holes from lighter ones and so on. And some string theory mechanisms related to black hole entropy and similar issues. Most of people are postdocs and students here. Among seniors, Holger Nielsen and Mina Aganagić are present to our benefit.

By the way, started a stub black hole. Please contribute.

• A query box has been added:

I suspect there is a variant of the definition involving a transformation $R^Z_{X Y} \colon [X,Y] \to [[Y,Z],[X,Z]]$ rather than $L$. Is this correct? If so, how do these two definitions relate? Can one of them be expressed in terms of the other? Or is there a refined definition which comprises both $L$ and $R$?

• A comparatively long and technical section “From hom-functors to units and counits” (on adjoint functors) was sitting inside the Idea-section of adjunction. It seemed plainly misplaced there, and distracting attention from what should be the content of this entry, as opposed to the entry adjoint functor. So I have moved it now to where it seems to belong: inside the Examples-section.

• typo list: - closed \$ for page rendering about 60% through file - invalid mathcal(G)

ccg

• Added to T-duality a section with the discussion of the usual path-integral heuristics for why the two sigma-models on T-dual backgrounds yield equivalent quantum field theories.

• Add a note about “Daniel’s answer” to the semantics-structure question. The discussion on this page should really be merged into the main text and archived at the forum.

• updated broken link to Reyes blog

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

• am giving this its own entry in order to record some reference which used to be at equivariant cohomotopy but didn’t really belong there. Thanks to David R. for pointing to Theorem 5 in

• Tammo tom Dieck, Orbittypen und äquivariante Homologie II, Arch. Math. (Basel)26(1975), no. 6, 650–662 (pdf)

which much improves in readability over theorem 3.11 in

• Arthur Wasserman, section 3 of Equivariant differential topology, Topology Vol. 8, pp. 127-150, 1969 (pdf)
• created black holes in string theory, since somebody asked me: a brief paragraph explaining how the entropy-counting works and some references.

• I added a bit to category of simplices, including the fact that the category of nondegenerate simplices is final and thus colimits can be computed using only that, and that the nerve of the category of simplices itself is colimit-preserving.

• I did not change anything, I would not like to do it without Urs’s consent and some opinion. The entry AQFT equates algebraic QFT and axiomatic QFT. In the traditional circle, algebraic quantum field theory meant being based on local nets – local approach of Haag and Araki. This is what the entry now describes. The Weightman axioms are somewhat different, they are based on fields belonging some spaces of distributions, and 30 years ago it was called field axiomatics, unlike the algebraic axiomatics. But these differences are not that important for the main entry on AQFT. What is a bigger drawback is that the third approach to axiomatic QFT if very different and was very strong few decades ago and still has some followers. That is the S-matrix axiomatics which does not believe in physical existence of observables at finite distance, but only in the asymptotic values given by the S-matrix. The first such axiomatics was due Bogoliubov, I think. (Of course he later worked on other approaches, especially on Wightman’s. Both the Wightman’s and Bogoliubov’s formalisms are earlier than the algebraic QFT.)

I would like to say that axiomatic QFT has 3 groups of approaches, and especially to distinguish S-matrix axiomatics from the “algebraic QFT”. Is this disputable ?

• Added material on diagonal maps and the product functor, mentioning for instance the fact that the product functor is right adjoint to a diagonal functor.

• Added the contents of the canonical isomorphism induced by some non-canonical isomorphism as coming from Lack’s proof.

lelf

• in order to have a good place to record the diagram:

$\array{ ( q_1, q_2 ) &\mapsto& (x \mapsto q_1 \cdot x \cdot \overline{q}_2) \\ Sp(1) \times Sp(1) &\overset{\simeq}{\longrightarrow}& Spin(4) \\ \big\downarrow && \big\downarrow \\ Sp(1)\cdot Sp(1) &\overset{\simeq}{\longrightarrow}& SO(4) }$
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• I have expanded a bit at bilinear map, trying to add a pedagigical comment on the difference to a group homomorphism $A \times B \to C$

• Added related concepts and remarks on complete metrization and metrization in approach spaces.

• Page created, but author did not leave any comments.

• am starting some minimum here. Have been trying to read up on this topic. This will likely become huge towards beginning of next year

• Page created, but author did not leave any comments.