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I noticed that the entry analysis is in a sad state. I now gave it an Idea-section (here), which certainly still leaves room for expansion; and I tried to clean up the very little that is listed at References – General
a bare list of references, to be !include-ed into the list of References at relevant entries (notably at logarithmic CFT and at WZW model)
maybe a question for Zoran: I am am not sure if I am identifying the correct personal webpage or similar of Fyodor Malikov. Could you help?
a bare list of references, to be !include-ed into lists of references of relevant entries (such as 2d CFT, 2d SCFT, conformal cobordism category, modular functor and maybe elsewhere)
created effects of foundations on "real" mathematics and linked to it from foundations.
This is to archive an exchange between Toby and James on the blog here. The hope would be that eventually the list of examples is expanded here.
created Homotopy Type System with an Idea-section
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 ]
adding source
Anonymous