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added to quiver a very brief remark on the Gabriel classification theorem
Added to quiver a few references, and a few comments.
Motivation was that in working on an exposition of a result of J. Power on plane digraphs I had occasion to refresh my memory on a few things and found that these additions to quiver are relevant.
Refrained from making a (perhaps too nonconstructive) footnote warning readers that in some parts of the combinatorial literature quivers/pseudographs are not formalized but only introduced verbally as an afterthough (along the lines of “oh, and and by the way, sometimes we allow our digraphs to have loops and multiple arcs, even though our subset-of-VxV-formalisation-of-digraphs-strictly-speaking-cannot-express-multiplicities”). Of course, the usual formalization with a head- and a tail-function is widespread in the combinatorial literature, too; not saying that in combinatorics people could not make head and tail of quivers too.
By the way, the title of Ringel’s nice monograph is quite a misnomer, since the “character” of his theorem is quite different from the Four-Color-Theorem, in a rather precise sense. I refrained from commenting on this in quiver, since that is not relevant. If someone likes to hear more about it, I could briefly comment on it here, but strictly speaking this is off-thread.
Added to quiver a reference to an article of William Lawvere, for three, rather independent, reasons:
(0) The article of Lawvere cited appears to be fundamental, influential and useful to be told about.
(1) Lawvere’s imaginative and unusual terminology should, it seems to me, be documented.
(2) This is a small, to my way of thinking rather clear-cut example of an justified use of footnotes: this is a piece of information which
(2.0) should be recorded
(2.1) should definitely not go into the main text, least of all into the advertised-to-be-“slick” definition.
I don’t know about 2.1, but that particular footnote seems pretty inoffensive (i.e., I won’t stand in the way, unless further arguments are produced to change my mind). This doesn’t mean I’m letting go of the whole issue of footnotes, though! :-)
Yes, that’s a nice article by Lawvere by the way.
I suppose I don’t mind that footnote. It’s not clear to me that that terminology of Lawvere is worth documenting, but I won’t object.
Looking at quiver again, though, I don’t really like the “slick definition” section, and especially the fact that it comes before the “nuts-and-bolts” definition. I would be inclined to put the “nuts-and-bolts” definition first as the “definition” section, and then later on mention that quivers are diagrams on the walking quiver.
I’m totally fine with reordering them as nuts-and-bolts followed by slick, but I’m also totally fine with keeping them both in the definition section.
Thanks for the comments.
Since the current state of the article seems acceptable, and there is more urgent work to do, I will for the time being not change anything, only briefly summarize something and make a few suggestions:
summarizing:
Mike was in #6 calling into question not only the order of the sections, but also the current state of the “slick definition” section inself. I guess that one reason is that currrently it is rather non-slick, rather long-ish. My
suggestion would be to within the slick section make more use of standard technical terms and existing knowledge and put something like Mike’s formulation
“a quiver is a diagram on the walking quiver”
front-and-center (as they say),
and only then detail them.
Suggestion:
Since personally this nuts-and-bolts metaphor, while very common, seems needlessly self-deprecating, I would suggest cutting it out, and replace it
either by something totally matter-of-fact, e.g. “Definition without the concept of category”, which is what it is (and without-the-concept-of-functor is a corollary),
or replace it by something more graceful and jocular (and usefully information-containing), namely
“Making heads and tails of the slick definition”
which can serve to add coherence (if the two functions are called something like and , that is)
Again, will not make edits to this, for the time being, it seems this entry can wait.
Nuts-and-bolts is an exceedingly common phrase. I think if anyone else felt it carried negative connotations, it would have been brought up a very long time ago. I happen to like it, and please don’t replace it.
Re 9: Thanks for response.
I happen to like it, and please don’t replace it.
I, for one, will not replace it.
I have added pointer to p. 1 of Gabriel 72 for the origin of the word “quiver”, and added the relevant quote to the beginning of the Idea-section. While I was at it, I streamlined, rearranged and slightly expanded that Idea-section a little.
We could maybe have a page concept with an attitude.
A concept with attitude is the same as some concept , but indicating that one is interested specifically in doing certain things to such .
For instance
a quiver is just a directed graph, but with the attitude that one wants to study its quiver representations;
a presheaf is just a contravariant functor, but with the attitute that one wants to sheafify it.
There are more such examples.
The name makes me chuckle a little (thinking of a concept with a bad attitude, or some bad-ass concept striking a pose with attitude), but this or something like it, why not?
Okay, I started something here
Added a disambiguation note. I think this is justified, because quiver (editor) has become widely used enough that it is frequently referred to without disambiguation in the context of writing LaTeX, but feel free to remove if you think this is not reasonable.
I’m going to delete this:
“While therefore, as concepts in themselves, quiver and directed graph are just the same, using the term quiver serves to indicate certain constructions that one is interested in (a concept with an attitude)…”
since among graph theorists “directed graph” is commonly used to mean a set with a subset of ordered pairs such that . See, for example, Wikipedia. Someone reading the above passage (Eric Forgy) just concluded that “quiver” is just a funny word for this sort of directed graph, missing the later warning that no, it’s not.
I’ve tried to improve this discussion a bit.
The valid point of the line you deleted was that “quiver” is just not a term used for graphs in graph theory, no matter the adjectives; it’s a term in representation theory strictly indicating the intention (“attitude”) to study quiver representations.
I have re-phrased again (here) and added this quote from Derksen & Weyman 2005, ftn 1:
The underlying motivations of quiver theory are quite different from those in the traditional graph theory. To emphasize this distinction, it is common in our context to use the word “quivers” instead of “graphs”.
(Incidentally, this footnote sits on the lead-in sentence which asserts that “A quiver is just a directed graph.” :-)
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