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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 25th 2013

added to quiver a very brief remark on the Gabriel classification theorem

• CommentRowNumber2.
• CommentAuthorPeter Heinig
• CommentTimeJul 23rd 2017
• (edited Jul 23rd 2017)

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.

• CommentRowNumber3.
• CommentAuthorPeter Heinig
• CommentTimeJul 23rd 2017

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.

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeJul 23rd 2017

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! :-)

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeJul 23rd 2017

Yes, that’s a nice article by Lawvere by the way.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeJul 23rd 2017

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.

• CommentRowNumber7.
• CommentAuthorTodd_Trimble
• CommentTimeJul 23rd 2017

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.

• CommentRowNumber8.
• CommentAuthorPeter Heinig
• CommentTimeJul 24th 2017

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 $h$ and $t$, that is)

Again, will not make edits to this, for the time being, it seems this entry can wait.

• CommentRowNumber9.
• CommentAuthorTodd_Trimble
• CommentTimeJul 24th 2017

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.

• CommentRowNumber10.
• CommentAuthorPeter Heinig
• CommentTimeJul 24th 2017
• (edited Jul 24th 2017)

Re 9: Thanks for response.

I happen to like it, and please don’t replace it.

I, for one, will not replace it.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeOct 1st 2018

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.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeOct 1st 2018

We could maybe have a page concept with an attitude.

A concept $X$ with attitude is the same as some concept $Y$, but indicating that one is interested specifically in doing certain things to such $Y$.

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.

• CommentRowNumber13.
• CommentAuthorTodd_Trimble
• CommentTimeOct 1st 2018

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?

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeOct 1st 2018

Okay, I started something here