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A quiver in representation theory is what is called in most other areas a directed graph. Does anybody know why Gabriel felt that a new name was needed for this object? I am more interested in why he might have felt graph or digraph was not a good choice of terminology than why he thought quiver is a good name. (I rather like the name myself.)

On a related note, does anybody know why quiver representations, resp. morphisms of quiver representations, are not commonly defined as functors from the free category on the quiver to the category of finite dimensional vector spaces, resp. natural transformations?

Added I made this community wiki in case this will garner more responses.

My motivation for asking this is that one of my students just defended her thesis, which involved quivers, and the Computer Scientist on the committee remarked that these are normally called directed graphs and using that term might make the thesis appeal to a wider community. Afterwards, some of us were wondering what prompted Gabriel to coin a new term for this concept.

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    $\begingroup$ I dont know what Gabriel thought but here is an answer why the term is used from: amazon.de/Elements-Representation-Theory-Associative-Algebras/… page 42 says:"...There are two main reasons for using the term quiver rather than graph:the first one is that the former has become generally accepted by specialists;the second is that the latter is used in so many different contexts and even senses( a graph can be oriented or not,with or without multiple arrows or loops) that it may lead,for our purposes at east to certain ambiguities. $\endgroup$
    – trew
    Commented Aug 14, 2011 at 17:45
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    $\begingroup$ For the first question, here is what I was told when I was doing my thesis at Paris in the 1980's (the oral tradition is important in maths!). It seems that, at Paris in the 1960's, it was a common trend to create new names for existing objects, to emphasize a new point of view. So Gabriel introduced quivers for something that contains arrows, Bruhat and Tits introduced buildings for simplicial complexes of a special kind (they contain lots of chambers), etc... Concerning Gabriel, see ncatlab.org/nlab/show/Pierre+Gabriel $\endgroup$
    – Alain Valette
    Commented Aug 14, 2011 at 17:48
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    $\begingroup$ For a generous display of how far mathematical neologism can take you, there is the book by Gabriel and Roiter on representations of finite dimensional algebras. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Aug 14, 2011 at 19:09
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    $\begingroup$ While I do not know what Gabriel's motivations were, I think it is useful to have both quiver and directed graphs available, because they differ in intent. I have heard representation theorists talk about the directed graph underlying a quiver, for example: while this is like talking about the topological space underlying a topological space, it shows pretty clearly that the two terms are notionally different independently of being accidentally synonymous. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Aug 15, 2011 at 5:57
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    $\begingroup$ @Mariano, this leads to the interesting side question, what other forgetful functors are isomorphisms of categories? $\endgroup$
    – Benjamin Steinberg
    Commented Aug 15, 2011 at 13:23

3 Answers 3

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Gabriel actually gave a short explanation himself in [Gabriel, Peter. Unzerlegbare Darstellungen. I. (German) Manuscripta Math. 6 (1972), 71--103]:

Für einen solchen 4-Tupel schlagen wir die Bezeichnung Köcher vor, und nicht etwa Graph, weil letzerem Wort schon zu viele verwandte Begriffe anhaften.

Attempt at translation: For such a 4-tuple we suggest the name quiver, rather than graph, since the latter word already has too many related concepts connected to it.

(This is community wiki, so anyone can add a proper English translation.)

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  • $\begingroup$ Since I don't speak a word of German I do hope somebody can do it. $\endgroup$
    – Benjamin Steinberg
    Commented Feb 27, 2014 at 22:02
  • $\begingroup$ Ok, I gave it a try (I'm neither a German nor English native speaker). $\endgroup$
    – Dag Oskar Madsen
    Commented Feb 27, 2014 at 22:08
  • $\begingroup$ Looks good, but I tweaked it anyway. Gerhard "Sometimes Speaks Like A Native" Paseman, 2014.02.27 $\endgroup$
    – Gerhard Paseman
    Commented Feb 27, 2014 at 22:15
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    $\begingroup$ @GerhardPaseman Perhaps attached is more precise than connected? $\endgroup$
    – Dag Oskar Madsen
    Commented Feb 27, 2014 at 23:05
  • $\begingroup$ Translation seems correct $\endgroup$
    – Patrick Da Silva
    Commented Mar 2, 2014 at 17:21
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Here is some speculation. Some abstractions in mathematics can be used to study so many different things that even when you use the same abstraction you might want to give it a different name to indicate what sort of thing you're actually studying.

In other words, the name you use declares an intention: when you say "quiver," you're declaring an intention to study quiver representations or quiver varieties, etc. When you say "graph," on the other hand, you might instead be declaring an intention to study algorithms for finding shortest paths or a million other classical graph-theoretic questions.

As another example, consider the different things we might call a functor $F : C \to D$:

  • A "diagram (of shape $C$)." This declares an intent to talk about the limit or colimit of the functor. Here the emphasis is on $D$, or perhaps $F$.
  • A "model," a "representation," a "module," or an "algebra." This declares an intent to study and emphasize $C$, or perhaps $F$, by drawing an analogy to, respectively, models of logical theories, representations of groups, modules over algebras, or algebras over operads.
  • A "presheaf" (when the functor is contravariant and lands in something like $\text{Set}$). This declares a few possible intents, like an eye towards sheafification or a perspective that $F$ should be thought of as a generalized object in $C$.

It's a shame that we don't have mathematical nomenclature explicitly dedicated to declaring intentions like this, but using different terms for the same thing is better than nothing.

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    $\begingroup$ I essentially agree, but maybe the habit of "declaring intentions" via naming the same object with different names could offuscate some analogies that are appearent at an abstract level? $\endgroup$
    – Qfwfq
    Commented Feb 27, 2014 at 22:41
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    $\begingroup$ @Qfwfq: that's certainly an unfortunate side effect (in particular the analogy between models and representations seems to be underappreciated for this reason). I haven't found a better fix for this other than trying to mention all of the names I know for a thing. $\endgroup$
    – Qiaochu Yuan
    Commented Feb 27, 2014 at 23:11
  • $\begingroup$ @QiaochuYuan: a relevant comment from the literature, more or less explicitly commenting on the expository problem you summarize so nicely above, is given by D. N. Yetter in "On Deformations of Pasting Diagrams",Theory and Applications of Categories, Vol. 22, No. 2, 2009, pp. 24–53, saying on p. 25 that he is using a "gentle method of exposition by initially restating the familiar in less-familiar but readily generalizable terms", with regard to his calling finite directed graphs by the name "1-computad". The intent there is of course to generalize to $n$-computads. $\endgroup$
    – Peter Heinig
    Commented Jul 18, 2017 at 12:24
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    $\begingroup$ @QiaochuYuan: re your "a shame that we don't have mathematical nomenclature explicitly dedicated to declaring intentions like this": we seem not to have that, but one could use an inner-mathematical metaphor and say that the choice of the particular synonym is a choice of tangent vector, tangent to the point-that-is-the-concept-to-be-named, and pointing in the direction-in-which-to-generalize. $\endgroup$
    – Peter Heinig
    Commented Jul 18, 2017 at 12:26
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When doing my thesis it seemed it was simply because multiple edges and loops are allowed when working with quivers and therefore it is easier to use this well defined term rather than referring to it as a directed graph which seems to have a much broader meaning.

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  • $\begingroup$ This is what @trew suggested above. That is certainly an advantage. $\endgroup$
    – Benjamin Steinberg
    Commented Jan 28, 2014 at 19:53

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