[FOM] 827: Tangible Incompleteness Restarted/1

[Joe Shipman]

That raises the question of whether any results from graph theory have been applied in category theory in interesting ways. Do graph theorists have anything to tell category theorists about categories?

— JS 

Sent from my iPhone

> On Sep 30, 2019, at 2:29 PM, Louis H Kauffman <kauffman at uic.edu> wrote:
> 
> It should be pointed out in this discussion that a category is a digraph with extra (compositional) structure.
> Category theorists would not care to be categorized as studying a subcategory of graph theory.
> 
>> On Sep 30, 2019, at 9:20 AM, Harvey Friedman <hmflogic at gmail.com> wrote:
>> 
>> From Chow https://cs.nyu.edu/pipermail/fom/2019-September/021693.html
>> 
>>> It is true that everywhere in the literature, a quiver is defined as a
>>> directed graph.  I find this unfortunate because it does raise the
>>> question of why you would introduce a new word for something that already
>>> has a name.
>>> 
>>> Associated to a quiver is something called its path algebra.  When people
>>> use the word "quiver," they're signalling the fact that they're primarily
>>> interested in the representation theory of the path algebra, and not in
>>> the directed graph for its own sake.  Graph-theoretic facts about the
>>> underlying directed graph are interesting only insofar as they lend
>>> insight into the representation theory of the path algebra.  That's a very
>>> narrow set of graph-theoretic facts, compared to the sort of things that
>>> graph theorists might be interested in.
>>> 
>>> In my opinion, it would have been better to use the word "quiver" to refer
>>> to the path algebra rather than to the directed graph itself, but it's too
>>> late to change established terminology.
>>> 
>> These circumstances are highly suggestive of a math culture war. I get
>> the feeling that the relevant core mathematicians are expressing their
>> willful ignorance of digraphs - or maybe even marginalization of those
>> interested in digraphs in and of themselves -  through inventing
>> "quivers" for digraphs as Chow explains.
>> 
>> This whole embarrassing scenario (I'm taking for granted Chow's
>> account) would have been avoided through an adherence of the
>> fundamental driving criterion of "general intellectual interest". It
>> is completely obvious that the notions of graph and digraph and
>> related concepts like clique and independent set, are of a high level
>> of gii. Just consider how many salient examples:
>> 
>> A likes B
>> A has sent a message to B via the internet
>> A,B are friends
>> A,B are in internet correspondence
>> {A_1,...,A_k} like each other
>> {A_1,...,A_k} do not like each other
>> A_1,...,A_k, forms a custody chain
>> and so forth.
>> 
>> And labels on digraphs: A has sent a message n times to B via the
>> internet, where the label is n for the edge going from A to B.
>> 
>> F.o.m. at its highest levels is motivated by matters of high gii. At
>> least the basics of graph theory is also motivated by matters of high
>> gii.
>> 
>> Try giving such an account, which is totally understandable by
>> "everyone", for path algebras and representation theory. This should
>> definitely decide the relevant culture wars under a common
>> understanding of the true nature of intellectual life. Such a common
>> understanding is obviously missing, and an interesting question that I
>> frequently think about is: why and what is to be done about it?
>> 
>> Harvey Friedman
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