[FOM] 827: Tangible Incompleteness Restarted/1
Louis H Kauffman
kauffman at uic.edu
Wed Oct 2 14:56:02 EDT 2019
Yes, a category is a digraph with extra structure. And one of the more important collections of categories are the categories generated from digraphs via the structure of paths on the given digraph.
My point is that the concept of a graph is involved through and through with just about all aspects of mathematics. People often feel that they can make a point by contrast, and it gets remembered for too long.
J. H. C. Whitehead famously said “Graph theory is the slums of topology.” I’ll not comment on that. But I think it was an intuitive avoidance of the sort of complexity in topological invariants that can come from
deep combinatorial definitions. And at the same time Whitehead made such definitions with the Whitehead Groups and the concept of CW complexes. The dam broke in the 1980’s with advent of the Jones polynomial, its relation with the Tutte polynomial and contraction/deletion relations, relations with statistical physics, topological quantum field theory and categorical formulations that span the gamut from graph theoretic structures to braided monodical categories, field theory and beyond into present day link homology and its relationships with graphs and with the homology of categories. Mathematics is a whole and
it is silly to argue about why some people have prejudices.
> On Oct 1, 2019, at 4:24 AM, Timothy Y. Chow <tchow at math.princeton.edu> wrote:
> On Mon, 30 Sep 2019, Joe Shipman wrote:
>> 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?
> This question has been raised before on MathOverflow, and to a first approximation, the answer seems to be "no":
>>> 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.
> I'm not sure if you're trying to draw an analogy between quivers and categories. If you are, then one place the analogy breaks down is that nobody defines a category as a digraph, pure and simple. A category has extra structure. On the other hand, a quiver *is* standardly defined as a digraph (or perhaps a multidigraph, if you want to emphasize the possibility of multiple edges). Period. No extra structure. So why introduce a new word? There's no logical need for it.
> Gabriel, in the paper where he introduced quivers, justified his introduction of a new word (Köcher) on the grounds that the word "graph" already had too many connotations and related concepts ("schon zu viele verwandte Begriffe anhaften"). Whether Gabriel was waging a "culture war" as Friedman suggested, or whether he just wanted to emphasize that he was only interested in a limited circle of questions about the digraph/quiver, is unclear to me. Either way, as I said, I believe that the terminology is unfortunate, and generates precisely the type of confusion that Gabriel said he was trying to avoid.
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