My own experience is `it depends' a proof chasing a commutative diagram is close to outlining a proof where the steps are each fairly easy defining cogebras by dualizing diagrams for algebras is very helpful but there are other times where one has to struggle to figure out what the string of equations is that are being claimed jim Dusko Pavlovic wrote:

There is a letter to the editor in the October issue of the Notices of the AMS that may be of interest. Pat Donaly (of Boneville Power Adminsitration) writes, e.g. that

"There is a strong technical argument that commutative diagrams are as inherently obfuscatory as the heavy sub- and superscripting in classical differential geometry..." And so on.

It is not hard to counter this guy's arguments, even the "strong technical" ones, and I am sure someone will respond. But what strikes me as more interesting than demonstrating to everyone that diagrams are helpful --- is the persistence of the same kind of the silly debates about categories as a tool of mystification, vs categories as the order of the world. There is something almost psychoanalytic about all that misunderstanding. Like a sister who becomes deaf as soon as her brother enters the room. Well, I'd say that they both must be doing something wrong.

So I am thinking: before we jump the gun to write letters to Notices, perhaps we could try to clarify the general issue, and make sure we don't perpetrate anyone's negative perceptions of categories as a little patronizing form of maths.

I always thought chasing diagrams replaces long sequences of equations. And a bit more, like in snake lemma, when you need to go back along an arrow. Is there more to it than that? (If not, perhaps we could keep "strong technical" arguments about diagrams at a minimum, and use the opportunity to address a wider issue of structuring math and computation.)

-- dusko

One wonders what an 18th C analyst would make of a late 20th C calculus text. Integers and rationals as equinumerous sets? The continuum as a set? Functions as sets of pairs? My god, is everything a set? Sounds like some sort of grand unified theory of mathematics. Newton's laws of motion constituted a Grand Unified Theory of cosmology in their day, and are still good enough for rocket science if not for Geiger counters and linear accelerators. Resolution was a Theory Of Everything for automated theorem proving for a few years. Every body of science has its GUTs and TOEs. They don't last forever but enjoy their fifteen years or fifteen decades of fame before the new improved GUT or TOE comes along. I wouldn't feel at all bad if some niche of mathematics or CS adopted Chu spaces as its TOE, even if just for a little while. Naturally I'd be delighted if chupology turned out to have legs. Vaughan Pratt On every plane The human brain Has GUTs and TOEs As all it knows From: Derek Ross <math@antiquark.com>

The impression I got from reading about categories (on the internet of course) is that category theory is a sort of grand unified theory of mathematics. ... I no longer think that's what category theory is about, but you might be interested in the first impressions of a layman.

David Yetter wrote:

The Notices quote which started this is particularly annoying because it seems both to take the myopic view of category theory as a "language" and then to claim that category theory doesn't do any better in that role than the "debauch of indices".

The discussion reminds me, however, of the incident involving Moshe Flato, Nicholai Reshetikhin and myself, which I related in the introduction to my book: at a Joint-Summer Research Conference in the early 1990's Reshetikhin and I had a long discussion about Shum's coherence theorem and the role of monoidal categories in "quantum knot invariants", Flato was persistently dismissive of category theory as "mere language". I retired for the evening, leaving the discussion to the others. In the morning before the first session, as he was going to his seat, Flato tapped me on the shoulder, and whispered with a thumbs up sign, "Hey, viva les categories. These new ones, the braided monoidal ones."

and now derived cats and more are all the rage in e.g. D-brane theory

Of course, I suppose it would only muddy the waters further to try to explain both old fashioned commutative diagrams and new fashioned "string diagrams".