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