FOM: What categories are good for

[jshipman@bloomberg.net]

Colin argues persuasively that category theory can form a foundation for large
areas of mathematics.  (For example, many of the "big" theorems of algebraic
topology can be *rigorously* developed without ever talking about sets (or even
thinking about them if one's training has been in the categorical tradition)).
I think some of the "set partisans" demand a little bit too much of the catgory
theorists because they overlook some history (i.e. that Natural Transformations
preceded functors which preceded categories) and some phenomenology (that
category theory "works" on its own terms and achieves results that can be
rigorously developed in a set-free formalism).  The big problem in "categorical
foundations" is that "large areas of mathematics" is not ALL of mathematics.
My impression is that the results achievable from categorical foundations can be
smoothly developed set-theoretically (i.e. as if categories had never been
invented) but that the converse is not true (that is, to *rigorously* develop,
say, the theorems of DeBranges or Wiles or Freedman can be done from categorical
foundations only by "inventing sets"). I wouldn't mind being wrong...Joe Shipman