FOM: "categorical foundations" -- an oxymoron

[Vaughan R. Pratt]

>Imagine teaching undergraduate engineering
>students the basics of linear transformations.  Are you going to
>begin by explaining that a linear transformation is any arrow in any
>category satisfying axioms L1-L4?

No, one wouldn't, for the same reason that one would not explain that a
permutation is any element in any group satisfying the axioms for a
group.

The axiomatization of module categories (i.e. Colin's axioms)
constitutes the abstract theory of linear transformations under
composition and sum, in the same sense that the group axioms constitute
the abstract theory of permutations under composition and inverse.  Via
that correspondence, each attack on module categories can be translated
into the corresponding attack on groups.  The advantage of such a
translation is that it makes clearer what if anything is wrong with the
attack.

Vaughan Pratt