On 7/14/09, Allan Wechsler <acwacw@gmail.com> wrote:
I am as skilled a category therist as I am a dentist. That having been said, I shall walk out on a limb here. Category Theory attempts to capture the notion of thingies and functions that map thingies to thingies, in the greatest possible generality. (Yes, I know that the thingies in the range might be of a different sort than the thingies in the domain.)
If Category Theory could not capture the category of "sets and functions", it would fail at the outset.
So a category can't be restricted to anything as paltry as a set. Any approach that starts out by saying that a category is a kind of set, is pretty much doomed.
But in fact "objects" are only in there for historical reasons, and to pander to our regressive preference for having (relatively) concrete "things" around for "relations" to hold between. The category definitions could instead be rephrased to start from "morphisms" as fundamental, omitting "objects" entirely. It's not at this stage clear to me whether we then have to consider the entire class of all morphisms, or might instead restrict ourselves to discussing sets of morphisms between given pairs of objects. This approach would chime with my growing conviction that ontological problems concerning "existence" of this or that "thing" are a linguistic artifact. Does one ever hear of a passionate discussion about whether such-and-such a relation exists (in general), rather than whether a particular instance of that relation happens to hold in some particular situation? So --- let's hear it for 0-ary relations! Fred Lunnon