FOM: Picturing categorical set theory, reply to Silver

[Vaughan Pratt]

From: Charles Silver <csilver at sophia.smith.edu>

>I think so.  But, I'm looking for something slightly different. 
>Perhaps what I'm looking for isn't there.  The kind of thing I'm looking
>for is an underlying *conception* of function that is *explicated* by
>category theory.

Think of a function f:X->Y not as mapping individual elements x in X to
individual elements f(x) in Y, but as a transformation of the object X
(e.g. a three-dimensional vector space) into the object Y (e.g. a 2D
vector space).

A category C is called *concrete* when it identifies these two views of
f, with the identification being made via a faithful functor (one that
does not identify morphisms of the same homset) U:C->Set.

In the absence of such a U interpreting f as a function U(f) from the set
U(X) to the set U(Y), the objects of C are to be understood abstractly,
with the morphism f being an abstract transformation of X into Y.
Abstract transformations can be pictured as simply motion from X to Y.
For a slightly more vivid picture one may imagine that what moves from
X to Y is a blend of X and Y which starts out being X and steadily turns
into Y as it approaches Y itself.  The transformation is steady because
there is no a priori reason for it to be anything else.

Like all mathematics, category theory itself is too dry to offer or even
endorse such colorful interpretations.  But for someone looking for just
such a colorful interpretation to make categories somehow more real I
think this picture should work ok without leading one too far from the
mathematical truth.

Vaughan Pratt