FOM: Picturing categorical set theory, reply to Silver
csilver at sophia.smith.edu
Sat Jan 24 07:57:20 EST 1998
On Fri, 23 Jan 1998, Vaughan Pratt wrote:
> 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.
> 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.
Thank you for your colorful interpretation. I will ponder these
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