Intuitionism's Limits

[categories <cat-dist@mta.ca>]

Date: Sun, 2 Mar 1997 15:07:12 +1030 (CST)
From: William James <wjames@arts.adelaide.edu.au>

Intuitionism's Limits: if C is a category sufficiently complex to
demonstrate that some C-arrow f:a-->b is monic and B is a subcategory
of C containing just f (and the requisite identity arrows), do we
still know that f is monic? Should we? (Or, in other words, which
view *should* dominate: Intuitionism, Realism, the category
theoretic...?) What if C is something (semi?)fundamental like a
category of all sets and functions, or a category of categories?

I suppose the answer is that monicity is relative to a category,
but what supports this as a claim? And doesn't it seem to contradict
the reasonable realist claim that we can somehow know f in B to be
monic? (Or am I missing something straightforward: that properties
can be granted to f by its relationship to C via an inclusion functor?)

This goes to the issue of the adequacy of category theory as a foundation
in more than the simply technical sense.

(I could be using the term "realism" incorrectly too: I take it to be
a positon, in maths at least, that mathematical entities can
have collections of properties beyond the constraints of a given
theoretical context.)

William James