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