On 26 Feb 2010, at 18:53, Richard Garner wrote:
I think it is more perspicuous to treat this, not as an overloading of \in, but as an "implicit conversion" associated to the notion of category; that is, we allow the forgetful functor Cat->Set to be applied silently in contexts which otherwise would not type-check. In fact the vast majority of "abuses of notation" are of this character, when applied to, for example, any forgetful functor from an Eilenberg-Moore category; the discrete category functor Set->Cat; the Yoneda embedding C -> [C^op, Set]; the forgetful functor from universal cones to their vertex, etc, etc. In principle this becomes problematic as soon as the category generated by all such implicit conversions has non-identity idempotents;
Why restrict this to idempotents? Surely the category needs to be a preorder for the usage to be unambiguous? Paul
in practice, this category is free on a graph and we hope to identify a shortest path between two vertices!
Richard
-- Paul Blain Levy School of Computer Science, University of Birmingham +44 (0)121 414 4792 http://www.cs.bham.ac.uk/~pbl [For admin and other information see: http://www.mta.ca/~cat-dist/ ]