The Goguen category of L-fuzzy sets on a lattice L (Objects are pairs (A,\alpha) where \alpha:A\to L and morphisms are functions f:A\to B such that \beta{f(a)) >= \alpha(a)) has all functions whose underlying set function is an isomorphism both epic and monic, but not, in general, isomorphisms, which must preserve the lattice valued membership on the nose. Since these monic, epic maps are the ones which give the right subobjects to consider for fuzzy logic they are of interest. They do not determine a "folding" like the one you describe. On Mar 6, 2007, at 7:11 PM, David Karapetyan wrote:
So does every monic, epic arrow determine such a "folding" or are there monic, epics that can't be characterized in such a way?
Lawrence Stout Professof of Mathematics Illinois Wesleyan University