Fuzzy logic and topos theory Why doesn't one develop fuzzy logic by extending topos theory, by simply extending the subobject classifier $\Omega$ to the unit interval [0,1]?  Have people done that?

Fuzzy set theory is a mathematical model of vagueness, which is assumed to be a basic property of our world. In different words, it is assumed that we live in a world with vague objects where nothing is actually crisp. Ordinary mathematics assumes that everything is crisp. Thus I find it at least strange to try to "embed" mathematics of vagueness into universes of crisp mathematics. So I do not really understand what a topos of fuzzy sets would be? Arrows should connect objects up to some degree but this does not happen when one deals with say categories of fuzzy sets. Since we have a new set theory, which was also formalized, we need a new catetory theory and why not a new type theory.
