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?
Here are some papers, which I have not read in detail. I'm somewhat troubled that there aren't any Math Reviews for most of them, and I certainly can't vouch for their correctness. (I know nothing of fuzzy logic, and only a miniscule bit of topos theory; I'm merely an enthusiast.)