Why doesn’t one develop fuzzy logic by extending topos theory, by simply extending the subobject classifier Ω 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.)
Pitts, Andrew M. (1982). Fuzzy sets do not form a topos. In: Fuzzy Sets and Systems 8.1, pp. 101–104. DOI: 10.1016/0165-0114(82)90034-3
Let H be a complete Heyting algebra. The main result is that the category of H-valued fuzzy sets Fuz(H) defined by Eytan is a topos if and only if H is a Boolean algebra. Apparently, the problem is that H-valued fuzzy sets are insufficiently ‘fuzzified’ (Pitt’s word!) to be well-behaved enough to form a topos. To be precise, H-valued fuzzy sets ‘fuzzify’ only the ∈ predicate, while H-valued sets ‘fuzzify’ both = and ∈. It can be shown that the category of H-valued sets is equivalent to the category of sheaves over H, and hence do form a topos H-Set. Pitt shows that H-valued fuzzy sets are equivalent to a certain full subcategory of H-Set
Stout, Lawrence Neff (1984). Topoi and categories of fuzzy sets. In: Fuzzy Sets and Systems 12.2 pp. 169–184. DOI: 10.1016/0165-0114(84)90036-8
Let L be a completely distributive lattice. (Then it is also a complete Heyting algebra H.) The paper starts by showing that Goguen’s characterisation of categories of sets having ‘fuzzy boundary with fuzziness measured in L’ Set(L) are essentially incompatible with the axioms of elementary toposes: Set(L) is a topos if and only if L is the trivial lattice, in which case the category is the usual category of sets. As above, Goguen suggested that a ‘nicer category may result by taking maps which are fuzzy as well’; apparently this is what Eytan did with his Fuz(H).
Stout continues with a brief examination of the internal logic of Set(H), Fuz(H), Sh(H), noting that the internal logic of these categories and the usual fuzzy logic operations do not necessarily coincide, and SetHop and concludes with some results about change-of-base for these categories.
Barr, Michael (1986) Fuzzy set theory and topos theory. In: Canad. Math. Bull. 29.4, pp.501–508. DOI: 10.4153/CMB-1986-079-9.
There is a Math Review for this paper: MR860861.
Let L be a locale. Like the above two papers, the main contention is that L-valued fuzzy sets are not fuzzy enough for their categories to be toposes. Barr describes one generalisation and shows that it is equivalent to a sheaf on a locale L+, obtained by extending L with a new bottom element, and also shows that Eytan’s Fuz(L) is equivalent to a certain subcategory of Sh(L+).
Stout, Lawrence Neff (1991). A survey of Fuzzy Set and topos theory. In: Fuzzy Sets and Systems 42.1, pp. 3–14. DOI: 10.1016/0165-0114(91)90085-5.
I’ll just quote the paper directly: ‘Our survey of the various categorical formulations of Fuzzy Set theory and a variety of kinds of topoi will suggest that a synthesis of the two fields, involving some refinement of the notion of a category of fuzzy sets to give something weaker than a topos but with a richer logic, is desirable.
‘Throughout this process we should remember that the object is to find a proper foundation for Fuzzy Set theory, so that the original interval-valued fuzzy sets should at least be embeddable into our structure. The choice of definition should be based on what is natural, what is powerful, and what gives an elegant and complete foundation for fuzzy mathematics.’