One of the canonical examples used by Barr & Wells in order to motivate the use of topoi is that we can construct a theory for fuzzy logic and fuzzy set theory as set-valued sheaves on a poset (Heyting algebra) of confidence values for the fuzziness. Doing this constructs a fuzzy theory where both the membership and equality relations have more truth values than just true and false.
How would one construct a ternary approach using this mindset? In other words, is there an easy way to see a sheaf on a poset with three values as a fuzzy set theory for the truth values (true, maybe, false)? Or is this formulation even the wrong approach? Do I want a different Heyting algebra, so that the subobject classifier ends up having three elements?