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Mike Stay (Dec 12 2024 at 22:06):I was looking at the "intermediate" truth value of the arrow topos of functions and commuting squares and had an idea how to generalize it to more intermediate truth values: use a Pos-enriched category C with two objects E, V and add more parallel morphisms from V to E, but totally order them. Then look at the Pos-enriched functor category [C^op, Pos].
Mike Stay (Dec 12 2024 at 22:06):Consider the case with two parallel arrows s,t:V->E. In the normal Set-enriched case, we can think of presheaves G:C^op -> Set as graphs where G(E) is the set of edges, G(V) the set of vertices, G(s)(e) the source of e, and G(t)(e) the target of e. The subobject classifier has two vertices T and F, and five edges: four for each way of including or excluding the source and target while excluding the edge, plus one more for including source, target, and edge.
Mike Stay (Dec 12 2024 at 22:06):But when s ≤ t, the edge corresponding to {s} from T to F doesn't satisfy the constraint. So there are four totally ordered sieves: {}, {t}, {s, t}, {s, t, E}.
Mike Stay (Dec 12 2024 at 22:07):If we add a third edge u, in the Set-enriched case there'd be 2^3 + 1 = 9 truth values. But when s ≤ t ≤ u, the sieves {s}, {t}, {s,t}, and {s,u} don't satisfy the constraint and we're left with five totally ordered sieves: {}, {u}, {t,u}, {s,t,u}, {s,t,u,E}.
Mike Stay (Dec 12 2024 at 22:07):With n parallel totally ordered morphisms, we get n+2 totally ordered truth values. So we can consider the Pos-enriched category with two objects and an open interval (0,1) of parallel morphisms. The resulting subobject classifier has the closed interval [0,1] as truth values.
Mike Stay (Dec 12 2024 at 22:07):Sets are models of the theory: G(E) can be taken to be a discrete poset and G(V) can be taken to be the terminal poset, making each G(p) trivial. A fuzzy subset H ⊆ G, thought of as a map into the subobject classifier, assigns to each element of G(E) a value in [0,1] saying with what weight the element is in the subset. As an object of the topos, I think H(E) is a poset where the elements are ordered by their weights, H(V) is the poset F ≤ T, and H(x)(e) is T if 1-x ≤ weight(e) and F otherwise. We say an element has weight 0 if H(x) is always F and weight 1 if it's always T.
Mike Stay (Dec 12 2024 at 22:07):We use 1-x above because we need the values to go from F to T as x increases, with most of the values being F for small weight(e). Alternately, we could reverse the order on the open interval.
Intersection is min, union is max, a -> b is 1 if a ≤ b and b otherwise.
In general, an object X of the topos is a pair of posets X(E) and X(V) where the image under X(x) increases monotonically as x goes from 0 to 1.
Mike Stay (Dec 12 2024 at 22:18):This all seems like a pretty obvious construction to me. Does it have a name?