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If can make valuations into functors, then Monotonicity is just the functor action on the morphism. However the functor would have to map to $([0,\infty], \ge)$ if we want to include $+$ as a monoidal product so the functor is contravariant?
Then a valuation on a category (think frame) is a presheaf $F : X^{op} \to V$ where $V$ is the monoidal category of positive real numbers and infinity pointing to the smaller one (the category that comes up in Lawvere metric spaces) with addition as the monoidal product. Subject to the extra condition that $F(x) \otimes F(y) = F(x \prod y) \otimes F(x \coprod y)$.
I don’t know if this is useful to think about in this way. Oh and continuity is quite literally continuity of the presheaf. The extra condition is somewhat bothersome however, I can’t think of anywhere that comes up in category theory.
@Alizter: concerning the extra condition, especially how it comes up in other contexts in category theory, see this MO discussion initiated by David Spivak.
@Tobias oh that is very nice! Thank you very much for the link!
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