Potential isomorphisms of generalized approach spaces

We study relativizations and potential isomorphisms of quantale-valued approach spaces. We show how the category of generalized approach spaces and injective maps can be encoded using higher order structures, but under such an encoding generalized approach spaces need not have a relativization. We then provide an alternate encoding of generalized approach spaces using higher order structures which does always relativize. We also consider potential isomorphisms between generalized approach spaces and give a necessary and sufficient condition for such a potential isomorphism to exist, which is absolute between models of set theory. We then consider an abstract notion of sentence for generalized approach spaces and prove a downward Loowenheim-Skolem theorem as well as a Tarski-Vaught theorem for such generalized sentences.