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Albert Visser introduced five different categories of interpretations between theories INT0 (the category of synonymy), INT1 (the category of homotopy), INT2 (the category of weak homotopy), INT3 (the category of equivalence), and INT4 (the category of mutual interpretability) [Vis04]. The objects in these categories are first order theories, the morphisms are interpretations up to some level of identification between interpretations. The category of synonymy has the strictest criteria for two interpretations to be the same, the category of mutual interpretability the weakest. Visser proved that INT1 6= INT4 [Vis04, § 4.8.4.], but apart from that no separation results are known. One particular question is [Vis04, Open Question 4.16]:

Set Theory With and Without Urelements and Categories of Interpretations