Abstract
It is shown that by relaxing the conventional notion of translation and introducing ‘comparisons’ of translations as 2-arrows, intuitionistic theories of types form a 2-category which is equivalent to the 2-category of toposes with left exact functors and natural transformations. In this equivalence, translations preserving disjunctions (resp. the existential quantifiers, logical translations) correspond to functors preserving unions (resp. images, logical functors).
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