Type space functors and interpretations in positive logic

We construct a 2-equivalence $\mathfrak{CohTheory}^\text{op} \simeq \mathfrak{TypeSpaceFunc}$. Here $\mathfrak{CohTheory}$ is the 2-category of positive theories and $\mathfrak{TypeSpaceFunc}$ type space functors. give precise definition interpretations for logic, which will be 1-cells in $\mathfrak{CohTheory}$. The 2-cells are definable homomorphisms. restricts to duality categories, making philosophy that theory `the same' as collection its spaces (i.e. functor). In characterising those functors arise functors, we find they specific instances (coherent) hyperdoctrines. This connects two different schools thought on logical structure theory. key ingredient, Deligne completeness theorem, arises from topos theory, where have been studied under name coherent theories.