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