  G\"odel's Dialectica interpretation was conceived as a tool to obtain the
consistency of Peano arithmetic via a proof of consistency of Heyting
arithmetic in the 40s. In recent years, several proof-theoretic
transformations, based on G\"odel's Dialectica interpretation, have been used
systematically to extract new content from classical proofs, following a
suggestion of Kreisel. Thus, the interpretation has found new relevant
applications in several areas of mathematics and computer science. Several
authors have explained the Dialectica interpretation in categorical terms. In
our previous work, we introduced an intrinsic categorical presentation of the
Dialectica construction via a generalisation of Hofstra's work, using the
notion of G\"odel fibration and its proof-irrelevant version, a G\"odel
doctrine. The key idea is that G\"odel fibrations can be thought of as
fibrations generated by some basic elements playing the role of quantifier-free
elements. This categorification of quantifier-free elements is crucial not only
to show that our notion of G\"odel fibration is equivalent to Hofstra's
Dialectica fibration in the appropriate way, but also to show how G\"odel
doctrines embody the main logical features of the Dialectica Interpretation. To
show that, we derive the soundness of the interpretation of the implication
connective, as expounded by Troelstra, in the categorical model. This requires
extra logical principles, going beyond intuitionistic logic, namely Markov
Principle and the Independence of Premise principle, as well as some choice. We
show how these principles are satisfied in the categorical setting,
establishing a tight correspondence between the logical system and the
categorical framework. Finally, to complete our analysis, we characterise
categories obtained as results of the tripos-to-topos of Hyland, Johnstone and
Pitts applied to G\"odel doctrines.

Category Theory;Logic