A Unified Approach to Algebraic Set Theory

Categorical logic studies the relation between category theory and logical languages, and provides a very efficient framework in which to treat the syntax and the model theory on an equal footing. For a given theory T formulated in a suitable language, both the theory itself and its models can be viewed as categories with structure, and the fact that the models are models of the theory corresponds to the existence of canonical functors between these categories. This applies to ordinary models of first order theories, but also to more complicated topological models, forcing models, realisability and dialectica interpretations of intuitionistic arithmetic, domain-theoretic models of the λ-calculus, and so on. One of the best worked out examples is that where T extends the theory HHA of higher order Heyting arithmetic [29], which is closely related to the Lawvere-Tierney theory of elementary toposes. Indeed, every elementary topos (always taken with a natural numbers object here) provides a categorical model for HHA, and the theory HHA itself also corresponds to a particular topos, the “free” one, in which the true sentences are the provable ones.