This thesis studies some constructions for building new models of Martin-Löf type theory out of old. We refer to the main techniques as gluing and idempotent splitting. For each we give general conditions under which type constructors exist in the resulting model. These techniques are used to construct some examples of Dialectica models of type theory. The name is chosen by analogy with de Paiva's Dialectica categories, which semantically embody Gödel's Dialectica functional interpretation and its variants. This continues a programme initiated by von Glehn with the construction of the polynomial model of type theory. We complete the analogy between this model and Gödel's original Dialectica by using our techniques to construct a two-level version of this model, equipping the original objects with an extra layer of predicates. In order to do this we have to carefully build up the theory of finite sum types in a display map category. We construct two other notable models. The first is a model analogous to the Diller-Nahm variant, which requires a detailed study of biproducts in categories of algebras. To make clear the generalization from the categories studied by de Paiva, we illustrate the construction of the Diller-Nahm category in terms of gluing an indexed system of types together with a system of predicates. Following this we develop the general techniques needed for the type-theoretic case. The second notable model is analogous to the Dialectica category associated to the error monad as studied by Biering. This model has only weak dependent products. In order to get a model with full dependent products we use the idempotent splitting construction, which generalizes the Karoubi envelope of a category. Making sense of the Karoubi envelope in the type-theoretic case requires us to face up to issues of coherence in our models. We choose the route of making sure all of the constructions we use preserve strict coherence, rather than applying a general coherence theorem to produce a strict model afterwards. Our chosen method preserves more detailed information in the final model.

In this licentiate thesis we present a proof of the initiality conjecture for Martin-Löf’s type theory with 0, 1, N, A+B, ∏AB, ∑AB, IdA(u,v), countable hierarchy of universes (Ui)iєN closed under these type constructors and with type of elements (ELi(a))iєN. We employ the categorical semantics of contextual categories. The proof is based on a formalization in the proof assistant Agda done by Guillaume Brunerie and the author. This work was part of a joint project with Peter LeFanu Lumsdaine and Anders Mörtberg, who are developing a separate formalization of this conjecture with respect to categories with attributes and using the proof assistant Coq over the UniMath library instead. Results from this project are planned to be published in the future. We start by carefully setting up the syntax and rules for the dependent type theory in question followed by an introduction to contextual categories. We then define the partial interpretation of raw syntax into a contextual category and we prove that this interpretation is total on well-formed input. By doing so, we define a functor from the term model, which is built out of the syntax, into any contextual category and we show that any two such functors are equal. This establishes that the term model is initial among contextual categories. At the end we discuss details of the formalization and future directions for research. In particular, we discuss a memory issue that arose in type checking the formalization and how it was resolved. / <p>Licentiate defense over Zoom.</p>