🤖 AI Summary
This work proposes an abstract and unified categorical semantics for Martin-Löf type theory with explicit universe polymorphism. By employing generalized algebraic theories, it constructs two classes of categorical structures—families of categories equipped with an external tower of universes and families indexed by universe levels—corresponding respectively to two variants of the type theory. The higher-order semantic structure of each is revealed through their initial models. This study is the first to incorporate explicit universe polymorphism into the framework of generalized algebraic theories, thereby abstracting away syntactic details and highlighting its intrinsic categorical nature. The results not only demonstrate the applicability of this framework to sophisticated type systems but also provide a crucial instance supporting Voevodsky’s initiality conjecture.
📝 Abstract
We present generalized algebraic theories corresponding to slightly modified versions of two of the type theories in our paper Type Theory with Explicit Universe Polymorphism. We first present a generalized algebraic theory for categories with families with extra structure corresponding to Martin-Lof type theory with an external tower of universes. We then present a generalized algebraic theory for level-indexed categories with families with extra structure corresponding to Martin-Lof type theory with explicit universe polymorphism: a theory with universe level judgments, internally indexed universes, and level-indexed products. In this way we get abstract characterizations of the two theories as initial models of their respective generalized algebraic theories. We thus abstract from details of the grammar and inference rules of the type theories and highlight their high-level structure. More broadly, the present work can be viewed as a case study of a uniform approach to categorical logic based on generalized algebraic theories and categories with families. We also discuss the relevance to Voevodsky's initiality conjecture project.