Syntactic categories for dependent type theory: sketching and adequacy

TL;DR

This paper advocates using locally Cartesian closed categories as a foundational framework for dependent type theories, demonstrating their adequacy and advantages over more complex structures like Uemura's representable map categories.

Contribution

It introduces a semantic approach to dependent type theories using locally Cartesian closed categories and proves their adequacy compared to Uemura's framework.

Findings

Locally Cartesian closed categories effectively model dependent type theories.

Semantic adequacy of these categories relative to Uemura's framework is established.

They provide a simpler alternative to more complex categorical structures.

Abstract

We argue that locally Cartesian closed categories form a suitable doctrine for defining dependent type theories, including non-extensional ones. Using the theory of sketches, one may define syntactic categories for type theories in a style that resembles the use of Martin-L\"of's Logical Framework, following the "judgments as types" principle. The concentration of type theories into their locally Cartesian closed categories of judgments is particularly convenient for proving syntactic metatheorems by semantic means (canonicity, normalization, etc.). Perhaps surprisingly, the notion of a context plays no role in the definitions of type theories in this sense, but the structure of a class of display maps can be imposed on a theory post facto wherever needed, as advocated by the Edinburgh school and realized by the %worlds declarations of the Twelf proof assistant. Uemura has proposed representable map categories together with a stratified logical framework for similar purposes. The stratification in Uemura's framework restricts the use of dependent products to be strictly positive, in contrast to the tradition of Martin-L\"of's logical framework and Schroeder-Heister's analysis of higher-level deductions. We prove a semantic adequacy result for locally Cartesian closed categories relative to Uemura's representable map categories: if a theory is definable in the framework of Uemura, the locally Cartesian closed category that it generates is a conservative (fully faithful) extension of its syntactic representable map category. On this basis, we argue for the use of locally Cartesian closed categories as a simpler alternative to Uemura's representable map categories.