Comparing semantic frameworks for dependently-sorted algebraic theories

TL;DR

This paper provides a comprehensive comparison of various categorical models for dependently-sorted algebraic theories, unifying them through comprehension categories and clarifying their interrelationships.

Contribution

It offers a unifying framework using comprehension categories to embed and compare different models of algebraic theories with dependencies.

Findings

Most models embed as sub-2-categories of comprehension categories

Clarifies relationships between categorical models like contextual categories and display map categories

Provides a big-picture overview of the categorical semantics for dependent algebraic theories

Abstract

Algebraic theories with dependency between sorts form the structural core of Martin-L\"of type theory and similar systems. Their denotational semantics are typically studied using categorical techniques; many different categorical structures have been introduced to model them (contextual categories, categories with families, display map categories, etc.) Comparisons of these models are scattered throughout the literature, and a detailed, big-picture analysis of their relationships has been lacking. We aim to provide a clear and comprehensive overview of the relationships between as many such models as possible. Specifically, we take *comprehension categories* as a unifying language and show how almost all established notions of model embed as sub-2-categories (usually full) of the 2-category of comprehension categories.