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arXiv · 2412.19946

Comparing semantic frameworks for dependently-sorted algebraic theories

Benedikt Ahrens·Peter LeFanu Lumsdaine·Paige Randall North
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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.

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BibTeXRIS

Benedikt Ahrens, Peter LeFanu Lumsdaine, Paige Randall North. 2024-12-27. Comparing semantic frameworks for dependently-sorted algebraic theories. https://doi.org/10.1007/978-981-97-8943-6_1

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