Constructive higher sheaf models with applications to synthetic mathematics

2026-05-14

AI Summary

This work addresses the absence of constructive semantic models for extensions of dependent type theory—such as the univalence axiom and higher inductive types—in synthetic mathematics. It presents the first construction of a higher topos model within a fully constructive metatheory, thereby providing a unified semantic foundation for emerging formal systems like simplicial homotopy type theory and synthetic algebraic geometry. By establishing this model, the paper not only fills a critical gap in constructive semantics but also demonstrates the broad applicability and expressive power of higher topos models across diverse frameworks of synthetic mathematics, confirming their viability as a general semantic infrastructure.

Abstract

There have recently been several developments in synthetic mathematics using extensions of dependent type theory with univalence and higher inductive types: simplicial homotopy type theory, synthetic algebraic geometry and synthetic Stone duality. We provide a foundation of higher sheaf models of type theory in a constructive metatheory and, in particular, build constructive models of these formal systems.

Problem

• higher sheaf models
• constructive metatheory
• dependent type theory
• univalence
• synthetic mathematics

Innovation

• constructive metatheory
• higher sheaf models
• dependent type theory
• univalence
• higher inductive types