First-order homotopical logic
Abstract
We introduce a homotopy-theoretic interpretation of intuitionistic first-order logic based on ideas from Homotopy Type Theory. We provide a categorical formulation of this interpretation using the framework of Grothendieck fibrations. We then use this formulation to prove the central property of this interpretation, namely homotopy invariance. To do this, we use the result from arXiv:1905.10690 that any Grothendieck fibration of the kind being considered can automatically be upgraded to a 2-dimensional fibration, after which the invariance property is reduced to an abstract theorem concerning pseudonatural transformations of morphisms into 2-dimensional fibrations.
Keywords
Cite
@article{arxiv.1908.08944,
title = {First-order homotopical logic},
author = {Joseph Helfer},
journal= {arXiv preprint arXiv:1908.08944},
year = {2025}
}
Comments
Major revision: added several sections, fixed an error in (what is now) Theorem 12.2, restructured, rewrote introduction and expository material, updated to reflect changes in arXiv:1905.10690. 54 pages, comments welcome!