Synthetic perspectives on spaces and categories
arXiv:2510.15795 · doi:10.1137/25M1806569
The paper surveys proof techniques from homotopy type theory and simplicial type theory, focusing on induction over paths or arrows and constructions of (directed) univalent universes, to give new perspectives on spaces and categories.
Abstract
Recently discovered domain-specific formal systems -- specifically homotopy type theory and simplicial type theory -- provide new perspectives on spaces and categories in a natively equivalence-invariant setting. In this note, we expose fundamental proof techniques from these parallel settings: describing induction principles over paths or arrows and constructions involving universes that are either univalent or directed univalent.
v1: originally submitted version; v2: final journal version with a typo corrected and updated references