Synthetic 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.

arXiv commentv1: originally submitted version; v2: final journal version with a typo corrected and updated references