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arXiv:2510.15795math.CT

Synthetic perspectives on spaces and categories

Emily Riehl

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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 comments
v1: originally submitted version; v2: final journal version with a typo corrected and updated references
Journal reference
International Congress of Mathematicians 2026, Volume 4: Invited Lectures: Sections 5--8, 414--433
DOI
doi:10.1137/25M1806569
Subjects
math.CTCategory Theory math.ATAlgebraic Topology math.LOLogic

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  1. Emily Riehl is rewriting the foundations of higher category theory (2020) 1 comment

    • Emily Riehl is one of the best category theory writers in the business. Lurie’s opus was basically unreadable for me until I found her notes on (inf, 1)-categories and enrichment. More recently, she wrote https://arxiv.org/abs/2510.15795 on how univalence drives some approaches…

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