2025/05/22 by Nima Rasekh, Rasekh, Nima
Mathematics · Physics and Astronomy · #18N40 #18N45 #18N50 #18N60 #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Noncommutative and Quantum Gravity Theories
paper · pdf · doi:10.48550/arxiv.2505.16342
openalex publication_date 2025/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The unstraightening construction due to Lurie establishes an equivalence between presheaves and fibrations, using one prominent model of (∞,1)-categories, namely quasi-categories. In this work we generalize this result by proving that for all ∞-cosmoi of (∞,1)-categories in the sense of Riehl and Verity, which includes quasi-categories but also complete Segal spaces or 1-complicial sets, their corresponding notions of fibrations and presheaves are biequivalent ∞-cosmoi via a natural zig-zag of cosmological biequivalences. The major idea that makes this possible is a lift of the quasi-categorical unstraightening construction to a cosmological biequivalence.