2019/09/25 by Jonathan Beardsley, Beardsley, Jonathan, Maximilien Péroux +1
Mathematics · Pharmacology, Toxicology and Pharmaceutics · #16T15 #55U30 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Alkaloids: synthesis and pharmacology #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1909.11724
openalex publication_date 2019/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that there is an equivalence in any n-topos X between the pointed and k-connective objects of X and the 𝔼k-group objects of the (n-k-1)-truncation of X. This recovers, up to equivalence of ∞-categories, some classical results regarding algebraic models for k-connective, (n-1)-coconnective homotopy types. Further, it extends those results to the case of sheaves of such homotopy types. We also show that for any pointed and k-connective object X of X there is an equivalence between the ∞-category of modules in X over the associative algebra Ωk X, and the ∞-category of comodules in X for the cocommutative coalgebra Ωk-1X. All of these equivalences are given by truncations of Lurie's ∞-categorical bar and cobar constructions, hence the terminology "Koszul duality".