2020/05/23 by Redi Haderi, Haderi, Redi
Mathematics · #55P99 #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2005.11597
openalex publication_date 2020/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work we propose a realization of Lurie's prediction that inner fibrations p: X → A are classified by A-indexed diagrams in a ``higher category" whose objects are ∞-categories, morphisms are correspondences between them and higher morphisms are higher correspondences. We will obtain this as a corollary of a more general result which classifies all simplicial maps between ordinary simplicial sets in a similar fashion. Correspondences between simplicial sets (and ∞-categories) are a generalization of the concept of profunctor (or bimodule) pertaining to categories. While categories, functors and profunctors are organized in a double category, we will exhibit simplicial sets, simplicial maps, and correspondences as part of a simplicial category. This allows us to make precise statements and provide proofs. Our main tool is the language of double categories, which we use in the context of simplicial categories as well.