2025/11/03 by Carles Casacuberta, Casacuberta, Carles, Javier J. Gutiérrez +3 · 1 citation
Computer Science · Mathematics · #18C30 #18C35 #18N60 #55U35 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic, programming, and type systems
paper · pdf · doi:10.48550/arxiv.2511.01497
openalex publication_date 2025/11/03 · openalex created_date 2025/11/06 · openalex updated_date 2026/07/28
A sketch is a category equipped with specified collections of cones and cocones. Its models are functors to the category of sets that send the distinguished cones and cocones to limit cones and colimit cocones, respectively. Sketches provide a categorical formalization of theories, interpreting logical operations in terms of limits and colimits. Gabriel and Ulmer showed that categories of models of sketches involving only cones (called limit sketches) are precisely the locally presentable categories, while Lair extended this correspondence to sketches including both cones and cocones, thereby characterizing accessible categories. In this article, we discuss a homotopy-coherent generalization of sketches in the context of ∞-categories and prove that presentable ∞-categories are the ∞-categories of models of limit sketches, whereas accessible ∞-categories arise as the ∞-categories of models of arbitrary sketches. As illustrations, we make the corresponding sketches explicit for a wide range of ∞-categories, including complete Segal spaces, ∞-operads, A_∞-algebras, E_∞-algebras, spectra, and higher sheaves.