Sketchable infinity categories
Carles Casacuberta, Javier J. Gutiérrez, David Martínez-Carpena
TL;DR
This work extends classical sketch theory to the world of $\infty$-categories by establishing a precise link between presentability, accessibility, and sketches. It introduces higher sketches and a higher flatness criterion, proving that presentable $\infty$-categories are exactly the $\infty$-categories of models for limit sketches, while accessible $\infty$-categories correspond to models of arbitrary sketches. It further develops the tensorial and localization structures (e.g., $\Sigma \boxtimes T$ and $\operatorname{Mod}(\Sigma) \otimes \mathcal{C}$) and provides explicit limit sketches for central objects such as spectra, Segal spaces, $A_\infty$- and $E_\infty$-algebras, dendroidal Segal spaces, and higher sheaves. The framework yields a unified approach to modeling theories and higher algebra in $\infty$-categories and clarifies how localization and sketchness interact with presentability, enabling concrete constructions across a wide range of higher-categorical contexts.
Abstract
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 $\infty$-categories and prove that presentable $\infty$-categories are the $\infty$-categories of models of limit sketches, whereas accessible $\infty$-categories arise as the $\infty$-categories of models of arbitrary sketches. As illustrations, we make the corresponding sketches explicit for a wide range of $\infty$-categories, including complete Segal spaces, $\infty$-operads, $A_\infty$-algebras, $E_\infty$-algebras, spectra, and higher sheaves.