Sketchable infinity categories

TL;DR

This paper extends the concept of sketches to homotopy-coherent $$-categories, showing that presentable $$-categories can be characterized as models of limit sketches, broadening the categorical framework for higher structures.

Contribution

It introduces a homotopy-coherent generalization of sketches in $$-categories and characterizes presentable and accessible $$-categories as models of these sketches.

Findings

Presentable $$-categories are models of limit sketches.

Accessible $$-categories are models of arbitrary sketches.

Explicit sketches are provided for various $$-categories.

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.