A sketch consists of a category equipped with specified collections of cones and cocones, and its models are functors to the category of sets that map these cones and cocones to limits and colimits. Sketches formalize the concept of a theory by describing logical operations via limits and colimits, with classical examples of models including groups, fields, and categories. Gabriel and Ulmer showed that a category is a model of a sketch with only limit cones if and only if it is presentable, while Lair extended this correspondence to models of sketches and accessible categories.
In this talk, we present a homotopy-coherent generalization of sketches in the setting of $\infty$\nobreakdash-categories. We show that numerous $\infty$\nobreakdash-categories, including complete Segal spaces, $\infty$\nobreakdash-operads, $E_\infty$\nobreakdash-algebras, spectra, and higher sheaves, can be constructed as $\infty$\nobreakdash-categories of models of limit sketches. Moreover, we establish higher-categorical analogues of the classical correspondences with presentable and accessible $\infty$\nobreakdash-categories.
