The Yoneda embedding in simplicial type theory

Riehl and Shulman [1] introduced simplicial type theory (STT), a variant of homotopy type theory which aimed to study not just homotopy theory, but its fusion with category theory: (∞, 1)-category theory. While notoriously technical, manipulating ∞-categories in simplicial type theory is often easier than working with ordinary categories, with the type theory handling infinite stacks of coherences in the background. We capitalize on recent work by Gratzer et al. [2] defining the (∞, 1)-category of ∞-groupoids in STT to define presheaf categories within STT and systematically develop their theory. In particular, we construct the Yoneda embedding, prove the universal property of presheaf categories, refine the theory of adjunctions in STT, introduce the theory of Kan extensions, and prove Quillen’s Theorem A. In addition to a large amount of category theory in STT, we offer substantial evidence that STT can be used to produce difficult results in ∞-category theory at a fraction of the complexity.