  Formalized $1$-category theory forms a core component of various libraries of
mathematical proofs. However, more sophisticated results in fields from
algebraic topology to theoretical physics, where objects have "higher
structure," rely on infinite-dimensional categories in place of $1$-dimensional
categories, and $\infty$-category theory has thusfar proved unamenable to
computer formalization.
  Using a new proof assistant called Rzk, which is designed to support
Riehl-Shulman's simplicial extension of homotopy type theory for synthetic
$\infty$-category theory, we provide the first formalizations of results from
$\infty$-category theory. This includes in particular a formalization of the
Yoneda lemma, often regarded as the fundamental theorem of category theory, a
theorem which roughly states that an object of a given category is determined
by its relationship to all of the other objects of the category. A key feature
of our framework is that, thanks to the synthetic theory, many constructions
are automatically natural or functorial. We plan to use Rzk to formalize
further results from $\infty$-category theory, such as the theory of limits and
colimits and adjunctions.
