A Formalization of Higher-Order Categories in Lean 4

Higher-order categories are a generalization of ordinary category theory in which morphisms can exist at multiple levels. They play a central role in modern mathematics, particularly in algebraic topology and homotopy theory. 

This work focuses on the single-sorted and many-sorted presentations of higher-order categories, both finite-dimensional ($n$-categories) and infinite-dimensional ($omega$-categories). We study the relations between categories of different dimensions, as well as the relations between these presentations.

On the formal side, we develop a formalization in Lean~4 of these definitions and of some of the relations between them. Along the way, we discuss implementation aspects of the formalization, including design choices and considerations specific to its underlying dependent type theory.