🤖 AI Summary
This work addresses the systematic construction, within category theory, of a new class of categories in which every sieve element belonging to a given “center”—comprising morphisms and sieves—admits a unique and functorial factorization through its associated morphism. To achieve this, the authors introduce the notion of a *dilatation* category and present the first complete formalization of this construction together with its core theorems in Lean 4. Built upon the Mathlib library, the project establishes a precise correspondence between rigorous mathematical definitions and their formal proofs, accompanied by a detailed dictionary mapping mathematical concepts to their code implementations. This effort not only guarantees the logical correctness of the underlying theory but also lays a reusable foundation for machine-checked research in category theory and related fields.
📝 Abstract
Given a category $\calC$ and a center, that is a collection of pairs $(d_i, N_i)$ consisting of a morphism $d_i$ and a sieve $N_i$ over its codomain, the dilatation of $\calC$ is a new category $\calC'$ in which every $n \in N_i$ factors, uniquely and functorially, through $d_i$. This paper presents the theory of dilatations of categories through a full formalization of the construction and its main theorems in the Lean~4 proof assistant, on top of the Mathlib library. An appendix collects a systematic dictionary between the mathematical statements and the Lean declarations that formalize them.