Node Deletion Theorem: a precise rule for pruning nodes in recursive data types

Abstract

Fixed-point constructions in category theory provide a unifying framework for understanding inductive and recursive definitions. This paper presents a self-contained study of fixed-point constructions for ω-continuous endofunctors on large categories, including the existence and uniqueness of fixed points (initial algebras), the construction of joint fixed points for pairs of commuting functors, and a "node deletion" property describing how removing an object from a stable category affects the fixed-point structure. All results are stated and proved in standard category-theoretic language with careful handling of size issues via Grothendieck universes.