Universal Properties In Algebraic Geometry

Universal properties beautifully describe localisations in theory, but in practice they can’t carry the full weight of algebraic geometry. When formalising proofs in Lean, mathematicians discovered that relying solely on universal properties forces tedious diagram chases and ad-hoc constructions. A more practical characterisation—focused on invertibility, representation, and kernel behaviour—turned out to be far more effective, revealing how foundational results sometimes require new mathematical ideas.