projectivity

Projectivity is a mathematical term used across several areas to describe a property that enables lifting, extension, or reconstruction of structures in a controlled way. While the precise definition depends on the context, a common thread is the existence of choices or splittings that make constructions behave nicely, such as finding sections of maps or expressing objects as simpler building blocks.

In algebra, a left module P over a ring R is projective if every surjective module homomorphism f: M -> N and every homomorphism g: P -> N admit a lift h: P -> M with f ∘ h = g. Equivalently, P is a direct summand of a free module. Projective modules generalize free modules and play a central role in homological algebra, where they are used to build projective resolutions. Over a principal ideal domain, every finitely generated projective module is free; over more general rings, projective modules need not be free. In category-theoretic terms, projective objects satisfy a lifting property with respect to epimorphisms.

In category theory, an object P in a category is called projective if, for every epimorphism e: B -> C and every morphism f: P -> C, there exists a morphism g: P -> B with e ∘ g = f. This generalizes the module-theory notion and applies to many categories, including sets, groups, and sheaves. The concept underpins the construction of resolutions and the study of representability and exactness in categorical contexts.

In geometry, projective mathematics studies properties invariant under projective transformations, which preserve incidence rather than distances. Real and complex projective spaces, such as RP^n and CP^n, are formed by taking lines through the origin in a vector space and modding out by scalar multiplication. A projectivity (or projective transformation) is a bijection of a projective space induced by a linear automorphism.

Together, projectivity serves as a unifying idea for understanding how complex structures can be decomposed, lifted, or transformed while preserving essential relations.