Boxprojective

Boxprojective is a term used in category theory and homological algebra to denote a variant of projectivity relative to a construction called Box. The Box construction is an endofunctor on a chosen category that provides a context for defining relative lifting properties. The idea is to replace ordinary epimorphisms with Box-epimorphisms, which are morphisms that become epimorphic after applying Box.

Formally, let C be a category with an endofunctor Box: C → C. A morphism e: E → B is called a Box-epimorphism if Box(e): Box(E) → Box(B) is an epimorphism in C. An object P in C is called Box-projective if for every Box-epimorphism e: E → B and every morphism f: P → B, there exists a morphism g: P → E such that e ∘ g = f. This lifting property mirrors the defining property of ordinary projective objects, but relative to the Box-structure.

Special cases clarify the concept. If Box is the identity functor, Box-epimorphisms are ordinary epimorphisms and Box-projective objects coincide with ordinary projective objects. In module categories, Box can be the tensor product with a fixed module K, so Box(M) = M ⊗K. Here, a Box-epimorphism is an epimorphism after tensoring with K, and a Box-projective module P satisfies the lifting property with respect to such Box-epimorphisms.

Properties and uses. Box-projective objects are typically closed under finite direct sums and retracts. They give rise to relative projective resolutions and are central in relative homological algebra, cotorsion theories, and certain model category structures, where the Box operation encodes a chosen context for measuring lifting properties.

Note that the exact definition and choice of Box can vary by literature; Boxprojective is a general framework rather than a single canonical notion.