Lambda-presentable morphisms, injectivity and (weak) factorization systems

Abstract

We show that in a locally lambda-presentable category, every lambda(m)-injectivity class (i.e., the class of all the objects injective with respect to some class of lambda-presentable morphisms) is a weakly reflective subcategory determined by a functorial weak factorization system cofibrantly generated by a class of lambda-presentable morphisms. This was known for small-injectivity classes, and referred to as the "small object argument". An analogous result is obtained for orthogonality classes and factorization systems, where lambda-filtered colimits play the role of the transfinite compositions in the injectivity case. Lambda-presentable morphisms are also used to organize and clarify some related results (and their proofs), in particular on the existence of enough injectives (resp. pure-injectives).