$\lambda$ -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). Finally, locally $\lambda$ -presentable categories are shown to be cellularly generated by the set of morphisms between $\lambda$ -presentable objects.