(lambda)-presentable Morphisms, Injectivity and (Weak) Factorization Systems

✍ Scribed by Michel Hébert

• Book ID: 106347013
• Publisher: Springer
• Year: 2006
• Tongue: English
• Weight: 530 KB
• Volume: 14
• Category: Article
• ISSN: 0927-2852
• DOI: 10.1007/s10485-006-9024-9

✦ Synopsis

We show that in a locally λ-presentable category, every λ m -injectivity class (i.e., the class of all the objects injective with respect to some class of λ-presentable morphisms) is a weakly reflective subcategory determined by a functorial weak factorization system cofibrantly generated by a class of λ-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 λ-filtered colimits play the role of the transfinite compositions in the injectivity case. λ-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 λ-presentable categories are shown to be cellularly generated by the set of morphisms between λ-presentable objects.