More on injectivity in locally presentable categories
Jiřı́ Rosický, Jiřı́ Adámek, Francis Borceux · Theory and applications of categories · 2002
Injectivity with respect to morphisms having #-presentable domains and codomains is characterized: such injectivity classes are precisely those closed under products, #-directed colimits, and #-pure subobjects. This sharpens the result of the first two authors (Trans. Amer. Math. Soc. 336 (1993), 785-804). In contrast, for geometric logic an example is found of a class closed under directed colimits and pure subobjects, but not axiomatizable by a geometric theory. A more technical characterization of axiomatizable classes in geometric logic is presented. 1.