Injectivity with respect to morphisms having $\lambda$-presentable domains and codomains is characterized: such injectivity classes are precisely those closed under products, $\lambda$-directed colimits, and $\lambda$-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.
Keywords: locally presentable category, injectivity class, geometric logic.
2000 MSC: 18C35, 03C99.
Theory and Applications of Categories, Vol. 10, 2002, No. 7, pp 148-161.