Injectivity in locally presentable categories

Jiri Adámek – 12 March 1997

In a joint work of J.Rosicky, F. Borceux and J. Adamek, full subcategories of locally presentable categories are characterized, which can be presented by injectivity with respect to a set M of morphisms. If all domains and codomains of M-morphisms are k-presentable, such full subcategories are called k-injectivity classes. The characterization is as follows: If K is a locally k-presentable category,then k-injectivity classes are precisely the classes closed under products, split subobjects and k-filtered colimits in K. This sharpens the previous result (published by J.Adamek and J.Rosicky in 1993 in the Transactions of AMS) characterizing small-injectivity classes in locally presentable categories as classes closed under products, split subobjects and,for some cardinal k ,under k-filtered colimits. The latter theorem has been recently generalized by H.Hu and M.Makkai to a characterization of small-cone-injectivity classes as classes closed under split subobjects and, for some cardinal k, under k-filtered colimits. We show that this last result cannot be sharpened in the above manner. This leaves the following question open: Characterize, for a given cardinal k, classes of structures axiomatizable by geometric theories. in k-ary logic.