Topos with enough projectives

## Abstract In this paper, we provide a site characterization of topoi with enough projectives. More concretely, we prove that a Grothendieck topos has enough projective objects if and only if it is a sheaf topos over a κ-extensive site for some regular cardinal κ. These observations lead us to generalize the notion of Gaeta topoi to κ-Gaeta topoi. We will see that 2-Gaeta topoi are precisely presheaf topoi.