In the study of iterative theories we use the concept of a strongly locally presentable category. This is an extensive, locally finitely presentable category such that a. hom-sets of finitely presentable objects are finite and b. a strong quitent of a finitely presentable object is finitely pres. I would appreaciate knowing whether the effective topos has all these properties. Thanks, Jiri Adamek
A locally finitely presentable category has all colimits the effective topos has not! Nor has the effective topos filtered colimits as otherwise it were cocomplete. Moreover, the global sections functor Gamma from Eff to Set doesn't have a left adjoint Delta but instead a right adjoint Nabla. Of course, one might ask nevertheless which objects are finitely presentable taking only account of existing filtered colimits. Alas, there the answer isn't easy at all (for me at least) and I don't even see why it should be interesting. Thomas Streicher