categories: Re: effective topos

[Thomas Streicher <streicher@mathematik.tu-darmstadt.de>]

> 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