Generators in a topos

Question

What is known about (sets of) generators in an elementary topos ? In particular, does an elementary/Grothendieck topos have a dense set of generators ?

Answer 1

The fact that any Grothendieck topos has a dense generating set is more or less a tautology: after all, for any Grothendieck topos $\mathcal{E}$, there exist a small category $\mathcal{C}$ and a fully faithful embedding $\mathcal{E} \to [\mathcal{C}^\mathrm{op}, \mathbf{Set}]$, and we can always choose $\mathcal{C}$ so that the image of $\mathcal{E}$ contains the representable presheaves.

More generally, the following is a consequence of Giraud's theorem:

A complete/cocomplete locally small elementary topos is a Grothendieck topos if and only if it has a (small) generating set.

Of course, that means there are elementary toposes without generating sets. For instance, if $G$ is a non-small group, then $[G, \mathbf{Set}]$ is a complete/cocomplete locally small elementary topos that does not have a generating set.

Answer 2

The effective topos is a counterexample. The topos of sets is embedded into the effective topos as its subcategory of indecomposable projectives. Meanwhile the effective topos lacks many infinite colimits.