Properties of the $[S,Sets]$, where $S$ is small

Question

I am very new to topos theory and am interested in a couple little properties of a certain elementary topos.

Suppose $S$ is a small concrete category.

Then I was wondering.. which of there properties does the topos $[S:Sets]$ have:

• (co)Wellpowered?
• (co)Complete?
• small (co)generating set *(particularly, what is it, would it be the initial object)?
• Has enough projective objects (particularly, is the initial object projective)?

So far, I know its locally small, so that's a good start...

I know this is a long question, but I don't know where to start..

Answer 1

Presheaf toposes are:

• well-powered and co-well-powered,
• complete and cocomplete,
• have a (dense) generating set (namely, the representables) and a coseperator (power object of the disjoint union of the representables), and
• have enough projective objects (namely, the coproducts of representables).

Conversely, a locally small complete/cocomplete elementary topos that satisfies a strong version of "enough projectives" must be a presheaf topos, by a theorem of Bunge.