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Added a reference.
Can we say exactly what kind of pretopos the category of small presheaves on a category C is?
Is it a ΠW-pretopos, provided that PC is complete?
What is PC?
PC is the notation used in the article for the category of small presheaves.
ok, thanks. I think the $\Pi$ part is difficult. The $W$ part is easier. I don’t know offhand what $PC$ being complete gets you.
Yeah, I’d be doubtful that you can get lccc. Usually that sort of thing requires an adjoint functor theorem with small-generation hypotheses.
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