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• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeMay 13th 2020
• (edited May 13th 2020)

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?

• CommentRowNumber2.
• CommentAuthorDavidRoberts
• CommentTimeMay 13th 2020

What is PC?

• CommentRowNumber3.
• CommentAuthorDmitri Pavlov
• CommentTimeMay 13th 2020

PC is the notation used in the article for the category of small presheaves.

• CommentRowNumber4.
• CommentAuthorDavidRoberts
• CommentTimeMay 14th 2020

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.

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeMay 19th 2020

Yeah, I’d be doubtful that you can get lccc. Usually that sort of thing requires an adjoint functor theorem with small-generation hypotheses.