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Ellis D. Cooper (Jul 08 2022 at 22:29):Aren't there general theorems about when a comma category (F,G) is a topos given conditions on the three categories involved?
Jon Sterling (Jul 09 2022 at 07:17):When F is the identity map, then the theory of Artin gluing applies. In that case, it is enough for the functor G to preserve binary pullbacks. I'm not sure what happens for general F.
Morgan Rogers (he/him) (Jul 09 2022 at 14:35):I think OP is asking for conditions on more general categories to produce a topos?
John Baez (Jul 09 2022 at 14:41):Yes: just to be ultra-explicit, Jon is implicitly assuming the categories are toposes, while Ellis was not.
Reid Barton (Jul 09 2022 at 15:06):Are there examples of interest where the categories involved are not topoi?
Mike Shulman (Jul 10 2022 at 04:14):If all three categories are toposes, and and are inverse images of geometric morphisms, then the comma category (the notation is somewhat deprecated, although the name "comma category" has stuck around) is a topos. Indeed, it is the cocomma object of the two geometric morphisms in the bicategory of topoi and geometric morphisms.
It seems unlikely to me that there would be a useful criterion for a comma category to be a topos without assuming the input categories are toposes.
John Baez (Jul 10 2022 at 04:48):Whew, so my naive sense of "you ain't gonna get something for nothing" is confirmed by someone who actually knows some topos theory.