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Morgan Rogers (he/him) (Apr 28 2023 at 11:32):Lurie developed higher topos theory to parallel 1-toposes, but didn't get as far as logic. People have developed internal logic for these objects, but what fragment of logic is needed to describe the theories classified by these objects? That is, which fragment of logic is preserved by the inverse images of -geometric morphisms, so that representable pseudofunctors into on the -category of -toposes can be expressed as respective internal models of theories in that logic? I presume it will be some HoTT extension of geometric logic, but is there something more precise that is already known?
Leopold Schlicht (Apr 28 2023 at 18:36):