Internal higher topos theory


We develop the theory of topoi internal to an arbitrary infinity-topos B. We provide several characterisations of these, including an internal analogue of Lurie’s characterisation of infinity-topoi, but also a description in terms of the underlying sheaves of infinity-categories, and we prove a number of structural results about these objects. Furthermore, we show that the infinity-category of topoi internal to B is equivalent to the infinity-category of infinity-topoi over B, and use this result to derive a formula for the pullback of infinity-topoi. Lastly, we use our theory to relate smooth geometric morphisms of infinity-topoi to internal locally contractible topoi.

ArXiv preprint
Louis Martini
Louis Martini
PhD fellow in mathematics

I am a fourth year PhD student in the Department of Mathematical Sciences at the Norwegian University of Science and Technology in Trondheim, Norway.