Fiberwise extensions of localizations and colocalizations

In this talk, I will discuss the problem of extending functors to fiberwise constructions in the setting of higher topoi. I shall explain the relevance of such extensions both from a practical point of view, and through the lens of the internal logic of the topos. After discussing some known results about fiberwise extensions of localization functors, we will focus on the dual situation of cellularizations. We will see that extending such colocalizations is almost never possible, and relate our result to classical statements by Chachólski and Hernández, and to a no-go theorem of Shulman in homotopy type theory. Along the way, emphasis will be put on the pertinence of higher categorical language to formalize the problem of fiberwise constructions.