An Elementary Approach to Truncations

We study truncated objects using elementary methods. Concretely, we use universes and the resulting natural number object to define internal truncation levels and prove they behave similar to standard truncated objects. Moreover, we take an elementary approach to localizations, giving various equivalent conditions that characterize localizations and constructing a localization out of a sub-universe of local objects via an internal right Kan extension. We then use this general approach, as well as an inductive approach, to construct truncation functors. We use the resulting truncation functors to prove classical results about truncations, such as Blakers-Massey theorem, in the elementary setting. We also give examples of non-presentable $(\infty, 1)$-categories where the elementary approach can be used to define and compute truncations. Finally, we turn around and use truncations to study elementary $(\infty, 1)$-toposes and show how they can help us better understand subobject classifiers and universes