The aim of this thesis is to revisit a selection of classical topics in homotopy theory from the abstract point of view of higher categories. We take care to allow ourselves only model-independent arguments. We prove generalizations of the Hilton-Milnor theorem, the Freudenthal suspension theorem and the Barrat-Priddy-Quillen theorem in an arbitrary infinity-topos. We study polyhedral products in Cartesian closed infinity-categories and prove a formula for fat joins originally due to Porter.

In a second part, we study separately localizations and colocalizations of $\infty$-topoi from the point of view of fiberwise extensions. We show that Cartesian localizations can be partially extended to fiberwise constructions on bundles. On the other hand, we show that no non-trivial colocalization extends to a coreflective subfibration. The argument is a semantic translation of a no-go theorem of Shulman in homotopy type theory. We use our results to extend a fragment of Farjoun's theory of null and cellular spaces to pointed objects in an infinity-topos. We hope that our standpoint can shed light on the foundations of the theory and demonstrate how it can be applied to contexts beyond the realm of topology.