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Understanding the homotopy type of a space often benefits from studying its values under homology theories. Every homology theory determines a Bousfield localization on the category of spaces — for example, rational cohomology leads to rationalization, while $F_p$-homology induces $p$-completion. In favorable cases, these localizations admit explicit descriptions via monadic resolutions. A key example is the $p$-completion $L_p X$ of a nilpotent space $X$, which can be expressed as an inverse limit: $L_p X = lim_n (\Omega^\infty F_p \otimes \Sigma^\infty)^{n+1}(X)$. Similarly, any nilpotent space $X$ can be expressed as the inverse limit $X = lim_n (\Omega^\infty MU \otimes \Sigma^\infty)^{n+1}(X)$ of its iterated $MU$-homology, giving rise to the unstable Adams—Novikov spectral sequence. A fundamental ingredient in proving this equivalence is Bousfield and Kan's celebrated principal fibration lemma.

In this talk, I will discuss how these ideas extend to oo-topoi and explain their application in motivic homotopy theory. In particular, I will sketch a proof of the principal fibration lemma in this broader context. This is joint work with Tom Bachmann and Anton Engelmann.