The Hilton–Milnor theorem in higher topoi

Abstract

In this note, we show that the classical theorem of Hilton–Milnor on finite wedges of suspension spaces remains valid in an arbitrary $\infty$-topos. Our result relies on a version of James' splitting proved in [Devalapurkar and Haine, Doc. Math. 26 (2021), 1423–1464] and uses only basic constructions native to any model of $\infty$-categories.