A generalized Blakers–Massey theorem

We prove a generalization of the classical connectivity theorem of Blakers–Massey, valid in an arbitrary higher topos and with respect to an arbitrary modality, that is, a factorization system (L,R) in which the left class is stable by base change. We explain how to rederive the classical result, as well as the recent generalization of Chachólski, Scherer and Werndli (Ann. Inst. Fourier 66 (2016) 2641–2665). Our proof is inspired by the one given in homotopy‐type theory in Favonia et al. (2016).