On Goodwillie calculus via model structures on excisive functors and model structure for n-excisive functors:
Georg Biedermann, Boris Chorny, Oliver Röndigs, Calculus of functors and model categories, Advances in Mathematics 214 (2007) 92-115 [arXiv:math/0601221, doi:10.1016/j.aim.2006.10.009]
Georg Biedermann, Oliver Röndigs, Calculus of functors and model categories II, Algebr. Geom. Topol. 14 (2014) 2853-2913 [arXiv:1305.2834v2, doi:10.2140/agt.2014.14.2853]
Proving the Blakers-Massey theorem in any -topos and with the (n-connected, n-truncated) factorization system allowed to be replaced by more general modalities:
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