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The (higher) topos classifying $\infty$ -connected objects Joyal, André


Joint work with Mathieu Anel, Georg Biedermann and Eric Finster. I will present an application of Goodwillieâ s calculus to higher topos theory. The (higher) topos which classifies $\infty$-connected objects is formally the "dual" of the (higher) logos $S[U_\infty]$ freely generated by an $\infty$-connected object $U_\infty$. The logos $S[U_\infty]$ is a left exact topological localisation of the logos $S[U] = Fun[Fin, S]$ freely generated by an object $U$. We show that a functor $ Fin \to S$ belongs to $S[U_\infty]$ if and only if it is $\infty$-excisive if and only if it is the right Kan extension of its restriction to the subcategory of finite n-connected spaces $C_n \subset Fin$ for every $n \geq 0$. There is a morphism of logoi from $S[U_\infty]$ to the category of Goodwillie towers of functors $Fin \to S$, but we do not know if it is an equivalence of categories. We also consider the logos $S[U_\inftyâ ² ]$ freely generated by a pointed $\infty$-connected object $U'_\infty$ .

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