This talk will be a report on my series of papers with G. Biedermann, E. Finster and A. Joyal revisiting Goodwillie calculus in terms of higher topoi.
I will recall a number of structural analogies between topoi and commutative rings (free construction, triple image factorization, Jacobson radical and nilradical, ideals and their products, completion towers…) and explain how this can be used to recovers many variations of Goodwillie calculus (unstable, stable, for algebras…) as well as Weiss’ orthogonal calculus.