Higher Topos Theory and Goodwillie Calculus

Explore a detailed lecture from the Topos Institute Colloquium that delves into the comparison between Lurie's higher topos theory and Grothendieck's topos theory, examining their key similarities and differences. Learn about the unique aspects of ∞-category of parametrised spectra as a higher topos and its significance in lacking an analog in Grothendieck topos theory. Understand the Goodwillie tower's relationship with left exact localization of higher topos and its parallel to completion tower of ideals in commutative rings. Discover how a topos exhibits dual characteristics as both a space and a ring, and examine the category of logoi's shared properties with commutative rings. Developed in collaboration with Mahieu Anel, Georg Biedermann, and Eric Finster, this advanced mathematical exploration provides deep insights into contemporary topos theory.