Ivan Di Liberti - Towards higher topology
Channel: Institut des Hautes Études Scientifiques (IHÉS)
Date: 01/07/2021

Talk at the school and conference “Toposes online” (24-30 June 2021):
https://aroundtoposes.com/toposesonline/

Slides:
https://aroundtoposes.com/wp-content/uploads/2021/07/DiLibertiSlidesToposesOnline.pdf

We categorify the adjunction between locales and topological spaces, this amounts to an adjunction between (generalized) bounded ionads and topoi. We show that the adjunction is idempotent, yielding a biequivalence between topoi with enough points and sober bounded ionads. We relate this adjunction to the Scott adjunction, which was discussed from a more categorical point of view in previous work. We hint at the logical implications of this result in terms of completeness/reconstruction theorems.