Nima Rasekh - Every Elementary Higher Topos has a Natural Number Object
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/NimaSlidesToposesOnline.pdf

One key aspect of elementary topos theory is the existence of a natural number object. While it does not exist in every elementary topos (such as finite sets) we often need it to study more advanced aspects of topos theory (such as free monoids). In this talk we see how in the higher categorical setting, the existence of a natural number object can in fact be deduced from a small list of axioms that any reasonable definition of elementary higher topos should satisfy, hence proving that every elementary higher topos has a natural number object. We will observe how the proof involves ideas from algebraic topology, elementary topos theory and homotopy type theory.