This will be the first in a series of talks on infinity-category theory and its applications.

My aim for this first talk is to expose the audience to some of the basic homotopy theory of infinity-categories. Our first goal will be to prove Joyal's fundamental theorem that an infinity-category is a Kan complex if all its morphisms are invertible. Time permitting, I may also prove that a functor of infinity-categories is an equivalence if it is essentially surjective on objects and an equivalence on hom-spaces.

This is an informal seminar on arithmetic algebraic geometry and related fields.

The purpose is to present work in progress or to learn about some existing piece of mathematics. Typically the talks are parts of a longer series.