Abstract: Given a reduced crystallographic root system $R$ with the associated Weyl group $W$, the Weyl chambers from a fan and then give out a complex toric variety and its real part $X_R$. We will see that the underlying topological space $X_R/W$ is contractible.

Abstract: Elementary toposes are categories that share many properties of the category of sets. Every elementary topos has an internal language which is a version of typed intuitionistic higher-order logic obtained from the lattices of subobjects. It thus makes sense to speak of an elementary topos as a local set theory. The notion of elementary infinity-topos generalises this concept to infinity-categories. It is conjectured that the internal language of an elementary infinity-topos is a version of Homotopy Type Theory. Thus, it makes sense to speak of elementary infinity toposes as univalent dependent type theories, where instead of the lattice of subobjects we have a universal infinity-groupoid of all (small) objects. In the talk, I will introduce and motivate the notion of elementary infinity-topos, focussing on the concept of object classifiers and I will sketch the progress that has been made so far towards proving the conjecture. I will not assume any prior knowledge about toposes, logic or type theory.