Synthetic Stone Duality: A Synthetic Approach to Condensed Mathematics

Synthetic Stone duality is an extension of Homotopy Type Theory (HoTT) by 5 well-chosen axioms. These axioms are validated by the interpretation of HoTT in the higher topos of light condensed anima. Therefore, any results we prove in synthetic Stone duality can be interpreted as a result about light condensed anima.
First, we will explain the general concept of synthetic mathematics, with an emphasis on geometry, HoTT, higher topoi and cohomology. Then we will present the 5 axioms of Synthetic Stone duality and give detailed proofs of some of their elementary consequences, to give a feeling of how it is to work with them. 
We will then give an overview of our synthetic version of Theorem 3.2 from Peter Scholze Lecture Notes on condensed mathematics (itself adapted from Roy Dyckhoff). Our version states that the cohomology of a compact Hausdorff space with countably presented coefficients can be computed from a cover of X by a Stone space.
If time permits, we will sketch applications of this result to the shape modality, which gives a convenient interface between topological spaces and their homotopy types.