For the next seminar of the Cosynus team, we are pleased to welcome Roman Kniazev, who will talk about Topos-theoretic point of view on directed spaces.
Abstract: Topos theory provides a very general framework for the investigation of connections between geometry and logic. For example, given a topological space X, category of sheaves on a category of open sets of X is a Grothendieck topos, whose internal logic can be applied for reasoning. We attempted to adapt the semantical counterpart for directed spaces: we considered a trace space as a base category for a sheaf topos and found some topos-theoretic invariants. Also, we studied the representation of time in a trace space and the additional structure it brings to the topos. Besides that, we will briefly discuss further research directions, such as types of the internal language and transition to a global description (like small/big topos). This is a report based on my internship in the team.