Arithmetic topology is an idea that goes back to ideas of Mumford and
Mazur from the 1960s, when they suggested that number fields behave as if they are 3-manifolds, and nonarchimedean places behave as if they were embedded knots therein. I want to describe a program to develop some homotopical "shadows" of arithmetic topology that are sufficient to define arithmetic field theories in the sense of Kim. More precisely, I want to issue the following challenge: Construct the stratified homotopy type of the Ran space of a compactification of Spec O_K for a number field K. I will try to explain how the story of "exodromy" offers some interesting  insights and hurdles already at Step 1.

Note: We will continue chatting with Clark after the talk and anyone who is interested is welcome to join.