Title: A Topology on Categories of Locally Ringed Space and Applications to Arithmetic Geometry

Abstract: In this talk we present some relevant background on locally ringed spaces and consider a problem that appears in arithmetic geometry. We then use this problem to motivate a class of functors that arise in arithmetic geometry, provide examples of these functors, and show how these functors can induce a Grothendieck topology on their codomains. We then use this topology and its topos of sheaves to make progress towards our motivating problem.