# Local Realizability Toposes and a Modal Logic for Computability

### S. Awodey, L. Birkedal, D.S. Scott

**Abstract**

This work is a step toward developing a logic for types and
computation that includes both the usual spaces of mathematics and
constructions and spaces from logic and domain theory. Using
realizability, we investigate a configuration of three toposes, which
we regard as describing a notion of relative computability. Attention
is focussed on a certain local map of toposes, which we study first
axiomatically, and then by deriving a modal calculus as its internal
logic. The resulting framework is intended as a setting for
the logical and categorical study of relative computability.