  We present a hybrid dynamical type theory equipped with useful primitives for
organizing and proving safety of navigational control algorithms. This type
theory combines the framework of Fu--Kishida--Selinger for constructing linear
dependent type theories from state-parameter fibrations with previous work on
categories of hybrid systems under sequential composition. We also define a
conjectural embedding of a fragment of linear-time temporal logic within our
type theory, with the goal of obtaining interoperability with existing
state-of-the-art tools for automatic controller synthesis from formal task
specifications. As a case study, we use the type theory to organize and prove
safety properties for an obstacle-avoiding navigation algorithm of
Arslan--Koditschek as implemented by Vasilopoulos. Finally, we speculate on
extensions of the type theory to deal with conjugacies between model and
physical spaces, as well as hierarchical template-anchor relationships.
