Automata in toposes, and general Myhill-Nerode theorems

We extend the functorial approach to automata by Colcombet and Petrişan [arXiv:1712.07121] from the category of sets to any elementary topos with a natural number object and establish general Myhill-Nerode theorems in our setting. As a special case we recover the result of Bojańczyk, Klin and Lasota [arXiv:1402.0897] for orbit-finite nominal automata by considering automata in the Myhill-Schanuel topos of nominal sets.