Continuous Functions on Final Coalgebras

It can be traced back to Brouwer that continuous functions of type StrA→B, where StrA is the type of infinite streams over elements of A, can be represented by well founded, A-branching trees whose leafs are elements of B. This paper generalises the above correspondence to functions defined on final coalgebras for power-series functors on the category of sets and functions. While our main technical contribution is the characterisation of all continuous functions, defined on a final coalgebra and taking values in a discrete space by means of inductive types, a methodological point is that these inductive types are most conveniently formulated in a framework of dependent type theory.