  Non-well-founded trees are used in mathematics and computer science, for
modelling non-well-founded sets, as well as non-terminating processes or
infinite data-structures. Categorically, they arise as final coalgebras for
polynomial endofunctors, which we call M-types. In order to reason about trees,
we need the notion of path, which can be formalised in the internal logic of
any locally cartesian closed pretopos with a natural number object. In such
categories, we derive existence results about M-types, leading to stability of
locally cartesian closed pretoposes with a natural number object and M-types
under slicing, formation of coalgebras (for a cartesian comonad), and sheaves
for an internal site.
