Peter Freyd showed that the real interval [0, 1] is a final coalgebra for a functor on sets equipped with two points, which sends such a set to the 'wedge' of two copies of itself, identifying the second point of the first copy with the first point of the second copy.

The hyperfinite II_1 factor has trace values in [0, 1] and arises from a process of completing a union of finite subalgebras by a form of doubling discussed here.

Is there then a similar coalgebraic characterisation of the factor?