##
Final coalgebras from corecursive algebras

### Paul Levy

We give a technique to construct a final coalgebra in which each element
is a set of formulas of modal logic. The technique works for both the
finite and the countable powerset functors. Starting with an
injectively structured, corecursive algebra, we coinductively obtain a
suitable subalgebra called the "co-founded part". We see---first with
an example, and then in the general setting of modal logic on a dual
adjunction---that modal theories form an injectively structured,
corecursive algebra, so that this construction may be applied. We also
obtain an initial algebra in a similar way.

We generalize the framework beyond Set to categories equipped with a
suitable factorization system, and look at the examples of Poset and Set^op.