Cofree frame given a suplattice

Question

Does the forgetful functor from frames to suplattices have a right adjoint? Does it have a left adjoint?

I am thinking it has a right adjoint, and that there is a "cofree" frame.

Answer 1

The existence of a left adjoint is almost trivial. To construct the free frame on a suplattice $X$, just take the free frame on the underlying set and then impose relations saying sups agree with the suplattice structure of $X$. (The only nontrivial input here is the fact that the free frame on a set exists, despite the fact that frames have operations of arbitrarily large arity. This is because you can use the distributive law to write any element as a join of meets of the generators.)

There is no right adjoint because the forgetful functor does not preserve colimits. In particular, it does not preserve initial objects: the initial frame has two elements and the initial suplattice has one.