For anyone other than Chris Townsend or Steve Vickers,
I think it is simpler just to ask about the forgetful and free functors
$\mathbf{Frm}\rightleftarrows\mathbf{Dcpo}$
between the categories of frames and directed complete partial orders.
Not altogether surprisingly, $\mathbf{Frm}$ is the category of algebras
for the monad over $\mathbf{Dcpo}$.
There is a general question
for any adjunction $\mathcal{C}\rightleftarrows\mathcal{D}$:
Suppose we replace $\mathcal C$ by the category of algebras
$\mathcal{C}'$ for the monad over $\mathcal D$ that is induced
by the adjunction,
then $\mathcal D$ by the category $\mathcal{D}'$ of coalgebras
for the comonad on $\mathcal{C}'$, and so on.
In fact, this stabilises with the algebras over the coalgebras
under the algebras, ie the next step is $\mathcal{D}''\cong\mathcal{D}'$.
Indeed, only two steps are needed ($\mathcal{C}''\cong\mathcal{C}'$)
if we started with categories in which idempotents split.
Steve Lack first gave me the proof of this.
Now suppose in addition that the base category $\mathcal{D}$, in Chris's
case $\mathbf{Dcpo}$, has finite products and
the monad has a strength, which it does in the case of $\mathbf{Frm}$.
Then the category $\mathcal{D}'$ of coalgebras under the algebras
also has finite products.
These are the coproducts of algebras over $\mathbf{Loc}$ in Chris's question.
At least so it says in some notes of mine called
underlyingset/universal possibly from March 2007.
The proof is quite complicated and works with strong monads in the abstract
rather than frames, though it was motivated by exactly this question.
I might be persuaded to re-upload these notes into my brain
to see whether the proof is correct.
The question of the existence of coequalisers of algebras for a monad
(or equalisers in $\mathcal{D}'$ in my setting)
is a notoriously difficult one.
The objects of the category $\mathbf{Dcpo}'$ in my notation above
(which is the opposite of the category of algebras in Chris's question)
are called localic locales by Steve Vickers and colocales by me.
If any progress is to be made with the investigation of this category
and the furtherance of the motivations behind it,
or even in getting more than three people interested,
I think some more concrete description is needed of the objects.
Even abstractly, an object of $\mathcal C'$ admits at most one morphism
that makes it a coalgebra (object of $\mathcal D'$).
Hence a colocale is a special kind of frame,
whilst of course a frame is a special dcpo.
The adjunctions that Chris considers in his question
contain the monadic one that gives my programme
Abstract Stone Duality
its name as full subcategories.
The latter relates continuous frames to locally compact locales,
so any continuous frame is a colocale.
In particular, the free frame on any semilattice qua semilattice
is continuous and therefore already a colocale.
My feeling is that Steve Vickers has already provided
enough tools in his investigations of presentations of frames
and of powerlocales, along with the methods of Formal Topology,
to give a useful characterisation of colocales.
Using this, the colocale structure on (the free frame on)
any semilattice would be described by some kind of modal logic.
Given the representation of a frame as a cover relation on a semilattice
(or a nucleus on the free frame on this semilattice),
the question would therefore appear to amount to asking
whether the cover relation is compatible with the modal logic.
How can colocales be special frames but also generalise
locally compact spaces?
The mis-match comes from the application of the functor:
the locally compact space corresponds to the colocale
structure on its continuous frame of opens.
Studying bases would eliminate this excessively free
structure.
I look forward to seeing the results of Chris's investigations of this.
