The category-theoretic nature of general frames for modal logic is
explored. A new notion of ``modal map'' between frames is defined,
generalizing the usual notion of bounded morphism/p-morphism. The
category $\Fm$ of all frames and modal maps has reflective
subcategories $\CHFm$ of compact Hausdorff frames, $\DFm$ of
descriptive frames, and $\UEFm$ of ultrafilter enlargements of frames.
All three subcategories are equivalent, and are dual to the
category of modal algebras and their homomorphisms.
The ultrafilter enlargement of a frame $A$ is shown to be the
free compact Hausdorff frame generated by $A$ relative to $\Fm$. The
monad $\E$ of the resulting adjunction is examined and its
Eilenberg-Moore category is shown to be isomorphic to $\CHFm$. A
categorical equivalence between the Kleisli category of $\E$ and $\UEFm$
is defined from a construction that assigns to each frame $A$ a frame
$A^*$ that is ``image-closed'' in the sense that every point-image
$\{b:aRb\}$ in $A$ is topologically closed. $A^*$ is the unique
image-closed frame having the same ultrafilter enlargement as $A$.
These ideas are connected to a category $\W$ shown by S. K. Thomason to
be dual to the category of complete and atomic modal algebras and their
homomorphisms. $\W$ is the full subcategory of the Kleisli category of
$\E$ based on the Kripke frames.