A pair of monads in Topology
Keywords:
monad, Eilenberg-Moore algebras, stably compact spaces, distributive lattices, frames and locales, coherent frames, compactifications, prime filters, idealsAbstract
Two monads of interest arise from the dual adjunction between the category of topological spaces and that of (bounded)
distributive lattices. These are the open prime filter monad and the ideal lattice monad. It is known that the ideal lattice monad
induces the ideal frame comonad on the category of frames. We show that this ideal frame comonad can be paired with the open
prime filter monad via the open set-spectrum adjunction. From this, we give a new proof of the equivalence between the category
of stably compact spaces and that of stably compact frames on one hand, and that of compact Hausdorff spaces and compact regular frames on the other. We show, among other things, how the Čech-Stone compactifications in Pointfree Topology and Pointset
Topology relate to each other in this particular context.
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