Proper maps and quasi-adjoints
Keywords:
Proper maps, quasi-adjoints, Smyth hyperspace, Hoare hyperspace, continuous valuations, projective limitsAbstract
We show that a continuous map is proper if only if it has a quasi-adjoint, namely a left adjoint in the Kleisli category of the Smyth hyperspace monad. As applications, we show that the Smyth and Hoare hyperspace functors, as well as the continuous valuation functor, preserve proper maps. We also show that any projective limit of consonant sober spaces with proper bonding maps is consonant and sober. No separation axiom is assumed.
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