The double contravariant powerset monad in the Goguen category of fuzzy sets

A monad is constructed in the Goguen category of fuzzy sets valued in a unital quantale, which is an analog of the double contravariant powerset monad in the category of sets. With help of this monad it is proved that the Goguen category of fuzzy sets is dually monadic over itself.Comment: 21 page

Hofmann-Mislove theorem for approach spaces

The Hofmann-Mislove theorem says that compact saturated sets of a sober topological space correspond bijectively to open filters of its open set lattice. This paper concerns an analogy of this result for approach spaces. To this end, the notion of compact functions of approach spaces is introduced. Such functions are an analogue of compact subsets in the enriched context. It is shown that for a sober approach space, the inhabited and saturated compact functions correspond bijectively to the proper open $[0,\infty]$-filters of the metric space of its upper regular functions, which is an analogy of the Hofmann-Mislove theorem for approach spaces.Comment: 24 page