## Open maps of preordered sets

### Soichiro Fujii – 31 May 2023

A monotone map f : P → Q between preordered sets is open if the direct image under f of any upper set in P is an upper set in Q. In this talk, we investigate relationships between the category Ord of preordered sets and monotone maps, and its wide subcategory Open of preordered sets and open maps. The category Open has all small colimits created by the inclusion Open → Ord, but the inclusion functor does not seem to have a right adjoint. We impose a suitable size condition (relative to a cardinal κ) on preordered sets and obtain a full subcategory Open(κ) of Open. Then the inclusion functor Open(κ) → Ord does have a right adjoint, whose value at a preordered set is constructed by quotienting a set of “Kripke models” under “bisimilarity”. [Joint work with Richard Garner]

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