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Just kind of messing around here, but this seems like a natural common generalization of coherence spaces and finiteness spaces.
Nice. I think Girard-Loader totality spaces are $\kappa=\{1\}$, right?
Also, I thought these kinds of spaces will be always be *-autonomous because they’re built by double-gluing-along-hom-functors + orthogonality, right? (Section 5 of Glueing and orthogonality for models of linear logic, Hyland + Schalk, doi).
Thanks for the pointer! Yes, it looks like totality spaces are unary spaces, except that totality spaces (in the reference I found) include the additional assumption that every element is contained in some unary subset and also some co-unary subset.
Regarding double gluing and orthogonality, maybe, I’m not familiar with that and don’t have the time to understand it right now. Feel free to add to the entry!
Thanks! Maybe it would be better to have a separate page all about this kind of orthogonality, with interlinks?
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