In the context of modal logic, by possible world semantics one means the interpretation of the possibility and necessity modalities as existential quantification and universal quantification over a context of “possible worlds”. Such a model is sometimes referred to as a Kripke frame.
For formalization of this in terms of hyperdoctrines/dependent type theory see at necessity and possibility – In first-order logic and dependent type theory.
Under categorical semantics such a context interprets as an object of some category whose interpretation is similar to that of a probability space in measure theory – which is one sensible formalization of possible worlds (e.g. Toronto-McCarthy 10, slide 23).
Stanford Encyclopedia of Mathematics Possible Worlds
Stanford Encyclopedia of Mathematics, Modal Logic – Possible worlds semantics
Neil Toronto, Jay McCarthy, From Bayesian Notation to Pure Racket, 2010 (pdf)
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