nLab coalgebraic model for modal logic

Idea

Modal logics are said to be the logic of relational structures. The Kripke frame semantics of modal logics provides a clear picture of this:

Recall: A frame for the basic modal language $\mathcal{L}_\omega(1)$ is a pair $\mathfrak{F} = (W,R)$ with $W$ a non-empty set and $R$ a binary relation on $W$.

At their ‘crudest’ the objects studied are just sets and binary relations on them. In the basic ‘many worlds’ interpretation, a world $w$ satisfies a modal formula $\Box\phi$ if all worlds $w\prime$ accessible from $w$ (that is satisfying $wRw\prime$) satisfy $\phi$.

Now convert $R$ into a mapping $\rho : W\to \mathcal{P}(W)$, where $\rho(w)$ is the set of worlds accessible from $w$. It is easy to rephrase the notion of ‘satisfies’ in terms of $\rho$, but $(W,\rho)$ is a coalgebra for the power set endofunctor on $Set$, so Kripke frames provide a simple example of a coalgebraic model for modal formulae. There are many other types of coalgebras for endofunctors and many lead to modal logics.

References

• Corina Cirstea, Alexander Kurz, Dirk Pattinson, Lutz Schröder and Yde Venema, Modal logics are coalgebraic (pdf)