Final Coalgebras from Corecursive Algebras

8 years 9 days ago

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We give a technique to construct a ﬁnal coalgebra in which each element is a set of formulas of modal logic. The technique works for both the ﬁnite and the countable powerset functors. Starting with an injectively structured, corecursive algebra, we coinductively obtain a suitable subalgebra called the “co-founded part”. We see – ﬁrst with an example, and then in the general setting of modal logic on a dual adjunction – that modal theories form an injectively structured, corecursive algebra, so that this construction may be applied. We also obtain an initial algebra in a similar way. We generalize the framework beyond Set to categories equipped with a suitable factorization system, and look at the examples of Poset and Set op .

Paul Blain Levy

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