MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

82 Robert Goldblatt7.7 Modal Logic for CoalgebrasThe mathematics of modality has recently been applied in theoretical computerscience to the category-theoretic notion of a coalgebra. This application is still“under construction” but can already be seen as a natural evolution of some ofthe trends that have been described in this article.If T : C → C is a functor on a category C, then an algebra for T is definedto be a pair (A, τ A ) comprising a C-object A and a C-arrow τ A from T A to A.A morphism from T -algebra T A τ A−→ A to T -algebra T Bτ B−→ B is a C-arrowA −→ f B such that f ◦ τ A = τ B ◦ T f. This is a categorization of the classical notionof a homomorphism of abstract algebras. To explain that properly is beyondour scope, and the interested reader should consult such sources as [Mac Lane,1971, especially §VI.8] and [Manes, 1976] for enlightenment. But the idea canbe illustrated by considering the category Malg of (normal) modal algebras andtheir homomorphisms (section 6.5), which is the category of algebraic models ofthe smallest normal modal logic K. There is a functor T K : Set → Set on thecategory of sets and functions such that T K A is the underlying set of the free modalalgebra F A generated by the set A. If A is itself the underlying set of some modalalgebra A, then there is a unique function T K A τ A−→ A that is a homomorphismfrom F A onto A leaving members of A fixed. The map A ↦→ (A, τ A ) then gives anisomorphism between Malg and the category of T K -algebras and their morphisms.Note that free modal algebras can be constructed as Lindenbaum algebras: ifa set A is viewed as a collection of propositional variables, then T K A is the setof equivalence classes of propositional modal formulas in these variables, withformulas α and β being equivalent when α ↔ β is a K-theorem. This constructionis important even when A = ∅, for there are infinitely many variable-free formulasconstructible from the constants ⊤ and ⊥ by the truth-functional connectives andthe modalities ✷ and ✸. The free algebra F ∅ is an initial object in the categoryMalg, because for each modal algebra A there a unique homomorphism from F ∅to A, since each constant formula has a uniquely determined value in A. TheT K -algebra corresponding to F ∅ is an initial object in the category of T K -algebras.Now category theory has a principle of duality that creates a new concept outof a given one by “reversing the arrows”, with the new concept being named byattaching the prefix “co” to the name of the old one. This leads to the notion of aT -coalgebra as an arrow of the form A τ A−→ T A, with a coalgebraic morphism fromcoalgebra A τ A−→ T A to coalgebra B τ B−→ T B being an arrow A −→ f B such thatτ B ◦ f = T f ◦ τ A , as inAτ A ↓T Af−→T f−→B↓ τ BAny modal frame can be viewed as a coalgebra for the powerset functor P : Set →T B