Category theory and applications to computing

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We propose studies of the categorical versions distributivity and distributive laws for monads, algebras of processes with sequential and parallel operations and their syntax via initial models and of semantic models for databases. It is also proposed to accelerate development of a Java based software system, the Java EA Sketch Editor, for editing database semantic models for automated generation of data description languages for XML and SQL(relational) databases. Complete distributivity for a complete lattice (ccd) is characterized by the existence of two adjoint functors to the left of the principal down set functor. We propose to study locally small categories with two adjoints left of the Yoneda functor, the completely distributive categories. The ccd lattices are the projective sup-complete lattices and the nuclear objects in their category. Analogous results are expected for completely distributive categories. `Lax' monads on a bicategory include indexed monoidal monads on the span bicategory. Under standard exactness conditions, algebras for a lax monad factor the monad into a bicategory homomorphism and a morphism of bicategories. Recent work on cartesian bicategories permits studying transfer of a tensor structure to algebras for a lax monad. Generic commutative separable algebras with actions are cospans of graphs. We will extend this to the situation in which the parallel operations extend to composite processes via distributive laws. Accounting for constraints in entity-relationship diagrams leads to  finite-limit, finite coproduct (EA) sketches and models describe database states. For database views and updates, using (two sided) fibrations gives a universal solution to the view update problem. New work will relate fibrational updates to view complements and reversible updates.