Category theory and applications to computing

我们建议通过初始模型和数据库的语义模型研究单单子、顺序和并行操作过程的代数及其语法的分类版本、分布性和分配律。本文还提出了一个基于Java的软件系统——Java EA Sketch Editor的开发，用于编辑数据库语义模型，以便为XML和SQL（关系）数据库自动生成数据描述语言。完备格（ccd）的完备分布性是由主下集函子左边存在两个伴随函子来表征的。我们提出研究在Yoneda函子左边有两个伴随的局部小范畴，即完全分布范畴。ccd格是射影超完备格及其范畴内的核对象。对于完全分布的类别，期望得到类似的结果。双范畴上的‘ Lax’单列包括跨双范畴上的索引单列。在标准精确条件下，将松弛单元的代数分解为双范畴同态和双范畴的态射。最近关于笛卡儿双范畴的工作允许研究张量结构到松弛单的代数的转移。带作用的一般可交换可分代数是图的共张。我们将把它扩展到并行运算通过分配律扩展到复合过程的情况。考虑实体关系图中的约束导致有限-限制，有限副积（EA）草图和模型描述数据库状态。对于数据库视图和更新，使用（双面）连接为视图更新问题提供了一个通用的解决方案。新的工作将涉及纤维性更新，以查看补体和可逆更新。

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.