Category theory and applications to computing

我们提出了研究的分类版本的分配和分配法律的单子，代数的进程与顺序和并行操作和它们的语法通过初始模型和语义模型的数据库。它还建议加速开发一个基于Java的软件系统，Java EA草图编辑器，用于编辑数据库语义模型，用于自动生成XML和SQL（关系）数据库的数据描述语言。完备格的完备分配性（ccd）的特征是在主下集函子的左边存在两个伴随函子。我们建议研究在米田函子左边有两个伴随的局部小范畴，即完全分配范畴。ccd格是投射超完备格，是投射超完备格范畴中的核对象。完全分配范畴也有类似的结果。bicategory上的“Lax”monad包括span bicategory上的索引monoidal monad。在标准正合性条件下，一个松弛单子的代数将单子分解为一个双范畴同态和一个双范畴态射。最近的工作carbohydrate bicategories允许研究转移的张量结构代数松弛单子。具有作用的一般交换可分代数是图的余扩张。我们将把这一点推广到并行运算通过分配律推广到复合过程的情况。考虑到实体关系图中的约束，导致有限限制，有限余积（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（双侧）纤维化给出了视图更新问题的通用解决方案。新的工作将涉及纤维更新，以查看补充和可逆的更新。

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.