Category theory and applications to computing

范畴理论及其在计算中的应用

• 批准号: 5401-2006
• 负责人: Rosebrugh, Robert
• 金额: $ 0.73万
• 依托单位: Mount Allison University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=366479
• 关键词: Category; theory; applications; computing

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.

我们通过初始模型和数据库的语义模型研究了单元格的范畴版本分布性和分布律、具有顺序和并行操作的进程的代数及其语法。还提议加快开发一个基于Java的软件系统--Java EA Sketch Editor，用于编辑数据库语义模型，以自动生成XML和SQL(关系)数据库的数据描述语言。完备格的完全分配性是由主下集函子左侧存在两个伴随函子来刻画的。我们建议研究带有Yoneda函子左两个伴随的局部小范畴，即完全分配范畴。CCD格是射影超完备格，是其范畴中的核对象。对于完全分配范畴，期望得到类似的结果。双范畴上的‘Lax’单峰包括跨度双范畴上的索引单形单峰。在标准的精确度条件下，一个松弛单子的代数将该单子分解成一个双范畴同态和一个双范畴的态射。最近关于笛卡尔双范畴的工作允许研究张量结构到松弛单子的代数的转换。带作用的一般交换可分代数是图的余弦。我们将把它扩展到通过分配律将并行操作扩展到组合过程的情况。在实体关系图中考虑约束会导致有限限制、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(双边)纤切为视图更新问题提供了通用的解决方案。新的工作将把纤维更新与查看补充和可逆更新联系起来。