Chu范畴上成对双代数的双范畴
The Double Category of Paired Dialgebras on the Chu Category
摘要：
21世纪的科学和实际需要导致人类将科学和技术的各种专门和不同分支汇集在一起。这种融合揭示了需要新的数学理论，能够提供共同的语言和框架，供来自不同领域的专业人士在解决跨学科和具有挑战性的问题。本论文也是沿着这个方向展开的。在这里，我们开发了一个新的形式主义的中心思想是“统一的各种数学分支”。为此目的，我们利用三个主要工具，从今天的数学，其中每一个拥有一个统一的性质本身：范畴论，特别是理论的“双重cateogries”，理论的“泛双代数”，和“楚建设”。借助这些工具，我们定义和研究了一个双重类别，它包含了数学和理论计算机科学体内常见的形式主义的重要部分。我们证明了这个双重范畴具有“水平自对偶”和“垂直自对偶”的性质。此外，我们还对这一范畴中的二元水平积和上积的存在性进行了初步的研究。最后，我们对未来的工作提出了一些建议。
英文摘要：
The scientific and practical needs of the twenty-first century lead humankind to convergence of the specialized and diverse branches of science and technology. This convergence reveals the need for new mathematical theories capable of providing common languages and frameworks to be utilized by professionals from different fileds in solving interdisciplinary and challenging problems. The present thesis is done in the same direction. Here, we develop a new formalism with the central idea of "unification of various mathematical branches". For this purpose, we utilize three major tools from today's mathematics, each of which possessing a unifying nature itself: category theory and especially the theory of "double cateogries", the theory of "universal dialgebra", and the "Chu construction". With the aid of these tools, we define and study a double category that subsumes a significant portion of the formalisms usual within the body of mathematics and theoretical computer science. We show that this double category possesses the properties of "horizontal self-duality" and "vertical self-duality". Also, we perform a primary investigation about existence of binary horizontal products and coproducts in this category. Finally, we give some suggestions for future work.
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