# The Double Category of Paired Dialgebras on the Chu Category

## TL;DR

This paper introduces a new double category framework on the Chu category that unifies various mathematical structures, demonstrating self-duality properties and exploring categorical products and coproducts.

## Contribution

It develops a novel formalism combining double categories, dialgebras, and Chu construction to unify diverse mathematical formalisms within a single framework.

## Key findings

- The double category exhibits horizontal and vertical self-duality.
- It investigates the existence of binary horizontal products and coproducts.
- Provides foundational insights for future categorical unification efforts.

## Abstract

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.