Pi-Calculus Modeling for Cyberworlds Systems using the Fibration and Cofibration Duality

Cyberworld systems are characterized by distributed functions and mobile communication. The pi-calculus gives theoretical background for designing and modeling such systems. In this paper, an original method for designing mobile communication systems executed in parallel in the cyberworlds theoretically and systematically is discussed using homotopy theory in the most modern field of mathematics. Homotopy theory gives computer science the theoretical back ground of designing and modeling in the most general way. The homotopy lifting property and homotopy extension property categorizing topological spaces in mathematics are effective in bottom-up / top-down development in computer science. By applying it for designing and modeling complicated systems in the cyberworlds, the paper shows incrementally modular abstraction hierarchy starting with homotopy theory and ending with program codes makes a system development theoretical and systematical.