Mathematical Foundation for Designing and Modeling Cyberworlds

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For designing and modeling complicated and sophisticated systems such as cyber worlds, their mathematical foundation is critical. To realize it, two important properties called the homotopy lifting property and homotopy extension property are applied for designing and modeling a system in a bottom-up way and a top-down way, respectively. Activities of Internet Company are described by $\pi$-calculus processes and a Petri net which are derived from system requirements in a bottom-up way and a top-down way using the homotopy lifting property and the homotopy extension property. Entities in both properties are specified by the incrementally modular abstraction hierarchy by climbing down the abstraction hierarchy from the most abstract homotopy level to the most specific view level, while keeping invariants such as homotopy equivalence or topological equivalence.

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