Algebraic topological modeling for cyberworld design
Tosiyasu L. Kunii · 2005
The diversity of cyberworlds makes it hard to see consistency in terms of invariants. The consistency requires for us to abstract the most essentials out of the diversity, and hence the most abstract mathematics. It has been true in science in general, and in the theory of universe in particular. What are the most essential invariants in modeling cyberworlds? A branch of the most abstract mathematics is topology. For topology to be computable, it has to be algebraic. So, the searches have been for over two decades in algebraic topology for cyberworld invariants. Equivalence relations define invariants at various abstraction levels. The paper solely serves as an initial summary of algebraic topological resources for studying cyberworlds starting from the very elementary set theoretical level. High social impact application cases of e-financing and emanufacturing are presented at the end. 0. Prologue: What are cyberworlds? Cyberworlds are being formed in cyberspaces as computational spaces. My discovery of cyberworlds goes back to 1969 [4]. Now cyberspaces are on the web either intentionally or spontaneously, with or without design. Widespread and intensive local activities are melting each other on the web globally to create cyberworlds. The major key players of cyberworlds include e-finance that trades a GDP-equivalent a day and e-manufacturing that is transforming industrial production into Web shopping of product components and assembly factories. Without proper modeling, cyberworlds have continued to grow chaotic and are now out of human understanding and control. A novel information model we named “an adjunction space model ” serves to globally integrate local models. As an information model, it is also applicable to the category of irregular data models that capture spatiotemporal aspects of information worlds. Mathematically