Algebraic topological modeling for cyberworld design

The diversity of cyberworlds makes it hard to seeconsistency in terms of invariants. The consistencyrequires for us to abstract the most essentials out of thediversity, and hence the most abstract mathematics. Ithas been true in science in general, and in the theory ofuniverse in particular. What are the most essentialinvariants in modeling cyberworlds? A branch of themost abstract mathematics is topology. For topology tobe computable, it has to be algebraic. So, the searcheshave been for over two decades in algebraic topology forcyberworld invariants. Equivalence relations defineinvariants at various abstraction levels. The paper solelyserves as an initial summary of algebraic topologicalresources for studying cyberworlds starting from the veryelementary set theoretical level. High social impactapplication cases of e-financing and e-manufacturing arepresented at the end.