A General Design Method Based on Algebraic Topology -- A Divide and Conquer Method

Incrementally modular abstraction hierarchy (IMAH) based on algebraic topology is a general design method that is applicable to many fields including computer science, information technology and software engineering. Despite the fact that many development examples using IMAH have been announced in a variety of application fields, the number of researchers utilizing IMAH in their application fields is still limited because homotopy theory is a difficult concept for non-mathematical people to gasp. In this paper we utilize push outs and pullbacks instead of homotopy theory to explain how IMAH can be employed to develop systems. Push outs and pullbacks are general concepts encompassing set theory, homotopy theory, and category theory. We describe utilization of IMAH with push outs and pullbacks in the environment of a divide-and-conquer method, where a complicated system is divided into two simple subsystems. When dividing a system or integrating two subsystems, the mathematical concepts of pushout and pullback can be employed. Push outs and pullbacks are also utilized in descending and ascending an abstraction hierarchy, which is an essential power of IMAH. We design a chair, a concurrent system, and Google maps using IMAH to illustrate its utility.