Conway's law, revised from a mathematical viewpoint

In this article, we revise Conway's Law from a mathematical point of view. By introducing a task graph, we first rigorously state Conway's Law based on the homomorphisms in graph theory for the software system and the organizations that created it. Though Conway did not mention it, the task graph shows the geometric structure of tasks, which plays a crucial role. Furthermore, due to recent requirements for high-level treatment of communication (due to security, knowledge hiding, etc.) in organizations and hierarchical treatment of organizations, we have reformulated these statements in terms of weakened homomorphisms, and the continuous maps in graph topology. In order to use graph topology and the continuous map in Conway's law, we have prepared them as mathematical tools, and then we show the natural expression of Conway's correspondences with hierarchical structures.