Division of maps: Sections and retractions
1. Determination problems
Many scientific investigations begin with the observation that one quantity f determines another quantity h. Here is an example. Suppose we have a cylinder, with a weighted piston pushing down on a trapped sample of gas. If we heat the system, the volume of the trapped gas will increase, raising the piston. If we then cool the system to its original temperature, the gas returns to its original volume, and we begin to suspect that the temperature determines the volume. (In the diagram below, f assigns to each state of the system its temperature, and h assigns to each state its volume.)
Our suspicion is that there is a map g which makes h = g f; such a g is called ...