Universal mapping properties Incidence relations
1. A special property of the category of sets
We want to discuss two related ideas:
1. universal mapping properties, and
2. detecting the structure of an object by means of figures and incidence relations.
An example of (1) is the property appearing in the definition of terminal object: to say that 1 is terminal means that for each object X, there is exactly one map X 1. The ‘for each’, ‘for every’, or ‘for all’ is what makes us call this a universal property: the object 1 is described by its relation to every object in the ‘universe’, i.e. the category under consideration.
The idea ...