Examples of categories
Directed graphs and other structures
We recall from Session 10:
1. Given an endomap of the ball with no fixed point, we can construct a retraction of the ball to its boundary.
2. Brouwer proved that no such retraction is possible.
We deduced by pure logic:
3. Every endomap of a ball has a fixed point.
We saw further that:
4. The sphere and the ball cannot be isomorphic (since the sphere does have a fixed point free endomap, for example, its antipodal map.)
It is critical that the category which we were discussing is not the category of abstract sets and arbitrary functions; it must rather be some