**ARTICLE III**

**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

category ...

Start Free Trial

No credit card required