O'Reilly logo

Conceptual Mathematics, Second Edition by Stephen H. Schanuel, F. William Lawvere

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

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 ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required