Products in categories
We want to make precise the notion of product of objects in a category. For this it will be helpful to remember the idea of Galileo that in order to study the motion of an object in space it is sufficient to study instead two simpler motions, the motion of its shadow on a horizontal plane and the motion of its level on a vertical line. The possibility of recombining these two motions to reconstruct the original motion in space is the basis for the notion of product. We are now in a position to make all these ideas precise.
By multiplying the disk times the segment we get their product, the cylinder. The basic ingredients that reveal the cylinder as that product are the two maps ‘shadow’ and ‘level’:
When we ...