19 - Pullbacks and pushouts

Summary

Pullbacks and pushouts are more advanced universal properties, showing some more of the general features that are missing from the universal properties we’ve seen so far. We describe the general idea of a cone over a diagram, and define a limit over a diagram as a universal cone over that diagram, and show that products are in fact limits over a discrete diagram. We define pullbacks as limits over a particular diagram which includes some arrows. We unravel pullbacks in Set, and show that intersections are an example. We also describe the category in which pullbacks are terminal objects, and show how pullbacks can be used to define composable pairs of arrows in the definition of a category. We define pushouts as the dual of pullbacks, and then unravel that as a direct definition. We examine pushouts in Set and show that, given two sets, the square involving their intersection and their union is both a pullback and a pushout. We briefly discuss pushouts of topological spaces as a way of glueing spaces together.