Abstract Nonsense - Examples

Examples

Typical instances are arguments involving diagram chasing, application of the definition of universal property, definition of natural transformations between functors, use of the Yoneda lemma, arguments exploiting classifying spaces, and so on.

To spell out a concrete example, consider a 3-manifold M with positive Betti number. One would like to show that M admits a map to the 2-sphere which is "non-trivial", i.e. non-homotopic to the constant map. By a general nonsense argument, there is a map

to the Eilenberg-MacLane space, corresponding to a non-trivial element in H2(M). Since K(Z,2) is a complex projective space and the latter admits a skeleton structure with no cells in odd dimensions, we can apply the cellular approximation theorem to conclude that the map f can be pushed down to the 2-skeleton, which happens to be the 2-sphere.

Though this proof establishes the truth of the statement in question, the proof technique has little to do with the topology or geometry of the 2-sphere, let alone 3-manifolds. The result is that the proof offers little geometric insight into the nature of such a map. On the other hand, the proof is surprisingly short and clean, and a “hands-on” approach involving the physical construction of such a map would be potentially laborious. A reader expecting a long, difficult proof might be surprised—or even delighted—by this bit of general nonsense.