Utility of the 2-Categorical Structure of $\mathsf{Top}$?

It's well known that $\mathsf{Top}$ is a 2-category with homotopy classes of homotopies as 2-arrows. I'm a bit afraid to ask this question, but what is the utility of this 2-categorical structure?

Certainly, homotopy is an invaluable notion, and one might say the 2-categorical structure is what enables homotopy theory, but I don't know enough to have a reply to that; It just doesn't seem like basic topology and algebraic topology make use of the "entire" 2-categorical structure of $\mathsf{Top}$ (or $\mathsf{Ch}_\bullet$).

Examples of constructions or results that really require the 2-categorical structure (something that actually requires the interchange law, for instance) would be much appreciated!

Your question mentions the interchange law. The book Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids (pdf available there) makes essential use of this law in applying certain kinds of double and higher groupoids to homotopy theory and basic algebraic topology, without using singular homology.

A rough idea is that we form an algebraically defined category crossed complexes, and show how to model certain topological situations

Not much use is made of $2$-categories, but lots of use is made of monoidal closed categories, in dealing with homotopy classification problems. Setting it all up is not so easy, but the underlying intuitions are explained.

Note that the workers in algebraic topology of the early 20th century were looking for higher dimensional nonabelian versions of the fundamental group. However, by the interchange law, double groups are just abelian groups, and the idea came to be viewed as a mirage.

But double groupoids are more complicated than groups, and so higher groupoids have a possibility of modelling some higher homotopy theory; this is done not for spaces but for filtered spaces in the above book.

March 21: For an introduction to the ideas of the book in relation to the history of algebraic topology, see this presentation given in Galway, Dec 2014. .

March 24: Here is a link to a paper on double semigroups.

February 23, 2017 Here is a link to a recent paper on Modelling and computing homotopy types: I.

• Thank you for the interesting recommendation. I'm actually studying fundamental groupoids from your other book as I'm writing this. Very conceptual and elegant! – user153312 Mar 7 '15 at 22:43