Who invented algebraic topology?

Here are my talk slides:

Abstract. As algebraic topology becomes more important in applied mathematics it is worth looking back to see how this subject has changed our outlook on mathematics in general. When Noether moved from working with Betti numbers to homology groups, she forced a new outlook on topological invariants: namely, they are often functors, with two invariants counting as the same if they are naturally isomorphic. To formalize this it was necessary to invent categories, and to formalize the analogy between natural isomorphisms between functors and homotopies between maps it was necessary to invent 2-categories. These are just the first steps in the homotopification of mathematics, a trend in which algebra more and more comes to resemble topology, and ultimately abstract spaces (for example, homotopy types) are considered as fundamental as sets. It is natural to wonder whether topological data analysis is a step in the spread of these ideas into applied mathematics, and how the importance of robustness in applications will influence algebraic topology.

I thank Mike Shulman with some help on model categories and quasicategories. Any mistakes are, of course, my own fault.