### The Rise and Spread of Algebraic Topology

#### Posted by John Baez

People have been using algebraic topology in data analysis these days, so we’re starting to see conferences like this:

- Applied Algebraic Topology 2017, August 8-12, 2017, Hokkaido University, Sapporo, Japan.

I’m giving the first talk at this one. I’ve done a lot of work on applied category theory, but only a bit on on applied algebraic topology. It was tempting to smuggle in some categories, operads and props under the guise of algebraic topology. But I decided it would be more useful, as a kind of prelude to the conference, to say a bit about the overall history of algebraic topology, and its inner logic: how it was inevitably driven to categories, and then 2-categories, and then $\infty$-categories.

This may be the least ‘applied’ of all the talks at this conference, but I’m hoping it will at least trigger some interesting thoughts. We don’t want the ‘applied’ folks to forget the grand view that algebraic topology has to offer!

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 oftenfunctors, with two invariants counting as ‘the same’ if they arenaturally 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.

## Re: The Rise and Spread of Algebraic Topology

I’ll be there too, and I’m very much looking forward to the conference, to my first ever visit to Japan, and to seeing old friends. I’ll be speaking on magnitude homology.

Magnitude itself is first of all an invariant of finite metric spaces (though it can be extended to most compact spaces). Topological data analysis is especially interested in finite metric spaces, which often arise as data sets. So, that’s how magnitude connects with applied topology. More specifically, magnitude seems to give us good information about the “effective number of points” and “effective dimension” of a finite space at different scales.

Magnitude

homologyis a homology theory of enriched categories, but it can be applied, in particular, to finite metric spaces. Another homology theory of finite metric spaces is persistent homology — which is a major theme in applied algebraic topology. So, it’s tempting to compare the two. No one has yet done a proper comparison as far as I know, but I hope my talk will tempt someone to do one.