#### John Baez

#### August 8, 2017

### The Rise and Spread of Algebraic Topology

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.

You can see the slides here. The first
slide uses a photograph which I believe was
taken by Henry Segerman.

© 2017 John Baez

baez@math.removethis.ucr.andthis.edu