Intuitive notion of functoriality in topological data analysis

Question

For school, we have to give a presentation about topological data analysis and I am in charge of motivating why topological data analysis is cool and useful. Most of what I say is based on "Topology and data" by Gunnar Carlsson. Specifically, I was planning to illustrate the four big advantages he lists by use of an example. However, I can't seem to really wrap my head around the fourth and seemingly most important advantage, namely functoriality.

I have followed a basic course on Category theory and Algebraic topology, but I am not sure what the biggest advantages of this functoriality in data analysis are. So far, I think consistency in randomly sampled subsets is important (because of the example by Gunnar Carlson), but my intuition seems to lag behind.

So, what I am asking is if there is some unifying intuition for the advantages of functoriality in topological data analysis?

Answer 1

I would suggest reading Robert Ghrist's Section 4.10 of his book, Elementary Applied Topology.

To give a brief excerpt from p. 75:

There is hardly a more important feature of homology than this functoriality. One implication in the sciences is to inference. It is sometimes the case that what is desired is knowledge of the homology of an important but unobserved space $X$; the observed data comprises the homology of a pair of spaces $Y_1$, $Y_2$, which are related by a map $f: Y_1 \rightarrow Y_2$ ...

[example] ... one can discern the presence of a true hole with two imprecise observations and a map between them.