[notthingnill]

Five minutes reading https://johnhw.github.io/umap_primes/index.md.html

Without using any category, topology of sheaf theory, this is what I believe is in this paper:

(1) the prior hypothesis is that data points in R^n are a sample from a uniform distribution in a Riemann space.

(2) Try to define a Riemann metric such that the number of sample points in any ball B is propotional to the volume of B.

(3) Since (2) doesn't define a global Riemann metric, they define a fuzzy membership relation. I suppose the role of the fuzzy tool is that local distance information is weighted according to the variance of the local distance estimations.

Disclaimer, I could be completely wrong.