"Grothendieck conjectured that the infinity groupoid captures all information about a topological space up to weak homotopy equivalence"
The homotopy hypothesis has something mystical about it.
in terry tao’s recent interview with lex fridman there’s an interesting bit on poincaré conjecture where he goes out of his way not to use these words.
vaguely related : synthetic homotopies visualisation tool - https://github.com/marcinjangrzybowski/cubeViz2
> In fact such a 2-sphere can be wrapped around the core an arbitrary number of times.
This is really hard for me to visualize. What does it look like for a 2-sphere to wrap around the core multiple times? Also, I would have expected it to be able to wrap around in multiple ways since there are more dimensions here, leading to pi^2(b^3 \ {0}) = Z^2. How would one even prove that this isn't the case?