§ Fundamental group functor does not preserve epis
- Epis in the category of topological spaces are continuous functions that have dense image.
- Take a circle S1 and pinch it in the middle to get S1∨S1. this map is an epi: f:S1→S1∨S1.
- See that this does not induce an epi π(Z)→π(Z)⋆π1(Z).
- Maybe even more simply, the map f:[0,1]→S1 is an epi
- Thus, fundamental group functor does not preserve epis.