## § Fundamental group functor does not preserve epis

• Epis in the category of topological spaces are continuous functions that have dense image.
• Take a circle $S^1$ and pinch it in the middle to get $S^1 \lor S^1$. this map is an epi: $f: S^1 \to S^1 \lor S^1$.
• See that this does not induce an epi $\pi(Z) \to \pi_(Z) \star \pi_1(Z)$.
• Maybe even more simply, the map $f: [0, 1] \to S^1$ is an epi
• Thus, fundamental group functor does not preserve epis.