Non-abelian fundamental group on a path-connected space I am doing a self-study of algebraic topology, and am having some difficulties comprehending the idea of a non-abelian fundamental group on a path connected space. (See for example Hatcher Exercise 1.1.3 page 37 which implies that such groups do exist.)
If anyone could supply some intuition for such spaces by examples, pictures, or explanation, it would be greatly appreciated.
 A: The whole Section 1.2 of Hatcher's book is dedicated to van Kampen's theorem, where nonabelian fundamental groups are abundant. The more-or-less canonical example of the wedge sum $S^1 \vee S^1$ is in particular described, and I find the explanation in the introduction of the section rather clear. You have two loops, $a$ and $b$, and it's not possible to find a homotopy that preserves endpoints between $ab$ and $ba$; just try it.
PS: There are tons of examples in this section, and it starts two pages after the exercise you mention, so... Maybe next time try to read a little further.
A: With regard to intuition, imagine tying together two broom handles A,B by a rope going once round A and then once round B; this is not the same as, i.e. cannot be deformed into,  going once round B and then round A. 
With regard to proof, I prefer to use the van Kampen theorem for the fundamental groupoid $\pi_1(X,S)$ on a set $S$ of base points found in Topology and Groupoids (T&G). This enables us to get away from the usual path-connectedness assumption in the van Kampen Theorem for $X= U \cup V$; one needs to assume only that $S$ meets each path component of $U,V, U \cap V$, and the proof by universal properties is no more difficult than  for the case $S$ is a point. I feel it is also more intuitive. 
Thus because the interval $[0,2]$ is convex it is easy to identify the groupoid $\pi_1([0,2], \{0,1,2\}) $ as being determined by two generators $\iota_1: 0 \to 1$ and $\iota_2:1 \to 2$.  Identifying $S=\{0,1,2\}$ to a single point $y$ gives a wedge $Y = S^1 \vee S^1$ of two circles. It follows that $\pi_1(Y,y)$ is the free group on two generators, namely the images of $\iota_1, \iota_2$ in $\pi_1(Y,y)$. (cf. T&G 9.1.2 (Corollary 2)). That is,  by allowing groupoids and many base points we have in dimension $1$ an algebraic model of the topological process of identifying a discrete set of points. 
