Fundamental groupoid

Definition
The fundamental groupoid of a topological space $$X$$ is defined as follows:


 * As a set, it is the set of all homotopy classes of defining ingredient::paths between points in $$X$$ (i.e., functions from the defining ingredient::closed unit interval to $$X$$), where two paths are homotopic if there is a homotopy between them that preserves endpoints at every stage of the homotopy.
 * The partial multiplication is defined by concatenation of paths where the right endpoint of the left path coincides with the left endpoint of the right path. Specifically, if $$f_1,f_2: [0,1] \to X$$ are paths, such that $$f_1(1) = f_2(0)$$, then $$f_1 * f_2$$ is defined as (up to homotopy):

$$\! (f_1 * f_2)(t) = \lbrace \begin{array}{rl} f_1(2t), & 0 \le t \le 1/2\\ f_2(2t - 1), & 1/2 < t \le 1 \\\end{array}$$