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  "body": "\n## Definition\n\nLet $X$ be a topological space.\nThe *fundamental groupoid* $\\Pi_1(X)$ is the category in which:\n\n1. the objects are the points of $X$;\n2. a morphism from $x$ to $y$ is a [[path homotopy|CCNKA8C711]] class $[\\gamma]$ of paths $\\gamma \\colon [0, 1] \\to X$ with $\\gamma(0) = x$ and $\\gamma(1) = y$;\n3. the composite of $[\\alpha] \\colon x \\to y$ and $[\\beta] \\colon y \\to z$ is $[\\alpha \\cdot \\beta] \\colon x \\to z$, where $(\\alpha \\cdot \\beta)(s)$ is $\\alpha(2s)$ for $s \\le 1/2$ and $\\beta(2s - 1)$ for $s \\ge 1/2$;\n4. the identity of $x$ is the class of the constant path at $x$.\n\n## Remarks\n\nEvery morphism $[\\gamma]$ has the inverse $[\\bar\\gamma]$, where $\\bar\\gamma(s) = \\gamma(1 - s)$, so $\\Pi_1(X)$ is a groupoid.\nThe group of morphisms from $x$ to itself is the fundamental group $\\pi_1(X, x)$.\nExample: for a convex set $C \\subseteq \\R^n$, there is exactly one morphism between any two points, since any two paths with the same ends are path homotopic by the straight-line homotopy.\nNon-example: if paths are taken without passing to path homotopy classes, composition is not associative, so clause 3 does not define a category.\nTwo points lie in the same connected component of $\\Pi_1(X)$ exactly when they lie in the same path component of $X$.\n",
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