The fundamental groupoid of a topological space

Take the points of $X$ as objects and define $\mathrm{Hom}(x,y)$ to be the endpoint-preserving homotopy classes of paths from $x$ to $y$. For $\alpha :x\to y$ and $\beta :y\to z$, put $[\beta ]\circ [\alpha ]=[\alpha \ast \beta ]$, and take the constant path at $x$ as $1_x$.

If $H$ deforms $\alpha$ to ${\alpha }^{\mathrm{\prime }}$ and $K$ deforms $\beta$ to ${\beta }^{\mathrm{\prime }}$ rel endpoints, paste $H(2s,t)$ for $s\le 1\mathrm{/}2$ to $K(2s-1,t)$ for $s\ge 1\mathrm{/}2$. The endpoint conditions agree along the seam, so the finite closed-pasting argument in [L2] gives an endpoint-preserving homotopy $\alpha \ast \beta \simeq {\alpha }^{\mathrm{\prime }}\ast {\beta }^{\mathrm{\prime }}$. Thus composition is well defined on classes.

Composing a path $\alpha$ with the straight-line homotopies from $s$ to $\max (0,2s-1)$ and to $\min (2s,1)$ proves the left and right identity laws. If $r(s)=2s$ for $s\le 1\mathrm{/}2$ and $r(s)=2-2s$ for $s\ge 1\mathrm{/}2$, then $\alpha ((1-t)r(s))$ contracts $\alpha \ast \bar{\alpha }$ rel endpoints; the analogous formula $\alpha (t+(1-t)(1-r(s)))$ contracts $\bar{\alpha }\ast \alpha$. Hence reversal supplies a two-sided inverse class.

For three composable paths, let $\delta :[0,3]\to X$ traverse them successively. The two bracketings are $\delta \circ p$ and $\delta \circ q$, where $p(s)=4s$ for $s\le 1\mathrm{/}2$ and $p(s)=2s+1$ for $s\ge 1\mathrm{/}2$, while $q(s)=2s$ for $s\le 1\mathrm{/}2$ and $q(s)=4s-1$ for $s\ge 1\mathrm{/}2$. The formula $\delta ((1-t)p(s)+tq(s))$ is an endpoint-preserving homotopy, so composition is associative on classes.

Steps 2.1, 3.1, and 2.2 make ${\mathrm{\Pi }}_1(X)$ a groupoid. Its automorphisms at $x$ are the based-loop classes, but $[\beta ]\circ [\alpha ]=[\alpha \ast \beta ]=[\alpha ][\beta ]$ in the traversal-order product of [L4]. Thus the identity on loop classes identifies ${\mathrm{Aut}}_{{\mathrm{\Pi }}_1(X)}(x)$ with ${\pi }_1(X,x{)}^{\mathrm{o}\mathrm{p}}$. The inversion map $[\alpha ]\mapsto [\bar{\alpha }]$ is therefore a canonical group isomorphism ${\mathrm{Aut}}_{{\mathrm{\Pi }}_1(X)}(x)\cong {\pi }_1(X,x)$.