Path

Let $X$ be a topological space.

Definition. A path in $X$ is a continuous map $\gamma: [0,1] \to X$. The point $\gamma(0)$ is the initial point and $\gamma(1)$ is the terminal point. A path from $x$ to $y$ is a path with $\gamma(0) = x$ and $\gamma(1) = y$. A loop at $x_0$ is a path with $\gamma(0) = \gamma(1) = x_0$.

Definition. The concatenation of a path $\alpha$ from $x$ to $y$ with a path $\beta$ from $y$ to $z$ is the path $\alpha * \beta: [0,1] \to X$ defined by

The reverse of a path $\gamma$ is $\bar{\gamma}(t) = \gamma(1 - t)$.

Definition. $X$ is path-connected if for every $x, y \in X$ there exists a path from $x$ to $y$.

Proposition. Two paths $\gamma, \delta$ from $x$ to $y$ are homotopic rel $\{0,1\}$ if and only if $\gamma * \bar{\delta}$ is null-homotopic as a loop at $x$.

Proposition. Concatenation of paths is associative, unital, and invertible up to homotopy rel $\{0,1\}$. That is, for paths $\alpha: x \to y$, $\beta: y \to z$, $\gamma: z \to w$:

1. $(\alpha * \beta) * \gamma \simeq \alpha * (\beta * \gamma)$ rel $\{0,1\}$,
2. $c_x * \alpha \simeq \alpha \simeq \alpha * c_y$ rel $\{0,1\}$, where $c_x$ is the constant path at $x$,
3. $\alpha * \bar{\alpha} \simeq c_x$ and $\bar{\alpha} * \alpha \simeq c_y$ rel $\{0,1\}$.

Proof sketch. Each statement is verified by an explicit reparametrization homotopy. For instance, associativity uses a homotopy that linearly shifts the “breakpoints” from $(1/4, 1/2)$ to $(1/2, 3/4)$. $\square$

Definition. The fundamental groupoid $\Pi_1(X)$ is the category whose objects are the points of $X$ and whose morphisms $\mathrm{Hom}(x,y)$ are homotopy classes (rel $\{0,1\}$) of paths from $x$ to $y$, with composition given by concatenation. The preceding proposition ensures this is a well-defined groupoid. The fundamental group $\pi_1(X, x_0)$ is the automorphism group of $x_0$ in $\Pi_1(X)$.

Examples.

• In $\mathbb{R}^n$, any two paths from $x$ to $y$ are homotopic rel $\{0,1\}$ via the linear homotopy $H(s,t) = (1-s)\gamma(t) + s\delta(t)$.
• In $S^1$, the homotopy class of a loop at $* = 1$ is determined by its winding number.
• In directed homotopy, paths carry a preferred direction and the reverse $\bar{\gamma}$ is a directed path only when $\bar{\gamma} \in dX$.