Fundamental groupoid

Definition

Let $X$ be a topological space. The fundamental groupoid $\Pi_1(X)$ is the category in which:

the objects are the points of $X$ ;
a morphism from $x$ to $y$ is a path homotopy class $[\gamma]$ of paths $\gamma \colon [0, 1] \to X$ with $\gamma(0) = x$ and $\gamma(1) = y$ ;
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 $s \ge 1/2$ ;
the identity of $x$ is the class of the constant path at $x$ .

Remarks

Every morphism $[\gamma]$ has the inverse $[\bar\gamma]$ , where $\bar\gamma(s) = \gamma(1 - s)$ , so $\Pi_1(X)$ is a groupoid. The group of morphisms from $x$ to itself is the fundamental group $\pi_1(X, x)$ . Example: 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. Non-example: if paths are taken without passing to path homotopy classes, composition is not associative, so clause 3 does not define a category. Two points lie in the same connected component of $\Pi_1(X)$ exactly when they lie in the same path component of $X$ .

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Path homotopy concept
Fundamental groupoid definition

Tag	`ZFEDKM4TJ7`
URI	`m2://ZFEDKM4TJ7`
Kind	definition
Writing	W0 stub
MSC	55Q05, 18B40
Defines	Fundamental groupoid
Uses	Path homotopy
Changed	2 November 2026

Fundamental groupoid

Definition

Remarks

Reading path

Further reading

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