Fundamental groupoid

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A groupoid (a category in which all morphisms are isomorphisms) defined from a topological space $X$; the objects are the points of $X$, and the morphisms from an object $x_0$ to $x_1$ are the homotopy classes $\mathrm{rel} \{0,1\}$ of paths starting at $x_0$ and ending at $x_1$; composition is the product of classes of paths. The group of automorphisms of an object $x_0$ is the same as the fundamental group $\pi_1(X,x_0)$.


A useful survey of the applications of fundamental groupoids can be found in [a1].


[a1] R. Brown, "From groups to groupoids: a brief survey" Bull. London Math. Soc. , 19 (1987) pp. 113–134
How to Cite This Entry:
Fundamental groupoid. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.V. Khokhlov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article