In mathematics , especially in algebraic topology , the fundamental groupoid of a topological space is supposed to summarize the set of path connection components and the fundamental groups (for all ) in a single algebraic object.
![\ pi_1 (X, x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/f51cddd9d7b2d6dba06932d169d8d748bce0b4f6)
![x \ in X](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d)
The fundamental groupoid is a groupoid , i.e. a category in which every morphism is an isomorphism . The objects are the points from , the morphisms from to are the homotopy classes (relative ) of continuous paths with .
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![y](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d)
![{\ displaystyle p \ colon \ left [0,1 \ right] \ to X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78ae949c8d9aa6581a31ffa0f2b6b8d3563e75ca)
![{\ displaystyle p (0) = x, p (1) = y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b46d7d743ed3ccae6ed251456f3a9fba5788777e)
In this groupoid, the set corresponds to the isomorphism classes of objects, while the automorphism group corresponds to the object .
![{\ displaystyle \ pi _ {0} (X)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/674b6a9c0bfedc5b8137473d63973f7074f6d7b8)
![\ pi_1 (X, x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/f51cddd9d7b2d6dba06932d169d8d748bce0b4f6)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
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