Fundamental groupoid

The fundamental groupoid $𝜋 1 𝑋$ of a topological space $𝑋$ is a groupoid where

$O b (𝜋 1 𝑋) = 𝑋$
$𝜋 1 𝑋 (𝑥, 𝑦)$ is the set of homotopy classes of paths from $𝑥$ to $𝑦$

which is a quotient category of the Category of paths.

Proof of groupoid

Let $𝑥, 𝑦, 𝑧 \in 𝑋$ . If $𝛼 : 𝑥 ⇝ 𝑦$ and $𝛽 : 𝑦 ⇝ 𝑧$ are paths, then their concatenation $𝛼 \cdot 𝛽 : 𝑥 ⇝ 𝑧$ is a continuous path given by

𝛼 \cdot 𝛽 (𝑡) = {𝛼 (2 𝑡) 𝑡 \leq 12 𝛽 (2 𝑡 - 1) 𝑡 \geq 12

Likewise the reverse traversal of a path $𝛼 - 1 : 𝑦 ⇝ 𝑥$ is a path given by $𝑡 \mapsto 𝛼 (𝑡 - 1)$ . Now consider homotopy classes of paths using the Path traversal lemma. Define

𝜙 (𝑡) = ⎧ { {⎨ { {⎩ 2 𝑡 0 \leq 𝑡 \leq 14 14 + 𝑡 14 \leq 𝑡 \leq 12 12 + 12 𝑡 12 \leq 𝑡 \leq 1

Then $(𝛼 \cdot 𝛽) \cdot 𝛾 = (𝛼 \cdot (𝛽 \cdot 𝛾)) \circ 𝜙 ≃ 𝛼 \cdot (𝛽 \cdot 𝛾)$ . Hence concatenation is associative up homotopy. Similar arguments can be made for $[𝛼] [𝛼 - 1]$ and $[𝛼] [𝑐 𝑇]$ . <span class="QED"/

Properties

Seifert-Van Kampen-Brown theorem

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