Homotopy theory MOC

Fundamental groupoid

The fundamental groupoid of a topological space is a groupoid where

which is a quotient category of the Category of paths.

Proof of groupoid

Let . If and are paths, then their concatenation is a continuous path given by

Likewise the reverse traversal of a path is a path given by . Now consider homotopy classes of paths using the Path traversal lemma. Define

Then . Hence concatenation is associative up homotopy. Similar arguments can be made for and . <span class="QED"/

Properties

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