# Definition:Fundamental Group Functor

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## Definition

Let $\mathbf{Top}^*$ be the category of pointed topological spaces.

Let $\mathbf{Grp}$ be the category of groups.

The **fundamental group functor** is the covariant functor $\pi_1 : \mathbf{Top}^* \to \mathbf{Grp}$ with:

Object functor: | A pointed topological space $(X, x_0)$ is sent to its fundamental group. | |

Morphism functor: | A continuous pointed mapping $f$ is sent to its induced homomorphism of fundamental groups $\pi_1(f) = f_*$. |