Fundamental group functor

Definition
The fundamental group functor $$\pi_1$$ is a functor from the category of based topological spaces with based maps to the category of groups. The functor is defined as follows:


 * It sends each topological space $$X$$ to its fundamental group $$\pi_1(X)$$
 * Given a map $$f:X \to Y$$ of topological spaces, the associated morphism of fundamental groups is defined as follows: $$\pi_1(f)(\gamma) = f \circ \gamma$$ upto homotopy.

If $$f,g: X \to Y$$ are homotopic maps, then they induce the same homomorphism on fundamental groups. Thus, the fundamental group functor descends to a functor from the homotopy category of based topological spaces to the category of groups.

Properties of the functor
The fundamental group of a product of based topological spaces is the product of their fundamental groups.

The fundamental group of a wedge sum is the direct sum of the fundamental groups.

The fundamental group functor is far from faithful. Firstly, two different maps which are homotopic, induce the same homomorphism on fundamental groups. Even if we go down to the homotopy category, there could be many non-homotopic maps that induce the same map on fundamental groups.

The fundamental group functor is not full either. In other words, given two topological spaces, every homomorphism between their fundamental groups need not be realized as arising from a continuous map between the topological spaces.

Surjectivity on objects
Every group can be realized as the fundamental group of a topological space. More strongly, associated with every group, there is a classifying space, viz a space whose universal covering space is contractible, and whose fundamental group is the given group.