Editing Fundamental group (section)

== Functoriality ==
If <math>f\colon X \to Y</math> is a [[continuous function (topology)|continuous map]], <math>x_0 \in X</math> and <math>y_0 \in Y</math> with <math>f(x_0) = y_0,</math> then every loop in <math>X</math> with base point <math>x_0</math> can be composed with <math>f</math> to yield a loop in <math>Y</math> with base point <math>y_0.</math> This operation is compatible with the homotopy equivalence relation and with composition of loops. The resulting [[group homomorphism]], called the [[Induced homomorphism (fundamental group)|induced homomorphism]], is written as <math>\pi(f)</math> or, more commonly,

:<math>f_* \colon \pi_1(X, x_0) \to \pi_1(Y, y_0).</math>

This mapping from continuous maps to group homomorphisms is compatible with composition of maps and [[identity morphism]]s. In the parlance of [[category theory]], the formation of associating to a topological space its fundamental group is therefore a [[functor]]

:<math>\begin{align}
   \pi_1 \colon \mathbf{Top}_* &\to \mathbf{Grp} \\
   (X, x_0) &\mapsto \pi_1(X, x_0)
 \end{align}</math>

from the [[category of pointed spaces|category of topological spaces together with a base point]] to the [[category of groups]]. It turns out that this functor does not distinguish maps that are [[homotopic]] relative to the base point: if <math>f,g:X\to Y</math> are continuous maps with <math>f(x_0) = g(x_0) = y_0</math>, and ''f'' and ''g'' are homotopic relative to {''x''<sub>0</sub>}, then ''f''<sub>∗</sub> = ''g''<sub>∗</sub>. As a consequence, two [[homotopy equivalent]] path-connected spaces have isomorphic fundamental groups:

:<math>X \simeq Y \implies \pi_1(X, x_0) \cong \pi_1(Y, y_0).</math>

For example, the inclusion of the circle in the punctured plane
:<math>S^1 \subset \mathbb{R}^2 \setminus \{0\}</math>
is a [[homotopy equivalence]] and therefore yields an isomorphism of their fundamental groups.

The fundamental group functor takes [[product topology|products]] to [[direct product|products]] and [[wedge sum|coproducts]] to [[free product of groups|coproducts]]. That is, if ''X'' and ''Y'' are path connected, then

:<math>\pi_1 (X \times Y, (x_0, y_0)) \cong \pi_1(X, x_0) \times \pi_1(Y, y_0)</math>

and if they are also [[locally contractible]], then

:<math>\pi_1(X \vee Y) \cong \pi_1(X)*\pi_1(Y).</math>

(In the latter formula, <math>\vee</math> denotes the [[wedge sum]] of pointed topological spaces, and <math>*</math> the [[free product]] of groups.) The latter formula is a special case of the [[Seifert–van Kampen theorem]], which states that the fundamental group functor takes [[pushout (category theory)|pushouts]] along inclusions to pushouts.