Editing Pushout (category theory) (section)

==Application: the Seifert–van Kampen theorem==
{{Main|Seifert-van Kampen theorem}}
The Seifert–van Kampen theorem answers the following question. Suppose we have a [[Connected_space#Path_connectedness|path-connected]] space <math>X</math>, covered by path-connected open subspaces <math>A</math> and <math>B</math> whose intersection <math>A \cap B</math> is also path-connected. (Assume also that the basepoint <math>\ast</math> lies in the intersection of ''A'' and ''B''.) If we know the [[fundamental group]]s of <math>A</math>, <math>B</math> and <math>A\cap B</math> can we recover the fundamental group of <math>X</math>? The answer is yes, provided we also know the induced homomorphisms
<math>\pi_1(A \cap B,*) \to \pi_1(A,*)</math>
and
<math>\pi_1(A \cap B,*) \to \pi_1(B,*).</math>
The theorem then says that the fundamental group of <math>X</math> is the pushout of these two induced maps. Of course, <math>X</math> is the pushout of the two inclusion maps of <math>A \cap B</math> into <math>A</math> and <math>B</math>. Thus we may interpret the theorem as confirming that the fundamental group functor preserves pushouts of inclusions. We might expect this to be simplest when <math>A \cap B</math> is [[simply connected]], since then both homomorphisms above have trivial domain. Indeed, this is the case, since then the pushout (of groups) reduces to the [[free product]], which is the coproduct in the category of groups. In a most general case we will be speaking of a [[free product with amalgamation]].

There is a detailed exposition of this, in a slightly more general setting ([[covering space|covering]] [[groupoid]]s) in the book by J. P. May listed in the references.