Let $\pi(U\cup V)$ be the push-out of the diagram $\pi(U)\leftarrow \pi(U\cap V) \rightarrow \pi(V)$ that appears when we apply Vam-Kampen Theorem to the open sets $U,V$ in a topological space $X$. Moreover, im working with a variety $X$, so i can assume that the morphisms are surjective and the groups finitely generated.
Now, consider the diagram * $\pi(U)\times G_1 \leftarrow \pi(U\cap V)\times G_{12}\rightarrow \pi(V)\times G_2$ induced by the above diagram, and the diagram $G_1\leftarrow G_{12}\rightarrow G_2$ (with push-out $G$) of surjections of finitely generated abelian groups.
- Is the push-out of * isomorphic to $\pi(U\cup V) \times G$?
I think that it is true, by proceding with generators and relations.
How much this argument can be genralized? For example, in the category of fundamental groups ( fundamental groups of topological spaces) the push-out commute with the product of the diagrams ( by applying Van-Kampen Theorem on the product of the spaces). In the category of abelian groups it is true again, but in the category of groups this is false.
- In which other subcategories of the category of groups is this true?
Thanks in advance.