Fibered Products of Hopf Algebras and Seifert-van Kampen Theorem for Semi-graphs of Tannakian Categories

It is known that Seifert-van Kampen theorem (for “good” topological spaces) can be showed by arguing the category of covering spaces. Similar arguments should be valid for abstract Galois categories (which Mochizuki calls ”connected anabelioids”) and neutral Tannakian categories. But when we try to state the theorem, the problem is the existence of amalgams (in other words, fibered coproducts) of profinite groups and that of affine group schemes (which is translated to the existence of fibered products of commutative Hopf algebras). A construction of amalgams of profinite groups can be found in Zalesskii [6]. We will construct fibered products of commutative Hopf algebras by using the explicit construction of cofree coalgebras which Hazewinkel gave in [3]. Another interest is the existence of so-called HNN extensions of affine group schemes, which we will also prove. By combining these two kinds of constructions, when we are given data of finitely many affine group schemes and a manner of composing them, we can describe the composite affine group scheme. The main theorem in this article is that, when we are given data of finitely many neutral Tannakian categories and a manner of glueing them, the fundamental group of the glued neutral Tannakian categories is isomorphic to the composition of the respective fundamental groups under the assumption that the data can be translated to the data of affine group schemes, which is not true in general unlike the case of Galois categories and profinite groups.