The geometry of anabelioids
The author makes a rather detailed exposition of his theory of anabelioids. The article is motivated by the question (following the introduction): ``To what extent can the fundamental group of a Galois category be constructed in a canonical fashion which is independent of a choice of a basepoint?'' The point of view adopted in the article is motivated by \textit{A. Grothendieck}'s anabelian philosophy [in: Geometric Galois Actions. 1. Around Grothendieck's Esquisse d'un Programme, London Math. Soc. Lect. Note Ser. 242, 49--58 (1997; Zbl 0901.14002)]. In the first two sections of the paper the author develops the general theory of anabelioids. An anabelioid is a finite product of categories \(X=\prod_iX_i\), where each \(X_i\) (a ``connected component'') is a category isomorphic to some category \(B(G)\) of finite sets with a continuous \(G\)-action for some fixed profinite group \(G\). One of the central notions of the paper is the ``faithful quasi-core'' of an anabelioid, a weakening of the ``core'' defined in a previous paper of the author [J. Pure Appl. Algebra 131, 227--244 (1998; Zbl 0965.14013)]. The condition for a quasi-core is essentially that a certain forgetful functor is an equivalence. The main result of this part of the paper is: when an anabelioid possesses a faithful quasi-core, then its fundamental group may be constructed in a canonical way as a profinite group. The last part of the paper is devoted to the case of hyperbolic curves: the theory of ``cores'' [\textit{S. Mochizuki}, loc. cit. and Doc. Math., J. DMV Extra Vol., 609--640 (2003; Zbl 1092.14507)] in the context of hyperbolic curves is translated into the language of anabelioids. The author proves that if a non-proper hyperbolic curve over a \(p\)-adic or a number field is a geometric core, then its associated anabelioid admits a faithful quasi-core.
scientific article; zbMATH DE number 2151104
scientific article