nLab anabelioid



An anabelioid is a category intended to play the role of a ‘generalised geometric object’ in algebraic/arithmetic geometry. Its definition is simple: a finite product of Galois categories, or in other words of classifying topoi of profinite groups. The significance comes from the fact that in anabelian geometry, an algebraic variety is essentially determined by its algebraic fundamental group, which arises from a Galois category associated to the algebraic variety. The idea, due to Shinichi Mochizuki, is that one can develop the geometry of these Galois categories themselves, and products of Galois categories in general; thus, develop a form of categorical algebraic geometry.

To quote from Remark of Mochizuki2004:

The introduction of anabelioids allows us to work with both “algebro-geometric anabelioids” (i.e., anabelioids arising from (anabelian) varieties) and “abstract anabelioids” (i.e., those which do not necessarily arise from an (anabelian) variety) as geometric objects on an equal footing.

The reason that it is important to deal with “geometric objects” as opposed to groups, is that:

We wish to study what happens as one varies the basepoint of one of these geometric objects.


The following definitions follow Mochizuki2004.


A connected anabelioid is exactly a Galois category.


An anabelioid is a category equivalent to a finite product of connected anabelioids, that is, to a finite product of Galois categories.


An anabelioid is also known as a multi-Galois category.

Associated notions


Created on April 17, 2020 at 22:29:54. See the history of this page for a list of all contributions to it.