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P66
§3.8. Inter-universality and logical AND relations
One fundamental aspect of inter-universal Teichm¨uller theory lies in the consideration of distinct universes that arise naturally when one considers Galois
categories ? i.e., ´etale fundamental groups ? associated to various schemes. Here,
it is important to note that, when phrased in this way,
this fundamental aspect of inter-universal Teichm¨uller theory is, at least
from the point of view of mathematical foundations, no different from
the situation that arises in [SGA1].
On the other hand, the fundamental difference between the situation considered
in [SGA1] and the situations considered in inter-universal Teichm¨uller theory lies
in the fact that, whereas in [SGA1], the various distinct schemes that appear are
related to one another by means of morphisms of schemes or rings,
the various distinct schemes that appear in inter-universal Teichm¨uller
theory are related to one another, in general, by means of relations ?
such as the log- and Θ-links ? that are non-ring/scheme-theoretic in
nature, i.e., in the sense that they do not arise from morphisms of schemes or rings.

Indeed, let us first observe that the “basepoints” of k and kN determined by k
allows us to regard Gk and GkN , respectively, as the ´etale fundamental groups of k and kN .

This leads one naturally to consider weaker structures such as
・ sets equipped with a topology and a continuous action of a topological
group, in the case of the log-link, or
・ realified Frobenioids or topological monoids equipped with a continuous
action of a topological group, in the case of the Θ-link,
which are indeed coric [with respect to the respective links]. Indeed, it is precisely
this sort of consideration that gave rise to the term “inter-universal”.

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