20/04/23 11:31:51 Ayy0wuIh.net
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P7
Finally, we observe that although, in the above discussion, we concentrated on
the similarities, from an “inter-universal” point of view, between the vertical and
horizontal arrows of the log-theta-lattice, there is one important difference between
these vertical and horizontal arrows: namely,
・ whereas the copies of the full arithmetic fundamental group - i.e., in
particular, the copies of the geometric fundamental group - on either
side of a vertical arrow are identified with one another,
・ in the case of a horizontal arrow, only the Galois groups of the local
base fields on either side of the arrow are identified with one another
- cf. the discussion of Remark 3.6.3, (ii).
P71
Remark 3.1.4. Note that because the data involved in a species is given by
abstract set-theoretic formulas, the mathematical notion constituted by the species
is immune to, i.e., unaffected by, extensions of the universe - i.e., such as
the ascending chain V0 ∈ V1 ∈ V2 ∈ V3 ∈ ... ∈ Vn ∈ ... ∈ V that appears in
the discussion preceding Definition 3.1 - in which one works. This is the sense
in which we apply the term “inter-universal”. That is to say, “inter-universal
geometry” allows one to relate the “geometries” that occur in distinct universes.
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