SEMI-GRAPHS OF ANABELIOIDS

Shinichi Mochizuki
October 2005



In this paper, we discuss various “general nonsense” aspects ofthe geometry of semi-graphs of proﬁnite groups [cf. [Mzk3], Appendix], by applying
the language of anabelioids introduced in [Mzk4]. After proving certain basic properties concerning various commensurators associated to a semi-graph of anabelioids,
we show that the geometry of a semi-graph of anabelioids may be recovered fromthe category-theoretic structure of certain naturally associated categories — e.g.,“temperoids” [in essence, the analogue of a Galois category for the “tempered fundamental groups” of [André]] and “categories of localizations”. Finally, we apply thesetechniques to obtain certain results in the absolute anabelian geometry [cf. [Mzk3],[Mzk8]] of tempered fundamental groups associated to hyperbolic curves over p-adic
local ﬁelds.

§0. Notations and Conventions§1. Zariski’s Main Theorem for Semi-graphs§2. Commensurability Properties§3. The Tempered Fundamental Group§4. Categories of Localizations§5. Arithmetic Semi-graphs of Anabelioids§6. Tempered Anabelian Geometry
Appendix: Quasi-temperoids
Index

Introduction
In this paper, we continue to pursue the theme of categorical representationof scheme-theoretic geometries, which played a central role in [Mzk6], [Mzk7], as
well as in the previous anabelian work of the author [e.g., [Mzk2], [Mzk3], [Mzk5],[Mzk8]]. The original motivation of the present work lies in the problem of ﬁnding
an appropriate and eﬃcient way of representing, via categories, the geometry of2000 Mathematical Subject Classiﬁcation. 14H30, 14H25.
Typeset by AMS-TEX
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SHINICHI MOCHIZUKI“formal localizations” of hyperbolic curves over p-adic local ﬁelds. Here, we use theterm “formal localizations” to refer to the localizations of the p-adic formal completion of a stable log curve over the ring of integers of a p-adic local ﬁeld obtained bycompleting along the irreducible components and nodes of the geometric logarithmic
special ﬁber speciﬁed by some sub-semi-graph of the “dual semi-graph with compact
structure” [cf. [Mzk3], Appendix] associated to this geometric logarithmic special
ﬁber. Since the geometry of such formal localizations is substantially reﬂected in
the geometry of localizations of the semi-graph of proﬁnite groups [cf. [Mzk3], Appendix] associated to this geometric logarithmic special ﬁber, it is thus natural,from the point of view of the goal of categorical representation of this geometry offormal localizations, to study the geometry of this semi-graph of proﬁnite groups.
Moreover, when working with proﬁnite groups as “geometric objects”, it is natural
to apply the language of anabelioids introduced in [Mzk4].
The main results of this paper may be summarized as follows:(1) In §1, we study the geometry of semi-graphs, and in particular, expose
a proof related to the author by M. Matsumoto of a sort of analogue forcertain types of morphisms of ﬁnite semi-graphs of “Zariski’s main theorem” in scheme theory [cf. Theorem 1.2]. This result has some interestinggroup-theoretic consequences related to the author by A. Tamagawa [cf.
Corollary 1.6]; in addition, it admits an interesting interpretation from amore “arithmetic” point of view [cf. Remark 1.5.1].
(2) In §2, we begin our study of the geometry of semi-graphs of anabeloids.
Our main result [cf. Corollary 2.7] concerns certain properties of thecommensurator in the proﬁnite fundamental group associated to a graphof anabelioids of the various subgroups associated to subgraphs of thegiven graph of anabelioids.
(3) In §3, we take up the study of “tempered fundamental groups” [i.e.,roughly speaking, fundamental groups that correspond to coverings dominated by the composite of an arbitrary ﬁnite covering and a [not necessarily ﬁnite] covering of “some” associated semi-graph — cf. [André]], byworking with “temperoids”, i.e., the analogue for tempered fundamentalgroups of Galois categories [in the case of proﬁnite groups]. Our main result [cf. Theorem 3.7; Corollary 3.9] states that for certain kinds of graphsof anabelioids, the vertices (respectively, edges) of the underlying graphmay be recovered from the associated tempered fundamental group as the[conjugacy classes of] maximal compact subgroups (respectively, nontrivial intersections of distinct maximal compact subgroups) of this tempered
fundamental group. We then apply this result to show, in the case of
hyperbolic curves over p-adic local ﬁelds, that the entire dual semi-graphwith compact structure may be recovered solely from the geometric tempered fundamental group of such a curve [cf. Corollary 3.11].
(4) Although the tempered fundamental group furnishes perhaps the mosteﬃcient way of reconstructing a graph of anabelioids from a naturally associated category, in §4, we examine another natural approach to this problem, via categories of localizations. This approach is motivated partly by

SEMI-GRAPHS OF ANABELIOIDS

the geometry of formal localizations of stable log curves referred to above,and partly by the naive observation that given a semi-graph of anabelioids, it is natural to “localize” not just by considering coverings, but also
by “physically localizing on the underlying semi-graph”. After studying
various basic properties of such categories of localizations [including someinteresting properties that follow from “Zariski’s main theorem for semigraphs” — cf. Proposition 4.4, (i), (ii)], we show that, given a graph ofanabelioids satisfying certain properties, the original graph of anabelioids
may be recovered functorially from its associated category of localizations[cf. Theorem 4.8]. One recurrent theme in the theory of §4 [which isconsistent with the more general theme of “categorical representation ofscheme-theoretic geometries” referred to above] is the idea that the geometry that is hidden in such a category of localizations may be developed,
using entirely category-theoretic notions, in a fashion that is remarkably
reminiscent of classical scheme theory — cf. the application of “Zariski’s
main theorem for semi-graphs” in Proposition 4.4, (i), (ii); the “valuative
criterion” of Proposition 4.6.
(5) In §3, 4, we considered semi-graphs of anabelioids that are not equippedwith “Galois actions”. Thus, in §5, we generalize the [more eﬃcient]
theory of §3 [instead of the theory of §4, since this becomes somewhat
cumbersome] to the “arithmetic” situation that arises in the case of a hyperbolic curve over a p-adic local ﬁeld, i.e., of a semi-graph of anabelioids
equipped with an “arithmetic action” by a proﬁnite group. The translation of the theory of §3 into its “arithmetic analogue” in §5 is essentiallyroutine, once one replaces, for instance, “maximal compact subgroups” by“arithmetically maximal compact subgroups” [cf. Theorem 5.4].
(6) In §6, we consider the tempered analogue of the absolute anabelian geometry developed in [Mzk8]. In particular, we show that in many respects,
this tempered analogue is essentially equivalent to the original proﬁnite
version [cf. Theorem 6.6], and, moreover, that the various absolute anabelian results of [Mzk8] concerning decomposition groups of closed points— in particular, a sort of “weak section conjecture” — also hold in thetempered case [cf. Theorem 6.8; Corollaries 6.9, 6.11]. This is particularlyinteresting in that the tempered version exhibits, in a very explicit way,
the geometry of this “weak section conjecture” in a fashion that is quite
reminiscent of the “discrete real section conjecture” of [Mzk5], §3.2 [cf.
Remark 6.9.1], i.e., relative to the well-known analogy between geodesics
on trees [cf., e.g., Lemma 1.8, (ii); [Serre]] and geodesics in Riemannian“straight line spaces” [i.e., Riemannian spaces satisfying the condition (*)
of [Mzk5], §3.2].
(7) In the Appendix, we discuss a slight generalization of the notions of“temperoids” and “anabelioids” that sometimes appears in practice, especially when one wishes to consider, from the point of view of the categories
discussed in the present paper, the “stack-theoretic analogue” of various“scheme-theoretic notions” [cf., e.g., Remarks 4.1.2, 4.8.4]. The main result of the Appendix [cf. Theorem A.4] states that a temperoid may be

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SHINICHI MOCHIZUKI

reconstructed category-theoretically from a certain type of subcategory of
the temperoid [i.e., a “quasi-temperoid”]. This sort of technical result isalso of interest, relative to the analogy between temperoids and anabelioids, in the context of the theory of cores of anabelioids developed in[Mzk4].
Finally, we remark that to a certain extent, this paper was conceived by the author as a piece of mathematical infrastructure, i.e., to develop the basic properties
and “general nonsense” of the very “primitive” [by comparison to many modern
mathematical notions] notion of a semi-graph of anabelioids in maximal possible
generality. Thus, although, for instance, the exposition of §2, §3 could be substantially simpliﬁed if one restricts oneself to the sort of semi-graphs of anabelioids thatarise from stable log curves, it seemed more natural to the author to develop thistheory under minimal possible hypotheses. As a result of this choice on the partof the author, the present paper contains a very large number of new terms, which
may be ignored to a substantial extent on a ﬁrst reading of the present paper, byassuming, for instance, that all semi-graphs of anabelioids are of the sort that arise
from stable log curves. Also, it is hoped that the Index provided at the end of thepaper may aid in the tracking down of unknown terminology.
Acknowledgements:
I would like to thank Akio Tamagawa and Makoto Matsumoto for many helpfulcomments concerning the material presented in this paper. In particular, I am
indebted to A. Tamagawa for informing me of Corollary 1.6 and the reference
quoted in Remark 1.7.1, and to M. Matsumoto for informing me of his proof of
Theorem 1.2.

SEMI-GRAPHS OF ANABELIOIDS

5

Section 0: Notations and Conventions
Topological Groups:
Let G be a Hausdorﬀ topological group, and H ⊆ G a closed subgroup. Let us
write
def
ZG (H) = {g ∈ G | g · h = h · g, ∀ h ∈ H}
for the centralizer of H in G;
NG (H) = {g ∈ G | g · H · g −1 = H}
deffor the normalizer of H in G; anddef

CG (H) = {g ∈ G | (g · H · g −1 )H has ﬁnite index in H, g · H · g −1 }for the commensurator of H in G. Note that: (i) ZG (H), NG (H) and CG (H) are
subgroups of G; (ii) we have inclusions
H, ZG (H) ⊆ NG (H) ⊆ CG (H)and (iii) H is normal in NG (H).
Note that ZG (H), NG (H) are always closed in G, while CG (H) is not necessarily closed in G. If H = CG (H), then we shall say that H is commensurably
terminal in G.
If G is center-free, then we have a natural exact sequence
1 → G → Aut(G) → Out(G) → 1[where Aut(G) denotes the group of automorphisms of the topological group G;
the injective [since G is center-free!] homomorphism G → Aut(G) is obtained byletting G act on G by inner automorphisms; Out(G) is deﬁned so as to render thesequence exact]. If J → Out(G) is a homomorphism of groups, then we shall write
out

def

G  J = Aut(G) ×Out(G) J
for the “outer semi-direct product of J with G”. Thus, we have a natural exact
out

sequence: 1 → G → G  J → J → 1.
Categories:
Let C be a category. We shall denote the collection of objects of C by:
Ob(C)

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SHINICHI MOCHIZUKI

If A ∈ Ob(C) is an object of C, then we shall denote byCA
the category whose objects are morphisms B → A of C and whose morphisms (froman object B1 → A to an object B2 → A) are A-morphisms B1 → B2 in C. Thus,
we have a natural functor(jA )! : CA → C(given by forgetting the structure morphism to A). Similarly, if f : A → B is amorphism in C, then f deﬁnes a natural functor
f! : CA → CB
by mapping an arrow (i.e., an object of CA ) C → A to the object of CB given by
the composite C → A → B with f. Also, we shall denote by
C[A] ⊆ C
the full subcategory determined by the objects of C that admit a morphism to A.
If the category C admits ﬁnite products, then (jA )! is left adjoint to the naturalfunctor∗jA: C → CA
given by taking the product with A, and f! is left adjoint to the natural functorf ∗ : CB → CA
given by taking the ﬁbered product over B with A. We shall call an object A ∈ Ob(C)terminal if for every object B ∈ Ob(C), there exists a unique arrow B → A in C.
We shall refer to a natural transformation between functors all of whose component morphisms are isomorphisms as an isomorphism between the functors inquestion. A functor φ : C1 → C2 between categories C1 , C2 will be called rigid if φ
has no nontrivial automorphisms. A category C will be called slim if the naturalfunctor CA → C is rigid, for every A ∈ Ob(C).
If G is a proﬁnite group, then we shall denote byB(G)the category of ﬁnite sets with continuous G-action. Thus, B(G) is a Galois category, or, in the terminology of [Mzk4], a connected anabelioid. Moreover, B(G) isslim if and only if, for every open subgroup H ⊆ G, we have ZG (H) = {1} [cf.
[Mzk4], Corollary 1.1.6, Deﬁnition 1.2.4].
A diagram of functors between categories will be called 1-commutative if thevarious composite functors in question are isomorphic. When such a diagram “commutes in the literal sense” we shall say that it 0-commutes. Note that when a diagram in which the various composite functors are all rigid “1-commutes”, it follows

SEMI-GRAPHS OF ANABELIOIDS

7

from the rigidity hypothesis that any isomorphism between the composite functors
in question is necessarily unique. Thus, to state that such a diagram 1-commutes
does not result in any “loss of information” by comparison to the datum of a speciﬁc
isomorphism between the various composites in question.
Given two functors Φi : Ci → Di (where i = 1, 2) between categories Ci , Di , weshall refer to a 1-commutative diagram
∼

C2
⏐
⏐Φ
 2

∼

D2

→
C1
⏐
⏐Φ
 1
D1

→

— where the horizontal arrows are equivalences of categories — as an abstractequivalence from Φ1 to Φ2 . If there exists an abstract equivalence from Φ1 to Φ2 ,then we shall say that Φ1 , Φ2 are abstractly equivalent.
We shall say that a nonempty [i.e., non-initial] object A ∈ Ob(C) is connected
if it is not isomorphic to the coproduct of two nonempty objects of C. We shallsay that an object A ∈ Ob(C) is mobile if there exists an object B ∈ Ob(C) suchthat the set HomC (A, B) has cardinality ≥ 2 [i.e., the diagonal from this set tothe product of this set with itself is not bijective]. We shall say that an object
A ∈ Ob(C) is quasi-connected if it is either immobile [i.e., not mobile] or connected.
Thus, connected objects are always quasi-connected. If every object of a category C
is quasi-connected, then we shall say that C is a category of quasi-connected objects.
We shall say that a category C is totally (respectively, almost totally) epimorphic if
every morphism in C whose domain is arbitrary (respectively, nonempty) and whose
codomain is quasi-connected is an epimorphism.
We shall say that C is of ﬁnitely (respectively, countably) connected type if it isclosed under formation of ﬁnite (respectively, countable) coproducts; every object of
C is a coproduct of a ﬁnite (respectively, countable) collection
of connected objects;
and, moreover, all ﬁnite (respectively, countable) coproducts
Ai in the category
satisfy the condition that the natural map


HomC (B, Ai ) → HomC (B,



Ai )

is bijective, for all connected B ∈ Ob(C). If C is of ﬁnitely or countably connected
type, then every nonempty object of C is mobile; in particular, a nonempty objectof C is connected if and only if it is quasi-connected.
If a mobile object A ∈ Ob(C) satisﬁes the condition that every morphism in C
whose domain is nonempty and whose
codomain is equal to A is an epimorphism,∼then A is connected. [Indeed, C1 C2 → A, where C1 , C2 are nonempty, implies
that the composite map
HomC (A, B) → HomC (A, B) × HomC (A, B) → HomC (C1 , B) × HomC (C2 , B)∼C2, B) → HomC (A, B)= HomC (C18SHINICHI MOCHIZUKI

is bijective, for all B ∈ Ob(C).]If C is a category of ﬁnitely or countably connected type, then we shall write
C0 ⊆ Cfor the full subcategory of connected objects. [Note, however, that in general, objectsof C 0 are not necessarily connected — or even quasi-connected — as objects of C 0 !]On the other hand, if, in addition, C is almost totally epimorphic, then C 0 is totally
epimorphic, and, moreover, an object of C 0 is connected [as an object of C 0 !] if andonly if [cf. the argument of the preceding paragraph!] it is mobile [as an object of
C 0 ]; in particular, [assuming still that C is almost totally epimorphic] every objectof C 0 is quasi-connected [as an object of C 0 ].
If C is a category, then we shall writeC ⊥ (respectively, C  )for the category formed from C by taking arbitrary “formal” [possibly empty] ﬁnite(respectively, countable) coproducts of objects in C. That is to say, we deﬁne the“Hom” of C ⊥ (respectively, C  ) by the following formula:


 
def
Ai ,
Bj ) =
HomC (Ai , Bj )
Hom(
i

j

i

j

[where the Ai , Bj are objects of C]. Thus, C ⊥ (respectively, C  ) is a categoryof ﬁnitely (respectively, countably) connected type. Note that objects of C deﬁneconnected objects of C ⊥ or C  . Moreover, there are natural [up to isomorphism]
equivalences of categories∼(C ⊥ )0 → C;∼(C  )0 → C;∼

(D0 )⊥ → D;∼

(E 0 ) → E

if D (respectively, E) is a category of ﬁnitely connected type (respectively, category of countably connected type). If C is a totally epimorphic category of quasiconnected objects, then C ⊥ (respectively, C  ) is an almost totally epimorphic category of ﬁnitely (respectively, countably) connected type.
In particular, the operations “0”, “⊥” (respectively, “”) deﬁne one-to-one
correspondences [up to equivalence] between the totally epimorphic categories ofquasi-connected objects and the almost totally epimorphic categories of ﬁnitely (respectively, countably) connected type.
If C is a [small] category, then we shall write G(C) for the graph associated to C.
This graph is the graph with precisely one vertex for each object of C and precisely
one edge for each arrow of C [joining the vertices corresponding to the domain andcodomain of the arrow]. We shall refer to the full subcategory of C determined by
the objects and arrows that compose a connected component of the graph G(C)as a connected component of C. In particular, we shall say that C is connected if
G(C) is connected. [Note that by working with respect to some “suﬃciently large”

SEMI-GRAPHS OF ANABELIOIDS

9envelopping universe, it makes sense to speak of a category which is not necessarily
small as being connected.]
If C is a category, then we shall say that an object A ∈ Ob(C) is indissectible
if, for every pair of arrows A1 → A, A2 → A of C, where A1 , A2 are nonempty,
there exists a pair of arrows ψ1 : B → A1 , ψ2 : B → A2 such that φ1 ◦ ψ1 = φ2 ◦ ψ2 ,where B is nonempty.
If C if a category and S is a collection of arrows in C, then we shall say thatan arrow A → B is minimal-adjoint to S if every factorization A → C → B of thisarrow A → B in C such that A → C lies in S satisﬁes the property that A → C is,
in fact, an isomorphism. Often, the collection S will be taken to be the collection ofarrows satisfying a particular property P; in this case, we shall refer to the property
of being “minimal-adjoint to S” as the minimal-adjoint notion to P.
Curves:Suppose that g ≥ 0 is an integer. Then if S is a scheme, a family of curves ofgenus gX →S
is deﬁned to be a smooth, proper, geometrically connected morphism of schemes
X → S whose geometric ﬁbers are curves of genus g.
Suppose that g, r ≥ 0 are integers such that 2g − 2 + r > 0. We shall denotethe moduli stack of r-pointed stable curves of genus g (where we assume the points
to be unordered) by Mg,r [cf. [DM], [Knud] for an exposition of the theory of such
curves; strictly speaking, [Knud] treats the ﬁnite étale covering of Mg,r determinedby ordering the marked points]. The open substack Mg,r ⊆ Mg,r of smooth curves
will be referred to as the moduli stack of smooth r-pointed stable curves of genus gor, alternatively, as the moduli stack of hyperbolic curves of type (g, r). The divisor
at inﬁnity Mg,r \Mg,r of Mg,r determines a log structure on Mg,r ; denote thelog

resulting log stack by Mg,r .
A family of hyperbolic curves of type (g, r)
X →S
is deﬁned to be a morphism which factors X → Y → S as the composite of anopen immersion X → Y onto the complement Y \D of a relative divisor D ⊆ Y
which is ﬁnite étale over S of relative degree r, and a family Y → S of curves ofgenus g. One checks easily that, if S is normal, then the pair (Y, D) is unique up
to canonical isomorphism. (Indeed, when S is the spectrum of a ﬁeld, this fact iswell-known from the elementary theory of algebraic curves. Next, we consider anarbitrary connected normal S on which a prime l is invertible (which, by Zariski
localization, we may assume without loss of generality). Denote by S  → S the ﬁnite étale covering parametrizing orderings of the marked points and trivializationsof the l-torsion points of the Jacobian of Y . Note that S  → S is independent of
the choice of (Y, D), since (by the normality of S), S  may be constructed as the

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SHINICHI MOCHIZUKI

normalization of S in the function ﬁeld of S  (which is independent of the choiceof (Y, D) since the restriction of (Y, D) to the generic point of S has already beenshown to be unique). Thus, the uniqueness of (Y, D) follows by considering theclassifying morphism (associated to (Y, D)) from S  to the ﬁnite étale covering of(Mg,r )Z[ 1l ] parametrizing orderings of the marked points and trivializations of thel-torsion points of the Jacobian [since this covering is well-known to be a scheme,
for l suﬃciently large].) We shall refer to Y (respectively, D; D; D) as the compactiﬁcation (respectively, divisor at inﬁnity; divisor of cusps; divisor of marked points)of X. A family of hyperbolic curves X → S is deﬁned to be a morphism X → S
such that the restriction of this morphism to each connected component of S is afamily of hyperbolic curves of type (g, r) for some integers (g, r) as above.
Write
C g,r → Mg,rfor the tautological curve over Mg,r ; D g,r ⊆ Mg,r for the corresponding tautologicaldivisor of marked points. The divisor given by the union of D g,r with the inverse
image in C g,r of the divisor at inﬁnity of Mg,r determines a log structure on C g,r ;log

denote the resulting log stack by C g,r . Thus, we obtain a morphism of log stacks
log

log

C g,r → Mg,rlog

which we refer to as the tautological log curve over Mg,r . If S log is any log scheme,
then we shall refer to a morphism
C log → S log
which is obtained as the pull-back of the tautological log curve via some [necessarily
log
uniquely determined — cf., e.g., [Mzk1], §3] classifying morphism S log → Mg,r as astable log curve. If C has no nodes, then we shall refer to C log → S log as a smoothlog curve.
If XK (respectively, YL ) is a hyperbolic curve over a ﬁeld K (respectively, L),then we shall say that XK is isogenous to YL if there exists a hyperbolic curve ZM
over a ﬁeld M together with ﬁnite étale morphisms ZM → XK , ZM → YL .

SEMI-GRAPHS OF ANABELIOIDS

11

Section 1: Zariski’s Main Theorem for Semi-graphs
In this §, we prove an analogue [cf. Theorem 1.2 below] for semi-graphs of“Zariski’s main theorem” (for schemes).
We begin with some general remarks concerning semi-graphs [a notion deﬁned
in [Mzk3], Appendix]. First, we recall that a semi-graph G consists of the following
collection of data:(1) a set V — whose elements we refer to as “vertices”;(2) a set E — whose elements we refer to as “edges” — each of whose
elements
e is a set of cardinality 2 satisfying the property “e = e ∈ E =⇒ 
e e = ∅”;
(3) a collection ζ of maps ζe [one for each edge e] — which we refer to as the“coincidencemaps” — such that ζe : e → V {V} [where we note that
V {V} = ∅ since V ∈/ V] is a map from the set e to the set V {V}.
We shall refer to the subset ζe−1 (V) ⊆ e [i.e., the inverse image of the subset
V ⊆ V {V} of elements = V] as the verticial portion of an edge e; to the restrictionof ζe to the verticial portion of e as the verticial restriction of ζe ; and to thecardinality of the verticial portion of e as the verticial cardinality of e. A graph G
is a semi-graph G for which every e ∈ E has verticial cardinality precisely 2. We
shall refer to an element b ∈ e as a branch of the edge e. A semi-graph will becalled ﬁnite (respectively, countable) if both its set of vertices and its set of edges
are ﬁnite (respectively, countable). A component of a semi-graph is deﬁned to be
the datum of either an edge or a vertex of the semi-graph.
Let G = {V, E, ζ} be a semi-graph. If e ∈ E is an edge of G of verticial
cardinality 2 whose image via ζe consists of (not necessarily distinct) elements v1 ,v2 of V, then we shall say that e joins v1 to v2 . If v = ζe (b), for some branch b ofan edge e [so v is a vertex], then we shall say that the edge e meets or abuts to thevertex v, and that the branch b of the edge e abuts to the vertex v. Thus, an edgeof a graph always abuts to at least one vertex, while an edge of a semi-graph may
abut to no vertices at all. A morphism between semi-graphs
G = {V, E, ζ} → G = {V  , E  , ζ  }is a collection of maps V → V  ; E → E  ; and for each e ∈ E mapping to e , a∼bijection e → e [or, equivalently — since both e and e are sets of cardinality 2 —an injection e → e ] — all of which are compatible with the verticial restrictionsof the respective coincidence maps. Thus, here, we allow an edge that abuts to no(respectively, precisely one) vertex to map to an edge that abuts to any number≥ 0 (respectively, ≥ 1) of vertices.
A semi-graph G may be thought of as a topological space as follows: We regard
each vertex v as a point [v]. If e is an edge, consisting of branches b1 , b2 , then we
regard e as the “interval” given by the set of formal sums λ1 · [b1 ] + λ2 · [b2 ], where

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SHINICHI MOCHIZUKI

λ1 , λ2 ∈ R [here, R denotes the topological ﬁeld of real numbers]; λ1 + λ2 = 1;
for i = 1, 2, λi ≤ 1 (respectively, λi < 1) if bi abuts (respectively, does not abut)
to a vertex; moreover, if bi abuts to a vertex v, then we identify the formal sum
1 · [bi ] + 0 · [b3−i ] with [v]. Thus, relative to this point of view, it is natural tothink of the branch bi as the portion of the interval just deﬁned consisting of formalsums such that λi > 12 . Also, we observe that this construction of an associatedtopological space is functorial: Every morphism of semi-graphs induces a continuous
morphism of the corresponding topological spaces. In the following discussion, weshall often invoke this point of view without further explanation.

e
e
v

b

b

v

A Typical Semi-graphv, v  : vertices;e: a closed edge;

b: a branch of e that abuts to v;e : an open edge that abuts to v b : a branch of e that abuts to v A sub-semi-graph H of a semi-graph G is a semi-graph satisfying the following
properties: (a) the set of vertices (respectively, edges) of H is a subset of the setof vertices (respectively, edges) of G; (b) every branch of an edge of H that abuts,
relative to G, to a vertex v of G lying in H also abuts to v, relative to H; (c) if abranch of an edge of H either abuts, relative to G, to a vertex v of G that does notlie in H, or does not abut to a vertex, relative to G, then this branch does not abut
to a vertex, relative to H. A morphism of semi-graphs will be called an embedding
if it induces an isomorphism of the domain onto a sub-semi-graph of the codomain.
Let G be a semi-graph. Then we shall refer to an edge of G that is of verticial
cardinality 2 (respectively, < 2; 0) as closed (respectively, open; isolated). We shallsay that two closed edges e and e of G are coverticial if the following condition

SEMI-GRAPHS OF ANABELIOIDS

13

holds: the edge e abuts to a vertex v of G if and only if the edge e abuts to v.
We shall say that G is locally ﬁnite if, for every vertex v of G, the set of edges
that abut to v is ﬁnite. We shall say that G is untangled if every closed edge ofG abuts to two distinct vertices. We shall refer to a connected semi-graph thathas precisely one vertex and precisely two edges, both of which are open, as a
joint. If a sub-semi-graph of a given semi-graph is a joint, then we shall refer tothis sub-semi-graph as a subjoint of the given semi-graph. We shall refer to thesub-semi-graph of G obtained by omitting all of the open edges as the maximal
subgraph of the semi-graph. We shall refer to as the compactiﬁcation of G thegraph obtained from G by appending to G, for each branch b of an edge of G that
does not abut to a vertex, a new vertex vb to which b is to abut. Thus, G forms
a sub-semi-graph of its compactiﬁcation. Moreover, any morphism of semi-graphs
induces a unique morphism between the respective compactiﬁcations. Finally, weobserve that every connected component of the topological space associated to the
maximal subgraph of G (respectively, G) is a deformation retract [in the sense ofalgebraic topology] of the corresponding connected component of the topologicalspace associated to G (respectively, the compactiﬁcation of G). A semi-graph whose
associated topological space is contractible [in the sense of algebraic topology] will
be referred to as a tree.
We recall in passing that there is a semi-graph that is naturally associatedto any pointed stable curve over an algebraically closed ﬁeld [cf. [Mzk3], Appendix]: That is to say, the vertices (respectively, closed edges; open edges; branchesof a closed edge) of this semi-graph are precisely the irreducible components (respectively, nodes; marked points; branches of a node) of the pointed stable curve.
The coincidence maps are determined in an evident fashion by the geometry of thepointed stable curve.
Let v (respectively, e; b) be a(n) vertex (respectively, edge; branch of an edge)of G. Then we deﬁne morphisms of semi-graphs
G[v] → G;

G[e] → G;

G[b] → G

as follows: G[v] consists of a single vertex v  , which maps to v, and, for each branch
bv of an edge ev of G that abuts to v, an edge ebv of verticial cardinality 1 thatmaps to ev in such a way that the branch of ebv lying over bv abuts to v  . G[e]consists of a single edge e , which maps to e, and, for each branch be of e abutting
to a vertex vbe of G, a vertex vb e [of G[e]] that maps to vbe and is the abutment of
the branch be of e that lies over be . If b is a branch of an edge eb that abuts to avertex vb [of G], then G[b] is the sub-semi-graph of G[eb ] consisting of the uniqueedge of G[eb ] and the vertex of G[eb ] which is the abutment of the branch of thisunique edge that lies over b. Thus, G[v], G[e], G[b] are all trees [even if G fails to
be untangled]; if the branch b is a branch of the edge e that abuts to v, then we
have natural morphisms G[b] → G[v], G[b] → G[e] over G.
A morphism
φ : GA → GB
between semi-graphs will be called an immersion [or an immersive morphism] (respectively, excision [or an excisive morphism]) if it satisﬁes the condition that, for14

SHINICHI MOCHIZUKI

every vertex vA of GA that maps to a vertex vB of GB , the induced map frombranches abutting to vA to branches abutting to vB is injective (respectively, bijective). Thus, if we think of GA and GB as topological spaces, then an immersion
φ : GA → GB is locally [in some small neighborhood of every point of GA ] anembedding (respectively, a homeomorphism) of topological spaces.
Observe that: the ﬁve classes of morphisms G[v] → G, G[b] → G[e], G[e] → G,
G[b] → G, G[b] → G[v], are all immersive; the ﬁrst two of these classes are always
excisive; the last three of these classes are not excisive in general.
Also, we observe that a morphism of sub-semi-graphs GA ⊆ GB is immersive(respectively, excisive) if and only if, for every vertex vA of GA mapping to a vertex
vB of GB , the induced morphism of semi-graphs GA [vA ] → GB [vB ] is an embedding(respectively, isomorphism).
A morphism of semi-graphs
φ : GA → GB
will be called proper if it preserves verticial cardinalities of edges. A proper excision
will be referred to as a graph-covering. A graph-covering with ﬁnite ﬁbers will bereferred to as a ﬁnite graph-covering. Note that if φ : GA → GB is a graph-covering,with GA , GB connected, then the associated map of topological spaces will be acovering in the sense of algebraic topology. Conversely, every covering, in the senseof algebraic topology, of the topological space associated to GB arises in this way.
Also, we observe that, just as in the case of coverings of topological spaces, it makes
sense to speak of a graph-covering as Galois [i.e., “arising from a normal subgroupof the fundamental group”] and to speak of the pull-back of a graph-covering by anarbitrary morphism of semi-graphs.
Proposition 1.1. Any immersion from a connected graph into a tree is, in fact,
an embedding.
Proof. Indeed, suppose that we are given an immersion φ : GA → GB into atree GB which is not an embedding. If φ is injective on vertices, then it follows
from the deﬁnition of an immersion that φ is injective on edges, hence that φ is anembedding. Thus, it suﬃces to show that φ is injective on vertices.
Suppose that there exist distinct vertices v1 , v2 of GA that map to the samevertex w of GB . Write γA for a path on GA that connects v1 to v2 . Without
loss of generality, we may assume that γA has minimal length among paths on
GA that join distinct vertices of GA that map to the same vertex of GB . WritedefγB = φ(γA ). Then note that the minimality condition (together with the factthat φ is an immersion) implies that γB does not intersect itself. Thus, γB is aloop, starting and ending at w, and deﬁned by a sequence of edges, all of which are
distinct. But this contradicts the fact that GB is a tree. This completes the proof.
SEMI-GRAPHS OF ANABELIOIDS

15

Thus, in particular, if we start with an arbitrary immersion of connected graphs[which are not necessarily trees]φ : GA → GBthen Proposition 1.1 implies that the induced morphism
GA → GBon universal graph-coverings [i.e., the associated topological coverings are universal
coverings of GA , GB , respectively, in the sense of algebraic topology] — which are
well-deﬁned up to composition with deck transformations — is an embedding [since
it is an immersion into a tree]. More generally, given an arbitrary graph-covering
GB  → GB
one can ask when the base-changed immersion
φ  : G A → G B 
is an embedding on each connected component of GA . Proposition 1.1 implies thatthe universal graph-covering GB → GB is suﬃcient to realize this condition.
In fact, however, when GA , GB are ﬁnite, this condition may be realized by aﬁnite graph-covering GB  → GB :Theorem 1.2.

(“Zariski’s Main Theorem for Semi-graphs”) Let
φ : GA → GB

be an immersion of ﬁnite semi-graphs. Then:(i) The morphism φ factors as the composite of an embedding
GA → GB and a ﬁnite graph-covering GB  → GB .
(ii) There exists a ﬁnite graph-covering GB  → GB such that the restrictionof the base-changed morphismφ  : G A → G B 
to each connected component of GA is an embedding.
Remark 1.2.1. The author is indebted to M. Matsumoto for the following elegant
graph-theoretic proof of Theorem 1.2.

16

SHINICHI MOCHIZUKI

Remark 1.2.2.
The general form of Theorem 1.2 is reminiscent of the wellknown result in algebraic geometry (“Zariski’s Main Theorem” — cf., e.g., [Milne],
Chapter I, Theorem 1.8) that any separated quasi-ﬁnite morphism
f :X→Ybetween noetherian schemes factors as the composite of an open immersion X → Y 
and a ﬁnite morphism Y  → Y — cf. also Lemma 1.5 below.
Proof. First, we observe that (ii) follows formally from (i) [by taking the ﬁnite
graph-covering of (ii) to be a Galois ﬁnite graph-covering of GB that dominates thegraph-covering of (i)]. Thus, it suﬃces to prove (i).
Next, let us observe that:(a) Any immersion of semi-graphs for which the induced morphism between
the respective compactiﬁcations is an embedding is itself an embedding(of semi-graphs).
(b) Restriction from the compactiﬁcation of GB to GB induces an equivalenceof categories between the respective categories of ﬁnite graph-coverings.
Moreover, the compactiﬁcation of a ﬁnite graph-covering of GB is naturally isomorphic to the corresponding ﬁnite graph-covering of the compactiﬁcation of GB .
In particular, by replacing the semi-graphs involved by their compactiﬁcations, it
suﬃces to prove (i) in the case where all of the semi-graphs are, in fact, graphs.
Thus, for the remainder of the proof, we assume that GA , GB are graphs.
Let us writeHn(where n ≥ 1 is an integer) for the graph consisting of one vertex vH and n edges
eH,1 ; . . . ; eH,n (all of which run from vH to vH ).
Next, let us observe that by Lemma 1.4 below, there exists an immersion
ζ : GB → Hn
which we may compose with φ to form an immersion:ψ : GA → Hn
Moreover, since pull-backs of ﬁnite graph-coverings of Hn via ζ form ﬁnite graphcoverings of GB , it follows that in order to prove that the assertion of Theorem
1.2, (i), is true for φ, it suﬃces to prove that it is true for ψ. On the other hand,
Theorem 1.2, (i), follows for ψ by Lemma 1.5 below. 
Note that, relative to the topological space point of view discussed above, thevertex vH of the graph Hn meets precisely 2n branches.

SEMI-GRAPHS OF ANABELIOIDS

Lemma 1.3.

17

Let G be a ﬁnite graph. Then:(i) To give a morphism
φ : G → Hn
is equivalent to assigning an orientation and a “color” ∈ {1, . . . , n} to each edgeof G.
(ii) The morphism φ is an immersion if and only if for each color i ∈{1, . . . , n}, and at each vertex v of G, the number of branches of color i that enter(respectively, leave) v — i.e., relative to the assigned orientations — is ≤ 1.
(iii) The morphism φ is an excision [or, equivalently, a ﬁnite graph-covering]
if and only if for each color i ∈ {1, . . . , n}, and at each vertex v of G, the number ofbranches of color i that enter (respectively, leave) v — i.e., relative to the assigned
orientations — is = 1.
Proof. First, we ﬁx an orientation on each edge eH,i of Hn , and regard the edge
eH,i as being of color i.
Now let us prove (i). Given a morphism φ : G → Hn , we obtain orientations
and colors on the edges of G by pulling back the orientations and colors of Hn via
φ. Conversely, given a choice of orientations and colors on the edges of G, we obtaina morphism φ : G → Hn by sending all the vertices of G to vH and mapping theedges of G to the edges of Hn in the unique way which preserves orientations andcolors.
Assertions (ii) and (iii) follow immediately by considering the local structureof Hn at vH . Note that in general, a morphism of ﬁnite graphs is always proper,
hence is a ﬁnite graph-covering if and only if it is excisive. 
Lemma 1.4.
integer n ≥ 1.

Every ﬁnite graph G admits an immersion G → Hn for some

Proof. Indeed, if we take n to be the number of edges of G and assign distinct
colors to distinct edges of G, then it is immediate from Lemma 1.3, (ii), that (forany assignment of orientations) the resulting morphism G → Hn is an immersion.
Lemma 1.5. Let φ : G → Hn be an immersion of ﬁnite graphs. Then φ extends
to a ﬁnite graph-covering φ : G → Hn for some embedding G → G .
Proof. We construct (G , φ ) from (G, φ) by adding edges (equipped with orientations and colors) to G until the resulting φ is excisive, i.e., satisﬁes the conditionof Lemma 1.3, (iii). Suppose that there exists a vertex v of G that does not satisfy
this condition. This means that there is some color i such that either there does
not exist a branch of color i entering v or there does not exist a branch of color i

18

SHINICHI MOCHIZUKI

leaving v (or both). If there do not exist any branches of color i meeting v, then we
simply add an edge of color i to G that runs from v to v. Now suppose (without
loss of generality) that there exists a branch of color i leaving v, but that there
does not exist a branch of color i entering v. Then we follow the i-colored edge
defleaving v1 = v to a new vertex v2 (necessarily distinct from v1 ). Now there are two
possibilities:(1) There exists an i-colored edge leaving this vertex.
(2) There does not exist an i-colored edge leaving this vertex.
If (1) holds, then we repeat the above procedure — i.e., we follow this i-colorededge out of v2 to another vertex v3 , which is necessarily distinct from v2 since theunique (by Lemma 1.3, (ii)) i-colored edge entering v2 originated from a vertex
which is distinct from v2 . Thus, continuing in this way, we obtain a sequence
v1 , v2 , v3 , . . .
of distinct (by Lemma 1.3, (ii)) vertices of G. Since G is ﬁnite, this sequence must
eventually terminate at some vertex vk satisfying (2). Then we add an i-colorededge to G running from vk to v1 to form a pair (G[2], φ[2]) extending the originaldef(G[1], φ[1]) = (G, φ).
Note that φ[2] is still an immersion, that G[2] has the same set of vertices as
G[1], that the set of “colors” [labeled 1, . . . , n] remains unchanged, and that the
cardinality of the [ﬁnite] set of [ordered] pairs consisting of a vertex and a color
which violate the condition of Lemma 1.3, (iii), relative to φ[2], is < the cardinalityof the [ﬁnite] set of [ordered] pairs consisting of a vertex and a color which violate
the condition of Lemma 1.3, (iii), relative to φ[1]. Thus, if we apply the procedure(G[1], φ[1]) → (G[2], φ[2])to (G[2], φ[2]) to obtain some (G[3], φ[3]), and so on, we obtain a sequence of pairs(G[1], φ[1]); (G[2], φ[2]); (G[3], φ[3]); . . .
which — by the ﬁniteness of the sets of vertices and colors — necessarily terminatesin a pair (G , φ ) such that φ is a ﬁnite graph-covering, as desired. 
Remark 1.5.1. Consider the case of an immersion
φ : G → H1
where G is a ﬁnite connected graph. Since the (topological) fundamental groupof H1 is equal to Z, the isomorphism class of a (connected) ﬁnite graph-coveringG → H1 of H1 is determined by its degree d (a positive integer) [in the sense ofalgebraic topology]. Then one can ask what conditions one must place on d for the

SEMI-GRAPHS OF ANABELIOIDS

19

corresponding ﬁnite graph-covering to satisfy the property of Theorem 1.2, (ii). In
some sense, there are essentially two phenomena that may occur:(1) The case where φ itself is a ﬁnite graph-covering, of degree n. In thiscase, the resulting condition on d is nonarchimedean, i.e.:d≡0(mod n)(2) The case where G consists of n vertices v1 , . . . , vn , and precisely one edge
joining vj to vj+1 , for j = 1, . . . , n − 1 (and no other edges). In this case,the resulting condition on d is archimedean, i.e.:d≥n

The above analysis suggests that there is some interesting arithmetic “hidden” inTheorem 1.2.
The following interesting consequence of Theorem 1.2 — which asserts, ineﬀect, that ﬁnitely generated subgroups of ﬁnite rank [discrete] free groups admitbases with properties reminiscent of their abelian counterparts — was pointed out
to the author by A. Tamagawa:
Corollary 1.6.
(Finitely generated Subgroups of Finite Rank Free
Groups) Let F be a ﬁnitely generated subgroup of a free group G of ﬁnite rank (soF is also free of ﬁnite rank). Then:(i) There exists an immersion of ﬁnite graphs φ : GA → GB whose induced
morphism on (topological) fundamental groups is isomorphic to the inclusion F →G.
(ii) There exists a ﬁnite index subgroup H ⊆ G such that H contains F , and,moreover, there exists a set of free generators γ1 , . . . , γr of H with the property thatfor some s ≤ r, γ1 , . . . , γs form a set of free generators of F .
Proof. First, observe that if GA → GB  is an embedding, then any set of free generators of the fundamental group of GA may be extended to a set of free generatorsof the fundamental group of GB  . In light of this observation, assertion (ii) follows
by applying Theorem 1.2, (i), to an immersion as in assertion (i) (of the present
Corollary).
Thus, it suﬃces to prove (i). Let GB be any graph whose fundamental groupis equal to G. Then the subgroup F ⊆ G deﬁnes an inﬁnite graph-covering
G A → G B

20

SHINICHI MOCHIZUKI

of GB . In particular, GA has fundamental group equal to F . Although, in general,the graph GA will not necessarily be ﬁnite, it follows from the fact that its fundamental group F is ﬁnitely generated that there exists a ﬁnite subgraph GA ⊆ GA
such that the natural injection of fundamental groups π1 (GA ) → π1 (GA ) is, infact, a bijection. Moreover, the composite GA → GA → GB is an immersion(since it is a composite of immersions). This completes the proof of (i). 
Another interesting consequence of Theorem 1.2 is the following well-known
result:Corollary 1.7.
(Residual Finiteness of Free Groups) Every discrete freegroup F injects into its proﬁnite completion.
Proof. Indeed, let G be a connected graph with π1 (G) = F . If H ⊆ F is the kernelof the map from F to its proﬁnite completion, write H → G for the corresponding
graph-covering. If H is not a tree, then one veriﬁes immediately that H contains aﬁnite connected subgraph H which is not a tree. In particular, H admits nontrivial
ﬁnite graph-coverings. Let G be a ﬁnite connected subgraph of G which contains
the image of H . Then if we apply Theorem 1.2, (i), to the immersion H → G , weobtain [since ﬁnite graph-coverings of a subgraph of a given graph always extend to
ﬁnite graph-coverings of the given graph] that there exists a ﬁnite graph-coveringK → G whose pull-back to H is nontrivial. Thus, if we extend K → G to a ﬁnite
graph-covering K → G, we obtain a ﬁnite graph-covering of G whose pull-back to
H is nontrivial. But this contradicts the deﬁnition of H. 
Remark 1.7.1. We recall in passing that there is also a pro-l version of thisresidual ﬁniteness result — cf., e.g., [RZ], Proposition 3.3.15.
Finally, before continuing, we note the following useful result concerning ﬁnite
group actions on semi-graphs, which is implicit in the theory of [Serre]:
Lemma 1.8.
(Finite Group Actions on Semi-graphs) Let G be a connected semi-graph, equipped with the action of a ﬁnite group G. Then:(i) Every ﬁnite sub-semi-graph G of G is contained in a ﬁnite connected subsemi-graph G of G that is stabilized by the action of G.
(ii) Suppose that G is a tree. Then: (a) there exists at least one vertex or edgeof G that is ﬁxed by G; (b) if G ﬁxes two distinct vertices w1 , w2 of G, then G
acts trivially on any “geodesic” [i.e., path of closed edges of minimal length] thatjoins w1 , w2 ; (c) if G ﬁxes three distinct vertices of G, then there exists at least
one subjoint of G on which G acts trivially.
Proof. First, we consider assertion (i). Since G is connected, we may assume
without loss of generality that G is connected and contains the G-orbit of some

SEMI-GRAPHS OF ANABELIOIDS

21

vertex. Then one veriﬁes easily that if we take G to be the G-orbit of G , thenthe desired properties are satisﬁed.
Next, we consider assertion (ii). First, we verify assertion (a). This follows
formally from [Serre], Chapter I, §6.5, Corollary 3 to Proposition 26, Proposition
27 — at least if one assumes, as in done in [Serre], that G ﬁxes some orientation on
the tree G. On the other hand, by “splitting” each edge of G which violates thisassumption into two new edges, corresponding to the two branches of the originaledge, one sees immediately that one still obtains assertion (a), even without thisassumption. This completes the proof of (ii), (a).
Next, to prove (ii), (b), recall from [Serre], Chapter I, §2.2, Proposition 8, that
there is a unique path of minimal length from w1 to w2 . Since G ﬁxes w1 , w2 ,it thus follows that G ﬁxes this path. Thus, [since it is evident that there are no
automorphisms of this path that ﬁx w1 , w2 ] we conclude that G acts trivially on
this path, as desired. This completes the proof of (ii), (b). Finally, we observe that(ii), (c) follows formally from (ii), (b). Remark 1.8.1. We observe, in passing, that Lemma 1.8, (ii), (a), implies [thewell-known fact — cf., e.g., [Serre], Chapter I, §3.4, Theorem 5] that a free group[which may be thought of as the fundamental group of some graph, hence admits
a free action on some tree] does not contain any nontrivial ﬁnite subgroups.

22

SHINICHI MOCHIZUKI

Section 2: Commensurability Properties
In this §, we begin our study of the geometry of semi-graphs of proﬁnite groups
by considering various topics concerning commensurators and slimness.
In the following, we shall use the language of anabelioids of [Mzk4]— i.e.,“multi-Galois categories” in the language of [SGA1] — which, in eﬀect, amounts toworking with proﬁnite groups up to inner automorphism [cf., e.g., [Mzk4], Proposition 1.1.4]. If X is a connected anabelioid [i.e., a Galois category], then we shalldenote the proﬁnite fundamental group [for some choice of basepoint] of X by:
π1 (X )
As is well-known, this proﬁnite group is, in a natural sense, independent of the
choice of basepoint, up to inner automorphism.
Deﬁnition 2.1.

We shall refer to the following data G:(a) a semi-graph G;(b) for each vertex v of G, a connected anabelioid Gv ;(c) for each edge e of G, a connected anabelioid Ge , together with, for eachbranch b ∈ e abutting to a vertex v, a morphism of anabelioids b∗ : Ge →Gv
as a semi-graph of (connected) anabelioids [cf. the notion of a “semi-graph of proﬁnite groups” introduced in [Mzk3], Appendix]. We shall refer to the various Gv , Ge
as the constituent anabelioids of G. Given two semi-graphs of anabelioids, there isan evident notion of morphism between semi-graphs of anabelioids [cf. also Remark
2.4.2 below]. If all of the b∗ ’s are π1 -monomorphisms [i.e., induce injective homomorphisms on associated fundamental groups — cf. [Mzk4], Deﬁnition 1.1.12], thenwe shall say that G is of injective type. When the underlying semi-graph G is agraph, we shall refer to a semi-graph of anabelioids G as a graph of (connected)
anabelioids.
Let G be a connected semi-graph of anabelioids [i.e., the underlying semi-graph
G is assumed to be connected]. If G has at least one vertex, then let us denote byB(G)the category of objects given by data
{Sv , φe }where v (respectively, e) ranges over the vertices (respectively, edges) of G; for eachvertex v, Sv ∈ Ob(Gv ); for each edge e, with branches b1 , b2 abutting to vertices

SEMI-GRAPHS OF ANABELIOIDS

23∼

v1 , v2 , respectively, φe : {(b1 )∗ }∗ Sv1 → {(b2 )∗ }∗ Sv2 is an isomorphism in Ge , andmorphisms given by morphisms [in the evident sense] between such data. If G has
no vertices — and hence precisely one edge e, which is necessarily isolated — thendef
we shall write B(G) = Ge . One veriﬁes immediately that this category B(G) is aconnected anabelioid.
Now let G ∈ Ob(B(G)); write
def

def

B  = B(G)G → B = B(G)for the corresponding ﬁnite étale covering of anabelioids [cf. [Mzk4], Deﬁnition
1.2.2, (i)]. Then it follows from the deﬁnition of the anabelioid B(G) associatedto the semi-graph of anabelioids G [i.e., in terms of ﬁnite étale coverings of eachof the constituent anabelioids of G, together with gluing isomorphisms] that B itself arises naturally as the B(−) of some connected semi-graph of anabelioids G equipped with a morphismG → Gof semi-graphs of anabelioids which lies over some proper morphism of semi-graphs:
G → G
Here, if v (respectively, e) is a vertex (respectively, edge) of G, then the vertices v (respectively, edges e ) of G are the elements of the set of connected components of
the [not necessarily connected!] anabelioid B  ×B Gv (respectively, B  ×B Ge ). The
connected anabelioid Gv  (respectively, Ge  ) is the connected component anabelioidof B  ×B Gv (respectively, B  ×B Ge ) determined by v  (respectively, e ).
Let us denote by
def

def

ΠG = π1 (G) = π1 (B(G))
the fundamental group of the connected anabelioid B(G) relative to some basepoint.
Put another way, if we choose basepoints for the constituent anabelioids of G, thenG determines a “semi-graph of proﬁnite groups” [cf. [Mzk3], Appendix, except that
here the underlying semi-graph is not necessarily ﬁnite], and one may think of ΠG
as the proﬁnite group associated to this semi-graph of proﬁnite groups.
For each vertex v (respectively, branch b of an edge e that abuts to the vertex v)of G, let us write Πv (respectively, Πb ) for the fundamental group of the anabelioid
Gv (respectively, Ge ) for some choice of basepoint. Thus, we have natural outerhomomorphisms:
Πv → ΠG ; Πb → ΠG
Moreover, the branch b determines an associated outer homomorphism:
Πb → Πv
If G is of injective type, then we shall also denote the image of Πb in Πv , which iswell-deﬁned up to conjugation in Πv , by Πb . [Here, we note that the use of thesubscript “b” in the notation “Πb ” — i.e., as opposed to “e” in, for instance, the24

SHINICHI MOCHIZUKI

notation “Ge ” — is useful in discussions concerning subgroups of Πv [as opposed tojust subgroups of ΠG ].]If H ⊆ G is a [not necessarily connected] sub-semi-graph of G, then we shallwrite GH for the semi-graph of anabelioids determined by restricting G to H. That
is to say, the underlying semi-graph of GH is H, and for each component c (i.e.,defeither an edge or vertex) of H, we let (GH )c = Gc ; if a branch b of an edge e of H
abuts to a vertex v of H, then we take the associated morphism b∗ : (GH )e → (GH )v
to be the morphism associated to the corresponding branch of G. If H is connected,
then we shall writeΠHfor the fundamental group of GH , for some choice of basepoint. Thus, we have anatural outer homomorphism ΠH → ΠG .
Deﬁnition 2.2.
(i) We shall refer to an arrow G  → G that arises as in the above discussion
def

def

from a ﬁnite étale covering of anabelioids B  = B(G)G → B = B(G) as a ﬁnite
étale covering [of G].
(ii) If a morphism of semi-graphs of anabelioids satisﬁes the property that eachof its induced morphisms between constituent anabelioids is an isomorphism (respectively, is ﬁnite étale; induces an homomorphism with open image between therespective π1 (−)’s), then we shall say that the morphism is locally trivial (respectively, locally ﬁnite étale; locally open).
Remark 2.2.1. Note that by considering collections of normal open subgroupsof ΠG whose intersection is trivial, one may think of ΠG as acting on a “universal
pro-ﬁnite étale covering” of G. In particular, ΠG acts naturally on the underlying“pro-semi-graph” of this “universal pro-ﬁnite étale covering”. Moreover, it follows
immediately from the deﬁnitions [cf. the discussion of the universal covering associated to a graph of groups in [Serre], Chapter I, p. 51] that the image of each Πv(respectively, Πb ) in ΠG is equal to the stabilizer of a compatible system of vertices(respectively, edges) of this pro-semi-graph.
Deﬁnition 2.3.
Let G be a semi-graph of anabelioids of injective type, with
underlying semi-graph G.
(i) We shall say that G is of bounded order if there exists an integer M ≥ 1such that all of the π1 (Gv )’s, where v ranges over the vertices of the underlying
semi-graph G, are ﬁnite groups of order dividing M.
(ii) We shall refer to a morphism of semi-graphs of anabelioids
G → GSEMI-GRAPHS OF ANABELIOIDS

25∼

which induces an isomorphism on underlying semi-graphs G → G [relative to which
we may identify G, G ] and for which G  is a semi-graph of anabelioids of boundedorder as an approximator for G. We shall say that an approximator is π1 -epimorphic
if each of the induced morphisms between the respective constituent anabelioids is
a π1 -epimorphism [i.e., induces a surjective homomorphism on associated fundamental groups — cf. [Mzk4], Deﬁnition 1.1.12].
(iii) We shall say that G is quasi-coherent if, for every integer M ≥ 1, and every
collection of ﬁnite étale coverings Hc → Gc of degree ≤ M, where c ranges over thecomponents of G, there exists an approximator
G → G
such that, for each component c of G, the pull-back to Gc of the “universal covering”
Hc → Gc of Gc [i.e., the ﬁnite étale covering determined by the trivial subgroup of
π1 (Gc )] splits Hc → Gc . In this situation, we shall say that this approximator splits
the given collection of coverings. We shall say that a quasi-coherent G is coherent
if, for each component c of G, the proﬁnite group π1 (Gc ) is topologically ﬁnitely
generated [which, as is well-known, implies that Out(π1 (Gc )) is equipped with anatural proﬁnite group structure].
Remark 2.3.1. Relative to the notation of Deﬁnition 2.3, (iii), by replacing theconstituent anabelioids of G  by the image anabelioids of the constituent anabelioidsof G [cf. [Mzk4], Deﬁnition 1.1.7, (i)], one may always take the approximator of
Deﬁnition 2.3, (iii), to be π1 -epimorphic.
Deﬁnition 2.4.
Let G be a semi-graph of anabelioids of injective type, with
underlying semi-graph G.
(i) Let v be a vertex of G. If, for every integer M ≥ 1, there exists a π1 epimorphic approximator
G → G
for G such that there exists a subgroup NM ⊆ π1 (Gv ) of order ≥ M which hastrivial intersection with all of the conjugates, in π1 (Gv ), of all of the π1 (Ge ) [wheree ranges over the edges abutting to v], then we shall say that v is elevated. If all of
the vertices of G are elevated, then we shall say that G is totally elevated.
(ii) If, for every vertex v of G, the anabelioid Gv is slim, then we shall say that
G is verticially slim.
(iii) Let e be a closed edge of G. If there exists a ﬁnite étale covering G  → G
of G such that the underlying graph G of G  contains a pair of distinct coverticial
edges ea , eb , both of which map to e in G, then we shall say that e is sub-coverticial.
If, for every ﬁnite étale covering G  → G of G, it holds that every edge e of theunderlying graph G of G  that maps to e is sub-coverticial, then we shall saythat e is universally sub-coverticial. If every closed edge of G is sub-coverticial

26

SHINICHI MOCHIZUKI(respectively, universally sub-coverticial), then we shall say that G is totally subcoverticial (respectively, totally universally sub-coverticial).
(iv) Let e be an edge of G. We shall say that e is aloof (respectively, estranged)
if, for every vertex v to which some branch b of e abuts and every g ∈ Πv , theintersection in Πv of Πb with any subgroup of the form g · Πb · g −1 , where either/ Πb , has inﬁnite indexb = b is a branch of an edge that abuts to v or b = b and g ∈in Πb (respectively, and is, in fact, trivial). If every edge of G is aloof (respectively,
estranged), then we shall say that G is totally aloof (totally estranged).
Remark 2.4.1.
It is immediate that “(totally) estranged” implies “(totally)
aloof”. Moreover, one veriﬁes easily that if G  → G is a ﬁnite étale covering, andv  (respectively, e ; e ; e ) is a(n) vertex (respectively, edge; edge; edge) of G thatmaps to a(n) elevated vertex v (respectively, universally sub-coverticial edge e;aloof edge e; estranged edge e) of G, then v  (respectively, e ; e ; e ) is itself elevated(respectively, universally sub-coverticial; aloof; estranged).
Remark 2.4.2. Let φ : G → H be a morphism between semi-graphs of anabelioidsof injective type. Concretely speaking, this means that we are given, for each vertexv (respectively, edge e) of G mapping to a vertex w (respectively, edge f) of H, a1-morphism of anabelioids
φv : Gv → Hw (respectively, φe : Ge → Hf )together with an isomorphism φb of the composite 1-morphism of anabelioids Ge →Gv → Hw with the composite 1-morphism of anabelioids Ge → Hf → Hw whenevera branch b of e abuts v. That is to say, strictly speaking, φ is a “1-morphism”; the1-morphisms from G to H form a category, of which φ is an object; one can thenspeak of isomorphisms between various objects of this category.
On the other hand, observe that if we restrict our attention to locally open morphisms φ between totally aloof semi-graphs of anabelioids, then it follows formallyfrom the deﬁnitions [cf. also [Mzk4], Corollary 1.1.6] that:
The isomorphism class of φ is completely determined by the isomorphismclass of the φv .
If, moreover, we restrict our attention to locally open morphisms φ between totally
aloof, verticially slim semi-graphs of anabelioids, then:The 1-morphism φ has no nontrivial automorphisms.
That is to say, as long as we restrict our attention to locally open morphisms [e.g.,
locally ﬁnite étale] between totally aloof, verticially slim semi-graphs of anabelioids,
we may work with such morphisms as if they are simply “morphisms in a category”,
rather than 1-morphisms in a 2-category [cf. the situation for ﬁnite étale morphisms

SEMI-GRAPHS OF ANABELIOIDS

27

between slim anabelioids: [Mzk4], Proposition 1.2.5]. In the following, we shall often
take this point of view without further mention.
Proposition 2.5. (Injectivity) Let G be a connected, quasi-coherent graphof anabelioids. Let H ⊆ G be a connected subgraph of the underlying graph G of
G. Then:(i) The natural morphisms Πb → ΠG , Πv → ΠG , ΠH → ΠG are injective.
By abuse of notation, we will denote their images, which are well-deﬁned up to
conjugation in ΠG , by Πb , Πv , ΠH , respectively.
(ii) Suppose that G is of bounded order. Then there exists a normal open
subgroup H ⊆ ΠG , such that all of the natural morphisms Πb → ΠG /H, Πv →
ΠG /H are injective.
Proof. First, we consider assertion (i). Since the natural morphism Πb → Πv is
injective, and the morphism Πv → ΠG may be considered as a special case of themorphism ΠH → ΠG [i.e., the case where H consists of a single vertex and no edges],
it suﬃces to show that any ﬁnite étale Galois covering of GH may be split by a ﬁnite
étale covering pulled back from G. Note that this is immediate in the locally trivial
case. Indeed, in this case, it suﬃces to extend the given ﬁnite étale covering fromGH to the remainder of G by gluing [which is always possible, by the local triviality
assumption!].
Thus, by our assumption of quasi-coherence, it suﬃces to construct, under thefurther assumption that G is of bounded order, a ﬁnite étale covering of G each ofwhose constituent anabelioids is trivial. We construct such a covering by gluing:
Let M ≥ 1 be an integer such all of the orders [π1 (Gv ) : 1] divide M. Over thevertex v, we take the covering to be the union of M/[π1 (Gv ) : 1] copies of some
“universal covering” of Gv [i.e., the ﬁnite étale covering determined by the trivial
subgroup of π1 (Gv )]. If e is an edge that abuts to the vertex v, then the restrictionof this covering to e is a union of M/[π1 (Ge ) : 1] copies of some universal coveringof Ge — i.e., a covering of Ge whose isomorphism class is independent of v! Thus,by choosing appropriate gluing isomorphisms, we obtain a covering of G having thedesired properties.
As for assertion (ii), the Galois closure of a covering such as that constructedin the preceding paragraph determines a normal open subgroup H ⊆ ΠG having
the properties asserted in assertion (ii). 
Remark 2.5.1. Note that the quasi-coherence hypothesis in the injectivity asserdeftion of Proposition 2.5, (i), is by no means superﬂuous: For instance, let G = Z/lZ
[where l is a prime number]; write GN for the proﬁnite group given by taking thedirect product of a collection of copies of G indexed by the set of natural numbers
N. Write α : GN → GN for the continuous injection induced by the injection N → N
given by N  n → n + 1. Then let us observe that there does not exist a proper
open subgroup H ⊆ GN such that H = α−1 (H). In particular, if one takes G to

28

SHINICHI MOCHIZUKI

be the graph with precisely one vertex and one edge [both of whose branches abutto the unique vertex], and G to be the graph of anabelioids with underlying graph
G such that the unique vertex and unique edge are equipped with the anabelioid
B(GN ) and the morphisms B(GN ) → B(GN ) corresponding to the two branches of
the unique edge are given by the identity and B(α), then one concludes from theabove observation that the proﬁnite fundamental group associated to this graph ofanabelioids G is trivial.
Proposition 2.6.
(Commensurability) Let G be a connected, quasicoherent graph of anabelioids. Let H, K ⊆ G be connected subgraphs of theunderlying graph G of G. Suppose that there exists a component c of H that does
not belong to K and which is either an elevated vertex or a sub-coverticial
edge [i.e., relative to G]. Then the intersection of ΠH with any conjugate of ΠK
has inﬁnite index in ΠH . In particular, no conjugate of ΠH is commensurable toa conjugate of ΠK .
Proof. First, we consider the case where c = v is a vertex. We may assume
without loss of generality that ΠK ΠH has ﬁnite index in ΠH . Then by taking“M” in Deﬁnition 2.4, (i), to beΠH ] + 1[ΠH : ΠK
and applying the fact that G is elevated at v, we obtain the existence of a π1 epimorphic approximator
G → G
such that, if we denote the various fundamental groups associated to G  by meansof a “dash”, then there is a subgroup NM ⊆ Πv of order ≥ M that has trivial
intersection with each of the conjugates of the Πb , for branches b that abut to v.
Since G  is of bounded order, it follows from Proposition 2.5, (ii), that there exists
a normal open subgroup K ⊆ ΠK such that Πw injects into ΠK /K, for all vertices
w of K; moreover, it follows that there exists an integer M  that is divisible bothby [ΠK : K] and by twice the orders of all of the Πw , as w ranges over the verticesof G.
Now we construct a [not necessarily connected!] ﬁnite étale coveringG  → G 
as follows: Over K, we take this covering to be the union of M  /[ΠK : K] copiesof the covering deﬁned by the normal open subgroup K ⊆ ΠK . Over vertices
w = v not contained in K, we take this covering to be a union of M  /[Πw : 1]
copies of a “universal covering” of Gw. Over v, we take this covering to be a unionof M /2[Πv : NM ] copies of the covering deﬁned by the Πv -set Πv /NM and ofM  /2[Πv : 1] copies of a “universal covering” of Gv . Note that the restriction of
any of these coverings over w or v to an abutting edge e is isomorphic to a unionof “universal coverings” of Ge . Thus, by choosing appropriate gluing isomorphisms,
we obtain a covering G  → G  .

SEMI-GRAPHS OF ANABELIOIDS

29

On the other hand, the existence of this covering leads to a contradiction,def
as follows: This covering determines a ﬁnite ΠG -set S; write H  = Ker(ΠG →Aut(S)); denote the images in ΠG /H  of the various fundamental groups underconsideration by means of a “double dash”. Thus, we have equalities ΠK = ΠK /K,Πv = Πv [because of the existence of “universal coverings” in the restriction of G 
to v] and inequalities as follows:[ΠH : ΠK



ΠH ] ≤ [ΠH : ΠKΠH ] < M
Moreover, since ΠK acts freely on S, it follows that ΠK ΠH also acts freely on S,
and hence that the isotropy subgroup I ⊆ ΠH associated to an element of S has
deforder ≤ [ΠH : ΠK ΠH ] < M. In particular, Iv = I Πv has order < M. On
the other hand, by the construction of G  , it follows that for some such isotropy
group Iv ⊆ Πv = Πv , we have NM ⊆ Iv , where we recall that NM has order ≥ M,
a contradiction.
Next, we consider the case where c = e is an edge. Since e is sub-coverticial,
it suﬃces [by replacing G by a ﬁnite étale covering of G] to show that ΠK ΠH
has inﬁnite index in ΠH under the assumption that H contains a pair of distinct
coverticial edges ea , eb , neither of which is contained in K. Thus, let us suppose
defthat ΠK ΠH has ﬁnite index in ΠH ; set M = [ΠH : ΠK ΠH ] + 1.
Now observe that by using the loop L of H constituted by ea , eb , we may
construct a ﬁnite graph-covering of degree M
G → G
which is trivial over K, but connected over L. Then considering the actions of ΠH ,
ΠK on the corresponding ﬁnite ΠG -set yields a contradiction. This completes theproof. 
Remark 2.6.1.
Note that the hypothesis in Proposition 2.6 concerning theexistence of an elevated vertex or a sub-coverticial edge is by no means superﬂuous:
Indeed, let G be a proﬁnite group, H ⊆ G an open subgroup. Suppose that G is
the graph with precisely three vertices, labeled 1, 2, 3, and precisely two edges, oneof which joins the vertices 1, 2, the other of which joins the vertices 2, 3. Supposethat the graph of anabelioids structure of G is given by equipping the vertex 1(respectively, 2; 3) with the anabelioid B(G) (respectively, B(H); B(H)), the twoedges with the anabelioid B(H); we take the morphisms B(H) → B(H), B(H) →B(G) corresponding to the various branches of edges to be the morphisms inducedby the identity and inclusion homomorphisms. Then one veriﬁes immediately thatthe proﬁnite fundamental group associated to this graph of anabelioids is naturally
isomorphic to G, but that if one takes H to be the subgraph determined by the [nonelevated!] vertex 1 and K to be the subgraph determined by the vertices 2, 3 andthe [non-sub-coverticial!] edge that joins them, then the conclusion of Proposition
2.6 is false [even if one interchanges H, K].

30

SHINICHI MOCHIZUKI

Corollary 2.7.
(Slimness and Commensurators) Let G be a connected,
quasi-coherent graph of anabelioids. Let H, K ⊆ G be connected subgraphs of
the underlying graph G of G. Suppose that every vertex of H, K is elevated [i.e.,
relative to G]. Then:(i) We have CΠG (ΠH ) = ΠH . In particular, if v is an elevated vertex of G,
then CΠG (Πv ) = Πv .
(ii) Suppose that H contains a vertex v such that Gv is slim. Then the naturalmorphism B(GH) → B(G) is relatively slim [cf. [Mzk4], Deﬁnition 1.2.9]. In
particular, B(G) is slim.
(iii) Suppose that every edge of H, K is sub-coverticial [i.e., relative to G]
and that ΠH is commensurable to a conjugate of ΠK in ΠG . Then H = K.
/ ΠH .
Proof. First, we consider assertion (i). Suppose that g ∈ CΠG (ΠH ), but g ∈Then there exists a connected ﬁnite Galois étale coveringG → Gdef— whose restriction to H = GH , we denote by H → H — satisfying the property
that there exists a connected component H of H such that g · H = H in G .
Since [as one
immediately] H injects into G as a subgraph, it thus follows
 veriﬁesthat g · H
H = ∅ [in G ]. Thus, assertion (i) follows from Proposition 2.6.
Next, we consider assertion (ii). By assertion (i), we have, for any open subgroup H ⊆ ΠH :ZΠG (H) ⊆ ZΠG (HΠv ) ⊆ CΠG (Πv ) = Πv

Thus, we conclude [since Gv is slim] that ZΠG (H) ⊆ ZΠv (HΠv ) = {1}.

Finally, assertion (iii) is a formal consequence of Proposition 2.6. 
Remark 2.7.1. There is a certain overlap between the content of Corollary 2.7
and the results obtained in [HR]. The techniques of [HR], however, are more grouptheoretic in nature and somewhat diﬀerent in spirit from those employed in thepresent exposition.
Remark 2.7.2. Note that Corollary 2.7, (i), implies in particular that [in thenotation of loc. cit.] if J ⊆ ΠG is any [not necessarily closed!] subgroup such thatΠv ⊆ J , then we have
CJ (Πv ) ⊆ CΠG (Πv ) = Πv
— i.e., Πv is commensurably terminal in J .
Next, we consider some concrete examples:SEMI-GRAPHS OF ANABELIOIDS

31

Example 2.8. Trivial Edge Anabelioids. One veriﬁes immediately that every
semi-graph of anabelioids G such that Ge is trivial for all edges e is quasi-coherent.
In this case, a vertex v of G is elevated if and only if Πv is inﬁnite. Similarly,
a closed edge abutting to vertices v, w is sub-coverticial (respectively, universally
sub-coverticial) if and only if both Πv and Πw are nontrivial (respectively, inﬁnite).
Deﬁnition 2.9.

Let Σ be a set of prime numbers.

(i) We shall refer to as a Σ-integer any positive integer each of whose prime
factors belongs to Σ.
(ii) Let A be a connected anabelioid. Then we shall refer to as the pro-Σcompletion of A the connected anabelioid constituted by the full subcategory of
A determined by the objects dominated by a Galois covering of the ﬁnal objectof A whose degree is a Σ-integer. Similarly, given a semi-graph of anabelioids, weshall refer to as the pro-Σ completion of the given semi-graph of anabelioids thesemi-graph of anabelioids obtained by replacing each constituent anabelioid by its
pro-Σ completion.
Example 2.10.
Stable Curves. Let Σ be a nonempty set of prime numbers.
Suppose that G is a semi-graph of anabelioids with the property that each Πv is the
maximal pro-Σ quotient of the fundamental group of a hyperbolic Riemann surface
of ﬁnite type, and that each Πb → Πv is the inclusion morphism of the inertia groupof one of the cusps. Now observe that a hyperbolic Riemann surface of ﬁnite typeadmits Galois coverings of degree a Σ-integer which are ramiﬁed with ramiﬁcationindex equal to an arbitrary given Σ-integer at a given cusp, but which are unramiﬁed at all other cusps. In light of this observation [together with various otherproperties that follow immediately from the well-known structure of fundamentalgroups of hyperbolic Riemann surfaces of ﬁnite type], one veriﬁes immediately that
G is coherent, totally elevated, totally universally sub-coverticial, totally estranged[cf. the proof of [Mzk3], Lemma 1.3.7], and verticially slim [cf. the proof of [Mzk3],
Lemma 1.3.1]. In particular, the pro-Σ completion of the semi-graph of anabelioids
determined by the semi-graph of proﬁnite groups with compact structure associatedto a pointed stable curve over an algebraically closed ﬁeld of characteristic 0 [cf.
[Mzk3], Appendix] satisﬁes these properties. Similarly, if Σ contains at least one
prime = p, then a hyperbolic curve over a separably closed ﬁeld of characteristic p
admits Galois coverings of degree a Σ-integer which are ramiﬁed with ramiﬁcationindex equal to an arbitrary given Σ-integer prime to p at a given cusp, but which
are unramiﬁed at all other cusps; in light of this observation [together with various
other well-known properties of fundamental groups of hyperbolic curves over separably closed ﬁeld of characteristic p], one veriﬁes easily that the pro-Σ completionof the semi-graph of anabelioids determined by the semi-graph of proﬁnite groups
with compact structure associated to a stable curve over a separably closed ﬁeld of
characteristic p > 0 is coherent, totally elevated, totally universally sub-coverticial,totally estranged [cf. the proof of [Mzk3], Lemma 1.3.12], and verticially slim [cf.
the proof of [Mzk3], Lemma 1.3.10].

32

SHINICHI MOCHIZUKI

Remark 2.10.1. In the case of Example 2.10, it is not diﬃcult to show, using
exactly the same techniques as those used in the proofs of Proposition 2.6, Corollary
2.7, that CΠG (Πb ) = Πb . Since, however, we shall not need this result in thefollowing, and, moreover, a precise description of the condition on edges in the caseof a more general G necessary to carry out such an argument [i.e., the analogue
for edges of the notion of an “elevated vertex”] would be rather technical to writeout in detail, we leave the task of working out the routine details to the interested
reader.
The case of Example 2.10 [cf. also Example 3.10 below] motivates the following
extension of the notion of a “morphism of semi-graphs of anabelioids”: First, weobserve that any semi-graph G may be regarded as a category
Cat(G)
as follows: The objects of this category are the components [i.e., vertices and edges]of G. The morphisms of this category are the identity morphisms of the components
and the branches of edges [i.e., if b is a branch of an edge e that abuts to a vertex
v, then we regard b as a morphism e → v] of G. Thus, if G is the underlying
semi-graph of a semi-graph of anabelioids G, then for every object c of Cat(G), we
have an anabelioid Gc , and for every morphism b : e → v of Cat(G), we have amorphism of anabelioids b∗ : Ge → Gv .
Deﬁnition 2.11.
Let G, H be semi-graphs of anabelioids. Then a generalized
morphism of semi-graphs of anabelioids
Φ:G→H
is deﬁned to be a collection of data, as follows:(a) a functor Cat(Φ) : Cat(G) → Cat(H);(b) for every object c of Cat(G) that is mapped by Cat(Φ) to an object d ofCat(H), a morphism of anabelioids Φc : Gc → Hd ;(c) for every arrow φ : c → c of Cat(G) that is mapped by Cat(Φ) to an∼arrow ψ : d → d of Cat(H), an isomorphism Φφ : ψ∗ ◦ Φc → Φc ◦ φ∗ , suchthat Φφ is the identity whenever φ is an identity morphism.

Remark 2.11.1. It is immediate from the deﬁnitions that every [non-generalized]
morphism of semi-graphs of anabelioids determines a generalized morphism of semigraphs of anabelioids. Also, just as in the non-generalized case, it is immediatefrom the deﬁnitions that every generalized morphism of semi-graphs of connectedanabelioids
Φ:G→H
determines, in a natural fashion, a morphism B(G) → B(H) between the associatedanabelioids.

SEMI-GRAPHS OF ANABELIOIDS

33

Section 3: The Tempered Fundamental GroupIn this §, we deﬁne and study the basic properties of the tempered fundamentalgroup of a semi-graph of anabelioids. The notion of the tempered fundamentalgroup is introduced in [André], §4. In the present manuscript, however, we wish to
study this notion from a more categorical point of view.
Let Π be a topological group. Then let us write
B temp (Π)
for the category whose objects are countable [i.e., of cardinality ≤ the cardinality ofthe set of natural numbers], discrete sets equipped with a continuous Π-action andwhose morphisms are morphisms of Π-sets.
Deﬁnition 3.1.
(i) If Π may be written as an inverse limit of an inverse system of surjectionsof countable discrete topological groups, then we shall say that Π is tempered.
(ii) Any category equivalent to a category of the form B temp(Π) for some tempered topological group Π will be referred to as a connected temperoid. Any category equivalent to a product [in the sense of a product of categories] of a countable[hence possibly empty!] collection of connected temperoids will be referred to as atemperoid.
(iii) Let T1 , T2 be temperoids. Then a morphism φ : T1 → T2 is deﬁned to
be a functor φ∗ : T2 → T1 that preserves ﬁnite limits and countable colimits. A
morphism φ will be called rigid if the functor φ∗ is rigid [cf. §0].
(iv) A connected object T of a temperoid T will be called Galois if, for any
two arrows ψ1 , ψ2 : S → T of T , where S is connected, there exists a [unique]automorphism α ∈ AutT (T ) of T such that ψ1 = α ◦ ψ2 .
Remark 3.1.1. Observe that every proﬁnite group is tempered. Moreover, justas in the case of proﬁnite groups, if a tempered group may be written as an inverse
limit of an inverse system indexed by a countable set of surjections of countablediscrete topological groups, then the group is countably generated [i.e., generated
as a topological group by a countable set of generators].
Remark 3.1.2. Suppose that Π is tempered. Then every open subgroup of Π isclosed and of countable index in Π. Moreover, the topology of Π admits a basis ofopen normal subgroups. If H ⊆ Π is an arbitrary subgroup, then the Π-set Π/Hforms an object of B temp (Π) if and only if H is open. If H1 , H2 ⊆ Π are open,
then there is a natural bijection between the morphisms Π/H1 → Π/H2 and thecosets h · H2 satisfying h−1 · H1 · h ⊆ H2 . In particular, if T1 , T2 are objects ofa temperoid T , and T1 is connected, then the set HomT (T1 , T2 ) is countable. If

34

SHINICHI MOCHIZUKIΠ is also tempered, then any continuous homomorphism Π → Π determines [bycomposing the action of Π on a Π -set with this homomorphism so as to obtain aΠ-set] a morphism of connected temperoids B temp (Π) → B temp (Π ).
Remark 3.1.3. Suppose that Π is tempered. Then an object of B temp (Π) is Galois
if and only if it is isomorphic to the object determined by a Π-set of the form Π/N,
where N ⊆ Π is an open normal subgroup. Alternatively, a connected object T
defof T = B temp (Π) is Galois if and only if the product T × T is isomorphic to the
coproduct of copies of T indexed by the elements of the [countable!] set AutT (T ),where the restriction to the copy labeled by σ ∈ AutT (T ) of the projection to theﬁrst (respectively, second) factor of T × T is given by the identity (respectively, σ).
Remark 3.1.4.
lioid), thenNote that if X is an anabelioid (respectively, connected anabedef

X  = (X 0 )[cf. §0] is a temperoid (respectively, connected temperoid). We shall refer to X as the temperiﬁcation of X . Just as in the case of anabelioids, the decompositionof a temperoid into a countable coproduct of connected temperoids — each ofwhich we shall refer to as a connected component of the original temperoid — may
be recovered completely category-theoretically from the categorical structure of theoriginal temperoid [i.e., by considering decompositions of the terminal object — cf.
[Mzk4], Deﬁnitions 1.1.8, 1.1.9]. Also, just as in the case of anabelioids, to givea morphism between two temperoids is equivalent to giving, for each connectedcomponent of the domain temperoid, a morphism [of connected temperoids] —which we shall refer to as a component morphism of the morphism — from thatconnected component to some connected component of the codomain temperoid.
Finally, we remark that, just as in the case of anabelioids, it is crucial in thegeometry of temperoids to allow for multiple connected components which arise, forinstance, when one pulls back an étale [cf. Deﬁnition 3.4, (i) below] morphism ofconnected temperoids via an arbitrary morphism of connected temperoids.
Remark 3.1.5. It is immediate from the deﬁnitions that every temperoid is analmost totally epimorphic category of countably connected type [cf. §0].
Remark 3.1.6. Observe that although every endomorphism of a connected objectof an anabelioid is an automorphism, temperoids do not, in general, satisfy thisproperty. Indeed, if Π is a [discrete] free group on generators e1 , e2 , and H ⊆ Π isthe subgroup generated by elements of the form en2 ·e1 ·e−n
2 , where n ranges over the
positive integers, then conjugation by e2 determines an endomorphism H → H [i.e.,
an endomorphism of the object determined by the Π-set Π/H of B temp (Π)] which is
not an automorphism. Nevertheless, it is immediate from the deﬁnitions that every
endomorphism of a Galois connected object of a temperoid is an automorphism.
Remark 3.1.7. In some situations, instead of considering temperoids or anabelioids, it is useful to consider slightly more general versions of these notions, which

SEMI-GRAPHS OF ANABELIOIDS

35

we shall refer to as “quasi-temperoids”, “quasi-anabelioids”, respectively. For moreon these “routine” generalizations, we refer to the Appendix.
Remark 3.1.8. For a more general treatment of “categories that behave likeanabelioids and temperoids”, we refer to [Dub].
Proposition 3.2.

(The “Grothendieck Conjecture” for Connected Temdef

peroids) For i = 1, 2, let Πi be a tempered [topological] group; write Ti =B temp (Πi ). Then the category of morphisms
T1 → T2

is equivalent to the category whose objects are continuous group homomorphisms
φ : Π1 → Π2 and whose morphisms φ → ψ are elements g ∈ Π2 such thatγg ◦ φ = ψ, where we write γg for the automorphism of Π2 given by conjugating by
g. In particular, there is a natural bijective correspondence between the setof isomorphism classes of morphisms T1 → T2 and the set of [continuous] outerhomomorphisms Π1 → Π2 .
Proof. Observe that, by thinking of a Galois object T ∈ Ob(T2 ) as an “AutT2 (T )torsor object” [cf. Remark 3.1.3], it follows that if φ : T1 → T2 is a morphism of
temperoids, then the object φ∗ (T ) of T1 is Galois. In light of this observation, theproof of Proposition 3.2 is formally entirely similar to that of [Mzk4], Proposition
1.1.4 [i.e. the “Grothendieck Conjecture” for connected anabelioids]. Remark 3.2.1. In particular, Proposition 3.2 implies that, if X is a connected
temperoid, then it makes sense to write π1temp (X ). We shall refer to this temperedgroup π1temp (X ) [which is well-deﬁned, up to inner automorphism] as the tempered
fundamental group of X .
Corollary 3.3. (Rigid Morphisms of Connected Temperoids) We maintain the notation of Proposition 3.2. Let φ : Π1 → Π2 be a continuous homomorphism that gives rise to a morphism of temperoids B temp (φ) : T1 → T2 . Then
B temp (φ) is rigid if and only if the centralizer ZΠ2 (Im(φ)) is trivial.
Proof. This is a formal consequence of Proposition 3.2 [cf. [Mzk4], Corollary
1.1.6]. 
Next, let us observe that, if T ∈ Ob(T ) is an object of a connected temperoid
T , then the categoryTTforms a temperoid [which is connected if and only if T is connected]. Moreover,forming the product with T yields a functor T → TT which determines a morphismof temperoids as follows:
TT → T

36

SHINICHI MOCHIZUKI

Thus, in the spirit of [Mzk4], §1.2, we make the following deﬁnition:
Deﬁnition 3.4.
(i) An étale morphism of connected temperoids is a morphism that is abstractly
equivalent to a morphism of the form TT → T considered in the above discussion.
An étale morphism of temperoids is a morphism each of whose component morphisms is étale.
(ii) A continuous homomorphism of tempered groups Π1 → Π2 will be called
relatively temp-slim if, for every open subgroup H ⊆ Π1 , the centralizer ZΠ2 (Im(H))
is trivial. A tempered group will be called temp-slim if its identity morphism isrelatively temp-slim. A temperoid will be called temp-slim if and only if each of its
connected components is equivalent to the “B temp (−)” of some temp-slim temperedgroup.
Remark 3.4.1. In light of Corollary 3.3 above [cf. also [Mzk4], Corollary 1.1.6],
one veriﬁes immediately that a temperoid is slim as a category [cf. §0] if and only if
it is temp-slim as a temperoid; similarly, an anabelioid is slim as an anabelioid [cf.
[Mzk4], Deﬁnition 1.2.4, (ii)] if and only if it is slim as a category [cf. §0]; moreover,an anabelioid is slim as an anabelioid if and only if its temperiﬁcation [cf. Remark
3.1.4] is temp-slim as a temperoid.
Remark 3.4.2. By Corollary 3.3, one may work with relatively temp-slim morphisms of temperoids as if they are “morphisms in a category” [not “1-morphisms
in a 2-category”]. In particular, if one works under this convention, then the category of étale objects over a given temp-slim temperoid T forms a category whichis equivalent to T itself [cf. [Mzk4], Proposition 1.2.5]. Moreover, just as in thecase of anabelioids [cf. [Mzk4], §1.2], one may work with “pro-temperoids” andhence consider universal coverings [which are pro-temperoids] of a given temp-slim
temperoid.
Now we return to semi-graphs of anabelioids. Let G be a connected, countable[i.e., the underlying semi-graph is countable] semi-graph of anabelioids. If G has atleast one vertex, then let us denote by
B cov (G)the category of objects given by data
{Sv , φe }where v (respectively, e) ranges over the vertices (respectively, edges) of G; for eachvertex v, Sv ∈ Ob(Gv ); for each edge e, with branches b1 , b2 abutting to vertices∼
v1 , v2 , respectively, φe : {(b1 )∗ }∗ Sv1 → {(b2 )∗ }∗ Sv2 is an isomorphism in Ge , andmorphisms given by morphisms [in the evident sense] between such data. If G has

SEMI-GRAPHS OF ANABELIOIDS

37

no vertices — and hence precisely one edge e, which is necessarily isolated — thendef
we shall write B cov (G) = Ge .
The deﬁnition of B cov (G) extends immediately to arbitrary semi-graphs of anabelioids, each connected component of which is countable. Moreover, for such G,
we have a natural full embedding:
B(G) → B cov (G)An object of B cov (G) that lies in the essential image of B(G) will be called ﬁnite.
Now observe that we may associate, in a natural way, to any object of B cov (G)a morphism of countable semi-graphs of anabelioids:G → G[cf. the discussion of §2 following Deﬁnition 2.1 in the case of B(G)].
Deﬁnition 3.5.
(i) We shall refer to a morphism of semi-graphs of anabelioids G  → G that
may be constructed in this way as a covering of semi-graphs of anabelioids [so, inthe terminology of §2, coverings of semi-graphs of anabelioids that arise from ﬁnite
objects of B cov (G) determine “ﬁnite étale coverings of semi-graphs of anabelioids”].
(ii) Suppose that G  → G satisﬁes the condition that there exists a ﬁnite étale
covering G  → G with the property that, for any component [i.e., vertex or edge] cof G, the restriction of G  → G to Gc splits the restriction of G  → G to Gc . Then
we shall say that the covering G  → G, as well as the object of B cov (G) that gave
rise to this covering, is tempered. Also, we shall write
B temp(G) ⊆ B cov (G)for the full subcategory determined by the tempered objects. Thus, we have natural
full embeddings:
B(G) → B temp(G) → B cov (G)(iii) An arrow φ : H1 → H2 of B cov (G) with connected domain and codomain
will be called Galois if, for any two arrows ψ1 , ψ2 : K → H1 such that φ◦ψ1 = φ◦ψ2 ,there exists a [unique] automorphism α ∈ AutH2 (H1 ) of H1 over H2 such that
ψ1 = α ◦ ψ2 .
Remark 3.5.1. It is immediate from the deﬁnitions that passing to the underlying
morphism of semi-graphs yields an equivalence between the datum of a locally trivial
covering of the semi-graph of anabelioids G and the datum of a graph-covering withcountable ﬁbers of the semi-graph G.

38

SHINICHI MOCHIZUKI

Remark 3.5.2.
It follows from Remark 2.4.2 that, if G is totally aloof andverticially slim, then the construction given above of a covering of semi-graphs ofanabelioids associated to an object of B cov (G) determines a natural full embedding of
B cov (G) into the category of totally aloof, verticially slim semi-graphs of anabelioids
and locally ﬁnite étale morphisms over G.
Now let us assume that the semi-graph of anabelioids G is connected, countable,quasi-coherent, totally elevated, totally aloof, and verticially slim. Let{G i → G}i∈I
be some coﬁnal [i.e., in B(G)] collection of connected ﬁnite étale Galois coverings,
indexed by a set I. Thus, π1 (G) may be constructed as the inverse limitilim←− Gal(G /G)iof the resulting inverse system of ﬁnite groups Gal(G i /G). Let us write
G ∞,i → G i
for the “combinatorial universal covering” of G i [i.e., the covering of G i determined by the universal graph-covering of the underlying semi-graph Gi ; thus,
Gal(G ∞,i /G i ) is a free discrete group]. One veriﬁes immediately that G ∞,i → G
is a Galois tempered covering. Then we set:def∞,i
π1temp (G) = ←lim− Gal(G /G)i

Note that π1temp (G) is independent, up to inner automorphism, of the choice of thecoﬁnal system {G i → G}i∈I .
Proposition 3.6. (The Tempered Fundamental Group of a Semi-graphof Anabelioids) Let G, G  be connected, countable, quasi-coherent, totally
elevated, totally aloof, verticially slim semi-graphs of anabelioids. Then:(i) The topological group π1temp (G) deﬁned above is tempered.
(ii) There is a natural equivalence of categories:∼

B temp (π1temp (G)) → B temp (G)In particular, the category B temp (G) is a connected temperoid. We shall referto π1temp (G) as the tempered fundamental group of G.
(iii) The full embedding B(G) → B temp (G) induces an injection π1temp (G) →
π1 (G) of topological groups.

SEMI-GRAPHS OF ANABELIOIDS

39(iv) Any morphism of semi-graphs of anabelioids G  → G induces a morphismof temperoids
B temp (G  ) → B temp (G)[by pulling back tempered coverings of G to tempered coverings of G  ]. Moreover,
if the original morphism of semi-graphs of anabelioids is locally open, then thismorphism of temperoids is relatively temp-slim. In particular, the temperoid
B temp (G) is temp-slim.
(v) Suppose that G is coherent, and that we are given a tempered coveringG  → G. Then the resulting morphism of temperoids
B temp (G  ) → B temp (G)is étale.
Proof. Assertion (i) follows from the deﬁnitions and the fact that G is countable.
Next, we consider assertion (ii). First, we observe that, by the deﬁnition of a tempered covering, it follows that every connected tempered covering H → G appears
as a subcovering of some G ∞,i → G; in particular, we may choose a factorization
G ∞,i → H → G. Thus, in light of the deﬁnition of G ∞,i as the “combinatorial
universal covering” of G i [together with the fact that G i → G is Galois], it follows
immediately that the ﬁber product covering G ∞,i ×G H [regarded as a covering of
G ∞,i via the projection to the ﬁrst factor] splits as a covering of G ∞,i. Similarly,[cf. also Remark 3.1.3] the covering G ∞,i ×G G ∞,i [regarded as a covering of G ∞,ivia the projection to the ﬁrst factor] splits as a covering of G ∞,i . Moreover, theaction of Gal(G ∞,i /G) on the ﬁrst factor of G ∞,i ×G G ∞,i determines an action ofGal(G ∞,i /G) on the set of components π0 (G ∞,i ×G G ∞,i) which is transitive andfree of ﬁxed points [cf. Remark 3.1.3]. On the other hand, since the domain andcodomain of the covering G ∞,i → H are connected, one veriﬁes immediately thatthe resulting map on sets of connected componentsξ : π0 (G ∞,i ×G G ∞,i) → π0 (G ∞,i ×G H)
is surjective and compatible with the natural actions of Gal(G ∞,i /G) on the domain
and codomain [arising from the ﬁrst factor of the ﬁber products]. In particular, theﬁber of the element of the codomain of ξ determined by the “graph” of the coveringmorphism G ∞,i → H is the H-orbit, for some subgroup H ⊆ Gal(G ∞,i /G), of theelement of the domain of ξ determined by the “graph” of the identity morphism
G ∞,i → G ∞,i . Now forming the quotient of G ∞,i by H yields an intermediate
covering G ∞,i → G H → G such that the morphism G H → G admits a factorization[by the deﬁnition of H!] G H → H → G. On the other hand, by forming theﬁber product with G ∞,i over G and looking at the induced morphisms on sets ofconnected components, we obtain surjections
π0 (G ∞,i ×G G ∞,i )  π0 (G ∞,i ×G G H )  π0 (G ∞,i ×G H)
the composite of which is equal to ξ. Moreover, by the construction of G H , it follows
formally that the ﬁrst of these two surjections factors through the quotient of the set

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SHINICHI MOCHIZUKI

π0 (G ∞,i ×G G ∞,i ) by the action of H; we thus conclude [from the fact that ξ inducesa bijection of the quotient of the set π0 (G ∞,i ×G G ∞,i) by the action of H onto theset π0 (G ∞,i ×G H)] that the surjection π0 (G ∞,i ×G G H )  π0 (G ∞,i ×G H) is, in fact,a bijection. But from this, one concludes immediately that the covering morphism
G H → H is an isomorphism. Thus, in summary, we conclude that the covering
H → G may be constructed as the quotient of the covering G ∞,i → G by the actionof Gal(G ∞,i /H) [i.e., H]; and [by applying the surjectivity of ξ to the “graphs” ofvarious covering morphisms] that every morphism of coverings [of G] from G ∞,i to
H may be obtained from the original morphism by composing with the action ofGal(G ∞,i /G). Conversely, one veriﬁes immediately that the quotient of G ∞,i → G
by the action of any subgroup of Gal(G ∞,i /G) yields a subcovering of G ∞,i → G.
This shows that there is a natural equivalence between B temp (Gal(G ∞,i /G)) and thefull subcategory of B temp(G) determined by the subcoverings of G ∞,i → G. Passing
to the limit over i then completes the proof of assertion (ii).
Assertion (iii) follows from the fact that discrete free groups [such as Gal(G ∞,i /G i )]are always residually ﬁnite [cf., e.g., Corollary 1.7]. The ﬁrst sentence of assertion(iv) follows from the deﬁnitions; the remainder of assertion (iv) follows, in lightof the injection of assertion (iii), formally from Corollary 2.7, (ii). Finally, to verify assertion (v), it suﬃces to show that the composite of any tempered coveringH → G  with the given tempered covering G  → G forms a tempered coveringH → G; moreover, [since composites of “combinatorial” [i.e., locally trivial] coverings manifestly form “combinatorial” coverings] we may assume without loss ofgenerality that the covering H → G  is ﬁnite étale. But then it follows from thethe coherence of G [cf. Deﬁnition 2.3, (iii)] that there exists a ﬁnite étale covering
H → G whose pull-back to G  splits the restrictions of H → G  to each of the Gc[where c is a component of G ]. Theorem 3.7.
(Maximal Compact Subgroups of the Tempered Fundamental Group) Let G be a connected, countable, quasi-coherent, totally
elevated, totally estranged, verticially slim semi-graph of anabelioids. Assume that G has at least one vertex. Then:(i) For each vertex v of G, there is a natural continuous, injective outer homomorphism π1 (Gv ) → π1temp (G). By abuse of notation, we shall write π1 (Gv ) for
the subgroup [well-deﬁned up to conjugation] determined by the image of this homomorphism. We shall refer to the [necessarily compact!] subgroups of π1temp (G)that arise in this way as the verticial subgroups.
(ii) Let us think of the verticial subgroups as being parametrized by a vertex
v of G and an element of the coset space π1temp (G)/π1 (Gv ). Then if H1 , H2 areverticial subgroups of π1temp (G) that arise from distinct parametrization data,then H1 H2 has inﬁnite index in H1 . In particular, verticial subgroups thatarise from distinct parametrization data are distinct.
(iii) Every compact subgroup of π1temp (G) is contained in at least one verticial
subgroup. If a nontrivial compact subgroup of π1temp (G) is contained in more than
one verticial subgroup, then it is contained in precisely two verticial subgroups,

SEMI-GRAPHS OF ANABELIOIDS

41

determined by a compatible system of pairs of vertices of the G∞,i [as i ranges overthe elements of I] joined to one another by a single [closed] edge. In particular,
in this case, this compact subgroup is contained in the image of some π1 (Ge ), forsome edge e of G. We shall refer to such images of “π1 (Ge )’s” as the edge-like
subgroups of π1temp (G).
(iv) The maximal compact subgroups of π1temp (G) are precisely the verticial subgroups. The nontrivial intersections of two distinct maximal
compact subgroups of π1temp (G) are precisely the edge-like subgroups.
Proof. In light of the injection of Proposition 3.6, (iii), assertion (i) (respectively,(ii)) follows from the deﬁnitions; Proposition 2.5, (i) (respectively, Proposition 2.6;
Corollary 2.7, (i) [cf. also Remark 2.7.2]).
Now suppose that H ⊆ π1temp (G) is a nontrivial compact subgroup. Then H
acts continuously on the semi-graph G∞,i , for each i ∈ I. Thus, this action factors
through a ﬁnite quotient. In particular, by Lemma 1.8, (ii), (a), H ﬁxes at least oneedge or vertex of the semi-graph G∞,i . Since the action of H is over G, it follows
that if H ﬁxes an edge, then it does not switch the branches of the edge. Since G,
hence also G∞,i , is connected and has at least one vertex, it thus follows that everyedge of G∞,i abuts to at least one vertex. In particular, we conclude that if H ﬁxes
an edge of G∞,i , then it ﬁxes a vertex, i.e., that H always ﬁxes at least one vertex
of G∞,i .
Now suppose that for some coﬁnal subset J ⊆ I, H ﬁxes ≥ 3 vertices of G∞,j ,for every j ∈ J . Then by Lemma 1.8, (ii), (c), we conclude that, for every j ∈ J ,H acts trivially on some subjoint of G∞,j . In particular, H acts trivially on some
subjoint of Gj . Since the semi-graphs Gj are all ﬁnite, we thus conclude that we
may choose a compatible system of such subjoints [i.e., on which H acts trivially] of
the Gj . But this implies [cf. Remark 2.2.1] that H is contained in some conjugate
in the proﬁnite fundamental group π1 (G) of some π1 (Gv ), and, moreover, that it
is in fact contained, for two distinct branches b, b abutting to v of edges e, e ,respectively [where e is not necessarily distinct from e ], in the intersection of theimages of π1 (Ge ), π1 (Ge ), via b, b . But since G is assumed to be totally estranged,
we thus conclude that H is trivial, in contradiction to our hypotheses.
Thus, in summary, we have shown that for some coﬁnal subset J ⊆ I, H ﬁxes
at least one, but no more than two vertices of G∞,j , for every j ∈ J . Moreover,by Lemma 1.8, (ii), (b), it follows that if H ﬁxes two vertices of G∞,j , then thesetwo vertices are joined to one another by a single [closed] edge. In particular, by
possibly replacing J by some smaller coﬁnal subset, we may assume that there
exists a compatible system of vertices of G∞,j , for j ∈ J , each of which is ﬁxed by
H. On the other hand, we may also conclude that there exist at most two suchcompatible systems. This completes the proof of assertion (iii). Finally, assertion(iv) follows formally from assertions (ii), (iii). 
Remark 3.7.1. The notion thatmaximal compact subgroups correspond to points

42

SHINICHI MOCHIZUKI

is a recurrent theme in the geometry of group actions. Classical well-known examples of this phenomenon include the theory of symmetric spaces obtained as
quotients of a real reductive group by a maximal compact subgroup or, in the padic case, of Qp -valued points of a reductive group by Zp -valued points. Anotherexample of this sort of situation is the “discrete real section conjecture” of [Mzk5],§3.2.
Deﬁnition 3.8. We shall refer to as quasi-geometric any continuous homomorphism of tempered groups
Π1 → Π2that satisﬁes the following condition: Any maximal compact subgroup K1 ⊆ Π1
(respectively, nontrivial intersection K1 H1 of two distinct maximal compact subgroups K1 , H1 ⊆ Π1 ) maps surjectively to an open subgroup of some maximal

compact subgroup K2 ⊆ Π2 (respectively, of some nontrivial intersection K2 H2of two distinct maximal compact subgroups K2 , H2 ⊆ Π2 ). A quasi-geometricmorphism of temperoids is a morphism of temperoids each of whose component
morphisms arises [cf. Proposition 3.2] from a quasi-geometric continuous homomorphism of tempered groups.
Remark 3.8.1.
geometric.

It is immediate that any isomorphism of temperoids is quasi-

Corollary 3.9.
(Reconstruction of the Underlying Semi-graph of Anabelioids) Let G, H be connected, countable, quasi-coherent, totally elevated, totally estranged, verticially slim graphs of anabelioids. Then applying“B temp(−)” determines a natural bijective correspondence between locally open
morphisms of semi-graphs of anabelioids
G →H
and quasi-geometric morphisms of temperoids B temp (G) → B temp (H).
Proof. First, we note that any locally open morphism of semi-graphs of anabelioids G → H determines a morphism of temperoids B temp(G) → B temp (H)[cf. Proposition 3.6, (iv)] whose quasi-geometricity follows by “substituting” theequivalences of Theorem 3.7, (iv), into Deﬁnition 3.8. Next, we observe thatby Proposition 3.2; Deﬁnition 3.8; Theorem 3.7, (ii), (iii), any quasi-geometricφ : B temp (G) → B temp (H) determines a map from the vertices to G to the verticesof H — i.e., by considering the unique [conjugacy class of] verticial subgroup(s)of π1temp (H) that contain(s) the image of a given verticial subgroup of π1temp (G).
Similarly, by considering nontrivial intersections of maximal compact subgroups,
one obtains [in light of the fact that, since H is totally elevated and totally aloof,
all of the edge-like subgroups of π1temp (H) are inﬁnite] that any quasi-geometricφ : B temp (G) → B temp(H) determines a map from the edges to G to the edges of H
which is compatible with the map obtained above on vertices. Thus, in summary,SEMI-GRAPHS OF ANABELIOIDS

43

we conclude that a quasi-geometric φ : B temp(G) → B temp (H) determines a map on
the underlying graphs G → H that is functorial in φ.
Next, we observe that if φ : B temp (G) → B temp(H) is quasi-geometric, then any
morphismφ : B temp (G  ) → B temp (H )induced by φ between étale coverings of the domain and codomain of φ [i.e., G  → G,
H → H are tempered coverings] is again quasi-geometric. Indeed, this follows
immediately from the characterization of nontrivial maximal compact subgroups(respectively, nontrivial intersections of two distinct maximal compact subgroups)
as the verticial subgroups (respectively, edge-like subgroups) — cf. Theorem 3.7,(iii), (iv). Thus, we obtain a morphism of graphs G → H , which is functorial inG  , H . Finally, by varying G  , H , we conclude that φ arises from a morphism ofgraphs of anabelioids which [again by Theorem 3.7, (iii), (iv)] is manifestly uniqueand locally open. This completes the proof of Corollary 3.9. 
Remark 3.9.1. Suppose that G is as in Theorem 3.7. Then observe that, if G
is a semi-graph which is not a graph, then the techniques developed here are not
suﬃcient, in general, to reconstruct the open edges of G from, say, the isomorphismclass of the tempered group π1temp (G) — cf. Remark 4.8.1.
Example 3.10. Pointed Stable Curves over p-adic Local Fields I. Let K
log
be a ﬁnite extension of Qp; K an algebraic closure of K; XK
a smooth log curve
def

log
log
= XK
×K K;
over K [cf. §0]. Let us write XK
log)
π1temp (XKlog) is a tempered
for the tempered fundamental group of [André], §4. Thus, π1temp (XK
topological group [in the sense of Deﬁnition 3.1, (i)] and ﬁts into a natural exact
sequence:log
log
1 → π1temp (XK) → π1temp (XK) → GK → 1
deflog) for the geometric tempered funda[where GK = Gal(K/K); we write π1temp (XKlog
log, i.e., the tempered fundamental group of XK×K (K)∧ ; themental group of XK“∧” denotes the p-adic completion]. To simplify the notation, let us write:
def

log
);
Π = π1temp (XK

def

log
Δ = π1temp (XK)

Note that Δ is also tempered, so we obtain temperoids:
B temp(Π);

B temp(Δ)

Now let us write
G (respectively, G c )44

SHINICHI MOCHIZUKI

for the graph of anabelioids (respectively, semi-graph of anabelioids) determined by
the semi-graph of proﬁnite groups [without compact structure!] (respectively, withcompact structure) associated to the geometric special ﬁber of the stable model oflog
XK[cf. [Mzk3], Appendix] over the ring of integers OK of K. Note that it follows∼

from the deﬁnitions that we have a natural equivalence B temp (G) → B temp (G c ) anda natural full embedding:B temp(G) → B temp(Δ)

Now suppose that we are given an exhaustive sequence of open characteristic [hence
normal] subgroups of ﬁnite index. . . ⊆ Ni ⊆ . . . ⊆ Δ
def[where i ranges over the positive integers] of Δ; write Δi = Δ/Ni . Thus, Ni
log
determines a ﬁnite log étale covering of XK, whose geometric special ﬁber gives
rise to semi-graphs of anabelioids
Gi ;

Gic

on which Δi acts faithfully. Recall from Example 2.10 that Gi , Gic are coherent,
totally elevated, totally universally sub-coverticial, totally estranged, and verticiallyslim. In particular, Gi satisﬁes the hypotheses of Theorem 3.7, Corollary 3.9.
Next, let us observe that we obtain a natural compatible system of generalized[cf. Deﬁnition 2.11] morphisms of graphs of anabelioids
Gi → Gj[where i ≥ j], which are compatible with the actions of Δi , Δj , as follows: Thefunctor
Cat(Gi ) → Cat(Gj )is obtained by mapping a vertex v (respectively, vertex v; edge e; edge e) of Gi toa(n) vertex v  (respectively, edge e ; edge v  ; edge e ) of Gj whenever the map on
geometric special ﬁbers between the coverings determined by Δi , Δj maps the irreducible component corresponding to v into the irreducible component corresponding
to v  in such a way that the image of the irreducible component corresponding to v
is not equal to a node (respectively, the irreducible component corresponding to v
into the node corresponding to e ; the node corresponding to e to a non-nodal point
lying in the irreducible component corresponding to v  ; the node corresponding to e
to the node corresponding to e ). The remainder of the data necessary to deﬁne thegeneralized morphism of graphs of anabelioids Gi → Gj is determined naturally byconsidering the map on geometric special ﬁbers between the coverings determinedby Δi , Δj . For a more group-theoretic description of these generalized morphisms
Gi → Gj , we refer to the discussion of Example 5.6 below.
Finally, we observe that these generalized morphisms of graphs of anabelioids
induce — by applying “B temp(−)” [cf. Remark 2.11.1] — natural morphisms oftemperoids. . . → B temp (Gi ) → . . . → B temp (Gj ) → . . . → B temp (G)SEMI-GRAPHS OF ANABELIOIDS

45

compatible with the actions of the Δi , hence also corresponding surjections oftempered groups [where we note that each of the “outer semi-direct product groups”[cf. §0] admits a natural topology with respect to which the resulting topologicalgroup is tempered]:
out

def

out

def

Δ  . . .  Δ[i] = π1temp (Gi )  Δi  . . .  Δ[j] = π1temp (Gj )  Δj
 . . .  π1temp (G)Note that each Δ[i] is temp-slim. [Indeed, this follows from the fact that Δi actsfaithfully on Gi , hence also faithfully on B temp (Gi ) [cf., e.g., the injectivity portion of
the bijection of Corollary 3.9]; the temp-slimness portion of Proposition 3.6, (iv).]
Since Δ is the inverse limit of the Δ[i], and GK is slim [cf. e.g., [Mzk3], Theorem
1.1.1], we thus conclude that both Δ and Π are temp-slim.
log
Remark 3.10.1. We maintain the notation of Example 3.10. Write B temp (XK
)
log
log
temp
(XK ) is a temperoid whose
for the category of tempered coverings of XK [so B
temp
log
tempered fundamental group is π1 (XK )]. Let Σ be a set of prime numbers.
Denote bylog Σlog
B temp (XK) ⊆ B temp (XK)

the full subcategory determined by the tempered coverings dominated by coverings
which arise as a combinatorial covering [i.e., a covering arising from a graph-coveringof the dual graph of the geometric special ﬁber] of a ﬁnite étale Galois covering of
log
log Σ
XK
whose degree is a Σ-integer. One veriﬁes immediately that B temp (XK) isa temperoid. We shall refer to the tempered fundamental group of this temperoidlog
as the pro-Σ tempered fundamental group of XK. Then, as long as Σ contains atleast one prime = p, the entire discussion of Example 3.10 may be carried out forlog
the pro-Σ tempered fundamental group of XK. [We leave the routine details to theinterested reader.] Also, we observe that the analogue of “G” (respectively, “G c ”)
in the pro-Σ case is precisely the pro-Σ completion [in the sense of Deﬁnition 2.9,(ii)] of the semi-graph of anabelioids G (respectively, G c ).
In the case of tempered fundamental groups of pointed stable curves, i.e.,
Example 3.10, we observe [cf. Remark 3.9.1] that, in the notation of Example
3.10, not only G, but also the semi-graph of anabelioids G c may be reconstructedgroup-theoretically from the tempered fundamental group:
Corollary 3.11.
(Reconstruction of Semi-graphs of Anabelioids Associated to Pointed Stable Curves) For  = α, β, let K be a ﬁnite extension
log)K a smooth log curve over K
of Qp ; K  an algebraic closure of K ; (X
def

def

log
log
log
[cf. §0]. Let us write (X)K  = (X)K ×K K  ; Δ[] = π1temp ((X)K  );

G[] (respectively, G c [])

46

SHINICHI MOCHIZUKI

for the graph of anabelioids (respectively, semi-graph of anabelioids) determined by the semi-graph of proﬁnite groups [without compact structure!] (respectively, with compact structure) associated to the geometric special ﬁber of the stable
log
model of (X)K  [cf. [Mzk3], Appendix] over the ring of integers OK  of K  [cf.
Example 3.10]. Then any isomorphism of topological groups∼

γ : Δ[α] → Δ[β]determines a compatible isomorphism of semi-graphs of anabelioids∼

G c [α] → G c [β]in a fashion that is functorial with respect to γ. Moreover, if such a γ exists, thenpα = pβ .
Proof. Let Σ be a set of prime numbers such that pα , pβ ∈ Σ. Write Δ[] for the
proﬁnite completion of Δ[]; Δ[]Σ for the pro-Σ tempered fundamental group of
Remark 3.10.1; Δ[]Σ for the maximal pro-Σ quotient of Δ[]; G c []Σ , G[]Σ forthe respective pro-Σ completions [in the sense of Deﬁnition 2.9, (ii)] of G c [], G[].
Moreover, since Galois coverings of degree a Σ-integer are necessarily admissible[cf., e.g., [Mzk1], §3], it follows that Δ[]Σ may be identiﬁed with the tempered
temp
fundamental group π1temp (G c []Σ ) ∼= π1 (G[]Σ ).
Next, let us observe that the kerneldefIΣ [] = Ker(Δ[]  Δ[]Σ)may be recovered as the kernel JΣ [] of the natural morphism Δ[] → Δ[]Σ .
Indeed, since it follows from the deﬁnitions that the morphism Δ[] → Δ[]Σ
factors through Δ[]Σ , we obtain that IΣ [] ⊆ JΣ []. On the other hand, the factthat JΣ [] ⊆ IΣ [] follows from the fact that free discrete groups inject into theirpro-Σ completions [cf. Remark 1.7.1]. This completes the proof of the equalityIΣ [] = JΣ [].
In particular, we conclude that the quotientstemp
Δ[]  Δ[]Σ ∼= π1 (G[]Σ )are compatible with γ. Thus, by Corollary 3.9, we conclude that γ induces a natural,functorial isomorphism of graphs of anabelioids∼

G[α]Σ → G[β]Σ
∼hence, in particular, an isomorphism of graphs G[α] → G[β].
Next, let us observe, that:

SEMI-GRAPHS OF ANABELIOIDS

47(i) The technique used to obtain an isomorphism of graphs of anabelioids inthe preceding paragraph may also be applied to open subgroups of ﬁnite
index Δ [α] ⊆ Δ[α], Δ [β] ⊆ Δ[β] that correspond via γ.
Moreover, let us recall that:(ii) The decomposition groups of cusps in Δ[]Σ are commensurably terminal[cf. [Mzk3], Lemma 1.3.7].
(iii) Every nontrivial image via the natural morphismΔ []Σ → Δ[]Σ
associated to some open subgroup of ﬁnite index Δ [] ⊆ Δ[] of the decomposition group of a node in Δ []Σ [i.e., in the terminology of Theorem
3.7, an “edge-like subgroup”] is either an open subgroup of an edge-like
subgroup of Δ[]Σ or an open subgroup of a decomposition group of acusp in Δ[]Σ [but not both, since G[]Σ is totally aloof — cf. Example
2.10, Remark 2.4.1].
(iv) Every decomposition group of a cusp in Δ[]Σ admits an open subgroupthat arises as the image via some morphism Δ []Σ → Δ[]Σ as in (ii) ofan edge-like subgroup of Δ []Σ .
Indeed, (ii) follows from [Mzk3], Lemma 1.3.7; (iii) is immediate from the deﬁnitions. On the other hand, (iv) may be veriﬁed as follows: Suppose thatΔ [] ⊆ Δ[]is an open normal subgroup of ﬁnite index that corresponds to a covering oflog)K  whose ramiﬁcation indices at the cusps are prime to p , but which is(Xramiﬁed over the irreducible component C of the special ﬁber of the stable modellog)K  that contains [the restriction to the special ﬁber of] the cusp of interof (Xest, which we shall denote by x . [Note that such a Δ [] always exists: Indeed,by passing to a Galois covering of degree a Σ-integer, one may assume that thenormalization of C is of genus ≥ 2; then one veriﬁes immediately that the covering arising from “multiplication by p on the Jacobian” satisﬁes the conditions
just stated.] Then the ramiﬁcation over C implies that this covering has ≥ p
distinct cusps lying over x which, nevertheless, map to the same point of the normalization of C in the [necessarily inseparable — by our ramiﬁcation assumption!]
ﬁeld extension of its function ﬁeld determined by the covering. Thus, we conclude
that [the restrictions to the special ﬁber of] these distinct cusps must lie on an
irreducible component of the special ﬁber [of the stable model] of the covering thatcollapses to [the restriction to the special ﬁber of] x [cf. also [Tama2], Theorem
0.2, for a more general result concerning the existence of coverings with collapsing
irreducible components]. Now, sorting through the deﬁnitions, we see that thiscompletes the proof of (iv).

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SHINICHI MOCHIZUKI

Thus, in summary, it follows formally from (i), (ii), (iii), (iv) that the natural,functorial isomorphism of graphs of anabelioids∼

G[α]Σ → G[β]Σ
induced by γ extends uniquely to a natural, functorial isomorphism of semi-graphsof anabelioids∼
G c [α]Σ → G c [β]Σ[which may also be regarded as being induced by γ], hence, in particular, an iso∼morphism of semi-graphs Gc [α] → Gc [β].
Next, let us observe that, if Δ [α] ⊆ Δ[α], Δ [β] ⊆ Δ[β] are normal open
subgroups of ﬁnite index that correspond via γ, and we write G [], G  [] for thecorresponding graphs/graphs of anabelioids [without compact structure], then thedecomposition groupDv ⊆ Δ[]/Δ []determined by a vertex v of G [] acts naturally on the anabelioid G  []Σv [i.e., thepro-Σ completion of the anabelioid G  []v ]. Thus, the inertia groupIv ⊆ Dv
at v — i.e., the subgroup that acts trivially on this anabelioid — is necessarily oforder a power of p . (Indeed, here we use the easily veriﬁed fact that any nontrivial
automorphism of an irreducible component of the special ﬁber [of the stable modelof the covering determined by Δ []] induces a nontrivial outer automorphism of
the tame pro-Σ fundamental group [i.e., where “tame” means that one only allows
tame ramiﬁcation at the nodes and cusps] of the open subscheme of this irreducible
component given by taking the complement [in this irreducible component] of thenodes and cusps.) In particular, [since there exist Δ [] for which Iv is nontrivial— cf., e.g., the proof of assertion (iv) given above] we obtain that pα = pβ . Thus,defin the following, we shall write p = pα = pβ .
Next, let us observe that the natural quotienttemp
Δ[]  π1temp (G[]) ∼= π1 (G c [])— i.e., the quotient determined by the “admissible quotient” of Δ[], in the senseof [Mzk3], §2 — may be characterized as follows: A normal open subgroup of ﬁnite
index Δ [] ⊆ Δ[] arises from this quotient if and only if no irreducible componentof the special ﬁber of the stable model of the corresponding covering collapses inlog
the stable model of (X)K  , and, moreover, the decomposition groups at thenodes and cusps (respectively, inertia groups at the irreducible components) of thecorresponding covering are prime to p (respectively, trivial). Indeed, this follows
immediately from well-known “purity of the branch locus” results and the wellknown “structure of local fundamental groups of stable curves” [cf., e.g., [Tama2],
Lemma 2.1]. Now let us write (G )c [], (G  )c [] for the semi-graph/semigraphof anabelioids [with compact structure!] associated to Δ []. Then observe that

SEMI-GRAPHS OF ANABELIOIDS

49

this characterization is equivalent to the following “group-theoretic” condition [i.e.,
condition compatible with γ]:
The natural map Δ []Σ → Δ[]Σ is quasi-geometric; the stabilizer ⊆Δ[]/Δ [] of any edge of the semi-graph (G )c [] has order prime to p;
the stabilizer ⊆ Δ[]/Δ  [] of any vertex v of the semi-graph (G )c []acts faithfully on the anabelioid (G  )c []Σv [i.e., the pro-Σ completion of
the anabelioid (G  )c []v ].
Thus, we conclude that γ induces an isomorphismtemp
π1temp (G[α]) ∼= π1 (G[β])hence, by Corollary 3.9, we conclude that γ induces an isomorphism of graphs ofanabelioids∼
G[α] → G[β]∼which — by applying the functorial isomorphisms “Gc [α] → Gc [β]” obtained above
to arbitrary normal open subgroups of ﬁnite index Δ [] ⊆ Δ[] that arise fromthe “admissible quotient” — induces a uniquely determined isomorphism of semigraphs of anabelioids∼
G c [α] → G c [β][which is, of course, functorial in γ], as desired. 
Remark 3.11.1. Note that for any set of primes Σ of cardinality ≥ 3 [i.e., sothat Σ contains at least one prime = pα , pβ ], the argument given above also yields
a “pro-Σ version” of Corollary 3.11, i.e., where one replaces the isomorphism∼

γ : Δ[α] → Δ[β]in the statement of Corollary 3.11 by an isomorphism∼

Δ[α]Σ → Δ[β]Σ
between the respective pro-Σ tempered fundamental groups [in the sense of Remark
3.10.1].
Remark 3.11.2. Once one recovers the “admissible quotients” Δ[]  π1temp (G c []),one may apply the results of [Tama1] to the various verticial subgroups of π1temp (G c [])to recover, in certain cases, the isomorphism class of the curve determined by thecomplement of the nodes and cusps in the irreducible component of the special ﬁber
corresponding to this verticial subgroup.

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SHINICHI MOCHIZUKI

Section 4: Categories of Localizations
In this §, we consider categories of localizations of a semi-graph of anabelioids
satisfying certain conditions.
Let G be a totally aloof, verticially slim semi-graph of anabelioids [so that we
may apply Remark 2.4.2]; Γ a ﬁnite group of automorphisms of G [i.e., Γ actsfaithfully on G].
If H → G is a morphism of semi-graphs, then we shall write GH for the semigraph of anabelioids obtained by pulling back [in the evident sense] the semi-graphof anabelioids structure of G via H → G. If v (respectively, e; b) is a(n) vertex(respectively, edge; branch of an edge) of G, then we shall writedefG[v] = GG[v] ;def

G[e] = GG[e] ;def

G[b] = GG[b][i.e., relative to the natural morphisms G[v] → G, G[e] → G, G[b] → G of §1].
Deﬁnition 4.1.
(i) We shall say that Γ acts piecewise faithfully on G if every element γ ∈ Γsatisﬁes the following condition: If there exists a vertex v of G such that γ ﬁxes
v as well as all of the branches of closed edges of G that abut to v, then γ is theidentity.
(ii) Any locally trivial morphism of totally aloof, verticially slim semi-graphsof anabelioids G  → G whose underlying morphism of semi-graphs is an immersion (respectively, excision; embedding) will also be referred to as an immersion(respectively, excision; embedding), or, alternatively, as an immersive (respectively,
excisive; embedding) morphism.
(iii) Let H be a totally aloof, verticially slim semi-graph of anabelioids. Then
we shall refer to as a (G, Γ)-structure on H any Γ-orbit [relative to the action of Γon G] of locally ﬁnite étale morphisms of totally aloof, verticially slim semi-graphsof anabelioids H → G. We shall refer to any of the morphisms in this orbit asa structure morphism [relative to this particular (G, Γ)-structure on H]. We shallsay that a (G, Γ)-structure on H is iso-immersive (respectively, iso-excisive) if some[or, equivalently, every] structure morphism H → G factors as the composite of an
immersion (respectively, excision) H → G  with a ﬁnite étale morphism G  → G suchthat G is untangled [so the composites of the immersion H → G with the various
H[v] → H, H[e] → H, H[b] → H are all embeddings]. If H → H is a locally ﬁnite
étale morphism between totally aloof, verticially slim semi-graphs of anabelioids
that are equipped with (G, Γ)-structures, then we shall say that this morphism
is compatible with the (G, Γ)-structures if the composite of this morphism with astructure morphism of H yields a structure morphism of H.
(iv) Let H be a totally aloof, verticially slim semi-graph of anabelioids. Then
we shall refer to as a local (G, Γ)-structure on H the datum of a (G, Γ)-structure

SEMI-GRAPHS OF ANABELIOIDS

51

for each H[c], where c varies among the components [i.e., vertices and edges] of theunderlying semi-graph H satisfying the property that if a branch b of an edge e of H
abuts to a vertex v of H, then the given (G, Γ)-structures on H[v], H[e] coincide on
H[b]. We shall say that a local (G, Γ)-structure is iso-immersive (respectively, isoexcisive) if each of its constituent (G, Γ)-structures is iso-immersive (respectively,
iso-excisive). We shall say that a local (G, Γ)-structure is verticially iso-excisive if
each of its constituent (G, Γ)-structures at a vertex is iso-excisive. If H → H is alocally ﬁnite étale morphism between totally aloof, verticially slim semi-graphs ofanabelioids that are equipped with local (G, Γ)-structures, then we shall say that
this morphism is compatible with the local (G, Γ)-structures if each of the induced
morphisms H[c] → H[c ] (where c is a component of H mapping to a component c
of H ) is compatible with the given (G, Γ)-structures.
Remark 4.1.1. Let H be a totally aloof, verticially slim semi-graph of anabelioids,
equipped with a local (G, Γ)-structure. Then each component c of H determines
a well-deﬁned Γ-orbit of components of G [by mapping c to G via a structure
morphism]. In particular, if c is an edge, then it makes sense to say that c lies overan open (respectively, closed) edge of G. In this case, we shall say that c is Gopen (respectively, G-closed). One veriﬁes immediately that any locally ﬁnite étalemorphism compatible with given local (G, Γ)-structures maps G-open (respectively,
G-closed) edges to G-open (respectively, G-closed) edges.
Remark 4.1.2. The reason for working with the action of Γ [which at times
may appear to be somewhat cumbersome — cf., e.g., Deﬁnition 4.1, (iii), (iv)] is
to “simulate” a stack-theoretic situation, of the sort which occurs naturally, forinstance, when one considers formal localizations of pointed stable “orbicurves”[cf. the discussion in the Introduction]. In particular, this circumvents the needto introduce [conceivably even more cumbersome] notions of “orbi-semi-graphs ofanabelioids”, etc.
Next, we assume further that Γ acts piecewise faithfully on G, and that G is
ﬁnite, connected, coherent, totally elevated, and totally universally sub-coverticial.
Then we may deﬁne a category of localizations of G
Loc(G, Γ)associated to the pair (G, Γ) as follows: The (ﬁnite) closed objects are the connectedﬁnite étale coverings G  of G, which we regard as being equipped with the resulting G- [i.e., (G, {1})-)] structure. The inﬁnite open objects are the semi-graphs ofanabelioids G  that appear as connected tempered coverings of G of inﬁnite degree.
We regard inﬁnite open objects as being equipped with the resulting G-structure.
An object that is either closed or inﬁnite open will be called tempered. A ﬁnite open
object H is a ﬁnite, connected, quasi-coherent, totally elevated, totally universally
sub-coverticial, totally aloof, verticially slim semi-graph of anabelioids, equippedwith an iso-immersive, verticially iso-excisive local (G, Γ)-structure, such that H

52

SHINICHI MOCHIZUKI

contains at least one non-isolated open edge which is, however, G-closed. The morphisms between tempered objects (respectively, morphisms from a ﬁnite open object
to an arbitrary object) are the locally ﬁnite étale morphisms of semi-graphs of anabelioids compatible with the G-structure (respectively, local (G, Γ)-structure). There
are no morphisms from a tempered to a ﬁnite open object. This completes thedeﬁnition of the category Loc(G, Γ).
Deﬁnition 4.2.
(i) The (possibly inﬁnite) verticial length (respectively, edge-wise length) of an
object of Loc(G, Γ) is deﬁned to be the cardinality of the set of vertices (respectively,
closed edges) of the underlying semi-graph. An open object of verticial length 1(respectively, 2) and edge-wise length 0 (respectively, 1) will be referred to as anuclear object (respectively, link). A locally trivial morphism from a nuclear object
to a link (respectively, an arbitrary object) will be referred to as an NL-morphism(respectively, a verticial morphism). The verticial degree of an arrow H → K in
Loc(G, Γ) at a vertex v of H mapping to a vertex w of K is deﬁned to be the[necessarily ﬁnite] degree of the ﬁnite étale morphism of anabelioids Hv → Kw
induced by the arrow.
(ii) A graph-localization morphism in Loc(G, Γ) is deﬁned to be a locally trivial morphism which satisﬁes the condition that it is an isomorphism wheneverits domain is closed. A strict graph-localization morphism is a graph-localization
morphism for which the induced arrow on underlying semi-graphs is injective on
vertices.
Remark 4.2.1. Note that a morphism in Loc(G, Γ) is locally trivial if and only if
it is excisive. [Indeed, suﬃciency follows from the deﬁnition of the term “excisive”[cf. Deﬁnition 4.1, (ii)]; as for necessity, the “local surjectivity” of branches follows
from the verticial iso-excisiveness of the local (G, Γ)-structures involved; the “local
injectivity” of branches follows, in light of the local triviality assumption, from thetotal aloofness of the semi-graphs of anabelioids involved.]
Remark 4.2.2. One veriﬁes immediately from the deﬁnitions that every embedding in Loc(G, Γ) is a strict graph-localization morphism. The converse to thisstatement, however, may easily be seen to be false in general.
Proposition 4.3.

(Basic Properties of the Category of Localizations)(i) The underlying semi-graph of anabelioids of an object of Loc(G, Γ) is connected, coherent, totally elevated, totally universally sub-coverticial, totally aloof, verticially slim, and of injective type; if, moreover, this object is
ﬁnite open, then it is of positive verticial length. The underlying morphism ofsemi-graphs of a morphism of Loc(G, Γ) is locally ﬁnite étale.
(ii) If H is a ﬁnite open object of Loc(G, Γ), then any excision H → H of ﬁnite
defconnected graphs of positive verticial length determines an excision H = HH → H
of Loc(G, Γ).

SEMI-GRAPHS OF ANABELIOIDS

53(iii) Let φ : H → K be a morphism in Loc(G, Γ) from a ﬁnite open object H toa tempered object K. Then φ is not an isomorphism of semi-graphs of anabelioids.
(iv) Let H be a ﬁnite open object of Loc(G, Γ) such that H is a tree. Then thelocal (G, Γ)-structure on H arises from a unique (G, Γ)-structure on H [hence,in particular, from a [not necessarily unique] G-structure].
(v) Every morphism in Loc(G, Γ) is an epimorphism. In particular, the category Loc(G, Γ) is totally epimorphic.
(vi) Every endomorphism of a ﬁnite [open or closed] object of Loc(G, Γ) is
an automorphism. Moreover, the automorphism group of any ﬁnite object ofLoc(G, Γ) is ﬁnite.
Proof. Assertion (i) (respectively, (ii)) is immediate from the deﬁnitions [cf. also
Remark 2.4.1] (respectively, [cf. also the fact that the domain of any nonproper
excision admits an edge that maps to an edge of strictly greater verticial cardinality]). As for assertion (iii), we may assume without loss of generality that K is
ﬁnite. Then assertion (iii) follows from the fact that φ necessarily maps an open,
G-closed edge of H [which always exists, by the deﬁnition of an open object] to an
open, G-closed edge of K. But this contradicts the easily veriﬁed fact that every
G-closed edge of K is closed in K.
Next, we consider assertion (iv). The desired morphism H → G may be constructed by gluing together local structure morphisms to G; the fact that sucha gluing operation may be performed — despite the “Γ-ambiguities” involved —follows from our assumption that H is a tree. Finally, the uniqueness of the (G, Γ)structure follows by reducing to the case of nuclear H by assertion (v) below [onechecks immediately that there are no vicious circles in the argument], in which casethe desired uniqueness is immediate from the deﬁnitions.
Next, we consider assertion (v). Let φ, ψ : H → K, ξ : H → H be morphisms
in Loc(G, Γ) such that φ ◦ ξ = ψ ◦ ξ. By localizing on H [and applying the fact thatany ﬁnite étale morphism of slim anabelioids is an epimorphism in the category ofﬁnite étale morphisms of slim anabelioids], one veriﬁes immediately that it suﬃces
to treat the case where ξ is an NL-morphism [so, in particular, H is nuclear; H is alink]. Write eH for the unique closed edge of H; bH for the branch of eH that abuts
to the unique vertex v of H . Since φ, ψ then coincide on the edges of H that abutto v, we conclude that φ, ψ coincide on H[eH ]. In particular, we may also assume
that K is a link. Write eK for the unique closed edge of K.
Now I claim that it suﬃces to show that φ, ψ coincide on H[eH ]. Indeed, since
H is a link, H[eH ] may be obtained from H by simply omitting the open edges.
Thus, it suﬃces to check that φ, ψ map each open edge e of H to the same openedge e of K and induce the same morphism He → Ke . On the other hand, sinceφ, ψ coincide on H[eH ], both of these assertions follow from the fact that K is
totally aloof. This completes the proof of the claim.
To show that φ, ψ coincide on H[eH ], we reason as follows: First, we observethat [by our deﬁnition of the category Loc(G, Γ); the piecewise faithfulness of Γ]

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H[bH ] ; H[eH ]; K[eK ] may be equipped with G-structures that are compatible withξ, φ, and ψ. Thus, these G-structures induce injective [by Proposition 2.5, (i)] outerhomomorphisms
π1 (H ) = π1 (H[bH ]) → π1 (G);π1 (H) = π1 (H[eH ]) → π1 (G);π1 (K) = π1 (K[eK ]) → π1 (G)which are compatible with the outer homomorphisms
π1 (ξ) : π1 (H ) → π1 (H);π1 (φ), π1 (ψ) : π1 (H) → π1 (K)induced by ξ, φ, ψ. This compatibility implies that π1 (φ), π1 (ψ) diﬀer by conjugation by some element g ∈ π1 (G). By the coincidence of φ, ψ on H , however, we
may assume that g centralizes [the image of] π1 (H ). Thus, by Corollary 2.7, (i),we conclude that g is the identity, and hence that π1 (φ), π1 (ψ) coincide. On theother hand, by Corollary 2.7, (i), and the fact that K is totally aloof, this implies
that φ, ψ coincide on H[eH ], as desired.
Finally, we consider assertion (vi). Let φ be an endomorphism of a ﬁnite [openor closed] object H. By ﬁniteness, it is immediate that, for some integer M ≥ 1,
φM ﬁxes some vertex v of H and induces the identity on the anabelioid Hv . Thus,by assertion (v), we conclude that φM is the identity, so φ is an automorphism,
as desired. The ﬁniteness of the automorphism group of H is immediate from theﬁniteness of H itself. 
Proposition 4.4.
(Associated Anabelioids) Let H → K, L → K be morphisms between ﬁnite [open or closed] objects in Loc(G, Γ). Then:(i) There exists a ﬁnite étale covering K → K of K such that the induced
morphism H → K from any connected component H of the pull-back of this ﬁnite
étale covering to H is an embedding.
(ii) There exists a ﬁnite étale covering K → K of K such that K embeds
into a ﬁnite étale covering G  → G of G.
(iii) The morphism H → K in Loc(G, Γ) induces a relatively slim π1 -monomorphism of slim anabelioids
B(H) → B(K)
which completely determines the original morphism H → K [among all morphisms in Loc(G, Γ) from H to K].
(iv) Suppose that H → K, L → K are ﬁnite étale. Then every morphism
H → L in Loc(G, Γ) lying over K is ﬁnite étale and induces a ﬁnite étale morphism
on associated anabelioids [cf. assertion (iii)]. Moreover, the full subcategory of suchﬁnite étale objects over K determines a full embedding:
B(K)0 → Loc(G, Γ)KSEMI-GRAPHS OF ANABELIOIDS

55

When K = G, we have Loc(G, Γ)K = Loc(G, Γ); the essential image of this embedding is the full subcategory of closed objects.
Proof. First, we consider assertion (i). Since K is quasi-coherent, it follows from
Proposition 2.5, (i), that we may reduce immediately to the case where the given
morphism H → K is locally trivial. Thus, we are reduced to a problem in graphtheory — a problem solved in §1 — cf. Theorem 1.2 [i.e., “Zariski’s main theorem
for semi-graphs”].
Assertion (ii) may be shown as follows: Let K → K be a combinatorial universal covering. Although K [i.e., K → K] will not, in general, determine an
object of Loc(G, Γ)K , it can, nevertheless, be thought of as as inductive limit ofKα ∈ Ob(Loc(G, Γ)K ) associated to connected ﬁnite sub-semi-graphs Kα of K . By
Proposition 4.3, (iv), these Kα admit compatible [i.e., as α varies] morphisms [inLoc(G, Γ)K ] to G. Moreover, the uniqueness, up to ﬁnitely many possibilities, [cf.
Proposition 4.3, (iv)] of such a compatible system implies that some ﬁnite index
subgroup of Gal(K /K) ﬁxes such a compatible system. In particular, we conclude
that, for some ﬁnite subcovering K → K of K /K, we obtain a morphism [inLoc(G, Γ)K ] K → G. Thus, the existence of a K → K as asserted follows by
applying assertion (i) to the morphism K → G.
Next, we consider of assertion (iii). The fact that B(H), B(K) are slim follows
from Corollary 2.7, (ii). This slimness implies that the proﬁnite fundamental groupsof B(H), B(K) have no nontrivial normal ﬁnite closed subgroups. Thus, by assertion(i), to show that B(H) → B(K) is a π1 -monomorphism, it suﬃces to show, under
the further assumption that H → K is an embedding, that any ﬁnite étale coveringH → H of H may be split by the pull-back of a ﬁnite étale covering K → K
of K; moreover, by the injectivity of Proposition 2.5, (i), we may even assume
further that H → H is locally trivial. But then the existence of a covering K → K
as desired follows from the [easily veriﬁed] assertion that “a ﬁnite graph-coveringof a connected sub-semi-graph of a given connected semi-graph may be extended
to a ﬁnite graph-covering of the given connected semi-graph” [cf. the proof of
Proposition 2.5, (i)]. This completes the proof of the fact that B(H) → B(K) is
a π1 -monomorphism. Next, we show that B(H) → B(K) is relatively slim anddetermines H → K. By Proposition 4.3, (v), we reduce immediately to the casewhere H is nuclear. Then our conclusion follows from Corollary 2.7, (i), (ii), (iii).
This completes the proof of assertion (iii).
Finally, assertion (iv) is a formal consequence of assertion (iii); Proposition
4.3, (iii); and the deﬁnitions. 
Proposition 4.5.
(The Subcategory of Tempered Objects) Let H, K be
tempered objects of Loc(G, Γ). Then every morphism H → K in Loc(G, Γ) is atempered covering. In particular, we have a natural full embedding
B temp (G)0 → Loc(G, Γ)
whose essential image is the full subcategory of tempered objects.

56

SHINICHI MOCHIZUKI

Proof. Indeed, by Proposition 3.6, (iv), (v), we obtain a morphism of temperoids
B temp (H) → B temp (K)that is compatible with the étale morphisms of temperoids B temp (H) → B temp (G),B temp (K) → B temp (G) induced by the G-structures. It thus follows formally thatthe morphism B temp (H) → B temp(K) is étale, hence corresponds to some tempered
covering H → K. But this tempered covering must coincide with the original
morphism H → K. Indeed, both morphisms induce the same arrow B temp (H) →B temp (K). Thus, if v is a vertex of H, and K → K is an excision with ﬁnite open
domain such that the restrictions to H[v] of the two morphisms in question bothfactor through K , then we conclude that these two morphisms both induce thesame arrow B(Hv ) → B(K ) [cf. Proposition 2.5, (i); Corollary 2.7, (i)], so weconclude by Proposition 4.3, (v); Proposition 4.4, (iii). 
Proposition 4.6.
(Valuative Criterion for Finite Étale Morphisms)
Let φ : H → K be a morphism between ﬁnite objects in Loc(G, Γ). Then φ is a
ﬁnite étale morphism if and only if the following condition is satisﬁed: for every
NL-morphism H0 → H1 and every commutative diagram
H0
⏐
⏐


−→

H1
⏐
⏐


H

−→

K

in Loc(G, Γ), there exists a commutative diagram
H0 −→
⏐
⏐
id

H2
⏐
⏐


−→

H1

H0

— where the horizontal arrows are NL-morphisms — such that the composite commutative diagram
H0 −→ H2
⏐
⏐
⏐
⏐


H

−→

K

admits a morphism H2 → H in Loc(G, Γ) that makes the resulting triangles in thisdiagram commute.
Proof. First, we consider necessity. By pulling back the ﬁnite étale morphism
H → K to H1 , we reduce immediately to the case where H1 = K. But then the factthat the condition in question is satisﬁed follows immediately from the deﬁnitionof a ﬁnite étale morphism.

SEMI-GRAPHS OF ANABELIOIDS

57

Next, we consider suﬃciency. By base-changing H → K by some ﬁnite étalemorphism as in Proposition 4.4, (i), we reduce immediately to the case where
H → K is an embedding. But then the condition in question implies that H → K
is surjective, which implies that H → K is an isomorphism [hence, in particular,ﬁnite étale], as desired. 
Proposition 4.7.
(Domination of Links) For i = 1, 2, let H0 → Hi bean NL-morphism. Suppose that the unique closed edge of Hi is the image of thesame [i.e., for i = 1, 2] edge of H0 . Then there exists an NL-morphism H0 → H3
which ﬁts into commutative diagrams
H0 −→
⏐
⏐
id

H3
⏐
⏐


H0

Hi

−→

for i = 1, 2.
Proof. First, we observe that we may choose G-structures on H1 , H2 that coincide
when restricted to H0 [cf. Proposition 4.3, (iv)]. In the following, “i” will always
range over the elements of the set {1, 2}. Now by the deﬁnition of Loc(G, Γ), there
exists a ﬁnite Galois étale coveringG → G
together with ﬁnite étale subcoverings G  → Gi → G [for i = 1, 2] such that G , Gi
are untangled; there exists an embedding Hi → Gi compatible with the G-structures.
Then [by conjugating Gi appropriately] we may assume that there exists a vertexv  (respectively, edge e ) of G  whose image in Gi is equal to the image of theunique vertex of H0 (respectively, the unique closed edge of Hi ) via the compositemorphism
H→
H→
G(respectively,
H
→
G
),
and,
moreover,
that
I
H1 =
0
i
i
i
i

I H2 , where we write
Hi ⊆ Gal(G  /G)
for the subgroup determined by the subcovering Gi → G andI ⊆ Gal(G  /G)
deffor the isotropy subgroup associated to v  . Set H3 = I H1 = I H2 ; write
G3 → G for the subcovering determined by H3 . Then if we take H3 to be the linkcontained in G3 which is determined by the images v3 , e3 of v  , e , respectively, then
the natural morphism G3 → Gi restricts to a morphism
H3 → Hi
with the desired properties. 

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SHINICHI MOCHIZUKI

Remark 4.7.1. Note that by applying Proposition 4.7 in an iterative fashion,
one may construct an NL-morphism with domain H0 that dominates an arbitrary
given ﬁnite collection of NL-morphisms H0 → Hi , for i = 1, . . . , n [i.e., such thatthe unique closed edge of each Hi is the image of the same edge of H0 ]. Moreover,given any pair of such dominating NL-morphisms H0 → Ha , H0 → Hb , there exists[again by Propsition 4.7] an NL-morphism H0 → Hc that dominates H0 → Ha ,H0 → Hb . Observe that in this situation, although a priori, we obtain two arrows
Hc → Hi [i.e., one passing through Ha , the other passing through Hb ], these twoarrows necessarily coincide [by Proposition 4.3, (v)]. Thus, in summary, for eachedge e of H0 , we obtain a natural system of dominating NL-morphisms with domain
H0 , each of whose codomains is a link with closed edge given by the image [via the
NL-morphism under consideration] of e.
Theorem 4.8.
(Category-Theoreticity of Categories of Localizations)
For i = 1, 2, let Gi be a ﬁnite, connected, coherent, totally elevated, totally
universally sub-coverticial, totally aloof, verticially slim graph of anabelioids, with underlying graph Gi ; let Γi be a ﬁnite group that acts piecewisefaithfully on Gi . Suppose that Gi has at least one edge. Write
Loc(Gi , Γi )ﬁn ⊆ Loc(Gi , Γi )for the full subcategory determined by the ﬁnite objects. Then the categories
Loc(Gi , Γi )ﬁn (respectively, Loc(Gi , Γi )) are slim; every equivalence of categories∼∼

Φ : Loc(G1 , Γ1 )ﬁn → Loc(G2 , Γ2 )ﬁn (respectively, Φ : Loc(G1 , Γ1 ) → Loc(G2 , Γ2 ))arises, up to unique isomorphism, from a unique isomorphism of graphs ofanabelioids∼
G1 → G2∼

together with a compatible isomorphism Γ1 → Γ2 .
Proof. First, we reconstruct the underlying semi-graph of anabelioids of an object
def
def
H of C = Loc(Gi , Γi )ﬁn (respectively, C = Loc(Gi , Γi )) [where i = 1, 2] categorytheoretically as follows: The objects of verticial length 1 are precisely the indissectible[cf. §0] objects. An object H of verticial length 1 is nuclear if and only if the domainof every morphism with codomain H is of verticial length 1. If H is nuclear, thenthe result of applying “⊥” to the category CH of objects and morphisms over H isa Galois category isomorphic to the anabelioid B(H). The verticial morphisms are
precisely the morphisms with nuclear domain which are, moreover, minimal-adjoint[cf. §0] to the morphisms with nuclear codomain. The vertices of the underlying
semi-graph of an object H are precisely the isomorphism classes, over H, of verticial
morphisms K → H. Thus, in particular, we conclude that Φ induces a bijection
between the sets of vertices of the underlying semi-graphs of corresponding objects,
together with compatible isomorphisms of the various constituent anabelioids at thevertices; moreover, these bijections and isomorphisms are compatible with arrows
in C. In particular, Φ preserves locally trivial morphisms.

SEMI-GRAPHS OF ANABELIOIDS

59

An object H is a link if and only if H is of verticial length 2, and, moreover,any locally trivial morphism K → H, where K is also of verticial length 2, is an
isomorphism. The closed edges of the underlying semi-graph of an object H are
precisely the isomorphism classes, over H, of locally trivial morphisms K → H,
where K is a link. An [open] edge of a nuclear object H is a system of compatible
closed edges of NL-morphisms H → K, as we vary the NL-morphism as describedin Remark 4.7.1. Thus, we conclude that Φ induces an isomorphism between theunderlying semi-graphs of corresponding objects; moreover, these isomorphisms arecompatible with arrows in C. Note that this implies, for instance, that Φ preserves
the embeddings.
In particular, Φ preserves the ﬁnite objects [i.e., objects of ﬁnite verticial
length], as well as the NL-morphisms. Moreover, by Proposition 4.6, Φ preserves
the ﬁnite étale morphisms between ﬁnite objects. Thus, by considering the isotropy
subgroups associated to the various vertices and edges in the Galois group of a ﬁnite étale Galois covering, one sees that Φ induces an isomorphism between theunderlying semi-graphs of anabelioids of corresponding ﬁnite objects in a fashionthat is compatible with arrows in C. Moreover, these induced isomorphisms may be
extended immediately to the case of inﬁnite objects [i.e., when such objects exist in
C] by representing such objects as inductive limits of inductive systems consistingof ﬁnite objects and embeddings.
Next, we observe that the closed objects H of C are precisely the ﬁnite objects
whose underlying semi-graph H is a graph; Gi is the unique closed object of C, up
to isomorphism, to which every other closed object maps.
Next, we observe that one may recover the various local (G, Γ)-structures onopen objects as follows: First, we note that we may reconstruct the automorphisms
Gi → Gi in Γi by localizing on Gi . That is to say, if φ : H → Gi is a morphism withﬁnite open domain [where we note that such a φ always exists, for instance, if H isa tree — cf. Proposition 4.3, (iv)], then by Proposition 4.3, (v), the Γi -span of φ isprecisely the set of all morphisms H → Gi in C. Moreover, since Γi acts piecewisefaithfully, it follows that the cardinality of this set is always equal to the order of Γi .
Thus, by taking H to be various localizations of Gi and then gluing, we recover ﬁrst
the set of morphisms of semi-graphs of anabelioids Gi → Gi arising from elementsof Γi and then the group structure [by composing morphisms of semi-graphs ofanabelioids].
Thus, in summary, we have shown that Φ induces an isomorphism of semigraphs of anabelioids∼
G1 → G2∼

together with a compatible isomorphism Γ1 → Γ2 . Moreover, we have shown that
Φ induces an isomorphism between the underlying semi-graph of anabelioids ofcorresponding objects in a fashion that is compatible with arrows in C, as well aswith the given local (G, Γ)-structures. Thus, it is an easily veriﬁed tautology thatthe equivalence Φ is isomorphic to the equivalence induced by the isomorphisms∼∼
G1 → G2 , Γ1 → Γ2 .
Finally, it remains to verify that C is slim. Let A ∈ Ob(C); suppose that ψ is an
automorphism of the natural functor CA → C. Concretely speaking, this means that

60

SHINICHI MOCHIZUKI

for every object β : B → A of CA , we are given an automorphism ψβ ∈ AutC (B)in such a way that the assignment β → ψβ is compatible with the image in C ofarrows of CA . Since A is arbitrary, it suﬃces [by considering, for given β : B → A,
the automorphism of the resulting composite functor CB → CA → C induced by ψ]
to show that ψA [i.e., the automorphism assigned to the identity A → A] is theidentity. By considering β with nuclear domain, we conclude immediately that ψβﬁxes the set of vertices of A. Thus, since Gi is totally aloof, we reduce to the casewhere A is nuclear.
Now since A is nuclear, to show that ψ is trivial, it suﬃces to show the following:If v be a vertex of Gi ; Γv ⊆ Γi is the corresponding isotropy subgroup; and Hv isthe extension of Γv by π1 ((Gi )v ) arising from the action of Γv on (Gi )v , then the
proﬁnite group Hv is slim. But this is an immediate formal consequence of the factthat Gi is verticially slim and totally aloof, together with our assumption that theaction of Γi on Gi is piecewise faithful. Remark 4.8.1.
Returning to the notation used in the discussion preceding
Theorem 4.8, suppose that the semi-graph G has at least one vertex. Write H ⊆ G
for the maximal subgraph [cf. §1] of G. Then observe that [whenever Loc(G, Γ) isdef

deﬁned] if we set H = GH , then Γ acts naturally and piecewise faithfully on H;
Loc(H, Γ) is deﬁned; and we have natural equivalences∼∼

Loc(G, Γ)ﬁn → Loc(H, Γ)ﬁn ;Loc(G, Γ) → Loc(H, Γ)[deﬁned by simply omitting all G-open edges]. Thus, there is no essential loss ofgenerality in restricting Theorem 4.8 to the case where G is a graph.
Remark 4.8.2.
One veriﬁes easily that [whenever Loc(G, Γ) is deﬁned] thefollowing ﬁve conditions are equivalent:(i) G has no closed edges.
(ii) G is a tree [cf. §1] with at most one vertex.
(iii) Loc(G, Γ)⊥ is a Galois category.
(iv) Every monomorphism of Loc(G, Γ) is an isomorphism.
(v) Every object of Loc(G, Γ) is closed.
Moreover, if any of these conditions is satisﬁed, then there is a natural equivalence∼

Loc(G, Γ)⊥ → B(G)— so, in particular, Loc(G, Γ)⊥ does not depend on the action of Γ [which is, at
any rate, trivial, if G admits at least one vertex]. Thus, since equivalences between
connected anabelioids are “well-understood”, there is no essential loss of generality
in excluding from Theorem 4.8 the case in which these conditions are satisﬁed.

SEMI-GRAPHS OF ANABELIOIDS

61

Remark 4.8.3. In the resp’d case of Theorem 4.8 [i.e., where one includes theinﬁnite objects], if one assumes further that Gi is totally estranged, then the proof of
Theorem 4.8 may be simpliﬁed somewhat, by applying Proposition 4.5, Corollary
3.9.
Remark 4.8.4. In the notation of the proof of Theorem 4.8, if an object A of C⊥
is nuclear, then the category CA
is easily veriﬁed to be a connected anabelioid; the⊥category C[A] [cf. §0] is easily veriﬁed to be a “connected quasi-anabelioid” [cf.
Remark A.4.2 of the Appendix].

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SHINICHI MOCHIZUKI

Section 5: Arithmetic Semi-graphs of Anabelioids
In this §, we consider semi-graphs of anabelioids equipped with a continuous
action of a proﬁnite group, which we think of as an “arithmetic structure” onthe semi-graph of anabelioids. We then proceed to study a certain “arithmetic
analogue” of the theory of maximal compact subgroups of §3.
Deﬁnition 5.1.
(i) Let G be a connected, coherent, totally aloof, verticially slim semi-graph ofanabelioids. Let A be a slim connected anabelioid, equipped with a basepoint, so
we may speak of π1 (A). We shall refer to as an action of π1 (A) on G the datum ofa homomorphismρG : π1 (A) → Aut(G)[where Aut(G) denotes the group of automorphisms of G as a totally aloof, verticiallyslim semi-graph of anabelioids]. Note that any such pair (G, ρG ) admits an “inneraction” by π1 (A) — i.e., by letting π1 (A) act on π1 (A) by conjugation and on G
via ρG . We shall say that an action of π1 (A) on G is continuous if, for some open
subgroup H ⊆ π1 (A), the following conditions are satisﬁed:(a) π1 (A) is topologically ﬁnitely generated.
(b) The semi-graph G is locally ﬁnite.
(c) The action of H on G is trivial; the resulting outer homomorphism
H → Out(π1 (Gv )), where v ranges over the vertices of G, is continuous[i.e., relative to the natural proﬁnite group topology on Out(π1 (Gv ))].
(d) There is a ﬁnite set V of vertices of G such that for every vertex w ofG, there exists a v ∈ V and an isomorphism of semi-graphs of anabelioids∼
G[v] → G[w] that is compatible with the action of H on both sides.
(ii) A triple G = (G, A, ρG ) as in (i), where ρG is a continuous action of π1 (A)on G, will be referred to as a connected arithmetic semi-graph of anabelioids (overA). Suppose that G = (G, A, ρG ) is a connected arithmetic semi-graph of anabelioids. Then we shall refer to G (respectively, A; ρG ) as the geometric component(respectively, arithmetic component; arithmetic action) of G. The arithmetic actionof an arithmetic semi-graph of groups induces [what, by abuse of terminology, we
shall also refer to as] “arithmetic actions” on various objects [e.g., the underlying
semi-graph, etc.] associated to this arithmetic semi-graph of groups.
(iii) A [not necessarily connected] arithmetic semi-graph of anabelioids G is deﬁned to be a formal collection of connected arithmetic semi-graphs of anabelioids;
each object in this collection will be referred to as a connected component of G.
Note that the geometric components of each of the connected components of G together determine a natural geometric component [i.e., a (not necessarily connected)

SEMI-GRAPHS OF ANABELIOIDS

63

semi-graph of anabelioids] of G. We shall say that an arithmetic semi-graph of anabelioids is ﬁnite (respectively, elevated at a vertex; totally elevated; sub-coverticial
at a closed edge; universally sub-coverticial at a closed edge; totally sub-coverticial;totally universally sub-coverticial; aloof at an edge; totally aloof; estranged at anedge; totally estranged) if its geometric component is so.
(iv) Given two connected arithmetic semi-graphs of anabelioids G = (G  , A , ρG  ),G = (G, A, ρG ), a morphism of connected arithmetic semi-graphs of anabelioids
G →Gconsists of a pair

π1 (A ) → π1 (A);G → G[i.e., a continuous homomorphism of proﬁnite groups and a morphism of semigraphs of anabelioids] which is compatible with ρG  , ρG , and which we regard up
to composition with the inner action of π1 (A) on (G, ρG ). We shall refer to themorphism G  → G (respectively, the induced morphism of anabelioids A → A) as
the geometric (respectively, arithmetic) component of the morphism. [In particular,
if we restrict our attention to G → G whose geometric component is locally open,
then we may work with such morphisms as if they are “morphisms in a category”— cf. Remark 2.4.2.] A morphism of [not necessarily connected] arithmetic semigraphs of anabelioids
G →G
is deﬁned to be a collection of morphisms, one from each connected component ofG to some connected component of G. We shall say that G → G is ﬁnite étale(respectively, tempered; locally trivial; locally open; locally ﬁnite étale; immersive;
excisive; an embedding; BC-ﬁnite étale) if each of its geometric components is ﬁnite
étale (respectively, a tempered covering [a term which we recall only makes sensewhen G is countable]; locally trivial; locally open; locally ﬁnite étale; immersive; excisive; an embedding; an isomorphism); each of its arithmetic components is ﬁnite
étale (respectively, ﬁnite étale; an isomorphism; a composite of a π1 -epimorphismwith a ﬁnite étale morphism; ﬁnite étale; an isomorphism; an isomorphism; an isomorphism; ﬁnite étale); and its induced map on connected components has ﬁnite[but possibly empty] ﬁbers (respectively, has countable ﬁbers; is arbitrary; is arbitrary; is arbitrary; is arbitrary; is arbitrary; is injective; has ﬁnite [but possibly
empty] ﬁbers). [Here, the abbreviation “BC” is to be understood to stand for thephrase “base of constants”.]
Proposition 5.2. (Arithmetic Tempered Coverings) Let G = (G, A, ρG ) bea connected arithmetic semi-graph of anabelioids. Then:(i) Every tempered covering of G appears as the geometric component of atempered covering of G.
(ii) Suppose that H → G, K → G are tempered coverings with isomorphic
geometric components. Then there exist BC-ﬁnite étale coverings H → H, K → Ksuch that H , K are isomorphic as tempered coverings over G.

64

SHINICHI MOCHIZUKI(iii) Let us denote by
B temp(G)

the category whose objects are tempered morphisms G → G and whose morphisms are tempered morphisms over G. Then B temp(G) is a connected temperoid. Similarly, the full subcategory
B(G) ⊆ B temp(G)determined by the ﬁnite étale coverings forms a connected anabelioid. If G is
totally elevated, then B temp(G) temp-slim, and B(G) is slim.
(iv) Write:
def

def

Πtemp
= π1temp (G) = π1temp (B temp (G));
G
def

def

ΠG = π1 (G) = π1 (B(G));

def

Πtemp
= π1temp (B temp (G)) = π1temp (G)GdefΠG = π1 (B(G)) = π1 (G)Then there are natural morphisms
B temp (G) → B temp (G) → A ;B(G) → B(G) → A

which induce natural exact sequences:
1 → Πtemp→ Πtemp→ ΠA → 1GG
1 → ΠG → ΠG → ΠA → 1
def

We shall refer to ΠG (respectively, Πtemp; ΠG ; Πtemp; ΠA = π1 (A)) as the geoGG
metric (respectively, geometric tempered; arithmetic; arithmetic tempered;BC-) fundamental group of G.
Proof. Assertions (i), (ii) follow from the various ﬁniteness assumptions in ourdeﬁnition of a “continuous action” [cf. Deﬁnition 5.1, (i); the fact that G is coherent].
[Note, in particular, that one must make use of the assumption of Deﬁnition 5.1,(i), (d), in order to verify assertions (i), (ii) for arbitrary inﬁnite G.] Except forthe ﬁnal sentence of assertion (iii), assertions (iii), (iv) follow immediately from thedeﬁnitions and assertions (i), (ii); the ﬁnal sentence of assertion (iii) follows from
Corollary 2.7, (ii). 
Remark 5.2.1. At this point, one could proceed to develop a theory of “categories of arithmetic localizations” of arithmetic semi-graphs of anabelioids, in thestyle of §4. Although this is quite possible [we leave the details to the enthusiastic
reader!], it is rather cumbersome, so instead we restrict ourselves [cf. Remark 4.8.3]
to considering the categorical representation of arithmetic semi-graphs of anabelioids aﬀorded by the “arithmetically maximal compact subgroups” [cf. Deﬁnition
5.3 below] of the tempered fundamental groups of Proposition 5.2, in the style ofCorollary 3.9.

SEMI-GRAPHS OF ANABELIOIDS

65

Let G be a connected, countable, totally elevated, totally estranged arithmetic
semi-graph of anabelioids, with underlying semi-graph G. In the notation of Proposition 5.2, we would like to consider compact subgroups of the arithmetic tempered
fundamental group Πtemp. Note that for every vertex v of G, we obtain an associG
ated decomposition group⊆ ΠtempΠtemp
G,vG[well-deﬁned up to conjugation in Πtemp], which [by Corollary 2.7, (i), (iii); theG

injection of Proposition 3.6, (iii)] may be thought of as the commensurator in Πtemp
G
temp def temp  temp
ΠG . Similarly, if b is a branch of an edge e of G that abutsof ΠG,v = ΠG,v
to v, then we obtain a decomposition group⊆ Πtemp⊆ Πtemp
Πtemp
G,b
G,v
G

[well-deﬁned up to conjugation in Πtemp], which [since G is totally estranged, hence,Gofin particular totally aloof] may be thought of as the commensurator in Πtemp
G,v
def temp  temp
ΠG .
= ΠG,bΠtempG,bDeﬁnition 5.3.

(i) A closed subgroup of Πtemp
will be called arithmetically ample if it surjectsGwill be called arithonto an open subgroup of ΠA . A compact subgroup of Πtemp
G
metically maximal if it is maximal among arithmetically ample compact subgroupsof Πtemp.
G(ii) Let e be an edge of G. We shall say that e is arithmetically estranged if,
for every vertex v to which some branch b of e abuts and every g ∈ Πtemp
, the
G,v

intersection in Πtempof Πtemp
with any subgroup of the form g · Πtemp· g −1 , where
G,v
G,b
G,b

either b = b is a branch of an edge that abuts to v or b = b and g ∈/ Πtemp, fails
G,b
to be arithmetically ample. If every edge of G is arithmetically estranged, then weshall say that G is totally arithmetically estranged.
(iii) We shall refer to subgroups of Πtempof the form “Πtemp” (respectively,GG,v“Πtemp”) as verticial (respectively, edge-like).
G,bRemark 5.3.1.

Note that all verticial and edge-like subgroups of ΠtempareGof a(n) verticial
compact and arithmetically ample. Also, the intersection with Πtemp
G
temp
is a(n) verticial (respectively, edge-like)(respectively, edge-like) subgroup of ΠG
in the sense of Theorem 3.7.
subgroup of Πtemp
G

The main result of the present §5 is the following “arithmetic analogue” ofTheorem 3.7, Corollary 3.9:

66

SHINICHI MOCHIZUKI

Theorem 5.4. (Arithmetically Maximal Compact Subgroups) Let G, H
be connected, countable, totally elevated, totally arithmetically estrangedarithmetic graphs of anabelioids, with the same arithmetic component A. Suppose, moreover, that the arithmetic actions on the underlying graphs G, H do notswitch the branches of any edge. Then:(i) Every arithmetically ample compact subgroup of π1temp (G) is contained in
at least one verticial subgroup. If an arithmetically ample compact subgroup of
π1temp (G) is contained in more than one verticial subgroup, then it is containedin precisely two verticial subgroups, whose intersection forms an edge-like subgroup.
(ii) The arithmetically maximal compact subgroups of π1temp (G) are precisely the verticial subgroups. The arithmetically ample intersections of
two distinct arithmetically maximal compact subgroups of π1temp (G) are precisely
the edge-like subgroups.
(iii) Applying “B temp(−)” determines a natural bijective correspondence between locally open morphisms of arithmetic semi-graphs of anabelioids
G →H
over A and “arithmetically quasi-geometric” morphisms of temperoids
B temp (G) → B temp(H) over A , i.e., morphisms that arise from a continuous mor→ Πtemp
that maps any arithmetically maximal compact subgroup
phism Πtemp
G
H

temp
K1 ⊆ ΠG(respectively, arithmetically ample intersection K1 H1 of two distinct) to an open subgroup ofarithmetically maximal compact subgroups K1 , H1 ⊆ Πtemp
G(respectively, of some
some arithmetically maximal compact subgroup K2 ⊆ ΠtempHarithmetically ample intersection K2 H2 of two distinct arithmetically maximal
compact subgroups K2 , H2 ⊆ Πtemp).
H
Proof. Modulo the evident “arithmetic translation” — e.g., “nontrivial” is to be
replaced by “arithmetically ample” and “estranged” by “arithmetically estranged”— the proofs are entirely parallel to those of Theorem 3.7, Corollary 3.9. 
Before proceeding, we review the following well-known result:Lemma 5.5.
(Decomposition Groups of Proper Hyperbolic Curves
over Finite Fields) Let X be a proper hyperbolic curve over a ﬁnite ﬁeld
k. Write ΠX for the étale fundamental group of X; ΠX  Gk for the naturalaugmentation to the absolute Galois group of k. Then a k-valued point x ∈ X(k)is determined by the outer homomorphism σx : Gk → ΠX that it induces.
Proof. Write J for the Jacobian of X; assume for simplicity that there exists
a point x0 ∈ X(k). Then x0 determines a closed embedding X → J whose induced morphism on étale fundamental groups ΠX  ΠJ may be identiﬁed with

SEMI-GRAPHS OF ANABELIOIDS

67

the quotient of ΠX by the commutator subgroup of the kernel of the surjection
ΠX  Gk . Thus, it suﬃces to show that a point a ∈ J (k) is determined by theouter homomorphism σa : Gk → ΠJ that it induces.
Write σ0 : Gk → ΠJ for the outer homomorphism induced by the identity
element of J (k). Then the diﬀerence between σa and σ0 may be thought of as
an element of ηa ∈ H 1 (k, T ), where we deﬁne T to be the kernel of the natural surjection ΠJ  Gk . Note, moreover, that we have a natural isomorphism∼
T → Hom(Q/Z, J (k)), where k is the algebraic closure of k determined by thebasepoint of k implicit in the discussion. On the other hand, by well-known general
nonsense [cf., e.g., [Naka], Claim (2.2); [NTs], Lemma (4.14); [Mzk2], the Remark∼
preceding Deﬁnition 6.2], there is a natural isomorphism H 1 (k, T ) → J (k), whichmaps ηa to a. In particular, ηa, hence also σa , is suﬃcient to determine a itself. 
Example 5.6. Pointed Stable Curves over p-adic Local Fields II. We workin the notation of Example 3.10. Also, Suppose that we are given an exhaustive
sequence of open characteristic [hence normal] subgroups of ﬁnite index. . . ⊆ Mi ⊆ . . . ⊆ Π[where i ranges over the positive integers] of Π such that Ni = MidefΔ; writelog
Thus, Mi determines a ﬁnite log étale covering of XK; we assume

Πi = Π/Mi .
that Mi has been chosen so that this covering has stable reduction over the ring ofintegers of the ﬁnite extension of K that it determines. Then the outer action of
Mi on Ni determines an arithmetic action on the semi-graphs of anabelioids Gi , Gic
of Example 3.10; that is to say, we obtain arithmetic semi-graphs of anabelioids
Gi;

c

Gi

with underlying semi-graphs of anabelioids Gi , Gic , respectively, equipped with natc
ural actions by Πi . Moreover, G i , G i are connected, ﬁnite, totally elevated, andtotally universally sub-coverticial [cf. Example 3.10]. Also, it follows immediatelycfrom Lemma 5.5 that G i , G i are totally arithmetically estranged. In particular, [atleast for i suﬃciently large] G i satisﬁes the hypotheses of Theorem 5.4.
Now I claim that the generalized morphisms of arithmetic graphs of anabelioids
Gi → G j[where i ≥ j] — i.e., the generalized morphisms of graphs of anabelioids of Example
3.10 considered together with the natural compatible arithmetic actions on thedomain and codomain — may be recovered group-theoretically from the associatedmorphisms of tempered fundamental groups
Πtemp→ ΠtempGGij

as follows: First, we consider the functor Cat(Gi ) → Cat(Gj ). Now observe that if v
(respectively, e) is a(n) vertex (respectively, edge) of Gi such that the image in Πtemp
G
j

68

SHINICHI MOCHIZUKI

of the verticial (respectively, edge-like) subgroup determined by v (respectively, e)
is contained in a [necessarily unique] edge-like subgroup H of Πtemp, then thisGjfunctor maps v (respectively, e) to the edge of Gj determined by H. On the other
hand, if v (respectively, e) is a(n) vertex (respectively, edge) of Gi such that theimage in Πtempof the verticial (respectively, edge-like) subgroup determined by v
Gj, but is contained(respectively, e) is not contained in an edge-like subgroup of ΠtempGjin a verticial subgroup H of Πtemp, then this functor maps v (respectively, e) toGjthe vertex of Gj determined by H [cf. Lemma 5.5 in the case where this image
fails to be an open subgroup of H]. That these characterizations make sense and,moreover, do indeed yield the map on objects determined by the functor in question
follows from Theorem 5.4, (i), (ii); Lemma 5.5. The remainder of the data necessary
to deﬁne the generalized morphism of arithmetic graphs of anabelioids G i → G j isdetermined naturally by considering the maps between the various verticial and, Πtemp. This completes the proof of the claim.
edge-like subgroups of Πtemp
G
G
i

j

Moreover, by a similar argument, together with the technique of Corollary
3.11, one may reconstruct the generalized morphisms of arithmetic semi-graphs ofanabelioidscc
Gi → G j[where i ≥ j] group-theoretically from the corresponding morphisms of temperedgroups Mi → Mj .
Remark 5.6.1.
There is an immediate proﬁnite generalization of the grouptheoretic reconstruction in Example 5.6 of the generalized morphism of arithmetic
graphs of anabelioidscc
Gi → G j[where i ≥ j] from the corresponding morphism of proﬁnite groups:
Mi∧ → Mj∧
[where the “∧” denotes proﬁnite completion]. Indeed, this follows by applying [inplace of Theorem 5.4, the technique of Corollary 3.11] the fact that the “dual semigraph with compact structure of the geometric special ﬁber” may be recovered evenin the proﬁnite case, from the Galois action on the geometric proﬁnite fundamentalgroup [cf. [Mzk3], Lemma 2.3].

SEMI-GRAPHS OF ANABELIOIDS

69

Section 6: Tempered Anabelian Geometry
In this §, we observe that the theory of Galois sections in absolute anabeliangeometry [cf. [Mzk8]] admits a fairly straightforward generalization to the case of
tempered fundamental groups.
Let K be a ﬁnite extension of Qp ; K an algebraic closure of K; XK a hyperbolic
def

curve over K. Let us write XK = XK ×K K;
π1temp (XK )log
for the tempered fundamental group of [André], §4 [cf. also the group “π1temp (XK)”
tempof Examples 3.10, 5.6]. Thus, π1 (XK ) is a tempered topological group [in the senseof Deﬁnition 3.1, (i)] and ﬁts into a natural exact sequence:

1 → π1temp (XK ) → π1temp (XK ) → GK → 1
def[where GK = Gal(K/K); we write π1temp (XK ) for the geometric tempered fundamental group of XK , i.e., the tempered fundamental group of XK ×K K; the “∧”denotes the p-adic completion]. To simplify the notation, let us write:
def

temp
Πtemp
(XK );XK = π1def

Δtemp= π1temp (XK )X

In the following discussion, we shall denote the proﬁnite completion of a group by
def
def
temp
. It follows frommeans of a “∧”. Also, we shall write ΠXK = Πtemp
XK ; Δ X = Δ X
the well-known residual ﬁniteness of discrete free groups [cf., e.g., Corollary 1.7]
temp
→ ΔX [cf. the discussion ofthat we have natural injections Πtemp
XK → ΠXK , ΔX
[André], §4.5].
Lemma 6.1.

(Proﬁnite Normalizers)(i) Let F be a ﬁnitely generated [discrete] free group of rank > 1. ThenNF (F ) = F .
(ii) We have: NΔX (Δtemp) = Δtemp
.
X
X
temp
(iii) We have: NΠXK (Πtemp
XK ) = ΠXK .

Proof. Assertion (i) (respectively, (ii)) is the content of [André], Lemma 3.2.1(respectively, [André], Corollary 6.2.2). Assertion (iii) follows immediately from
assertion (ii). Deﬁnition 6.2.

Let F , F1 , F2 be tempered groups [cf. Deﬁnition 3.1, (i)]. Then:(i) We shall refer to a [not necessarily closed] subgroup H ⊆ F as being of DFGtype [i.e., “dense, ﬁnitely generated type”] if it is dense in some open subgroup of

70

SHINICHI MOCHIZUKI

the proﬁnite completion F , and, moreover, for any open normal subgroup J ⊆ F ,the image of H in F/J is ﬁnitely generated.
(ii) We shall refer to a [not necessarily closed] subgroup H ⊆ F as being of
DOF-type [i.e., “dense in an open subgroup of ﬁnite index type”] if it is dense insome open subgroup of F of ﬁnite index.
(iii) A continuous homomorphism F1 → F2 will be said to be of DFG-type (respectively, of DOF-type) if its image is a subgroup of F2 of DFG-type (respectively,of DOF-type).
Lemma 6.3.

(Dense Subgroups)(i) Let F be a ﬁnitely generated [discrete] free group of rank > 1. Supposethat H ⊆ F is a ﬁnitely generated subgroup which is dense in F . Then H = F .
temp(ii) Let F be either Πtemp; write F for the proﬁnite completion of F .
XK or ΔX
Then a subgroup H ⊆ F is of DFG-type if and only if it is of DOF-type.

(iii) Let F , F be as in (ii). Suppose that F1 , F2 ⊆ F are subgroups of DOFtype which are dense in F . Then, for any f ∈ F such that f · F1 · f −1 = F2 , itfollows that f ∈ F .
Proof. Assertion (i) follows immediately from the “structure theory of ﬁnitely
generated subgroups of free groups of ﬁnite rank” [cf., e.g., Corollary 1.6, (ii)]. As
for assertion (ii), let us ﬁrst observe that by replacing F by an open subgroup of F
of ﬁnite index containing H, we may assume that H is dense in F . Now suﬃciency
is immediate. To prove necessity, we note that it follows from assertion (i), together
with the assumption that H is dense in F , that the image of H in each F/J is equal
to F/J , i.e., that H is dense in F , as desired. Finally, assertion (iii) follows from
Lemma 6.1, (i) [cf. the proofs of Lemma 6.1, (ii), (iii)]. 
Now suppose that L ⊆ K is also a ﬁnite extension of Qp ; YL is a hyperbolic
curve over L. We shall use similar notation for the various fundamental groups[i.e., tempered, proﬁnite étale, etc.] associated to YL to the notation used thus far
for XK . Now we have the following result [cf. [Mzk8], Theorem 1.2]:
Theorem 6.4.
(Tempered Anabelian Theorem for Hyperbolic Curves
over Local Fields) The tempered fundamental group functor determines a bijection between the set of dominant morphisms of schemes
XK → YL
temp
and the set of outer homomorphisms of DOF-type φ : Πtemp
that ﬁt
XK → ΠYL
into a commutative diagram

Πtemp
XK
⏐
⏐


−→

φ

Πtemp
YL
⏐
⏐


GK

−→

GL

SEMI-GRAPHS OF ANABELIOIDS

71

for which the induced morphism GK → GL is an open immersion [i.e., an isomorphism onto an open subgroup of GL ] which arises from an embedding of ﬁelds
L → K.
Proof. One veriﬁes immediately from the deﬁnition of the “tempered fundamental
temp
group” that any Πtemp
that arises geometrically is of DFG-type, hence,
XK → ΠYL
by Lemma 6.3, (ii), of DOF-type. On the other hand, given a homomorphism φ :tempof DOF-type, proﬁnite completion yields an open homomorphismΠtemp
XK → ΠYL
φ : ΠXK → ΠYL , so by [Mzk8], Theorem 1.2 [i.e., in essence, [Mzk2], Theorem A],
we obtain that φ arises, up to inner automorphism, from a dominant morphismof schemes XK → YL . In particular, this dominant morphism of schemes induces
a homomorphism ψ : Πtemp→ Πtempof DOF-type, whose proﬁnite completion
XKYLψ : ΠXK → ΠYL diﬀers from φ by composition with an inner automorphism of
ΠYL . On the other hand, by Lemma 6.3, (iii), we thus conclude that φ diﬀers fromψ by composition with an inner automorphism of Πtemp
YL , as desired. 
Next, let us write XK → X K for the compactiﬁcation [cf. §0] of XK . Letx ∈ XK
be a closed point. Thus, x determines, up to conjugation by an element of Πtemp
XK ,a decomposition group:Dx ⊆ Πtemp
XK
We shall refer to a closed subgroup of Πtemp
XK which arises in this way as a decomtemp
position group of ΠXK . If x is a cusp, then we shall refer to the decompositiongroup Dx as cuspidal. Note that Dx always surjects onto an open subgroup of GK .
Moreover, the subgroupdef
Ix = Dx Δtemp
X
is isomorphic to Z(1) [i.e., the proﬁnite completion of Z, Tate twisted once] (respectively, {1}) if x is (respectively, is not) a cusp. We shall refer to a closed subgroup ofΠtemp
XK which is equal to “Ix ” for some cusp x as a cuspidal geometric decompositiongroup.
Theorem 6.5.

(Tempered Decomposition Groups)(i) (Determination of the Point) The closed point x is completely determined by the conjugacy class of the closed subgroup Dx ⊆ Πtemp
XK . If x is acusp, then x is completely determined by the conjugacy class of the closed subgroupIx ⊆ Πtemp
XK .
(ii) (Commensurable Terminality) The subgroup Dx is commensurably ter(H) for any open subgroupminal in Πtemp
XK . If x is a cusp, then Dx = CΠtemp
XK
H ⊆ Ix .

72

SHINICHI MOCHIZUKI(iii) (Absoluteness of Cuspidal Decomposition Groups) Every isomorphism of tempered groups∼
temp
α : Πtemp
XK → ΠYL
preserves cuspidal decomposition groups and cuspidal geometric decomposition groups.
(iv) (Cuspidal and Noncuspidal Decomposition Groups) No noncuspidal decomposition group of Πtemp
XK is contained in a cuspidal decomposition grouptempof ΠXK .
Proof. Assertions (i), (ii), (iv) follow formally from [Mzk8], Theorem 1.3, (i), (ii),(iv), respectively. Assertion (iii) follows from Corollary 3.11. 
To a large extent, the absolute anabelian geometry of tempered fundamentalgroups is essentially equivalent to the absolute anabelian geometry of proﬁnite fundamental groups. Indeed, we have the following result:
Theorem 6.6.
(Tempered and Proﬁnite Outer Isomorphisms) Everyouter isomorphism∼
ΠXK → ΠYL
of proﬁnite groups arises from a unique outer isomorphism
∼

temp
Πtemp
XK → ΠYL

of tempered groups.
∼

Proof. Indeed, it is immediate that every outer isomorphism Πtemp→ Πtemp
XK
YL∼
determines an outer isomorphism ΠXK → ΠYL . Now let∼

α : ΠXK → ΠYL
be an arbitrary outer isomorphism. Let HX ⊆ ΠXK , HY ⊆ ΠYL be open normal
subgroups [of ﬁnite index] that correspond via α. Then by [Mzk3], Lemma 2.3, αdetermines a natural isomorphism∼cc
αH : G H
→ GH
X
Y
c
c
between the “semi-graphs of anabelioids with compact structure” GH
, GH
[cf.
X
Y
Example 2.10; [Mzk3], Appendix] associated to the geometric special ﬁbers of thecoverings corresponding to HX , HY . Moreover, αH is compatible with the natu

def
 def
= ΠXK /(HX ΔX ), HY = ΠYL /(HY ΔY ), relative to theral actions of HX ∼→ HY induced by α.
isomorphism αH  : HX⊆
In particular, we conclude that the closed subgroups JX ⊆ HX Δtemp
X temptemp
temp
ΔX , JY ⊆ HY ΔY⊆ ΔYdetermined by considering the pro-tempered

SEMI-GRAPHS OF ANABELIOIDS

73

ccoverings of XK , YK arising from the various tempered coverings [cf. §3] of GH
,
X
c
∧
∧
satisfyα(J)=
J[wherethe“∧”denotes
proﬁnitecompletion,or,equivaGH
X
YY,lently, closure in ΠXK , ΠYL ]. Also, we note that the natural outer actions of HX
temp
temp

c
c
HY on π1 (GHX ), π1 (GHY ), respectively, determine natural isomorphisms
∼

out

∼

out

temp
c

(GH
)  HXΠtemp
XK /JX → π1
X
temp
c
Πtemp
(GH
)  HYYL /JY → π1
Ywhich we shall use in the following discussion to identify the quotients on the left
with the “outer semi-direct products” [cf. §0] on the right.
Now let us write
out

∼

out

c

c
)  HX→ π1temp (GH)  HYβH : π1temp (GHXY

for the isomorphism induced by αH , αH  . The functoriality of the construction of
αH in [Mzk3], Lemma 2.3, implies that the proﬁnite completion βH diﬀers from∧ ∼→ ΠYL /JY∧ induced by α by composition with an inner
the isomorphism ΠXK /JX
automorphism. Also, we observe that, as one varies HX , HY , consideration of theresulting “generalized morphisms of arithmetic graphs of anabelioids” [cf. Example
5.6, Remark 5.6.1] shows that the resulting βH ’s are compatible [up to inner automorphism]. Thus, by passing to the corresponding inverse limit, we conclude thatthe various βH determine an isomorphism of tempered fundamental groups∼

temp
β : Πtemp
XK → ΠYL

whose proﬁnite completion β diﬀers from α by an inner automorphism, as desired.
That such a β is unique, up to inner automorphism, follows from Lemma 6.1, (iii).
Remark 6.6.1. One veriﬁes easily that the technique used in the proof of Theorem
6.6 may also be applied to give another proof of Theorem 6.5, (iii) [i.e., without
resorting to the theory of §3].
Now that we have the tempered versions — i.e., Theorems 6.4, 6.5 — of [Mzk8],
Theorems 1.2, 1.3, the theory of [Mzk8], §2, concerning the category of dominantlocalizations DLocK (XK ) [cf. loc. cit.] generalizes in a fairly straightforward
fashion to the tempered case:First, we deﬁne the category
DLocGK (Πtemp
XK )as follows: An object of this category is a surjection of tempered groups
HJ

74

SHINICHI MOCHIZUKI

where H ⊆ Πtemp
is an open subgroup of ﬁnite index; J is the quotient of H
XK
by the closed normal subgroup generated by some collection of cuspidal geometricdecomposition groups;
and we assume that J is “hyperbolic”, in the sense that theimage of Δtemp
Hin
J is nonabelian. Given two objects Hi  Ji , where i = 1, 2,
Xof this category, a morphism in this category is deﬁned to be a diagram of the form
H1
⏐
⏐

J1

H2
⏐
⏐

−→

J2

where the vertical morphisms are the given morphisms, and the horizontal morphism is an outer homomorphism of DOF-type that is compatible with the various
natural [open] outer homomorphisms from the Hi , Ji to GK .
Next, letDx ⊆ Πtemp
XK
be a decomposition group associated to some closed point x ∈ X K .
Deﬁnition 6.7. We shall say that x or Dx is of tempered DLoc-type if Dx admits
an open subgroup that arises as the image via a morphism Z → XK of DLocK (XK ).
of some cuspidal decomposition group of Πtemp
Z
Theorem 6.8. (Tempered Group-theoreticity of the Category of Dominant Localizations) Let K, L be ﬁnite extensions of Qp ; XK (respectively, YL )a hyperbolic curve over K (respectively, L). Then:(i) The tempered fundamental group functor determines equivalences of categories∼∼

DLocK (XK ) → DLocGK (Πtemp
XK );DLocL (YL ) → DLocGL (Πtemp
YL )(ii) Every isomorphism of tempered groups∼

temp
α : Πtemp
XK → ΠYL

induces an equivalence of categories∼

temp
DLocGK (Πtemp
XK ) → DLocGL (ΠYL )hence also [by applying the equivalences of (i)] an equivalence of categories∼

DLocK (XK ) → DLocL (YL )in a fashion that is functorial, up to unique isomorphisms of equivalences of categories, with respect to α. Moreover, α preserves the decomposition groups oftempered DLoc-type.

SEMI-GRAPHS OF ANABELIOIDS

75(iii) In the situation of (ii) above, suppose further that XK , YL are oncepunctured elliptic curves. Then α preserves the decomposition groups of the“torsion closed points” — i.e., the closed points that arise from torsion pointsof the underlying elliptic curve. Moreover, the resulting bijection between torsion
closed points of XK , YL is compatible with the isomorphism on abelianizations of∼abgeometric fundamental groups Δab
X → ΔY — i.e., “Tate modules” — induced byα.
(iv) In the situation of (ii) above, suppose further that XK , YL are isogenous[cf. §0] to hyperbolic curves of genus zero. Then the isomorphism α preserves thedecomposition groups of the algebraic closed points. In particular, XK is deﬁned
over a number ﬁeld if and only if YL is.
Proof. In light of Theorems 6.4, 6.5, the present Theorem 6.8 follows by exactly
the same arguments as those applied in [Mzk8] to prove [Mzk8], Theorem 2.3;
Corollaries 2.5, 2.6, 2.8. 
Remark 6.8.1. Just as in the case of [Mzk8], Corollaries 2.6, 2.8, the proofs ofTheorem 6.8, (iii), (iv), only require the isomorphism version of Theorem 6.4 [cf.
[Mzk8], Remark 2.8.1].
Corollary 6.9.
(Tempered Absoluteness of Decomposition Groups for
Genus Zero) In the situation of Theorem 6.8, (iv), suppose further both XK and
YL are deﬁned over a number ﬁeld. Then the isomorphism α preserves thedecomposition groups of all the closed points.
Proof. Corollary 6.9 follows from Theorem 6.8, (iv), by applying a similar argument to the argument used in the proof of [Mzk8], Corollary 3.2. In the present
tempered case, one must therefore verify the tempered analogue of [Mzk8], Lemma
3.1. We do this as follows: First, we choose a sequence of characteristic open
subgroups [cf., e.g., [André], Lemma 6.1.2, (i)]. . . ⊆ Δtemp[j + 1] ⊆ Δtemp[j] ⊆ . . . ⊆ Δtemp
X
X
X[where j ranges over the positive integers] of Δtemp
such that the Δtemp[j] form aXX
tempbase of the topology of Δtemp
.
Next,
let
us
observe
that
if
H
⊆
Δ
is
a
characterXX
istic open subgroup of ﬁnite index, then the open subgroup of H determined by the
combinatorial universal covering of the dual graph of the geometric special ﬁber ofthe covering corresponding to H is still characteristic [cf. [André], Lemma 6.1.1, or,indeed, Corollary 3.11 of the present paper]. Thus, we may assume, without lossof generality, that [for all positive integers j] the dual graph Gj of the [geometric][j] isspecial ﬁber of each of the tempered covering of XK corresponding to Δtemp
X
a tree. In particular, given any section
σ : GK → Πtemp
XK

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SHINICHI MOCHIZUKI

we obtain open subgroups
def

temp
Πtemp
[j] ⊆ Πtemp
XK
XK [j,σ] = Im(σ) · ΔX

[where Im(σ) denotes the image of σ in Πtemp
XK ] corresponding to a tower of tempered
coverings of XK :. . . → XK [j + 1, σ] → XK [j, σ] → . . . → XK
Also, we observe that the natural action of Πtemp
XK on the tree Gj factors throughtemp
ΔX[j] .
Now suppose that Im(σ) is not contained in any cuspidal decomposition groupof Πtemp
XK . Then the following conditions on σ are equivalent:(i) σ arises from a point x ∈ XK (K) [i.e., “Im(σ) = Dx ”].
(ii) For every integer j ≥ 1, XK [j, σ](K) = ∅.
(iii) For every integer j ≥ 1, XK [j, σ](K)alg = ∅ [where the superscript “alg”
denotes the subset of algebraic K-rational points, i.e., K-rational points that map
to algebraic points of XK (K)].
(iv) For every integer j ≥ 1, Πtemp
XK [j,σ] contains a decomposition group [i.e.,

relative to Πtemp
XK ] of an algebraic closed point of XK that surjects onto GK .

Indeed, the implications (i) =⇒ (ii); (iii) =⇒ (ii), (iv); and (iv) =⇒ (iii) follow
formally as in the proof of [Mzk8], Lemma 3.1. Moreover, the implication (ii) =⇒(iii) — i.e., “approximation via Krasner’s lemma” [cf. the proof of [Mzk8], Lemma
3.1] — follows as in loc. cit., since given any point xj ∈ XK [j, σ](K) with imagex ∈ XK (K), the completion at [the OK -valued point determined by] xj of thenormalization in XK [j, σ] of some proper model of XK over OK is ﬁnite over thecompletion of this proper model at [the OK -valued point determined by] x.
Finally, we consider the implication (ii) =⇒ (i). In the case of loc. cit., this
implication followed formally from the fact that the topological space

X K [j, σ](K)j≥1

was [in the case of loc. cit.] manifestly compact. In the present tempered case,although this compactness is not immediate, we may nevertheless conclude, atleast for some coﬁnal set of j, the compactness of X K [j, σ](K) by observing thatthe points of X K [j, σ](K) always determine components [i.e., vertices or edges] of
temp
temp
Gj that are ﬁxed by the natural action of the image Im(Πtemp
XK [j,σ] ) ⊆ ΠXK /ΔX[j] ,i.e., by the natural action of GK on Gj via σ. On the other hand, it follows from
our assumption that the Gj are trees [cf. Theorem 3.7, Theorem 5.4, and their
proofs; Lemma 1.8, (ii)] that, at least for some coﬁnal set of j, this set of ﬁxed

SEMI-GRAPHS OF ANABELIOIDS

77

components of Gj is ﬁnite, thus implying the desired compactness of X K [j, σ](K).
Remark 6.9.1. As observed in Remark 3.7.1, the argument used in the ﬁnal
portion of the proof of Corollary 6.9 is reminiscent of the argument used in the“discrete real section conjecture” of [Mzk5], §3.2. This is interesting since Corollary
6.9 itself may be regarded as a weak form of the “section conjecture” for Πtemp
XK 
GK [i.e., roughly speaking, the assertion that all sections of this surjection arise
geometrically]. This state of aﬀairs suggests that:
Perhaps it is more natural to consider the section conjecture for the tempered fundamental group Πtemp
XK than for its proﬁnite completion ΠXK .
Indeed, if one considers the section conjecture for the tempered fundamental group
Πtemp
XK , it seems natural to expect that the theory of arithmetically maximal compact
subgroups discussed in §5 could provide useful insights that are not available in the
proﬁnite case.
Next, letDx ⊆ Πtemp
XK
be a decomposition group associated to some closed point x ∈ X K (K). Then one
has tempered analogues of the various notions of “absoluteness” given in [Mzk8],
Deﬁnition 4.1, (iv); [Mzk8], Deﬁnition 4.8 — which we denote by means of a preﬁx“temp-”. Observe that Dx also forms a “proﬁnite Dx ⊆ ΠXK ” in the sense of[Mzk8]. Moreover, since ﬁnitely generated free groups are well-known to be “good”[i.e., the natural map from the cohomology of the proﬁnite completion of this groupwith coeﬃcients in a ﬁnite [i.e., as a set] module to the cohomology of the original
group in the same module is an isomorphism], it is immediate that Πtemp
XK is also
good. We thus conclude [cf. Theorem 6.6] the following:
Corollary 6.10.

(Tempered Absoluteness)(i) The point x is a discretely absolute cusp (respectively, an integrally
absolute cusp) if and only if it is a discretely temp-absolute cusp (respectively, an integrally temp-absolute cusp).
(ii) The hyperbolic curve XK is unitwise absolute if and only if it is unitwise temp-absolute.
Corollary 6.11.
(Unitwise and Integral Temp-absoluteness for Genus
Zero) Let XK be a hyperbolic curve over K, with stable reduction over OK ,which is isogenous to a hyperbolic curve of genus zero. Then XK is unitwise
temp-absolute, and every cusp of XK is integrally temp-absolute.

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SHINICHI MOCHIZUKI

Proof. Indeed, this follows by formally “substituting” the equivalences of Corollary
6.10, (i), (ii), into [Mzk8], Corollary 4.11. 
Finally, we also observe that it is immediate that the tempered analogue of[Mzk8], Theorem 4.3, holds:
Theorem 6.12. (Rigidity of Cuspidal Geometric Decomposition Groups)
In the notation of Theorem 6.8, (ii), suppose that α induces isomorphisms∼

Ix → Iy ;∼μZ (K) → μZ (L)where x ∈ X K (K) (respectively, y ∈ Y L (L)) is a cusp; μZ (−) is as in [Mzk8],
Theorem 4.3. Then these isomorphisms are compatible with the natural iso∼∼morphisms μZ (K) → Ix ; μZ (L) → Iy .
Remark 6.12.1. Note that unlike Corollaries 6.10, 6.11; Theorem 6.12, results
such as Theorem 6.8, (iii), (iv); Corollary 6.9 do not follow formally from their
proﬁnite analogues, since, in the latter case, it is by no means clear that any ΠXK conjugate of the decomposition group of a closed point that happens to be containedtemp
in Πtemp
XK ⊆ ΠXK is necessarily a ΠXK -conjugate of the decomposition group of aclosed point.

SEMI-GRAPHS OF ANABELIOIDS

79

Appendix: Quasi-temperoids
In this Appendix, we discuss a certain minor generalization of the notion of atemperoid introduced in §3.
Let T be a temperoid. If A is an object of T , write π0 (A) for the set ofconnected components of A andT [A] ⊆ T
for the full subcategory determined by the objects of T that admit a morphism toA [cf §0]. This Appendix is devoted to the resolution [cf. Theorem A.4 below] of
the following question:
Given the abstract category T [A], is it possible to reconstruct the category
T category-theoretically from T [A]?
This sort of issue occurs naturally, for instance, in the study of cores of anabelioids,
and, indeed, is addressed implicitly in [Mzk4], Proposition 2.1.1, (iii). Since, however, this sort of technical issue is a bit more technically complicated in the caseof temperoids than in the case of anabelioids, and, moreover, even in the case ofanabelioids, this sort of issue is not addressed explicitly or in detail in [Mzk4], it
seemed appropriate to the author to give the details in the present Appendix of
how this sort of issue may be resolved.
Deﬁnition A.1.
(i) Any category equivalent to a category of the form
T [A]— where A is a connected object, and T is a connected temperoid — will be referred
to as a connected quasi-temperoid.
(ii) A category equivalent to a product [in the sense of a product of categories]
of a countable [hence possibly empty!] collection of connected quasi-temperoids
will be referred to as a quasi-temperoid. An object A of a quasi-temperoid Q will
be called nondegenerate if, for every connected object B of Q, there exist arrows
C → B, C → A, for some connected object C of Q.
(iii) Let Q1 , Q2 be quasi-temperoids. Then a quasi-morphism φ : Q1 → Q2
is deﬁned to be a functor φ∗ : Q2 → Q1 that preserves ﬁnite limits and countable
colimits. A quasi-morphism φ will be called rigid (respectively, a morphism) if thefunctor φ∗ is rigid [cf. §0] (respectively, preserves nondegenerate objects).
Remark A.1.1. One veriﬁes immediately that a quasi-temperoid is an almost
totally epimorphic category of countably connected type [cf. §0].

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Remark A.1.2. Unlike the situation with temperoids, a quasi-temperoid does
not, in general, admit a terminal object. Thus, it is not in general the case that aquasi-morphism of quasi-temperoids is necessarily a morphism. Indeed, if Q is aconnected quasi-temperoid that does not admit a terminal object, then one veriﬁes
immediately that the functor Q → Q that maps all objects of Q to some empty
object of Q preserves ﬁnite limits and countable colimits, but fails to preserve
nondegenerate objects.
Proposition A.2. (Connected Components of Quasi-temperoids) Let E,
E  be countable sets; for each e ∈ E (respectively, e ∈ E  ), let Qe (respectively,Qe ) be a connected quasi-temperoid; set:
def

Q =



Qe ;

Q =

def



Qe

e ∈E e∈E

Also, let φ : Q → Q be a quasi-morphism of quasi-temperoids. Then:(i) For each e ∈ E, the natural projection functorπe∗ : Q → Qe
determines a morphism of quasi-temperoids Qe → Q.
(ii) For each e ∈ E, write

ιe : Qe → Q

for the natural inclusion functor [i.e., the functor whose composite with πf∗ , wheref ∈ E, maps all objects of Qe to empty [i.e., initial] objects of Qf if f = e, andis the identity if f = e]. If  is a connected component of the full subcategory of
connected objects Q0 ⊆ Q [cf. §0], then writeQ ⊆ Qfor the full subcategory determined by the objects A of Q such that all of the connected components of A belong to . Then the essential image of ιe is equal to Q
for a unique , and, moreover, the resulting correspondencee → 
determines a bijection between E and the set of connected components of Q0 .
(iii) The nondegenerate objects of Q are precisely the objects each of whose
component objects ∈ Ob(Qe ) is nonempty.
(iv) If φ is a morphism of quasi-temperoids, and both E and E  are ofcardinality one, then the functor φ∗ is faithful.
(v) If φ is a morphism of quasi-temperoids, then φ induces a map ψ :E → E  , and, for each e ∈ E, a morphism of quasi-temperoids φe : Qe → Qψ(e)SEMI-GRAPHS OF ANABELIOIDS

81

such that φ coincides with the morphism of quasi-temperoids formed by “taking the
product” [in the evident sense] of the φe .
Proof. Assertion (i) (respectively, (iii)) follows immediately from the deﬁnitions(respectively, and assertion (ii)). To prove assertion (ii), we argue as follows: By
unraveling the deﬁnitions, one veriﬁes immediately that every connected object of
Q lies in the essential image of a unique ιe , and that two connected objects of Q lie
in the essential image of the same ιe if and only if they belong to the same connectedcomponent of Q0 . Now assertion (ii) follows formally from these observations.
Next, we verify assertion (iv). First, in light of Remark A.1.1, it suﬃces tocheck faithfulness on arrows A → B between connected objects A, B of Q . But
since A, B are connected, it follows immediately that there exists a connected
object C of Q such that the products A × C, B × C split as coproducts of copies
of C. Since A, B, C are connected [hence nonempty], they are nondegenerate [cf.
assertion (iii)]. Moreover, again by Remark A.1.1, arrows A → B are represented
faithfully by the maps π0 (A × C) → π0 (B × C) they induce. But, since φ∗ (A),φ∗ (B), φ∗ (C) are nondegenerate, hence, in particular, nonempty, this implies thatφ∗ itself is faithful, as desired.
Finally, we verify assertion (v). Let us consider the various composite functorsdefκ[e , e] = πe∗ ◦ φ∗ ◦ ιe : Qe → Qe
where e ∈ E, e ∈ E  . Now observe that, if we ﬁx e and allow e to vary, then since
the product of objects belonging to distinct Q ’s will always be an empty objectof Q , it follows from the fact that πe , φ are quasi-morphisms of quasi-temperoids[together with the easily veriﬁed observation that any product of nonempty objectsof a connected quasi-temperoid will always be nonempty] that there is at most onee ∈ E  such that the essential image of κ[e , e] contains nonempty objects. On theother hand, since πe∗ ◦ φ∗ preserves nondegenerate objects, it follows [cf. assertion(iii)] that there exists at least one e ∈ E  such that the essential image of κ[e , e]contains nonempty objects. Thus, in summary, for each e ∈ E, there exists a uniquee ∈ E  such that the essential image of κ[e , e] contains nonempty objects; setdefψ(e) = e . Then it follows immediately again from the fact that the functor πe∗ ◦ φ∗preserves nondegenerate objects that the functor κ[ψ(e), e] preserves nondegenerate
objects, hence determines a morphism of quasi-temperoids Qe → Qψ(e) . Moreover,unraveling the deﬁnitions, we see that the remainder of assertion (v) follows formallyfrom what we have done thus far. Remark A.2.1. Thus, Proposition A.2 serves, in eﬀect, to reduce the theory ofarbitrary quasi-temperoids to the theory of connected quasi-temperoids.
Deﬁnition A.3.

Let Q be a connected quasi-temperoid. Then:(i) Any pair (A, ΓA ), where A is an object of Q, and ΓA ⊆ AutQ (A) is a
subgroup, will be referred to as a QD- [or quotient data] pair [of Q]. If ΓA acts

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SHINICHI MOCHIZUKI

transitively on π0 (A), then we shall say that this QD-pair is weakly connected; if
A is connected, then we shall say that this QD-pair is strongly connected. [Thus,
every strongly connected QD-pair is weakly connected.](ii) A morphism of QD-pairs of Q
(A, ΓA ) → (B, ΓB )is deﬁned to be a morphism φ : A → B such that, for every γA ∈ ΓA , there
exists a γB ∈ ΓB such that γB ◦ φ = φ ◦ γA . If, moreover, φ induces a surjection
π0 (A)  π0 (B), then we shall say that this morphism is 0-proper. Thus, in the 0proper case, it follows from the fact that the category Q is almost totally epimorphic
that γB is unique, hence that the correspondence γA → γB determines an associatedgroup homomorphism ΓA → ΓB .
(iii) Let (A, ΓA ) be a QD-pair. Then we shall say that an arrow φ : A → B of
Q forms a quotient of this QD-pair — and write B ∼= A/ΓA — if the following two
properties are satisﬁed: (a) φ ◦ γA = φ, ∀ γA ∈ ΓA ; (b) for every arrow ψA : A → C
satisfying ψA ◦ γA = ψA , ∀ γA ∈ ΓA , there exists a unique arrow ψB : B → C suchthat ψB ◦ φ = ψA . [Thus, one veriﬁes immediately that the quotient of a QD-pair is
unique, up to unique isomorphism, if it exists, and that the quotient of a QD-pairis connected if and only if the QD-pair is weakly connected.](iv) A 0-proper morphism of QD-pairs of Q
(A, ΓA ) → (B, ΓB )will be called 1-proper if the associated homomorphism ΓA → ΓB is surjective, and,moreover, the arrow A → B forms a quotient of the QD-pair (A, Ker(ΓA  ΓB )).
[Thus, under the 1-properness assumption, one veriﬁes immediately that if B → C
forms a quotient of (B, ΓB ), then the composite arrow A → B → C forms a quotientof (A, ΓA ).]Remark A.3.1.
One veriﬁes immediately that the following conditions on amorphism of QD-pairs (A, ΓA ) → (B, ΓB ) are equivalent:(a) (A, ΓA ) → (B, ΓB ) is 0-proper.
(b) A → B is an epimorphism in Q [cf. Remark A.1.1].
(c) B is the colimit in Q of the diagram formed by the two projections
A ×B A → A.
Moreover, the notion of a “quotient” given in Deﬁnition A.3, (iii), may also be
stated in terms of colimits. In particular, the condition that (A, ΓA ) → (B, ΓB ) be
1-proper may be stated entirely in terms of colimits.
Theorem A.4.
(Connected Quasi-temperoids) For i = 1, 2, let Ti be aconnected temperoid; let Ai be a connected object of Ti ; writedef

λi : Qi = Ti [Ai ] → Ti

SEMI-GRAPHS OF ANABELIOIDS

83

for the natural functor. Then any morphism of quasi-temperoids φ : Q1 → Q2
ﬁts into a 1-commutative diagram
φ

Q2
⏐
⏐
λ2

ψ

T2

Q1 −→
⏐
⏐
 λ1
T1

−→

— where the morphism of [quasi-]temperoids ψ : T1 → T2 that makes this diagram
1-commute is unique, up to unique isomorphism.
Proof. WriteDi
for the category whose objects are QD-pairs of Qi and whose morphisms are morphisms of QD-pairs. Note that it follows immediately from the deﬁnitions that if(B, ΓB ) is a QD-pair of Qi , then the QD-pair λi (B, ΓB ) of Ti admits a quotient in
Ti . Thus, we obtain a natural functor
qi : Di → Ti
given by the assignment (B, ΓB ) → B/ΓB . Note that this functor maps 1-propermorphisms of Di to isomorphisms of Ti . Moreover, one veriﬁes immediately that
this functor qi is essentially surjective; that qi ((B, ΓB )) is connected if and only if(B, ΓB ) is weakly connected; and that every connected object of Ti is isomorphic to
the image via qi of a strongly connected QD-pair.
Thus, to reconstruct Ti from Di , it suﬃces to reconstruct the morphisms
B/ΓB → C/ΓCin Ti between the images via qi of two objects (B, ΓB ), (C, ΓC ) of Di . To this end,
we deﬁnedef

Hom((B, ΓB ), (C, ΓC )) = HomDi ((B, ΓB ), (C, ΓC ))/ΓC[i.e., where ΓC acts by composition from the right], so that the functor qi induces
a natural map:
Hom((B, ΓB ), (C, ΓC )) → HomTi (B/ΓB , C/ΓC )Since Ti is an almost totally epimorphic category countably connected type, it suﬃces
to describe the morphisms between connected objects of Ti .
Next, suppose that (B, ΓB ), (C, ΓC ) are strongly connected QD-pairs. Thenobserve that the functor qi induces an injection:
Hom((B, ΓB ), (C, ΓC )) → HomTi (B/ΓB , C/ΓC )84

SHINICHI MOCHIZUKI

Indeed, this follows immediately by considering the natural splitting [in Qi or Ti ]of C ×C/ΓC C into a coproduct of copies of C indexed by ΓC [together with thefact that (B, ΓB ), (C, ΓC ) are strongly connected QD-pairs]. Moreover, one veriﬁes
immediately that every element of HomTi (B/ΓB , C/ΓC ) arises from some morphismof strongly connected QD-pairs(B  , ΓB  ) → (C, ΓC )where (B  , ΓB  ) → (B, ΓB ) is a 1-proper morphism of strongly connected QD-pairs[so we obtain a morphism B/ΓB → C/ΓC by composing the induced morphism∼
B  /ΓB  → C/ΓC with the inverse of the induced isomorphism B  /ΓB  → B/ΓB ].
Now one veriﬁes easily that any two 1-proper morphisms of strongly connected QDpairs (B  , ΓB  ) → (B, ΓB ), (B  , ΓB  ) → (B, ΓB ) ﬁt into a commutative diagram of
1-proper morphisms of strongly connected QD-pairs of Di :(B  , ΓB  ) −→⏐⏐

(B  , ΓB  )⏐⏐

(B  , ΓB  )(B, ΓB )−→Thus, we conclude that HomTi (B/ΓB , C/ΓC ) may be reconstructed as the following
ﬁltered inductive limit:
def

Hom((B, ΓB ), (C, ΓC )) = lim Hom((B  , ΓB  ), (C, ΓC ))−→[i.e., over 1-proper morphisms of strongly connected QD-pairs (B  , ΓB  ) → (B, ΓB )and transition morphisms (B  , ΓB  ) → (B  , ΓB  ) over (B, ΓB )]. Moreover, one veriﬁes immediately that this reconstruction is compatible with composition of arrows[i.e., composition of arrows in Di induces composition of “Hom-arrows”].
Next, suppose that (B, ΓB ), (C, ΓC ) are weakly connected QD-pairs. Then observe that each connected component B  of B determines a strongly connected QDpair (B  , ΓB  ) [i.e., where we take ΓB  ⊆ ΓB to be the subgroup of automorphismsthat ﬁx the element [B  ] ∈ π0 (B) determined by B  ] such that qi ((B , ΓB  )) ∼=qi ((B, ΓB )); a similar statement holds for (C, ΓC ). Moreover, if B , B (respectively, C  , C  ) are connected components of B (respectively, C), then one veriﬁes
immediately that any choice of elements γB ∈ ΓB , γC ∈ ΓC such that γB (B  ) = B  ,γC (C  ) = C  determines a bijection∼

Hom((B  , ΓB  ), (C  , ΓC  )) → Hom((B , ΓB  ), (C  , ΓC  ))which is, in fact, independent of the choice of γB , γC . Thus, if we deﬁne
Hom((B  , ΓB  ), (C  , ΓC  ))Hom((B, ΓB ), (C, ΓC )) ⊆B  ,C [where the product ranges over all choices of connected components B  , C  of B, C,
respectively] to be the subset of collections of elements that correspond via theseSEMI-GRAPHS OF ANABELIOIDS

85

bijections, then the natural projections of this direct product determine bijections
as follows:∼Hom((B, ΓB ), (C, ΓC )) → Hom((B  , ΓB  ), (C  , ΓC  ))∼→ HomTi (qi ((B , ΓB  )), qi ((C  , ΓC  )))∼→ HomTi (qi ((B, ΓB )), qi ((C, ΓC )))Moreover, it follows immediately from the deﬁnitions that we have a natural map:
Hom((B, ΓB ), (C, ΓC )) → Hom((B, ΓB ), (C, ΓC ))Finally, as observed above, since Ti is an almost totally epimorphic category countably connected type, the deﬁnition of “Hom”, as well as the resulting bijection of“Hom” with “HomTi ” and the natural map from “Hom” to “Hom” extend immediately to pairs of objects of Di that are not necessarily weakly connected.
Thus, in summary, if we writePi
for the category whose objects are the objects of Di and whose morphisms are givenby the “Hom’s”, then we obtain natural functors∼

Qi → Di → Pi → Ti
— i.e., the ﬁrst functor maps an object B of Qi to the QD-pair (B, {1}); the second
functor arises from the construction of Pi ; the third functor is the equivalence
induced by the natural functor Di → Ti considered above.
Now observe that the functor φ∗ : Q2 → Q1 induces a 1-commutative diagram
Q2 −→ D2
⏐
⏐
⏐ ∗
⏐
φ


−→

P2
⏐
⏐


−→ D1

−→

P1

Q1

Indeed, the construction of the second vertical arrow is immediate from the deﬁnitions. The construction of the third vertical arrow follows by observing that sinceφ∗ preserves countable colimits, it follows that the functor D2 → D1 preserves 0and 1-proper morphisms [cf. Remark A.3.1]. Thus, by combining this diagram with∼the equivalences Pi → Ti , we obtain a diagram as in the statement of Theorem A.4.
The fact that the resulting functor ψ ∗ : T2 → T1 preserves countable colimits(respectively, ﬁbered products) follows by a routine argument from the fact thatφ∗ preserves countable colimits (respectively, countable colimits and ﬁnite limits).
Thus, to show that ψ is a morphism of temperoids [i.e., that ψ ∗ preserves ﬁnite
limits], it suﬃces to show that ψ ∗ preserves terminal objects. Now let B be aconnected object of Q2 such that B × B splits as a coproduct of copies of B [i.e., inother words, B is a connected Galois object of T2 that admits a morphism in T2 to
A2 ]. Note that such a B always exists. Then if we let Aut(B) act on, say, the second

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SHINICHI MOCHIZUKI

factor of B × B, then the resulting QD-pair (B × B, Aut(B)) maps via φ∗ to a QDpair (φ∗ (B) × φ∗ (B), φ∗ (Aut(B))) of Q1 . Moreover, since the ﬁrst projection B ×
B → B forms a quotient of the former QD-pair, it follows that the ﬁrst projectionφ∗ (B) × φ∗ (B) → φ∗ (B) forms a quotient of the latter QD-pair. It thus follows
immediately that φ∗ (Aut(B)) acts transitively on π0 (φ∗ (B)). Next, observe thatB × B is a coproduct of copies of B. Thus, we conclude that φ∗ (B) × φ∗ (B) isa coproduct of copies of φ∗ (B). But this implies that the connected componentsof φ∗ (B), all of which are isomorphic to one another [by the transitivity observed
above], form Galois objects of T1 . Thus, the preceding observation concerning thequotient of the QD-pair in Q1 implies that the stabilizer in φ∗ (Aut(B)) of any
connected component φ∗ (B) is necessarily equal to the entire automorphism groupof this connected component. Moreover, since φ∗ preserves nondegenerate objects, itfollows that φ∗ (B) is nonempty [i.e., φ∗ (B) has at least one connected component].
Thus, in summary, φ∗ maps the QD-pair (B, Aut(B)) [any quotient of which forms
a terminal object in T2 ] to a QD-pair (φ∗ (B), φ∗ (Aut(B))) any quotient of whichforms a terminal object in T1 . That is to say, we have shown that ψ ∗ preserves
terminal objects, as desired.
Finally, the asserted uniqueness of ψ [up to unique isomorphism] follows immediately from the fact that arbitrary objects of Ti may be obtained as quotientsof QD-pairs of Qi . 
Remark A.4.1. Thus, Theorem A.4 serves, in eﬀect, to reduce the theory ofarbitrary connected quasi-temperoids to the theory of connected temperoids.
Remark A.4.2. One veriﬁes immediately that, by replacing the term “temperoid” by the term “anabelioid” [and the terms “countable/countably” by the terms“ﬁnite/ﬁnitely”], one obtains an entirely analogous [but, in fact, slightly easier]theory of “quasi-anabelioids” to the theory of quasi-temperoids developed above.
We leave the routine details to the reader.

SEMI-GRAPHS OF ANABELIOIDS

Index

0-commutative, §0, Categories
0-proper, A.3, ii
1-commutative, §0, Categories
1-proper, A.3, iv
abstract equivalence, §0, Categories
abut, §1
action, 5.1, ialmost totally epimorphic, §0, Categories
aloof (totally), 2.4, iv
anabelioids, before 2.1
approximator, 2.3, ii
arithmetic action, 5.1, ii
arithmetically ample, 5.3, iarithmetically estranged (totally), 5.3, ii
arithmetically maximal, 5.3, iarithmetic component, 5.1, ii
arithmetic semi-graph of anabelioids(connected, connected component of, geometric component of), 5.1, ii, iii
BC (base of constants), 5.1, iv
bounded order, 2.3, ibranch, §1category of dominant localizations, after 6.6.1
category of localizations, after 4.1.2
category of quasi-connected objects, §0, Categories
centralizer, §0, Topological Groups
closed edge, §1closed object, after 4.1.2
coherent, 2.3, iii
coincidence map, §1
combinatorial universal covering, after 3.5.2
commensurably terminal, §0, Topological Groupscommensurator, §0, Topological Groups
compactiﬁcation (of a curve), §0, Curves
compactiﬁcation of a semi-graph, §1component of a semi-graph, §1connected (component of a) category, §0, Categories
connected semi-graph of anabelioids, after 2.1
constituent anabelioids, 2.1
continuous action, 5.1, icountable semi-graph, §1countable semi-graph of anabelioids, after 3.4.2
countably connected type, §0, Categories

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covering of a semi-graph of anabelioids, 3.5, icoverticial edges, §1
cuspidal (geometric) decomposition group, after 6.4
DFG-type, 6.2, i, iii
divisor at inﬁnity, §0, Curves
divisor of cusps, §0, Curves
divisor of marked points, §0, Curves
DOF-type, 6.2, ii, iii
edge, §1edge-like subgroup, 3.7, iii
edge-like subgroup (arithmetic case), 5.3, iii
edge-wise length, 4.2, ielevated (totally), 2.4, iembedding (of semi-graphs), §1
embedding (of semi-graphs of anabelioids), 4.1, ii
estranged (totally), 2.4, iv
étale morphism of temperoids, 3.4, i
excision (of semi-graphs), §1
excision (of semi-graphs of anabelioids), 4.1, ii
family of (hyperbolic) curves, §0, Curves
ﬁnite étale covering of semi-graph of anabelioids, 2.2, i
ﬁnitely connected type, §0, Categories
ﬁnite open object, after 4.1.2
ﬁnite semi-graph, §1
Galois covering of a semi-graph of anabelioids, 3.5, iii
Galois object (of a temperoid), 3.1, iv
G-closed, 4.1.1
generalized morphism of semi-graphs of anabelioids, 2.11
geometric component, 5.1, ii(G, Γ)-structure, 4.1, iii
G-open, 4.1.1
graph, §1graph-covering (ﬁnite, Galois), §1
graph-localization morphism (strict), 4.2, ii
immersion (of semi-graphs), §1
immersion (of semi-graphs of anabelioids), 4.1, ii
immobile, §0, Categories
indissectible , §0, Categories
inﬁnite open object, after 4.1.2
injective type, 2.1
inner action, 5.1, i
iso-excisive, 4.1, iii, iv
iso-immersive, 4.1, iii, iv
isolated edge, §1isomorphism between functors, §0, Categories

SEMI-GRAPHS OF ANABELIOIDS

joint, §1link, 4.2, i
local (G, Γ)-structure, 4.1, iv
locally ﬁnite étale morphism of semi-graphs of anabelioids, 2.2, ii
locally ﬁnite semi-graph, §1
locally open morphism of semi-graphs of anabelioids, 2.2, ii
locally trivial morphism of semi-graphs of anabelioids, 2.2, ii
log curve (stable, smooth), §0, Curves
maximal subgraph, §1minimal-adjoint, §0, Categories
mobile, §0, Categories
moduli stack, §0, Curves
morphism of arithmetic semi-graphs of anabelioids (properties of), 5.1, iv
morphism of QD-pairs, A.3, ii
morphism of quasi-temperoids, A.1, iii
morphism of (semi-)graphs of anabelioids, 2.1
morphism of semi-graphs, §1morphism of temperoids, 3.1, iii
NL-morphism, 4.2, inondegenerate object, A.1, ii
normalizer, §0, Topological Groups
nuclear object, 4.2, iopen edge, §1
open object, after 4.1.2
π1 -epimorphic approximator, 2.3, ii
π1 -monomorphism, 2.1
piecewise faithfully, 4.1, iproper morphism of semi-graphs, §1properties of an arithmetic semi-graph of anabelioids, 5.1, iii
pro-Σ-completion, 2.9, ii
pro-temperoids, 3.4.2
QD-pair, A.3, i
quasi-anabelioid, A.4.2
quasi-coherent, 2.3, iii
quasi-connected, §0, Categories
quasi-geometric, 3.8
quasi-morphism of quasi-temperoids, A.1, iii
quasi-temperoid (connected), A.1, i, ii
quotient of a QD-pair, A.3, iii
rigid, §0, Categories
semi-graph, §1(semi-)graph of anabelioids, 2.1
Σ-integer, 2.9, i

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slim, §0, Categories
strongly connected, A.3, istructure morphism, 4.1, iii
sub-coverticial (totally, universally), 2.4, iii
subjoint, §1
sub-semi-graph, §1
temp-, after 6.9.1
tempered covering of a semi-graph of anabelioids, 3.5, ii
tempered DLoc-type, 6.7
tempered fundamental group of a temperoid, 3.2.1
tempered object, after 4.1.2
tempered topological group, 3.1, itemperiﬁcation, 3.1.4
temperoid, 3.1, ii
temp-slim (relatively), 3.4, ii
topological space associated to a semi-graph, §1
totally epimorphic, §0, Categories
tree, §1
universal graph-covering, §1
universal pro-ﬁnite étale covering, 2.2.1
untangled semi-graph, §1vertex, §1
verticial cardinality, §1
verticial degree, 4.2, iverticial length, 4.2, iverticially iso-excisive, 4.1, iv
verticially slim, 2.4, ii
verticial morphism, 4.2, iverticial portion, §1
verticial restriction, §1
verticial subgroup, 3.7, iverticial subgroup (arithmetic case), 5.3, iii
weakly connected, A.3, iSEMI-GRAPHS OF ANABELIOIDS

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Research Institute for Mathematical Sciences
Kyoto University
Kyoto 606-8502, Japan
Fax: 075-753-7276
motizuki@kurims.kyoto-u.ac.jp