RESPONSES TO QUESTIONS ON FROBENIOIDS

Shinichi Mochizuki
December 2015

The Geometry of Frobenioids I: The General Theory ([FrdI])
Question 1: Deﬁnition 1.2, (i): Since the validity of the condition that two arrows
be “metrically equivalent” depends solely on the images via “Div” of the two arrows, might it not be better to call such arrows “Div-equivalent” and then, to avoid
confusion, to use the term “Φ-equivalent” for arrows that, in the current terminology, are called “Div-equivalent”?
Response 1:
First of all, the terminology “Φ-equivalent” is not very good since it depends on
the notation “Φ” for a speciﬁc monoid on a base category. The sense underlying
the terminology “base-equivalent”/“Div-equivalent” is the idea that the pair of morphisms induce the same morphism on the base object/collection of divisors of thebase object. That is to say, these two terms (i.e., base-/Div-equivalent) are mostnaturally regarded as being “parallel”. Thus, the term “divisor-equivalent” would
be one reasonable alternative to “Div-equivalent”. By contrast, the sense underlying the terminology “metrically equivalent”/“isometric” is that it corresponds, inthe case of Frobenioids that arise from metrized line bundles, to the idea that themorphism(s) involved induce(s) the same/no discrepancy in metrics. That is tosay, from the point of view of maintaining the close relationship to the term “isometric”, it does not seem to me that it would be desirable to replace “metricallyequivalent” by “Div-equivalent”.
Question 2: Deﬁnition 1.2, (ii): Are there any other equivalent deﬁnitions of thenotion of a pull-back morphism?
Response 2:
I am not aware of any natural equivalent deﬁnitions. It seems that the deﬁnition given is suﬃciently natural, relative to the way in which the term “pull-backmorphism” is typically applied in arithmetic geometry.
Question 3: Deﬁnition 1.2, (iii): Is it true that, if one restricts one’s attention to
the portion of the theory of Frobenioids that is used in the IUTeich papers, thenthe only Frobenioids in which there exist morphisms that are not co-angular are
the archimedean Frobenioids?
Response 3:
Yes. Indeed, the only Frobenioids that are used in the IUTeich papers are thefollowing:
Typeset by AMS-TEX

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SHINICHI MOCHIZUKI(F1) tempered Frobenioids, i.e., a generalization developed in [EtTh], §3, §4,§5, of the geometric example given in [FrdI], Example 6.1, to the case of
the tempered coverings that arise in the theory of the theta function;(F2) Frobenioids associated to number ﬁelds as in [FrdI], Example 6.3 (cf.,e.g., [IUTchIII], Example 3.6), together with the realiﬁcations associatedto (certain of) such Frobenioids;(F3) certain special cases of the p-adic Frobenioids discussed in [FrdII], Example 1.1 (cf., e.g., the Frobenioids “Cv ”, “Cv ” considered in [IUTchI],
Example 3.3, (i), which also appear in R11 and R13 — i.e., the responses
to questions Q11 and Q13 — below), together with various related Frobenioids obtained by forming associated realiﬁcations or by forming thequotient “O×μ = O× /Oμ ” of the units “O× ” by the torsion subgroup“Oμ ⊆ O× ”;(F4) copies of the archimedean Frobenioid discussed in [FrdII], Example 3.3,(ii) (i.e., the Frobenioid denoted “C”).
Here, we note that (F3) and (F4) are inessential since a Frobenioid as in (F3)essentially amounts to (i.e., may be replaced by) a suitable topological monoid witha continuous action by a topological group, while a Frobenioid as in (F4) essentiallyamounts to a copy of the topological monoid OC (i.e., the multiplicative topologicalmonoid of nonzero complex numbers of absolute value ≤ 1). In this context, it
should, however, be pointed out that, in fact, (F3) is a bit “less inessential” than(F4) since (F3) occurs in the theory of (F1). At any rate, from the point of viewof studying IUTeich,(I1) in [FrdI], one may assume that all Frobenioids are model Frobenioids (cf.
[FrdI], Theorem 5.2, (ii)), which implies, in particular, that every objectof a Frobenioid is isotropic, and that every morphism of a Frobenioid is
co-angular;(I2) one may in fact ignore [FrdII], §3, §4, §5.
Here, we note that (I1) results in a substantial simpliﬁcation of the theory of [FrdI].
Question 4: Deﬁnition 1.2 and the Chart of Types of Morphisms in a Frobenioid atthe end of [FrdI]: The deﬁnitions of many types of morphisms concern properties of1) the projection to the base category, 2) the zero divisor image, 3) the Frobenius
degree. Might it not be easier to keep track of the terms (and, moreover, allow
one to simplify the Chart of Types of Morphisms) if one instead refers to linear
morphisms as 3-morphisms, to isometries as 2-morphisms, to base-isomorphisms
as 1-morphisms, to pre-steps as 13-morphisms, to morphisms of Frobenius type(in light of later results) as co-angular 12-morphisms, to pull-back morphisms (inlight of later results) as co-angular 23-morphisms, and to isometric pre-steps as123-morphisms? Then co-angular 123-morphisms are isomorphisms, etc.
Response 4:
The point of the naming conventions introduced in Deﬁnition 1.2 was precisely
to give “combinatorial”, or “mechanical” (i.e., involving “1”, “2”, and “3”) characterizations of various types of morphisms that typically appear in arithmetic
geometry. That is to say, the terminology was chosen so as to suggest well-known

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situations familiar from conventional arithmetic geometry. The content of the definitions of this terminology was then intended to be a sort of translation of terminology that was close to familiar wording in conventional arithmetic geometry into
combinatorial/mechanical/“123”-type language. If, however, you have a proposal
for a slightly diﬀerent sort of table involving “123”, then you are certainly welcome
to construct such a table.
Question 5: Deﬁnition 1.3 contains 7 parts, some of which consist of subparts.
Might it be possible, if one restricts one’s attention to the applications to IUTeich,to reduce the number of deﬁning properties?
Response 5:
In light of (I1) of R3, properties (iii), (a), (b); (v); and (vii) of Deﬁnition 1.3 may
be omitted.
Question 6: Remark 3.1.2: “the image of 1 ∈ Z≥0 ”: Does this expression in factrefer to “the image of (1, 1) ∈ F”?
Response 6:
Yes. I believe that the intended meaning is clear from the context, but this issue isaddressed in the newly released version of the Comments for [FrdI].
Question 7: Theorem 5.2, (i), (b): Should “to as the projection to D to φ” bereplaced by “to as the projection to D of φ?
Response 7:
Thank you. This is indeed a misprint which is corrected in the newly released
version of the Comments for [FrdI].
Question 8: I have not read the IUTeich papers yet. I wonder if there are Theorems
and Propositions of [FrdI] whose statements hold by obvious reasons for all the
Frobenioids that appear in the papers on IUTeich (and which therefore may beomitted in one’s preparation for studying the IUTeich papers)?
Response 8:
As discussed in R3, once one assumes (I1), the entire theory of [FrdI] become mucheasier. One approach (cf. R3) to presenting the theory of Frobenioids to people
whose main interest is to prepare for studying the IUTeich papers might be(A1) to start by discussing the fundamental example of [FrdI], Deﬁnition 1.1,(iii);(A2) then proceed to discussing the explicit construction of model Frobenioids
as in [FrdI], Theorem 5.2 (cf. also Proposition T in R23 below);(A3) explain how, by specifying appropriate divisor monoids and rational function monoids, one may obtain [FrdI], Examples 6.1, 6.3; [FrdII], Example
1.1, as special cases of the notion of a model Frobenioid;(A4) then proceed to discussing the [FrdI], Corollary 4.11 (which, in fact,
may, from the point of view of studying IUTeich, simply be accepted onfaith!), in the case of model Frobenioids, explaining a few of the main,
representative ideas of the proofs (such as [FrdI], Remark 3.1.2).
In this context, I should perhaps mention that, other than [FrdI], Corollary 4.11,
perhaps the most important property of the theory of Frobenioids that is used in[EtTh], §3, §4, §5 (and hence also in IUTeich) is [FrdI], Proposition 5.6.

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Question 9: Remark 6.3.1: How much is known about the use of Frobenioids tostudy arithmetic line bundles on regular proper models of curves over number ﬁelds?
Should one look at Frobenioids whose base category is a Frobenioid? Or should
one perhaps develop a theory of two-dimensional Frobenioids, where the monoid ofpositive integers is replaced by the monoid of positive elements of Z × Z equippedwith the lexicographical ordering?
Response 9:
Note that in Example 6.1, there is no restriction on the dimension of the variety V .
In particular, the nontriviality of extending Example 6.3 to the case of arithmetic
line bundles on (two-dimensional) regular proper models of curves over number
ﬁelds does not lie in the two-dimensionality of such models. In particular, there is
no issue of being having to contend with “two-dimensional Frobenioids” or monoids
more complicated than N that arise from lexicographic orderings. Rather, thenontriviality (i.e., the “complications” referred to in Remark 6.3.1) of extending
Example 6.3 as discussed in Remark 6.3.1 lies in the issue of how to represent,
in a category-theoretic fashion, the notion of an arbitrary (say, C ∞ -class) metricon the archimedean localization of an arithmetic line bundle. I have not thought
about this issue in much detail, but it appears to involve various nontrivial technicalcomplications. On the other hand, another possibility for further development of thetheory of Frobenioids that is not particularly related to archimedean localizations,but is related to more complicated monoids is the following:
One could try to construct some sort of more general type of Frobenioidthat corresponds to the “tautological system of monoids corresponding tovaluations on Berkovich spaces”.
That is to say, since each point of a Berkovich space corresponds to a valuation,
each such point has a natural monoid (i.e., the monoid determined by the valuation) attached to it. One could then try to consider the abstract topological space(i.e., determined by a Berkovich space) equipped with this tautological system of
monoids, construct some sort of associated category (i.e., in the spirit of the notionof a Frobenioid), and then see to what extent various properties of this category
can be reconstructed category-theoretically, i.e., without using, for instance, somegiven p-adic scheme that gave rise to the Berkovich space.

RESPONSES TO QUESTIONS ON FROBENIOIDS

5The Geometry of Frobenioids II: Poly-Frobenioids ([FrdII])
Question 10: Example 1.1, (i): The object ΦZ0 is not deﬁned in [FrdII]. As a result,in the ﬁrst displayed line of (ii), ΦΛ
0 is not deﬁned for the type Λ = Z, i.e., there is
no explicitly stated restriction there to the eﬀect that Λ = Z.
Response 10:
Thank you. This is indeed a slight oversight in the use of notation which is addressedin the newly released version of the Comments for [FrdII].
Question 11: Example 1.1, (ii): In the third line of this example, is it assumed thatthe output monoprime monoids of Φ are of type Λ?
Response 11:No, this is certainly not assumed. For instance, in the case of the p-adic Frobenioid“Cv ” considered in [IUTchI], Example 3.3, (i), the output monoprime monoids ofΦ are of type Q = Λ = Z.
Question 12: Example 1.1, (ii): What is Φ(K)? Probably “K” is used as a shorthand for “Spec(K)”, an object of D0 , but even then Φ is a functor deﬁned on D,
not on D0 .
Response 12:
Thank you. This is indeed a misprint which is corrected in the newly released
version of the Comments for [FrdII].
Question 13: Example 1.1, (ii): I do not understand the meaning of “absolutely
primitive” and “ﬁeldwise saturated”. (Here, I took into account the correction in
the Comments for [FrdII], replacing Φ(K) by Φ(K)gp in the deﬁnition of “ﬁeldwise
saturated”.) The current deﬁnitions appear to imply the following conclusions: if
Λ = Z (resp. Q, resp. R), then Φ is absolutely primitive iﬀ the output monoprime
monoids of Φ are of type Z (resp. Z or Q, resp. Z or Q or R); Φ is always ﬁeldwise
saturated. Is this what was intended?
Response 13:
First of all, it is true that Φ is always absolutely primitive if Λ = R. Also, you
are correct in asserting that, when Λ = Q, Φ is absolutely primitive if and only
if the output monoprime monoids of Φ are of type ≤ Λ = Q (i.e., relative to theordering on {Z, Q, R} induced by the relation of inclusion). In particular, the notionof being absolutely primitive is only interesting when Λ = Z. On the other hand,
I was entirely unable to understand the remaining cases of your assertion. Writeord(p) ∈ ord(Q×p ) for the element determined by p ∈ Qp . Let n ∈ N≥1 . Consider
the case where D = D0 , and Φ is given by the assignment
Spec(K) → n1 · N · ord(p)(for Spec(K) ∈ Ob(D0 )). (Thus, the output monoprime monoids of Φ are of type
Z.) Then Φ is not ﬁeldwise saturated for any n ∈ N≥1 . (Indeed, if the ramiﬁcationindex eK of K over Qp does not divide n, then ord(K × ) = e1K · Z · ord(p) ⊆1
gp
n · Z · ord(p) = Φ(Spec(K)) .) Moreover, when Λ = Z, it holds that Φ is absolutely
primitive if and only if n = 1. (Indeed, Φ(Spec(K)) = n1 · N · ord(p) ⊆ Z · ord(p) =Λ · ord(Q×p ) if and only if n = 1.) Finally, I should perhaps mention that this sortof Φ (for Λ = Z) in the case n = 1 appears in the case of the p-adic Frobenioid“Cv ” considered in [IUTchI], Example 3.3, (i).

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Question 14: Remark 1.2.2, the last sentence: If one works with idèles, it is notunnatural to consider situations where p is mapped to p times a unit.
Response 14:
Your assertion does not appear to contradict the assertion that the situation considered in Remark 1.2.2 “never arises in conventional scheme theory”. That is to say,“conventional scheme theory” refers to the situation where one only considers morphisms of schemes, i.e., (when the schemes involved are aﬃne, such as Spec(Qp ))ring homomorphisms. The point here is that it is never the case (at least when×u = 1) that an assignment Q×p  p → p · u ∈ Qp of the sort considered here arises
from a ring homomorphism Qp → Qp .
Question 15: Example 1.3, (i): What is B temp (Π)Π/Π0 , i.e., what is the role of thesubindex Π/Π0 ?
Response 15:
This is a special case of the notational convention “CA ” (i.e., where A is an objectof the category C) discussed at the beginning of the discussion of “Categories” in[FrdI], §0.
Question 16: Deﬁnition 2.2, (i), and Example 2.2, (ii), (b), “AD is Galois”: Whatis the deﬁnition of a Galois object? I was unable to ﬁnd the deﬁnition of a Galois
object in [FrdI], [FrdII].
Response 16:
Here, the category “D” under consideration is, as discussed in Example 1.3, (i), a
subcategory of a category (i.e., “temperoid”) of the sort discussed in [SemiAnbd], §3(cf. [SemiAnbd], Deﬁnition 3.1, (ii)). The term “Galois” is deﬁned in [SemiAnbd],
Deﬁnition 3.1, (iv).
Question 17: Deﬁnition 2.2, (i): H is a normal open subgroup of G, and, in particular, it is possible that H = G. Why then is the homomorphism G → GAdetermined up to composition with an inner automorphism, while H → HA is only
well-deﬁned up to composition with conjugation by an element of G?
Response 17:
The arrow H  HA of the third display of Deﬁnition 2.2, (i), is determined precisely by restricting the arrow G  GA of the second display of Deﬁnition 2.2,(i). Thus, the G-conjugacy indeterminacies of the latter arrow G  GA necessarily induce G-conjugacy indeterminacies on the former arrow H  HA . There is
no reason to expect that these indeterminacies may be “reduced” to H-conjugacy
indeterminacies.
Question 18: Deﬁnition 2.2, (ii): I could not ﬁnd any explicit deﬁnition of the action of H on μN (A). Does it act via the homomorphism H → HA ?
Response 18:
Yes. My understanding was that this is implicit in the phrasing “the natural surjective homomorphism H  HA induces isomorphisms” that immediately precedes
the ﬁrst display of Deﬁnition 2.2, (ii).
Question 19: Deﬁnition 2.2, (iii): The wording here seems to imply that the notionsof “ﬁeldwise saturated” and “absolutely primitive” are complementary to one other.
As I wrote in Q13, I was unable to understand the deﬁnitions of these terms.

RESPONSES TO QUESTIONS ON FROBENIOIDS

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Response 19:
The assertion that (in the situation under consideration) if Φ is absolutely primitive,then it is not ﬁeldwise saturated follows immediately from an argument of the sortdiscussed in R13: that is to say, it suﬃces to consider the case where eK = 1, soord(K × ) = e1K · Z · ord(p) ⊆ Φ(Spec(K))gp ⊆ Z · ord(p) = Λ · ord(Q×p ).
Question 20: Beginning of the proof of Theorem 2.4, the second sentence: could
you provide a veriﬁcation of this fact?
Response 20:
This fact follows immediately from the deﬁnitions (cf. the arguments used in R13
and R19).
Question 21: Deﬁnition 3.1, (v): Should AutFA (B) be replaced by AutFA (B → A)?
Response 21:
This is not a misprint, but instead reﬂects my understanding that in this context(i.e., where one considers “B” as an object of “FA ”), the “object of FA ” denoted by“B” is precisely the object determined by the arrow B → A under consideration.
Nevertheless, this issue is addressed in the newly released version of the Comments
for [FrdII].
Question 22: Example 3.3: Is the category C0 equivalent to some model Frobenioid?
Response 22:
No. It is clear from the discussion of Example 3.3, (i), (ii) (cf. also Deﬁnition 3.1,
(iii)) that C0 contains non-isotropic objects. Thus, the fact that C0 is not equivalent
to a model Frobenioid follows formally from the fact (cf. Theorem 5.2, (ii)) that amodel Frobenioid is necessarily of isotropic type.

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Both [FrdI] and [FrdII]:
Question 23: [FrdI], Examples 6.1 and 6.3, “one veriﬁes immediately that a morphism L → M ... may be thought of as consisting ...”, and [FrdII], Example 1.1,
“we observe in passing that an object of C0 lying over Spec(K) may be thought ofas a metrized line bundle on Spec(K) ...”: It seems that both are special cases of ageneral property of model Frobenioids that allows one to think of the objects of amodel Frobenioid as metrized line bundles. If this is the case, then it seems that it
would be preferable to provide, for ease of reference, an explicit statement of sucha general property and its proof.
Response 23:
Both instances of this pattern are immediate from the deﬁnitions and simply
amount to the well-known observation that a (metrized) line bundle may be understood as a rational equivalence class of divisors. It does not appear to be feasible toformulate this sort of pattern as a “general property” of Frobenioids since, in general, only the “rational equivalence class of divisors” aspect appears in the general
theory of Frobenioids. That is to say, the “(metrized) line bundle” aspect does notappear in the general theory of Frobenioids; it only appears in speciﬁc situations
where one constructs a Frobenioid as a suitably deﬁned category of (metrized) line
bundles on a(n) (arithmetic) scheme. In particular, since it appears diﬃcult to formulate such constructions from scheme theory in any sort of generality, it does notappear realistic (at least relative to the theory of Frobenioids in its present form)
to expect a formulation of a “general property” of the sort that you appear to havein mind. I suppose, however, that one tautological approach (i.e., as in [IUTchIII],
Example 3.6, (i)) to making the construction of [FrdI], Theorem 5.2, (i), look more
like a category of (metrized) line bundles is the following result (whose proof isimmediate from the deﬁnitions):
Proposition T. (Tautological Torsor-theoretic Approach to Model Frobenioids) Let D, Φ, B, DivB : B → Φgp , and C be as in [FrdI], Theorem 5.2. Write
C tor for the category deﬁned as follows:· An object of C tor is a triple(AD , TA , τA )where AD ∈ Ob(D); TA is a B(AD )-torsor; τA is a trivialization of theΦ(AD )gp -torsor obtained from TA by executing the “change of structuregroup” operation determined by the homomorphism DivB (AD ) : B(AD ) →Φ(AD )gp . Thus, for any d ∈ Z, we obtain an object(AD , TA⊗d , τA⊗d )by executing the “change of structure group” operation determined by thehomomorphism B(AD ) → B(AD ) given by multiplication by d.
· a morphism of C tor
def

def

φ : A = (AD , TA , τA ) → B = (BD , TB , τB )is a collection of data as follows: (a) an element d ∈ N≥1 ; (b) a morphism
Base(φ) : AD → BD , which determines [i.e., by executing the “change of

RESPONSES TO QUESTIONS ON FROBENIOIDS

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structure group” operations determined by the homomorphisms B(BD ) →B(AD ), Φ(BD )gp → Φ(AD )gp ] an object φ∗ B = (AD , φ∗ TB , φ∗ τB ); (c) an
isomorphism of B(AD )-torsors∼

TA⊗d → φ∗ TB
that maps the trivialization τA⊗d to an element in the Φ(AD )-orbit of thetrivialization φ∗ τB . Composites of morphisms are deﬁned in the evident
fashion.
Then there is a natural equivalence of categories∼

C → C tordeﬁned as follows: An object (AD , α) of C [cf. [FrdI], Theorem 5.2, (i)] is mapped
to the object (AD , TA , τA ) of C tor , where TA is the trivial B(AD )-torsor, and τA is
the trivialization of TA obtained by shifting the tautological trivialization of TA bythe element −α ∈ Φ(AD )gp . A morphism (AD , α) → (BD , β) of C determined bydata as in [FrdI], Theorem 5.2, (i), (a), (b), (c), (d), is mapped to the morphism(AD , TA , τA ) → (BD , TB , τB ) of C tor for which the data (a), (b) [as in the deﬁnitionof C tor ] is determined [in the evident way] by the data of [FrdI], Theorem 5.2,(i), (a), (b), and the data (c) [as in the deﬁnition of C tor ] is the isomorphism of∼trivial B(AD )-torsors TA⊗d → φ∗ TB determined by multiplication by the elementuφ ∈ B(AD ) of [FrdI], Theorem 5.2, (i), (d).
Question 24: In view of Q23, might it not be possible to replace the current deﬁnition of a model Frobenioid by some sort of alternative deﬁnition which uses metrized
line bundles or some variant of metrized line bundles as in [FrdII], Example 3.3,
and would then include both this example and the examples referred to in Q23
as special cases? One could then discuss the relationship between this alternative
deﬁnition and the current deﬁnition of a model Frobenioid.
Response 24:
As discussed in R22, the category C0 of [FrdII], Example 3.3, is not equivalent
to a model Frobenioid. In fact, the original deﬁnition of a Frobenioid given in[FrdI], Deﬁnition 1.3, was intended/arrived at precisely as an approach to giving a
uniﬁed treatment of the examples that you refer to (i.e., [FrdI], Examples 6.1 and
6.3; [FrdII], Examples 1.1 and 3.3), together with the tempered Frobenioids treatedin [EtTh]. The price of insisting on such a uniﬁed approach may be seen in the
numerous technicalities that appear in the theory of [FrdI], [FrdII], e.g., involving
objects that are not necessarily isotropic and morphisms that are not necessarily
co-angular. I do not know at the present time of another approach to obtaining sucha uniﬁed formulation. (It would be interesting, however, if you are able to ﬁnd analternative approach to obtaining such a uniﬁed formulation.) I suppose that, fromthe point of view of using model Frobenioids, perhaps the closest kind of mathematical object that appears in [FrdI], [FrdII] to the sort of alternative version of a model
Frobenioid that you seem to have in mind may be seen in the poly-Frobenioids of[FrdII], §5, i.e., in the case where one takes the global and nonarchimedean portionsof the poly-Frobenioid to be model Frobenioids and the archimedean portion of thepoly-Frobenioid to be a Frobenioid as in [FrdII], Example 3.3 — cf., e.g., [FrdII],
Example 5.6.

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Question 25: To what extent could one use the theory of adèles and idèles tosimplify the portion of the theory of Frobenioids, as it is currently formulated, thatis used in the papers on IUTeich?
Response 25:
As explained in R3, the only types of Frobenioids that appear in IUTeich are thetypes discussed in (F1), (F2), (F3), and (F4). Of these, the Frobenioids of (F3)and (F4) may be replaced by very simple objects. Also, the Frobenioids of (F2)
are rather elementary and may be treated to a substantial extent without invoking
the general theory of Frobenioids. On the other hand, the theory developed in[EtTh], §3, §4, §5, concerning the tempered Frobenioids of (F1) is rather nontrivial.
I do not see any reason to believe that this theory may be simpliﬁed by usingadèles/idèles. Here, let us recall that model Frobenioids (i.e., such as tempered
Frobenioids) essentially amount to the data consisting of a divisor monoid and arational function monoid (as in [FrdI], Theorem 5.2). This data is, in fact, alreadyin a form that is very close, in spirit, to objects that occur in the theory of idèles.
On the other hand, one fundamental diﬀerence between the conventional theory ofidèles and the theory of Frobenioids is that the former starts from a ﬁxed number
ﬁeld and concerns the study of various properties of idèles and related objects in a
fashion that makes essential use of the way in which such objects were originallyconstructed from a number ﬁeld. By contrast, the theory of Frobenioids starts
from abstract “combinatorial objects” (such as abstract monoids) that are treated
without reference to any rings/schemes (such as number ﬁelds). That is to say,the goal of the theory of Frobenioids is to see just how much of the structure thatmight arise, in various conventional situations, from some sort of ring or scheme,
may in fact be reconstructed purely category-theoretically, i.e., without reference toany rings/schemes (cf. [FrdI], §I2). I do not see any reason to expect that the useof adèles/idèles would result in a meaningful simpliﬁcation of the theory of suchreconstruction algorithms.