Theory and Applications of Categories,
Vol. 29, No. 7, 2014, pp. 198214.
A GALOIS THEORY FOR MONOIDS
Dedicated to Manuela Sobral on the occasion of her seventieth birthday
ANDREA MONTOLI, DIANA RODELO AND TIM VAN DER LINDEN
We show that the adjunction between monoids and groups obtained via the
Grothendieck group construction is admissible, relatively to surjective homomorphisms,
in the sense of categorical Galois theory. The central extensions with respect to this
Galois structure turn out to be the so-called special homogeneous surjections.
Abstract.
Introduction
B on a monoid X can be dened as a monoid homomorphism
EndpX q is the monoid of endomorphisms of X . These actions
An action of a monoid
B
Ñ
EndpX q,
where
were studied in [15], where it is shown that they are equivalent to a certain class of
split epimorphisms, called
Schreier split epimorphisms
in the recent paper [13].
Some
properties of Schreier split epimorphisms, as well as the closely related notions of
special
Schreier surjection
and
Schreier reexive relation,
were then studied in [2] and [3], where
the foundations for a cohomology theory of monoids are laid.
of the category of groups, such as the
Many typical properties
Split Short Five Lemma
or the fact that any
internal reexive relation is transitive, remain valid in the category of monoids when,
in the spirit of relative homological algebra, those properties are restricted to Schreier
B Ñ EndpX q factors
AutpX q of automorphisms of X , the corresponding split epimorphism
is called homogeneous [2]. Some properties of homogeneous split epimorphisms and of
the related notions of special homogeneous surjection and homogeneous reexive relation
split epimorphisms and Schreier reexive relations. When an action
through the group
were also studied in [2] and [3].
The rst and the second author were supported by the Centro de Matemática da Universidade de
Coimbra (CMUC), funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCTFundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0324/2013 and grants number PTDC/MAT/120222/2010 and
SFRH/BPD/69661/2010. The third author is a Research Associate of the Fonds de la Recherche
ScientiqueFNRS. He would like to thank CMUC for its kind hospitality during his stays in Coimbra.
Received by the editors 2014-02-05 and, in revised form, 2014-04-25.
Transmitted by Walter Tholen. Published on 2014-05-01.
2010 Mathematics Subject Classication: 20M32, 20M50, 11R32, 19C09, 18F30.
Key words and phrases: categorical Galois theory; homogeneous split epimorphism; special homogeneous surjection; central extension; group completion; Grothendieck group.
c Andrea Montoli, Diana Rodelo and Tim Van der Linden, 2014. Permission to copy for private use
granted.
198
A GALOIS THEORY FOR MONOIDS
199
The aim of the present paper is to approach the concept of homogeneous split epimorphism from the point of view of categorical Galois theory [6, 7].
classical
junction
instance
Grothendieck group
Recall that the
group completion construction [10, 11, 12] gives an adbetween the categories Mon of monoids and Gp of groups, which is relevant for
in K -theory, where it is used in the denition of K0 . We prove that this ador
junction is admissible in the sense of categorical Galois theory, when it is considered with
respect to the class of surjective homomorphisms both in
Mon
and in
Gp.
We further
show that the central extensions with respect to this adjunction are the special homogeneous surjections. This gives a positive answer to the question whether homogeneous split
epimorphisms can be characterised in a way which does not refer to the underlying split
epimorphism of sets.
The paper is organised as follows. In Section 1 we recall some basic notions of categorical Galois theory. In Section 2 we prove that the Grothendieck group adjunction is
part of an admissible Galois structure (Theorem 2.2). In Section 3 we recall the denitions of Schreier split epimorphism and homogeneous split epimorphism, special Schreier
surjection and special homogeneous surjection together with some of their properties. In
Section 4 we show that the central extensions with respect to the Galois structure under
consideration are exactly the special homogeneous surjections (Theorem 4.3).
1. Galois structures
We recall the denition of
extension
Galois structure
and the concepts of
trivial, normal
and
central
arising from it, as introduced in [6, 7, 8]. For the sake of simplicity we restrict
ourselves to the context of Barr-exact categories [1].
pC , X , H, I, η, , E , F q
Γ
1.1. Definition. A Galois structure
consists of an
adjunction
,2
I
C lr K X
H
with unit
η : 1C
ñ HI and counit : IH ñ 1X
as well as classes of morphisms
E
in
C
and
F
(1)
E
and
F
contain all isomorphisms;
(2)
E
and
F
are pullback-stable;
(3)
E
and
F
are closed under composition;
(4)
H pF q „ E ;
(5)
I pE q „ F .
We will follow [7] and call the morphisms in
E
between Barr-exact categories
in
X
and
such that:
F
fibrations.
C
and
X,
ANDREA MONTOLI, DIANA RODELO AND TIM VAN DER LINDEN
200
1.2. Definition. A trivial extension is a bration
square
ηA
A
f
,2
B
,2
ηB
p is a trivial extension.
in
C
such that the
HI pAq
pq
HI f
HI pB q
is a pullback. A central extension is a bration
bration
f: AÑB
f
whose pullback
p pf q
along
some
A normal extension is a bration such that its kernel
pair projections are trivial extensions.
It is well known and easy to see that trivial extensions are always central extensions
and that any normal extension is automatically central.
Given an object
B
in
C
we consider the induced adjunction
pE Ó B q lr
IB
K
,2
HB
pF Ó I pB qq,
pE Ó B q for the full subcategory of the slice category pC Ó B q determined
B
B
by morphisms in E ; similarly for pF Ó I pB qq. Here I
is the restriction of I , and H
sends a bration g : X Ñ I pB q to the pullback
where we write
,2 H
A
pq
pX q
HB g
B
of
H pg q
along
ηB
,2 HI
pq
H g
pB q
ηB .
1.3. Definition. A Galois structure
ble when all functors
H
B
Γ pC , X , H, I, η, , E , F q
is said to be admissi-
are full and faithful.
[9, Proposition 2.4] If Γ is admissible, then I : C Ñ X preserves
pullbacks along trivial extensions. In particular, the trivial extensions are pullback-stable,
so that every trivial extension is a normal extension.
1.4. Proposition.
2. The Grothendieck group of a monoid
pM, , 1q is given by
Ñ GppM q which is universal
The Grothendieck group (or group completion) of a monoid
a group
GppM q
together with a monoid homomorphism
with respect to monoid homomorphisms from
we have
GppM q M
M
to groups [10, 11, 12]. More precisely,
GpFpM q
,
NpM q
A GALOIS THEORY FOR MONOIDS
201
GpFpM q denotes the free group on M and NpM q is the normal subgroup generated
1 (from now on, we simply write m1 m2 instead
by elements of the form rm1 srm2 srm1 m2 s
of m1 m2 ). This gives us an equivalence relation on GpFpM q generated by rm1 srm2 s rm1m2s with equivalence classes rm1srm2s rm1m2s. Thus, an arbitrary element in
GppM qan equivalence class of wordscan be represented by a word of the form
where
rm1srm2s1rm3srm4s1 rmnsιpnq or rm1s1rm2srm3s1rm4s rmnsιpnq,
where ιpnq 1, n P N, m1 , . . . , mn P M and no further cancellation is possible.
Let
Mon
and
Gp
represent the categories of monoids and of groups, respectively. The
group completion of a monoid determines an adjunction
Gp
,2
A)
Mon lr K Gp,
(
Mon
Mon is the forgetful functor. To simplify notation, we write GppM q instead of
MonGppM q when referring to the monoid structure of GppM q. The counit is 1Gp and
the unit is dened, for any monoid M , by
where
ηM : M
Ñ GppM q : m ÞÑ rms.
2.1. Remark. It is well known that in general
ηM
is neither surjective nor injective. For
example:
The additive monoid of natural numbers is such that
fact,
ηM
is injective whenever
The monoid
M
M
ηN :
N Ñ Z is an injection.
In
is a monoid with cancellation.
pt0, 1u, , 1q has a trivial Grothendieck group and therefore ηM is
surjective.
The product
N M,
for
M
as above, is such that
GppN M q
Z (in fact,
it
is not dicult to see that the group completion functor preserves products) and
ηNM :
N M Ñ Z is neither surjective nor injective.
By choosing the classes of morphisms
E
and
F
to be the surjections in
Mon
and
Gp,
respectively, we obtain a Galois structure
ΓMon
pMon, Gp, Mon, Gp, η, , E , F q.
Since this is the only Galois structure we shall consider in detail, without further mention
we take all normal, central and trivial extensions in this paper with respect to
ΓMon .
ANDREA MONTOLI, DIANA RODELO AND TIM VAN DER LINDEN
202
2.2. Theorem.
The Galois structure ΓMon is admissible.
Proof. For any monoid
M,
we must prove that the functor
MonM : pF
Ó GppM qq Ñ pE Ó M q
is fully faithful. Given a morphism α : pA, f q Ñ pB, g q in pF Ó GppM qq, its image through
MonM
is dened by the universal property of the front pullback below:
M
GppM q A
πA
MonM
M
,2
pαq
A
α
!)
GppM q B
!),2
πB
B
B)
pq
(
MonM f
f
pq
g
MonM g
| |
M
ηM
,2
| |
GppM q.
β : pA, f q Ñ pB, g q such
p q β paq by induction on
M
First we prove that Mon
is faithful. Consider morphisms α,
M
M
that Mon
α
Mon β . For any a A, we prove that α a
p q
pq
P
n (supposing that no cancellations are
the class f paq.
If f paq rms, then pm, aq P M GppM q A and
the length
possible) of the word that represents
MonM pαqpm, aq MonM pβ qpm, aq
pq pq
p q rms1, then f pa1q rms and we nd αpa1q pq
αpaq β paq.
1
1
1
1
Suppose that αpa q β pa q for those a P A which have f pa q represented by a word of
length n 1 or smaller. Suppose that f paq is represented by a word of length n, n ¥ 2.
implies that α a
β a . If f a
1
β a
as in the previous case; hence
It can be written as the product (= concatenation) of a word of length one and a word of
n 1.
1
f pa
1 aq,
length
By the surjectivity of
and
for some
f,
their corresponding classes can be written as
P A. Then
1
1
αpaq αpa1 qαpa
1 aq β pa1 qβ pa1 aq β paq
a1
f p a1 q
by the induction hypothesis.
Now we have to show that
by induction in the case when
MonM is full. The proof goes in two steps:
M is a free monoid (Lemma 2.3 below), then
from the free case to the general case (the subsequent Lemma 2.4).
rst a proof
an extension
A GALOIS THEORY FOR MONOIDS
2.3. Lemma.
Proof. Let
203
The functor MonM is full for all free monoids M .
M
be a free monoid. To simplify notation, we identify the classes in
with their representatives. Consider group surjections
f
and
g
GppM q
B)
as in Diagram (
and a
monoid homomorphism
γ : pM
GppM q A, MonM pf qq Ñ pM GppM q B, MonM pgqq.
We dene a group homomorphism α : pA, f q Ñ pB, g q as follows. For any a P A, we dene
αpaq by decomposing a into a product of elements in the image of πA . The main diculty
lies in proving that the result is independent of the chosen decomposition.
pm, aq P M GppM q A and we dene
αpaq πB pγ pm, aqq.
1
1
If f paq rms , then f pa q rms and we dene
αpaq πB pγ pm, a1 qq1 .
ιpnq
1
Suppose that a a1 a2 an
such that f pai q rmi s, with mi P M , and n is the
smallest number for which such a decomposition in GppM q exists. Then we must put
αpaq πB pγ pm1 , a1 qqπB pγ pm2 , a2 qq1 πB pγ pmn , an qqιpnq ;
(C)
ιpnq
1
the case a a1 a2 an
can be treated similarly.
If
f paq rms
To prove that
That is to say, if
for some
M
then
α is a homomorphism,
ιpkq
1
a x1 x
such
2 xk
agree with
Since
m P M,
it now suces to show that it is well dened.
that
f pxi q
rlis, with li P M , then (C) must
πB pγ pl1 , x1 qqπB pγ pl2 , x2 qq1 πB pγ plk , xk qqιpkq .
is free, and hence the group
GppM q
is free, if the words
rm1srm2s1 rmnsιpnq and rl1srl2s1 rlk sιpkq
are both of minimal length, then k n and li mi . Thus we only have to prove the
result for decompositions of equal length mapping down to the same word in GppM q. We
do this by induction on
1.
n.
a x1 and f pa1q rm1s f px1q for some m1 P M .
1 Then obviously αpa1 q αpx1 q. The same happens if a1 a x1 and f pa1 q rm1 s
f px1 q for some m1 P M .
More generally, let a, x P A be such that f paq rms f pxq for some m P M . Then
f px1 aq r1s, so
αpaq πB pγ pm, aqq πB pγ pm, xqqπB pγ p1, x1 aqq αpxqαpx1 aq
Case n
Suppose that
which implies that
a1
αpx1 aq αpxq1 αpaq.
204
ANDREA MONTOLI, DIANA RODELO AND TIM VAN DER LINDEN
This formula will be useful in the sequel of the proof.
Case n 2. Now consider a A such that a1 a2 1
a x1 x2 1 and f ai
f xi with mi M . Then α xi 1 ai
α xi 1 α ai by the formula above. Hence
x2 1 a2 implies α x1 1 α a1
α x2 1 α a2 , so that
P
p q rmis 1
p q
P
p q p q p q
x
1 a1 p q p q p q p q
αpa1 qαpa2 q1 αpx1 qαpx2 q1 .
1
1
The case in which a1 a2 a x1 x2 and f pai q rmi s f pxi q is similar.
1
1
Case n 3. Suppose a1 a
2 a3 a x1 x2 x3 such that f pai q rmi s f pxi q, with
mi P M . Then
1
1
1
x
1 a1 a2 a3 x2 x3
gives
1
1
1
αpa2 a
1 x1 q αpa3 q αpx2 q αpx3 q
rm2s1rm3s. Similarly,
1
1 1
a1 a
2 x 1 x 2 x 3 a3
because they both map to the same word
gives
because they both map to
αpa1 qαpa2 q1
αpx1qαpa3x3 1x2q1
rm1srm2s1. As a consequence, the equality
1
1
a3 x 3 x 2 a2 a1 x 1
rm2s of length one gives
1
1
αpa3 x
3 x2 q αpa2 a1 x1 q
1
1 αpx2 q1 αpx3 qαpa3 q1 and thus
so that αpx1 q αpa1 qαpa2 q
αpa1 qαpa2 q1 αpa3 q αpx1 qαpx2 q1 αpx3 q.
1 1
1 1
Again, the case a1 a2 a3 a x1 x2 x3 can be treated analogously.
Case n ¥ 4. Suppose that the result holds for all decompositions which map down to
ιpnq
1
words of minimal length n 1 or shorter in GppM q. Suppose that a1 a2 a3 an
a
ιpnq
1
x1 x2 x3 xn such that f pai q rmi s f pxi q, with mi P M . Then
px1 1a1a2 1qa3 aιnpnq x2 1x3 xnιpnq
1 rmn sιpnq , so by the induction hypothesis we nd
both map to rm2 s
1
1
ιpnq
αpa2 a
αpx2q1αpx3q αpxnqιpnq.
1 x1 q αpa3 q αpan q
1
1
Furthermore, αpa2 a1 x1 q αpa2 qαpa1 q αpx1 q as shown above (case n 3) so that
αpa1 qαpa2 q1 αpa3 q αpan qιpnq αpx1 qαpx2 q1 αpx3 q αpxn qιpnq .
ιpnq
1 1
The case a1 a2 a3 an
a x1 1x2x3 1 xιnpnq being similar, this concludes the proof.
above the word
A GALOIS THEORY FOR MONOIDS
2.4. Lemma.
205
The functor MonM is full for all monoids M .
Proof. As in the previous lemma, we simplify notation by identifying the classes in
GppM q
with their representatives.
Consider group surjections
ism
f
B) as well as a group homomorph-
g
and
as in Diagram (
γ : pM
GppM q A, MonM pf qq Ñ pM GppM q B, MonM pgqq.
We cover the monoid M with the free monoid FpM q on M , then apply the Grothendieck
group functor to obtain the following commutative diagram with exact columns:
Kerp_rM q
,2
Np_M q
ηFpM q
2, GpFpM q
FpM q ,2
rM
M
We pull back
rM .
,2
qM
GppM q.
MonM pf q, MonM pg q and the morphism γ
between them along the surjection
We thus obtain a diagram
FpM q GppM q A
rM
GppM q 1A
,2 ,2 M
pq
γ
rM
FpM q GppM q B
p
p qq
rM
MonM f
rM
GppM q 1B
,2 ,2
M
pq
,2 A
$,
GppM q B
πB
,2
pgq
x x
,2 ,2 M
rM
B
f
MonM
x x
FpM q
g
,2
ηM
x x
GppM q.
qM ηF pM q, the left hand side triangle of this diagram can also be obtained
ηM rM
by taking the pullbacks of
FpM q GppM q A ,2
η Fp M q
f
g
and
GppM q 1A
pγ q
rM
,2
qM ηF pM q :
along
GpFpM q GppM q A
FpM q GppM q B ,2
ηFpM q
GppM q 1B
pq
f
qM
p qq
MonM f
rM
pMonM pgqq
rM
x x
FpM q ,2
Since the functor
arrow above. It
,2
A
pA
β
$,
πA
MonM f
pMonM pgqq
rM
Since
GppM q A
γ
$,
p
ηM
_
ηFpM q
,2
$,
pM q GppM q B
,2 GpF
,2
B
pB
f
pq
g
qM
g
x x
GpFpM q
qM
,2
Gp
x x
pM q.
MonFpM q is full by Lemma 2.4, we nd a morphism β given by the dotted
suces to show that β keeps the elements of the kernel NpM q (of qM ,
ANDREA MONTOLI, DIANA RODELO AND TIM VAN DER LINDEN
206
thus also) of
pA
pB
and
xed, because then it induces the needed
the universal property of
The group
GpFpM q.
in
NpM q
pA
α : pA, f q
Ñ pB, gq by
as a cokernel of its kernel.
is generated by words
rm1srm2srm1m2s1 as a normal subgroup of
Hence it suces to prove for elements of the type
GpFpM q GppM q A
prm1srm2srm1m2s1, 1q
that
β prm1 srm2 srm1 m2 s1 , 1q prm1 srm2 srm1 m2 s1 , 1q P GpFpM q GppM q B.
Since
f
is a surjection, there exists an element
aPA
such that
prM GppM q 1Aqprm1srm2s, aq pm1m2, aq prM GppM q 1Aqprm1m2s, aq.
For some b P B we have γ pm1 m2 , aq pm1 m2 , bq, so using the commutativity of the
second diagram, we see that
pγ qprm srm s, aq prm srm s, bq
rM
1
2
1
2
and
pγ qprm m s, aq prm m s, bq.
rM
1 2
1 2
On the other hand, using the commutativity of the third diagram we nd
β prm1 srm2 s, aq β pηFpM q GppM q 1A qprm1 srm2 s, aq
pηFpM q GppM q 1B qprM pγ qprm1srm2s, aqq
prm1srm2s, bq
and, similarly,
β prm1 m2 s, aq
prm1m2s, bq, for some b P B as above.
Since
β
is a group
homomorphism, we obtain
β prm1 srm2 srm1 m2 s1 , 1q β prm1 srm2 s, aqβ prm1 m2 s, aq1
prm1srm2s, bqprm1m2s1, b1q
prm1srm2srm1m2s1, 1q
which concludes the proof.
2.5. Remark. We can restrict the group completion to commutative monoids: it is easily
seen that then
ΓMon
restricts to an admissible Galois structure
ΓCMon
pCMon, Ab, CMon, Gp|CMon, η1, 1, E 1, F 1q
induced by the (co)restriction
|
Gp CMon
,2
CMon lr K Ab,
CMon
A) to commutative monoids and abelian groups.
of the adjunction (
We end this section with an example showing that the adjunction (
A)
is not semi-
left-exact [5]: it is not admissible with respect to all morphisms, instead of just the
surjections [4].
A GALOIS THEORY FOR MONOIDS
2.6. Example. Consider
where
ηN2 :
A is the subgroup of Z
2
N2 Ñ Z2
with morphisms
generated by
f
207
and
g
as in Diagram (
B),
p1, 1q, f is determined by f p1, 1q p1, 1q
and
Z3 Ñ Z2 : pk, l, mq ÞÑ pk, lq.
M
2
2
3
2
Then N Z A 0 while N Z Z N Z, so that the functor Mon
g:
2
2
it maps, for instance, both A
2
zero morphism 0
.
ÑN Z
is not faithful:
Ñ B : p1, 1q ÞÑ p1, 1, 0q and p1, 1q ÞÑ p1, 1, 1q to the
3. Schreier split epimorphisms and homogeneous split epimorphisms
In this section we recall some denitions and results from [2] and [3].
category
Mon
of monoids.
3.1. Definition. Consider a split epimorphism
N ,2
k
,2
X lr
f
s
pf, sq with its kernel:
,2 ,2
such that
x n sf pxq.
D)
Y.
It is called a Schreier split epimorphism when, for any
nPN
We work in the
(
x P X,
there exists a unique
Note that when we say split epimorphism we consider the chosen splitting as part of
the structure; and for the sake of simplicity, we take canonical kernelsso
of
N
is a subset
X.
3.2. Definition. The split epimorphism (D) is said to be right homogeneous when,
y P Y , the function µy : N Ñ f 1 py q dened through multiplication
on the right by spy q, so µy pnq n spy q, is bijective. Similarly, by duality, we can dene
1 py q : n ÞÑ spy q n
a left homogeneous split epimorphism: now the function N Ñ f
must be a bijection for all y P Y . A split epimorphism is said to be homogeneous when
for every element
it is both right and left homogeneous.
[2, Propositions 2.3 and 2.4] Consider a split epimorphism pf, sq as
in (D). The following statements are equivalent:
3.3. Proposition.
(i)
(ii)
(iii)
(iv)
pf, sq is a Schreier split epimorphism;
there exists a unique function q : X Ñ N such that q pxqsf pxq x, for all x P X ;
there exists a function q : X Ñ N such that q pxqsf pxq x and q pn spy qq n, for
all n P N , x P X and y P Y ;
pf, sq is right homogeneous.
ANDREA MONTOLI, DIANA RODELO AND TIM VAN DER LINDEN
208
Y and N , an action of Y on N is a monoid homowhere EndpN q is the monoid of endomorphisms of N .
3.4. Definition. Given monoids
morphism
ϕ: Y
Ñ EndpN q,
Actions correspond to Schreier split epimorphisms via a semidirect product construction:
3.5. Semidirect products. It is shown in [13] that any Schreier split epimorphism (D)
corresponds to an action
ϕ
of
Y
N
on
dened by
ϕpy qpnq y n q pspy q nq
for
y
PY
and
n P N.
Thus
pf, sq is isomorphic, as a split epimorphism, to
N
where
N
ϕ Y
product of sets
x1,0y
ϕ Y
,2 N
is the semidirect product of
N
Y
N
lr
πY
x0,1y
and
,2
Y,
Y
with respect to
ϕ:
the cartesian
equipped with the operation
px1, y1q px2, y2q px1 ϕy px2q, y1y2q,
ϕpy1q P AutpN q. See [13], [2] or Chapter 5 in [3] for more details.
1
where
ϕy1
3.6. Proposition. [2, Proposition 3.8] A Schreier split epimorphism (D) is homogeneous if and only if the corresponding action ϕ : Y Ñ EndpN q factors through the group
AutpN q of automorphisms of N .
3.7. Lemma.
[2, Lemma 4.1] Consider the morphism of Schreier split epimorphisms
N lr 2,
u
r
q
k
,2
X lr
f
s
,2 ,2
Y
u
v
1
f1
lr q
1
1
N ,2 1 ,2 X rl 1 ,2 ,2 Y 1
k
s
r of u to N . Then the left hand side square consisting
and their kernels, and the restriction u
1
of the functions q and q also commutes: q 1 u u
rq .
This lemma has the following useful consequence.
Given a morphism of Schreier split epimorphisms as in Lemma 3.7,
r preserves the action of the object Y on N : for all y P Y and n P N ,
the homomorphism u
3.8. Corollary.
rpy nq vpyq urpnq.
u
Proof.
rpy nq urq pspy q nq q 1 upspy q nq q 1 puspy q upnqq q 1 ps1 v py q urpnqq vpyq urpnq
u
A GALOIS THEORY FOR MONOIDS
209
We now extend these concepts to surjections which are not necessarily split.
3.9. Definition. Given a surjective homomorphism
π1
Eqpg q lr
,2
∆
π2
g
,2 X
g
of monoids and its kernel pair
E)
,2 ,2 Y,
(
g is called a special Schreier surjection when pπ1 , ∆q is a Schreier split epimorphism.
It is called a special homogeneous surjection when pπ1 , ∆q is a homogeneous split
epimorphism.
As a consequence of Theorem 5.5 in [2], if
g
is a special Schreier surjection, then its
kernel is necessarily a group.
3.10. Remark. The name
Schreier extension
was used in [16, 14] to describe a dierent,
but closely related concept.
3.11. Remark. A special Schreier (resp. homogeneous) surjection which is a split epimorphism is always a Schreier (resp. homogeneous) split epimorphism.
Yet a Schreier
(resp. homogeneous) split epimorphism is not necessarily a special Schreier (resp. homogeneous) surjection. Indeed, according to Proposition 3.1.12 in [3], a Schreier (resp.
homogeneous) split epimorphism is a special Schreier (resp. homogeneous) surjection if
and only if its kernel is a group. In fact, by Proposition 2.3.4 in [3], taking the kernel pair
of a Schreier split epimorphism pf, sq as in (D), we do obtain a Schreier split epimorphism
pπ1, xsf, 1X yq. Nevertheless, the split epimorphism pπ1, ∆q need not be Schreier.
As a consequence of Theorem 5.5 in [2] and of the remark above we have:
A surjective homomorphism g as in (E) is a special Schreier (resp.
special homogeneous) surjection if and only if the kernel pair projection π1 is a special
Schreier (resp. special homogeneous) surjection.
3.12. Corollary.
[3, Proposition 7.1.4] Special Schreier and special homogeneous surjections are stable under products and pullbacks.
3.13. Proposition.
3.14. Proposition.
[3, Proposition 7.1.5] Given any pullback
X
f
g
Y
h
,2 ,2
,2 ,2
X1
1
f
Y1
with g and h surjective homomorphisms, if f is a special Schreier (resp. special homogeneous) surjection, then so is f 1 .
[2, Proposition 3.4] Any split epimorphism (D) such that Y is a
group is a homogeneous split epimorphism.
3.15. Proposition.
ANDREA MONTOLI, DIANA RODELO AND TIM VAN DER LINDEN
210
3.16. Remark. According to the proposition above and to Remark 3.11, a split epimorphism (
its kernel
D)
N
such that
Y
is a group is a special homogeneous surjection if and only if
X
is a group. Furthermore, in that case also
is a group [3, Corollary 2.2.5].
Conversely, every surjective homomorphism between groups is a special homogeneous
surjection.
4. Normal extensions and central extensions
In this section we characterise the trivial split extensions, the central and the normal
ΓMon .
extensions in the Galois structure
The central extensions turn out to be precisely
the special homogeneous surjections, while a split epimorphism of monoids is a trivial
extension if and only if it is a special homogeneous surjection. This gives a characterisation
which does not refer to the underlying split epimorphism of sets: Denition 3.2 in terms
of elements, Proposition 3.3 where the splitting
q
is a
function
rather than a morphism
of monoids.
4.1. Lemma.
Any morphism of homogeneous split epimorphisms and their kernels
N ,2
r
u
,2
k
f
X lr
u
N 1 2, 1 2, X 1 lr
k
,2 ,2
s
f1
,2 ,2
s1
Y
ηY
GppY q
factors into the composite
N ,2
,2
k
f
X lr
ηY
,2 ,2
s
f2
Y
ηY
N 2, 2 2, X 2 lr 2 2, ,2 GppY q
k
s
u
r
N1 ,2
u
k1
,2
X 1 lr
f1
s1
,2 ,2
GppY q
of morphisms of homogeneous split epimorphisms and their kernels, where ϕ is as in
Proposition 3.6 and X 2 N ϕ GppY q for ϕ : GppY q Ñ AutpN q, the unique group homomorphism satisfying ϕ ϕηY .
X N ϕ Y for ϕ : Y Ñ AutpN q. By adjointness,
ϕ gives rise to a unique group homomorphism ϕ : GppY q Ñ AutpN q
ϕηY ϕ. Note that ϕ is necessarily given by
Proof. As mentioned above, we have
this monoid morphism
for which
and
1
ιpnq
ϕpry1 sry2 s1 ryn sιpnq q ϕy1 ϕ
y2 ϕyn
P AutpN q
(
1
ιpnq
ϕpry1 s1 ry2 s ryn sιpnq q ϕ
y1 ϕy2 ϕyn
P AutpN q.
(
F)
G)
A GALOIS THEORY FOR MONOIDS
211
Via the functoriality of the semidirect product construction this already yields the upper
ηY 1N ηY . This leaves us with nding u : X 2 Ñ X 1 .
1
The needed morphism u : N ϕ GppY q Ñ N ψ GppY q, where ψ is the action for
1
1
r is a morphism of GppY qwhich X N ψ GppY q, is induced once we prove that u
part of the diagram, where
actions. More precisely, we have to show that
u
rpϕz pnqq ψz purpnqq
for all
z
P GppY q and n P N .
extends to all elements of
and (
G).
ϕ ϕηY tell us precisely
ηY pyq of GppY q, so it suces to check that it
Corollary 3.8 and the fact that
that this equality holds for generators
GppY q.
z
F)
This needs a straightforward verication based on (
Consider a split epimorphism pf, sq as in (D). The following statements are equivalent:
4.2. Proposition.
(i)
(ii)
f is a trivial extension;
f is a special homogeneous surjection.
Proof. (i)
ñ (ii) If f is a trivial extension, then by denition the diagram
f
X lr
,2 ,2
s
Y
ηX
H)
ηY
(
p q,2 ,2
GppX q lr
GppY q
Gppsq
Gp f
is a pullback. By Remark 3.16, the group homomorphism
surjection; hence so is
(ii)
f
Gppf q is a special homogeneous
by Proposition 3.13.
ñ (i) Given a split epimorphism pf, sq which is a special homogeneous surjection,
H) is a pullback.
we have to show that the square (
Taking kernels we obtain the morphism
of special homogeneous surjections and their kernels
N ,2
η€
X
in [3], the square (
X lr
ηX
K 2, 1
k
where, in particular, the kernel
,2
k
N
of
f
,2 ,2
s
Y
ηY
p q,2 ,2
,2 GppX q lr
GppY q
Gppsq
f
Gp f
is a group by Remark 3.11.
H) is a pullback precisely when η€
X
By Theorem 2.3.7
is an isomorphism.
ANDREA MONTOLI, DIANA RODELO AND TIM VAN DER LINDEN
212
Lemma 4.1 gives us the diagram of solid arrows
N ,2
k
,2 X lr
ηY
N ,2
η€
X
k2
,2 X 2 lr
LR
ηX
g
f
,2 ,2
s
f2
s2
,2 ,2
Y
ηY
GppY q
pq
,2 ,2
K 2, 1 ,2 GppX q lr
GppY q.
k
Gppsq
On the other hand, since
X2
Gp f
is a group (thanks to Remark 3.16), the universal property
GppX q makes ηY induce a unique group homomorphism g : GppX q Ñ X 2
gηX ηY . Note that this g is actually a morphism of split epimorphisms:
of
f 2 gηX
f 2ηY ηY f Gppf qηX
Gppf q by the universal property of ηX , while
gGppsqηY gηX s ηY s s2 ηY
2
and thus gGppsq s .
Finally, we have ηX g 1GppX q since ηX gηX ηX ηY so that
such that
f 2g
ηX .
On the other hand,
using Lemma 2.1.6 in [3]which says that Schreier split epimorphisms are strongly split
epimorphisms, that is, the kernel and the section are jointly strongly epimorphicfrom
gηX k 2
gηX ηY k gηX k ηY k k2 and gηX s2 gGppsq s2
€
we conclude that gηX 1X 2 . In particular, the arrow η
X is an isomorphism, hence the
square (
H) is a pullback.
4.3. Theorem.
are equivalent:
For a surjective homomorphism of monoids g , the following statements
(i)
g is a central extension;
(ii)
g is a normal extension;
(iii)
g is a special homogeneous surjection.
Proof. Consider a surjective homomorphism and its kernel pair (E). Then
extension
ô
ô
ô
(1.2)
(4.2)
(3.12)
π1
is a trivial extension
π1
is a special homogeneous surjection
g
is a special homogeneous surjection.
g
is a normal
A GALOIS THEORY FOR MONOIDS
213
A normal extension is always a central extension by denition. To prove that (i) implies
(iii), let us suppose that g is a central extension. Then there exists a bration p such that
p pg q is a trivial extension, which makes it a normal extension by Proposition 1.4, hence a
special homogeneous surjection by (ii) ñ (iii). Since p is a surjective homomorphism, we
can apply Proposition 3.14 to conclude that g is a special homogeneous surjection, too.
4.4. What about special Schreier surjections? A natural question that arises is,
whether the special Schreier surjections admit a similar characterisation. More precisely,
A) factorise in such a way that the special Schreier surjections become
does the reection (
the central extensions with respect to this new adjunction? As explained in the proof of
Proposition 4.2, any split epimorphism of groups is necessarily special homogeneous, which
implies that so are the central extensions. Hence we would need a reective subcategory
of
X
Mon in which all spit epimorphisms are Schreier split epimorphisms. By Corollary 3.1.7
X is contained in the category of groups, which
in [3] though, this would mean that
defeats the purpose.
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CMUC, Universidade de Coimbra, 3001501 Coimbra, Portugal
Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade do Algarve, Campus de Gambelas, 8005139 Faro, Portugal
Institut de Recherche en Mathématique et Physique, Université catholique de Louvain,
chemin du cyclotron 2 bte L7.01.02, B1348 Louvain-la-Neuve, Belgium
Email: montoli@mat.uc.pt
drodelo@ualg.pt
tim.vanderlinden@uclouvain.be
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