Normality and quotient in crossed modules, cat$^1$-groups and internal groupoids within groups with operations

0.61 MB·

by  Tunçar Şahan

Normality and quotient in crossed modules,
1
cat -groups and internal groupoids within groups
with operations
Tunçar S¸ahan∗a and Osman Mucuk†b
6
1 aDepartmentofMathematics,AksarayUniversity,Aksaray,TURKEY
0
2 bDepartmentofMathematics,ErciyesUniversity,Kayseri,TURKEY

n
a
J

6
2

Abstract
]
T
C Inthispaperwedefinethenotionsofnormalsubcrossedmoduleandquotientcrossed
.
h module within groups with operations; and using the equivalence of crossed modules
t
a over groups with operations and internal groupoids we prove how normality and quo-
m
tient concepts are related in these two categories. Further we prove an equivalence of
[
crossedmodulesovergroupswithoperationsandcat1-groupswithoperationsforacer-
1
v tainalgebraiccategory;andthenbythisequivalencewedeterminenormalandquotient
5
9 objectsinthecategoryofcat1-groupswithoperations. Finallywecharacterizethecover-
0 ingsofcat1-groupswithoperations.
7
0
.
1 KeyWords: Groupwithoperations,quotientcrossedmodule,internalgroupoid.
0
Classification: 20L05,57M10,22AXX,22A22
6
1
:
v
i
X 1 Introduction
r
a
Crossed modules as defined by Whitehead [30, 31] have been widely used in homotopy
theory [7], the theory of group representation (see [9] for a survey), in algebraic K-theory
[20], and homological algebra [19, 22]. Crossed modules can be viewed as 2-dimensional
groups[5].
∗E-mail: [email protected]
†E-mail: [email protected]
1
The notions of subcrossed module and normal subcrossed module were defined in [28].
In[11]BrownandSpencerprovedthatthecategoryofinternalgroupoidswithinthegroups,
which are also called in [11] under the name of G-groupoids and alternative names, quite
generally used are group-groupoid [10] or 2-group (see for example [3]) is equivalent to the
categoryofcrossedmodulesofgroups. Usingtheequivalencein[11],recentlyin[24]normal
andquotientobjectsinthecategoryofgroup-groupoidshavebeenobtained.
In [21] Loday defined an algebraic object called cat1-group as a group G with two endo-
morphismss,tofGsuchthatst = t,ts = sand[Kers,Kert] = 0,where[Kers,Kert]represents
the commutator subgroup of G; and proved that the categories of cat1-groups and crossed
modulesareequivalent.
In [29] Porter proved a similar result to one in [11] holds for certain algebraic categories,
introduced by Orzech [27], which definition was adapted by him and called category of
groupswithoperations. ApplyingPorter’sresult,thestudyofinternalcategorytheorywas
continued in the works of Datuashvili [15] and [16]. Moreover, she developed cohomology
theory of internal categories, equivalently, crossed modules, in categories of groups with
operations [13] and [14]. The equivalences of the categories in [11] and [29] enable us to
generalize some results on group-groupoids which are internal categories within groups to
the more general internal groupoids for a certain algebraic category C (see for example [2],
[23],[25]and[26]).
In this paper for an algebraic category C we define normal subcrossed module and quo-
tient crossed module for groups with operations; and then obtain normal subgroupoid and
quotientgroupoidininternalgroupoidscorrespondingrespectivelytoanormalsubcrossed
module and a quotient crossed module of groups with operations. Further we prove an
equivalence of crossed modules over groups with operations and cat1-groups with opera-
tions for a certain algebraic category. This equivalence enables us to determine normal and
quotientobjects;andcoveringsinthecategoryofcat1-groupswithoperations.
MainresultsofthispaperconstitutesomepartsofthePhDthesisoffirstauthoratErciyes
Universityin2014.
2 Preliminaries
Let G be a groupoid. We write G for the set of objects of G and write G for the set
0 1
of morphisms. We also identify G with the set of identities of G and so an element of G
0 0
may be written as x or 1 as convenient. We write d ,d : G → G for the source and target
x 0 1 1 0
maps, and, as usual, write G(x,y) for d−1(x) ∩ d−1(y), for x,y ∈ G . The composition h ◦ g
0 1 0
of two elements of G is defined if and only if d (h) = d (g), and so the map (h,g) (cid:55)→ h ◦ g
0 1
2
is defined on the pullback G × G of d and d . The inverse of g ∈ G(x,y) is denoted by
1d0 d1 1 0 1
g−1 ∈ G(y,x).
If x ∈ G , we write St x for d−1(x) and call the star of G at x. Similarly we write Cost x
0 G 0 G
for d−1(x) and call costar of G at x. The set of all morphisms from x to x is a group, called
1
objectgroupatx,anddenotedbyG(x).
A groupoid G is totally intransitive if G(x,y) = ∅ for all x,y ∈ G such that x (cid:54)= y. Such
0
a groupoid is determined entirely by the family {G(x) | x ∈ G } of groups. This totally
0
intransitive groupoid is sometimes called totally disconnected or bundle of groups (Brown [4,
pp.218]).
LetGbeagroupoid. AsubgroupoidH ofGisapairofsubsetsH ⊆ G andH ⊆ G such
1 1 0 0
that d (H ) ⊆ H , d (H ) ⊆ H , 1 ∈ H for each x ∈ H and H is closed under the partial
0 1 0 1 1 0 x 1 0 1
multiplicationandtheinversioninG. AsubgroupoidH ofGiscalledwideifH = G .
0 0
Definition2.1 LetGbeagroupoid. AsubgroupoidN ofGiscallednormalifitiswideinG
andg ◦N(x) = N(y)◦g forobjectsx,y ∈ G andg ∈ G(x,y).
0
Quotient groupoid is formed as follows (Higgins [17, pp.86] and Brown [4, pp.420]). Let
N be a normal subgroupoid of the groupoid G. The components of N define a partition on
G and we write [x] for the class containing x. Then N also defines an equivalence relation
0
on G by g ∼ h for a,b ∈ G if and only if g = n ◦ h ◦ m for some m,n ∈ N . A partial
1 1 1
composition [h]◦[g] on the morphisms is defined if and only if there exist g ∈ [g], h ∈ [h]
1 1
suchthath ◦g isdefinedinG andthen[h]◦[g] = [h ◦g ]. Thispartialcompositiondefines
1 1 1 1 1
agroupoidonclasses[x]’sasobjects. Thegroupoiddefinedinthismanneriscalledquotient
groupoidanddenotedbyG/N.
As it is stated in Brown [4, pp.218], in the case where the normal subgroupoid N is
totally intransitive, we have that (G/N) = G and G/N(x,y) consists of all cosets g ◦N(x)
0 0
forx,y ∈ G andg ∈ G(x,y). Thegroupoidcompositionbecomes
0
(h◦N(y))◦(g ◦N(x)) = (h◦g)◦N(x).
forg ∈ G(x,y)andh ∈ G(y,z).
WerecallthatacrossedmoduleofgroupsoriginallydefinedbyWhitehead[30,31],con-
sists of two groups A and B, an action of B on A denoted by b·a for a ∈ A and b ∈ B; and
a morphism α: A → B of groups satisfying the following conditions for all a,a ∈ A and
1
b ∈ B
(i) α(b·a) = b+α(a)−b,
(ii) α(a)·a = a+a −a.
1 1
3
We will denote such a crossed module by (A,B,α). Let (A,B,α) and (A(cid:48),B(cid:48),α(cid:48)) be two
crossedmodules. Amorphism(f ,f )from(A,B,α)to(A(cid:48),B(cid:48),α(cid:48))isapairofmorphismsof
1 2
groupsf : A → A(cid:48) andf : B → B(cid:48) suchthatf α = α(cid:48)f andf (b·a) = f (b)·f (a)fora ∈ A
1 2 2 1 1 2 1
andb ∈ B.
It was proved by Brown and Spencer in [11, Theorem 1] that the category XMod(Grp) of
crossedmodulesovergroupsisequivalenttothecategoryGrpGpdofgroup-groupoids.
3 Normal and quotient crossed modules in groups with op-
erations
The idea of the definition of categories of groups with operations comes from Higgins
[18] and Orzech [27]; and the definition below is from Porter [29] and Datuashvili [12,
pp.21],whichisadaptedfromOrzech[27].
Definition3.1 From now on C will be a category of groups with a set of operations Ω and
with a set E of identities such that E includes the group laws, and the following conditions
hold: IfΩ isthesetofi-aryoperationsinΩ,then
i
(a)Ω = Ω ∪Ω ∪Ω ;
0 1 2
(b) The group operations written additively 0,− and + are respectively the elements of
Ω ,Ω andΩ . LetΩ(cid:48) = Ω \{+},Ω(cid:48) = Ω \{−}andassumethatif(cid:63) ∈ Ω(cid:48),then(cid:63)◦ definedby
0 1 2 2 2 1 1 2
a(cid:63)◦ b = b(cid:63)aisalsoinΩ(cid:48). AlsoassumethatΩ = {0};
2 0
(c)Foreach(cid:63) ∈ Ω(cid:48),Eincludestheidentitya(cid:63)(b+c) = a(cid:63)b+a(cid:63)c;
2
(d) For each ω ∈ Ω(cid:48) and (cid:63) ∈ Ω(cid:48), E includes the identities ω(a + b) = ω(a) + ω(b) and
1 2
ω(a)(cid:63)b = ω(a(cid:63)b).
Acategorysatisfyingtheconditions(a)-(d)iscalledacategoryofgroupswithoperations.
Remark3.2 The set Ω contains exactly one element, the group identity; hence for instance
0
thecategoryofassociativeringswithunitisnotacategoryofgroupswithoperations.
Example3.3 Thecategoriesofgroups,ringsgenerallywithoutidentity,R-modules,associa-
tive, associative commutative, Lie, Leibniz, alternative algebras are examples of categories
ofgroupswithoperations.
A morphism between any two objects of C is a group homomorphism, which preserves
theoperationsinΩ(cid:48) andΩ(cid:48).
1 2
Thetopologicalversionofthisdefinitioncanbestatedasfollows:
4
Definition3.4 Let X be an object in C. If X has a topology such that all operations in Ω are
continuous,thenX iscalledatopologicalgroupwithoperationsinC.
In particular if C is the category of groups, then a topological group with operations
just becomes a topological group and if C is the category of R-modules, then it becomes a
topologicalR-module.
WewilldenotethecategoryoftopologicalgroupswithoperationsbyTopC.
FortheobjectsAandB ofC,thedirectproductA×B withtheusualoperationsbecomes
agroupwithoperations. HencethecategoryChasfiniteproducts.
ThesubobjectinthecategoryCcanbedefinedasfollows.
Definition3.5 Let A be an object in C. A subset B ⊆ A is called a subgroup with operations of
Aifthefollowingconditionsaresatisfied:
(i) b(cid:63)b ∈ B forb,b ∈ B and(cid:63) ∈ Ω ;
1 1 2
(ii) ω(b) ∈ B forb ∈ B andω ∈ Ω .
1
ThenormalsubobjectinthecategoryCofgroupswithoperationsisdefinedasfollows.
Definition3.6 [27,Definition1.7]LetAbeanobjectinCandN asubgroupwithoperations
of A. N is called a normal subgroup with operations or an ideal of A and written N (cid:1) A if the
followingconditionsaresatisfied:
(i) (N,+)isanormalsubgroupof(A,+);
(ii) a(cid:63)n ∈ N fora ∈ A,n ∈ N and(cid:63) ∈ Ω(cid:48).
2
Foramorphismf: A → B inC,Kerf = {a ∈ A|f(a) = 0}isanidealofA.
AquotientobjectinCisconstructedasfollows: LetAbeanobjectinCandN anidealof
A. ThentherelationonAdefinedby
a ∼ a iff a−a ∈ N
1 1
isanequivalencerelation. ThenthequotientsetA/N alongwiththeoperationsdefinedby
[a](cid:63)[a ] = [a(cid:63)a ]
1 1
ω([a]) = [ω(a)]
for(cid:63) ∈ Ω ,ω ∈ Ω becomesanobjectinCandcalledquotientgroupwithoperationsofAbyN.
2 1
InthefollowingpropositionweprovethatthecategoryChaskernelsincategoricalsense.
5
Proposition3.7 Let A be an object in C. Then N is an ideal of A if and only if it is a kernel of a
morphisminC.
Proof: WehavealreadyseenthatthekernelofamorphisminCisanidealofthedomain.
Conversely if N is an ideal of A, then the quotient A/N becomes an object in C and
quotientmorphismp: A → A/N hasN askernel. (cid:4)
Let A and B be two groups with operations in C. An extension of A by B is an exact
sequence
ı p
0 −→ A −→ E −→ B −→ 0 (1)
in which p is surjective and ı is the kernel of p. It is split if there is a morphism s: B → E
such that ps = ıd . A split extension of B by A is called a B-structure on A. Given such a
B
B-structure on A we get actions of B on A corresponding to the operations in C. For any
b ∈ B,a ∈ Aand(cid:63) ∈ Ω(cid:48) wehavetheactionscalledderivedactionsbyOrzech[27,pp.293]
2
b·a = s(b)+a−s(b)
(2)
b(cid:63)a = s(b)(cid:63)a.
Given a set of actions of B on A (one for each operation in Ω ), let A(cid:111)B be a universal
2
algebrawhoseunderlyingsetisA×B andwhoseoperationsare
(a(cid:48),b(cid:48))+(a,b) = (a(cid:48) +b(cid:48) ·a, b(cid:48) +b),
(a(cid:48),b(cid:48))(cid:63)(a,b) = (a(cid:48) (cid:63)a+a(cid:48) (cid:63)b+b(cid:48) (cid:63)a, b(cid:48) (cid:63)b).
Theorem3.8 [27, Theorem 2.4] A set of actions of B on A is a set of derived actions if and only if
A(cid:111)B isanobjectofC.
We recall that for groups with operations A and B, in [13, Proposition 1.1] all necessary
andsufficientconditionsfortheactionsofB onAtobederivedactionsaredetermined.
Lemma3.9 LetA,B,S andT beobjectsinC. SupposethatwehaveasetofderivedactionsofB on
AandasetofderivedactionsofT onS. IfS(cid:111)T isanidealofA(cid:111)B thenthefollowingsaresatisfied.
(a) S andT areidealsofAandB,respectively,
(b) b·s ∈ S forallb ∈ B,s ∈ S,
(c) (t·a)−a ∈ S forallt ∈ T,a ∈ A,
(d) b(cid:63)s ∈ S forallb ∈ B,s ∈ S,
6
(e) t(cid:63)a ∈ S forallt ∈ T,a ∈ A.
Proof: InthefollowingproofsweassumethatS (cid:111)T isanidealofA(cid:111)B and(cid:63) ∈ Ω(cid:48).
2
(a) For(s,0) ∈ S (cid:111)T and(a,0) ∈ A(cid:111)B,wehave
(a,0)+(s,0)−(a,0) = (a+s−a,0) ∈ S (cid:111)T
and
(a,0)(cid:63)(s,0) = (a(cid:63)s,0) ∈ S (cid:111)T.
Hencea+s−a ∈ S andS isanidealofA. Similarlyif(0,t) ∈ S(cid:111)T and(0,b) ∈ A(cid:111)B,
then
(0,b)+(0,t)−(0,b) = (0,b+t−b) ∈ S (cid:111)T
and
(0,b)(cid:63)(0,t) = (0,b(cid:63)t) ∈ S (cid:111)T.
HenceT isanidealofB.
(b) If(s,0) ∈ S (cid:111)T and(0,b) ∈ A(cid:111)B,then
(0,b)+(s,0)−(0,b) = (b·s,0) ∈ S (cid:111)T
andhenceb·s ∈ S.
(c) For(0,−t) ∈ S (cid:111)T and(t·a,0) ∈ A(cid:111)B,itimpliesthat
(t·a,0)+(0,−t)−(t·a,0) = ((t·a)−a,−t) ∈ S (cid:111)T
andso(t·a)−a ∈ S.
(d) For(s,0) ∈ S (cid:111)T and(0,b) ∈ A(cid:111)B,wehave
(0,b)(cid:63)(s,0) = (b(cid:63)s,0) ∈ S (cid:111)T
andthereforeb(cid:63)s ∈ S.
(e) If(0,t) ∈ S (cid:111)T and(a,0) ∈ A(cid:111)B,then
(0,t)(cid:63)(a,0) = (t(cid:63)a,0) ∈ S (cid:111)T.
andt(cid:63)a ∈ S.
7
(cid:4)
LetE beaB-structureonAandletF beaT-structureonS asbelow
(cid:117)(cid:117) s
E : 0 (cid:47)(cid:47)A ι (cid:47)(cid:47)E (cid:47)(cid:47)B (cid:47)(cid:47)0
p
(cid:117)(cid:117) s(cid:48)
F : 0 (cid:47)(cid:47)S ι(cid:48) (cid:47)(cid:47)F (cid:47)(cid:47)T (cid:47)(cid:47)0.
p(cid:48)
IfF isanidealofE thenF isanormalsubstructureofE. Thenwecanconstructthequotient
splitextensionasfollows.
(cid:114)(cid:114) s∗
E/F : 0 (cid:47)(cid:47)A/S ι∗ (cid:47)(cid:47)E/F (cid:47)(cid:47)B/T (cid:47)(cid:47)0
p∗
Thatis,E/F isaB/T-structureonA/S.
Definition3.10 [29]AcrossedmoduleinCisatriple(A,B,α),whereAandBaretheobjects
ofC,B actsonA,i.e.,wehaveaderivedactioninC,andα: A → B isamorphisminCwith
theconditions:
CM1. α(b·a) = b+α(a)−b;
CM2. α(a)·a(cid:48) = a+a(cid:48) −a;
CM3. α(a)(cid:63)a(cid:48) = a(cid:63)a(cid:48);
CM4. α(b(cid:63)a) = b(cid:63)α(a),α(a(cid:63)b) = α(a)(cid:63)b
foranyb ∈ B,a,a(cid:48) ∈ A,and(cid:63) ∈ Ω(cid:48).
2
A morphism (A,B,α) → (A(cid:48),B(cid:48),α(cid:48)) between two crossed modules is a pair f: A → A(cid:48)
andg: B → B(cid:48) ofthemorphismsinCsuchthat
(i) gα(a) = α(cid:48)f(a),
(ii) f(b·a) = g(b)·f(a),
(iii) f(b(cid:63)a) = g(b)(cid:63)f(a)
foranyb ∈ B,a ∈ Aand(cid:63) ∈ Ω(cid:48).
2
Definition3.11 We call a crossed module (S,T,σ) in C as a subcrossed module of a crossed
module(A,B,α)inCif
8
SCM1. S isasubobjectofA;andT isasubobjectofB;
SCM2. σ istherestrictionofα toS;
SCM3. theactionofT onS isinducedbytheactionofB onA.
Definition3.12 A subcrossed module (S,T,σ) of (A,B,α) in the sense of Definition 3.11 is
callednormalif
NCM1. T isanidealofB,
NCM2. b·s ∈ S forallb ∈ B,s ∈ S,
NCM3. (t·a)−a ∈ S forallt ∈ T,a ∈ A,
NCM4. b(cid:63)s ∈ S forallb ∈ B,s ∈ S,
NCM5. t(cid:63)a ∈ S forallt ∈ T,a ∈ A.
Remark3.13 Here we note that if (S,T,σ) is a normal subcrossed module of (A,B,α) then
by the conditions [CM2] of Definition 3.10; and [NCM2] and [NCM4] of Definition 3.12, S
becomesanidealofA.
As an example if (f,g): (A,B,α) → (A(cid:48),B(cid:48),α(cid:48)) is a morphism of crossed modules in C,
then(Kerf,Kerg,α ),thekernelof(f,g),isanormalsubcrossedmoduleof(A,B,α).
|Kerf
AsacorollaryofLemma3.9anormalcrossedmoduleinCcanbecharacterizedasfollow.
Corollary3.14 Asubcrossedmodule(S,T,σ)of(A,B,α)isnormalifandonlyifS(cid:111)T isanideal
ofA(cid:111)B.
WenowobtainquotientcrossedmoduleinCasfollows:
Theorem3.15 Let (S,T,σ) be a normal subcrossed module of (A,B,α) in C. Then we have a
crossed module(A/S,B/T,α∗)calledquotient crossed modulewhereA/S andB/T are quotient
groupswithoperations.
Proof: TheactionsofB/T onA/S aredefinedby
[b]·[a] = [b·a]
[b](cid:63)[a] = [b(cid:63)a].
9
These actions are well defined. If [b] = [b ] ∈ B/T and [a] = [a ] ∈ A/S, then b−b ∈ T and
1 1 1
a−a ∈ S. Hence
1
b·a−b ·a = b·(a−a +a )−b ·a
1 1 1 1 1 1
= b·(a−a )+b·a −b ·a
1 1 1 1
= b·(a−a )+(b−b +b )·a −b ·a
1 1 1 1 1 1
= b·(a−a )+(b−b )·(b ·a )−b ·a
1 1 1 1 1 1
andbythesubstitutionss = a−a ∈ S,t = b−b ∈ T anda = b ·a ∈ A,wegetthat
1 1 2 1 1
b·a−b ·a = b·(a−a )+(b−b )·(b ·a )−b ·a
1 1 1 1 1 1 1 1
= b·s+(t·a )−a .
2 2
Since b·s,(t·a )−a ∈ S by the conditions [NCM2] and [NCM3], we have that b·s+(t·
2 2
a )−a = b·a−b ·a ∈ S. Thismeans[b·a] = [b ·a ]inB/T. Moreover
2 2 1 1 1 1
b(cid:63)a−b (cid:63)a = b(cid:63)a−b(cid:63)a +b(cid:63)a −b (cid:63)a
1 1 1 1 1 1
= b(cid:63)(a−a )+(b−b )(cid:63)a
1 1 1
andbythesamesubstitutionsaboveweget
b(cid:63)a−b (cid:63)a = b(cid:63)s+t(cid:63)a .
1 1 1
andsobytheconditions[NCM4]and[NCM5]b(cid:63)a−b (cid:63)a ∈ S. Hence[b(cid:63)a] = [b (cid:63)a ]and
1 1 1 1
thereforetheseactionsarewelldefined.
Thesearederivedactionssincetheconditionsof[13,Proposition1.1]aresatisfied.
Ontheotherhanditisclearthattheboundarymap,α∗: A/S → B/T definedbyα∗([a]) =
[α(a)],iswelldefinedandtheconditions[CM1]-[CM4]ofDefinition3.10aresatisfied. (cid:4)
The following result is useful for some proofs (see for example the proofs of Theorem
5.11andTheorem6.7). Theproofisclearandsoitisomitted.
Proposition3.16 Let
(cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47)
1 S A P 1
σ α π
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
(cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47)
1 T B Q 1
be a short exact sequence of crossed modules in C. Then (S,T,σ) is a normal subcrossed module of
10