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COLIMITS IN THE CATEGORY OF QUADRATIC MODULES
HASAN AT·IK
Abstract. To use of colimits to put structures together is not uncommon in mathematics; coproducts particularly of structures such as groups and vector spaces have been known for a long time. Colimits have also been used in computer science for example to put together labeled graphs, in system theory etc. The importance of category for theoretical computer scientists is everyday increasing. In order to contribute usage of colimits, we show existence of …nite (co)limits in the category of quadratic modules of groups by careful construction of (co)product ob ject and (co)equilaser of morphisms of quadratic modules. Moreover, we give some examples of coproduct.
Introduction
Crossed modules were initially de…ned by Whitehead [12] as an algebraic model for homotopy connected 2-types. Corresponding de…nitions were given for some varieties of algebras in [8, 11, 10]. Brown, Higgins and Siviera [5] gave a construction of a coproduct object in the category of crossed modules over a group R: Then they obtained the colimits for the category of crossed R-modules of groups by using the equivalence of categories of Cat1-groups and that of crossed modules. In his thesis,
Nizar [9] has shown the existence of …nite limits and colimits in the category of crossed R-modules of commutative algebras. The Lie algebra case of the similar work has been done by Ladra and Casas in [7].
Recently, Baues [2] de…ned the notion of a quadratic module of groups as an alge-braic model for homotopy connected 3-types and gave a relation between quadratic modules and simplicial groups. In [1] Arvasi and Ulualan have explored relations among some algebraic models for homotopy 3-types such as quadratic modules, 2-crossed module,crossed square and simplicial groups.
Received by the editors: Feb. 25, 2015, Accepted: Nov. 18, 2015. 2010 Mathematics Subject Classi…cation. 18D35, 18G30, 18G50, 18G55. Key words and phrases. Colimit, coproduct, quadratic module.
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To understand the working category of quadratic modules we need the descrip-tion of its very remarkable construcdescrip-tions. One of the important categorical con-structions is colimit. In this paper, we show the existence of …nite colimits of category of quadratic modules. The interest in computing colimits of quadratic modules for algebraic topology is the 2-dimensional Van Kampen theorem due to Brown and Higgins [4, 3, 6, 7]. The following proposition [5] leads us to …nd …nite colimit in the category of quadratic modules.
Proposition: If functors S ! C admit (co)products and (co)equilasers, then they admit (co)limits.
Therefore, we construct the notions of products and coproducts for quadratic R-modules and we explore equilaser and coequilaser of morphisms of quadratic modules with the same domain and codomain. Hence we show the existence of …nite limits and colimits in the category of quadratic R-modules.
1. Quadratic Modules
Recall that a pre-crossed module is a group homomorphism @ : M ! Q together with an action of Q on M , written mq for q 2 Q and m 2 M, satisfying the
condition @(mq) = q 1@(m)q for all m 2 M and q 2 Q.
Let @ : M ! Q be a pre-crossed module. The Pei¤er commutator is de…ned as hm; m0i = m 1m0 1mm0@(m): The pre-crossed modules in which all Pei¤er commu-tators are trivial are precisely the crossed modules. Namely, a crossed module is a pre-crossed module @ : M ! Q satisfying the extra condition: m0@(m)= m 1m0m for all m; m02 M. We will denote the category of crossed modules by XMod.
Since the Pei¤er commutators are always de…ned in a pre-crossed module, it is a natural idea to factor out by the normal subgroup that they generate and consider the induced map from the quotient. The Pei¤er subgroup P2(@) = hM; Mi of M
is the subgroup of M generated by all Pei¤er commutators. For any pre-crossed module @ : M ! Q, the Pei¤er subgroup P2(@) of M is an N -invariant normal
subgroup. Let Mcr = M=P
2(@): This quotient group is a Q-group. Then we can
say that for the pre-crossed module @ : M ! Q, the induced map gives a crossed module @cr : Mcr! Q which is called the crossed module associated to @.
A nil(2)-module [2] is a pre-crossed module @ : M ! Q with an additional “nilpotency”condition. This condition is P3(@) = 1, where P3(@) is the subgroup
We shall denote the category of nil(2)-modules by Nil(2). Now we give the following de…nition from [2].
De…nition 1.1. A quadratic module (!; @2; @1) is a diagram
of homomorphisms between groups such that the following axioms are satis…ed. QM1) The homomorphism @1 : C1 ! C0 is a nil(2)-module with Pei¤ er
com-mutator map w de…ned above. The quotient map C1 C = (C1cr)ab is given by
x 7! fxg; where fxg 2 C denotes the class represented by x 2 C1 and C = (C1cr)ab
is the abelianization of the associated crossed module Ccr
1 ! C0.
QM2) The boundary homomorphisms @2and @1 satisfy @1@2= 1 and the quadratic
map ! is a lift of the Pei¤ er commutator map w, that is @2! = w.
QM3) C2 is a C0-group and all homomorphisms of the diagram are equivariant
with respect to the action of C0. Moreover, the action of C0 on C2 satis…es the
formula (a 2 C2; x 2 C1)
a@1x= !((fxg f@
2ag) (f@2ag fxg))a:
QM4) Commutators in C2 satisfy the formula (a; b 2 C2)
!(f@2ag f@2bg) = [b; a]:
A morphism ' : (!; @2; @1) ! (!0; @20; @10) between quadratic modules is given by
a commutative diagram, ' = (f2; f1; f0)
where (f1; f0) is a morphism between nil(2)-modules which induces ' : C ! C0and
modules by Quad. With a …xed group R, consider the category of quadratic R-modules
We will denote the category of such quadratic modules by Quad=R. A morphism between quadratic R-modules is a quadratic module morphism ' = (f2; f1; f0) as
de…ned above in which f0 is the identity homomorphism on the group R.
2. Finite Limits in Quad=R
In this section, we will show that the category of quadratic R-modules has …nite limits by constructing the product of two quadratic modules and equaliser of two morphisms in the category of quadratic R-modules.
Proposition 2.1. In Quad=R every pair of morphisms with common domain and codomain has an equaliser.
Proof: Let (f; g) : (L1; C; R) ! (L2; D; R) be two morphisms of quadratic modules
where
f = (f2; f1) and g = (g2; g1), f2; g2 : L1 ! L2, f1; g1 : C ! D. Let E and
F be sets as follows E = fc 2 C : f1(c) = g1(c)g and F = fl1 2 L1 : f2(l1) =
g2(l1)g: It is clear that (F; E; "2; "1) is a subquadratic module of (L1; C; @2; @1) in
which "2 and "1 are induced from @2 and @1 respectively. The inclusion (i; j) :
(F; E; R) ! (L1; C; R) is a morphism of quadratic R-modules. Suppose that there
exist a quadratic module (F0; E0; "0
2; "01) and a morphism (i0; j0) : (F0; E0; R) !
(L1; C; R) of quadratic R-modules such that f2i0(y) = g2i0(y), f1j0(x) = g1j0(x)
for all x 2 E0; y 2 F0. Hence i0(y) 2 F; j0(x) 2 E: Thus we de…ne : E0 ! E as
(x) = j0(x) and : F0 ! F as (y) = i0(y). The fact (i0; j0) being a quadratic
unique for the commutative diagram
Namely, j = j0; i = i0: Thus the morphism (i; j) is an equalizer of (f; g):
Proposition 2.2. The category Quad=R has pullbacks.
Proof: Let (f2; f1) : (L1; C; R) ! (L2; B; R) and (g2; g1) : (L3; D; R) ! (L2; B; R)
be two morphisms of quadratic modules where
are quadratic R-modules. We form the groups X; Y such that X = f(c; d) : f1(c) =
g1(d)g C D and Y = f(l1; l3) : f2(l1) = g2(l3)g L1 L3 and morphisms x; y
such that x : X ! B is given by (c; d) 7! f1(c) = g1(d) and y : Y ! X is given
by (l1; l3) 7! (@2(l1); 2(l3)):
Thus we obtain the following commutative diagram.
Since
1x((c; d)r) =
for all r 2 R; (c; d) 2 X, the map 1x is a pre-crossed module. For (c; d); (c0; d0); (c00; d00) 2 X we obtain hh(c; d); (c0; d0)i; (c00; d00 1(c0; d0 1(c; d)(c0; d0 1x(c;d); (c00; d00)i = h(c 1c0 1c; d 1d0 1d)(c0; d0 1f1c; (c00; d00)i = h(c 1c0 1cc0 1f1c; d 1d0 1dd0 1f1c); (c00; d00)i = h(c 1c0 1cc0@1c; d 1d0 1dd0 1g1d); (c00; d00)i = h(hc; c0i; hd; d0i); (c00; d00)i = (hc; c0 1; hd; d0 1)(c00 1; d00 1)(hc; c0i; hd; d0i)(c00; d00 1x(hc;c0i;hd;d0i) = (hc; c0 1c00 1hc; c0ic00 1f1(hc;c0i); hd; d0 1d00 1hd; d0id00 1g1(hd;d0i)) = (hhc; c0i; c00i; hhd; d0i; d00i) = (1; 1): Therefore, 1x : X ! R is a nil(2)-module. Now, we will show that
is a quadratic R-module.
QM1) We know that 1x : X ! R is a nil(2)-module. We have also
1xy(l1; l3) = 1x(@2(l1); 2(l3)) = 1(f2(@2(l1))) = 1( 2(f1(l1))) = 1 2(f1l1) = 1:
QM2) For f(c; d)g f(c0; d0)g 2 M00 M00; we de…ne !00: M00 M00! Y as follows:
Then we have y!00(f(c; d)g f(c0; d0)g) = y(!1(fcg fc0g); !2(fdg fd0g)) = (@2!1(fcg fc0g); 2!2(fdg fd0g)) = (w1(fcg fc0g); w2(fdg fd0g)) = (hc; c0i; hd; d0i) = h(c; d); (c0; d0)i = w00(f(c; d)g f(c0; d0)g): QM3) For(l1; l3) 2 Y; (c; d) 2 X, we get !00(fy(l1; l3)g f(c; d)gf(c; d)g fy(l1; l3)g) = !00(f(@2l1; 2l3)g f(c; d)gf(c; d)g f(@2l1; 2l3)g = (!1(f@2l1g fcg); !2(fdg f 2l3g))(!1(fcg f@2l1g); !2(f 2l3g fdg)) = (!1(f@2l1g fcg)(fcg f@2l1g); !2(fdg f 2l3g))((f 2l3g fdg)) = (l11l@1(c) 1 ; l 1 3 l 1(d) 3 ) = (l 1 1 l 1 f1(c) 1 ; l 1 3 l 1 g1(d) 3 ) = (l11l 1f1(c) 1 ; l31l 1 f1(c) 3 ) = (l1; l3) 1(l1; l3) 1f1(c) = (l1; l3) 1(l1; l3) 1x(c;d): QM4) For (l1; l3); (l01; l02) 2 Y , we obtain !00(fy(l1; l3)g fy(l10; l03)g) = !00(f(@2l1; 2l3)g f(@2l10; 2l03)g) = (!1(f@2l1g f@2l01g; !2(f 2l3g f 2l30g)) = ([l1; l01]; [l3; l03]) = [(l1; l3); (l01; l03)]:
Thus the diagram
There are two induced morphisms p1; p2 : 00 ! 1 and q1; q2 : 00 ! 3 given
by projections; note that f p = gq; and this shows that the diagram
is commutative and the morphisms p and q satisfy the universal property: let (p0
1; p02) : 0 ! 1 and (q10; q20) : 0 ! 3 be any morphisms of quadratic modules
with f p0 = gq0 and 0 : L0 ! A ! R; then there exists a unique morphism (h1; h2) : 0 ! 00 given by h1(x0) = (p01(x0); q01(x0)) and h2(y0) = (p20(y0); q02(y0))
for x0 2 A; y02 L0 such that the diagram
is commutative; i.e. p1h1 = p01, p2h2 = p02, q1h1= q01 and q2h2= q20. Then we say
that Quad=R has pullbacks.
We note that the category of quadratic modules has terminal object, t, thus
the proof of the following is easy.
Proposition 2.3. Quad=R has …nite products.
Proof: It is of course su¢ cient to prove the proposition for a family having just two members say 1 and 3. The product 1u 3 will be the pullback over the
and the diagram
is commutative i.e. f2p1= g2p2; @1p01= 1p02. Then it is easy to check that
is a quadratic module where 1 : C u D ! R is given by 1(c; d) = @1p01(c; d) = 1p02(c; d) and 2 : L1u L3 ! C u D is given by 2(l1; l3) = (@2l1; 2l3) and
!13: ((C uD)cr)ab ((C uD)cr)ab ! L1uL3is given by !13(f(c; d)g f(c0; d0)g) =
(!1(fcg fc0g); !3(fdg fd0g)):
Then by induction, Quad=R has …nite products.
Proposition 2.4. Quad=R has a limit for any functor F : ! Quad=R with …nite.
Proof: As Quad=R has …nite products and equilasers, the result follows.
Therefore the category Quad=R is …nitely complete, i.e. it has all …nite limits. 3. Finite Colimits in Quad=R
This section will describe the construction of …nite colimits of quadratic modules over a group R: First we will give the coequalizer of two morphisms then the construction of the coproducts of two quadratic R-modules.
Proposition 3.1. In Quad=R every pair of morphisms of crossed modules with common domain and codomain has a coequilaser.
Proof: Let f = (f1; f2); g = (g1; g2) : 1 ! 2 be two quadratic R-module
Let I be a normal subgroup of M2generated by elements of the form f1(m) g1(m),
for all m 2 M1 and J be the normal subgroup of L2 generated by elements of the
form f2(l) g2(l), for all l 2 L1: Note that I ker 1 and Im 2 is the normal
subgroup of I. Set the factor groups M = M2=I and L = L2=J . De…ne 1: M ! R
as 1(m2I) = 1(m2) and 2: L ! M as 2(l2J ) = 2(l2)I. In this case
is a quadratic module with the quadratic map ! : (M )cr ab (M )cr ab ! L which is given by fm
2Ig fm02Ig 7 ! !(fm2g fm02g):
Since 1((m2I)r) = 1(m2rI) = 1( m2r) = r 1 1(m2)r = r 1 1(m2I)r
and for m2I; m02I; m002I 2 M,
hhm2I; m02Ii; m002Ii = h(m21I)(m0 12 I)(m2I)(m0 1 (m0 2I) 2 ; (m002I)i = hm21m0 12 m2m02 1(m02)I; (m00 2I)i = hhm2; m2iI; m002Ii = hhm2; m02i; m002iI = I;
the map 1is a nil(2)-module. We leave as an exercise to the reader the veri…cation
of remaining quadratic module axioms. Moreover, the induced map p = (p1; p2) : 2! is a quadratic module morphism. Namely the diagram
is commutative. Finally we will check the universal property of p: If there exist a quadratic module and a quadratic module morphism p0 = (p0
1; p02) : 2 ! 0 then
there exists a unique quadratic module morphism ' = ('0
1; '02) : ! 0 which
is '0
Then p is universal morphism so we get the following commutative diagram
Therefore, p is an coequilaser of f and g. Construction of Coproduct
We give the construction of coproduct of two quadratic modules in the category of quadratic R-modules. Let
be two quadratic modules. Suppose that X acts on C via 1, so we can form the
semidirect product
X n C = f(x; c) : x 2 X; c 2 Cg
with the multiplication (x; c)(x0; c0) = (xx0 1x0c0) for (x; c); (x0; c0) 2 X n C, where
an action of R on X n C is given by (x; c)r = (xr; cr) for r 2 R: We get the
injections i1 : X ! X n C, i1(x) = (x; 1) and j1 : C ! X n C, j1(c) = (1; c): We
de…ne 01 : X n C ! R by 01(x; c) = 1x@1c: It is clear that 01 is a well-de…ned
homomorphism. Let P be a normal subgroup of X n C generated by elements of the forms:
(1) hh(x; c); (x0; c0)i; (x00; c00)i
(2) h(x; c); h(x0; c0); (x00; c00)ii
for (x; c); (x0; c0); (x00; c00) 2 X n C.
Thus we can form the factor group X nC=P and we get an induced morphism 1:
X nC=P ! R as 1((x; c)P ) = 1x@1c: Clearly 1is a nil(2)-module. Furthermore,
L2 acts on L1 via 1 2: Since 1 2 = 1, the semidirect product of L2 and L1 is
direct product of L2 and L1; that is L2n L1 = L2 L1: We get the injections
de…ne the map 02: L2 L1! X n C by 02(l2; l1) = ( 2l2; @2l1): 02is also a
well-de…ned group homomorphism. Let P0 be a normal subgroup of L2 L1 generated
by elements of the forms
(1) (!2(fx1g fhx2; x3ig); !1(fc1g fhc2; c3ig));
(2) (!2(fhx1; x2ig fx3g); !1(fhc1; c2ig fc3g))
for all (x1; c1); (x2; c2); (x3; c3) 2 X n C.
We can form the factor group L2 L1=P0and we de…ne a map 2: L2 L1=P0!
X n C=P as 2((l2; l1)P0) = ( 2l2; @2l1)P: An action of X n C=P on L2 L1=P0
is given via 1 such that ((l2; l1)P0(x;c)P = ((l2; l1)P0 1(x;c)P = (l2 1x; l1@1c)P0 for
(x; c)P 2 X n C=P and (l2; l1)P02 L2 L1=P0. Then we get the following result.
Proposition 3.2.
is a quadratic R-module.
Proof: Firstly, we de…ne the quadratic map ! by
!(f(x; c)P g f(x0; c0)P g) = (!2(fxg fx0g); !1(fcg fc0g))P0
for (x; c)P; (x0; c0)P 2 X n C=P:
QM1) We know that 1 is a nil(2)-module and
1 2((l2; l1)P ) = 1(( 2(l2); @2(l1))P ) = 1 2(l2)@1@2l1= 1: QM2) For (x; c)P; (x0; c0)P 2 X n C=P we get 2!(f(x; c)P g f(x0; c0)P g) = 2((!2(fxg fx0g); !1(fcg fc0g))P0) = ( 2!2(fxg fx0g); @2!1(fcg fc0g))P = (w2(fxg fx0g); w1(fcg fc0g)P ) = (hx; x0i; hc; c0i)P = h(x; c); (x0; c0)iP = w(f(x; c)P g f(x0; c0)P g):
QM3) For (x; c)P 2 X n C=P and (l2; l1)P0 2 L2 L1=P0, we obtain !(f 2((l2; l1)P0)g f(x; c)P gf(x; c)P g f 2((l2; l1)P0)g) =!(f( 2l2; @2l1)P g f(x; c)P gf(x; c)P g f( 2l2; @2l1)P g) =(!2(f 2l2g fxgfxg f 2l2g); !1(f 2l2g fcgfcg f 2l2g))P0 =(l2 1l2 1(x); l1 1l1@1(c))P0= (l2; l1) 1(l2 1x; l1@1c)P0 =(l2; l1) 1(l2; l1) 1(x;c)PP0: QM4) For (l2; l1)P0; (l02; l01)P02 L2 L1=P0, we have !(f 2((l2; l1)P0)g f 2((l20; l01)P0)g) = !(f( 2l2; @2l1)P g f( 2l20; @2l01)P g) = (!2(f 2l2g f 2l20g); !1(f@2l1g f@2l01g))P0 = ([l2; l20]; [l1; l10])P0 = [(l2; l1)P0; (l02; l10)P0]:
Theorem 3.3. The constructed quadratic module
where L2 L1= L2 L1=P0 and X C = X n C=P with the morphisms i = (i1; i2),
j = (j1; j2) is the coproduct of the quadratic modules
Proof: We will check the universal property of morphisms (i1; j1) into (X n
Consider an arbitrary quadratic R-module
and morphisms of quadratic R-modules f = (f2; f1) : 2! B, g = (g2; g1) : 1! B, i.e.
Then there is a map h = (h1; h2) : ! B given by h1((l2; l1)P0) = f1(l2)g1(l1),
h2((x; c)P ) = f2(x)g2(c):
It is a unique morphism of quadratic modules for the diagram
to commute. Actually we obtain
h2i2(l2) = h2(l2; 1) = f2(l2)g2(1) = f2(l2);
h1i1(x) = h1(x; 1) = f1(x)g1(1) = f1(x);
h2j2(l1) = h2(1; l1) = f2(1)g2(l1) = g2(l1);
h1j1(c) = h1(1; c) = f1(1)g1(c) = g1(c):
The construction of coproducts in Quad=R will give us a functor
which is left adjoint to the diagonal functor
M : Quad=R ! Quad=R Quad=R:
Proposition 3.4. Quad=R has all colimits for any functor J : ! Quad=R; i.e. Quad=R is cocomplete.
Proof: Since Quad=R has coproduct and coequilaser, it is clear. 4. Example for Coproduct of Quadratic Modules
We will give a description of the coproduct of quadratic modules in the particular case of two quadratic modules 1 : L1 ! C ! R and 2 : L2 ! X ! R in the
useful case when v1(X) 1(C) and there is a P -equivariant section : 1c ! C
of 1:
De…nition 4.1. If C acts on the group X we de…ne [X; C] to be the subgroup of X generated by the elements x 1xc for all x 2 X; c 2 C: This subgroup is called the
displacement subgroup.
Proposition 4.2. The displacement subgroup [X; C] is a normal subgroup of X: Proof: Let c 2 C; x; x12 X: We easily check that
x11(x 1xc)x1= (xx1) 1(xx1)c(x11xc1) 12 [X; C]
De…nition 4.3. We de…ne X=[X; C] as a quotient of X by displacement subgroup. The elements of X=[X; C] is written by [x]. It is clear that X=[X; C] is a trivial C-module since [xc] = [x]:
Proposition 4.4. Let 1: C ! R; v1: X ! R be nil(2)-modules, so that C acts
on X via 1. Then R acts on X=[X; C] by [x]r = [x]r for r 2 R: Moreover this
action is trivial when restricted to 1C:
Proof: It is because (x 1xc)r = (x 1)r(xc)r = (xr) 1(xr)cr
for all x 2 X; c 2 C; r 2 R: The action is trivial since [x] 1c= [x 1c] = [xc] = [x]:
Proposition 4.5. Let 1 : C ! R; v1 : X ! R be nil(2)-modules, such that
v1(X) 1(C). Then X=[X; C] is Abelian and therefore " : C X=[X; C] ! R
Proof: Let x1; x2; x32 X: Choose c 2 C such that v1x1= 1c: Then hh[x1]; [x2]i; [x3]i = h[x11][x21][x1][xv21x1]; [x3]i = [(xv1x1 2 ) 1][x11][x2][x1][x31][x11][x21][x1][xv21x1][x v1(x11x 1 2 x1xv1x12 ) 3 ] = ([x2 1c]) 1[x11][x2][x1][x31][x 1 1 ][x 1 2 ][x1][x21c][x3] = ([x21][x11][x2][x1])[x3] 1([x21][x 1 1 ][x2][x1]) 1[x3] = 1:
therefore X=[X; C] is Abelian. Now we will show that " is a nil(2)-module
hh(c1; [x1]); (c2; [x2])i; (c3; [x3])i = h(c1; [x1]) 1(c2; [x2]) 1(c1; [x1])(c2; [x2])"(c1;[x1]); (c3; [x3])i = h(c11c21c1c21c1; [x1] 1[x2] 1[x1][x2]); (c3; [x3])i = h(hc1; c2i; 1); (c3; [x3])i = (hc1; c2i; 1) 1(c3; [x3]) 1(hc1; c2i; 1)(c3; [x3])"(hc1;c2i;1) = (hc1; c2i 1c31hc1; c2ic31(hc1;c2i); 1) = (hhc1; c2i; c3i; 1) = (1; 1)
Proposition 4.6. Let 1 : L1 ! C2 ! R and1 2 : L2 v2
! X v1
! R be two quadratic modules with v1(X) 1(C) and L2 is an Abelian group with trivial
action of 1C: Let 1 : 1(C) ! C be an R-equivariant section of 1: Then the
morphisms of quadratic modules
i1: C ! C X=[X; C]; c 7! (c; 1)
i2: L1 ! L1 L2; l17! (l1; 1)
j1: X ! C X=[X; C]; x 7! ( 1v1x; [x])
j2: L2 ! L1 L2; l27! (1; l2)
give a coproduct of quadratic modules. Hence the canonical morphism of quadratic modules C X ! C X=[X; C]; c x 7! (c 1v1x; [x])
L1 L2 ! L1 L2; l1 l27! (l1; l2)
Proof: It can be shown easily that
is a quadratic module where
"1: C X=[X; C] ! R; (c; [x]) 7! 1c
"2: L1 L2 ! C X=[X; C]; "2(l1; l2) 7! ( 2l1; 1)
! : C C ! L1 L2; f(c1; [x1])g f(c2; [x2])g 7! (!1(fc1g fc2g); 1)
Then one can show that pairs (i1; j1) and (i2; j2) satis…es the universal property of
the coproduct of quadratic modules.
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