C om mun.Fac.Sci.U niv.A nk.Series A 1 Volum e 66, N umb er 2, Pages 225–242 (2017) D O I: 10.1501/C om mua1_ 0000000814 ISSN 1303–5991
http://com munications.science.ankara.edu.tr/index.php?series= A 1
ON PULLBACK AND INDUCED CROSSED MODULES OF R-ALGEBROIDS
OSMAN AVCIO ¼GLU AND ·IBRAHIM ·ILKER AKÇA
Abstract. In this paper we study the pullback and induced crossed mod-ules of R-algebroids, prove that the related induced crossed module functor is the left adjoint of the related pullback crossed module functor and give some consequences of the adjunction.
1. Introduction
Crossed modules, algebraic models of two types, were …rstly invented by White-head [22, 23] in his study on homotopy groups. Various studies on crossed modules of groups and groupoids can be found in papers and books such as [8, 9, 19], and those of algebras in [4, 5, 18, 20, 21] and in [11, 12, 13] with di¤erent names. G.H. Mosa [17] has studied crossed modules of R-algebroids and double R-algebroids.
Pullback crossed modules of groups is introduced in [8, 10] and induced crossed modules of groups in [7, 8, 10]. It’s proved in [8] that, in the category of crossed modules of groups, the induced crossed module functor is the left adjoint of the pullback crossed module functor.
R-algebroids were especially studied by B. Mitchell, [14, 15, 16], and by S. M. Amgott, [3]. B. Mitchell has given a categorical de…nition of R-algebroids. G.H. Mosa has de…ned crossed modules of R-algebroids and proved the equivalence of crossed modules of algebroids and special double algebroids with connections in [17]. M. Alp has constructed the pullback and pushout crossed modules of algebroids in [1] and [2], respectively.
After the introduction, in the second section of this study we give some basic data on algebroids, modules over algebroids and (pre)crossed modules of R-algebroids.
In the third section, …rst we study the pullback crossed modules of R-algebroids, whose construction is done by M. Alp in [1]. Then we prove the ‘naturality property’ of this construction (Proposition 2).
Received by the editors: May 18, 2016; Accepted: January 25, 2017.
2010 Mathematics Subject Classi…cation. Primary 18A40; Secondary 18A30, 18B99. Key words and phrases. Algebroids, crossed modules of algebroids, adjoint functors.
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In the fourth section …rst we give a construction of induced crossed modules of R-algebroids using the construction of free precrossed modules of R-algebroids in [6]. Then we prove the ‘naturality property’ of this construction (Proposition 4). Finally, as the basic goal of this paper, we prove in the category of crossed modules of R-algebroids that the induced crossed module functor is the left adjoint of the pullback crossed module functor (Theorem 1).
In the …fth section we explore some consequences of the adjunction given in Theorem 1.
Throughout the paper R is a commutative ring. 2. Preliminaries The following data can be found in [3, 14, 15, 16, 17]:
De…nition 1. A category of which each homset has an R-module structure and of which composition is R-bilinear is called an R-category. A small R-category is called an R-algebroid. Moreover if we omit the axiom of the existence of identi-ties from an R-algebroid structure then the remaining structure is called a pre-R-algebroid.
Remark 1. If A is an R-algebroid then:
1: A has an object set Ob (A) = A0, a morphism set Mor (A) and two functions
s; t : Mor (A) ! Ob (A), the source and target functions, respectively.
2: For any a 2 Mor (A) if sa = x and ta = y then x and y are called as the source and target of a, respectively, and a is said to be from x to y.
3: For all x; y 2 A0 the set of all morphisms from x to y, which is denoted by
A (x; y) and called a homset, is an R-module.
Throughout this paper, for any R-algebroid A, a 2 A will mean that a is a morphism of A and the composition of any a; b 2 A with ta = sb will be denoted by ab. Moreover the identity morphism on any x 2 A0 will be denoted by 1x or
only by 1 if there is no ambiguity.
De…nition 2. An R-linear functor between two R-categories is called an R-functor and an R-functor between two R-algebroids is called an R-algebroid morphism. Moreover a map between two pre-R-algebroids obtained by omitting the axiom of identity preservation from an R-algebroid morphism is called a pre-R-algebroid morphism.
Note from the De…nitions 1 and 2 that an R-algebroid is a pre-R-algebroid and an R-algebroid morphism is a pre-R-algebroid morphism.
De…nition 3. Let A be an R-algebroid and I = fI (x; y) A (x; y) j x; y 2 A0g be
a family of R-submodules. For all w; x; y; z 2 A0, a02 A (w; x) ; a002 A (y; z) and
a 2 I (x; y) if a0a 2 I (w; y) and aa002 I (x; z) then I is said to be a two sided ideal
De…nition 4. Let A be an-R-algebroid and M be a pre-R-algebroid with the same object set A0 as A. A family of maps de…ned for all x; y; z 2 A0 as
M (x; y) A (y; z) ! M (x; z)
(m; a) 7 ! ma
is called a right action of A on M; if the conditions 1: (ma)a0
= maa0, 4: (m
1+ m2)a = ma1+ ma2,
2: ma1+a2= ma1+ ma2, 5: (r m)a = r ma= mr a,
3: (m0m)a = m0ma, 6: m1tm = m
are satis…ed for all r 2 R; a; a0; a
1; a2 2 A, m; m0; m1; m2 2 M with tm0 = sm,
tm = tm1= tm2= sa = sa1= sa2, ta = sa0.
A left action of A on M is de…ned in the same manner:
De…nition 5. Let A be an R-algebroid and M be a pre-R-algebroid with the same object set A0 as A. If A has a right and a left action on M and if the condition
(am)a0 =a ma0
is satis…ed for all m 2 M, a; a02 A with ta = sm, tm = sa0 then A is said to have
an associative action on M.
De…nition 6. Let A be an R-algebroid and M be a pre-R-algebroid with the same object set A0 as A. If A has an associative action on M then M is called an
A-module. In this case we write (M; A) and call this pair an (A-)module over R-algebroids. Moreover, for any two modules (M; A) and (N; B) over R-algebroids, a pair (f; g) : (M; A) ! (N; B) is called a module morphism over R-algebroids if f : M ! N is a pre-R-algebroid morphism, g : A ! B is an R-algebroid morphism and the conditions
1: f m 2 N (g (sm) ; g (tm)) ,
2: f (am) =ga(f m) and f ma0 = (f m)ga0 are satis…ed for all m 2 M, a; a0 2 A with ta = sm, tm = sa0.
All modules over R-algebroids and their morphisms form a category denoted by ModAlg (R). Moreover, all A-modules with the identity morphism on A form a subcategory ModAlg(R) =A of ModAlg(R).
De…nition 7. Let A be an R-algebroid, M be a pre-R-algebroid with the same object set A0 as A and A have an associative action on M. A pre-R-algebroid morphism
: M ! A is called a precrossed (A-)module of R-algebroids if the condition
CM1) (am) = a ( m) and ma0
= ( m) a0
is satis…ed and a precrossed (A-)module : M ! A of R-algebroids is called a crossed (A-)module of R-algebroids if the condition
CM2) m m0
is satis…ed for all a; a02 A and m; m02 M with ta = sm, tm = sa0= sm0.
Let M = ( : M ! A) and N = ( : N ! B) be two (pre)crossed modules of R-algebroids, f : M ! N be a pre-R-algebroid morphism and g : A ! B be an R-algebroid morphism. The pair (f; g) : M ! N is called a ( pre)crossed module morphism if the conditions
1: f (am) = ga(f m) and f ma0
= (f m)ga0,
2: f = g
are satis…ed for all a; a0 2 A, m 2 M with ta = sm, tm = sa0. Note that if
: M ! A is a (pre)crossed module then M is an A-module and a (pre)crossed module morphism is a module morphism satisfying the second condition.
All precrossed modules of R-algebroids and their morphisms form a category denoted by PXAlg(R). Moreover, all precrossed A-modules of R-algebroids with the identity morphism on A form a subcategory, PXAlg(R) =A; of PXAlg(R). Similarly, all crossed modules of R-algebroids form the category XAlg(R) and all crossed A-modules of R-algebroids form the subcategory XAlg(R) =A of XAlg(R) : Obviously, XAlg(R) is a full subcategory of PXAlg(R) and XAlg(R) =A is a full subcategory of PXAlg(R) =A:
Example 1. If A is an R-algebroid and I is a two sided ideal of A, then the inclusion morphism
i : I ! A
is a crossed module with the action of A on I de…ned by
ab = ab and ba0= ba0
for all a; a02 A, b 2 I with ta = sb, tb = sa0.
3. Pullback crossed modules of R-algebroids
M. Alp has given a construction of the pullback crossed modules of R-algebroids in [1]. In this section, in the proof of the following proposition after giving a brief summary of his construction in part 1 we show that the pullback crossed module satis…es the related universal property in part 2. Moreover we also specify the pullback crossed module functor.
Proposition 1. Let A and B be two R-algebroids, f : A ! B be an R-algebroid morphism and N = ( : N ! B) be a crossed B-module of R-algebroids. There exists a crossed A-module f N = f : f N ! A of R-algebroids and a crossed module morphism f ; fb : f N ! N such that for any crossed A-module M = ( : M ! A) of R-algebroids and crossed module morphism ( ; f) : M ! N there exists a unique crossed A-module morphism (h; idA) : M ! f N such that
f N , with the morphism f ; f , is called the pullback crossed module of N alongb f . The pullback crossed module is unique up to isomorphism.
Proof. 1: i: De…ne the source and target of each (n; a) 2 N A as s (n; a) = sa and t (n; a) = ta, respectively, and for all x; y 2 A0 form the subset f N (x; y) =
f(n; a) j s (n; a) = x, t (n; a) = y and fa = ng of N A.
ii: For all x; y 2 A0 the set f N (x; y) has an R-module structure with the
addition de…ned as (n1; a1) + (n2; a2) = (n1+ n2; a1+ a2) and the R-action de…ned
as r (n; a) = (r n; r a).
iii: The family f N = ff N (x; y) j x; y 2 A0g is a pre-R-algebroid with the
composition de…ned as (n; a) (n0; a0) = (nn0; aa0) and an A-module with the asso-ciative A-action de…ned asa00(n; a) = f a00n; a00a and (n; a)a0
= nf a0; aa0 under
the conditions ta00= s (n; a), t (n; a) = sa0.
iv: The map f : f N ! A de…ned as f (n; a) = a is a crossed module. (for details see [1])
2: De…ne bf : f N ! A as bf (n; a) = n. Clearly bf is a pre-R-algebroid morphism and f ; fb is a crossed module morphism since
b
f (n; a) = n 2 N (sn; tn) = N (s ( n) ; t ( n)) = N (s (fa) ; t (fa)) = N (f (sa) ; f (ta)) = N (f (s (n; a)) ; f (t (n; a))) and b f (n; a)a0 = f nb f a0; aa0 = nf a0 = f (n; a)b f a 0 , b f a00(n; a) = fb f a00n; a00a =f a00n =f a00 f (n; a)b
Now, for any crossed A-module M = ( : M ! A) and crossed module morphism ( ; f ) : M ! N , de…ne h : M ! f N as h (m) = ( m; m) . h is well de…ned since = f and so ( m) = f ( m) for all m 2 M. By a direct calculation, it can
be shown that (h; idA) : M ! f N is a crossed A-module morphism. Moreover
m = bf ( m; m) = bf hm for all m 2 M, which means = bf h, as required. Let (h0; id
A) : M ! f N be a crossed A-module morphism satisfying = bf h0.
Then h0 must be de…ned as h0m = (h0
1m; h02m) where h01m 2 N and h02m 2 A. But,
in this case, h0
1m = bf (h01m; h02m) = bf h0m = m and h02m = f (h01m; h02m) = f h0m = m since f h0 = . So h0m = (h01m; h02m) = ( m; m) = hm for all
m 2 M, which means h is unique.
Finally, assume that ef N = fe : ef N ! A is a crossed module of R-algebroids and f ; fe : ef N ! N is a crossed module morphism which together satisfy the same conditions as f N and f ; f . Then there exists unique mor-b phisms eh; idA : f N ! ef N and (h; idA) : ef N ! f N making related
universal diagrams commutative. So f = feeh = f heh and fe = f h = feehh which together requires heh = idf N and ehh = idf Ne . Thus eh is an isomorphism
and the pullback crossed module f N , with the morphism f ; f , is unique up tob isomorphism.
So, we get a pullback crossed module functor f : XAlg(R)=B ! XAlg(R)=A which gives a crossed A-module f N for any crossed B-module N and is de…ned as f (g; idB) = (f g; idA) on morphisms such that (f g) (n; a) = (gn; a).
Now we prove an important property of pullback crossed module:
Proposition 2. If A, B, C are R-algebroids and f : A ! B, f0 : B ! C are
R-algebroid morphisms then the functor f f0 is naturally isomorphic to (f0f ) .
Proof. For any N = ( : N ! C) 2 XAlg(R)=C, the B-module f0 N; the A-module
(f f0 ) N = f (f0 N) and the A-module (f0f ) N is formed by the pairs (n; b),
((n; b) ; a) and (n; a), respectively, under the conditions n = f0b,
f0 (n; b) = f a
and n = (f0f ) a, the second of which means b = f a since f0 (n; b) = b by
de…nition. So, any element ((n; b) ; a) of (f f0 ) N is, in fact, of the form ((n; f a) ; a).
It can easily be shown that, for all crossed modules N = ( : N ! C), the map
N: (f f0 ) N ! (f0f ) N, de…ned as N((n; f a) ; a) = (n; a), is an isomorphism.
crossed module morphisms (g; idC) : N ! N0 and for all ((n; f a) ; a) 2 (f f0 ) N (f0f ) g N ((n; f a) ; a) = (f0f ) g (( N) ((n; f a) ; a)) = (f0f ) g (n; a) = (gn; a) = N0((gn; f a) ; a) = N0((f0 g) (n; f a) ; a) = N0((f (f0 g)) ((n; f a) ; a)) = N0(((f f0 ) g) ((n; f a) ; a)) = ( N0((f f0 ) g)) ((n; f a) ; a) ,
i.e. the diagram in Figure 2 is commutative:
That means (f0f ) g
N= N0((f f0 ) g) and the family
( N; idA) : (f f0 ) N ! (f0f ) N j N = ( : N ! C) 2 XAlg(R)=C
is a natural isomorphism between f f0 and (f0f ) .
4. Induced crossed modules of R-algebroids
Although a similar crossed module construction might be possible to that in [2] given by M. Alp, for the construction of the induced crossed module, we prefer to use the free precrossed modules of R-algebroids constructed in [6], to provide an application. The summary of the construction, in [6], of the free precrossed A-module FP(!) = (!P : FP(!) ! A) of R-algebroids determined by the function
! : K ! A where K is a set and A is an R-algebroid such that !k is a morphism of A for all k 2 K, is as follows:
1: The building blocks are elements of the form aka0 with ta = s (!k) and t (!k) = sa0, and words of the form a
1k1a01a2k2a02:::ankna0nwith ta01= sa2,:::,ta0n 1=
san, where n 2 N+, a; a1; :::; an; a0; a01; :::; a0n 2 A and k; k1; :::; kn 2 K. The source
and the target of any word p = a1k1a01a2k2a20:::ankna0n are sp = sa1 and tp = ta0n,
respectively.
2: For all x; y 2 A0, FP(!) (x; y) is the quotient group obtained by dividing the
its normal subgroup generated by all elements of the form
a1k1a01::: ai+ a00i kia0i:::ankna0n a1k1a01:::aikia0i:::ankna0n a1k1a10:::a00ikiai0:::ankna0n, a1k1a01:::aiki a0i+ ai000 :::ankna0n a1k1a01:::aikia0i:::ankna0n a1k1a01:::aikia000i :::ankna0n,
(r a1) k1a01:::aikia0i:::ankna0n a1k1a01::: (r ai) kia0i:::ankna0n, (r a1) k1a01:::aikia0i:::ankna0n a1k1a01:::aiki r a0i :::ankna0n.
So, the elements of FP(!) (x; y) are of the form
P
i
[pi] where pi is a word with
spi= x and tpi= y, and [pi] is the coset of pi.
3: FP(!) (x; y) has an R-module structure with the R-action de…ned as r [pi] =
(r ai1) ki1a0i1:::ainkina0in and as r P i [pi] = P i r [pi].
4: FP(!) = fFP(!) (x; y) j x; y 2 A0g is a pre-R-algebroid on A0with the
com-position de…ned as P i [pi]P j [pj] = P i;j
[pipj] where if pi= ai1ki1a0i1:::ainkina0in and
pj= aj1kj1a0j1:::ajn0kjn0a0jn0 then pipj = ai1ki1a
0
i1:::ainkina0inaj1kj1a0j1:::ajn0kjn0a0jn0
under the condition t [pi] = ta0in= saj1= s [pj].
5: FP(!) is an A-module with the associative A-action de…ned asa P i [pi] = P i [ap i] and P i [pi] a0 =P i h pa0 i i whereap
i = (aai1) ki1a0i1:::ainkina0in and p
a0 i =
ai1ki1a0i1:::ainkin a0ina
0 with the condition that ta = sp
i, tpi= sa0.
6: !P : FP(!) ! A is de…ned as !P[aka0] = a (!k) a0 on generators and
as !P
P
i
[pi] = P i
!P[pi] on elements where !P[pi] = !P ai1ki1a0i1:::ainkina0in =
!P ai1ki1a0i1 :::!P ainkina0in .
Proposition 3. Let A and B be two R-algebroids, f : A ! B be an R-algebroid morphism and N = ( : N ! A) be a crossed A-module of R-algebroids. There exists a crossed B-module f N = f : f N ! B of R-algebroids and a crossed
module morphism f ; f : N ! f N such that for any crossed B-module M = ( : M ! B) of R-algebroids and crossed module morphism ( ; f) : N ! M there exists a unique crossed B-module morphism (h; idB) : f N ! M such that
f N , with the morphism f; f , is called the crossed module induced from N by f . The induced crossed module is unique up to isomorphism.
Proof. As summarised above, there exists a free precrossed B-module FP f m =
f m
P : FP f m ! B determined by f m where m : Mor (N) ! A is
de…ned as m(n) = n. Let I be the two sided ideal of FP f m generated by all
elements of the form
[b1n1b01b2n2b02] [(b1(f n1) b01b2) n2b02] , [b1n1b01b2n2b02] [b1n1(b01b2(f n2) b02)] , [bnb0] + [bn1b0] [b (n + n1) b0] , [b (an) b0] [(b (f a)) nb0] , h b na0 b0i [bn ((f a0) b0)] , [(r b) nb0] [b (r n) b0] , [bn (r b0)] [b (r n) b0] .
Obviously, I is closed under the actions of R and B. Now, construct the family f N =
FP(f m)
I = f N (x; y) =
FP(f m)(x;y)
I(x;y) j x; y 2 B0 of quotient R-modules. For
any word b1n1b01:::btntb0tlet’s denote the coset of [b1n1b01:::btntb0t] by b1n1b01:::btntb0t.
Then, note that, any coset b1n1b01:::btntb0t is, in fact, of the form b1n1b001, where
b00
1 = b01b2(f n2) b02:::bt(f nt) b0t 2 B. Thus each element of f N is of the form
P
i
binib0i for some bi; b0i 2 B, ni 2 N: Clearly, f N is an R-algebroid B-module
thanks to the addition, composition, R-action and associative B-action induced by those de…ned on FP f m .
Moreover f m
P induces a precrossed module f N = f : f N ! B de…ned
as f bnb0 = f m P[bnb
0] = b f
mn b
0 = b (f n) b0 on generators and the
precrossed module f N is also a crossed module thanks to the …rst two generators of I.
De…ne the function f : N ! f N as fn = 1n1 where 1n1 = 1s(f n)n1t(f n). By
this de…nition the pair f ; f is a crossed module morphism since
1: f (n1+ n2) = 1 (n1+ n2) 1 = 1n11 + 1n21 = 1n11 + 1n21 = f n1+ f n2, 2: f (nn0) = 1 (nn0) 1 = 1 (n n0 ) 1 = 1n ((f n0) 1) = 1n (11 (f n0) 1) = 1n11n01 = 1n1 1n01 = f (n) f (n0) , 3: f (r n) = 1 (r n) 1 = (r 1) n1 = r 1n1 = r f n, 4: f (an) = 1 (an) 1 = (1 (f a)) n1 = ((f a) 1) n1 = f a(1n1) = f a1n1 = f a f n , 5: f na0 = 1 (na0 ) 1 = 1n ((f a0) 1) = 1n (1 (f a0)) = (1n1)f a0 = 1n1f a0 = f n f a 0 , 6: f f n = f 1n1 = 1 (f n) 1 = f n
for all n; n1; n2; n0 2 N, a; a0 2 A and r 2 R such that sn1= sn2, tn1 = tn2; ta =
sn; tn = sn0= sa0.
Now for all crossed B-module M = ( : M ! B) of R-algebroids and for all crossed module morphism ( ; f ) : N ! M de…ne the function h : f N ! M as h bnb0 =b( n)b0
on generators. It can easily be shown that h preserves the addition, R-action and B-action. Moreover
h b1n1b01b2n2b02 = h b1n1(b01b2(f n2) b02) = b1( n1)b 0 1b2( n2)b02 = b1( n 1)b 0 1(b2( ( n2))b02) = b1( n 1)b 0 1 b2( n2)b02 = b1( n 1)b 0 1 b2( n 2)b 0 2 = h b 1n1b01 h b2n2b02
for all generators b1n1b01, b2n2b02 of f N with tb01 = sb2, which means h preserves
the composition. Besides,
( h) bnb0 = h bnb0 = b( n)b0
= b ( ( n)) b0 = b (( ) (n)) b0 = b ((f ) (n)) b0= f bnb0
= idB f bnb0
on generators. That is, (h; idB) is a crossed B-module morphism. Finally
hf (n) = h 1n1 =1( n)1= n
for all n 2 N, i.e. h makes the universal diagram in Figure 3 commutative. It can also directly be shown that (h; idB) is the unique morphism satisfying = hf , and
f N , with the morphism f; f , is unique up to isomorphism.
Thus we get an induced crossed module functor f : XAlg(R)=A ! XAlg(R)=B which gives a crossed B-module f N for any crossed A-module N and is de…ned as
f (g; idA) = (f g; idB) on morphisms such that (f g) bnb0 = b (gn) b0 on
genera-tors.
Proposition 4. If A, B, C are R-algebroids and f : A ! B, f0 : B ! C are
R-algebroid morphisms then the functor f0f is naturally isomorphic to (f0f ) .
Proof. For any N = ( : N ! A) 2 XAlg(R)=A, generators of the B-module f N; the C-module (f0f ) N = f0(f N) and the C-module (f0f ) N are of the forms bnb0,
c bnb0 c0 and cnc0, respectively.
For all crossed module N = ( : N ! A) de…ne N : (f0f ) N ! (f0f ) N as N c bnb0 c0 = (c (f0b)) n ((f0b0) c0) on generators. Obviously N preserves the
addition, R-action and C-action. It also preserves the composition since
N c1 b1n1b01 c10 c2 b2n2b02 c02 = N c1 b1n1b01 c01c2 f0 f b2n2b02 c02 = (c1(f0b1)) n1((f0b01) (c01c2(f0(b2((f ) (n2)) b02)) c02) ) = (c1(f0b1)) n1((f0b01) c01c2((f0b2) ((f0f ) (n2)) (f0b02)) c02) = (c1(f0b1)) n1(((f0b01) c01) (c2(f0b2)) (((f0f ) ) (n2)) ((f0b02) c02) ) = (c1(f0b1)) n1((f0b01) c01) (c2(f0b2)) n2((f0b02) c02) = N c1 b1n1b01 c10 N c2 b2n2b02 c02
for all generators c1 b1n1b01 c01, c2 b2n2b02 c02of (f0f ) N with tc01= sc2.
Moreover (f0f ) N c bnb0 c0 = (f0f ) N c bnb0 c0 = (f0f ) (c (f0b)) n ((f0b0) c0) = (c (f0b)) ((f0f ) n) ((f0b0) c0) = c ((f0b) (f0((f ) (n))) (f0b0)) c0 = c (f0(b ((f ) (n)) b0)) c0 = c f0 f bnb0 c0 = c f0 f bnb0 c0 = f f0 c bnb0 c0 = idC f f0 c bnb0 c0
on generators, which means ( N; idC) : (f0f ) N ! (f0f ) N is a crossed
C-module morphism.
Now, for all crossed modules N = ( : N ! A) de…ne N : (f0f ) N !
(f0f ) N as
Npreserve the addition, R-action and C-action. It also preserves the composition since N c1n1c01 c2n2c02 = N c1n1(c01c2(((f0f ) ) (n2)) c02) = c11n11 (c01c2(f0((f ) (n2))) c02) = c11n11 (c01c2(f0(1 ((f ) (n2)) 1)) c02) = c11n11 c01c2 f0 f 1n21 c02 = c11n11 c01c2 f0 f 1n21 c02 = c11n11c01c21n21c02 = N c1n1c10 N c2n2c02
for all generators c1n1c01, c2n2c02 of (f0f ) N with tc01= sc2. Moreover
f f0 N cnc0 = f f0 N cnc0 = f f0 c1n1c0 = c f0 f 1n1 c0 = c f0 f 1n1 c0 = c (f0(1 ((f ) (n)) 1)) c0 = c (f0(f n)) c0 = c (((f0f ) ) (n)) c0 = (f0f ) cnc0 = idC (f0f ) cnc0
on generators, which means ( N; idC) : (f0f ) N ! (f0f ) N is a crossed
C-module morphism.
Now, for all generators c bnb0 c0 of (f0f ) N
( N N) c bnb0 c0 = N N c bnb0 c0 = N (c (f0b)) n ((f0b0) c0) = (c (f0b)) 1n1 ((f0b0) c0) = c b1n1b0 c0 = c (b1) n (1b0) c0 = c bnb0 c0,
i.e. N N= id(f0f )N and for all generators cnc0 of (f0f ) N ( N N) cnc0 = N N cnc0 = N c1n1c0 = c f01s((f )(n)) n f01t((f )(n)) c0 = c 1s((f0f )(n)) n 1t((f0f )(n)) c0 = (c (1tc)) n ((1sc0) c0) = cnc0,
i.e. N N= id(f0f ) N. That is, Nis an isomorphism from (f0f ) N to (f0f ) N.
Moreover, for all N = ( : N ! A) ; N0 = ( 0 : N0 ! A) 2 XAlg(R)=A, for all
crossed module morphisms (g; idA) : N ! N0 and for all generators c bnb0 c0 of
(f0f ) N (((f0f ) g) N) c bnb0 c0 = ((f0f ) g) N c bnb0 c0 = ((f0f ) g) (c (f0b)) n ((f0b0) c0) = (c (f0b)) (gn) ((f0b0) c0) = N0 c b (gn) b0 c0 = N0 c (f g) bnb0 c0 = N0 (f0(f g)) c bnb0 c0 = ( N0((f0f ) g)) c bnb0 c0
which means the diagram in Figure 4 is commutative:
So, we can conclude that
f( N; idC) : (f0f ) N ! (f0f ) N j N = ( : N ! A) 2 XAlg(R)=Ag
is a natural isomorphism between f0f and (f0f ) .
Theorem 1. For any R-algebroids A and B, and any R-algebroid morphism f : A ! B the induced crossed module functor f is the left adjoint of the pullback crossed module functor f .
Proof. We must …nd a natural equivalence
: (XAlg(R)=B) (f ( ) ; ) = (XAlg(R)=A) ( ; f ( )) which is required to give a map
: Ob (XAlg(R)=A) Ob (XAlg(R)=B) ! Sets (M; N ) 7 ! (M; N )
where (M; N ), from (XAlg(R)=B) (f M; N ) to (XAlg(R)=A) (M; f N ), is a bi-jection and natural in both M and N .
For all crossed modules M = ( : M ! A) and N = ( : N ! B) de…ne (M; N ) as (M; N ) (h; idB) = ( (M; N ) (h) ; idA) such that ( (M; N ) (h)) (m)
= h 1m1 ; m for all (h; idB) 2 (XAlg(R)=B) (f M; N ) and m 2 M: (M; N ) (h)
is well de…ned since h 1m1 = f 1m1 = 1 ((f ) m) 1 = f ( m) for all
m 2 M. Moreover, it can easily be seen that ( (M; N ) (h) ; idA) is a crossed
A-module morphism and (M; N ) is 1-1.
For any (g; idA) 2 (XAlg(R)=A) (M; f N ) the morphism g : M ! f N must
be de…ned as gm = (g1m; g2m), for all m 2 M, such that g1m 2 N, g2m 2 A and
g1m = f g2m. But g2m = f (g1m; g2m) = f (gm) = f g m = m since
(g; idA) is a crossed A-module morphism. So we can write gm = (g1m; m) where
g1m = f m. De…ne h : f M ! N as h bmb0 =bg1mb
0
on generators. Clearly h is an R-algebroid morphism preserving B-action and (h; idB) is a crossed B-module
morphism since ( h) bmb0 = bg 1mb 0 = b ( g1m) b0 = b (f m) b0 = b ((f ) (m)) b0= f bmb0 = idB f bmb0
on generators. That is (h; idB) 2 (XAlg(R)=B) (f M; N ). Moreover
( (M; N ) (h)) (m) = h 1m1 ; m = 1g1m1; m = (g1m; m) = gm
for all m 2 M which means (M; N ) is onto and so is a bijection.
Moreover, provided that ( ) is composition with ( ) from right, for all crossed module M0 = ( 0: M0 ! A), for all (g; idA) 2 (XAlg(R)=A) (M; M0), (h; idB) 2
(XAlg(R)=B) (f M0; N ) and m 2 M (M; N ) (f g) (h) (m) = (M; N ) (f g) h (m) = ( (M; N ) (h (f g))) (m) = (h (f g)) 1m1 ; m = h 1 (gm) 1 ; ( 0g) (m) = h 1 (gm) 1 ; 0(gm) = ( (M0; N ) (h)) (gm) = (( (M0; N ) (h)) g) (m) = (g ( (M0; N ) (h))) (m) = ((g (M0; N )) (h)) (m)
which means the diagram in Figure 5 is commutative and so (M; N ) is natural in M:
Finally, provided that ( ) is composition with ( ) from left, for all crossed module N0 = ( 0 : N0 ! B), for all (g; id
B) 2 (XAlg(R)=B) (N ; N0), (h; idB) 2 (XAlg(R)=B) (f M; N ) and m 2 M (( (M; N0) g ) (h)) (m) = ( (M; N0) (g h)) (m) = ( (M; N0) (gh)) (m) = (gh) 1m1 ; m = g h 1m1 ; m = (f g) h 1m1 ; m = (f g) (( (M; N ) (h)) (m)) = ((f g) ( (M; N ) (h))) (m) = ((f g) ( (M; N ) (h))) (m) = (((f g) (M; N )) (h)) (m)
which means the diagram in Figure 6 is commutative and so (M; N ) is natural in N :
5. Consequences of the Adjunction Theorem 1 has some consequences:
1. In the proof of the Theorem 1, since (M; N ) is a bijection, its inverse
1(M; N ) is also a bijection from (XAlg(R)=A) (M; f N ) to (XAlg(R)=B) (f M; N )
and de…ned for all (g; idA) 2 (XAlg(R)=A) (M; f N ) as 1(M; N ) (g; idA) = 1(M; N ) (g) ; id
B such that 1(M; N ) (g) bmb0 =bf gmb b
0
on generators. 2. The family, called the unit of the adjunction,
f( M; idA) = (M; N ) (idf M; idB) : M ! f f M j M 2 Ob (XAlg(R)=A)g
is a natural transformation from 1XAlg(R)=Ato f f where 1XAlg(R)=Ais the identity
functor on XAlg(R)=A. Moreover M = ( M; idA) is universal for each M =
( : M ! A) 2 Ob (XAlg(R)=A), i.e. for each N 2 Ob (XAlg(R)=B) and for each morphism (g; idA) : M ! f N there exists a unique morphism (g0; idB) :
f M ! N making the universal diagram in Figure 7 commutative:
It can be shown that M(m) = 1m1; m for all m 2 M and (g0; idB) = 1
(M; N ) (g; idA) which requires g0 to be de…ned on generators as g0 bmb0 = bf gmb b0.
3. The family, called the counit of the adjunction,
( N; idB) = 1(M; N ) (idf N; idA) : f f N ! N j N 2 Ob (XAlg(R)=B)
is a natural transformation from f f to 1XAlg(R)=Bwhere 1XAlg(R)=Bis the
iden-tity functor on XAlg(R)=B. Moreover N = ( N; idB) is universal for each N =
( : N ! B) 2 Ob (XAlg(R)=B), i.e. for each M 2 Ob (XAlg(R)=A) and for each morphism (h; idB) : f M ! N there exists a unique morphism (h0; idA) : M !
It can be shown that N b (n; a) b0 = bnb0
on generators and (h0; idA) =
(M; N ) (h; idB) which requires h0 to be de…ned as h0(m) = h 1m1 ; m for
all m 2 M .
4. For each M 2 Ob (XAlg(R)=A) and for each N 2 Ob (XAlg(R)=B)
f Mf ( M) = idf M and f ( N) f N = idf N
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Current address : Osman Avc¬o¼glu: Usak University, Faculty of Arts and Sciences, Department of Mathematics, 64200 - Usak, Turkey.
E-mail address : [email protected]
Current address : ·Ibrahim ·Ilker Akça: Eskisehir Osmangazi University, Faculty of Science and Letters, Department of Mathematics and Computer Sciences, 26480 - Eskisehir, Turkey.
