(Co)-Induced Two Crossed Modules

(Co)-Ind uced Tw o-Cross ed Mo dul es U. E. Arslan, Z. Arv asi and G. Onarlı Abstract W e introduce the notion of an (co)-ind uced 2-crossed mo dule, which extends th e notion of an (co)- induced crossed mod ule (Brow n and Hig- gins). In tro duction Induced crossed mo dules were defined b y Brown and Higgins [2] and studied further in pap er by Bro wn and W ensley [4, 5],[1]. This is lo oked at in detail in a bo o k by Brown, Higgins and Siv era [3]. Induced cr o ssed mo dules a llow detailed computations of non-Abe lia n information on second relative groups. T o obtain analog o us result in dimension 3, we make essential use of a 2- crossed mo dule defined by Co nduch ´ e [6]. A ma jor aim o f this pap er is given a ny θ = ( φ ′ , φ ) : ( M , P, α ) → ( N , Q , β ) pre-cro s sed module mor phis m to in tro duce induced ( N → Q )-2-cro s sed mo dule { θ ∗ ( L ) , N , Q , ∂ ∗ , β } which ca n b e used in applications of the 3-dimensio nal V an Kamp en Theorem. The metho d of Brown and H igg ins [2] is generalized to give results on { θ ∗ ( L ) , N , Q , ∂ ∗ , β } . Also we co nstruct pullback (co -induced) 2-c r ossed mo d- ules in terms of the concept of pullbac k pre-cr ossed modules in [3]. How ever; Brown, Higgins and Sivera [3] indicate a bifibratio n fr o m crossed squar es, so leading to the notion of induced c r ossed square, which is relev a n t to a triadic Hurewicz theorem results in dimension 3. 1 Preliminaries Throughout this pa per all actions will b e left. The right actions in some r efer- ences will be rewritten by using left actions. 1 W e are grateful to referee and D. Conduc h´ e for helpful comments. 1 (Co)-Induced Tw o- Crossed Modules 2 1.1 Crossed Mo dules Crossed mo dules o f gr oups were initially de fined by Whitehead [11] as mo dels for (homotopy) 2-types. W e recall from [9] the definition of cro ssed mo dules of groups. A c rossed mo dule, ( M , P, ∂ ) , consists of gro ups M and P with a left a ction of P on M , wr itten ( p, m ) 7→ p m and a group ho momorphism ∂ : M → P satisfying the following c onditions: C M 1) ∂ ( p m ) = p∂ ( m ) p − 1 and C M 2) ∂ ( m ) n = mnm − 1 for p ∈ P , m, n ∈ M . W e say that ∂ : M → P is a pre-cro ssed module, if it satisfies CM1 . If ( M , P , ∂ ) and ( M ′ , P ′ , ∂ ′ ) are cross e d mo dules, a morphism, ( µ, η ) : ( M , P , ∂ ) → ( M ′ , P ′ , ∂ ′ ) , of crossed modules consis ts of gro up homomo r phisms µ : M → M ′ and η : P → P ′ such that ( i ) η ∂ = ∂ ′ µ and ( ii ) µ ( p m ) = η ( p ) µ ( m ) for a ll p ∈ P, m ∈ M . Crossed mo dules a nd their mor phis ms fo rm a ca tegory , of c o urse. It will usually be denoted by XMo d . W e also ge t o b vio usly a categ ory PXMo d o f pre- crossed mo dules. There is, for a fixed gro up P , a sub catego ry XMo d /P of XMo d , which has as ob jects those crossed mo dules with P a s the “base”, i.e., a ll ( M , P , ∂ ) for this fixed P , and having a s mor phisms from ( M , P, ∂ ) to ( M ′ , P ′ , ∂ ′ ) those ( µ, η ) in XMo d in which η : P → P ′ is the identit y homo mo rphism on P . Some standart examples of cros sed mo dules are: (i) normal subgroup cro s sed mo dules ( i : N → P ) where i is an inclus ion of a no rmal s ubgroup, and the action is given by conjuga tion; (ii) automor phism crosse d mo dules ( χ : M → Aut ( M )) in which ( χm ) ( n ) = mnm − 1 ; (iii) Ab elian cros sed mo dules 0 : M → P where M is a P -module; (iv) central extensio n crosse d mo dules ∂ : M → P where ∂ is a n epimorphis m with kernel contained in the center o f M . 1.2 Pullbac k Crossed Mo dules W e re call fro m [3] b elow a presentation of the pullba ck (co-induced) cros s ed mo dule. Let φ : P → Q b e a homo mo rphism o f gro ups and let N = ( N , Q, v ) be a crossed mo dule. W e define a subgr oup φ ∗ ( N ) = N × Q P = { ( n, p ) | v ( n ) = φ ( p ) } (Co)-Induced Two-Cros sed Mo dules 3 of the pr o duct N × P . This is the usual pullback in the ca tegory o f gro ups . There is a commutativ e diagr am φ ∗ ( N ) ¯ φ / / ¯ v   N v   P φ / / Q where ¯ v : ( n, p ) 7→ p, ¯ φ : ( n, p ) 7→ n. Then P acts on φ ∗ ( N ) via φ and the diagonal, i.e. p ′ ( n, p ) = ( φ ( p ′ ) n, p ′ pp ′− 1 ) . This gives a P -action. Since ( n, p )( n 1 , p 1 )( n, p ) − 1 =  nn 1 n − 1 , pp 1 p − 1  =  v ( n ) n 1 , pp 1 p − 1  =  φ ( p ) n 1 , pp 1 p − 1  = ¯ v ( n,p ) ( n 1 , p 1 ) , we get a crossed mo dule φ ∗ ( N ) = ( φ ∗ ( N ) , P, ¯ v ) which is called the pu l l b ack crosse d modul e of N a long φ . This constr uction satisfies a universal pr op- erty , analogous to that of the pullback of groups. T o state it, we use also the morphism of cr ossed mo dules  ¯ φ, φ  : φ ∗ ( N ) → N . Theorem 1 F or any cr osse d mo dule M = ( M , P, µ ) and any morphism of cr osse d mo dules ( h, φ ) : M → N , ther e is a unique morphism of cr osse d P -mo dules h ′ : M → φ ∗ ( N ) such that the fol lowing diagr am c ommutes M h % % µ   h ′ " " ❋ ❋ ❋ ❋ ❋ φ ∗ ( N ) ¯ φ / / ¯ v   N v   P φ / / Q. This can b e expressed functor ially: φ ∗ : XMo d /Q → XMo d /P is a pullback functor. This functor has a left a djoin t φ ∗ : XMo d /P → XMo d /Q which gives a n induced cr ossed mo dule as follows. Induced crossed mo dules were defined b y Brown and Higgins in [2] and studied further in pap e rs by Br own a nd W ensley [4, 5]. W e recall from [3] b elow a pr esentation of th e induced cross ed mo dule which is helpful for the calculation of co limits. (Co)-Induced Two-Cros sed Mo dules 4 1.3 Induced Crossed Mo dules In this sectio n we will briefly ex pla in Br own and Higgins’ cons truction o f induced (pre-)cross ed mo dules in [2] to c o mpare 3 -dimensional constructio n g iven in section 4. Definition 2 F or any cr osse d P -m o dule M = ( M , P , µ ) and any homomor- phism φ : P → Q the cr osse d mo dule induced by φ fr om µ should b e given by: 1 ( i ) a cr osse d Q -mo dule φ ∗ ( M ) = ( φ ∗ ( M ) , Q, φ ∗ µ ) , ( ii ) a morphism of cr osse d mo dules ( f , φ ) : M → φ ∗ ( M ) , satisfying the dual u niversal pr op ert y that for any morphism of cr osse d mo dules ( h, φ ) : M → N ther e is a u nique morphism of cr osse d Q -mo dules h ′ : φ ∗ ( M ) → N su ch t hat the diagr am N v   M f / / h 2 2 µ   φ ∗ ( M ) φ ∗ µ   h ′ < < ① ① ① ① ① P φ / / Q c ommu tes. The cross ed module φ ∗ ( M ) = ( φ ∗ ( M ), Q, φ ∗ µ ) is called the induced cr ossed mo dule of M = ( M , P , µ ) along φ . 1.3.1 Construction of Induced Crossed Mo dules The free Q -gro up Q M genera ted b y M is the kernel of the c anonical mor phism M ∗ Q → Q where M ∗ Q is the free pro duct. Hence it is g enerated by elements q mq − 1 with q ∈ Q and m ∈ M . It is equiv alent to say that Q M is gener ated by the set Q × M with rela tion ( q , m 1 )( q , m 2 ) = ( q, m 1 m 2 ) (1 . 3 . 1 . 1) for m 1 , m 2 ∈ M , q ∈ Q . Then Q a cts on Q M by q ′ ( q , m ) = ( q ′ q , m ) for m ∈ M and q ′ , q ∈ Q. Let µ : M → P be a crossed mo dule a nd φ : P → Q b e a morphism of groups. As Q M is the fr ee Q - group, we have the commutativ e diagr am (Co)-Induced Two-Cros sed Mo dules 5 M / / µ   φµ " " ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ Q M µ ′   P φ / / Q. That is the morphism φµ ex tends to the mor phism µ ′ : Q M → Q given by µ ′ ( q , m ) = q φµ ( m ) q − 1 for m ∈ M , q ∈ Q. Thu s µ ′ is the free Q -pre- crossed mo dule genera ted by M . T o ge t a free crossed mo dule we hav e to divide Q M by the Peiffer subgr o up, that is by rela- tion: ( q 1 , m 1 )( q 2 , m 2 )( q 1 , m 1 ) − 1 = ( q 1 φµ ( m 1 ) q − 1 1 q 2 , m 2 ) (1 . 3 . 1 . 2) for m 1 , m 2 ∈ M a nd q 1 , q 2 ∈ Q. Next step, to obtain the induc e d pre-cr ossed mo dule, we must identify the action of P on M with the action of φ ( P ) on Q M , that is we must divide by relation ( q , p m ) = ( qφ ( p ) , m ) for m ∈ M , p ∈ P and q ∈ Q. Then we hav e the induced cr ossed mo dule denoted by φ ∗ ( M ), dividing by relatio n (1.3.1.2). Thu s Brown and Higgins proved the following res ult in [2]. Prop ositio n 3 L et µ : M → P b e a cr osse d P -mo dule and let φ : P → Q b e a morphism of gr oups. T hen the induc e d cr osse d Q - mo dule φ ∗ ( M ) is gener ate d, as a gr oup, by the set Q × M with defining re lations ( i ) ( q , m 1 )( q , m 2 ) = ( q , m 1 m 2 ) , ( ii ) ( q , p m ) = ( q φ ( p ) , m ) , ( iii ) ( q 1 , m 1 )( q 2 , m 2 )( q 1 , m 1 ) − 1 = ( q 1 φµ ( m 1 ) q − 1 1 q 2 , m 2 ) for m , m 1 , m 2 ∈ M , q , q 1 , q 2 ∈ Q a nd p ∈ P . The morphism φ ∗ µ : φ ∗ ( M ) → Q is given by φ ∗ µ ( q , m ) = q φµ ( m ) q − 1 , the action o f Q on φ ∗ ( M ) by q ( q 1 , m ) = ( q q 1 , m ) , a nd the canonica l morphism φ ′ : M → φ ∗ ( M ) by φ ′ ( m ) = (1 , m ) . If φ : P → Q is an epimo rphism, the induced c r ossed mo dule ( φ ∗ ( M ) , Q, φ ∗ µ ) has a simpler desc r iption. Prop ositio n 4 ([ 2 ] , Pr op osition 9) If φ : P → Q is an epimorphi sm, and µ : M → P is a cr osse d mo dule, t hen φ ∗ ( M ) ∼ = M / [ K, M ] , wher e K = Ker φ , and [ K, M ] denotes the su b gr oup of M gener ate d by al l k mm − 1 for al l m ∈ M , k ∈ K . (Co)-Induced Two-Cros sed Mo dules 6 Prop ositio n 5 ([ 2 ] , Pr op osition 10) If φ : P → Q is an inje ction and µ : M → P i s a cr osse d mo dule, let T b e a left tr ansversal of φ ( P ) in Q, and let B b e the fr e e pr o duct of gr oups T M ( t ∈ T ) e ach isomorphi c with M by an isomorphi sm m 7→ t m ( m ∈ M ) . L et q ∈ Q acts on B by the rule q ( t m ) = u ( p m ) wher e p ∈ P , u ∈ T , and q t = u φ ( p ) . L et δ : B → Q b e define d by t m 7→ t ( φµm ) t − 1 . Then µ ∗ ( M ) = B /S wher e S is the normal closur e in B of the elements bcb − 1 ( δb c − 1 ) for b, c ∈ B . 2 Tw o-Crossed M o dules Conduch ´ e [6] describ ed the notion of a 2-cr ossed mo dule as a mo del of connected homotopy 3 - t yp es. A 2 -cr osse d mo dule is a nor mal complex of gr oups L ∂ 2 → M ∂ 1 → P , that is ∂ 2 ( L ) E M , ∂ 1 ( M ) E P and ∂ 1 ∂ 2 = 1 , tog ether with a n actio n of P on all thr e e groups a nd a ma pping {− , −} : M × M → L which is o ften called the Peiffer lifting such that the action o f P o n itself is by conjugation, ∂ 2 and ∂ 1 are P -equiv ariant. PL1 : ∂ 2 { m 0 , m 1 } = m 0 m 1 m − 1 0  ∂ 1 m 0 m − 1 1  PL2 : { ∂ 2 l 0 , ∂ 2 l 1 } = [ l 0 , l 1 ] PL3 : { m 0 , m 1 m 2 } = m 0 m 1 m − 1 0 { m 0 , m 2 } { m 0 , m 1 } { m 0 m 1 , m 2 } =  m 0 , m 1 m 2 m − 1 1   ∂ 1 m 0 { m 1 , m 2 }  PL4 : a ) { ∂ 2 l , m } = l  m l − 1  b ) { m, ∂ 2 l } = m l  ∂ 1 m l − 1  or { ∂ 2 l , m } { m, ∂ 2 l } = l  ∂ 1 m l − 1  PL5 : p { m 0 , m 1 } = { p m 0 , p m 1 } for a ll m, m 0 , m 1 , m 2 ∈ M , l , l 0 , l 1 ∈ L and p ∈ P . Note that we hav e not sp ecified how M acts on L . In [6], that as follows: if m ∈ M and l ∈ L , define m l = l  ∂ 2 l − 1 , m  . F r om this equa tion ( L, M , ∂ 2 ) bec omes a cr ossed mo dule . W e deno te such a 2- crossed mo dule of g r oups by { L, M , P , ∂ 2 , ∂ 1 } . A mo rphism o f 2 -crosse d mo dules is given by a diag ram L ∂ 2 / / f 2   M ∂ 1 / / f 1   P f 0   L ′ ∂ ′ 2 / / M ′ ∂ ′ 1 / / P ′ where f 0 ∂ 1 = ∂ ′ 1 f 1 , f 1 ∂ 2 = ∂ ′ 2 f 2 f 1 ( p m ) = f 0 ( p ) f 1 ( m ) , f 2 ( p l ) = f 0 ( p ) f 2 ( l ) (Co)-Induced Two-Cros sed Mo dules 7 and {− , −} ( f 1 × f 1 ) = f 2 {− , −} for a ll m ∈ M , l ∈ L a nd p ∈ P . These co mpo s e in an obvious wa y giving a categor y whic h we will denote by X 2 Mo d . There is, for a fixed g r oup P , a sub category X 2 Mo d /P of X 2 Mo d which has as ob jects those cr ossed mo dules with P as the “bas e”, i.e., all { L, M , P , ∂ 2 , ∂ 1 } for this fixed P , and having as morphism from { L, M , P , ∂ 2 , ∂ 1 } to { L ′ , M ′ , P ′ , ∂ ′ 2 , ∂ ′ 1 } those ( f 2 , f 1 , f 0 ) in X 2 Mo d in which f 0 : P → P ′ is the ident ity homomorphism on P . Similar ly we get a sub categor y X 2 Mo d / ( M , P ) of X 2 Mo d for a fixed pr e-crosse d mo dule M → P . Some remar k s o n trivial Peiffer lifting of 2-cr ossed mo dules given by Porter in [9] a re: Suppo se we have a 2-cr ossed mo dule L ∂ 2 → M ∂ 1 → P , with e xtra condition tha t { m, m ′ } = 1 for a ll m, m ′ ∈ M . The o b vio us thing to do is to see what each o f the defining prop erties of a 2 - crossed mo dule g ive in this case. (i) There is an action of P on L and M and the ∂ s are P -equiv ariant. (This gives nothing new in our sp ecia l cas e .) (ii) {− , −} is a lifting o f the Peiffer commutator so if { m , m ′ } = 1 , the Peiffer ident ity holds for ( M , P , ∂ 1 ) , i.e. tha t is a cro ssed mo dule; (iii) if l , l ′ ∈ L, then 1 = { ∂ 2 l , ∂ 2 l ′ } = [ l , l ′ ] , so L is Ab elian and, (iv) as {− , −} is trivial ∂ 1 m l − 1 = l − 1 , so ∂ M has tr ivial a ction o n L. Axioms P L3 a nd PL5 v anish. Examples of 2-Cross e d Mo dules 1 . Let M ∂ 1 → P be a pre-cr ossed mo dule . Consider the Peiffer subgr oup h M , M i ⊂ M , gener ated by the Peiffer commutators h m, m ′ i = mm ′ m − 1 ( ∂ 1 ( m ) m ′− 1 ) for a ll m, m ′ ∈ M . Then h M , M i ∂ 2 → M ∂ 1 → P is a 2- c rossed mo dule with the Peiffer lifting { m, m ′ } = h m, m ′ i , [10]. 2 . Any crossed mo dule gives a 2-cro ssed mo dule. If ( M , P , ∂ ) is a crossed mo dule, the resulting sequence L → M → P is a 2-cr o ssed mo dule by taking L = 1 . This is functorial and XMo d ca n b e con- sidered to be a full categor y o f X 2 Mo d in this way . It is a reflective s ub ca teg ory since ther e is a re fle c tio n functor o btained as follows: If L ∂ 2 → M ∂ 1 → P is a 2-cr ossed mo dule, then Im ∂ 2 is a no r mal subgr o up o f M and there is a n induced cro ssed mo dule structur e on ∂ 1 : M Im ∂ 2 → P , (c.f. [9]). (Co)-Induced Two-Cros sed Mo dules 8 Remark 6 1 . Anothe r way of enc o ding 3 -typ es is u sing the noting of a cr osse d squar e by Guin-Wal´ ery and L o day, [8] . 2 . A cr osse d squar e c an b e c onsider e d as a c omplex of cr osse d mo dules of length one and thus, Conduch ´ e, gave a dir e ct pr o of fr om cr osse d squ ar es to 2 -cr osse d mo dules. F or this c onst ruction se e [7]. 3 Pullbac k T w o-Crossed Mo dules In this section w e in tro duce the notion of a pullbac k 2-cr ossed mo dule, which extends a pullback cr ossed module defined by Brown-Higgins, [2]. The imp or- tance of the “pullback” is that it enables us to mov e from cros sed Q - mo dule to crossed P - mo dule, when a morphis m of gr oups φ : P → Q is given. Definition 7 Given a 2 -cr osse d mo dule { H , N , Q , ∂ 2 , ∂ 1 } and a morphism of gr ou ps φ : P → Q, t he pul lb ack 2 -cr osse d mo dule is given by (i) a 2 - cr osse d mo dule φ ∗ { H, N , Q, ∂ 2 , ∂ 1 } = { φ ∗ ( H ) , φ ∗ ( N ) , P , ∂ ∗ 2 , ∂ ∗ 1 } (ii) given any morphism of 2 -cr osse d mo dules ( f 2 , f 1 , φ ) : { B 2 , B 1 , P, ∂ ′ 2 , ∂ ′ 1 } → { H , N , Q, ∂ 2 , ∂ 1 } , ther e is a unique ( f ∗ 2 , f ∗ 1 , id P ) 2 -cr osse d mo dule morphism that makes t he fol- lowing diagr am c ommute: ( B 2 , B 1 , P, ∂ 2 ′ , ∂ 1 ′ ) ( f ∗ 2 ,f ∗ 1 ,id P ) s s ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ( f 2 ,f 1 ,φ )   ( φ ∗ ( H ) , φ ∗ ( N ) , P , ∂ ∗ 2 , ∂ ∗ 1 ) ( φ ′′ ,φ ′ ,φ ) / / ( H, N , Q, ∂ 2 , ∂ 1 ) or mor e simply as B 2 f ∗ 2 $ $ ❏ ❏ ❏ ❏ f 2 ( ( ∂ 2 ′   φ ∗ ( H ) ∂ ∗ 2   φ ′′ / / H ∂ 2   B 1 f ∗ 1 $ $ ❏ ❏ ❏ ❏ f 1 ( ( ∂ 1 ′   φ ∗ ( N ) ∂ ∗ 1   φ ′ / / N ∂ 1   P id P ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ φ ( ( P φ / / Q. (Co)-Induced Two-Cros sed Mo dules 9 3.0.2 Construction of Pul l bac k 2 -Crossed Mo dules W e can construct pullback 2- crossed mo dules by using the notio n of the pullback pre-cro s sed mo dule in [3]. Let ( f 2 , f 1 , φ ) : { B 2 , B 1 , P, ∂ ′ 2 , ∂ ′ 1 } → { H , N , Q, ∂ 2 , ∂ 1 } be a 2-cr ossed mo dule mo r phism a nd we consider ( φ ∗ ( N ) , P , ∂ ∗ 1 ) the pullback pre-cro s sed module o f N → Q by φ given in subsection 1.2. Since ∂ 1 ∂ 2 f 2 = φ∂ ′ 1 ∂ ′ 2 = 1 , the image f 2 ( B 2 ) must be contained in ∂ − 1 2 ( K e r ∂ 1 ) . Whence the pullback 2-cros sed mo dule a sso ciated to φ is ∂ − 1 2 ( K e r ∂ 1 ) ∂ ∗ 2 → φ ∗ ( N ) ∂ ∗ 1 → P where ∂ ∗ 2 ( h ) = ( ∂ 2 ( h ) , 1 ) fo r h ∈ ∂ − 1 2 ( K e r ∂ 1 ). Theorem 8 If H ∂ 2 → N ∂ 1 → Q is a 2 -cr osse d mo dule and φ : P → Q is a morphism of gro ups then ∂ − 1 2 ( K e r ∂ 1 ) ∂ ∗ 2 → φ ∗ ( N ) ∂ ∗ 1 → P is a pul lb ack 2 -cr osse d mo dule wher e ∂ ∗ 2 ( h ) = ( ∂ 2 ( h ) , 1 ) ∂ ∗ 1 ( n, p ) = p, the action of P on φ ∗ ( N ) and ∂ − 1 2 ( K e r ∂ 1 ) by p ( n, p ′ ) =  φ ( p ) n, pp ′ p − 1  and p h = φ ( p ) h, r esp e ctively. Pro of. (i) Since ∂ ∗ 1 ∂ ∗ 2 ( h ) = ∂ ∗ 1 ( ∂ 2 ( h ) , 1 ) , ∂ − 1 2 ( K e r ∂ 1 ) ∂ ∗ 2 → φ ∗ ( N ) ∂ ∗ 1 → P is a normal co mplex of groups. ∂ ∗ 2 is P -equiv ariant with the action p ( n, p ′ ) =  φ ( p ) n, pp ′ p − 1  . p ∂ ∗ 2 ( h ) = p ( ∂ 2 ( h ) , 1 ) = ( φ ( p ) ∂ 2 ( h ) , p 1 p − 1 ) = ( φ ( p ) ∂ 2 ( h ) , 1 ) = ( ∂ 2  φ ( p ) h  , 1) = ( ∂ 2 ( p h ) , 1 ) = ∂ ∗ 2 ( p h ) It is clea r that ∂ ∗ 1 is P -equiv ariant. The Peiffer lifting {− , −} : φ ∗ ( N ) × φ ∗ ( N ) → ∂ − 1 2 ( K e r ∂ 1 ) is g iven by { ( n, p ) , ( n ′ , p ′ ) } = { n, n ′ } . (Co)-Induced Two-Cros sed Mo dules 10 PL1: ( n, p ) ( n ′ , p ′ ) ( n, p ) − 1  ∂ ∗ 1 ( n,p ) ( n ′ , p ′ ) − 1  = ( n, p ) ( n ′ , p ′ )  n − 1 , p − 1  p  n ′− 1 , p ′− 1  = ( n, p ) ( n ′ , p ′ )  n − 1 , p − 1   φ ( p ) n ′− 1 , pp ′− 1 p − 1  =  nn ′ n − 1 , pp ′ p − 1   ∂ 1 ( n ) n ′− 1 , pp ′− 1 p − 1  =  nn ′ n − 1  ∂ 1 ( n ) n ′− 1  , pp ′ p − 1 pp ′− 1 p − 1  =  nn ′ n − 1  ∂ 1 ( n ) n ′− 1  , 1  = ( ∂ 2 { n, n ′ } , 1 ) = ∂ ∗ 2 { n, n ′ } = ∂ ∗ 2 { ( n, p ) , ( n ′ , p ′ ) } . PL2: { ∂ ∗ 2 h, ∂ ∗ 2 h ′ } = { ( ∂ 2 h, 1) , ( ∂ 2 h ′ , 1) } = { ∂ 2 h, ∂ 2 h ′ } = [ h, h ′ ] . The rest of axioms of 2-cr o ssed mo dule is g iven in a ppendix. (ii) ( φ ′′ , φ ′ , φ ) :  ∂ − 1 2 ( K e r ∂ 1 ) , φ ∗ ( N ) , P , ∂ ∗ 2 , ∂ ∗ 1  → { H , N , Q, ∂ 2 , ∂ 1 } or diag rammatically , ∂ − 1 2 ( K e r ∂ 1 ) ∂ ∗ 2   φ ′′ / / H ∂ 2   φ ∗ ( N ) ∂ ∗ 1   φ ′ / / N ∂ 1   P φ / / Q is a mor phism o f 2-c r ossed mo dules. (See app endix.) Suppo se that ( f 2 , f 1 , φ ) : { B 2 , B 1 , P, ∂ ′ 2 , ∂ ′ 1 } → { H , N , Q, ∂ 2 , ∂ 1 } is a ny 2- crossed mo dule morphis m B 2 ∂ ′ 2 / / f 2   B 1 f 1   ∂ ′ 1 / / P φ   H ∂ 2 / / N ∂ 1 / / Q. Then we will show that there is a unique 2 - crossed mo dule morphis m ( f ∗ 2 , f ∗ 1 , id P ) : { B 2 , B 1 , P, ∂ ′ 2 , ∂ ′ 1 } →  ∂ − 1 2 ( K e r ∂ 1 ) , φ ∗ ( N ) , P , ∂ ∗ 2 , ∂ ∗ 1  (Co)-Induced Two-Cros sed Mo dules 11 B 2 ∂ ′ 2 / / f ∗ 2   B 1 f ∗ 1   ∂ ′ 1 / / P id P ∂ − 1 2 ( K e r ∂ 1 ) ∂ ∗ 2 / / φ ∗ ( N ) ∂ ∗ 1 / / P where f ∗ 2 ( b 2 ) = f 2 ( b 2 ) and f ∗ 1 ( b 1 ) = ( f 1 ( b 1 ) , ∂ ′ 1 ( b 1 )) which is an element in φ ∗ ( N ) . First let us chec k that ( f ∗ 2 , f ∗ 1 , id P ) is a 2-cro ssed mo dule mor phism. F o r b 1 , b ′ 1 ∈ B 1 , b 2 ∈ B 2 , p ∈ P id P ( p ) f ∗ 2 ( b 2 ) = p f 2 ( b 2 ) = φ ( p ) f 2 ( b 2 ) = f 2 ( p b 2 ) = f ∗ 2 ( p b 2 ) . Similarly id P ( p ) f ∗ 1 ( b 1 ) = f ∗ 1 ( p b 1 ) , also ab ov e diagr am is commutativ e a nd {− , −} ( f ∗ 1 × f ∗ 1 ) ( b 1 , b ′ 1 ) = {− , −} ( f ∗ 1 ( b 1 ) , f ∗ 1 ( b ′ 1 )) = {− , −} (( f 1 ( b 1 ) , ∂ ′ 1 ( b 1 )) , ( f 1 ( b ′ 1 ) , ∂ ′ 1 ( b ′ 1 )) = { f 1 ( b 1 ) , f 1 ( b ′ 1 ) } = {− , −} ( f 1 × f 1 ) ( b 1 , b ′ 1 ) = f 2 {− , −} ( b 1 , b ′ 1 ) = f 2 { b 1 , b ′ 1 } = f ∗ 2 { b 1 , b ′ 1 } = f ∗ 2 {− , −} ( b 1 , b ′ 1 ) for b 1 , b ′ 1 ∈ B 1 . F ur ther more; the verification of the following equations a re immedia te. φ ′′ f ∗ 2 = f 2 and φ ′ f ∗ 1 = f 1 . Thu s w e get a functor φ ∗ : X 2 Mo d / Q → X 2 Mo d / P which gives o ur pullba ck 2 -crossed mo dule and its left adjoint functor φ ∗ : X 2 Mo d /P → X 2 Mo d /Q gives a n induced 2- crossed mo dule w hich will b e mentioned in section 4. 3.1 Example of Pullbac k Two-Crossed Mo dules Given 2-cro ssed mo dule {{ 1 } , G, Q , 1 , i } where i is a n inclusion of a normal subgroup a nd a morphism φ : P → Q of g roups, the pullback 2- crossed mo dule is φ ∗ {{ 1 } , G, Q , 1 , i } = {{ 1 } , φ ∗ ( G ) , P , ∂ ∗ 2 , ∂ ∗ 1 } = n { 1 } , φ − 1 ( G ) , P , ∂ ∗ 2 , ∂ ∗ 1 o (Co)-Induced Two-Cros sed Mo dules 12 as φ ∗ ( G ) = { ( g , p ) | φ ( p ) = i ( g ) , g ∈ G, p ∈ P } ∼ = { p ∈ P | φ ( p ) = g } = φ − 1 ( G ) E P. The pullback diag ram is { 1 } ∂ ∗ 2 =1   { 1 }   φ − 1 ( G ) ∂ ∗ 1   / / G i   P φ / / Q. Particularly if G = { 1 } , then φ ∗ ( { 1 } ) ∼ = { p ∈ P | φ ( p ) = 1 } = ker φ ∼ = { 1 } and so {{ 1 } , { 1 } , P , ∂ ∗ 2 , ∂ ∗ 1 } is a pullback 2-cro ssed mo dule. Also if φ is a n isomorphism and G = Q, then φ ∗ ( Q ) = Q × P . Similarly when we consider ex a mples given in Section 1 , the following dia- grams a re pullbacks. { 1 } ∂ ∗ 2   { 1 }   φ ∗ ( Q ) ∂ ∗ 1   / / Q   Aut ( P ) Autφ / / Aut ( Q ) { 1 } ∂ ∗ 2   { 1 }   N × K er φ ∂ ∗ 1   / / N 0   P φ / / Q 4 Induced Tw o-Crossed Mo dules In this section we will construct induced 2-cro ssed modules by extending the discussion ab out induced crossed mo dules in subsection 1.3. Definition 9 F or any 2 -cr osse d mo dule L ∂ → M α → P and any pr e-cr osse d mo dule morphism ( M α → P ) ( φ ′ ,φ ) − → ( N β → Q ) , the 2 -cr osse d mo dule induc e d by θ = ( φ ′ , φ ) fr om { L, M , P , ∂ , α } is given by (i) a 2 - cr osse d mo dule { θ ∗ ( L ) , N , Q , ∂ ∗ , β } (ii) given any morphism of 2 -cr osse d mo dules ( f , φ ′ , φ ) : { L, M , P , ∂ , α } − → { B , N , Q, ∂ ′ , β } (Co)-Induced Two-Cros sed Mo dules 13 then ther e is a u nique morphism of 2 -cr osse d mo dules ( f ∗ , id N , id Q ) that makes the fol lowing diagr am c ommute: ( L, M , P, ∂ , α ) ( φ ′′ ,φ ′ ,φ ) * * ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ( f ,φ ′ ,φ )   ( B , N , Q, ∂ ′ , β ) ( θ ∗ ( L ) , N , Q , ∂ ∗ , β ) ( f ∗ ,id N ,id Q ) o o ❴ ❴ ❴ ❴ ❴ ❴ or mor e simply as B ∂ ′   L ∂   φ ′′ / / f 0 0 θ ∗ ( L ) f ∗ 7 7 ♥ ♥ ♥ ♥ ♥ ♥ ∂ ∗   ( N → Q ) ( M → P ) θ / / ( φ ′ ,φ ) 1 1 ( N → Q ) . ( id N ,id Q ) 7 7 ♦ ♦ ♦ ♦ ♦ The 2-cr o ssed mo dule { φ ∗ ( L ) , N , Q , ∂ ∗ , β } is called the induced 2-cros sed mo d- ule of { L, M , P, ∂ , α } a long θ = ( φ ′ , φ ). 4.1 Construction of Induced 2 -Crossed Mo dules The idea o f the construction of induced 2-cros s ed mo dules is the same a s that of induced crossed mo dule; w e put all the data in a free gr o up and divide this group by all r elations which we need to hav e the prop erties we wan t. Given a morphism θ = ( φ ′ , φ ) : ( M α → P ) − → ( N β → Q ) of pre-cro ssed mo dules, we c a n define the 2-cro ssed mo dule induced b y θ , then we put things together. T hus we hav e the commutativ e diag ram L / / ∂   φ ′ ∂ & & ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ Q L   ( M → P ) ( φ ′ ,φ ) / / ( N → Q ) where the co mplex L ∂ → M α → P is a 2-cro ssed mo dule, Q L is a fr ee gr oup generated by L a nd the first mor phism of the complex Q L → N → Q (Co)-Induced Two-Cros sed Mo dules 14 of Q - groups is induced by φ ′ ∂ . W e co nsider the free pr o duct Q L ∗ h N × N i where h N × N i is the fr e e group gener ated by the set N × N . One can see that the action of Q on N induces an action on h N × N i by q ( n 1 , n 2 ) = ( q n 1 , q n 2 ) for n 1 , n 2 ∈ N a nd q ∈ Q . Thus we get a morphism ¯ ∂ : Q L ∗ h N × N i → N given by ¯ ∂ (( q , l )( n 1 , n 2 )) = q φ∂ ( l ) h n 1 , n 2 i where h n 1 , n 2 i = n 1 n 2 n − 1 1  β ( n 1 ) n − 1 2  for n 1 , n 2 ∈ N . It is clear tha t the Peiffer lifting is g iven by { n 1 , n 2 } = ( n 1 , n 2 ) for n 1 , n 2 ∈ N . Thu s we get the free ( N → Q )-2-cr o ssed mo dule g enerated by L denoted { θ ∗ ( L ) , N , Q , ∂ ∗ , β } by dividing the gr oup Q L ∗ h N × N i by S genera ted by all elements o f the following rela tions  ¯ ∂ (( q , l )( n 1 , n 2 )) , ¯ ∂ (( q ′ , l ′ )( n ′ 1 , n ′ 2 ))  = [( q , l )( n 1 , n 2 ) , ( q ′ , l ′ )( n ′ 1 , n ′ 2 )] { n 0 , n 1 n 2 } = n 0 n 1 n − 1 0 { n 0 , n 2 } { n 0 , n 1 } { n 0 n 1 , n 2 } =  n 0 , n 1 n − 1 2 n 1  { n 0 , n 1 }  ¯ ∂ (( q , l )( n 1 , n 2 )) , n  = ( q , l )( n 1 , n 2 ) n (( q , l )( n 1 , n 2 )) − 1  n, ¯ ∂ (( q , l )( n 1 , n 2 ))  = n (( q , l )( n 1 , n 2 )) β ( n ) (( q , l )( n 1 , n 2 )) − 1            (4 . 1 . 1) for l , l ′ ∈ L, q , q ′ ∈ Q a nd n, n 0 , n 1 , n 2 , n ′ 1 , n ′ 2 ∈ N . W e no te that relations PL1 and PL5 are g iven a s follows: ∂ ∗ { n 1 , n 2 } = ∂ ∗ ( n 1 , n 2 ) = h n 1 , n 2 i and q { n 1 , n 2 } = q ( n 1 , n 2 ) = ( q n 1 , q n 2 ) = { q n 1 , q n 2 } . T o ge t the 2 -cross e d mo dule induced by θ , we add the rela tio ns { m 1 , m 2 } = { φ ′ ( m 1 ) , φ ′ ( m 2 ) } ( φ ( p ) , l ) = (1 , p l )  (4 . 1 . 2) for m 1 , m 2 ∈ M , l ∈ L a nd p ∈ P . Thu s w e hav e the following commutativ e dia g ram L φ ′′ / / ∂   ( Q L ∗ h N × N i ) /S ∂ ∗   M φ ′ / / α   N β   P φ / / Q (Co)-Induced Two-Cros sed Mo dules 15 where S is the normal subgr oup ge ne r ated by the rela tion (4.1.1) and φ ′′ is defined as φ ′′ ( l ) = (1 , l ) . Also , since we hav e relation (4 . 1 . 2) , θ = ( φ ′ , φ ) is a morphism of 2- crossed mo dules and β ∂ ∗ (( q , l ) ( n 1 , n 2 )) = β ( q ( φ ′ ∂ ( l )) h n 1 , n 2 i ) = q ( φα∂ ( l )) β ( n 1 n 2 n − 1 β ( n 1 ) 1 n − 1 2 ) = 1 so { θ ∗ ( L ) , N , Q , ∂ ∗ , β } is a c omplex of gr oups. Thus { θ ∗ ( L ) , N , Q , ∂ ∗ , β } is an induced ( N → Q )-2 -crossed mo dules. Then we obta in a nalogous res ult in dimension 3 which extends pr op osition 3 a s follows: Theorem 10 L et L ∂ → M α → P b e a 2 - cr osse d mo dule and let θ = ( φ ′ , φ ) : ( M α → P ) − → ( N β → Q ) b e a pr e-cr osse d m o dule morphism and let θ ∗ ( L ) b e the quotient of the fr e e pr o duct Q L ∗ h N × N i by S wher e Q L is gener ate d by the set Q × L with defining re lations ( q , l 1 )( q , l 2 ) = ( q , l 1 l 2 ) ( φ ( p ) , l ) = (1 , p l ) and the fr e e gr oup h N × N i is gener ate d by the set N × N and S is the normal sub gr oup gener ate d by the r elations PL 2 , PL 3 and PL 4 in the defin ition of a 2 -cr osse d mo dule. Then θ ∗ ( L ) ∂ ∗ → N α → Q is an induc e d ( N → Q ) - 2 - cr osse d mo dule wher e ∂ ∗ is given by ∂ ∗ (( q , l ) ( n 1 , n 2 )) = ( q φ ′ ∂ ( l )) h n 1 , n 2 i to gether with the Peiffer lifting { n 1 , n 2 } = ( n 1 , n 2 ) . Pro of. W e hav e only to chec k the universal prop erty . F or any mo r phism of 2-cros sed mo dules ( f , φ ′ , φ ) : { L, M , P , ∂ , α } → { B , N , Q, ∂ ′ , β } there is a unique morphism o f ( N → Q )-2-cro s sed mo dules ( f ∗ , id N , id Q ) : { θ ∗ ( L ) , N , Q , ∂ ∗ , β } → { B , N , Q , ∂ ′ , β } given by f ∗ (( q , l ) ( n 1 , n 2 )) = q f ( l ) { n 1 , n 2 } : ∂ ′ f ∗ (( q , l )( n 1 , n 2 )) = ∂ ′ ( q f ( l ) { n 1 , n 2 } ) = q ( ∂ ′ f ( l )) ∂ ′ { n 1 , n 2 } = q ( φ ′ ∂ ( l )) n 1 n 2 n − 1 1 β ( n 1 ) n − 1 2 = q ( φ ′ ∂ ( l )) h n 1 , n 2 i = ∂ ∗ (( q , l )( n 1 , n 2 )) . (Co)-Induced Two-Cros sed Mo dules 16 F ur ther more the verification of the following equatio ns ar e immediate: f ∗  q ′ (( q , l )( n 1 , n 2 ))  = q f ∗ (( q , l )( n 1 , n 2 )) and f ∗ { n 1 , n 2 } = { n 1 , n 2 } for l ∈ L, n 1 , n 2 ∈ N and q , q ′ ∈ Q a nd f ∗ φ ′′ = f is satisfied as requir ed. In the ca s e when ( φ ′ , φ ) : ( M , P , α ) → ( N , Q , β ) is an epimor phism o r a monomorphism of pre-cr o ssed mo dules , we get prop os itio n 11 a nd pr op osition 12 in dimens io n 3 in terms of prop ositio n 4 and prop ositio n 5 given by Br own for c r ossed mo dules. Prop ositio n 1 1 If L ∂ → M α → P is a 2 -cr osse d mo dule and θ = ( φ ′ , φ ) : ( M , P , α ) → ( N , Q, β ) is an epimorphism of pr e-cr osse d m o dules Ker φ = K and Ker φ ′ = T , then θ ∗ ( L ) ∼ = L/ [ K , L ] wher e [ K , L ] denotes the sub gr oup of L gener ate d by  k l l − 1 | k ∈ K , l ∈ L  for al l k ∈ K , l ∈ L. Pro of. As θ = ( φ ′ , φ ) : ( M , P , α ) → ( N , Q , β ) is an epimo rphism of pre-cro s sed mo dules, Q ∼ = P /K a nd N ∼ = M / T . Since K acts o n tr ivially on L/ [ K , L ] , Q ∼ = P /K acts on L/ [ K , L ] by q ( l [ K , L ]) = pK ( l [ K , L ]) = ( p l ) [ K , L ] . L/ [ K , L ] ∂ ∗ → N β → Q is a 2- c rossed mo dule wher e ∂ ∗ ( l [ K , L ]) = ∂ ( l ) T , β ( mT ) = α ( m ) K. As β ∂ ∗ ( l [ K , L ]) = β ( ∂ ( l ) T ) = α ( ∂ ( l )) K = K , L/ [ K , L ] ∂ ∗ → N β → Q is a co mplex of g roups. The Peiffer lifting N × N → L/ [ K , L ] is g iven by { mT , m ′ T } = { m, m ′ } [ K , L ] . PL1: ∂ ∗ { mT , m ′ T } = ∂ ∗ ( { m, m ′ } [ K , L ]) = ( ∂ { m, m ′ } ) T = ( mm ′ m − 1 α ( m ) m ′− 1 ) T = mT m ′ T ( m − 1 T ) ( α ( m ) m ′− 1 ) T = mT m ′ T ( mT ) − 1 α ( m ) K ( m ′ T ) − 1 = mT m ′ T ( mT ) − 1 β ( mT ) ( m ′ T ) − 1 PL2: { ∂ ∗ ( l [ K , L ]) , ∂ ∗ ( l ′ [ K, L ]) } = { ∂ ( l ) T , ∂ ( l ′ ) T } = { ∂ ( l ) , ∂ ( l ′ ) } [ K, L ] = [ l, l ′ ][ K, L ] = [ l [ K, L ] , l ′ [ K, L ]] (Co)-Induced Two-Cros sed Mo dules 17 for m, m ′ ∈ M and l , l ′ ∈ L . The rest o f axioms of the 2-cr ossed mo dule is given in app endix. ( φ ′′ , φ ′ , φ ) : { L , M , P, ∂ , α } → { L/ [ K , L ] , N , Q, ∂ ∗ , β } or diag rammatically , L φ ′′ / / ∂   L/ [ K , L ] ∂ ∗   M φ ′ / / α   N β   P φ / / Q is a mor phism o f 2-c r ossed mo dules. (See app endix.) Suppo se that ( f , φ ′ , φ ) : { L, M , P , ∂ , α } → { B , N , Q, ∂ ′ , β } is an y 2-cros sed mo dule mor phism. Then we will show that there is a unique 2-cros sed mo dule mo rphism ( f ∗ , id N , id Q ) : { L/ [ K, L ] , N , Q , ∂ ∗ , β } → { B , N , Q, ∂ ′ , β } L/ [ K , L ] ∂ ′ 2 / / f ∗   N id N   β / / Q id Q   B ∂ ′ / / N β / / Q where f ∗ ( l [ K , L ]) = f ( l ) such that ( f ∗ , id N , id Q )( φ ′′ , φ ′ , φ ) = ( f , φ ′ , φ ). Since f ( k l l − 1 ) = f ( k l ) f ( l − 1 ) = φ ( k ) f ( l ) f ( l ) − 1 = 1 B , f ∗ is well-defined. First let us check that ( f ∗ , id N , id Q ) is a 2- crossed mo dule morphism, for l [ K, L ] ∈ L / [ K, L ] and q ∈ Q, f ∗ ( q ( l [ K , L ])) = f ∗ ( pK ( l [ K , L ])) = f ∗ ( p l [ K, L ]) = f ( p l ) = φ ( p ) f ( l ) = pK f ∗ ( l [ K , L ]) = q f ∗ ( l [ K , L ]) , ∂ ′ f ∗ = ∂ ∗ and f ∗ { mT , m ′ T } = f ∗ ( { m, m ′ } [ K , L ]) = f { m, m ′ } = { mT , m ′ T } . (Co)-Induced Two-Cros sed Mo dules 18 Thu s ( f ∗ , id N , id Q ) is a morphism of 2 - crossed mo dules. F urthermor e the fol- lowing equa tion is verified f ∗ φ ′′ = f . So g iven any mor phism of 2-cr ossed mo dules ( f , φ ′ , φ ) : { L, M , P , ∂ , α } → { B , N , Q, ∂ ′ , β } , then there is a unique ( f ∗ , id N , id Q ) 2-cro ssed mo dule mo rphism that makes the following diag ram commutes: { L, M , P , ∂ , α } ( φ ′′ ,φ ′ ,φ ) * * ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ( f ,φ ′ ,φ )   { B , N , Q , ∂ ′ , β } { L/ [ K, L ] , N , Q, ∂ ∗ , β } ( f ∗ ,id N ,id Q ) o o ❴ ❴ ❴ ❴ ❴ ❴ or mor e simply a s B ∂ ′   L ∂   φ ′′ / / f 0 0 L/ [ K , L ] f ∗ 9 9 s s s s s ∂ ∗   N β   M α   φ ′ / / φ ′ 0 0 N id N r r r r r r r r r r r r r r r r r r r r β   Q P φ 0 0 φ / / Q. id Q t t t t t t t t t t t t t t t t t t t t Prop ositio n 1 2 L et θ = ( φ ′ , φ ) : ( M , P , α ) → ( N , Q, β ) b e a pr e-cr osse d mo d- ule morphism wher e φ is a monomorphism and { L, M , P, ∂ , α } b e a 2 - cr osse d mo dule, let T b e a left tr ansversal of φ ( P ) in Q, and T L ∗ h N × N i b e the fr e e pr o duct of T L and h N × N i wher e T L is the fr e e pr o duct of gr oups ( t ∈ T ) e ach isomorphic with L by an isomorphism l 7→ t l ( l ∈ L ) and h N × N i is the fr e e gr oup gener ate d by the set N × N . If q ∈ Q acts on T L ∗ h N × N i by the rule q ( t l ) = u ( p l ) wher e p ∈ P , u ∈ T and q t = uφ ( p ) , then θ ∗ ( L ) = ( T L ∗ h N × N i ) /S and  θ ∗ ( L ) , N , Q , ¯ ∂ , β  is a 2 -cr osse d mo dule with the Peiffer lifting { n 1 , n 2 } = ( n 1 , n 2 ) wher e S is t he normal closur e in T L ∗ h N × N i of the (Co)-Induced Two-Cros sed Mo dules 19 elements  ¯ ∂ ( t l ( n 1 , n 2 )) , ¯ ∂ ( t ′ l ′ ( n ′ 1 , n ′ 2 ))  = [ t l ( n 1 , n 2 ) , t ′ l ′ ( n ′ 1 , n ′ 2 )] { n 0 , n 1 n 2 } = n 0 n 1 n − 1 0 { n 0 , n 2 } { n 0 , n 1 } { n 0 n 1 , n 2 } =  n 0 , n 1 n − 1 2 n 1  { n 0 , n 1 }  ¯ ∂ ( t l ( n 1 , n 2 )) , n  = t l ( n 1 , n 2 ) n ( t l ( n 1 , n 2 )) − 1  n, ¯ ∂ ( t l ( n 1 , n 2 ))  = n ( t l ( n 1 , n 2 )) β ( n ) ( t l ( n 1 , n 2 )) − 1 { m 1 , m 2 } = { φ ′ ( m 1 ) , φ ′ ( m 2 ) } for l , l ′ ∈ L, n, n 0 , n 1 , n 2 ∈ N , t, t ′ ∈ T and m 1 , m 2 ∈ M . Now co nsider an a rbitrary push-out s quare { L 0 , M 0 , P 0 , ∂ 0 , α 0 }   / / { L 1 , M 1 , P 1 , ∂ 1 , α 1 }   (4 . 1 . 3) { L 2 , M 2 , P 2 , ∂ 2 , α 2 } / / { L, M , P , ∂ , α } of 2-cro ssed mo dules. In o rder to describ e { L, M , P , ∂ , α } , we first no te tha t ( M α → P ) is the push-out of the pr e - crossed mo dule mor phisms ( M 1 , P 1 , α 1 ) ← ( M 0 , P 0 , α 0 ) → ( M 2 , P 2 , α 2 ) . (This is b eca us e the functor { L, M , P , ∂ , α } 7→ ( M , P , α ) from 2 -crosse d mo dules to pre-cro ssed mo dules has a r ight adjoint ( M , P, ∂ ) 7→ {h M , M i , M , P , i, α } . ) The pre- crossed mo dule morphisms θ i = ( φ ′ i , φ i ) : ( M i , P i , α i ) → ( M , P , α ) ( i = 0 , 1 , 2) in (4 . 1 . 3) and can b e used to form induced ( M α → P )-2-cr ossed mo dules B i = ( θ i ) ∗ ( L i ) . L is the push-out in X 2 Mo d / ( M , P ) of the r esulting ( M α → P )-2 - crossed mo dule mor phis ms ( B 1 → M → P ) ( β 1 ,id M ,id P ) ← − ( B 0 → M → P ) ( β 2 ,id M ,id P ) − → ( B 2 → M → P ) and ca n b e describ ed a s follows. Prop ositio n 1 3 L et B i b e a ( M α → P ) - 2 -cr osse d mo dule for i = 0 , 1 , 2 and let L b e the push-out in X 2 Mo d / ( M , P ) of ( M α → P ) - 2 - cr osse d mo dule morphisms B 1 β 1 ← − B 0 β 2 − → B 2 . L et B b e the push-out of β 1 and β 2 in the c ate gory of gr oups, e quipp e d with the induc e d morphism B ∂ → ( M α → P ) s uch that αβ = 1 and the induc e d action of (Co)-Induced Two-Cros sed Mo dules 20 P on B . Then L = B / S, wher e S is the normal closur e in B of the elements { ∂ ( b ) , ∂ ( b ′ ) } [ b, b ′ ] − 1 { m, m ′ m ′′ } { m, m ′ } − 1  mm ′ m − 1 { m, m ′′ }  − 1 { mm ′ , m ′′ }  α ( m ) { m ′ , m ′′ }  − 1  m, m ′ m ′′ m ′− 1  − 1 { ∂ ( b ) , m }  m b − 1  − 1 b − 1 { m, ∂ ( b ) }  α ( m ) b − 1  − 1 ( m b ) − 1 p { m, m ′ } { p m, p m ′ } − 1 for b, b ′ ∈ B , m, m ′ , m ′′ ∈ M and p ∈ P . In the case when { L 2 , M 2 , P 2 , ∂ 2 , α 2 } is the trivial 2-cr ossed mo dule { 1 , 1 , 1 , id, i d } the push-out { L, M , P , ∂ 2 , ∂ 1 } in (4 .1.3) is the co kernel of the mor phism { L 0 , M 0 , P 0 , ∂ 0 , α 0 } → { L 1 , M 1 , P 1 , ∂ 1 , α 1 } . 5 App endix The pro of of Theorem 8 PL3: a ) ( n,p ) ( n ′ ,p ′ ) ( n,p ) − 1 { ( n, p ) , ( n ′′ , p ′′ ) } { ( n, p ) , ( n ′ , p ′ ) } = ( nn ′ n − 1 ,pp ′ p − 1 ) { n, n ′′ } { n, n ′ } = nn ′ n − 1 { n, n ′′ } { n, n ′ } = { n, n ′ n ′′ } = { ( n, p ) , ( n ′ n ′′ , p ′ p ′′ ) } = { ( n, p ) , ( n ′ , p ′ ) ( n ′′ , p ′′ ) } b ) n ( n, p ) , ( n ′ , p ′ ) ( n ′′ , p ′′ ) ( n ′ , p ′ ) − 1 o  ∂ ∗ 1 ( n,p ) { ( n ′ , p ′ ) , ( n ′′ , p ′′ ) }  =  ( n, p ) ,  n ′ n ′′ n ′− 1 , p ′ p ′′ p ′− 1  p { ( n ′ , p ′ ) , ( n ′′ , p ′′ ) } =  n, n ′ n ′′ n ′− 1  p { n ′ , n ′′ } =  n, n ′ n ′′ n ′− 1  φ ( p ) { n ′ , n ′′ } =  n, n ′ n ′′ n ′− 1  ∂ 1 ( n ) { n ′ , n ′′ } = { nn ′ , n ′′ } = { ( nn ′ , pp ′ ) , ( n ′′ , p ′′ ) } = { ( n, p ) ( n ′ , p ′ ) , ( n ′′ , p ′′ ) } for ( n, p ) , ( n ′ , p ′ ) , ( n ′′ , p ′′ ) ∈ φ ∗ ( N ). PL4: (Co)-Induced Two-Cros sed Mo dules 21 { ∂ ∗ 2 h, ( n, p ) } { ( n, p ) , ∂ ∗ 2 h } = { ( ∂ 2 h, 1) , ( n, p ) } { ( n, p ) , ( ∂ 2 h, 1) } = { ∂ 2 h, n } { n, ∂ 2 h } = h ∂ 1 ( n ) h − 1 = h φ ( p ) h − 1 = h p h − 1 = h ∂ ∗ 1 ( n,p ) h − 1 for ( n, p ) ∈ φ ∗ ( N ) and h ∈ ∂ − 1 2 ( K e r ∂ 1 ) . PL5: n p ′′ ( n, p ) , p ′′ ( n ′ , p ′ ) o = n φ ( p ′′ ) n, p ′′ p ( p ′′ ) − 1  ,  φ ( p ′′ ) n ′ , p ′′ p ′ ( p ′′ ) − 1 o = n φ ( p ′′ ) n, φ ( p ′′ ) n ′ o = φ ( p ′′ ) { n, n ′ } = p ′′ { n, n ′ } = p ′′ { ( n, p ) , ( n ′ , p ′ ) } for ( n, p ) , ( n ′ , p ′ ) ∈ φ ∗ ( N ) and p ′′ ∈ P . φ ′′ ( p h ) = p h and φ ′ ( p ( n, p ′ )) = φ ′  φ ( p ) n, pp ′ p − 1  = φ ( p ) h = φ ( p ) n = φ ( p ) φ ′′ ( h ) = φ ( p ) φ ′ ( n, p ′ ) φ ′ ( ∂ ∗ 2 h ) = φ ′ ( ∂ 2 h, 1) and ∂ 1 ( φ ′ ( n, p ′ )) = ∂ 1 ( n ) = ∂ 2 ( h ) = φ ( p ′ ) = ∂ 2 ( φ ′′ h ) = φ ( ∂ ∗ 1 ( n, p ′ )) {− , −} ( φ ′ × φ ′ ) (( n, p ) , ( n ′ , p ′ )) = {− , −} ( φ ′ ( n, p ) , φ ′ ( n ′ , p ′ )) = {− , −} ( n, n ′ ) = { n, n ′ } = φ ′′ ( { n, n ′ } ) = φ ′′ ( { ( n, p ) , ( n ′ , p ′ ) } ) = φ ′′ {− , −} (( n, p ) , ( n ′ , p ′ )) for h ∈ ∂ − 1 2 ( K e r ∂ 1 ) , ( n, p ) , ( n ′ , p ′ ) , ( n, p ′ ) ∈ φ ∗ ( N ) and p ∈ P . The pro of of prop o sition 11: PL3: { mT , m ′ T m ′′ T } = { m, m ′ m ′′ } [ K , L ] = ( mm ′ m − 1 { m, m ′′ } { m, m ′ } )[ K, L ] = mm ′ m − 1 { m, m ′′ } [ K , L ] { m, m ′ } [ K , L ] = ( mm ′ m − 1 ) T ( { m, m ′′ } [ K, L ]) { m, m ′ } [ K , L ] = mT m ′ T m − 1 T { m, m ′′ } [ K, L ] { m, m ′ } [ K, L ] = mT m ′ T ( m T ) − 1 { mT , m ′′ T } { mT , m ′ T } (Co)-Induced Two-Cros sed Mo dules 22 { mT m ′ T , m ′′ T } = { mm ′ , m ′′ } [ K , L ] = (  m, m ′ m ′′ m ′− 1  α ( m ) { m ′ , m ′′ } )[ K, L ] =  m, m ′ m ′′ m ′− 1  [ K, L ] ( α ( m ) K { m ′ , m ′′ } )[ K, L ] =  mT , m ′ T m ′′ T ( m ′ T ) − 1  β ( mT ) { m ′ T , m ′′ T } for m , m ′ , m ′′ ∈ M . PL4: a ) { ∂ ∗ ( l [ K , L ]) , mT } = { ∂ ( l ) T , mT } = { ∂ ( l ) , m } [ K , L ] = ( l m l − 1 )[ K, L ] = l [ K, L ] m l − 1 [ K, L ] = l [ K, L ] mT ( l [ K , L ]) − 1 b ) { mT , ∂ ∗ ( l [ K , L ]) } = { mT , ∂ ( l ) T } = { m, ∂ ( l ) } [ K, L ] = ( m l α ( m ) l − 1 )[ K, L ] = m l [ K, L ] α ( m ) l − 1 [ K, L ] = mT ( l [ K , L ]) α ( m ) K ( l − 1 [ K, L ]) = mT ( l [ K , L ]) β ( mT ) ( l [ K , L ]) − 1 for l ∈ L and m ∈ M . PL5: pK { mT , m ′ T } = pK ( { m, m ′ } [ K, L ]) = p { m, m ′ } [ K , L ] = { p m, p m ′ } [ K, L ] = { p mT , p m ′ T } =  pK ( mT ) , pK ( m ′ T )  for m , m ′ ∈ M a nd p ∈ P. φ ′′ ( p l ) = p l [ K, L ] = pK ( l [ K , L ]) = φ ( p ) φ ′′ ( l ) and φ ′ ( p m ) = p mT = pK ( mT ) = φ ( p ) φ ′ ( m ) ∂ ∗ φ ′′ ( l ) = ∂ ∗ ( l [ K , L ]) = ∂ ( l ) T = φ ′ ∂ ( l ) and β φ ′ ( m ) = β ( mT ) = α ( m ) K = φα ( m ) and simila rly φ ′′ {− , −} = {− , − } ( φ ′ × φ ′ ) for l ∈ L , m ∈ M and p ∈ P. References [1] R. 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Onarlı Department o f Ma thematics-Computer Eski¸ sehir O smangazi University 26480 E ski¸ sehir/T urkey e-mails: { uege,zar v as i, go narli } @ ogu.edu.tr