Pointed homotopy and pointed lax homotopy of 2-crossed module maps
aa r X i v : . [ m a t h . C T ] S e p Pointed homotopy and pointed lax homotopy of 2-crossed module maps
Bj¨orn Gohla a , Jo˜ao Faria Martins b a Centro de Matem´atica da Universidade do Porto, Departamento de Matem´atica da FCUP;Rua do Campo Alegre, 687 ; 4169-007 Porto, Portugal b Departamento de Matem´atica and Centro de Matem´atica e Aplica¸c˜oesFaculdade de Ciˆencias e Tecnologia (Universidade Nova de Lisboa), Quinta da Torre, 2829-516 Caparica, Portugal
Abstract
We address the (pointed) homotopy theory of 2-crossed modules (of groups), which are known to faithfullyrepresent Gray 3-groupoids, with a single object, and also connected homotopy 3-types. The homotopyrelation between 2-crossed module maps will be defined in a similar way to Crans’ 1-transfors between strictGray functors, however being pointed, thus this corresponds to Baues’ homotopy relation between quadraticmodule maps. Despite the fact that this homotopy relation between 2-crossed module morphisms is not,in general, an equivalence relation, we prove that if A and A ′ are 2-crossed modules, with the underlyinggroup F of A being free (in short A is free up to order one), then homotopy between 2-crossed module maps A → A ′ yields, in this case, an equivalence relation. Furthermore, if a chosen basis B is specified for F , thenwe can define a 2-groupoid HOM B ( A , A ′ ) of 2-crossed module maps A → A ′ , homotopies connecting them,and 2-fold homotopies between homotopies, where the latter correspond to (pointed) Crans’ 2-transforsbetween 1-transfors.We define a partial resolution Q ( A ), for a 2-crossed module A , whose underlying group is free, witha canonical chosen basis, together with a projection map proj: Q ( A ) → A , defining isomorphisms at thelevel of 2-crossed module homotopy groups. This resolution (which is part of a comonad) leads to a weakernotion of homotopy (lax homotopy) between 2-crossed module maps, which we fully develop and describe.In particular, given 2-crossed modules A and A ′ , there exists a 2-groupoid HOM
LAX ( A , A ′ ) of (strict) 2-crossed module maps A → A ′ , and their lax homotopies and lax 2-fold homotopies, leading to the questionof whether the category of 2-crossed modules and strict maps can be enriched over the monoidal category G ray.The associated notion of a (strict) 2-crossed module map f : A → A ′ to be a lax homotopy equivalencehas the two-of-three property, and it is closed under retracts. This discussion leads to the issue of whetherthere exists a model category structure in the category of 2-crossed modules (and strict maps) where weakequivalences correspond to lax homotopy equivalences, and any free up to order one 2-crossed module iscofibrant. Keywords:
Crossed module, 2-crossed module, quadratic module, homotopy 3-type, tricategory, Graycategory, Peiffer lifting, Simplicial group
1. Introduction and simplicial group background / context
Let G = (cid:0) G n , d ni , s ni ; i ∈ { , , . . . , n } , n = 0 , , , . . . (cid:1) be a simplicial group; [45, 27, 37, 18]. As usual,see for example [40, 25], we say that G is free if each group G n of n -simplices is a free group, with a Email addresses: [email protected] (Bj¨orn Gohla), [email protected] (Jo˜ao Faria Martins)
Preprint submitted to Elsevier September 25, 2018 hosen basis, and these basis are stable under the degeneracy maps s ni : G n → G n +1 . Recall that theMoore complex [43, 44, 16] N ( G ) of a simplicial group G is given by the (normal) complex of groups( · · · → A n ∂ n −→ A n − → · · · → A = G ), where: A n = n − \ i =0 ker( d ni ) , and ∂ n : A n → A n − is the restriction of the boundary map d nn : G n → G n − . We say that the Moore complexof G has length n if the unique (possibly) non trivial components of N ( G ) are A n − → A n − → · · · → A .(Here “length” correspond to the number of groups, rather than the number of arrows, which is the usualconvention). Not surprisingly, this Moore complex has a lot of extra structure, defining what a hyper crossedcomplex is [13], which retains enough information to recover the original simplicial group, up to isomorphism.This contains two well known results, stating that the categories of simplicial groups with Moore complexesof length two and three (respectively) are equivalent to the categories of crossed modules and of 2-crossedmodules of groups (respectively), see [43, 44, 16], the latter being exactly hyper crossed complexes of lengthtwo and three (respectively); we will go back to this issue below. Hyper crossed complexes therefore generaliseboth crossed modules and 2-crossed modules.Looking at the last two stages of the Moore complex N ( G ) of a simplicial group G , namely ∂ = ∂ : N ( G ) → N ( G ), one has an induced action of N ( G ) on N ( G ) by automorphisms, and also the ac-tion of N ( G ) on itself by conjugation, and the boundary map ∂ : N ( G ) → N ( G ) preserves these actions;in other words one has a pre-crossed module ([9, 3, 4, 40]), called the pre-crossed module associated to thesimplicial group G .The homotopy groups of a simplicial group G are, by definition, given by the homology groups of itsMoore complex N ( G ) (which is a normal complex of, not necessarily abelian, groups). These correspond tothe homotopy groups of the simplicial group G seen as a simplicial set (despite the fact that π ( S ), for S a simplicial set, is not in general a group but a set). Note that simplicial groups are Kan complexes, andtherefore their homotopy groups are well defined [37, 18].It is a fundamental result of Quillen [45, 27] that the category of simplicial groups is a model category,where weak equivalences are the simplicial group maps f : G → G ′ , inducing isomorphisms at the level ofhomotopy groups, and fibrations are the simplicial groups maps f : G → G ′ whose induced map on Moorecomplexes ( f i , i = 0 , , . . . ) : N ( G ) → N ( G ′ ) is surjective for all i >
0, and in particular any object is fibrant.Notice that, for f = ( f i ) to be a fibration, we do not impose that the induced map f : A = G → A ′ = G ′ be surjective; however if f is a weak equivalence and a fibration then certainly f is surjective. Cofibrationsare defined as being the maps that have the left lifting property with respect to all acyclic fibrations.In particular any free simplicial group is cofibrant [45, 25]. The pre-crossed module associated to a freesimplicial group is of the form F → F where F is a free group and F → F is a free pre-crossed module,[40]. Such a pre-crossed module is what is called in [40] a totally free pre-crossed module.Let SG denote the category of simplicial groups and SG n denote the full subcategory of simplicial groupswith Moore complex of length n . The former is a reflexive subcategory of SG and we denote the reflexionfunctor (the n -type or n th -Postnikov section) by P n : SG → SG n , a left adjoint to the inclusion functor SG n → SG . At the level of Moore complexes ( A m , ∂ m ) is sent, via P n , to (see [25]): A n − /∂ ( A n ) → A n − → A n − → · · · → A . This adjunction induces a closed model category structure on SG n where fibrations (weak equivalences) arethe maps whose underlying simplicial group map is a fibration (weak equivalence), and cofibrations are themaps which have the left lifting property with respect to all acyclic fibrations; all of this is explained forexample in [14]. Therefore the n -type functor P n preserves cofibrations, [15]. This will give model categorystructures in the categories of crossed modules and of 2-crossed modules (of groups); [14, 15]. The caseof crossed modules of groupoids in treated in [42, 39, 7], not appealing directly to simplicial group(oid)techniques.A crossed module ( ∂ : E → G, ⊲ ) is given by a group morphism ∂ : E → G , together with a left action ⊲ of G on E by automorphisms, so that ∂ preserves the actions, where G is given the adjoint action (thus2 ∂ : E → G, ⊲ ) is a pre-crossed module), such that, furthermore, for each e, f ∈ E the Peiffer pairing h e, f i . = (cid:0) ef e − (cid:1) (cid:0) ∂ ( e ) ⊲ f − (cid:1) ∈ E vanishes. The category of crossed modules is equivalent to the categoryof simplicial groups with Moore complex of length two, where morphisms of crossed modules ( ∂ : E → G, ⊲ ) → ( ∂ : E ′ → G ′ , ⊲ ) are given by chain maps ( ψ : E → E ′ , φ : G → G ′ ), preserving the actions. Crossedmodules form a model category [14, 39], where weak equivalences are the maps inducing isomorphisms onhomotopy groups (the homology groups of the underlying complexes) and fibrations ( E → G ) → ( E ′ → G ′ )are the crossed module maps ( ψ : E → E ′ , φ : G → G ′ ), such that ψ : E → E ′ is surjective (note that we donot require φ : G → G ′ to be surjective). The homotopy category of crossed modules is equivalent to thehomotopy category of pointed connected 2-types [36, 44, 3, 4] (where an n -type is a space X , homotopicto a CW-complex, such that π i ( X ) = { } , if i > n ). Since the reflexion functor P : SG → SG preservescofibrations, a crossed module ( ∂ : E → G ) is cofibrant when G is a free group, since in this case one canprove that there exists a free simplicial group (with Moore complex of length three) whose second Postnikovsection is ( ∂ : E → G ).The fact that, if ( ∂ : E → G, ⊲ ) is a crossed module, with G a free group, then it is cofibrant in the modelcategory of crossed modules of groupoids is directly proven for example in [42]. We note however that,considering the inclusion of the category of crossed modules of groups into the model category [42, 6, 39] ofcrossed modules of groupoids, it does not hold that a fibration of crossed modules of groups is necessarily afibration of crossed modules of groupoids, even though the same is true for weak equivalences. In fact a map( ψ, φ ) : ( E → G ) → ( E ′ → G ′ ), of crossed modules of groups, is a fibration of crossed modules of groupoidsif, and only if, both ψ : E → E ′ and φ : G → G ′ are surjective. This apparent contradiction between the twodefinitions of fibrations of crossed modules of groups should not come as a surprise, since crossed modulesof groups (and their maps) model pointed homotopy classes of maps connecting pointed 2-types, whereascrossed modules of groupoids model non-pointed 2-types and homotopy classes of maps between them. Wenote that a pointed fibration of pointed simplicial sets is also not necessarily a fibration of simplicial sets;[27, V-Lemma 6.6]. Considering the nerve functor [9], from the category of crossed modules to the categoryof pointed simplicial sets, the nerve of a fibration ( ψ, φ ) : ( E → G ) → ( E ′ → G ′ ), of crossed modules ofgroups, is a fibration of simplicial sets if, and only if, φ ( G/∂ ( E )) = G ′ /∂ ( E ′ ), by [27, V - Corollary 6.9],which is the same as saying that φ ( G ) = G ′ .The 2-crossed modules were defined by Conduch´e [16], who proved that the category of 2-crossed modulesis equivalent to the category of simplicial groups whose Moore complex has length three. A 2-crossed moduleis given by a complex of groups ( L δ −→ E ∂ −→ G ), with a given action of G on E and L by automorphisms,such that ( ∂ : E → G ) is a pre-crossed module, and also we have a map { , } : E × E → L (the Peiffer lifting),lifting the Peiffer commutator map <, > : E × E → E , where, as before, h e, f i = ( ef e − ) ( ∂ ( e ) ⊲ f − ). Thislifting has to satisfy very natural properties, satisfied by the Peiffer pairing itself.The category of 2-crossed modules is a Quillen model category, [14, 15], where a map ( µ, ψ, φ ) : ( L → E → G ) → ( L ′ → E ′ → G ′ ) is a fibration if, and only if, µ : L → L ′ and ψ : E → E ′ each are surjective, and aweak equivalence if it induces isomorphisms of homotopy groups. Since the reflection functor P : SG → SG preserves cofibrations, one can see that any 2-crossed module A = ( L → E → G ), with ( E → G ) being atotally free pre-crossed module (in short A is free up to order two), is cofibrant in this model category. Similarresults are proved in [35], where an analogous model category structure for Gray categories is constructed,and it is proven that a Gray category is cofibrant if, and only if, its underlying sesquicategory [46] is free ona computad. We will go back to this issue below. The homotopy category of 2-crossed modules is equivalentto the homotopy category of 3-types: pointed CW-complexes X such that π i ( X ) = { } if i >
3, where theWhitehead products π ( X ) × π ( X ) → π ( X ) are encoded in the Peiffer lifting.Crossed squares, defined in [29], were proven in [36] to be models for homotopy 3-types, furthermorebeing equivalent to cat -groups. Ellis constructed in [20] the fundamental crossed square of a pointedCW-complex, containing all relevant 3-type information. By using Brown-Loday’s theorem [11, 12], thisfundamental crossed square can be proven to be totally free, thus giving a combinatorial description of thefirst three homotopy groups of a CW-complex.The quadratic modules, defined by Baues in [4] (being models for homotopy 3-types) are a special caseof 2-crossed modules, satisfying however additional nilpotency conditions, ensuring for example that the3eiffer liftings { a, h b, c i} and {h a, b i , c } are trivial. This does simplify the calculations in some cases, sincethe main difficulty in working with 2-crossed modules has to do with performing complicated computationswith Peiffer liftings. It is proven in [2] that the category of quadratic modules is a reflexive subcategoryof the category of 2-crossed modules. Even though quadratic modules retain all of the necessary flexibilityfrom 2-crossed modules, in order to model homotopy 3-types, they have the drawback that the functorconstructed in [4, page 216] from the category of CW-complexes (and cellular maps) to the category ofquadratic modules is somehow un-intrinsic, being inductively defined while attaching cells. On the otherhand, the functor constructed in [22], which associates to a pointed CW-complex X its fundamental 2-crossed module (constructed via Ellis’ fundamental crossed square [20] of a pointed CW-complex), whichalso models the homotopy 3-type of X , was intrinsically defined by using usual relative and triadic homotopygroups. This fundamental 2-crossed module of a reduced CW-complex is totally free. Therefore it has acombinatorial description, permitting (as done in [4, 20]) a combinatorial description of the first threehomotopy groups of a reduced CW-complex, and the remaining 3-type information, such as the Whiteheadproducts π × π → π . In addition, maps between 3-types can be completely described, up to homotopy,by maps between fundamental 2-crossed modules (up to homotopy); see [22]. The fundamental quadraticmodule of a CW-complex is a quotient of this fundamental 2-crossed module.An advantage that 2-crossed modules have, in comparison with the (very similar) crossed squares, is thatthere is only one group appearing in dimension two, which facilitates the definition of homotopy betweenmaps. On the model category theoretical side, since the category of 2-crossed modules is equivalent to thecategory of simplicial groups with Moore complex of length three, this means that it comes with a wealthof homotopical algebra structures, transported from the category of simplicial groups [45].In addition to simplicial groups with Moore complex of length three, there are several categories equiv-alent to the category of 2-crossed modules, for example the category of braided regular crossed modules [6]and the category of Gray 3-groupoids [33], mentioned below. We emphasise that the category of 2-crossedmodules is not equivalent to the category of crossed squares, even though there is a mapping cone functorfrom the latter to the former. The comparison between some of these algebraic models for 3-types appearsin [2].Recall, see [17, 30, 28], that a Gray 3-category C (or Gray enriched category) is a category enriched overthe monoidal category of 2-categories, with the Gray tensor product. These can be given a more explicitdefinition, see [17, 33]. In particular, one has sets C , C , C and C of objects, morphisms, 2-morphisms and3-morphisms, and several operations between then. In particular, objects, 1-morphisms and 2-morphismsof C form a sesquicategory [46], a structure similar to a 2-category, but where the interchange law [33, 31]does not hold. Nevertheless, the interchange law holds up to a chosen tri-morphism: the “interchanger”.Gray 3-categories correspond to the strictest version of tri-categories, in the sense that any tricategory canbe strictified to a tri-equivalent Gray-category, [30, 28]. For this reason Gray 3-categories are also calledsemistrict tri-categories.Gray 3-groupoids can be defined analogously: there exists an inclusion of the category of 2-groupoidsinto the category of ω -groupoids [9] (equivalent to crossed complexes of groupoids), which has a left adjoint T (the cotruncation functor) similar to the 2-type functor. The tensor product of ω -groupoids is treatedin [9, 8], and from now on called the Brown-Higgins tensor product. Composing with the cotruncationfunctor yields a tensor product in the category of 2-groupoids (which is simply the restriction of the Graytensor product of 2-categories to 2-groupoids), part of a monoidal closed structure. A Gray 3-groupoid is agroupoid enriched over this monoidal category of 2-groupoids. These are therefore Gray 3-categories whereany i -cell ( i ≥
1) is strictly invertible. It is folklore, and explicitly proven for example in [33, 6], that any2-crossed module defines a Gray 3-groupoid with a single object (a “Gray 3-group”), and conversely. Inparticular the interchanger is derived from the Peiffer lifting in the given 2-crossed module.The notion of a homotopy between ω -groupoid maps is treated in [8, 9, 5]. Considering n -fold homotopiesbetween ω -groupoids maps defines an internal-hom “HOM” in the category of ω -groupoids, which, togetherwith the Brown-Higgins tensor product of ω -groupoids, induces a monoidal closed structure. By applyingthe cotruncation functor this yields a monoidal closed structure in the category of 2-groupoids. If A and B are 2-groupoids then HOM( A, B ) is simply the 2-groupoid of strict functors A → B , pseudo-naturaltransformations between them and their modifications. Considering 2-groups (2-groupoids with a single4bject) A and B , and pointed pseudo natural transformations and modifications, yields simply a groupoidwith objects the maps A → B , and morphisms their pointed pseudo-natural transformations. The lattergroupoid is what we want to generalise for Gray 3-groups.In [17], the notion of a pseudo-natural transformation (a 1-transfor) between strict functors of Gray3-categories was addressed. Unlike the notion of pseudo-natural transformations between strict functors of2-categories, this does not define (in general) an equivalence relation between Gray functors, the problembeing that we cannot [17], in general, concatenate, or invert, 1-transfors, due to the lack of the interchangecondition in Gray 3-categories. To have an equivalence relation between strict Gray functors one needs thefull force of tricategories, leading to a less restrictive notion of 1-transfors, see [30, 28, 26]. In particulargiven Gray 3-categories A and B one has a Gray 3-category HOM ( A, B ) of strict Gray functors and theirlax 1-, 2- and 3-transfors.Given that strict 1-transfors between Gray 3-category maps f, g : A → B can be modeled by maps fromthe tensor product (of Gray categories [17]) A ⊗ I into B , where A ⊗ I is a cylinder object for A , the factthat two maps being related by a 1-transfor does not yield, in general, an equivalence relation is not at allsurprising, given that in a model category such a construction would only be an equivalence relation, ingeneral, if A were cofibrant and B were fibrant. As we have mentioned before A is cofibrant if, and only if,the underlying sesquicategory of A is free on a computad, [35].In this article, we will analyse the notion of 1-transfors connecting Gray 3-groupoid maps, as well as 2-transfors connecting these, in context of 2-crossed modules. Similarly to [4, 22], we will restrict our discussionto the pointed case, and, since we are working with complexes of groups, we will name (strict) 1-transforsand 2-transfors as being homotopies and 2-fold homotopies. These are very similar to the usual notions ofhomotopies between chain complex maps, and 2-fold homotopies connecting them, however adapted to thenon-abelian case.To this end, given a 2-crossed module A ′ , we will define a (functorial) path-space P ∗ ( A ′ ) for it, togetherwith two fibrations Pr A ′ , Pr A ′ : P ∗ ( A ′ ) → A ′ . This will be a good path space [19], in the model category of2-crossed modules. We use P ∗ ( A ′ ) to define the homotopy relation between 2-crossed module maps A → A ′ .This coincides with the notion defined ad-hoc in [22]. Dually we note that cylinder objects were constructedin the category of quadratic modules in [4] and in the category of crossed chain algebras in [47]. The lattergeneralise 2-crossed modules.Let A = ( L → E → G ) and A ′ = ( L ′ → E ′ → G ′ ) be 2-crossed modules (or more precisely theunderlying complexes of them, since we also have actions ⊲ of G on E and L and a lifting { , } : E × E → L ofthe Peiffer pairing, and the same for A ′ .) A homotopy between the 2-crossed module maps f, f ′ : A → A ′ ,explicitly f = ( µ : L → L ′ , ψ : E → E ′ , φ : G → G ′ ) and f ′ = ( µ ′ : L → L ′ , ψ ′ : E → E ′ , φ ′ : G → G ′ ), is givenby two set maps s : G → E ′ and t : E → L ′ , satisfying appropriate properties (defining what we called aquadratic f -derivation), resembling the notion of homotopy between quadratic module maps, treated in [4].For example, for all g, h ∈ G we must have s ( gh ) = φ ( h ) − ⊲ s ( g ) s ( h ), thus s : G → E ′ is to be a derivation.As expected by the discussion above, the notion of homotopy between 2-crossed module maps A → A ′ is not an equivalence relation (we give in 3.1.1 an explicit example to show that this is the case). This issuecan be fixed, for example, if we consider the case when the underlying pre-crossed module ( ∂ : E → G ) of A is totally free, thus A is cofibrant in the model category of 2-crossed modules, which renders all objectsfibrant. This point of view was the one considered in [4, Lemma 4.7], while proving that the homotopyrelation between quadratic chain complex maps is an equivalence relation, in the totally free case.What is surprising (and will be the main result of this paper) is that if A = ( L → E → F ) is a 2-crossedmodule, such that F is a free group (in short A is free up to order one), then homotopy between 2-crossedmodule maps A → A ′ defines an equivalence relation. Moreover, if a free basis B of F is specified, then wecan define a groupoid, whose objects are the 2-crossed module maps A → A ′ and the morphisms are theirhomotopies, represented as (for ( s, t ) a quadratic f -derivation): f ( f,s,t ) −−−−→ f ′ . This process can be continued, to define a 2-groupoid HOM B ( A , A ′ ), with objects being the 2-crossedmodule maps A → A ′ , the morphisms being the homotopies between then, and the 2-morphisms being5-fold homotopies between homotopies. We will present very detailed calculations.Consider two 2-crossed modules: A = (cid:16) L δ −→ E ∂ −→ F, ⊲, { , } (cid:17) and A ′ = (cid:16) L ′ δ −→ E ′ ∂ −→ G ′ , ⊲, { , } (cid:17) , such that F is a free group, with a chosen basis B . If the quadratic f -derivation ( s : F → E ′ , t : E → L ′ )connects f : A → A ′ and f ′ : A → A ′ and ( s ′ : F → E ′ , t ′ : E → L ′ ) connects f ′ : A → A ′ and f ′′ : A → A ′ ,diagrammatically: f ( f,s,t ) −−−−→ f ′ and f ′ ( f ′ ,s ′ ,t ′ ) −−−−−→ f ′′ , then we explicitly construct a quadratic f -derivation ( s ⊗ s ′ , f ⊗ f ′ ), such that: f ( f,s ⊗ s ′ ,t ⊗ s ′′ ) −−−−−−−−−→ f ′′ . This concatenation “ ⊗ ” of homotopies is associative and it has inverses. The derivation s ⊗ s ′ : F → E ′ is theunique derivation F → E ′ which, on the chosen basis B of F , has the form b s ( b ) s ′ ( b ). In the case when( ∂ : E ′ → G ′ ) is a crossed module, and the Peiffer lifting of A ′ is trivial, we have that ( s ⊗ s ′ )( g ) = s ( g ) s ′ ( g )for each g ∈ F . Otherwise one has a map ω ( s,s ′ ) : F → L ′ , measuring the difference between s ⊗ s ′ and thepointwise product of s and s ′ ; namely we have:( s ⊗ s ′ )( g ) = s ( g ) s ′ ( g ) δ (cid:0) ω ( s,s ′ ) ( g ) (cid:1) − , for each g ∈ F. This map ω ( s,s ′ ) : F → L ′ has a prime importance in the construction of the concatenation of homotopies.Let G be a group. The free group on the underlying set of G is denoted by F group ( G ). The set inclusion G → F group ( G ) is written as g ∈ G [ g ] ∈ F group ( G ). We have the obvious basis [ G ] = { [ g ] , g ∈ G } of F group ( G ). Consider the obvious projection group map p : F group ( G ) → G , thus p ([ g ]) = g . There existsa very natural partial resolution functor Q , from the category of 2-crossed modules to itself, which, to a2-crossed module A = (cid:16) L δ −→ E ∂ −→ G, ⊲, { , } (cid:17) , associates the 2-crossed module: Q ( A ) = (cid:16) L δ ′ −→ E ∂ × p F group ( G ) ∂ ′ −→ F group ( G ) , ⊲, { , } (cid:17) , where: E ∂ × p F group ( G ) = { ( e, u ) ∈ E × F group ( G ) : ∂ ( e ) = p ( u ) } . Therefore Q ( A ) is free up to order one. Moreover there is a projection proj = ( r, q, p ) : Q ( A ) → A whichyields isomorphisms at the level of homotopy groups. It has the form:proj = L δ ′ / / r (cid:15) (cid:15) E ∂ × p F group ( G ) ∂ ′ / / q (cid:15) (cid:15) F group ( G ) p (cid:15) (cid:15) L δ / / E ∂ / / G where r = id and q ( e, u ) = e . Therefore proj: Q ( A ) → A is surjective and a weak equivalence (thus anacyclic fibration in the model category of 2-crossed modules).The map proj: Q ( A ) → A , resembling a cofibrant replacement, is proven to be part of a comonad in [26](we note that the results in this paper are independent from the ones in [26]). Its co-Kleisli category [38] leadsto weaker notions of maps A → A ′ between 2-crossed modules [24, 26], and of homotopies between strict 2-crossed module maps A → A ′ , as well as of their 2-fold homotopies, yielding a 2-groupoid HOM
LAX ( A , A ′ ),of strict maps A → A ′ , lax homotopies and lax 2-fold homotopies, which we fully describe. Let us be specific.Let A and A ′ be 2-crossed modules. Let hom( A , A ′ ) denote the set of 2-crossed module maps A → A ′ . Giventhat proj: Q ( A ) → A is surjective, the map f ∈ hom( A , A ′ ) f ◦ proj ∈ hom( Q ( A ) , A ′ ) is an injection.A 2-crossed module map Q ( A ) → A ′ is said to be strict if it factors (uniquely) through proj: Q ( A ) → A .6hen we define HOM
LAX ( A , A ′ ) as being the full sub-2-groupoid of HOM [ G ] ( Q ( A ) , A ′ ), with objects beingthe strict maps Q ( A ) → A ′ (and we call the 1- and 2-morphisms of HOM
LAX ( A , A ′ ) lax 1- and 2-foldhomotopies). After presenting Q ( A ) combinatorially (by generators and relations), we will give a fullycombinatorial description of lax homotopies between strict 2-crossed module maps, and their lax 2-foldhomotopies, therefore explicitly constructing the 2-groupoid HOM
LAX ( A , A ′ ).Lax homotopies between strict 2-crossed module maps behave well with respect to composition by strict2-crossed module maps. Therefore it is natural to conjecture that the category of 2-crossed modules, strict2-crossed module maps, lax homotopies and their 2-fold homotopies is a Gray 3-category.We say that a 2-crossed module map f : A → A ′ is a lax homotopy equivalence if there exists a 2-crossedmodule map g : A ′ → A such that f ◦ g and g ◦ f each are lax homotopic to id A ′ and id A , respectively. Sincewe can concatenate lax homotopies between 2-crossed module maps, the class of lax homotopy equivalenceshas the two-of-three property; [19]. Given that we can compose lax homotopies with strict 2-crossed modulemaps, any retract of a lax homotopy equivalence is a lax homotopy equivalence. All of this discussion leadsto the issue of whether there exists a model category structure in the category of 2-crossed modules (differentfrom the one already referred to, which has as cofibrant objects the retracts of free up to order two 2-crossedmodules) where weak equivalences correspond to lax homotopy equivalences, and where free up to orderone 2-crossed modules are cofibrant, and so that the path-space constructed in this paper is still a goodpath-space.
2. Preliminaries on pre-crossed modules and 2-crossed modules
Definition 1 (Pre-crossed module).
A pre-crossed module ( ∂ : E → G, ⊲ ) is given by a group morphism ∂ : E → G , together with a left action ⊲ of G on E by automorphisms, such that the following relation, called“first Peiffer relation”, holds: ∂ ( g ⊲ e ) = geg − , for each g ∈ G and each e ∈ E. A crossed module ( ∂ : E → G, ⊲ ) is a pre-crossed module satisfying, further, the “second Peiffer relation”: ∂ ( x ) ⊲ y = xyx − , for each x, y ∈ E. Note that in a crossed module ( ∂ : E → G, ⊲ ) the subgroup ker( ∂ ) ⊂ E is central in E .Let ( ∂ : E → G, ⊲ ) be a pre-crossed module. Given x, y ∈ E , their Peiffer commutator is given by h x, y i = (cid:0) xyx − (cid:1)(cid:0) ∂ ( x ) ⊲ y − (cid:1) . Thus a pre-crossed module is a crossed module if, and only if, all of its Peiffer commutators are the identityof E . In any pre-crossed module it holds that for each x, y ∈ E (and where 1 G is the identity of G ): ∂ ( h x, y i ) = 1 G . A morphism f = ( ψ, φ ) between the pre-crossed modules ( ∂ : E → G, ⊲ ) and ( ∂ ′ : E ′ → G ′ , ⊲ ′ ) is givenby a pair of group morphisms ψ : E → E ′ and φ : G → G ′ making the diagram: E ∂ −−−−→ G ψ y y φ E ′ ∂ ′ −−−−→ G ′ commutative, and such that ψ ( g ⊲ e ) = φ ( g ) ⊲ ′ ψ ( e ) , for each e ∈ E and g ∈ G. Morphisms of crossed modules are defined analogously, and therefore the category of crossed modules is afull subcategory of the category of pre-crossed modules.7 xample 2 (The underlying group functor Gr ). There is an underlying group functor Gr sending apre-crossed module ( E → G, ⊲ ) to the group G . This has a right adjoint R sending a group G to the crossedmodule (id : G → G, ad) , where we consider the adjoint action ad of G on G . The unit of this adjunction,yields a pre-crossed module map η G = ( ∂, id) : ( ∂ : E → G, ⊲ ) → (id : G → G, ad) . = η ( ∂ : E → G, ⊲ ) , for eachpre-crossed module G = ( ∂ : E → G, ⊲ ) . Definition 3 (The principal group functor Gr ). There is a principal group functor Gr from the cat-egory of pre-crossed modules to the category of groups, sending a pre-crossed module ( E → G, ⊲ ) to E . The category of pre-crossed modules is fibered [38, 9] over the category of groups, by considering theunderlying group functor Gr . Cartesian maps are easy enough to describe, [1]: Lemma 4.
Let G = ( ∂ : E → G, ⊲ ) and P = ( ∂ : M → P, ⊲ ) be pre-crossed modules. Consider a pre-crossedmodule map ( ψ : E → M, φ : G → P ) : G → P . Suppose that E is isomorphic to the obvious pullback, namelythat there exists a group isomorphism: f : E → G φ × ∂ M . = { ( g, m ) ∈ G × M : φ ( g ) = ∂ ( m ) } , and that ∂ : E → G and ψ : E → M are the compositions of f with the obvious projections G φ × ∂ M → G and G φ × ∂ M → M . Moreover, suppose that f ( g ⊲ e ) = g ⊲ f ( e ) , where g ∈ G and e ∈ E , where we put g ⊲ ( h, m ) = ( ghg − , φ ( g ) ⊲ m ) , for each g ∈ G and ( h, m ) ∈ G φ × ∂ M . Consider another pre-crossed module H = ( ∂ : W → H ) . Suppose we are given group maps ζ : H → G and φ ′ : H → P with φ ◦ ζ = φ ′ and a groupmap ψ ′ : W → E , making ( ψ ′ , φ ′ ) : H → P a pre-crossed-module morphism. Then there exists a uniquegroup morphism ζ ∗ : W → E , making the diagram: E ψ / / ∂ (cid:15) (cid:15) M ∂ (cid:15) (cid:15) W ζ ∗ > > ⑥⑥⑥⑥ ψ ′ ♥♥♥♥♥♥♥♥♥♥♥♥♥♥ ∂ (cid:15) (cid:15) G φ / / PH ζ > > ⑥⑥⑥⑥⑥⑥⑥⑥ φ ′ ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ (1) commutative. Moreover ( ζ ∗ , ζ ) : H → G is a pre-crossed module map.
Remark 5.
Note that, given the form of pull-backs in the category of groups, there exists a unique set map ζ ∗ : W → E that makes the diagram (1) commutative. Therefore a set map ζ ∗ : W → E that makes (1) commutative will immediately give a pre-crossed module map ( ζ ∗ , ζ ) : H → G . Proof.
That the group map ζ ∗ : W → E exists and is unique follows immediately since E is a pull-back.In particular ∂ ◦ ζ ∗ = ζ ◦ ∂ . Given g ∈ H and w ∈ W then ψ ( ζ ∗ ( g ⊲ w )) = ψ ′ ( g ⊲ w ) = φ ′ ( g ) ⊲ ψ ′ ( w ) = φ ( ζ ( g )) ⊲ ψ ′ ( w ) .∂ ( ζ ∗ ( g ⊲ w )) = ζ ( ∂ ( g ⊲ w )) = ζ ( g ∂ ( w ) g − ) = ζ ( g ) ζ ( ∂ ( w )) ζ ( g ) − . Therefore if g ∈ H and w ∈ W then f (cid:0) ζ ∗ ( g ⊲ w ) (cid:1) = ζ ( g ) ⊲ f ( ζ ∗ ( w )) . ⊲ ′ of a 2-crossedmodule We will follow the conventions of [16, 23] for the definition of a 2-crossed module. Important referenceson 2-crossed modules are [40, 33, 6, 43]. 8 efinition 6 (2-crossed module).
A 2-crossed module (of groups) is given by a chain complex of groups: L δ −→ E ∂ −→ G together with left actions ⊲ , by automorphisms, of G on L and E (and on G by conjugation), and a G -equivariant function { , } : E × E → L (called the Peiffer lifting). Here G -equivariance means: g ⊲ { e, f } = { g ⊲ e, g ⊲ f } , for each g ∈ G and e, f ∈ E. These are to satisfy: L δ −→ E ∂ −→ G is a chain complex of G -modules (in other words ∂ and δ are G -equivariant and ∂ ◦ δ = 1 .) δ ( { e, f } ) = h e, f i , for each e, f ∈ E . Recall that h e, f i = ( ef e − )( ∂ ( e ) ⊲ f − ) .
3. [ l, k ] = { δ ( l ) , δ ( k ) } , for each l, k ∈ L . Here [ l, k ] = lkl − k − . { δ ( l ) , e } { e, δ ( l ) } = l ( ∂ ( e ) ⊲ l − ) , for each e ∈ E and l ∈ L . { ef, g } = (cid:8) e, f gf − (cid:9) ∂ ( e ) ⊲ { f, g } , for each e, f, g ∈ E . { e, f g } = { e, f } ( ∂ ( e ) ⊲ f ) ⊲ ′ { e, g } , where e, f, g ∈ E .Here we have put: e ⊲ ′ l = l (cid:8) δ ( l ) − , e (cid:9) , where l ∈ L and e ∈ E. (2)The following is very well know; see [16, 43]. Lemma 7 (Secondary action ⊲ ′ of a 2-crossed module). Let A = ( L δ −→ E ∂ −→ G, ⊲, { , } ) be a 2-crossedmodule. The map ( e, l ) ∈ G × E e ⊲ ′ l of equation (2) is a left action of E on L by automorphisms, calledthe “secondary action of A ”. Together with the map δ : L → E , the action ⊲ ′ defines a crossed module. Note that in particular it follows that ker( δ ) ⊂ L is central in L . We also have: (cid:8) δ ( l ) − , e (cid:9) − l − = ( e ⊲ ′ l ) − = e ⊲ ′ l − = l − { δ ( l ) , e } , (3) e ⊲ ′ l = { δ ( l ) , e } − l, ∂ ( e ) ⊲ l = ( e ⊲ ′ l ) { e, δ ( l ) − } , ∂ ( e ) ⊲ l = { e, δ ( l ) } − e ⊲ ′ l. (4)Therefore ∂ ( a ) ⊲ l = a ⊲ ′ l if a ∈ E and l ∈ ker δ ; equation (6). For each a, b, c ∈ E we have: a ⊲ ′ { b, c } = ∂ ( a ) ⊲ { b, c } (cid:8) a, h b, c i − (cid:9) − = { ∂ ( a ) ⊲ b, ∂ ( a ) ⊲ c }{ a, ( ∂ ( b ) ⊲ c ) bc − b − } − . (5)A morphism f = ( µ, ψ, φ ) between the 2-crossed modules A = ( L → E → G , ⊲ , { , } ) and A =( L → E → G , ⊲ , { , } ) is given by group morphisms µ : L → L , ψ : E → E and φ : G → G , defininga chain map between the underlying complexes, such that, for each e, f ∈ E , g ∈ G and k ∈ L : µ ( { e, f } ) = { ψ ( e ) , ψ ( f ) } , µ ( g ⊲ k ) = φ ( g ) ⊲ µ ( k ) and ψ ( g ⊲ e ) = φ ( g ) ⊲ ψ ( e ) . The set of 2-crossed module morphisms A → A is denoted by hom( A , A ).For a proof of the following lemma we refer to [22, 23, 16]. Lemma 8.
In a 2-crossed module ( L δ −→ E ∂ −→ G, ⊲, { , } ) we have, for each e, f, g ∈ E , a ∈ G and k ∈ L : { e, E } = { E , e } = 1 L , a ⊲ ( e ⊲ ′ k ) = ( a ⊲ e ) ⊲ ′ ( a ⊲ k ) , (6) { ef, g } = ( e ⊲ ′ { f, g } ) { e, ∂ ( f ) ⊲ g } , { e, f g } = (cid:0) ( ef e − ) ⊲ ′ { e, g } (cid:1) { e, f } , (7) { e, f } − = ∂ ( e ) ⊲ { e − , ef e − } , { e, f } − = ( ef e − ) ⊲ ′ { e, f − } , (8) { e, f } − = ( ∂ ( e ) ⊲ f ) ⊲ ′ { e, f − } , { e, f } − = e ⊲ ′ { e − , ∂ ( e ) ⊲ f } . (9)9 xample 9. Given a pre-crossed module E → G , consider the Peiffer subgroup h E, E i ⊂ E , generated bythe Peiffer commutators h a, b i ; see subsection 2.1. Then h E, E i → E → G is a 2-crossed module, where thePeiffer lifting is { a, b } = h a, b i . Example 10 (The underlying pre-crossed module functor T ). There is a truncation functor T , orunderlying pre-crossed module functor, sending a 2-crossed module A = ( L δ −→ E ∂ −→ G, ⊲, { , } ) to its underly-ing pre-crossed module ( ∂ : E → G, ⊲ ) . This has a right adjoint sending a pre-crossed module ( ∂ : E → G, ⊲ ) to (ker( ∂ ) → E → G, ⊲ ) , where we considered the inclusion map ker( ∂ ) → E , and the Peiffer lifting is as inthe previous example. A left adjoint to the truncation functor was constructed in [22]. Example 11 (The underlying group Gr and principal group Gr functors). There is an underly-ing group functor Gr sending a 2-crossed module ( L → E → G ) to the group G . This has a right adjointsending a group G to the 2-crossed module ( { } → G → G ) , considering the identity map G → G , theadjoint action of G on G ; and the trivial Peiffer lifting. Compare with example 2. On the other hand theprincipal group functor Gr sends a a 2-crossed module ( L → E → G ) to the group E . Definition 12 (Freeness up to order one).
We say that a 2-crossed module A = ( L δ −→ E ∂ −→ F, ⊲, { , } ) is free up to order one if F = Gr ( A ) is a free group. In this paper, free up to order one 2-crossed modules A will always come equipped with a specified chosen basis of the underlying group Gr ( A ) . Definition 13 (Homotopy groups of a 2-crossed module).
Given a 2-crossed module A = ( L δ −→ E ∂ −→ G, ⊲, { , } ) then both im( ∂ ) ⊂ G and im( δ ) ⊂ E are normal subgroups. This permits us to define the homotopygroups π i ( A ) , where i = 1 , , as the first three homology groups of the underlying complex of A .2.3. The path space of a 2-crossed module Let A = ( L δ −→ E ∂ −→ G, ⊲, { , } ) be a 2-crossed module. Let us define the path space P ∗ ( A ) of it,together with two surjective 2-crossed module morphisms P ∗ ( A ) Pr A −−→−−→ Pr A A , and an inclusion i A : A → P ∗ ( A ),with Pr A ◦ i A = Pr A ◦ i A = id A . ∗ and the first and second lifted actions • of a 2-crossed module Most of this discussion appeared in [22].
Remark 14 (Convention on semidirect products).
Given a left action ⊲ of the group G on the group E by automorphisms, the convention for the semidirect product G ⋉ ⊲ E is: ( g, e )( g ′ , e ′ ) = (cid:0) gg ′ , ( g ′− ⊲ e ) e ′ (cid:1) . In particular given g ∈ G and e ∈ E we have: ( g, e ) − = (cid:0) g − , g ⊲ e − (cid:1) . We resume the notation of 2.2. Consider the left action of E on L (the secondary action ⊲ ′ of A ) givenby e ⊲ ′ k . = k { δ ( k ) − , e } , where e ∈ E and k ∈ L ; lemma 7. For the following see [16, 22]. Lemma 15 (Derived action).
Let A = ( L δ −→ E ∂ −→ G, ⊲, { , } ) be a 2-crossed module. There exists a leftaction ∗ of E on E ⋉ ⊲ ′ L , by automorphisms (called the “derived action of A ”), with the form: b ∗ ( e, k ) = (cid:0) ∂ ( b ) ⊲ e, (cid:0) b ⊲ ′ { b − , ∂ ( b ) ⊲ e − } (cid:1) b ⊲ ′ k (cid:1) = (cid:0) ∂ ( b ) ⊲ e, { b, e − } − b ⊲ ′ k (cid:1) , where e, b ∈ E and k ∈ L, see equation (9). Note that if e ∈ E and k ∈ L : b ∗ ( δ ( k ) , k − ) = (cid:0) δ ( ∂ ( b ) ⊲ k ) , ∂ ( b ) ⊲ k − (cid:1) . (10)10onsider the group E ⋉ ∗ ( E ⋉ ⊲ ′ L ), whose group law is (remark 14):( a, e, k )( a ′ , e ′ , k ′ ) = (cid:16) aa ′ , ( ∂ ( a ′− ) ⊲ e ) e ′ , (cid:16) ( a ′ e ′ ) − ⊲ ′ (cid:0) { a ′ , ∂ ( a ′ ) − ⊲ e − } k (cid:1) k ′ (cid:17) = (cid:16) aa ′ , ( ∂ ( a ′− ) ⊲ e ) e ′ , (cid:16) e ′− ⊲ ′ (cid:0) { a ′− , e − } − a ′− ⊲ ′ k (cid:1) k ′ (cid:17) . (11)Particular cases of the multiplication are:( a, , k )( a ′ , , k ′ ) = (cid:0) aa ′ , , ( a ′− ⊲ k ) k ′ (cid:1) , where a, a ′ ∈ E and k, k ′ ∈ L, (12)(1 , e, k )(1 , e ′ , k ′ ) = (cid:0) , ee ′ , ( a ′− ⊲ k ) k ′ (cid:1) , where e, e ′ ∈ E and k, k ′ ∈ L. (13)Thus since ( δ : L → E, ⊲ ′ ) is a crossed module: (cid:0) δ ( l ) , , k (cid:1)(cid:0) δ ( l ′ ) , , k ′ (cid:1) = (cid:0) δ ( l ) δ ( l ′ ) , , l ′− kl ′ k ′ (cid:1) , where k, k ′ , l, l ′ ∈ L ; (14) (cid:0) , δ ( l ) , k (cid:1)(cid:0) , δ ( l ′ ) , k ′ (cid:1) = (cid:0) , δ ( l ) δ ( l ′ ) , l ′− kl ′ k ′ (cid:1) , where k, k ′ , l, l ′ ∈ L. (15)Put a = ( a, , e = (1 , e,
1) and k = (1 , , k ), and the same for their images under ⊲ and ⊲ ′ . We have:( a, e, k ) = aek, aka − = a ⊲ ′ k, eke − = e ⊲ ′ k. (16)Putting l = (1 , , l ), for l ∈ L : klk − = δ ( k ) ⊲ ′ l = ( δ ( k ) , ,
1) (1 , , l ) ( δ ( k ) , , − = (1 , δ ( k ) ,
1) (1 , , l ) (1 , δ ( k ) , − . (17)Consider the semidirect product G ⋉ ⊲ E . The following essential lemma appeared in [22]. Lemma 16 (First lifted action).
Let A = ( L δ −→ E ∂ −→ G, ⊲, { , } ) be a 2-crossed module of groups. Thereexists a left action by automorphisms • of G ⋉ ⊲ E on E ⋉ ∗ ( E ⋉ ⊲ ′ L ) (the first lifted action of A ), with: ( g, x ) • ( a, e, k ) = (cid:16) g ⊲ a, g ⊲ (cid:0) ( ∂ ( a ) − ⊲ x ) e x − (cid:1) , g ⊲ (cid:16) ( xe − ) ⊲ ′ n a − , x − o − (cid:17) g ⊲ n x, e − a − o(cid:0) g∂ ( x ) (cid:1) ⊲ k (cid:17) . = g ⊲ (cid:16) a, ( ∂ ( a ) − ⊲ x ) e x − , ( xe − ) ⊲ ′ n a − , x − o − n x, e − a − o ∂ ( x ) ⊲ k (cid:17) . Put (1 G , x ) = x and ( g, E ) = g . Particular cases of the first lifted action, which will be important later are: g • ( a, e, k ) = ( g ⊲ a, g ⊲ e, g ⊲ k ) . (18) x • ( a, e, k ) = (cid:16) a, ( ∂ ( a ) − ⊲ x ) e x − , (cid:16) ( xe − ) ⊲ ′ n a − , x − o − (cid:17)n x, e − a − o ∂ ( x ) ⊲ k (cid:17) , (19) x • ( δ ( k ) , , l ) = ( δ ( k ) , , k − ∂ ( x ) ⊲ ( kl )) , where x ∈ E, and k, l ∈ L, (20) x • (1 , e, k ) = (cid:16) a, xex − , { x, e − } ∂ ( x ) ⊲ k (cid:17) , (21)( g, x ) • (1 , e, k ) = (cid:16) , g ⊲ ( xex − ) , g ⊲ (cid:0) { x, e − } ∂ ( x ) ⊲ k (cid:1)(cid:17) , (22) x • (1 , δ ( k ) , k − ) = (cid:16) , xδ ( k ) x − , { x, δ ( k ) − } ∂ ( x ) ⊲ k − (cid:17) = (cid:16) , xδ ( k ) x − , x ⊲ ′ k − (cid:17) , (23)( g, x ) • (1 , δ ( k ) , k − ) = (cid:16) , g ⊲ ( xδ ( k ) x − ) , g ⊲ (cid:0) { x, δ ( k ) − } ∂ ( x ) ⊲k − (cid:1)(cid:17) = (cid:16) , g ⊲δ ( x⊲ ′ k ) , g ⊲ (cid:0) x⊲ ′ k − (cid:1)(cid:17) , (24) x • ( a, ,
1) = (cid:16) a, ∂ ( a ) − ⊲ x x − , (cid:0) x ⊲ ′ (cid:8) a − , x − (cid:9) − (cid:1)(cid:8) x, a − (cid:9)(cid:17) , (25) δ ( k ) • ( a, ,
1) = (cid:16) a, ∂ ( a ) − ⊲ δ ( k ) δ ( k ) − , k ∂ ( a ) ⊲ k − (cid:17) , (26)If x is such that { x, e } = { e, x } = 1, for all e ∈ E , we have: x • ( a, e, k ) = (cid:16) a, ( ∂ ( a ) − ⊲ x ) ex − , ∂ ( x ) ⊲ k (cid:17) . (27)Let A = ( L δ −→ E ∂ −→ G, ⊲, { , } ) be a 2-crossed module. If ad is the adjoint action, it follows easily that:11 emma 17 (Second lifted action). There is a left action • of G ⋉ ⊲ E in L ⋉ ad L , by automorphisms(called the second lifted action of A ), which has the form (where x ∈ E , g ∈ G and ( k, l ) ∈ L ⋉ ad L ): x • ( k, l ) = ( k, k − ∂ ( x ) ⊲ ( kl )) and g • ( k, l ) = ( g ⊲ k, g ⊲ l ) . P ∗ ( A ) of a 2-crossed module A By using (20), (14) and (11), we can see that the maps( a, e, k ) ∈ E ⋉ ∗ ( E ⋉ ⊲ ′ L ) β ( ∂ ( a ) , e ) ∈ G ⋉ ⊲ E ( k, l ) ∈ L ⋉ ad L α ( δ ( k ) , E , l ) ∈ E ⋉ ∗ ( E ⋉ ⊲ ′ L ) (28)are ( G ⋉ ⊲ E )-equivariant group morphisms, with respect to the lifted actions • of A and the adjoint actionof G ⋉ ⊲ E on itself. This defines a chain complex of groups: L ⋉ ad L α −→ E ⋉ ∗ ( E ⋉ ⊲ ′ L ) β −→ G ⋉ ⊲ E. (29)The Peiffer pairing in the pre-crossed module E ⋉ ∗ ( E ⋉ ⊲ ′ L ) β −→ G ⋉ ⊲ E was calculated in [22]: h ( a, e, k ) , ( a ′ , e ′ , k ′ ) i = (cid:0) h a, a ′ i , , { a, a ′ } − { aeδ ( k ) , a ′ e ′ δ ( k ′ ) } (cid:1) . (30)By using example 9, and the structure of the group complex (29), namely equations (14) and the form ofthe product in L ⋉ ad L , there follows that there exists a 2-crossed module structure in (29), whose Peifferlifting | , | : (cid:0) E ⋉ ∗ ( E ⋉ ⊲ ′ L ) (cid:1) × (cid:0) E ⋉ ∗ ( E ⋉ ⊲ ′ L ) (cid:1) → L ⋉ ad L takes the following form: | ( a, e, k ) , ( a ′ , e ′ , k ′ ) | = (cid:0) { a, a ′ } , { a, a ′ } − { aeδ ( k ) , a ′ e ′ δ ( k ′ ) } (cid:1) . (31) Definition 18.
For a 2-crossed module A = ( L → E → G, ⊲, { } ) , the 2-crossed module P ∗ ( A ) = (cid:16) L ⋉ ad L α −→ E ⋉ ∗ ( E ⋉ ⊲ ′ L ) β −→ G ⋉ ⊲ E, • , | , | (cid:17) (32) just defined will be called the (pointed) path-space of A . Clearly the path-space construction P ∗ is functorialwith respect to 2-crossed module morphisms. By straightforward calculations we conclude that:
Theorem 19.
Let A = ( L δ −→ E ∂ −→ G, ⊲, { , } ) be a 2-crossed module. The maps ( k, l ) ∈ L ⋉ ad L p ′ kl ∈ L and ( a, e, k ) ∈ E ⋉ ∗ ( E ⋉ ⊲ ′ L ) q ′ aeδ ( k ) ∈ E, ( g, e ) ∈ G ⋉ E r ′ g∂ ( e ) ∈ G ; and also: ( k, l ) ∈ L ⋉ ad L p k ∈ L, ( a, e, k ) ∈ E ⋉ ∗ ( E ⋉ ⊲ ′ L ) q a ∈ E, ( g, e ) ∈ G ⋉ E r g ∈ G are group morphisms. Moreover the triples Pr A . = ( p, q, r ) and Pr A . = ( p ′ , q ′ , r ′ ) define surjective morphisms P ∗ ( A ) → A of 2-crossed modules. We also have an inclusion map i A : A → P ∗ ( A ) such that: g ( g, , e ( e, , , k ( k, . Therefore Pr A ◦ i A and Pr A ◦ i A each are the identity of A . Moreover the map (Pr A , Pr A ) : P ∗ ( A ) → A × A is a fibration of 2-crossed modules, considering the model category structure in the category of 2-crossedmodules defined in [14], see the introduction. This is because both maps ( p, p ′ ) : L ⋉ ad L → L × L and ( q, q ′ ) : E ⋉ ∗ ( E ⋉ ⊲ ′ L ) → E × E are surjective. Therefore P ∗ ( A ) is a good path-space for A , [19], sinceclearly i A : A → P ∗ ( A ) induces isomorphism at the level of homotopy groups.
12t is convenient to “visualise” the path space of a 2-crossed module A in the following way, putting emphasison the projection maps Pr A , Pr A : P ∗ ( A ) → A : A Pr A P ∗ ( A ) Pr A / / A . (33)The elements ( g, e ) ∈ Gr ( P ∗ ( A )) and ( a, e, k ) ∈ Gr ( P ∗ ( A )) (example 11) will several times be denoted as:( g, e ) = (cid:0) g e −→ g∂ ( e ) (cid:1) and ( a, e, k ) = (cid:0) a ( e,k ) −−−→ aeδ ( k ) (cid:1) . (34)The 2-crossed module boundary map β : Gr ( P ∗ ( A )) → Gr ( P ∗ ( A )) being (in this notation): (cid:0) a ( e,k ) −−−→ aeδ ( k ) (cid:1) β (cid:0) ∂ ( a ) e −→ ∂ ( a ) ∂ ( e ) (cid:1) . Note that (in the notation of theorem 19), if ( a, e, k ) ∈ Gr ( P ∗ ( A )), ( g, e ) ∈ Gr ( P ∗ ( A )), g ∈ G and a ∈ E :Pr A (cid:0) a ( e,k ) −−−→ aeδ ( k ) (cid:1) = a Pr A (cid:0) a ( e,k ) −−−→ aeδ ( k ) (cid:1) = aeδ ( k ) (35) i A ( g ) = (cid:0) g E −−→ g (cid:1) i A ( a ) = (cid:0) a (1 E , L ) −−−−−→ a (cid:1) (36) P ∗ ( P ∗ ( A )) of a 2-crossed module A Given a 2-crossed module A = ( L δ −→ E ∂ −→ G, ⊲, { , } ), we can iterate the path-space construction,obtaining a 2-crossed module P ∗ ( P ∗ ( A )), the double path-space of A , with underlying group complex:( L ⋉ ad L ) ⋉ ad ( L ⋉ ad L ) α ♯ −→ ( E ⋉ ∗ ( E ⋉ ⊲ ′ L )) ⋉ ∗ (( E ⋉ ∗ ( E ⋉ ⊲ ′ L ) ⋉ • ′ ( L ⋉ ad L )) β ♯ −→ ( G ⋉ ⊲ E ) ⋉ • ( E ⋉ ∗ ( E ⋉ ⊲ ′ L )) , (37)and lifted actions (now denoted by (cid:3) ) of the first group on the remaining ones. Here • ′ denotes the secondaryaction of P ∗ ( A ); lemma 7, whereas ∗ denotes the derived actions either of A or of P ∗ ( A ); lemma 15.There are four natural 2-crossed module maps P ∗ ( P ∗ ( A )) → P ∗ ( A ). Namely the maps Pr P ∗ ( A )1 andPr P ∗ ( A )0 of theorem 19, which have the form (respectively):( k, l, k ′ , l ′ ) ( kk ′ , k ′− lk ′ l ′ ) , ( a, e, k, a ′ , e ′ , k ′ , l, l ′ ) ( a, e, k )( a ′ , e ′ , k ′ )( δ ( l ) , , l ′ ) , ( g, x, a, e, k ) ( g, x )( ∂ ( a ) , e ) = ( g∂ ( a ) , ∂ ( a ) − ⊲ x e )and ( k, l, k ′ , l ′ ) ( k, l ) , ( a, e, k, a ′ , e ′ , k ′ , l, l ′ ) ( a, e, k ) , ( g, x, a, e, k ) ( g, x ) . Also, by applying the path-space functor P ∗ to the 2-crossed module maps Pr A and Pr A , from P ∗ ( A ) to A ,yields 2-crossed module maps P ∗ (Pr A ) , P ∗ (Pr A ) : P ∗ ( P ∗ ( A )) → P ∗ ( A ), which have the form, respectively:( k, l, k ′ , l ′ ) ( kl, k ′ l ′ ) , ( a, e, k, a ′ , e ′ , k ′ , l, l ′ ) ( aeδ ( k ) , a ′ e ′ δ ( k ′ ) , ll ′ ) , ( g, x, a, e, k ) ( g∂ ( x ) , aeδ ( k )) , ( k, l, k ′ , l ′ ) ( k, k ′ ) , ( a, e, k, a ′ , e ′ , k ′ , l, l ′ ) ( a, a ′ , l ) , ( g, x, a, e, k ) ( g, a ) . Let us now change notation and put (all of these are maps P ∗ ( P ∗ ( A )) → P ∗ ( A )): d = P ∗ (Pr A ); d = Pr P ∗ ( A )1 ; d = Pr P ∗ ( A )0 ; d = P ∗ (Pr A ) .
13t is convenient to visualise the double path space in the form below, emphasising the boundary maps tothe path space (compare with (33)): A P ∗ ( A ) / / A P ∗ ( P ∗ ( A )) d / / d O O d o o d (cid:15) (cid:15) A P ∗ ( A ) O O P ∗ ( A ) / / A P ∗ ( A ) O O (38)In conformity with this notation, we can denote ( a, e, k, a ′ , e ′ , k ′ , l, l ′ ) ∈ Gr (cid:0) P ∗ ( P ∗ ( A )) (cid:1) as: aa ′ δ ( l ) (cid:16) ( ∂ ( a ′− ) ⊲e ) e ′ , (cid:0) ( ∂ ( a ′− ) ⊲e ) e ′ (cid:1) − ⊲ ′ l − e ′− ⊲ ′ (cid:0) { a ′− ,e − } − a ′− ⊲ ′ k (cid:1) k ′ l l ′ (cid:17) / / aeδ ( k ) a ′ e ′ δ ( k ′ ) δ ( l ) δ ( l ′ ) a ( a ′ ,l ) O O ( e,k ) / / aeδ ( k ) ( a ′ e ′ δ ( k ′ ) ,ll ′ ) O O (39)and also ( g, x, a, e, k ) ∈ ( G ⋉ ⊲ E ) ⋉ • ( E ⋉ ∗ ( E ⋉ ⊲ ′ L )) = Gr (cid:0) P ∗ ( P ∗ ( A )) (cid:1) as:( g, x, a, e, k ) = g∂ ( a ) ( ∂ ( a ) − ⊲x ) e / / g ∂ ( x ) ∂ ( a ) ∂ ( e ) g ka O O x / / g∂ ( x ) a e δ ( k ) O O (40)Compare with the analogous notation for elements in the path-space in subsection 2.3. T ( A ) of a 2-crossed module A Let G = ( L δ −→ E ∂ −→ G, ⊲, { , } ) be a 2-crossed module. The triangle space T ( A ) of A , will (not surprisingly)have a primary importance in the construction of the concatenation of 2-crossed module homotopies. Thistriangle space is included inside the double path space as the limit of the diagram: P ∗ ( P ∗ ( A )) P ∗ (Pr A )= d (cid:15) (cid:15) A i A / / P ∗ ( A ) (41)Looking at (38), (39), (40), we are thus making the left pointing upwards arrows to consist only of identities.Let us make this more explicit in dimensions one and two. Clearly:Gr ( T ( A )) = ( E ⋉ ∗ ( E ⋉ ⊲ ′ L )) ⋉ ∗ (cid:0) ( { } ⋉ ∗ ( E ⋉ ⊲ ′ L ) ⋉ • ′ ( { } ⋉ ad L ) (cid:1) Gr ( T ( A )) = ( G ⋉ ⊲ E ) ⋉ • (cid:0) { } ⋉ ∗ ( E ⋉ ⊲ ′ L ) (cid:1) . The 2-crossed module boundary map β ′ : Gr ( T ( A )) → Gr ( T ( A )) being: β ′ (cid:0) ( a, e, k ) , (1 , f, l ) , (1 , m ) (cid:1) = ( ∂ ( a ) , e, , f, l ) . (42)14he restrictions of the projection maps d = Pr P ∗ ( A )0 and d = Pr P ∗ ( A )1 : P ∗ ( P ∗ ( A )) → P ∗ ( A ) to T ( A ) are: d ( a, e, k, , f, l, , m ) = ( a, e, k ) , d ( g, x, , e, k ) = ( g, x ) , (43) d ( a, e, k, , f, l, , m ) = (cid:0) a, ef, ( f − ⊲ ′ k ) lm (cid:1) , d ( g, x, , e, k ) = ( g, xe ) . (44)On the other hand the restrictions of d = P ∗ (Pr A ) and d = P ∗ (Pr A ) to T ( A ) and are given by, respectively: d ( a, e, k, , f, l, , m ) = ( aeδ ( k ) , f δ ( l ) , m ) , d ( g, x, , e, k ) = ( g∂ ( x ) , eδ ( k )) , (45) d ( a, e, k, , f, l, , m ) = ( a, , , d ( g, x, , e, k ) = ( g, . (46)As we did for the double path space, we will visualize the triangle space T ( A ) in the form below (47),emphasising the three non-trivial boundary maps T ( A ) → P ∗ ( A ) : A T ( A ) d / / d o o d (cid:15) (cid:15) A P ∗ ( A ) : : ✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈ P ∗ ( A ) / / A P ∗ ( A ) O O (47) 20 @ @ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) / / O O (48)In (48) we present the numbering of the vertices in (47). The consistent notation for elements of Gr ( T ( A ))and of Gr ( T ( A )) is, respectively:( g, x, , e, k ) = g∂ ( x ) ∂ ( e ) kg xe = = ③③③③③③③③③③③③③③③③③③③③ x / / g∂ ( x ) eδ ( k ) O O (49)and ( a, e, k, , e ′ , k ′ , , l ′ ) = aeδ ( k ) e ′ δ ( k ′ ) δ ( l ′ ) a (cid:0) ee ′ ,e ′− ⊲ ′ k k ′ l ′ (cid:1) : : ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ( e,k ) / / aeδ ( k ) ( e ′ δ ( k ′ ) ,l ′ ) O O (50)The boundary map β ′ : Gr ( T ( A )) → Gr ( T ( A )) in the 2-crossed module T ( A ) being: aeδ ( k ) e ′ δ ( k ′ ) δ ( l ′ ) a (cid:0) ee ′ ,e ′− ⊲ ′ k k ′ l ′ (cid:1) : : ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ( e,k ) / / aeδ ( k ) ( e ′ δ ( k ′ ) ,l ′ ) O O β ′ ∂ ( a ) ∂ ( e ) ∂ ( e ′ ) k ′ ∂ ( a ) ee ′ < < ①①①①①①①①①①①①①①①①①①①① e / / ∂ ( a ) ∂ ( e ) e ′ δ ( k ′ ) O O . (51)15onsider the pull back 2-crossed modules P ∗ ( A ) Pr A × Pr A P ∗ ( A ), P ∗ ( A ) Pr A × Pr A P ∗ ( A ) and P ∗ ( A ) Pr A × Pr A P ∗ ( A ).Looking at (47) and (33), we naturally denote these as (respectively): AA P ∗ ( A ) / / A P ∗ ( A ) O O AA P ∗ ( A ) ? ? ⑦⑦⑦⑦⑦⑦⑦⑦ A P ∗ ( A ) O O AA P ∗ ( A ) ? ? ⑦⑦⑦⑦⑦⑦⑦⑦ P ∗ ( A ) / / A c.f. (33). We have obvious boundary maps from the triangle space T ( A ) of A to each of these 2-crossedmodules. Call these 2-crossed module maps d + − , d ++ and d −− . Now a simple, however essential, result: Lemma 20.
There exists a pull-back diagram of groups, where the boundary maps in the relevant crossedmodules are denoted by β ′ : Gr ( T ( A )) β ′ u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ Gr ( d + − ) * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ Gr ( T ( A )) Gr ( d + − ) ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ Gr (cid:16) P ∗ ( A ) Pr A × Pr A P ∗ ( A ) (cid:17) β ′ t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ Gr (cid:16) P ∗ ( A ) Pr A × Pr A P ∗ ( A ) (cid:17) The same statement is true for P ∗ ( A ) Pr A × Pr A P ∗ ( A ) and P ∗ ( A ) Pr A × Pr A P ∗ ( A ) , with the obvious adaptations. Proof.
This is a simple inspection. Clearly (by (34) and (35)) the group Gr (cid:16) P ∗ ( A ) Pr A × Pr A P ∗ ( A ) (cid:17) isgiven by all pairs of elements in Gr ( P ∗ ( A )) of the form: (cid:0) ( a, e, k ) , ( aeδ ( k ) , e ′ , k ′ ) (cid:1) , being β ′ (cid:0) ( a, e, k ) , ( aeδ ( k ) , e ′ , k ′ ) (cid:1) = (cid:0) ( ∂ ( a ) , e ) , ( ∂ ( ae ) , e ′ (cid:1) , by (28). On the other hand the group pullback:Gr ( T ( A )) Gr ( d + − ) × β ′ Gr (cid:16) P ∗ ( A ) Pr A × Pr A P ∗ ( A ) (cid:17) is given by all pairs of elements in Gr ( T ( A )) × Gr (cid:16) P ∗ ( A ) Pr A × Pr A P ∗ ( A ) (cid:17) of the form: (cid:16) ( ∂ ( a ) , e, , x, l ) , (cid:0) ( a, e, k ) , ( aeδ ( k ) , xδ ( l ) , k ′ ) (cid:1)(cid:17) . Clearly the map:Gr ( T ( A )) (cid:0) β ′ , Gr ( d + − ) (cid:1) −−−−−−−−−−→ Gr (cid:0) T ( A ) (cid:1) Gr ( d + − ) × β ′ Gr (cid:16) P ∗ ( A ) Pr A × Pr A P ∗ ( A ) (cid:17) , made explicit below, is a group bijection:( a, e, k, , e ′ , k ′ , , l ′ ) (cid:16) ( ∂ ( a ) , e, , e ′ , k ′ ) , (cid:0) ( a, e, k ) , ( aeδ ( k ) , e ′ δ ( k ′ ) , l ′ ) (cid:1)(cid:17) . emark 21 (Explicit operations in the triangle space). By using (22) and (11) we can easily obtainan explicit form for the multiplication in Gr ( T ( A )) = ( G ⋉ ⊲ E ) ⋉ • (cid:0) { } ⋉ ∗ ( E ⋉ ⊲ ′ L ) (cid:1) . On the other handthe product in Gr ( T ( A )) = ( E ⋉ ∗ ( E ⋉ ⊲ ′ L )) ⋉ ∗ (( { } ⋉ ∗ ( E ⋉ ⊲ ′ L ) ⋉ • ′ ( { } ⋉ ad L )) is: (cid:0) ( a, e, k ) , (1 , f, l ) , (1 , m ) (cid:1)(cid:0) ( a ′ , e ′ , k ′ ) , (1 , f ′ , l ′ ) , (1 , m ′ ) (cid:1) = (cid:16) ( a, e, k )( a ′ , e ′ , k ′ ) , (cid:0) ( ∂ ( a ′ ) , e ′ ) − • (1 , f, l ) (1 , f ′ , l ′ ) (cid:1) , (1 , ( f ′ δ ( l ′ )) − ⊲ ′ { ( a ′ e ′ δ ( k ′ )) − , ( f δ ( l )) − } − (cid:1)(cid:0) , ( f ′ δ ( l ′ )) − ⊲ ′ ( a ′ e ′ δ ( k ′ )) − ⊲ ′ m (cid:1) (1 , m ′ ) (cid:17) . (52) The restriction △ of the lifted action of P ∗ ( A ) to an action of Gr ( T ( A )) on Gr ( T ( A )) is: ( g, x, , z, w ) △ (cid:0) ( a, e, k ) , (1 , f, l ) , (1 , m ) (cid:1) = ( g, x ) • (cid:16) ( a, e, k ) , ( ∂ ( a ) , e ) − • (1 , z, w ) (1 , f, l ) (1 , z, w ) − , (1 , ( zδ ( wl − ) f − ) ⊲ ′ ( { ( aeδ ( k )) − , δ ( w ) − z − } − ) { zδ ( w ) , ( aeδ ( k ) f δ ( l )) − } ∂ ( z ) ⊲ m (cid:1)(cid:17) (53) None of this formulae will be explicitely used.2.4.2. Definition of the disk space D ∗ ( A ) of a 2-crossed module A Let A be a 2-crossed module. Recall the construction of the triangle space T ( A ), the path-space P ∗ ( A )and also (47). The disk space D ∗ ( A ) of A is defined as being the limit of the diagram: T ( A ) d (cid:15) (cid:15) P ∗ ( A ) A i A o o We are thus making the pointing upwards arrow of (47) to consist only of identities. On the contrary ofthe triangle space, the operations on the disk space are quite simple. Let A = ( L δ −→ E ∂ −→ G, ⊲, { , } ) be a2-crossed module. The disk space D ∗ ( A ) is an embedded 2-crossed module, within the double path space P ∗ ( P ∗ ( A )) of A . In the last bit of (37), instead of E ⋉ ∗ ( E ⋉ ⊲ ′ L ), we put the subgroup isomorphic to L , ofelements of the form (1 , δ ( k ) , k − ), where k ∈ L . The group law is(1 , δ ( k ) , k − )(1 , δ ( l ) , l − ) = (1 , δ ( kl ) , ( kl ) − );see (15). Under the identification k = (1 , δ ( k ) , k − ), the restriction of the first lifted action • of A to thissubgroup is, by (24): ( g, x ) • k = g ⊲ ( x ⊲ ′ k ) . (54)As far as the second group of the disk D ∗ ( A ) is concerned, we consider the subgroup of ( E ⋉ ∗ ( E ⋉ ⊲ ′ L )) ⋉ • ′ ( L ⋉ ad L ), isomorphic to L , of elements of the form:(1 , δ ( k ) , k − , , . By equations (31) and (6) , we have, where | , | is the Peiffer lifting in P ∗ ( A ): | ( a, e, k ) , (1 , δ ( l ) , l − ) | = (1 L , L ) = | (1 , δ ( l ) , l − ) , ( a, e, k ) | . This will be used several times in the following calculations. The restriction of the derived action ∗ of E ⋉ ∗ ( E ⋉ ⊲ ′ L )) on ( E ⋉ ∗ ( E ⋉ ⊲ ′ L )) ⋉ • ′ ( L ⋉ ad L ) to this isomorphic image of L is, given by( a, e, l ) ∗ (cid:0) , δ ( k ) , k − , , (cid:1) = (cid:0) β ( a, e, l ) • (1 , δ ( k ) , k − ) , , (cid:1) = (cid:0) , ∂ ( a ) ⊲ ( eδ ( k ) e − ) , ∂ ( a ) ⊲ ( e ⊲ ′ k − ) , , (cid:1) . k = (1 , δ ( k ) , k − , ,
1) we have( a, e, l ) ∗ k = ∂ ( a ) ⊲ ( e ⊲ ′ k ) . We now describe the restriction of the lifted action (cid:3) of P ∗ ( A ). By (27):(1 , δ ( k ) , k − ) (cid:3) (cid:0) a, e, l, , δ ( m ) , m − , , (cid:1) = (cid:16) a, e, l, , e − ( ∂ ( a ) − ⊲ δ ( k )) e δ ( m ) δ ( k ) − , km − e − ⊲ ′ ( ∂ ( a ) − ⊲ k − ) , , (cid:17) . (55)Denoting the Peiffer lifting in D ∗ ( A ) and P ∗ ( A ) by { , } and | , | , respectively, we have, by (31): { ( a, e, k, , δ ( l ) , l − , , , ( a ′ , e ′ , k ′ , , δ ( l ′ ) , l ′− , , } = (cid:0) | aek, a ′ e ′ k ′ | , | aek, a ′ e ′ k ′ | − | aekδ ( l ) l − , a ′ e ′ k ′ δ ( l ′ ) l ′− | (cid:1) , where we used the notation of (16) and (17). Now note that, by (13), where { , } is the Peiffer lifting in A : | aekδ ( l ) l − , a ′ e ′ k ′ δ ( l ′ ) l ′− | = | aeδ ( l ) ( δ ( l − ) ⊲ ′ k ) l − , a ′ e ′ δ ( l ′ ) ( δ ( l ′ ) − ⊲ ′ k ′ ) l ′− | = ( { a, a ′ } , { a, a ′ } − { aeδ ( l ) δ ( δ ( l − ) ⊲ ′ kl − ) , a ′ e ′ δ ( l ′ ) δ ( δ ( l ′ ) − ⊲ ′ k ′ l ′− ) | = ( { a, a ′ } , { a, a ′ } − { aeδ ( k ) , a ′ e ′ δ ( k ′ ) } ) = | aek, a ′ e ′ k ′ | . Therefore { ( a, e, k, , δ ( l ) , l − , , , ( a ′ , e ′ , k ′ , , δ ( l ′ ) , l ′− , , } = (cid:0) | aek, a ′ e ′ k ′ | , , (cid:1) We thus have the following essential theorem:
Theorem 22.
Given a 2-crossed module A = ( L δ −→ E ∂ −→ G, ⊲, { , } ) , there exists a 2-crossed module D ∗ ( A ) ,called the disk-space of A , with underlying complex of groups: L ⋉ ad L α −→ ( E ⋉ ∗ ( E ⋉ ⊲ ′ L )) ⋉ ∗ L β −→ ( G ⋉ ⊲ E ) ⋉ • L, where (recall remark 14): ( g, e ) • k = g ⊲ ( e ⊲ ′ k ) ( a, e, k ) ∗ l = ∂ ( a ) ⊲ ( e ⊲ ′ l ) . (56) The underlying action (denoted by (cid:3) ) of ( G ⋉ ⊲ E ) ⋉ • L on ( E ⋉ ∗ ( E ⋉ ⊲ ′ L )) ⋉ ∗ L is given by: ( g, e ) (cid:3) ( a, f, l, l ′ ) = (cid:0) ( g, e ) • ( a, f, l ) , g ⊲ ( e ⊲ ′ l ′ ) (cid:1) , where ( g, e ) = ( g, e, L ) ∈ ( G ⋉ ⊲ E ) ⋉ • L (57) k (cid:3) ( a, e, l, l ′ ) = ( a, e, l, e − ⊲ ′ ( ∂ ( a ) − ⊲ k ) l ′ k − ) , where k = (1 G , E , k ) ∈ ( G ⋉ ⊲ E ) ⋉ • L. (58) The action (cid:3) of ( G ⋉ ⊲ E ) ⋉ • L on L ⋉ ad L has the form: ( g, e, l ) (cid:3) ( k, k ′ ) = ( g, e ) • ( k, k ′ ) . (59) The boundary maps are: α ( k, l ) = ( δ ( k ) , , l, and β ( a, e, k, l ) = ( ∂ ( a ) , e, l ) . And finally the Peiffer lifting is (recall (31) ): { ( a, e, k, l ) , (( a, e, k, l ) } = | ( a, e, k ) , ( a ′ , e ′ , k ′ ) | = (cid:0) { a, a ′ } , { a, a ′ } − { aeδ ( k ) , a ′ e ′ δ ( k ′ ) } (cid:1) . (60) The disk space D ∗ ( A ) has an inclusion map into P ∗ ( P ∗ ( A )) , factoring through T ( A ) , with the form: ( g, x, k ) ( g, x, , δ ( k ) , k − ) , ( a, e, k, l ) ( a, e, k, , δ ( l ) , l − , , , ( k, l ) ( k, l, , . oreover the maps d = ( p, q, r ) : D ∗ ( A ) → P ∗ ( A ) and d = ( p ′ , q ′ , r ′ ) : D ∗ ( A ) → P ∗ ( A ) , where: p ( k, l ) = ( k, l ) , q ( a, e, k, l ) = ( a, e, k ) , r ( g, e, k ) = ( g, e ) , (61) p ′ ( k, l ) = ( k, l ) , q ( a, e, k, l ) = ( a, eδ ( l ) , l − k ) , r ′ ( g, e, k ) = ( g, eδ ( k )) , (62) are morphisms of 2-crossed modules, and are obtained from the composition of the inclusion map D ∗ ( A ) → P ∗ ( P ∗ ( A )) and the projection maps d = Pr P ∗ ( A )1 , d = Pr P ∗ ( A )0 : P ∗ ( P ∗ ( A )) → P ∗ ( A ) ; see (33) . Note: ( a, e, k )(1 , δ ( l ) , l − ) = ( a, eδ ( l ) , ( δ ( l ) − ⊲ k ) l − ) = ( a, eδ ( l ) , l − k ) . Note that the maps d , d : D ∗ ( A ) → P ∗ ( A ) can (compare with (33) and (47)) be visualised as: A P ∗ ( A ) " " P ∗ ( A ) = = D ∗ ( A ) d O O d (cid:15) (cid:15) A (63) Let A = ( L δ −→ E ∂ −→ G, ⊲, { , } ) be a 2-crossed module. We freely use the notation of 2.4.1 and 2.3.2.The tetrahedron space T ET ( A ) of A (or more precisely its underlying group Gr ( T ET ( A )) will be used forproving that the concatenation of homotopies is associative, and several other places. Consider the 2-crossedmodule P ∗ ( T ( A )), the path-space of the triangle space of A . Its underlying group is:Gr ( T ( A )) ⋉ ∆ Gr ( T ( A )) = (cid:0) ( G ⋉ ⊲ E ) ⋉ • ( { } ⋉ ∗ ( E ⋉ ⊲ ′ L )) ⋉ ∆ (cid:0) E ⋉ ∗ ( E ⋉ ⊲ ′ L )) ⋉ ∗ (( { } ⋉ ∗ ( E ⋉ ⊲ ′ L ) ⋉ • ′ ( { } ⋉ ad L )) (cid:1) By looking at (47), we have three maps P ∗ ( T ( A )) → P ∗ ( P ∗ ( A )) being P ∗ ( d i ) , i = 0 , ,
2, and twoadditional maps Pr T ( A )0 , Pr T ( A ))1 : P ∗ ( T ( A )) → T ( A ); see theorem 19 and definition 18. This corresponds tothe five faces of a triangular prism. Let T ET ( A ) be the 2-crossed module which is the limit of the diagram: P ∗ ( T ( A )) P ∗ ( d ) (cid:15) (cid:15) P ∗ ( A ) i P ∗ ( A ) / / P ∗ ( P ∗ ( A ))We will only make use of the underlying group Gr ( T ET ( A )) of it, which is: (cid:0) ( G ⋉ ⊲ E ) ⋉ • ( { } ⋉ ∗ ( E ⋉ ⊲ ′ L )) (cid:1) ⋉ △ (cid:0) { } ⋉ ∗ ( { } ⋉ ⊲ ′ { } ) ⋉ ∗ (( { } ⋉ ∗ ( E ⋉ ⊲ ′ L ) ⋉ • ′ ( { } ⋉ ad L ) (cid:1) . The restrictions of the maps P ∗ ( d ) , P ∗ ( d ) , Pr T ( A )0 and Pr T ( A )1 to Gr ( T ET ( A )) are, respectively, and re-naming them with a simplicial notation: (cid:0) ( g, x, , z, w ) , (1 , , , , f, l, , m ) (cid:1) d (cid:0) g, xz, , f, lm (cid:1) (64) (cid:0) ( g, x, , z, w ) , (1 , , , , f, l, , m ) (cid:1) d (cid:0) g∂ ( x ) , zδ ( w ) , , f δ ( l ) , m (cid:1) (65) (cid:0) ( g, x, , z, w ) , (1 , , , , f, l, , m ) (cid:1) d (cid:0) g, x, , z, w (cid:1) (66) (cid:0) ( g, x, , z, w ) , , , , , f, l, , m ) (cid:1) d (cid:0) g, x, , zf, f − ⊲ ′ w l (cid:1) (67)19s we did for the triangle and disk space, we will represent the elements of Gr ( T ET ( A )) graphically,emphasising their boundary maps to Gr ( T ( A )). We therefore put: (cid:0) ( g, x, , z, w ) , (1 , , , , f, l, , m ) (cid:1) = g∂ ( x ) ∂ ( z ) ∂ ( f ) g∂ ( x ) ∂ ( z ) fδ ( l ) δ ( m ) O O g xzf A A ☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ xz ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ x / / wlm g∂ ( x ) zδ ( w ) i i ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ zδ ( w ) fδ ( l ) ^ ^ ❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂ m (68)(we added squares to the graphical notation (49) for elements of Gr ( T ( A )), in order to emphasise theelements of L assigned to the faces.) We consider the following numbering of the vertices of the tetrahedron:32 O O / / ♣♣♣♣♣♣ @ @ ✁✁✁✁✁✁✁✁ g g ◆◆◆◆◆◆ ^ ^ ❂❂❂❂❂❂❂❂ Equation (68) leaves out the d boundary of an element of (cid:0) ( g, x, , z, w ) , (1 , , , , f, l, , m ) (cid:1) ∈ Gr ( T ET ( A )),which is: g∂ ( x ) ∂ ( zf ) g f − ⊲ ′ w l xzf < < ②②②②②②②②②②②②②②②②②②② x / / g∂ ( x ) zfδ ( f − ⊲ ′ w l ) c c ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ (69)
3. Pointed homotopy of 2-crossed module maps
Suppose we have two pre-crossed modules ( E → G, ⊲ ) and ( E ′ → G ′ , ⊲ ). Definition 23.
Let φ : G → G ′ be a group morphism. A φ -derivation s : G → E ′ is a set map such that,for each g, h ∈ G , we have: s ( gh ) = φ ( h ) − ⊲ s ( g ) s ( h ) . Note that if s : G → E ′ is a derivation then: s (1 G ) = 1 E ′ and s ( g − ) = φ ( g ) ⊲ s ( g ) − . (70) Remark 24.
By looking at remark 14, φ -derivations are in one-to-one correspondence with group maps G → G ′ ⋉ ⊲ E ′ , where g ( φ ( g ) , s ( g )) . In particular if G is free, a φ derivation s : G → E ′ can be specified(and uniquely) by its values on a free basis B of G . Therefore a set map s : B → E ′ uniquely extends to a φ -derivation s : B → E ′ . We will very frequently use this line of thinking. This was pointed out to us by Ronnie Brown. A = (cid:16) L δ −→ E ∂ −→ G, ⊲, { , } (cid:17) and A ′ = (cid:16) L ′ δ −→ E ′ ∂ −→ G ′ , ⊲, { , } (cid:17) be 2-crossed modules. Let also f = ( µ, ψ, φ ) : A → A ′ be a 2-crossed module morphism. Definition 25.
A pair ( s, t ) of maps s : G → E ′ and t : E → L ′ will be called a quadratic f -derivation if s and t satisfy, for each g, h ∈ G and a, b ∈ E : s ( gh ) = (cid:0) φ ( h ) − ⊲ s ( g ) (cid:1) s ( h ) , (71) t ( ab ) = (cid:0) ψ ( b )(( s ◦ ∂ ))( b ) (cid:1) − ⊲ ′ (cid:0)(cid:8) ψ ( b ) , φ ( ∂ ( b ) − ) ⊲ s ( ∂ ( a )) − (cid:9) t ( a ) (cid:1) t ( b ) , (72) or alternatively, (9) : t ( ab ) = (cid:0) ( s ◦ ∂ )( b ) (cid:1) − ⊲ ′ (cid:16)(cid:8) ψ ( b ) − , s ( ∂ ( a )) − (cid:9) − ψ ( b ) − ⊲ ′ t ( a ) (cid:17) t ( b ) , (73) and also (for each g ∈ G and a ∈ E ): t ( g ⊲ a ) = φ ( g ) ⊲ (cid:16) s ( g ) s ( ∂ ( a )) − ⊲ ′ n ψ ( a ) − , s ( g ) − ) o − (cid:17) φ ( g ) ⊲ n s ( g ) , s ( ∂ ( a )) − ψ ( a ) − o(cid:0) φ ( g )( ∂ ◦ s )( g ) (cid:1) ⊲ t ( a ) . (74) Lemma 26 (Pointed homotopy of 2-crossed module maps).
In the condition of the previous defini-tion, if ( s, t ) is a quadratic f -derivation, and if we define f ′ = ( µ ′ , ψ ′ , φ ′ ) : A → A ′ as: µ ′ ( l ) = µ ( l ) ( t ◦ δ )( l ) , where l ∈ L (75) ψ ′ ( a ) = ψ ( a ) (( s ◦ ∂ ))( a ) ( δ ◦ t )( a ) , where a ∈ E (76) φ ′ ( g ) = φ ( g ) ( ∂ ◦ s )( g ) , where g ∈ G (77) then f ′ a morphism of 2-crossed modules A → A ′ . In this case we put: f ( f,s,t ) −−−−→ f ′ , and say that ( f, s, t ) is a homotopy (or 1-fold homotopy) connecting f and f ′ . This is proved by using the following lemma, noting that f ′ = Pr A ′ ◦ H and f = Pr A ′ ◦ H ; see theorem 19. Lemma 27.
Given a 2-crossed module morphism f = ( µ, ψ, φ ) : A → A ′ , the pair of maps t : E → L ′ and s : G → E ′ is a quadratic f -derivation if, and only if, H = ( i , i , i ) : A → P ∗ ( A ′ ) is a 2-crossed modulemorphism, where l ∈ L i (cid:0) µ ( l ) , t ◦ δ ( l ) (cid:1) ∈ L ′ ⋉ ad L ′ a ∈ E i (cid:0) ψ ( a ) , ( s ◦ ∂ )( a ) , t ( a ) (cid:1) ∈ E ′ ⋉ ∗ ( E ′ ⋉ ⊲ ′ L ′ ) g ∈ G i (cid:0) φ ( g ) , s ( g ) (cid:1) ∈ G ′ ⋉ ⊲ E ′ . Also, H = ( i , i , i ) is a 2-crossed module map if, and only if, ( i , i ) is a pre-crossed module map. Proof.
Conditions (71), (72), (74) express exactly that ( i , i ) is a pre-crossed module morphism. That H defines a morphism of complexes A → P ∗ ( A ′ ), which is equivariant with respect to the actions of G and G ′ ⋉ ⊲ E ′ , if and only if, ( t, s ) is a quadratic f - derivation follows from the explicit construction of P ∗ ( A ′ ). Wenow need to prove that H always preserves the Peiffer lifting, which follows immediately from the equation: t (cid:0) h a, b i (cid:1) = (cid:8) ψ ( a ) , ψ ( b ) (cid:9) − (cid:8) ψ ( a ) s ( ∂ ( a )) δ ( t ( a )) , ψ ( b ) s ( ∂ ( b )) δ ( t ( b )) (cid:9) , for each a, b ∈ E. (78)This equation was proven in [22]. 21 emark 28. Note that if ( s, t ) is a quadratic f derivation, where f : A → A ′ , connecting f and f ′ , and if g : A ′ → A ′′ is 2-crossed module map, then ( g ◦ s, g ◦ t ) is a quadratic ( g ◦ f ) -derivation connecting g ◦ f and g ◦ f ′ . Analogously if h : A ′′′ → A is a 2-crossed module map then ( s ◦ h, t ◦ h ) is a quadratic ( f ◦ h ) -quadraticderivation connecting f ◦ h and f ′ ◦ h . Remark 29.
Given f, f ′ : A → A ′ , consider a homotopy f = ( µ, ψ, η ) ( f,s,t ) −−−−→ ( µ ′ , ψ ′ , η ′ ) = f ′ . The under-lying maps G → Gr ( P ∗ ( A ′ )) and E → Gr ( P ∗ ( A ′ )) can be visualised (see (33) ) in the form: g ∈ G (cid:16) φ ( g ) s ( g ) −−→ φ ( g ) ∂ ( s ( g )) (cid:17) and e ∈ E (cid:16) ψ ( e ) (cid:0) s ( ∂ ( e )) ,t ( e ) (cid:1) −−−−−−−−−→ ψ ( e ) s ( ∂ ( e )) δ ( t ( e )) (cid:17) . As mentioned in the Introduction, given 2-crossed modules A and B , and considering 2-crossed modulemaps f, g : A → B , saying that f ∼ g if there exists a quadratic f -derivation ( s, t ) with f ( f,s,t ) −−−−→ g , doesnot define an equivalence relation. This is in contrast with the crossed complex case treated in [8, 9], butit is however not surprising for model category theoretical reasons [19], given that this should only be thecase (in general) when A is cofibrant and B is fibrant. Note that by Theorem 19 and Lemma 27, homotopybetween 2-crossed module maps is defined via a good path space in the model category of 2-crossed modules.Nevertheless, let us give an explicit example to show that 2-crossed module homotopy does not definean equivalence relation between 2-crossed module maps. In the example below, symmetry fails to hold.Let E = 2 Z be the group of even integers. Let Z = Z / E = { , } . Let ∂ : Z → Z be the quotient map.Let δ : E → Z be the inclusion. We have an action of Z on Z and on E by automorphisms, being 0 ⊲ n = n and 1 ⊲ n = − n , for each n ∈ Z .Clearly ( ∂ : Z → Z , ⊲ ) is a pre-crossed module. The Peiffer pairing is (given m, n ∈ Z ): h m, n i = (cid:26) , if m is even2 n , if m is oddConsider now the 2-crossed modules: A = (cid:0) { } → { } → Z (cid:1) and B = (cid:0) E δ −→ Z ∂ −→ Z , ⊲, { , } (cid:1) , where for A all actions are trivial, as is the Peiffer lifting, and the Peiffer lifting in B has the form: { m, n } = h m, n i , for m, n ∈ Z . See Example 9 for a similar construction.Let us consider f = ( η, ψ, φ ) : A → B , such that η and ψ are trivial, and φ : Z → Z is the identity map.This clearly is a 2-crossed module map. Let us also consider f ′ = ( η, ψ, φ ′ ) : A → B , such that η , ψ and φ ′ all are trivial, thus their images consist solely of the identity element.We now consider the f -quadratic derivation ( s, t ) such that s (1) = 1 , s (0) = 0 and t (0) = 0. It is clearthat (72) and (74) are satisfied. As for condition (71), we check it case by case:0 = s (1 + 1) = ( − φ (1)) ⊲ s (1) + s (1) = − s (1) + s (1) = 01 = s (0 + 1) = ( − φ (1)) ⊲ s (0) + s (1) = − s (0) + s (1) = 11 = s (1 + 0) = ( − φ (0)) ⊲ s (1) + s (0) = s (1) + s (0) = 10 = s (0 + 0) = ( − φ (0)) ⊲ s (0) + s (0) = − s (0) + s (0) = 0We clearly have that (by Lemma 26): f ( f,s,t ) −−−−→ f ′ . Let us see that there does not exist an f ′ -quadratic derivation ( s ′ , t ′ ) such that f ′ ( f ′ ,s ′ ,t ′ ) −−−−−→ f. Indeed, for thisto happen, we would need a φ ′ -derivation s ′ : Z → Z , such that ∂ ( s (1)) = 1. However, since φ ′ ( Z ) = { } , s ′ would need to be (by (71)) a non-trivial group morphism Z → Z , which is impossible.22 .2. Quadratic 2-derivations and 2-fold homotopy of 2-crossed modules Let A = (cid:16) L δ −→ E ∂ −→ G, ⊲, { , } (cid:17) and A ′ = (cid:16) L ′ δ −→ E ′ ∂ −→ G ′ , ⊲, { , } (cid:17) be 2-crossed modules. Let f = ( µ, ψ, φ ) : A → A ′ be a 2-crossed module morphism. Consider a quadratic f -derivation ( s, t ). Recall the construction of the disk space D ∗ ( A ′ ) of A ′ ; 2.4.2. Definition 30.
We say that a map k : G → L ′ is a quadratic ( f, s, t ) g, h ∈ G : k ( gh ) = (cid:16) s ( h ) − ⊲ ′ (cid:0) φ ( h ) − ⊲ k ( g ) (cid:1)(cid:17) k ( h ) . (79) Lemma 31.
The map k : G → L ′ is a quadratic ( f, s, t ) g ( φ ( g ) , s ( g ) , k ( g )) is a group morphism G → ( G ⋉ ⊲ E ) ⋉ • L = Gr ( D ∗ ( A ′ )) . In particular k (1 G ) = 1 K and for each g ∈ G : k ( g − ) = φ ( g ) ⊲ ( s ( g ) ⊲ ′ k ( g ) − ) . Moreover if g, h, i ∈ G : k ( g − hi ) = Ξ ( φ,s,k ) ( g, h, i ) , where we have defined: Ξ ( φ,s,k ) ( g, h, i ) = s ( i ) − ⊲ ′ (cid:16) φ ( i ) − ⊲ (cid:16) s ( h ) − ⊲ ′ (cid:0) ( φ ( h − ) φ ( g )) ⊲ ( s ( g ) ⊲ ′ k ( g ) − ) (cid:1) k ( h ) (cid:17)(cid:17) k ( i ) . (80) Proof.
The first assertion is immediate from remark 14 and (54). Also: k ( g − hi ) = s ( i ) − ⊲ ′ (cid:16) φ ( i ) − ⊲ (cid:0) s ( h ) − ⊲ ′ ( φ ( h ) − ⊲ k ( g − )) k ( h ) (cid:1)(cid:17) k ( i )= s ( i ) − ⊲ ′ (cid:16) φ ( i ) − ⊲ (cid:0) s ( h ) − ⊲ ′ ( φ ( h − g ) ⊲ ( s ( g ) ⊲ ′ k ( g ) − )) k ( h ) (cid:1)(cid:17) k ( i ) Corollary 32.
In the condition of the previous lemma, if G is a free group on the basis B a quadratic ( f, s, t ) k : G → L ′ can be specified, uniquely, by its valued in the basis B . More precisely givena set map k : B → L ′ there exists a unique quadratic ( f, s, t ) k : G → L ′ extending k . Lemma 33.
The set map k : G → L ′ is a quadratic ( f, s, t ) H =( j , j , j ) , below, is a 2-crossed module morphism from A into D ∗ ( A ′ ) . j ( g ) = (cid:0) φ ( g ) , s ( g ) , k ( g ) (cid:1) j ( e ) = (cid:0) ψ ( e ) , ( s ◦ ∂ )( e ) , t ( e ) , ( k ◦ ∂ )( e ) (cid:1) j ( l ) = (cid:0) l, ( t ◦ ∂ ( l ) (cid:1) Proof.
We already know by lemma 27 that, forgetting the last component of j and j , we have a 2-crossed module map A → P ∗ ( A ′ ). Given the form of D ∗ ( A ′ ) we clearly get a morphism of group complexes A → D ∗ ( A ′ ), which is compatible with the action of the first groups of the complexes on the remaining. Toprove this note the following calculation, and compare with (57), (58), (54) and (59): k ( ∂ ( g ⊲ e )) = k ( g∂ ( e ) g − ) = ( φ ( g ) ⊲ s ( g )) ⊲ ′ (cid:0) φ ( g ) ⊲ k ( g∂ ( e )) (cid:1) φ ( g ) ⊲ ( s ( g ) ⊲ ′ k ( g ) − )= φ ( g ) ⊲ (cid:16) s ( g ) ⊲ ′ (cid:0) k ( g∂ ( e )) k ( g ) − (cid:1)(cid:17) = φ ( g ) ⊲ (cid:0) s ( g ) ⊲ ′ (cid:16)(cid:16) s ( ∂ ( e )) − ⊲ ′ (cid:0) φ ( ∂ ( e )) − ⊲ k ( g ) (cid:1)(cid:17) k ( ∂ ( e )) k ( g ) − (cid:17) (cid:1) . We now need to prove that H preserves the Peiffer lifting. This follows from lemma 27 and the form ofthe Peiffer lifting on D ∗ ( A ′ ) and P ∗ ( A ′ ), see (31) and (60).23herefore, looking at the maps d , d : D ∗ ( A ′ ) → P ∗ ( A ′ ) of (61) and (62) (and their composition with H : A → D ∗ ( A ′ )), lemma 27 and the previous one, given a quadratic ( f, s, t ) 2-derivation k then s ′ : G → E ′ and t ′ : E → L ′ , defined as: s ′ ( g ) = s ( g ) ( δ ◦ k )( g ) , and t ′ ( e ) = ( k ◦ ∂ )( e ) − t ( e ) (81)yield a quadratic f -derivation ( s ′ , t ′ ), and in this case we put:( f, s, t ) ( f,s,t,k ) −−−−−→ ( f, s ′ , t ′ ) . Remark 34.
Note that if f ( f,s,t ) −−−−→ f ′ then we also have f ( f,s ′ ,t ′ ) −−−−−→ f ′ . Definition 35.
We say that the quadratic f -derivations ( s, t ) and ( s ′ , t ′ ) are 2-fold homotopic if there existsa quadratic ( f, s, t ) ( f, s, t ) ( f,s,t,k ) −−−−−→ ( f, s ′ , t ′ ) . The quadruple ( f, s, t, k ) will be calleda 2-fold homotopy connecting ( f, s, t ) and ( f, s ′ , t ′ ) . Remark 36.
Looking at the definition of the disk space 2-crossed module from the triangle space 2-crossedmodules, a quadratic ( f, s, t ) k → L ′ , such that g (cid:0) φ ( g ) , s ( g ) , , δ ( k ( g )) , k ( g ) − (cid:1) is a group morphism G → ( G ′ ⋉ ⊲ E ′ ) ⋉ • ( { } ⋉ ∗ ( E ′ ⋉ ⊲ ′ L ′ )) = Gr ( T ( A ′ )) . This also follows directly from (24) . (We have put f = ( µ, ψ, φ ) .) We can thus visualise a quadratic ( f, s, t ) k : G → L ′ as amap: g ∈ G φ ( g ) ∂ ( s ( g )) φ ( g ) k ( g ) − s ( g ) δ ( k ( g )) ; ; ✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇ s ( g ) / / φ ( g ) ∂ ( s ( g )) e e ❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑ ∈ Gr ( T ( A ′ )) In this subsection, let us fix two 2-crossed modules: A ′ = (cid:16) L ′ δ −→ E ′ ∂ −→ F, ⊲, { , } (cid:17) and A = (cid:16) L δ −→ E ∂ −→ G, ⊲, { , } (cid:17) . Suppose also that F is a free group, with a chosen (free) basis B ⊂ F . Let us define a groupoid [ A ′ , A ] B ,with objects the 2-crossed module maps A ′ → A and morphisms their homotopies. This will explicitlydepend on the chosen basis B of F . We will freely use the notation of subsections 2.3, 2.4.1 and 3.1. Consider homotopies of 2-crossed module maps A ′ → A : f = ( µ, ψ, φ ) ( f,s,t ) −−−−→ f ′ = ( µ ′ , ψ ′ , φ ′ ) and ( µ ′ , ψ ′ , φ ′ ) ( f ′ ,s ′ ,t ′ ) −−−−−→ f ′′ = ( µ ′′ , ψ ′′ , φ ′′ )(where f, f ′ , f ′′ : A ′ → A ). We therefore have pre-crossed module maps H, H ′ : A ′ → P ∗ ( A ); lemma 27. Weconsequentely have an induced map to the group pull-back:( H, H ′ ) : A ′ → P ∗ ( A ) Pr A × Pr A P ∗ ( A ) . Let us define the concatenation of homotopies: f = ( µ, ψ, φ ) ( f,s ⊗ s ′ ,t ⊗ t ′ ) −−−−−−−−→ f ′′ = ( µ ′′ , ψ ′′ , φ ′′ ) .
24t suffices to define the associated pre-crossed module map (lemma 27) H ⊗ H ′ = ( f, s ⊗ s ′ , t ⊗ t ′ ) : A ′ → P ∗ ( A ).The derivation ( s ⊗ s ′ ) : F → E is the unique φ -derivation (see definition 23 and remark 24) which on thechosen basis B of F has the form b ∈ B ( s ⊗ s ′ )( b ) = s ( b ) s ′ ( b ) ∈ E. (82)There is another piece of information that we will use, namely a set map ω ( s,s ′ ) : F → L , measuring thedifference (for each g ∈ F ) between s ( g ) s ′ ( g ) and ( s ⊗ s ′ )( g ); this difference is null in a crossed module.Recall the construction of the triangle space 2-crossed module 2.4.1. Given that F is free, there exists aunique group map X ( s,s ′ ) : F → ( G ⋉ ⊲ E ) ⋉ • ( { } ⋉ ∗ (cid:0) E ⋉ ⊲ ′ L ) (cid:1) = Gr ( T ( A )), say: g ∈ F X ( s,s ′ ) (cid:0) φ ( g ) , s ( g ) , , ζ ( g ) , ω ( s,s ′ ) ( g ) (cid:1) = φ ( g ) ∂ ( s ( g )) ∂ ( ζ ( g )) ω ( s,s ′ )( g ) φ ( g ) s ( g ) ζ ( g ) : : ✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈ s ( g ) / / φ ( g ) ∂ ( s ( g )) ζ ( g ) δ ( ω ( s,s ′ )( g ) ) O O which on the chosen basis B of F takes the form: b X ( s,s ′ ) (cid:0) φ ( b ) , s ( b ) , , s ′ ( b ) , (cid:1) = φ ( b ) ∂ ( s ( b )) ∂ ( s ′ ( b ))1 φ ( b ) s ( b ) s ′ ( b ) : : ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ s ( b ) / / φ ( b ) ∂ ( s ( b )) s ′ ( b ) O O (83)Looking, see (47), at the three P ∗ ( A ) sides of the triangle space T ( A ), we have maps from F to theunderlying group G ′ ⋉ ⊲ E ′ = Gr ( P ∗ ( A ′ )) of the path space P ∗ ( A ′ ). By remark 24, since there are uniquegroup maps F → G ′ ⋉ E ′ extending b ( φ ( b ) , s ( b ) s ′ ( b )), b ( φ ( b ) ∂ ( s ( b )) , s ′ ( b )) and b ( φ ( b ) , s ( b )),it follows at once that X ( s,s ′ ) has the form: g X ( s,s ′ ) (cid:0) φ ( g ) , s ( g ) , , ζ ( g ) , ω ( s,s ′ ) ( g ) (cid:1) = φ ( g ) ∂ ( s ( g )) ∂ ( s ′ ( g )) ω ( s,s ′ ) ( g ) φ ( g ) ( s ⊗ s ′ )( g ) ; ; ✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈ s ( g ) / / φ ( g ) ∂ ( s ( g )) s ′ ( g ) O O Therefore ( s ⊗ s ′ )( g ) = s ( g ) ζ ( g ) , and ζ ( g ) δ ( ω ( s,s ′ ) ( g )) = s ′ ( g ), or s ( g ) ζ ( g ) δ ( ω ( s,s ′ ) ( g )) = s ( g ) s ′ ( g ) , for each g ∈ F. (84)We have proven: 25 emma 37. There exists a unique group morphism X ( s,s ′ ) : F → Gr ( T ( A )) , with: g (cid:0) φ ( g ) , s ( g ) , , s ′ ( g ) δ ( ω ( s,s ′ ) ( g )) − , ω ( s,s ′ ) ( g ) (cid:1) = φ ( g ) ∂ ( s ( g )) ∂ ( s ′ ( g )) ω ( s,s ′ ) ( g ) φ ( g ) s ( g ) s ′ ( g ) δ ( ω ( s,s ′ ) ( g )) − : : ✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈ s ( g ) / / φ ( g ) ∂ ( s ( g )) s ′ ( g ) O O for each g in F , such that ω ( b ) = 1 (see (83) ) on the free generators b ∈ B of F . In particular by (84) : ( s ⊗ s ′ )( g ) = s ( g ) s ′ ( g ) δ ( ω ( s,s ′ ) ( g )) − , for each g ∈ F. (85)We now define ( t ⊗ t ′ ) : E ′ → L . For e ∈ E ′ put:( t ⊗ t ′ )( e ) = (cid:0) ω ( s,s ′ ) ( ∂ ( e )) (cid:1) (cid:0) s ′ ( ∂ ( e )) − ⊲ ′ t ( e ) (cid:1) t ′ ( e ) . (86)We prove that ( s ⊗ s ′ , t ⊗ t ′ ) is a quadratic ( µ, ψ, φ )-derivation. Consider the commutative triangle: F X ( s,s ′ ) / / Gr ( H,H ′ ) ( ( PPPPPPPPPPPPPP Gr ( T ( A )) Gr ( d + − ) (cid:15) (cid:15) Gr (cid:16) P ∗ ( A ) Pr A × Pr A P ∗ ( A ) (cid:17) Define Y ( s,s ′ ) : E ′ → Gr ( T ( A )) as being the unique group map making the diagram below commutative: E ′ Y ( s,s ′ ) / / ❴❴❴❴❴❴❴❴❴ ∂ r r ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ ❙❙❙❙❙❙ Gr ( H,H ′ ) ) ) ❙❙❙❙❙❙❙❙❙❙ Gr ( T ( A )) Gr ( d + − ) (cid:15) (cid:15) β ′ r r ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ F X ( s,s ′ ) / / Gr ( H,H ′ ) ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ Gr ( T ( A )) Gr ( d + − ) (cid:15) (cid:15) Gr (cid:16) P ∗ ( A ) Pr A × Pr A P ∗ ( A ) (cid:17) β ′ r r ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ Gr (cid:16) P ∗ ( A ) Pr A × Pr A P ∗ ( A ) (cid:17) (87)(We are using lemmas 4 and 20.) In particular ( X ( s,s ′ ) , Y ( s,s ′ ) ) will be a pre-crossed module map. Bycomposing with d in (47) will give H ⊗ H ′ = ( f, s ⊗ s ′ , t ⊗ t ′ ) : A ′ → P ∗ ( A ).To determine Y ( s,s ′ ) we only need (remark 5) to find a set map that makes the diagram (87) commutative.Looking at the construction of the triangle pre-crossed module (subsection 2.4.1), consider the set map: Y ( s,s ′ ) : E ′ → ( E ⋉ ∗ ( E ⋉ ⊲ ′ L )) ⋉ ∗ (( { } ⋉ ∗ ( E ⋉ ⊲ ′ L ) ⋉ • ′ ( { } ⋉ ad L )) = Gr ( T ( A )) , definition 3, of the form: e Y ( s,s ′ ) (cid:16) ψ ( e ) , s ( ∂ ( e )) , t ( e ) , , s ′ ( ∂ ( e )) δ ( ω ( s,s ′ ) ( ∂ ( e ))) − , ω ( s,s ′ ) ( ∂ ( e )) , , t ′ ( e ) (cid:17) . Y ( s,s ′ ) ( e ) = ψ ( e ) s ( ∂ ( e )) δ ( t ( e )) s ′ ( ∂ ( e )) δ ( t ′ ( e )) ψ ( e ) (cid:16) s ( ∂ ( e )) s ′ ( ∂ ( e )) (cid:0) δ ( ω ( s,s ′ ) ( ∂ ( e )) (cid:1) − ,ω ( s,s ′ ) ( ∂ ( e )) s ′ ( ∂ ( e )) ⊲ ′ t ( e ) t ′ ( e ) (cid:17) ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ (cid:0) s ( ∂ ( e )) ,t ( e ) (cid:1) / / ψ ( e ) s ( ∂ ( e )) δ ( t ( e )) (cid:0) s ′ ( ∂ ( e )) ,t ′ ( e ) (cid:1) O O Note that β ′ ◦ Y ( s,s ′ ) = X ( s,s ′ ) ◦ ∂, where β : Gr ( T ( A )) → Gr ( T ( A )) is the boundary map; (42) or (51).Clearly this makes the diagram (87) commute thus ( Y ( s,s ′ ) , X ( s,s ′ ) ) is a pre-crossed module map.In particular, by looking at the oblique side of Y ( s,s ′ ) and X ( s,s ′ ) it follows by remark 27 and (47) that( s ⊗ s ′ , t ⊗ t ′ ) is a quadratic ( µ, ψ, φ )-derivation. Lemma 38.
Consider homotopies of 2-crossed module maps f = ( µ, ψ, φ ) ( f,s,t ) −−−−→ f ′ = ( µ ′ , ψ ′ , φ ′ ) and ( µ ′ , ψ ′ , φ ′ ) ( f ′ ,s ′ ,t ′ ) −−−−−→ f ′′ = ( µ ′′ , ψ ′′ , φ ′′ ) . Then the ( µ, ψ, φ ) -quadratic derivation (cid:0) ( s ⊗ s ′ ) , ( t ⊗ t ′ ) (cid:1) connects f and f ′′ . Proof.
At the pre-crossed module level follows by construction. Let us give full details however. Let¯ f = (¯ µ, ¯ ψ, ¯ φ ) be the 2-crossed module morphism defined from f and the quadratic derivation ( s ⊗ s ′ , t ⊗ t ′ );lemma 26. We must prove that ¯ f = f ′′ . Let b be a free generator of F . Then¯ φ ( b ) = φ ( b ) ∂ ( s ⊗ s ′ ( b )) = φ ( b ) ∂ ( s ( b ) s ′ ( b )) = φ ( b ) ∂ ( s ( b )) ∂ ( s ′ ( b ))) = φ ′ ( b ) ∂ ( s ′ ( b )) = φ ′′ ( b ) . In particular it follows that ¯ φ ( g ) = φ ′′ ( g ), for each g ∈ F .Given e ∈ E ′ we have (we use (85) and the rule δ ( e ⊲ ′ k ) = eδ ( k ) e − , for each e ∈ E and k ∈ L ):¯ ψ ( e ) = ψ ( e ) ( s ⊗ s ′ )( ∂ ( e )) δ (( t ⊗ t ′ )( e ))= ψ ( e ) s ( ∂ ( e )) s ′ ( ∂ ( e )) δ ( ω ( ∂ ( e ))) − δ (cid:16)(cid:0) ( ω ◦ ∂ )( e ) (cid:1) (cid:0) s ′ ( ∂ ( e )) − ⊲ ′ t ( e ) (cid:1) t ′ ( e ) (cid:17) = ψ ( e ) s ( ∂ ( e )) δ ( t ( e )) s ′ ( ∂ ( e )) δ ( t ′ ( e )) = ψ ′ ( e ) s ′ ( ∂ ( e )) δ ( t ′ ( e )) = ψ ′′ ( e ) . Finally, given k ∈ L ′ we have (since ∂ ◦ δ ( k ) = 1 for each k ∈ L ):¯ µ ( k ) = µ ( k ) ( t ⊗ t ′ )( δ ( k )) = µ ( k ) t ( δ ( k )) t ′ ( δ ( k )) = µ ′′ ( k ) . Note s ′ (1 F ) = 1 E and ω (1 F ) = 1 L . Remark 39 (Some properties of ω ( s,s ′ ) : F → L ). Clearly ω ( s,s ′ ) (1 F ) = 1 L and ω ( s,s ′ ) ( b ) = 1 L , for each b ∈ B. Also, by remark 14 and equation (22) , for each g, h ∈ F : ω ( s,s ′ ) ( gh ) = ω ( s,s ′ ) ( h ) s ′ ( h ) − ⊲ ′ (cid:16) φ ( h ) − ⊲ (cid:8) φ ( h ) ⊲ s ( h ) − , δ ( ω ( s,s ′ ) ( g )) s ′ ( g ) − (cid:9) φ ′ ( h ) − ⊲ ω ( s,s ′ ) ( g ) (cid:17) and ω ( s,s ′ ) ( g − ) = φ ( g ) ⊲ (cid:8) s ( g ) , s ′ ( g )( ω ( s,s ′ ) ( g )) − (cid:9) φ ′ ( g ) ⊲ (cid:0) s ′ ( g ) ⊲ ′ ( ω ( s,s ′ ) ( g )) − (cid:1) . In particular, if b, b ′ , b ′′ are free generators of F : ω ( s,s ′ ) ( bb ′ ) = s ′ ( b ′ ) − ⊲ ′ (cid:0) { s ( b ′ ) − , φ ( b ′ ) − ⊲ s ′ ( b ) − } (cid:1) , and ω ( s,s ′ ) ( b − ) = φ ( b ) ⊲ (cid:8) s ( b ) , s ′ ( b ) (cid:9) . hus: ω ( s,s ′ ) ( b − b ′ b ′′ ) = Θ ( s,s ′ ) ( b, b ′ , b ′′ ) , where by definition: Θ ( s,s ′ ) ( b, b ′ , b ′′ ) = s ′ ( b ′′ ) − ⊲ ′ (cid:0) { s ( b ′′ ) − , φ ( b ′′ ) − ⊲ s ′ ( b ′ ) − } (cid:1)(cid:0) φ ′ ( b ′′ ) − ⊲ s ′ ( b ′ ) s ′ ( b ′′ ) (cid:1) − ⊲ ′ (cid:16) n(cid:0) φ ( b ′′ ) − ⊲ s ( b ′ ) s ( b ′′ ) (cid:1) − , ( φ ( b ′ b ′′ ) − φ ( b )) ⊲ (cid:0) s ( b ) s ′ ( b ) s ( b ) − (cid:1)o(cid:0) φ ′ ( b ′ b ′′ ) − φ ( b ) (cid:1) ⊲ { s ( b ) , s ′ ( b ) } (cid:17) . (88) We freely use the notation of subsection 2.4.1, and we resume the notation and context of 3.3.1.
Proposition 40.
The concatenation of homotopies is associative.
Proof.
Choose a chain of homotopies of 2-crossed module maps A ′ → A : f = ( µ, ψ, φ ) ( f,s,t ) −−−−→ f ′ = ( µ ′ , ψ ′ , φ ′ ) ( f ′ ,s ′ ,t ′ ) −−−−−→ f ′′ = ( µ ′′ , ψ ′′ , φ ′′ ) ( f ′′ ,s ′′ ,t ′′ ) −−−−−−−→ f ′′′ = ( µ ′′′ , ψ ′′′ , φ ′′′ ) . It is immediate that ( s ⊗ s ′ ) ⊗ s ′′ = s ⊗ ( s ′ ⊗ s ′′ ), since this is true in a free basis of F .Let us now see that ( t ⊗ t ′ ) ⊗ t ′′ = t ⊗ ( t ′ ⊗ t ′′ ). We now have, for each e ∈ E ′ :( t ⊗ t ′ )( e ) = (cid:0) ( ω ( s,s ′ ) ◦ ∂ )( e ) (cid:1) (cid:0) s ′ ( ∂ ( e )) − ⊲ ′ t ( e ) (cid:1) t ′ ( e ) , ( t ′ ⊗ t ′′ )( e ) = (cid:0) ( ω ( s ′ ,s ′′ ) ◦ ∂ )( e ) (cid:1) (cid:0) s ′′ ( ∂ ( e )) − ⊲ ′ t ′ ( e ) (cid:1) t ′′ ( e ) , (cid:0) t ⊗ ( t ′ ⊗ t ′′ ) (cid:1) ( e ) = ( ω ( s,s ′ ⊗ s ′′ ) ◦ ∂ )( e ) ( s ′ ⊗ s ′′ )( ∂ ( e )) − ⊲ ′ t ( e ) (cid:0) ( ω ( s ′ ,s ′′ ) ◦ ∂ )( e ) (cid:1) (cid:0) s ′′ ( ∂ ( e )) − ⊲ ′ t ′ ( e ) (cid:1) t ′′ ( e )= ( ω ( s,s ′ ⊗ s ′′ ) ◦ ∂ )( e ) ω ( s ′ ,s ′′ ) ( ∂ ( e )) ( s ′ ( ∂ ( e ) s ′′ ( ∂ ( e )) − ⊲ ′ t ( e ) (cid:0) s ′′ ( ∂ ( e )) − ⊲ ′ t ′ ( e ) (cid:1) t ′′ ( e ) , (cid:0) ( t ⊗ t ′ ) ⊗ t ′′ (cid:1) ( e ) = ( ω ( s ⊗ s ′ ,s ′′ ) ◦ ∂ )( e ) s ′′ ( ∂ ( e )) − ⊲ ′ (cid:16)(cid:0) ( ω ( s,s ′ ) ◦ ∂ )( e ) (cid:1) (cid:0) s ′ ( ∂ ( e )) − ⊲ ′ t ( e ) (cid:1) t ′ ( e ) (cid:17) t ′′ ( e ) . To prove associativity, we therefore need to prove that for each e ∈ E ′ : ω ( s,s ′ ⊗ s ′′ ) ( ∂ ( e )) ω ( s ′ ,s ′′ ) ( ∂ ( e )) = ω ( s ⊗ s ′ ,s ′′ ) ( ∂ ( e )) s ′′ ( ∂ ( e )) − ⊲ ′ ω ( s,s ′ ) ( ∂ ( e )) . We will prove that for each g ∈ F we have: ω ( s,s ′ ⊗ s ′′ ) ( g ) = ω ( s ⊗ s ′ ,s ′′ ) ( g ) (cid:0) s ′′ ( g ) − ⊲ ′ ω ( s,s ′ ) ( g ) (cid:1) ω ( s ′ ,s ′′ ) ( g ) − . (89)(It is a nice exercise to prove that this is coherent with (85).) To prove (89), consider the unique group map: W : F → (cid:0) ( G ⋉ ⊲ E ) ⋉ • ( { } ⋉ ∗ ( E ⋉ ⊲ ′ L )) (cid:1) ⋉ △ (cid:0) { } ⋉ ∗ ( { } ⋉ ⊲ ′ { } ) ⋉ ∗ (( { } ⋉ ∗ ( E ⋉ ⊲ ′ L ) ⋉ • ′ ( { } ⋉ ad L ) (cid:1) = Gr ( T ET ( A )) , which on the (chosen) free basis B of F is b ( φ ( b ) , s ( b ) , , s ′ ( b ) , , , , , , s ′′ ( b ) , , , W ( b ) = φ ( b ) ∂ ( s ( b )) ∂ ( s ′ ( b )) ∂ ( s ′′ ( b )) φ ( b ) ∂ ( s ( b )) ∂ ( s ′ ( b )) s ′′ ( b ) O O φ ( b ) s ( b ) s ′ ( b ) s ′′ ( b ) = = ④④④④④④④④④④④④④④④④④④④④④④④④④④④④④ s ( b ) s ′ ( b ) ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ s ( b ) / / φ ( b ) ∂ ( s ( b )) s ′ ( b ) k k ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ s ′ ( b ) s ′′ ( b ) c c ❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋ g ∈ F (by looking at the value of the compositions of thesemorphisms with W , in the chosen free basis of F ): W ( g ) = φ ( g ) ∂ ( s ( g )) ∂ ( s ′ ( g )) ∂ ( s ′′ ( g ))) φ ( g ) ∂ ( s ( g )) ∂ ( s ′ ( g )) s ′′ ( g ) O O φ ( g ) ( s ⊗ s ′ ⊗ s ′′ )( g ) : : tttttttttttttttttttttttttttttttttt ( s ⊗ s ′ )( g ) ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ s ( g ) / / ω ( s,s ′ ) ( g ) ω ( s ⊗ s ′ ,s ′′ ) ( g ) φ ( g ) ∂ ( s ( g )) s ′ ( g ) k k ❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ ( s ′ ⊗ s ′′ )( g ) e e ❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑ ω ( s ′ ,s ′′ ) ( g )By composing W : F → Gr ( T ET ( A )) with d : Gr ( T ET ( A )) → Gr ( T ( A )), see (67) and (69), yields agroup map Z : F → Gr ( T ( A )), whose value (for each g ∈ F ) is: Z ( g ) = φ ( g ) ∂ ( s ( g )) ∂ ( s ′ ( g ) ∂ ( s ′ ( g )) φ ( g ) ω ( s ⊗ s ′ ,s ′′ ) ( g ) ( s ′′ ( g ) − ⊲ ′ ω ( s,s ′ ) ( g )) ω ( s ′ ,s ′′ ) ( g ) − s ⊗ s ′ ⊗ s ′′ )( g ) ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ s ( g ) / / φ ( g ) ∂ ( s ( g )) ( s ′ ⊗ s ′′ )( g ) g g PPPPPPPPPPPPPPPPPPPPPPPPP
On free generators b ∈ B ⊂ F , the map Z has the form: Z ( b ) = ( φ ( b ) , s ( b ) , , s ′ ( b ) s ′′ ( b ) ,
1) = φ ( b ) ∂ ( s ( b )) ∂ ( s ′ ( b ) s ′′ ( b )) φ ( b ) 1 s ( b ) s ′ ( b ) s ′′ ( b ) rrrrrrrrrrrrrrrrrrrrrrr s ( b ) / / φ ( b ) ∂ ( s ( b )) s ′ ( b ) s ′′ ( b ) g g ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ thus (by using lemma 37 again) for each g ∈ F : Z ( g ) = φ ( g ) ∂ ( s ( g )) ∂ ( s ′ ( g ) ∂ ( s ′ ( g )) (cid:0) φ ( g ) ω ( s,s ′ ⊗ s ′′ ) ( g ) ( s ⊗ s ′ ⊗ s ′′ )( g ) ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ s ( g ) / / (cid:0) φ ( g ) ∂ ( s ( g )) ( s ′ ⊗ s ′′ )( g ) g g PPPPPPPPPPPPPPPPPPPPPPPPP from which we have ω ( s,s ′ ⊗ s ′′ ) ( g ) = ω ( s ⊗ s ′ ,s ′′ ) ( g ) s ′′ ( g ) − ⊲ ′ ω ( s,s ′ ) ( g ) ω ( s ′ ,s ′′ ) ( g ) − , for each g ∈ F . Consider a homotopy between 2-crossed module maps: f = ( µ, ψ, φ ) ( f,s,t ) −−−−→ f ′ = ( µ ′ , ψ ′ , φ ′ ) . s f , t f ) be the trivial quadratic f -derivation, such that s f ( g ) = 1 for each g ∈ F and t f ( e ) = 1, if e ∈ E ′ . Lemma 41.
We have that ω ( s f ,s ) ( g ) = 1 and ω ( s,s f ′ ) ( g ) = 1 , for each g ∈ F .
Proof.
Consider the unique group map X ( s f ,s ) : F → ( G ⋉ ⊲ E ) ⋉ • ( { } ⋉ ∗ ( E ⋉ ⊲ ′ L )) = Gr ( T ( A )) , of lemma 37, with, for each free generator b ∈ B ⊂ F : X ( s f ,s ) ( b ) = ( φ ( b ) , s f ( b ) , , s ( b ) , . Thus for each g ∈ F : X ( s f ,s ) ( g ) = (cid:0) φ ( g ) , s f ( g ) , , s ( g ) δ ( ω ( s f ,s ) ( g )) − , ω ( s f ,s ) ( g ) (cid:1) . Since g ∈ F (cid:0) φ ( g ) , , , s ( g ) , (cid:1) ∈ Gr ( T ( A )) is also a group morphism, by (18) and remark 24, whichextends the value of X ′ in B , it follows in particular that ω ( s f ,s ) ( g ) = 1 for each g ∈ F .Similarly, consider the map X : F → ( G ⋉ ⊲ E ) ⋉ • ( { } ⋉ ∗ ( E ⋉ ⊲ ′ L )) of lemma 37 with s ′ = s f ′ . On freegenerators X ( b ) = ( φ ( b ) , s ( b ) , , , g ∈ F ( φ ( g ) , s ( g ) , , , ∈ Gr ( T ( A )), there follows that X ( g ) = ( φ ( g ) , s ( g ) , , , ω ( s,s f ′ ) ( g ) = 1, for each g ∈ F .By (86) it follows that ( s, t ) ⊗ ( s f ′ , t f ′ ) = ( s, t ) and that ( s f , t f ) ⊗ ( s, t ) = ( s, t ) . Remark 42.
By the proof of lemma 41, we in particular, for a φ -derivation s : F → E , have maps: g ∈ F φ ( g ) ∂ ( s ( g )) φ ( g ) 1 s ( g ) ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ / / φ ( g ) s ( g ) h h ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ ∈ Gr ( T ( A )) and also: g ∈ F φ ( g ) ∂ ( s ( g )) φ ( g ) 1 s ( g ) ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ s ( g ) / / φ ( g ) s ( g ) i i ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ ∈ Gr ( T ( A )) We freely use the notation of subsections 2.4.1 and 2.4.3. Let f, f ′ : A ′ = (cid:16) L ′ δ −→ E ′ ∂ −→ F, ⊲, { , } (cid:17) → A = (cid:16) L δ −→ E ∂ −→ G, ⊲, { , } (cid:17) be 2-crossed module maps. Consider a 2-crossed module homotopy: f = ( µ, ψ, φ ) ( f,s,t ) −−−−→ f ′ = ( µ ′ , ψ ′ , φ ′ ) . Let us define its inverse: f ′ = ( µ ′ , ψ ′ , φ ′ ) ( f ′ , ¯ s, ¯ t ) −−−−→ f = ( µ, ψ, φ ) . The derivation ¯ s : F → E is the unique φ ′ -derivation which on the chosen basis B of F takes the form:¯ s ( b ) = s ( b ) − .
30e clearly have ( s ⊗ ¯ s )( g ) = 1 = (¯ s ⊗ s )( g ) for each g in F , for this is true in a free basis of F , remark 24.Looking at equation (85), for each g ∈ F we thus have:¯ s ( g ) = (cid:0) s ( g ) (cid:1) − δ ( ω ( s,s ) ) = δ ( ω ( s,s ) ) (cid:0) s ( g ) (cid:1) − . (91) Lemma 43.
For each g ∈ F we have ω ( s,s ) ( g ) = s ( g ) − ⊲ ′ ω ( s,s ) ( g ) . Proof.
Consider the map W : F → Gr ( T ET ( A )), in equation (90), for s ′ = ¯ s and s ′′ = s . Thus, if b ∈ B : W ( b ) = φ ( b ) ∂ ( s ( b )) φ ( b ) s ( b ) O O φ ( b ) s ( b ) A A ☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ s ( b ) / / φ ( b ) ∂ ( s ( b )) s ( b ) − i i ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ _ _ ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ s ⊗ s )( g ) = 1, for each g ∈ F , and lemmas 41 and 37, together with remark 42 we have: W ( g ) = φ ( g ) ∂ ( s ( g )) φ ( g ) s ( g ) O O φ ( g ) s ( g ) A A ✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄ ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ s ( g ) / / ω ( s,s ) ( g )1 φ ( g ) ∂ ( s ( g )) s ( g ) i i ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ _ _ ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅ ω ( s,s ) ( g ) . Composing W : F → Gr ( T ET ( A )) with d : Gr ( T ET ( A )) → Gr ( T ( A )), see (67) and (69), gives a map: g ∈ F φ ( g ) ∂ ( s ( g )) φ ( g ) ( s ( g ) − ⊲ ′ ω ( s,s ) ( g )) ω ( s,s ) ( g ) − s ( g ) ; ; ✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇ s ( g ) / / φ ( g ) ∂ ( s ( g )) e e ❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑ ∈ Gr ( T ET ( A ))which on the chosen basis B of F has the form: b φ ( b ) ∂ ( s ( b )) φ ( b ) 1 s ( b ) ; ; ✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇ s ( b ) / / φ ( b ) ∂ ( s ( b )) e e ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏
31y lemmas 41 and 37, and remark 42, it follows that s ( g ) ⊲ ′ ω ( s,s ) ( g ) (cid:0) ω ( s,s ) ( g ) (cid:1) − = 1, for each g ∈ F .We now define t : E ′ → L. For an e ∈ E ′ , put:¯ t ( e ) = (cid:0) ω ( s,s ) ( ∂ ( e )) (cid:1) − ( s∂ ( e )) ⊲ ′ t ( e ) − . (92) Lemma 44.
The pair (¯ s, ¯ t ) is an f ′ -quadratic derivation. We will use lemmas 4 and 20, similarly to the construction of the concatenation of homotopies.
Proof.
Consider the group map (lemma 37) M : F → ( G ⋉ ⊲ E ) ⋉ • ( { } ⋉ ∗ ( E ⋉ ⊲ ′ L )) = Gr ( T ( A )) with g M (cid:0) φ ( g ) , s ( g ) , , s ( g ) δ ( ω ( s,s ) ( g )) − , ω ( s,s ) ( g ) (cid:1) = (cid:0) φ ( g ) , s ( g ) , , ( s ( g )) − , ω ( s,s ) ( g ) (cid:1) , thus: g ∈ F M φ ( g ) ∂ ( s ( g )) ∂ (( s ( g )) − ) ω ( s,s ) ( g ) φ ( g ) s ( g )( s ( g )) − : : ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ s ( g ) / / φ ( g ) ∂ ( s ( g )) ( s ( g )) − δ ( ω ( s,s ) ( g )) O O = φ ( g ) ω ( s,s ) ( g )) φ ( g ) : : ✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈ s ( g ) / / φ ( g ) ∂ ( s ( g )) s ( g ) O O Consider the set map: N : E ′ → ( E ⋉ ∗ ( E ⋉ ⊲ ′ L )) ⋉ ∗ (( { } ⋉ ∗ ( E ⋉ ⊲ ′ L ) ⋉ • ′ ( { } ⋉ ad L )) = Gr ( T ( A )) with: e ∈ E ′ N (cid:16) ψ ( e ) , s ( ∂ ( e )) , t ( e ) , , ( s ( ∂ ( e ))) − , ω ( s,s ) ( ∂ ( e )) , , (cid:0) ω ( s,s ) ( ∂ ( e )) (cid:1) − ( s ( ∂ ( e )) ⊲ ′ t ( e ) − (cid:17) , thus: e ∈ E ′ N ψ ( e ) ψ ( e ) (1 , = = ④④④④④④④④④④④④④④④④④ (cid:0) s ( ∂ ( e ) ,t ( e ) (cid:1) / / ψ ( e ) s ( ∂ ( e )) δ ( t ( e )) (cid:0) s ( e ) , ( ω ( s,s ) ( ∂ ( e ))) − ( s ( ∂ ( e )) ⊲ ′ t ( e ) − (cid:1) e e ❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑ Let us see that N is a group morphism and also that ( N, M ) is a pre-crossed module map from ( ∂ : E ′ → F, ⊲ ) to the underlying pre-crossed module of T ( A ). By lemmas 4 and 20 it suffices to check that the diagrambelow in the category of sets commutes, which is straightforward (we note (91)): E ′ N / / ❴❴❴❴❴❴❴❴❴ ∂ r r ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ ❙❙❙❙❙❙ Gr ( H ′ ,H ) ) ) ❙❙❙❙❙❙❙❙❙❙ Gr ( T ( A )) Gr ( d −− ) (cid:15) (cid:15) β ′ r r ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ F M / / Gr ( H ′ ,H ) ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ Gr ( T ( A )) Gr ( d −− ) (cid:15) (cid:15) Gr (cid:16) P ∗ ( A ) Pr A × Pr A P ∗ ( A ) (cid:17) β ′ r r ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ Gr (cid:16) P ∗ ( A ) Pr A × Pr A P ∗ ( A ) (cid:17) H : A ′ → P ∗ ( A ) is associated to the quadratic f -derivation ( s, t ) and H ′ is associated with the trivialquadratic f -derivation ( s f , t f ); see lemma 27 and 3.3.3. We therefore have an associated map to the 2-crossedmodule pull-back: ( H ′ , H ) : A → P ∗ ( A ) Pr A × Pr A P ∗ ( A ).Thus ( N, M ) is a pre-crossed module map. Composing (
N, M ) with the pre-crossed module map d : T ( A ) → P ∗ ( A ) (see (45)), yields a pre-crossed module map from ( ∂ : E ′ → F, ⊲ ) to the underlyingpre-crossed module of P ∗ ( A ), which has the form: e ∈ E ′ (cid:16) ψ ( e ) s ( ∂ ( e )) δ ( t ( e )) , ( s ( ∂ ( e ))) − δ ( ω ( s,s ) ( ∂ ( e ))) , ω ( s,s ) ( ∂ ( e )) (cid:1) − ( s∂ ( e )) ⊲ ′ t ( e ) − (cid:17) = (cid:0) ψ ′ ( e ) , ¯ s ( ∂ ( e )) , t ( e ) (cid:1) ∈ Gr ( P ∗ ( A ))and g ∈ F (cid:0) φ ( g ) ∂ ( s ( g )) , s ( g ) (cid:1) = (cid:0) φ ′ ( g ) , s ( g ) (cid:1) ∈ Gr ( P ∗ ( A )) , hence (by lemma 27) it follows that (¯ s, ¯ t ) an f ′ -quadratic derivation.Now note that obviously s ⊗ s = s f and s ⊗ s = s f ′ , since the same is true in a free basis of F . On theother hand (by (91)), if e ∈ E ′ :( t ⊗ t )( e ) = ω ( s,s ) ( ∂ ( e )) (cid:0) s ( ∂ ( e )) − ⊲ ′ t ( e ) (cid:1) (cid:0) ω ( s,s ) ( ∂ ( e )) (cid:1) − s ( ∂ ( e )) ⊲ ′ t ( e ) − (cid:1) = ω ( s,s ) ( ∂ ( e )) (cid:0) ( δ ( ω ( s,s ) ( ∂ ( e ))) − s ( ∂ ( e )) (cid:1) ⊲ ′ t ( e ) (cid:0) ω ( s,s ) ( ∂ ( e )) (cid:1) − s ( ∂ ( e )) ⊲ ′ t ( e ) − = 1 , where we used the crossed module rule δ ( k ) ⊲ ′ l = klk − . Also (we use (91) again):( t ⊗ t )( e ) = ω ( s,s ) ( ∂ ( e )) s − ( ∂ ( e )) ⊲ ′ t ( ∂ ( e )) t ( e )= s ( ∂ ( e )) − ⊲ ′ ω ( s,s ) ( ∂ ( e )) ( s ( ∂ ( e ))) − ⊲ ′ (cid:16)(cid:0) ω ( s,s ) ( ∂ ( e )) (cid:1) − ( s ( ∂ ( e )) ⊲ ′ t ( e ) − (cid:17) t ( e ) = 1 . Thus we proved that ( s, t ) is an inverse of ( s, t ). We have finished proving the main result of this subsection:
Theorem 45.
Let A ′ = (cid:16) L ′ δ −→ E ′ ∂ −→ F, ⊲, { , } (cid:17) and A = (cid:16) L δ −→ E ∂ −→ G, ⊲, { , } (cid:17) be 2-crossed modules, where F is a free group, with a chosen basis B . We can define a groupoid [ A ′ , A ] B of 2-crossed module maps A ′ → A , and their homotopies. In the next subsection we will see that this construction can be expanded to be a 2-groupoid HOM B ( A ′ , A ),by considering 2-fold homotopies between 2-crossed module homotopies. Corollary 46.
Let A and A ′ be 2-crossed modules. If A ′ is free up to order one then homotopy between2-crossed module maps A ′ → A yields an equivalence relation.3.4. A 2-groupoid of 2-crossed module maps, 1- and 2-fold homotopies (in the free up to order one case) For the definition of a 2-groupoid see [31].
Let A ′ = (cid:16) L ′ δ −→ E ′ ∂ −→ G ′ , ⊲, { , } (cid:17) and A = (cid:16) L δ −→ E ∂ −→ G, ⊲, { , } (cid:17) be 2-crossed modules. (In this sub-subsection, only, we will not need to suppose that G ′ is a free group). Consider two 2-crossed module maps f, f ′ : A ′ → A . Let us define a groupoid [ f, f ′ ], with objects the homotopies ( f, s, t ) connecting f and f ′ (so the pair ( s, t ) is a quadratic f -derivation), the 2-morphisms being constructed from 2-fold homotopies(quadratic 2-derivations) k : G ′ → L . For nomenclature and notation we refer to subsection 3.2.The set of object of [ f, f ′ ] is the set of triples ( f, s, t ) where ( s, t ) is a quadratic f -derivation connecting f and f ′ . The set of 1-morphisms ( f, s, t ) → ( f, s ′ , t ′ ) is made out of quadruples ( f, s, t, k ), where k : G ′ → L is a quadratic ( f, s, t ) 2-derivation, such that ( f, s, t ) ( f,s,t,k ) −−−−−→ ( f, s ′ , t ′ ) . If we have a chain of arrows:( f, s, t ) ( f,s,t,k ) −−−−−→ ( f, s ′ , t ′ ) ( f,s ′ ,t ′ ,k ′ ) −−−−−−→ ( f, s ′′ , t ′′ ) , k ⋄ k ′ : G ′ → L , such that,for each g ∈ G ′ :( k ⋄ k ′ )( g ) = k ( g ) k ′ ( g ) . Lemma 47.
The map ( k ⋄ k ′ ) : G ′ → L is a quadratic ( f, s, t ) Proof.
By equations (79) and 81, and since ( δ : L → E, ⊲ ′ ) is a crossed module, we have, for each g, h ∈ G ′ :( k ⋄ k ′ )( gh ) . = k ( gh ) k ′ ( gh )= (cid:16) s ( h ) − ⊲ ′ (cid:0) φ ( h ) − ⊲ k ( g ) (cid:1)(cid:17) k ( h ) (cid:16) s ′ ( h ) − ⊲ ′ (cid:0) φ ( h ) − ⊲ k ′ ( g ) (cid:1)(cid:17) k ′ ( h )= (cid:16) s ( h ) − ⊲ ′ (cid:0) φ ( h ) − ⊲ k ( g ) (cid:1)(cid:17) k ( h ) (cid:16) ( δ ( k ( h )) − s ( h ) − ) ⊲ ′ (cid:0) φ ( h ) − ⊲ k ′ ( g ) (cid:1)(cid:17) k ′ ( h )= (cid:16) s ( h ) − ⊲ ′ (cid:0) φ ( h ) − ⊲ k ( g ) (cid:1)(cid:17) (cid:16) s ( h ) − ⊲ ′ (cid:0) φ ( h ) − ⊲ k ′ ( g ) (cid:1)(cid:17) k ( h ) k ′ ( h ) . = (cid:16) s ( h ) − ⊲ ′ (cid:0) φ ( h ) − ⊲ ( k ⋄ k ′ )( g ) (cid:1)(cid:17) ( k ⋄ k ′ )( h ) . Note that by equation (81), it follows that( f, s, t ) ( f,s,t,k ⋄ k ′ ) −−−−−−−→ ( f, s ′′ , t ′′ ) . This concatenation of quadratic ( f, s, t ) 2-derivations is clearly associative and it has units; the quadratic( f, s, t )-derivation such that k ( g ) = 1 L , ∀ g ∈ G ′ . The fact we have a groupoid [ f, f ′ ] follows from: Lemma 48. If ( f, s, t ) ( f,s,t,k ) −−−−−→ ( f, s ′ , t ′ ) , then the map k : G ′ → L such that k ( g ) = k ( g ) − for each g ∈ G ′ is a quadratic ( f, s ′ , t ′ ) Proof.
By equation (79), and since ( δ : L → E, ⊲ ′ ) is a crossed module, we have, for each g, h ∈ G ′ : k ( gh ) . = k ( gh ) − = (cid:16)(cid:0) s ( h ) − ⊲ ′ (cid:0) φ ( h ) − ⊲ k ( g ) (cid:1)(cid:1) k ( h ) (cid:17) − = (cid:16) k ( h ) (cid:0) ( δ ( k ( h )) − s ( h ) − ) ⊲ ′ (cid:0) φ ( h ) − ⊲ k ( g ) (cid:1)(cid:1)(cid:17) − = (cid:0) s ′ ( h ) − ⊲ ′ (cid:0) φ ( h ) − ⊲ k ( g ) (cid:1)(cid:1) k ( h ) . Given two 2-crossed modules A ′ and A we have a groupoid [ A ′ , A ] , whose objects are arbitrary 2-crossed module homotopies f ( f,s,t ) −−−−→ f ′ , where f, f ′ : A → A ′ , the morphisms being the 2-crossed module2-fold homotopies. Suppose that A ′ is free up to order one, with a chosen basis. To define a 2-groupoidHOM B ( A ′ , A ), we now need compatible left and right actions of the groupoid [ A ′ , A ] B on [ A ′ , A ] (seetheorem 45); in other words we need whiskering operators. Let A ′ = (cid:16) L ′ δ −→ E ′ ∂ −→ F, ⊲, { , } (cid:17) and A = (cid:16) L δ −→ E ∂ −→ G, ⊲, { , } (cid:17) be 2-crossed modules. We now go backto assuming F to be a free group over the chosen basis B . Let f, f ′ : A ′ → A be 2-crossed module maps.Suppose that we have two homotopies ( f, s, t ) and ( f, s ′ , t ′ ) connecting f and f ′ . Suppose that we have aquadratic ( f, s, t ) 2-derivation k : F → L connecting ( s, t ) and ( s ′ , t ′ ), thus ( f, s, t ) ( f,s,t,k ) −−−−−→ ( f, s ′ , t ′ ) . Sincethe latter will be a 2-morphism in HOM B ( A ′ , A ), we now represent it as: f ( f,s ′ ,t ′ ) ) ) ( f,s,t ) ⇑ ( f, s, t, k ) f ′ . f ′′ : A ′ → A be another 2-crossed module map. Suppose we also have a homotopy: f ′ = ( µ ′ , ψ ′ , φ ′ ) ( f ′ ,s ′′ ,t ′′ ) −−−−−−→ f ′′ = ( µ ′′ , ψ ′′ , φ ′′ ) , so what we have diagrammatically is: f ( f,s ′ ,t ′ ) ) ) ( f,s,t ) ⇑ ( f, s, t, k ) f ′ ( f ′ ,s ′′ ,t ′′ ) / / f ′′ . Let us define the whiskering: ( f, s, t, k ) ⊗ ( f ′ , s ′′ , t ′′ ) = ( f, s ⊗ s ′′ , t ⊗ t ′′ , k ⊗ s ′′ ) , such that k ⊗ s ′′ connects ( s ⊗ s ′′ , t ⊗ t ′′ ) and ( s ′ ⊗ s ′′ , t ′ ⊗ t ′′ ); diagrammatically: f ( f,s ′ ⊗ s ′′ ,t ′ ⊗ t ′′ ) + + ( f,s ⊗ s ′′ ,t ⊗ t ′′ ) ⇑ ( f, s, t, k ) ⊗ ( f ′ , s ′′ , t ′′ ) f ′′ . By definition (see corolary 32), k ⊗ s ′′ is the unique quadratic ( f, s ⊗ s ′′ , t ′ ⊗ t ′′ ) 2-derivation F → L , which onthe chosen basis B of F has the form: ( k ⊗ s ′′ )( b ) = s ′′ ( b ) − ⊲ ′ k ( b ) . Then we have for each g ∈ F : ( s ⊗ s ′′ )( g ) δ (( k ⊗ s ′′ )( g )) = ( s ′ ⊗ s ′′ )( g ); (93)c.f equation (81). This is because (remark 24), on the free generators b ∈ B ⊂ F , we have:( s ⊗ s ′′ )( b ) δ (( k ⊗ s ′′ )( b )) = s ( b ) s ′′ ( b ) δ ( s ′′ ( b ) − ⊲ ′ k ( b )) = s ( b ) δ ( k ( b )) s ′′ ( b ) = s ′ ( b ) s ′′ ( b ) = ( s ′ ⊗ s ′′ )( b ) . Lemma 49.
The following holds for each e ∈ E ′ (c.f. equation (81) ): ( k ⊗ s ′′ )( ∂ ( e )) − ( t ⊗ t ′′ )( e ) = ( t ′ ⊗ t ′′ )( e ) . (94) Proof.
We freely use the notation introduced in 2.4.1, 2.4.2 and 2.4.3. We have, for each e ∈ E ′ :( k ⊗ s ′′ )( ∂ ( e )) − ( t ⊗ t ′′ )( e ) = ( k ⊗ s ′′ )( ∂ ( e )) − ω ( s,s ′′ ) ( ∂ ( e )) s ′′ ( ∂ ( e )) − ⊲ ′ t ( e ) t ′′ ( e ) , whereas (putting s δ ( k ) : F → L as being the derivation g ∈ F s ( g ) δ ( k ( g ) ∈ L ):( t ′ ⊗ t ′′ )( e ) = ω ( s ′ ,s ′′ ) ( ∂ ( e )) s ′′ ( ∂ ( e )) − ⊲ ′ t ′ ( e ) t ′′ ( e )= ω ( s δ ( k ) ,s ′′ ) ( ∂ ( e )) s ′′ ( ∂ ( e )) − ⊲ ′ (( k ◦ ∂ ( e )) − t ( e )) t ′′ ( e ) . Therefore (94) is equivalent to:( k ⊗ s ′′ )( ∂ ( e )) − ω ( s,s ′′ ) ( ∂ ( e )) = ω ( sδ ( k ) ,s ′′ ) ( ∂ ( e )) s ′′ ( ∂ ( e )) − ⊲ ′ (( k ◦ ∂ ( e )) − , for each e ∈ E ′ . Let us then prove that for any g ∈ F we have:( k ⊗ s ′′ )( g ) − ω ( s,s ′′ ) ( g ) = ω ( s ′ ,s ′′ ) ( g ) s ′′ ( g ) − ⊲ ′ k ( g ) − . (95)35he technique of proof is entirely analogous to the proof that the concatenation of homotopies is associative.Consider the unique map K : F → Gr ( T ET ( A )), which on the free basis B of F is: K ( b ) = ( φ ( b ) , s ( b ) , , s ′′ ( b ) , , , , , , δ ( s ′′ ( b ) − ⊲ ′ k ( b )) , s ′′ ( b ) − ⊲ ′ k ( b ) − , , , that is, for each b ∈ B : K ( b ) = φ ( b ) ∂ ( s ( b )) ∂ ( s ′′ ( b )) φ ( b ) ∂ ( s ( b )) ∂ ( s ′′ ( b )) O O φ ( b ) s ( b ) s ′′ ( b ) δ ( s ′′ ( b ) − ⊲ ′ k ( b )) : : ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ s ( b ) s ′′ ( b ) ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ s ( b ) / / s ′′ ( b ) − ⊲ ′ k ( b ) − ) φ ( b ) ∂ ( s ( b )) s ′′ ( b ) k k ❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ s ′′ ( b ) e e ❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑ K ( g ) = φ ( g ) ∂ ( s ( g )) ∂ ( s ′′ ( g )) φ ( g ) ∂ ( s ( g )) ∂ ( s ′′ ( g )) O O φ ( g ) ( s ⊗ s ′′ )( g ) δ ( k ⊗ s ′′ )( g ) : : ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ( s ⊗ s ′′ )( g ) ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ s ( g ) / / ω ( s,s ′′ ) ( g )( k ⊗ s ′′ )( g ) − φ ( g ) ∂ ( s ( g )) s ′′ ( g ) k k ❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ s ′′ ( g ) e e ❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑ g ∈ G . Consider the unique map K ′ : F → Gr ( T ET ( A )), which on the free basis B of F is: K ′ ( b ) = ( φ ( b ) , s ( b ) , , δ ( k ( b )) , k ( b ) − , , , , , s ′′ ( b ) , , , , that is, if b ∈ B : K ′ ( b ) = φ ( b ) ∂ ( s ( b )) ∂ ( s ′′ ( b )) φ ( b ) ∂ ( s ( b )) ∂ ( δ ( k ( b ))) s ′′ ( b ) O O φ ( b ) s ( b ) δ ( k ( b )) s ′′ ( b ) > > ⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦ s ( b ) δ ( k ( b )) ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ s ( b ) / / k ( b ) − φ ( b ) ∂ ( s ( b )) j j ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ s ′′ ( b ) b b ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ g ∈ F : K ′ ( g ) = φ ( g ) ∂ ( s ( g )) ∂ ( s ′′ ( g )) φ ( g ) ∂ ( s ( g )) s ′′ ( g ) O O φ ( g ) s ( g ) δ ( k ( g )) s ′′ ( g ) δ ( ω ( s ′ ,s ′′ ) ( g )) − ? ? ⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦ s ( g ) δ ( k ( g )) ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ s ( g ) / / k ( g ) − ω ( s ′ ,s ′′ ) ( g ) φ ( g ) ∂ ( s ( g )) j j ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ s ′′ ( g ) a a ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ K and K ′ with the group morphism d : T ET ( A ) → T ( A ), of (67) and (69), yields twogroup morphisms F → Gr ( T ( A )), namely: g ∈ F φ ( g ) ∂ ( s ( g )) ∂ ( s ′′ ( g )) φ ( g ) ( k ⊗ s ′′ )( g ) − ω ( s,s ′′ ) ( g )) s ( g ) s ′′ ( g ) δ ( ω ( s,s ′′ ) )( g ) − δ ( k ⊗ s ′′ )( g ) sssssssssssssssssssss s ( g ) / / φ ( g ) ∂ ( s ( g )) s ′′ ( g ) f f ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ and g ∈ F φ ( g ) ∂ ( s ( g )) ∂ ( s ′′ ( g )) φ ( g ) ω ( s ′ ,s ′′ ) ( g ) s ′′ ( g ) − ⊲ ′ k ( g ) − s ( g ) δ ( k ( g )) s ′′ ( g ) δ ( ω ( s ′ ,s ′′ ) ( g )) − sssssssssssssssssssss s ( g ) / / φ ( g ) ∂ ( s ( g )) δ ( k ( g )) s ′′ ( g ) δ (cid:0) s ′′ ( g ) − ⊲ ′ k ( g ) − (cid:1) f f ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ (It is an instructive exercise to check that the last two equations are coherent with (93) and (85).) Sincethese morphisms agree on the chosen basis B of F , they coincide, thus equation (95) follows.Therefore, by (93) and (94), we have that (by equation (81)):( f, s ⊗ s ′′ , t ⊗ t ′′ ) ( f,s ⊗ s ′′ ,t ⊗ t ′′ ,k ⊗ s ′′ ) −−−−−−−−−−−−−→ ( f, s ′ ⊗ s ′′ , t ′ ⊗ t ′′ ) . Lemma 50 (Functoriality of the right whiskering).
Let f, f ′ , f ′′ : A ′ → A be 2-crossed module maps.Suppose that we are given homotopies ( f, s, t ) , ( f, s ′ , t ′ ) and ( f, s ′′ , t ′′ ) , connecting f and f ′ , as well as 2-fold homotopies ( f, s, t ) ( f,s,t,k ) −−−−−→ ( f, s ′ , t ′ ) and ( f, s ′ , t ′ ) ( f,s ′ ,t ′ ,k ′ ) −−−−−−→ ( f, s ′′ , t ′′ ) . Suppose we are also given ahomotopy f ′ ( f ′ ,u,v ) −−−−−→ f ′′ . Diagrammatically we have: f ( f,s ′ ,t ′ ) / / ( f,s,t ) G G ( f,s ′′ ,t ′′ ) (cid:23) (cid:23) ⇑ ( f,s,t,k ) ⇑ ( f,s ′ ,t ′ ,k ′ ) f ′ ( f ′ ,u,v ) / / f ′′ . hen: (cid:0) ( f, s, t, k ) ⋄ ( f, s ′ , t ′ , k ′ ) (cid:1) ⊗ ( f ′ , u, v ) = (cid:0) ( f, s, t, k ) ⊗ ( f ′ , u, v ) (cid:1) ⋄ (cid:0) ( f, s ′ , t ′ , k ′ ) ⊗ ( f ′ , u, v ) (cid:1) . Proof.
In the left-hand-side, since ( f, s, t, k ) ⋄ ( f, s ′ , t ′ , k ′ ) = ( f, s, t, kk ′ ), the underlying quadratic ( f, s ⊗ u, t ⊗ v ) 2-derivation ( kk ′ ) ⊗ u is the unique quadratic ( f, s ⊗ u, t ⊗ v ) 2-derivation F → L which on the chosenbasis B of F is b u ( b ) − ⊲ ′ (cid:0) k ( b ) k ′ ( b ) (cid:1) . On the right hand side we have the quadratic ( f, s ⊗ u, t ⊗ v )2-derivation F → L , which is the product of k ⊗ u and k ′ ⊗ u . On the chosen basis B of F it takes the form: b (cid:0) u ( b ) − ⊲ ′ k ( b ) (cid:1) (cid:0) u ( b ) − ⊲ ′ k ′ ( b ) (cid:1) = u ( b ) − ⊲ ′ (cid:0) k ( b ) k ′ ( b ) (cid:1) . Therefore: ( kk ′ ) ⊗ u = ( k ⊗ u ) ( k ′ ⊗ u ) , since the same is true in a free basis of F .Analogously: Lemma 51.
Suppose we have a 2-fold ( f, s, t ) homotopy ( f, s, t ) ( f,s,t,k ) −−−−−→ ( f, s ′ , t ′ ) . Consider also a chainof homotopies: f ′ ( f ′ ,u,v ) −−−−−→ f ′′ ( f ′′ ,u ′ ,v ′ ) −−−−−−→ f ′′′ ; diagrammatically: f ( f,s ′ ,t ′ ) ) ) ( f,s,t ) ⇑ ( f, s, t, k ) f ′ ( f ′ ,u,v ) / / f ′′ ( f ′ ,u ′ ,v ′ ) / / f ′′′ . Then: ( f, s, t, k ) ⊗ ( u ⊗ u ′ , v ⊗ v ′ ) = (cid:0) ( f, s, t, k ) ⊗ ( u, v ) (cid:1) ⊗ ( u ′ , v ′ ) . Proof. In B , the underlying quadratic ( f, s ⊗ u ⊗ u ′ , t ⊗ v ⊗ v ′ ) 2-derivation in the left-hand-side is: b ( u ⊗ u ′ )( b ) − ⊲ ′ k ( b ) = ( u ( b ) u ′ ( b )) − ⊲ ′ k ( b ) , whereas the underlying quadratic ( f, s ⊗ u ⊗ u ′ , t ⊗ v ⊗ v ′ ) 2-derivation on the right-hand-side restricts to: b u ′ ( b ) − ⊲ ′ ( u ( b ) − ⊲ ′ k ( b )) . Now apply corollary 32.Therefore
Proposition 52.
Let A ′ = (cid:16) L ′ δ −→ E ′ ∂ −→ F, ⊲, { , } (cid:17) and A = (cid:16) L δ −→ E ∂ −→ G, ⊲, { , } (cid:17) be 2-crossed modules,where F is a free group over the chosen basis B . Whiskering on the right gives a right action (by groupoidmorphisms) of the groupoid [ A ′ , A ] B of maps A → A ′ and their homotopies, on the groupoid [ A ′ , A ] of2-crossed module homotopies and their 2-fold homotopies.3.4.3. Left whiskering 2-fold homotopies by 1-fold homotopies (in the free up to order one case) We freely use the notation introduced in 2.4.1, 2.4.2 and 2.4.3. The discussion is very similar to the onein 3.4.2. Let A ′ = (cid:16) L ′ δ −→ E ′ ∂ −→ F, ⊲, { , } (cid:17) and A = (cid:16) L δ −→ E ∂ −→ G, ⊲, { , } (cid:17) be 2-crossed modules, where F isa free group over the chosen basis B . If we have 2-crossed module maps f, f ′ : A ′ → A , homotopies ( f, s, t )and ( f, s ′ , t ′ ) and a 2-fold homotopy ( f, s, t, k ), all fitting into the diagram: f ( f,s ′ ,t ′ ) ) ) ( f,s,t ) ⇑ ( f, s, t, k ) f ′ , f ′′ = ( µ ′′ , ψ ′′ , φ ′′ ) ( f ′′ ,s ′′ ,t ′′ ) −−−−−−−→ f = ( µ, ψ, φ ) , so what we have is: f ′′ ( f ′′ ,s ′′ ,t ′′ ) / / f ( f,s ′ ,t ′ ) ) ) ( f,s,t ) ⇑ ( f, s, t, k ) f ′ , let us define the whiskering: ( f ′ , s ′′ , t ′′ ) ⊗ ( f, s, t, k ) = ( f, s ′′ ⊗ s, t ′′ ⊗ t, s ′′ ⊗ k ) , such that we have: f ′′ ( f ′′ ,s ′′ ⊗ s ′ ,t ′′ ⊗ t ′′ ) + + ( f ′′ ,s ′′ ⊗ s,t ′′ ⊗ t ) ⇑ ( f ′′ , s ′′ , t ′′ ) ⊗ ( f, s, t, k ) f ′ . We put s ′′ ⊗ k as being unique quadratic ( f ′′ , s ′′ ⊗ s, t ′′ ⊗ t ) 2-derivation which on the basis B of F is:( s ′′ ⊗ k )( b ) = k ( b ) . Then we have for each g ∈ F : ( s ′′ ⊗ s )( g ) δ (( s ′′ ⊗ k )( g )) = ( s ′ ⊗ s ′′ )( g ); (98)c.f. equation (81). This is because on the free generators b ∈ F we have:( s ′′ ⊗ s )( b ) δ (( s ′′ ⊗ k )( b )) = s ′′ ( b ) s ( b ) δ ( k ( b )) = s ′′ ( b ) s ′ ( b ) = ( s ′′ ⊗ s ′ )( b ) . Lemma 53.
We have, for each e ∈ E ′ (c.f. equation (81) ): ( s ′′ ⊗ k )( ∂ ( e )) − ( t ′′ ⊗ t )( e ) = ( t ′′ ⊗ t ′ )( e ) . (99) Proof.
Equation 99 follows if we prove that for each g ∈ F we have:( s ′′ ⊗ k )( g ) − ω ( s ′′ ,s ) ( g ) = ω ( s ′′ ,s ′ ) ( g ) k ( g ) − . (100)Consider the unique map: K : F → Gr ( T ET ( A )) , which on the free basis B of F is: K ( b ) = ( φ ′′ ( b ) , s ′′ ( b ) , , s ( b ) , , , , , , δ ( k ( b )) , , , k ( b ) − ) , or K ( b ) = φ ′′ ( b ) ∂ ( s ′′ ( b )) ∂ ( s ( b )) φ ′′ ( b ) ∂ ( s ′′ ( b )) ∂ ( s ( b )) O O φ ′′ ( b ) s ′′ ( b ) s ( b ) δ ( k ( b )) > > ⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥ s ′′ ( b ) s ( b ) ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ s ′′ ( b ) / / k ( b ) − φ ′′ ( b ) ∂ ( s ′′ ( b )) s ( b ) j j ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ s ( b ) δ ( k ( b )) b b ❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊ k ( b ) − g ∈ F (by lemma 37 and remarks 42, 36 and 24): g ∈ F K φ ′′ ( g ) ∂ ( s ′′ ( g )) ∂ ( s ( g )) φ ′′ ( g ) ∂ ( s ′′ ( g )) ∂ ( s ( g )) O O φ ′′ ( g ) s ′′ ( g ) s ( g ) δ ( ω ( s ′′ ,s ) )( g ) − δ ( s ′′ ⊗ k ( g )) rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr ( s ′′ ⊗ s )( g ) ❢❢❢❢❢❢❢❢❢❢❢❢❢ ❢❢❢❢❢❢❢❢❢ s ′′ ( g ) / / ω ( s ′′ ,s ) ( g ) (cid:0) ( s ′′ ⊗ k )( g ) (cid:1) − φ ′′ ( g ) ∂ ( s ′′ ( g )) s ( g ) ❨❨❨❨❨❨❨❨❨❨❨❨❨ l l ❨❨❨❨❨❨❨❨❨❨❨❨ s ( g ) k ( g ) g g ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ k ( g ) − By composing K : F → Gr ( T ET ( A ) with the 2-crossed module morphism d : T ET ( A ) → T ( A ) in (67)yields a group morphism F → Gr ( T ( A )) with the form: g ∈ F d ◦ K φ ′′ ( g ) ∂ ( s ′′ ( g )) ∂ ( s ( g )) φ ′′ ( g ) ( s ′′ ⊗ k ( g )) − ω ( s ′′ ,s ) ( g ) k ( g ) s ′′ ( g ) s ( g ) δ ( ω ( s ′′ ,s ) )( g ) − δ ( s ′′ ⊗ k ( g )) rrrrrrrrrrrrrrrrrrrrrr s ′′ ( g ) / / φ ′′ ( g ) ∂ ( s ′′ ( g )) s ( g ) k ( g ) g g ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ By lemma 37, we have another group morphism F → Gr ( T ( A )) with the form: g φ ′′ ( g ) ∂ ( s ′′ ( g )) ∂ ( s ( g )) φ ′′ ( g ) ω ( s ′′ ,s ′ ) ( g ) s ′′ ( g ) s ′ ( g ) δ ( ω ( s ′′ ,s ′ ) )( g ) − rrrrrrrrrrrrrrrrrrrrrr s ′′ ( g ) / / φ ′′ ( g ) ∂ ( s ′′ ( g )) s ′ ( g ) g g ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ Since these morphisms agree on a basis of F , they coincide, from which (100), thus (99), follows.From equation (81), by (98) and (99) we therefore have:( f ′′ , s ′′ ⊗ s, t ′′ ⊗ t ) ( f ′′ ,s ′′ ⊗ s,t ′′ ⊗ t,s ′′ ⊗ k ) −−−−−−−−−−−−−−→ ( f, s ′′ ⊗ s ′ , t ′′ ⊗ t ′ ) . (We have used the crossed module rule δ ( k ) ⊲ ′ l = klk − for all k, l ∈ L .) As before we can easily prove that Proposition 54.
Whiskering on the left gives a left action (by groupoid morphisms) of the groupoid [ A ′ , A ] B of maps A → A ′ and their homotopies on the groupoid [ A ′ , A ] of 2-crossed module homotopies and their2-fold homotopies. Proposition 55.
Whiskering on the right commutes with wiskering on the left. If A ′ is free up to order one, with a chosen basis, we have therefore constructed a sesquigroupoid[46] HOM B ( A ′ , A ) with objects being the 2-crossed module maps f : A ′ → A , and the morphisms and 2-morphisms being homotopies and 2-fold homotopies. To prove that HOM B ( A ′ , A ) is a 2- groupoid we nowneed to prove that it satisfies the interchange law. 40 .4.4. The interchange law Let A ′ = (cid:16) L ′ δ −→ E ′ ∂ −→ F, ⊲, { , } (cid:17) and A = (cid:16) L δ −→ E ∂ −→ G, ⊲, { , } (cid:17) be 2-crossed modules, where F is afree group over the chosen basis B . Suppose we have the following diagram of 2-crossed module maps f, f ′ , f ′′ : A ′ → A ′ , homotopies and 2-fold homotopies: f ( f,s ′ ,t ′ ) ) ) ( f,s,t ) ⇑ ( f, s, t, k ) f ′ ( f ′ ,u ′ ,v ′ ) * * ( f ′ ,u,v ) ⇑ ( f, u, v, k ′ ) f ′′ . Let us prove the interchange law: (cid:0) ( f, s, t, k ) ⊗ ( f ′ , u, v ) (cid:1) ⋄ (cid:0) ( f, s ′ , t ′ ) ⊗ ( f, u, v, k ′ ) (cid:1) = (cid:0) ( f, s, t ) ⊗ ( f, u, v, k ′ ) (cid:1) ⋄ (cid:0) ( f, s, t, k ) ⊗ ( f ′ , u ′ , v ′ ) (cid:1) . In the left hand side we have the quadratic ( f, s ⊗ u, t ⊗ v ) 2-derivation, which on the basis B of F is: b ( u − ( b ) ⊲ ′ k ( b )) k ′ ( b ) . In the right hand side we have the ( f, s ⊗ u, t ⊗ v ) 2-derivation, which on the chosen basis B of F is: b k ′ ( b ) u ′ ( b ) − ⊲ ′ k ( b ) = k ′ ( b ) ( δ ( k ′ ( b ) − ) u ( b ) − ) ⊲ ′ k ( b ) = ( u − ( b ) ⊲ ′ k ( b )) k ′ ( b ) , where we have used (81) and the crossed module rules, recalling that ( δ : L → E, ⊲ ′ ) is a crossed module.We therefore proved that: Theorem 56 (Mapping space 2-groupoid).
Given two 2-crossed modules A = (cid:16) L δ −→ E ∂ −→ G, ⊲, { , } (cid:17) and A ′ = (cid:16) L ′ δ −→ E ′ ∂ −→ F, ⊲, { , } (cid:17) , with F free up to order 1, with a chosen basis B of F , there exists a2-groupoid: HOM B ( A ′ , A ) of 2-crossed module maps, 1-fold homotopies between 2-crossed module maps, and 2-fold homotopies between1-fold homotopies. We note that HOM B ( A ′ , A ) explicitly depends on the chosen basis B of F .
4. Lax homotopy of 2-crossed modules Q ( A ) and lax homotopy Consider a 2-crossed module of groups A = (cid:16) L δ −→ E ∂ −→ G, ⊲, { , } (cid:17) . We will consider a very naturalpartial resolution Q ( A ) of it, which is free up to order one, with a chosen basis, together with a surjectiveprojection proj: Q ( A ) → A , defining isomorphisms at the level of 2-crossed module homotopy groups. Itis proven in [26] that Q (clearly functorial by its construction) is a part of a comonad. We will then use Q ( A ) to define lax homotopy of (strict) 2-crossed module maps. Q ( A ) and abstract definition of Q -lax homotopy Let G be a group. The free group on the underlying set of G is denoted by F group ( G ). The inclusion(set) map G → F group ( G ) is denoted by g ∈ G [ g ] ∈ F group ( G ). The projection (group) map sending[ g ] ∈ F group ( G ) to g ∈ G is denoted by p : F group ( G ) → G . Note that we do not take [1], where 1 is theidentity of G , to be the identity of F group ( G ), the latter being the empty word, denoted by ∅ .For a 2-crossed module A = (cid:16) L δ −→ E ∂ −→ G, ⊲, { , } (cid:17) , we put: Q ( A ) = (cid:16) L δ ′ −→ E ∂ × p F group ( G ) ∂ ′ −→ F group ( G ) , ⊲, { , } (cid:17) , E ∂ × p F group ( G ) = { ( e, u ) ∈ E × F group ( G ) : ∂ ( e ) = p ( u ) } . Moreover ∂ ′ ( e, u ) = u , and, for all u ∈ F group ( G ), we put u ⊲ ( e, u ′ ) = ( p ( u ) ⊲ e, uu ′ u − ). Clearly the Peifferpairing is h ( e, u ) , ( e ′ , u ′ ) i = ( h e, e ′ i , ∅ ). We also put δ ′ ( k ) = ( δ ( k ) , ∅ ) and also u ⊲ k = p ( u ) ⊲ k , for all k ∈ L and u ∈ F group ( G ). It is immediate that with the Peiffer lifting: { ( e, u ) , ( e ′ , u ′ ) } = { e, e ′ } , this defines a 2-crossed module of groups (a very similar construction appears in [1]). Also, Q ( A ) is freeup to order one, and we choose the free basis [ G ] = { [ g ] , g ∈ G } of F group ( G ).There is a projection proj = ( r, q, p ) : Q ( A ) → A , which, rather clearly, yields isomorphisms at the levelof 2-crossed module homotopy groups (the homology groups of the underlying complexes). It has the form:proj = L δ ′ / / r (cid:15) (cid:15) E ∂ × p F group ( G ) ∂ ′ / / q (cid:15) (cid:15) F group ( G ) p (cid:15) (cid:15) L δ / / E ∂ / / G (101)where r = id and q ( e, u ) = e , for each ( e, u ) ∈ E ∂ × p F group ( G ).Consider 2-crossed modules A = (cid:16) L δ −→ E ∂ −→ G, ⊲, { , } (cid:17) and A ′ = (cid:16) L ′ δ −→ E ′ ∂ −→ G ′ , ⊲, { , } (cid:17) . If we have a2-crossed module morphism f : A → A ′ then f ◦ proj is a 2-crossed module morphism Q ( A ) → A ′ . Thisyields an injective map proj ∗ : hom( A , A ′ ) → hom( Q ( A ) , A ′ ), since proj : Q ( A ) → A is surjective. Herehom( A , A ′ ) denotes the set of 2-crossed module maps A → A ′ . Definition 57 (Lax mapping space).
A morphism Q ( A ) → A ′ is said to be a strict map A → A ′ if itfactors (uniquely) through proj : Q ( A ) → A . The lax mapping space 2-groupoid: HOM
LAX ( A , A ′ ) is the full sub-2-groupoid of HOM [ G ] ( Q ( A ) , A ′ ) , theorem 56, with objects the strict maps f : A → A ′ , eachuniquely identified with f ◦ proj: Q ( A ) → A ′ . The objects of HOM
LAX ( A , A ′ ) are therefore in one-to-onecorrespondence with 2-crossed module maps A → A ′ , and we call the 1- and 2-morphisms of HOM
LAX ( A , A ′ ) “lax homotopies” and “lax 2-fold homotopies”.4.1.2. The structure of Q ( A )We now want to completely unpack the definition of HOM
LAX ( A , A ′ ), definition 57. To unravel thestructure of Q ( A ), we prove an auxiliary lemma, describing the kernel of the projection map proj: Q ( A ) →A in (101). To characterize this kernel it suffices to elucidate the kernel ker( p ) of the obvious projection p : F group ( G ) → G . Then the kernel of the projection map proj: Q ( A ) → A is the 2-crossed module:ker(proj) = { → { E } × ker( p ) → ker( p ) } , with action by conjugation and trivial Peiffer lifting.For details on the definition of free crossed modules and free pre-crossed modules (possibly with ulteriorrelations), we refer to [21, 9]. Let G be a group. Given g, h ∈ G put:[ g, h ] = [ gh ] − [ g ][ h ] ∈ ker( p ) ⊂ F group ( G ) . (102)Note that we always have: [ gh, i ] [ i ] − [ g, h ] [ i ] = [ g, hi ] [ h, i ] , where g, h, i ∈ G. (103)42lso if g, h ∈ G :[ g ] [ h ] [ g, h ] − = [ gh ] , [1] = [1 , , [ g − ] = [ g ] − [1] [ g, g − ] . (104)Moreover: [ ghg − ] = [ g ] [ hg − ] [ g, hg − ] − = [ g ] [ h ] [ g − ] [ h, g − ] − [ g, hg − ] − = [ g ] [ h ] [ g ] − [1] [ g, g − ] [ h, g − ] − [ g, hg − ] − . (105) Lemma 58.
The inclusion map ι : ker( p ) → F group ( G ) , together with the action ⊲ of F group ( G ) on ker( p ) ⊂F group ( G ) by conjugation, is isomorphic to the crossed module, over F group ( G ) , formally generated by theelements ( g, h ) , where g, h ∈ G , with: ι ( g, h ) = [ g, h ] , for each g, h ∈ G, modulo the relations: ( gh, i ) [ i ] − ⊲ ( g, h ) = ( g, hi ) ( h, i ) , where g, h, i ∈ G. (106) In particular, ker( p ) → F group ( G ) is isomorphic to the pre-crossed module, formally generated by the symbols ( g, h ) , for all g, h ∈ G , with ι ( g, h ) = [ g, h ] , modulo the relations (106) , as well as the following relations(where g, g ′ , h, h ′ ∈ G and k, k ′ ∈ F group ( G ) ), enforcing the second Peiffer condition in definition 1: (cid:0) ι ( k ⊲ ( g, h )) (cid:1) ⊲ (cid:0) k ′ ⊲ ( g ′ , h ′ ) (cid:1) = (cid:0) k ⊲ ( g, h ) (cid:1) (cid:0) k ′ ⊲ ( g ′ , h ′ ) (cid:1) (cid:0) k ⊲ ( g, h ) (cid:1) − . (It suffices to consider the case k = ∅ .)Also, we have that ker( p ) ⊂ F group ( G ) , as a group, is generated by all conjugates (under F group ( G ) ) ofelements [ g, h ] ∈ F group ( G ) , where g, h ∈ G , and the relations (106) are the only relations between these. We will give a topological proof of this lemma. Recall [9] that, if (
X, Y ) is a pair of path-connected spaces,then the boundary map π ( X, Y ) → π ( Y ), together with the standard action of π on π , defines a crossedmodule Π ( X, Y ), a result due to Whitehead. If X is obtained from Y by attaching 2-cells, then Π ( X, Y )is the free crossed module on the attaching maps of the 2-cells of X in π ( Y ), a fact usually known asWhitehead theorem [48, 49, 50]. If X is a CW-complex, then X i denotes the i -skeleton of X . Proof.
Let K be the simplicial set which is the nerve of G , thus the geometric realisation of K is the usualclassifying space of G ; see for example [9]. We thus have a unique 0-simplex, the 1-simplices of K are inone-to-one correspondence with elements of G , and we denote these by [ g ], where g ∈ G . The 2-simplices of K are in one-to-one correspondence with pairs ( g, h ) , where g, h ∈ G , being: ∂ ( g, h ) = [ h ] ∂ ( g, h ) = [ gh ] ∂ ( g, h ) = [ g ] . The 3-simplices of K are triples ( g, h, i ) of elements of G , being: ∂ ( g, h, i ) = ( h, i ) ∂ ( g, h, i ) = ( gh, i ) ∂ ( g, h, i ) = ( g, hi ) ∂ ( g, h, i ) = ( g, h ) . Let us consider the fat geometric realization X of K (forgetting about the degeneracy maps of K , thuslooking at K merely as being a △ -complex, [32].) It is well known that the fat and standard geometricrealisations of a simplicial set are homotopy equivalent (see for example [10]), thus X is an asphericalCW-complex with π ( X ) ∼ = G .Consider the exact sequence { } → π ( X, X ) → π ( X ) → π ( X ) ∼ = G . These are the final groups of thelong homotopy exact sequence of the pair ( X, X ), where X is the 1-skeleton of X . Then π ( X ) → π ( X )is exactly the map p : F group ( G ) → G , so we only need to determine π ( X, X ). The crossed module( π ( X, X ) → π ( X )) = Π ( X, X ) is a quotient of the crossed module ( π ( X , X ) → π ( X )) =Π ( X , X ), which, by Whitehead theorem, is the free crossed module on the boundary maps of the 2-cells of X in π ( X ). Thus π ( X , X ) is the principal group of the pre-crossed module, over π ( X ),43enerated by all pairs ( g, h ) where g, h ∈ G , with ι ( g, h ) = [ g, h ], modulo the relations (for g, g ′ , h, h ′ ∈ G and k, k ′ ∈ F group ( G )), enforcing the second Peiffer condition in definition 1: (cid:0) ι ( k ⊲ ( g, h )) (cid:1) ⊲ (cid:0) k ′ ⊲ ( g ′ , h ′ ) (cid:1) = (cid:0) k ⊲ ( g, h ) (cid:1) (cid:0) k ′ ⊲ ( g ′ , h ′ ) (cid:1) (cid:0) k ⊲ ( g, h ) (cid:1) − . To obtain π ( X, X ) from π ( X , X ), we now need to add one extra relation for each 3-cell of X (see [21]),yielding that we should have:( gh, i ) [ i ] − ⊲ ( g, h ) = ( g, hi ) ( h, i ) , where g, h, i ∈ G, which arise from all 3-simplices of K . We use the notation of subsection 3.1 and 4.1.1. Consider 2-crossed modules A = (cid:16) L δ −→ E ∂ −→ G, ⊲, { , } (cid:17) and A ′ = (cid:16) L ′ δ −→ E ′ ∂ −→ G ′ , ⊲, { , } (cid:17) . Recall the construction of HOM
LAX ( A , A ′ ), definition 57, from Q ( A ) = (cid:16) L δ ′ −→ E ∂ × p F group ( G ) ∂ ′ −→ F group ( G ) , ⊲, { , } (cid:17) . Let f = ( µ , ψ , φ ) and f = ( µ , ψ , φ ) be 2-crossedmodule maps A → A ′ . Consider a lax homotopy connecting f and f , namely:( µ ′ , ψ ′ , φ ′ ) = f ′ . = ( f ◦ proj) ( f ◦ proj ,s,t ) −−−−−−−−→ ( f ◦ proj) . = f ′ = ( µ ′ , ψ ′ , φ ′ ) , thus ( s, t ) is a quadratic f ′ -derivation connecting f ′ = ( f ◦ proj) and f ′ = ( f ◦ proj). Therefore φ ′ ([ g ]) = φ ( g ) and φ ′ ([ g ]) = φ ( g ), for each g ∈ G . Also φ ′ ([ g, h ]) = φ ′ ([ g, h ]) = 1 G and ψ ′ (1 , [ g, h ]) = ψ ′ (1 , [ g, h ]) =1 E , for each g, h ∈ G . Following the notation of lemma 58, let us put ( g, h ) = (1 , [ g, h ]) ∈ E ∂ × p F group ( G ),thus ∂ ′ ( g, h ) = [ g, h ] = [ gh ] − [ g ][ h ].Let us look at the associated group map (lemma 27):( φ ′ , s ) : F group ( G ) → G ′ ⋉ ⊲ E ′ . Note the following equation, which will be used several times (we use (70) and φ ′ ([ gh ]) = φ ( gh )):( s ◦ ∂ ′ )( g, h ) = s ([ gh ] − [ g ][ h ])) = φ ( gh ) − ⊲ s ([ gh ] − ) φ ( h ) − ⊲ s ([ g ]) s ([ h ])= s ([ gh ]) − φ ( h ) − ⊲ s ([ g ]) s ([ h ]) . (107)The fact that φ ′ (ker( p )) = 1 and φ ′ (ker( p )) = 1 tells us that for each g, h ∈ G :1 = φ ′ ([ g, h ]) = φ ′ ([ gh ] − [ g ][ h ]) = φ ′ ([ gh ] − [ g ][ h ]) ∂ ( s ([ gh ] − [ g ][ h ])) = ∂ (cid:0) s ([ gh ] − [ g ][ h ]) (cid:1) = ∂ (cid:0) s ([ gh ]) − φ ( h ) − ⊲ s ([ g ]) s ([ h ]) (cid:1) . Thus we must have that, for each g, h ∈ G : ∂ (cid:0) s ([ gh ]) (cid:1) = ∂ (cid:0) φ ( h ) − ⊲ s ([ g ]) s ([ h ]) (cid:1) . (108)Let us now look at the group map (lemma 27):( ψ ′ , s ◦ ∂ ′ , t ) : E ∂ × p F group ( G ) → E ′ ⋉ ∗ ( E ′ ⋉ ⊲ ′ L ′ ) . Let Π( g, h ) = t ( g, h ) , where g, h ∈ G. Noting that ψ ′ ( g, h ) = 1, we have:( ψ ′ , s ◦ ∂ ′ , t )( g, h ) = (cid:0) , ( s ◦ ∂ ′ )( g, h ) , t ( g, h ) (cid:1) = (cid:0) , s ([ gh ]) − φ ( h ) − ⊲ s ([ g ]) s ([ h ]) , t ( g, h ) (cid:1) = (cid:0) , s ([ gh ]) − φ ( h ) − ⊲ s ([ g ]) s ([ h ]) , Π( g, h ) (cid:1) . ψ ′ ( g, h ) = 1 tells us that:1 = ( s ◦ ∂ ′ )( g, h ) δ (Π( g, h )) = s ([ gh ]) − φ ( h ) − ⊲ s ([ g ]) s ([ h ]) δ (Π( g, h )) . Therefore, for each g, h ∈ G , we have: s ([ gh ]) = φ ( h ) − ⊲ s ([ g ]) s ([ h ]) δ (Π( g, h )) , (109)thus also for g, h ∈ G : ( ψ ′ , s ◦ ∂ ′ , t )( g, h ) = (cid:0) , δ (Π( g, h )) − , Π( g, h ) (cid:1) , (110)and s ([ g, h ]) = δ (Π( g, h )) − , (111)thus in particular: s ([1]) = s ([1 , δ (Π(1 , − . (112)Since (cid:0) ( ψ ′ , s ◦ ∂ ′ , t ) , ( φ ′ , s ) (cid:1) is, by lemma 27, a pre-crossed module map into ( β : E ′ ⋉ ∗ ( E ′ ⋉ ⊲ ′ L ′ ) → G ′ ⋉ ⊲ E ′ , • ), we have:( ψ ′ , s ◦ ∂ ′ , t )(1 , k ⊲ [ g, h ]) = ( ψ ′ , s ◦ ∂ ′ , t ) (cid:0) k ⊲ ( g, h ) (cid:1) = (cid:0) φ ′ ( k ) , s ( k ) (cid:1) • ( ψ ′ , s ◦ ∂ ′ , t )( g, h )= (cid:0) φ ( p ( k )) , s ( k )) • (cid:0) , s ([ gh ]) − φ ( h ) − ⊲ s ([ g ]) s ([ h ]) , t ( g, h )) (cid:1) , (113)where k ∈ F group ( G ) and g, h ∈ G . The fact that ψ ′ ( k ⊲ ( g, h )) = 1 is equivalent (by (113) and lemmas 26and 27) to: φ ( p ( k )) ∂ ( s ( k )) ⊲ (cid:16) s ([ gh ]) − φ ( h ) − ⊲ s ([ g ]) s ([ h ]) δ (Π( g, h )) (cid:17) = 1 , therefore this fact is implied by (109).Let us now find necessary and sufficient conditions for k ∈ F group ( G ) (cid:0) φ ( p ( k ) , s ( k ) (cid:1) ∈ G ′ ⋉ ⊲ E ′ and ( e, k ) ∈ E ∂ × p F group ( G ) ( ψ ( q ( e, k )) , s ( ∂ ′ ( e, k )) , t ( e, k )) ∈ E ′ ⋉ ∗ ( E ′ ⋉ ⊲ ′ L ′ )to be a pre-crossed module morphism, yielding a lax homotopy between the strict 2-crossed module maps f and f . As far as (cid:0) φ ( p ( k ) , s ( k ) (cid:1) is concerned, since F group ( G ) is free on the underlying set of G , thereare no conditions on s to add to (108).Given ( e, k ) ∈ E ∂ × p F group ( G ) we have( e, k ) = (cid:0) e, [ ∂ ( e )] (cid:1) (cid:0) , [ ∂ ( e )] − k (cid:1) , where clearly [ ∂ ( e )] − k ∈ ker( p ). By using this and lemma 58, we can easily see that the group E ∂ × p F group ( G )is the principal group of the F group ( G ) pre-crossed module, formally generated by the symbols [ e ] . = ( e, [ ∂ ( e ]),with e ∈ E , and ( g, h ) . = (1 , ( g, h )), with g, h ∈ G , modulo the relations (114), below (for g, h ∈ G , e, f ∈ E and l ∈ F group ( G )), which (we leave the reader to verify this) do hold in E ∂ × p F group ( G ):[ e ][ f ] = [ ef ] ( ∂ ( e ) , ∂ ( f )) g ⊲ [ e ] = [ g ⊲ e ] ( g, ∂ ( e ) g − ) ( ∂ ( e ) , g − ) ( g, g − ) − (1 , − ( gh, i ) [ i ] − ⊲ ( g, h ) = ( g, hi ) ( h, i ) ∂ ( g, h ) ⊲ ( l ⊲ ( g ′ , h ′ )) = ( g, h ) ( l ⊲ ( g ′ , h ′ )) ( g, h ) − or ([ g, h ] l ) ⊲ ( g ′ , h ′ ) = ( g, h ) ( l ⊲ ( g ′ , h ′ )) ( g, h ) − . (114)Let t ( a ) = t ([ a ]) and s ( g ) = s ([ g ]), where g ∈ G and a ∈ E . Note that ψ ′ ([ e ]) = ψ ( e ) and ψ ′ ([ e ]) = ψ ( e ).Then t : E ∂ × p F group ( G ) → L ′ can be specified by t ( a ) and t ( g, h ) = Π( g, h ), which must satisfy relations(114). These translate into (in order of appearance): (cid:0) ψ ( a ) , s ( ∂ ( a )) , t ( a ) (cid:1) (cid:0) ψ ( b ) , s ( ∂ ( b )) , t ( b ) (cid:1) = (cid:0) ψ ( ab ) , s ( ∂ ( ab )) , t ( ab ) (cid:1) (cid:0) , δ (Π( ∂ ( a ) , ∂ ( b )) − , Π( ∂ ( a ) , ∂ ( b )) (cid:1) , a, b ∈ G , (by (11)): (cid:16) ψ ( a ) ψ ( b ) , φ ( ∂ ( b )) − ⊲ s ( a ) s ( ∂ ( b )) , (cid:0) ( s ◦ ∂ )( b ) (cid:1) − ⊲ ′ (cid:8) ψ ( b ) − , s ( ∂ ( a )) − (cid:9) − (cid:16)(cid:0) ψ ( b )(( s ◦ ∂ ))( b ) (cid:1) − ⊲ ′ t ( a ) (cid:17) t ( b ) (cid:17) = (cid:16) ψ ( ab ) , s ( ∂ ( ab )) δ (Π( ∂ ( a ) , ∂ ( b )) − , Π( ∂ ( a ) , ∂ ( b )) t ( ab ) (cid:17) . At the level of the first two components, equality always hold, given the calculations above. ThereforeΠ( ∂ ( a ) , ∂ ( b )) t ( ab ) = (cid:0) ( s ◦ ∂ )( b ) (cid:1) − ⊲ ′ (cid:16)(cid:8) ψ ( b ) − , s ( ∂ ( a )) − (cid:9) − ψ ( b ) − ⊲ ′ t ( a ) (cid:17) t ( b ) . (115)The second relation translates into (for any g ∈ G and a ∈ E ):( φ ( g ) , s ( g )) • ( ψ ( a ) , s ( ∂ ( a )) , t ( a )) = (cid:0) ψ ( g ⊲ a ) , s ( ∂ ( g ⊲ a )) , t ( g ⊲ a ) (cid:1) (cid:0) , δ (Π( g, ∂ ( a ) g − )) − , Π( g, ∂ ( a ) g − )) (cid:1)(cid:0) , δ (Π( ∂ ( a ) , g − )) − , Π( ∂ ( a ) , g − ) (cid:1) (cid:0) , δ (Π( g, g − )) , Π( g, g − ) − (cid:1) (cid:0) , δ (Π(1 , , Π(1 , − (cid:1) which gives: φ ( g ) ⊲ (cid:16) s ( g ) s ( ∂ ( a )) − ⊲ ′ n ψ ( a ) − , s ( g ) − ) o − (cid:17) φ ( g ) ⊲ n s ( g ) , s ( ∂ ( a )) − ψ ( a ) − o (cid:0) φ ( g )( ∂ ◦ s )( g ) (cid:1) ⊲ t ( a )= Π(1 , − Π( g, g − ) − Π( ∂ ( a ) , g − ) Π( g, ∂ ( a ) g − )) t ( g ⊲ a ) . (116)By using (23) and (18), the third relation translates into the cocycle type identity (for each g, h, i ∈ G ): s ( i ) ⊲ ′ (cid:0) φ ( i ) − ⊲ Π( g, h ) (cid:1) Π( gh, i ) = Π( h, i ) Π( g, hi ) . (117)Let us see that the fourth relation is void in this case. Note that, for g, h ∈ G and k ∈ F group ( G ), we have: (cid:0) ψ ′ , ( s ◦ ∂ ′ ) , t (cid:1) ( k⊲ ( g, h )) = (cid:0) φ ( p ( k )) , s ( k ) (cid:1) • (cid:0) ψ , ( s ◦ ∂ ′ ) , t (cid:1) ( g, h ) = (cid:0) φ ( p ( k )) , s ( k ) (cid:1) • (cid:0) , δ (Π( g, h )) − , Π( g, h ) (cid:1) . Applying (cid:0) ψ ′ , ( s ◦ ∂ ) , t (cid:1) to the left hand side of the last relation of (114) yields (since φ ( p ([ g, h ])) = 1 andby using (23)): (cid:0) φ ( p ( l )) , s ([ g, h ] l ) (cid:1) • (cid:0) , δ (Π( g ′ , h ′ )) − , Π( g ′ , h ′ ) (cid:1) = φ ( p ( l )) ⊲ (cid:0) , s ([ g, h ] l ) δ (Π( g ′ , h ′ )) − s ([ g, h ] l ) − , s ([ g, h ] l ) ⊲ ′ Π( g ′ , h ′ ) (cid:1) . Applying (cid:0) ψ ′ , ( s ◦ ∂ ) , t (cid:1) to the right-hand-side of the last relation of (114) gives: (cid:0) , δ (Π( g, h )) − , Π( g, h ) (cid:1) (cid:0) φ ( p ( l )) , s ( l ) (cid:1) • (cid:0) , δ (Π( g ′ , h ′ )) − , Π( g ′ , h ′ ) (cid:1) (cid:0) , δ (Π( g, h )) − , Π( g, h ) (cid:1) − = (cid:0) , δ (Π( g, h )) − , Π( g, h ) (cid:1) φ ( p ( l )) ⊲ (cid:0) , s ( l ) δ (Π( g ′ , h ′ )) − s ( l ) − , s ( l ) ⊲ ′ Π( g ′ , h ′ ) (cid:1) (cid:0) , δ (Π( g, h )) − , Π( g, h ) (cid:1) − . Both sides applied by (cid:0) ψ ′ , ( s ◦ ∂ ) , t (cid:1) coincide since (note the fact that ( δ : L → E, ⊲ ′ ) is a crossed module):Π( g, h ) − φ ( p ( l ) ⊲ ( s ( l ) ⊲ ′ Π( g ′ , h ′ )) Π( g, h ) = φ ( p ( l )) ⊲ (cid:16)(cid:0) δ ( φ ( p ( l )) − ⊲ Π( g, h ) − ) s ( l ) (cid:1) ⊲ ′ Π( g ′ , h ′ ) (cid:17) = φ ( p ( l )) ⊲ (cid:16)(cid:0) φ ( p ( l )) − ⊲ s ([ g, h ]) s ( l ) (cid:1) ⊲ ′ Π( g ′ , h ′ ) (cid:17) = φ ( p ( l )) ⊲ (cid:16)(cid:0) s ([ g, h ] l ) (cid:1) ⊲ ′ Π( g ′ , h ′ ) (cid:17) . We have (almost) proven:
Theorem 59.
Consider 2-crossed modules A = (cid:16) L δ −→ E ∂ −→ G, ⊲, { , } (cid:17) and A ′ = (cid:16) L ′ δ −→ E ′ ∂ −→ G ′ , ⊲, { , } (cid:17) .Let f = ( µ , ψ , φ ) and f = ( µ , ψ , φ ) be 2-crossed module maps A → A ′ . A lax homotopy connecting f and f , definition 57, which we write as: f f , ˆ s, ˆ t, Π) −−−−−−→ f , is given by: A (set) map ˆ s : G → E ′ , A (set) map ˆ t : E → L ′ , A (set) map
Π : G × G → L ′ . These are to satisfy, for each g, h ∈ G and a, b ∈ E , that: ∂ (ˆ s ( gh )) = ∂ (cid:0) φ ( h ) − ⊲ ˆ s ( g ) ˆ s ( h ) (cid:1) , (118)ˆ s ( gh ) = φ ( h ) − ⊲ ˆ s ( g ) ˆ s ( h ) δ (Π( g, h )) , (119)Π( ∂ ( a ) , ∂ ( b )) ˆ t ( ab ) = (cid:0) (ˆ s ◦ ∂ )( b ) (cid:1) − ⊲ ′ (cid:16)(cid:8) ψ ( b ) − , ˆ s ( ∂ ( a )) − (cid:9) − ψ ( b ) − ⊲ ′ ˆ t ( a ) (cid:17) ˆ t ( b ) , (120) φ ( g ) ⊲ (cid:16) ˆ s ( g )ˆ s ( ∂ ( a )) − ⊲ ′ n ψ ( a ) − , ˆ s ( g ) − o − (cid:17) φ ( g ) ⊲ n ˆ s ( g ) , ˆ s ( ∂ ( a )) − ψ ( a ) − o (cid:0) φ ( g )( ∂ ◦ ˆ s )( g ) (cid:1) ⊲ ˆ t ( a )= Π(1 , − Π( g, g − ) − Π( ∂ ( a ) , g − ) Π( g, ∂ ( a ) g − )) ˆ t ( g ⊲ a ) , (121)ˆ s ( i ) ⊲ ′ (cid:0) φ ( i ) − ⊲ Π( g, h ) (cid:1) Π( gh, i ) = Π( h, i ) Π( g, hi ) . (122) And, moreover, if g ∈ G , e ∈ E and l ∈ L : φ ( g ) = φ ( g ) ∂ (ˆ s ( g )) ,ψ ( e ) = ψ ( e ) ˆ s ( ∂ ( e )) δ (ˆ t ( e )) ,µ ( l ) = µ ( l ) Π(1 , − ˆ t ( δ ( l )) . (123) Moreover, the corresponding (strict) homotopy between strict 2-crossed module maps Q ( A ) → A ′ , namely: f ′ = f ◦ proj ( f ◦ proj ,s,t ) −−−−−−−−→ f ◦ proj = f ′ is given by the ( f ◦ proj) -quadratic derivation ( s, t ) , where s : F group ( G ) → E ′ is the unique ( φ ◦ p ) -derivation (definition 23) extending ˆ s : G → E ′ , and on the group generators of E ∂ × p F group ( G ) we havethat t ( e, [ ∂ ( e )]) = ˆ t ( e ) and t ( g, h ) = Π( g, h ) , where e ∈ E and g, h ∈ G . Recall that p : F group ( G ) → G is theobvious projection. We thus have a lax analogue of the strict homotopy relation treated in subsection 3.1.
Proof.
We just need to check the last equation of (123). Note that if l ∈ L : δ ′ ( l ) = ( δ ( l ) , ∅ ) = ( δ ( l ) , [1]) (1 , [1] − ) = [ δ ( l )] (1 , − . Therefore, by using (73) and (112), and noting ψ ′ ([1]) = 1 E ′ :ˆ t ( δ ′ ( l )) = ˆ s ([1] − ) − ⊲ ′ ˆ t ( δ ( l )) ˆ t (cid:0) (1 , − (cid:1) = ˆ s ([1] − ) − ⊲ ′ (cid:0) ˆ t ( δ ( l )) (ˆ t (1 , − (cid:1) . Thus µ ( l ) = µ ′ ( l ) = µ ( l ) ˆ t ( δ ′ ( l )) = µ ( l ) ˆ s ([1] − ) − ⊲ ′ (cid:0) ˆ t ( δ ( l )) Π(1 , − (cid:1) . And now note that ˆ s ([1] − ) = φ ′ ([1]) ⊲ ˆ s ([1]) − = ˆ s ([1]) − , since φ ′ ([1]) = 1, together with (112), and thesecond Peiffer condition in definition 1.Note that it follows by construction that f = ( µ , ψ , φ ), defined in (123), is a 2-crossed modulemorphism A → A ′ , if equations (118) to (122) are satisfied. This can easily be proven directly.47 .1.4. Composition and inverses of lax homotopies We now freely use the notation of 3.3.1 and 3.3.4, as well as definition 57.
Theorem 60.
Consider 2-crossed modules A = (cid:16) L δ −→ E ∂ −→ G, ⊲, { , } (cid:17) and A ′ = (cid:16) L ′ δ −→ E ′ ∂ −→ G ′ , ⊲, { , } (cid:17) .Given lax homotopies of 2-crossed module maps A → A ′ , say: f = ( µ, ψ, φ ) ( f, ˆ s, ˆ t, Π) −−−−−→ f ′ = ( µ ′ , ψ ′ , φ ′ ) ( f ′ , ˆ s ′ , ˆ t ′ , Π ′ ) −−−−−−−→ f ′′ = ( µ ′′ , ψ ′′ , φ ′′ ) , the explicit form of their concatenation, denoted by ( f, ˆ s, ˆ t, Π) ˆ ⊗ ( f ′ , ˆ s ′ , ˆ t ′ , Π ′ ) , is: f ( f, ˆ s ˆ ⊗ ˆ s ′ , ˆ t ˆ ⊗ ˆ t ′ , Π ˆ ⊗ Π ′ ) −−−−−−−−−−−−→ f ′′ , where (ˆ s ˆ ⊗ ˆ s ′ )( g ) = ˆ s ( g ) ˆ s ′ ( g ) , for each g ∈ G, (124)(ˆ t ˆ ⊗ ˆ t ′ )( g ) = ˆ s ′ ( ∂ ( e )) − ⊲ ′ ˆ t ( e ) ˆ t ′ ( e ) , for each e ∈ E, (125) and where (Π ˆ ⊗ Π ′ )( g, h ) = Θ (ˆ s, ˆ s ′ ) (cid:0) [ gh ] , [ g ] , [ h ] (cid:1) Π ′ ( g, h ) Π( g, h ) , (126) where Θ (ˆ s, ˆ s ′ ) was defined in (88) .Moreover, the inverse of f = ( µ, ψ, φ ) ( f, ˆ s, ˆ t, Π) −−−−−→ f ′ = ( µ ′ , ψ ′ , φ ′ ) is f ′ = ( µ ′ , ψ ′ , φ ′ ) ( f ′ , ˆ s, ˆ t, Π) −−−−−−→ f = ( µ, ψ, φ ) , where if g, h ∈ G and e ∈ E : ˆ s ( g ) = ˆ s ( g ) − , ˆ t ( e ) = ˆ s ( ∂ ( e )) ⊲ ′ ˆ t ( e ) − , Π( g, h ) = Θ (ˆ s, ˆ s ) (cid:0) [ gh ] , [ g ] , [ h ] (cid:1) − Π( g, h ) − . (127) Proof.
As far as the concatenation of lax homotopies is concerned, we just need to consider the correspond-ing chain of strict homotopies, given by the previous theorem: f ◦ proj ( f ◦ proj ,s,t ) −−−−−−−→ f ′ ◦ proj ( f ′ ◦ proj ,s ′ ,t ′ ) −−−−−−−−−→ f ′′ ◦ proj , and look at the construction of their concatenation in 3.3.1, noting that the underlying set of G is a free(chosen) basis of F group ( G ).We know that s ([ g ]) = ˆ s ( g ), s ′ ([ g ]) = ˆ s ′ ( g ) and ( s ⊗ s ′ )([ g ]) = s ([ g ]) s ′ ([ g ]) for each g ∈ G . Thus(ˆ s ˆ ⊗ ˆ s ′ )( g ) = ( s ⊗ s ′ )([ g ]) = ˆ s ( g ) ˆ s ′ ( g ) for each g ∈ G .Analogously, if e ∈ E , then ˆ t ( e ) = t ( e, [ ∂ ( e )]), ˆ t ′ ( e ) = t ′ ( e, [ ∂ ( e )]) and (ˆ t ˆ ⊗ ˆ t ′ )( e ) = ( t ⊗ t ′ )( e, [ ∂ ( e )]), and( t ⊗ t ′ )( e, [ ∂ ( e )]) = ω ( s,s ′ ) ( ∂ ′ ( e, [ ∂ ( e )])) s ′ ( ∂ ′ ( e, [ ∂ ( e )])) − ⊲ ′ t (( e, [ ∂ ( e )]) t ′ (( e, [ ∂ ( e )]))= ω ( s,s ′ ) ([ ∂ ( e )])) s ′ ([ ∂ ( e )])) − ⊲ ′ t (( e, [ ∂ ( e )]) t ′ (( e, [ ∂ ( e )]))= s ′ ([ ∂ ( e )])) − ⊲ ′ t (( e, [ ∂ ( e )]) t ′ (( e, [ ∂ ( e )])) = ˆ s ′ ( ∂ ( e )) − ⊲ ′ ˆ t ( e ) ˆ t ′ ( e ) , where we used remark 39. Thus (125) follows.Recall that, given g, h ∈ G , then Π( g, h ) = t (1 , [ g, h ]) and Π ′ ( g, h ) = t ′ (1 , [ g, h ]). We have:(Π ˆ ⊗ Π ′ )( g, h ) = ( t ⊗ t ′ )(1 , [ g, h ]) = ω ( s,s ′ ) ( ∂ ′ (1 , [ g, h ])) s ′ ( ∂ ′ (1 , [ g, h ])) − ⊲ ′ t (1 , [ g, h ]) t ′ (1 , [ g, h ])= ω ( s,s ′ ) ([ gh ] − [ g ][ h ])) s ′ ([ gh ] − [ g ][ h ]) − ⊲ ′ t ([1 , [ g, h ]) t ′ (1 , [ g, h ])= ω ( s,s ′ ) ([ gh ] − [ g ][ h ])) δ (Π ′ ( g, h )) ⊲ ′ t ([1 , [ g, h ]) t ′ (1 , [ g, h ])= ω ( s,s ′ ) ([ gh ] − [ g ][ h ])) Π ′ ( g, h ) Π( g, h )= Θ ( s,s ′ ) (cid:0) [ gh ] − , [ g ] , [ h ] (cid:1) Π ′ ( g, h ) Π( g, h ) = Θ (ˆ s, ˆ s ′ ) (cid:0) [ gh ] − , [ g ] , [ h ] (cid:1) Π ′ ( g, h ) Π( g, h ) , s ( g ) = s ([ g ]) and ˆ s ′ ( g ) = s ′ ([ g ]), for each g ∈ G .Inverses are handled in the same way. For instance if g, h ∈ G (we use (111)):Π( g, h ) = t (1 , [ g, h ]) = ( ω ( s,s ) ( ∂ ′ (1 , [ g, h ])) − s ( ∂ ′ (1 , [ g, h ])) ⊲ ′ t (1 , [ g, h ]) − = ω ( s,s ) ([ gh ] − [ g ][ h ]) − s ([ gh ] − [ g ][ h ]) ⊲ ′ Π( g, h ) − = Θ ( s,s ) (cid:0) [ gh ] , [ g ] , [ h ] (cid:1) − δ (Π( g, h )) − ⊲ ′ Π( g, h ) − = Θ ( s,s ) (cid:0) [ gh ] , [ g ] , [ h ] (cid:1) − Π( g, h ) − = Θ (ˆ s, ˆ s ) (cid:0) [ gh ] , [ g ] , [ h ] (cid:1) − Π( g, h ) − . We now discuss lax 2-fold homotopy. Consider 2-crossed modules A = (cid:16) L δ −→ E ∂ −→ G, ⊲, { , } (cid:17) and A ′ = (cid:16) L ′ δ −→ E ′ ∂ −→ G ′ , ⊲, { , } (cid:17) . Given two lax homotopies between the 2-crossed module maps f, f ′ : A → A ′ , say: f ( f, ˆ s ′ , ˆ t ′ , Π ′ ) & & ( f, ˆ s, ˆ t, Π) f ′ , a lax 2-fold homotopy connecting them, say: f ( f, ˆ s ′ , ˆ t ′ , Π ′ ) * * ( f, ˆ s, ˆ t, Π) ⇑ ( f, ˆ s, ˆ t, Π , ˆ k ) f ′ , is given by a map ˆ k : G → L ′ , without any restrictions, apart from that it should relate the two laxhomotopies as in (128), (129) and (130), below. This is because to ˆ k we associate the unique (strict)quadratic ( f ◦ proj , s, t ) 2-derivation k : F group ( G ) → L ′ such that k ([ g ]) = ˆ k ( g ) for each g ∈ G (corollary32). Therefore: ˆ s ′ ( g ) = ˆ s ( g ) δ (ˆ k ( g )) , for each g ∈ G, (128)and for each e ∈ E :ˆ t ′ ( e ) = t ′ (( e, [ ∂ ( e )]) = k ( ∂ ′ ( e, [ ∂ ( e )])) − t (( e, [ ∂ ( e )])) = k ([ ∂ ( e )]) − t (( e, [ ∂ ( e )])= ˆ k ( ∂ ( e )) − ˆ t ( e ) . (129)Also, by using equation (80) in lemma 31, for each g, h ∈ G :Π ′ ( g, h ) = t ′ ((1 , [ g, h ])) = k ( ∂ ′ ((1 , [ g, h ]))) − t ((1 , [ g, h ])) = k ([ gh ] − [ g ][ h ]) − Π( g, h )= (cid:16) Ξ ( φ,s,k ) ([ gh ] , [ g ] , [ h ]) (cid:17) − Π( g, h ) . (130)Thus, by using equation (80):Π ′ (1 ,
1) = (cid:16) Ξ ( φ,s,k ) ([1] , [1] , [1]) (cid:17) − Π(1 ,
1) = k ([1]) − Π(1 ,
1) = ˆ k (1) − Π(1 , . In particular, we can prove that, if (128), (129) and (130) are satisfied, then if f ( f, ˆ s, ˆ t, Π) −−−−−→ f ′ , we must alsohave that f ( f, ˆ s ′ , ˆ t ′ , Π ′ ) −−−−−−−→ f ′ , by using (123). 49f we have a chain of 2-fold homotopies ( f, ˆ s, ˆ t, Π) ( f, ˆ s, ˆ t, Π , ˆ k ) −−−−−−−→ ( f, ˆ s ′ , ˆ t ′ , Π ′ ) ( f, ˆ s ′ , ˆ t ′ , Π ′ , ˆ k ′ ) −−−−−−−−→ ( f, ˆ s ′′ , ˆ t ′′ , Π ′′ ) , diagrammatically: f ( f, ˆ s ′ , ˆ t ′ , Π ′ ) / / ( f, ˆ s, ˆ t, Π) C C ( f, ˆ s ′′ , ˆ t ′′ , Π ′′ ) (cid:27) (cid:27) ⇑ ( f, ˆ s, ˆ t, Π , ˆ k ) ⇑ ( f, ˆ s ′ , ˆ t ′ , Π ′ , ˆ k ′ ) f ′ Then their vertical concatenation is given by the map ˆ k ˆ ⋄ ˆ k ′ : G → L such that:(ˆ k ⋄ ˆ k ′ )( g ) = ˆ k ( g ) ˆ k ′ ( g ) , for each g ∈ G . By construction, ˆ k ˆ ⋄ ˆ k ′ does connect ( f, ˆ s, ˆ t, Π) and ( f, ˆ s ′′ , ˆ t ′′ , Π ′′ ).Suppose that we have a lax 2-fold homotopy say ( f, ˆ s, ˆ t, Π) ( f, ˆ s, ˆ t, Π , ˆ k ) −−−−−−−→ ( f, ˆ s ′ , ˆ t ′ , Π ′ ) , which we write as: f ( f, ˆ s ′ , ˆ t ′ , Π) * * ( f, ˆ s, ˆ t, Π) ⇑ ( f, ˆ s, ˆ t, Π , ˆ k ) f ′ , and that we also have a lax homotopy: f ′ = ( µ ′ , ψ ′ , φ ′ ) ( f ′ , ˆ s ′′ , ˆ t ′′ , Π ′′ ) −−−−−−−−−→ f ′′ = ( µ ′′ , ψ ′′ , φ ′′ ) , so what we have diagrammatically is: f ( f, ˆ s ′ , ˆ t ′ , Π ′ ) * * ( f, ˆ s, ˆ t, Π) ⇑ ( f, ˆ s, ˆ t, Π , ˆ k ) f ′ ( f ′ , ˆ s ′′ , ˆ t ′′ , Π ′′ ) / / f ′′ . The whiskering: ( f, ˆ s, ˆ t, Π , ˆ k ) ˆ ⊗ ( f ′ , ˆ s ′′ , ˆ t ′′ , Π ′′ ) = ( f, ˆ s ˆ ⊗ ˆ s ′′ , ˆ t ˆ ⊗ ˆ t ′′ , Π ˆ ⊗ Π ′′ , k ˆ ⊗ s ′′ )is given by the map ˆ k ˆ ⊗ ˆ s ′′ : G → L ′ , which has the form, for each g ∈ G :(ˆ k ˆ ⊗ ˆ s ′′ )( g ) = ˆ s ′′ ( g ) − ⊲ ′ ˆ k ( g ) . By construction we have: f ( f, ˆ s ′ ˆ ⊗ ˆ s ′′ , ˆ t ′ ˆ ⊗ ˆ t ′′ , Π ′ ˆ ⊗ Π ′′ ) ( ( ( f, ˆ s ˆ ⊗ ˆ s ′′ , ˆ t ˆ ⊗ ˆ t ′′ , Π ˆ ⊗ Π ′′ ) ⇑ ( f, ˆ s ˆ ⊗ ˆ s ′′ , ˆ t ˆ ⊗ ˆ t ′′ , Π ˆ ⊗ Π ′′ , ˆ k ˆ ⊗ ˆ s ′′ ) f ′ . Similarly, suppose that we have 2-crossed module maps f, f ′ : A ′ → A , lax homotopies ( f, ˆ s, ˆ t, Π) and( f, ˆ s ′ , ˆ t ′ , Π ′ ), connecting f and f ′ , a lax 2-fold homotopy ( f, ˆ s, ˆ t, Π , ˆ k ′ ), connecting them, and also a lax50omotopy f ′′ ( f ′′ , ˆ s ′′ , ˆ t ′′ , Π ′′ ) −−−−−−−−−→ f, diagrammatically: f ′′ ( f ′′ , ˆ s ′′ , ˆ t ′′ , Π ′′ ) / / f ( f, ˆ s ′ , ˆ t ′ , Π ′ ) * * ( f, ˆ s, ˆ t, Π) ⇑ ( f, ˆ s, ˆ t, Π , ˆ k ) f ′ . The whiskering: ( f ′′ , ˆ s ′′ , ˆ t ′′ , Π ′′ ) ˆ ⊗ ( f, ˆ s, ˆ t, Π , ˆ k ′ ) = ( f ′′ , ˆ s ′′ ˆ ⊗ ˆ s, ˆ t ′′ ˆ ⊗ ˆ t, Π ′′ ˆ ⊗ Π , ˆ s ′′ ⊗ ˆ k ) , is such that, for each g ∈ G we have ( ˆ s ′′ ˆ ⊗ ˆ k )( g ) = ˆ k ( g ) . By construction: f ′′ ( f ′′ ,s ′′ ⊗ s ′ ,t ′′ ⊗ t ′′ ) ( ( ( f ′′ , ˆ s ′′ ˆ ⊗ ˆ s, ˆ t ′′ ˆ ⊗ ˆ t, Π ′′ ˆ ⊗ Π) ⇑ ( f ′′ , s ′′ ˆ ⊗ ˆ s, ˆ t ′′ ˆ ⊗ ˆ t, Π ′′ ˆ ⊗ Π , ˆ s ′′ ⊗ ˆ k ) f ′ . By definition (since this is simply an unpacked version of definition 57), it follows:
Theorem 61.
Let A and A ′ be 2-crossed modules. There exists a 2-groupoid HOM
LAX ( A , A ′ ) , whoseobjects are the (strict) 2-crossed module maps A → A ′ , the morphisms are the (pointed) lax homotopiesbetween 2-crossed module maps, and the 2-morphisms are the (pointed) lax 2-fold homotopies between laxhomotopies, whose explicit descriptions, and various concatenations, are described in 4.1.3, 4.1.4 and 4.1.5.4.2. Composition of lax homotopies with strict 2-crossed module maps Theorem 62.
Let A , A ′ and A ′′ be 2-crossed modules. Let f, f ′ : A → A ′ be 2-crossed module maps. Letalso h = ( µ, ψ, φ ) : A ′ → A ′′ be another 2-crossed module map. If we have a lax 2-crossed module homotopy ( f, ˆ s, ˆ t, Π) connecting f and f ′ , then ( h ◦ f, ψ ◦ ˆ s, µ ◦ ˆ t, µ ◦ Π) . = h ◦ ( f, ˆ s, ˆ t, Π) is a lax homotopy connecting g ◦ f and g ◦ f ′ . Proof.
Equations (118) to (123) are satisfied since h preserves all 2-crossed module operations, strictly. Theorem 63.
Let A , A ′ and A ′′ be 2-crossed modules. Let f, f ′ : A → A ′ be 2-crossed module maps. Letalso h ′ = ( µ ′ , ψ ′ , φ ′ ) : A ′′ → A be a 2-crossed module morphism. If we have a lax 2-crossed module homotopy ( f, ˆ s, ˆ t, Π) connecting f and f ′ . Then ( f ◦ h ′ , ˆ s ◦ φ ′ , ˆ t ◦ ψ ′ , Π ◦ ( φ ′ × φ ′ )) . = ( f, ˆ s, ˆ t, Π) ◦ h ′ is a lax homotopy connecting f ◦ h ′ and f ′ ◦ h ′′ . The operators defined in theorems 62 and 63 will be called composition operators.
Theorem 64.
The composition operators preserve concatenations and inverses of lax homotopies.
Proof.
Immediate from the explicit form of the concatenations and inverses of lax homotopies, and the factthat we only compose homotopies with strict 2-crossed module morphisms.We thus have a sesquicategory [46], whose objects are the 2-crossed modules, the morphisms are the2-crossed module maps, and the 2-morphisms are the lax homotopies between then. It is important to notethat this is not a 2-category, since the interchange law does not hold in general.The composition operators are also defined, in the obvious way, and with the obvious properties, for lax2-fold homotopies and strict 2-crossed module maps. Given that for any two 2-crossed modules A and A ′ wehave a 2-groupoid HOM
LAX ( A , A ′ ), we expect that this will give a Gray category [17, 28, 30] of 2-crossedmodules, (strict) 2-crossed module maps, lax homotopies and lax 2-fold homotopies.51 .3. Lax homotopy equivalence of 2-crossed modules In this subsection we make use of subsection 4.2. Let A and A ′ be 2-crossed modules. Definition 65.
We say that f : A → A ′ is a lax homotopy equivalence if there exists a 2-crossed modulemap g : A ′ → A , and lax homotopies: id A (id A , ˆ s, ˆ t, Π) −−−−−−−→ g ◦ f and id A ′ (id A′ , ˆ u, ˆ v,J ) −−−−−−−−→ f ◦ g. In such a case g is said to be a homotopy inverse of f . Lemma 66.
The composition of lax homotopy equivalences is a lax homotopy equivalence.
Proof.
This result is almost immediate from the fact that we can concatenate lax homotopies betweenstrict 2-crossed module maps. Let us give details. Let A , A ′ and A ′′ be 2-crossed modules. Supposethat f : A → A ′ and f ′ : A ′ → A ′′ are lax homotopy equivalences. Let us see that f ′ ◦ f is a lax homotopyequivalence. Choose inverses up to homotopy g and g ′ of f and g ′ , respectively. We thus have lax homotopiesid A (id A , ˆ s, ˆ t, Π) −−−−−−−→ g ◦ f, id A ′ (id A′ , ˆ u, ˆ v,J ) −−−−−−−−→ f ◦ g, id A ′ (id A′ , ˆ s ′ , ˆ t ′ , Π ′ ) −−−−−−−−−→ g ′ ◦ f ′ , id A ′′ (id A′′ , ˆ u ′ , ˆ v ′ ,J ′ ) −−−−−−−−−→ f ′ ◦ g ′ . We prove that g ◦ g ′ is a lax homotopy inverse of f ′ ◦ f . This follows by considering the concatenationsbelow of lax homotopies:id A (id A , ˆ s, ˆ t, Π) −−−−−−−→ g ◦ f = g ◦ id A ′ ◦ f g ◦ (id A′ , ˆ s ′ , ˆ t ′ , Π ′ ) ◦ f −−−−−−−−−−−−→ g ◦ g ′ ◦ f ′ ◦ f, id A ′′ (id A′′ , ˆ u ′ , ˆ v ′ ,J ′ ) −−−−−−−−−→ f ′ ◦ g ′ = f ′ ◦ id A ′ ◦ g ′ f ′ ◦ (id A′ , ˆ u, ˆ v,J ) ◦ g ′ −−−−−−−−−−−−→ f ′ ◦ f ◦ g ◦ g ′ . Proposition 67.
Lax homotopy equivalences of 2-crossed modules have the two-of-three property [19].
Proof.
The complete proof is analogous to the proof of the particular case above.By using the composition operators, we can easily see that a retract of a lax homotopy equivalence is againa lax homotopy equivalence.
Acknowledgements
J. Faria Martins was partially supported by CMA/FCT/UNL, under the grant PEst-OE/MAT/UI0297/2011.B. Gohla was supported by FCT (Portugal) through the doctoral grant SFRH/BD/33368/ 2008. This workwas supported by FCT, by means of the projects PTDC/MAT/098770/2008 and PTDC/MAT/101503/2008.We are grateful to the referee for the comments and suggestions made.
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