aa r X i v : . [ m a t h . C T ] J a n HIGHER COVERINGS OF RACKS AND QUANDLES – PART II
FRANÇOIS RENAUD
Abstract.
This article is the second part of a series of three articles, in which we developa higher covering theory of racks and quandles. This project is rooted in M. Eisermann’swork on quandle coverings, and the categorical perspective brought to the subject by V. Even,who characterizes coverings as those surjections which are categorically central, relatively totrivial quandles. We extend this work by applying the techniques from higher categoricalGalois theory, in the sense of G. Janelidze, and in particular we identify meaningful higher-dimensional centrality conditions defining our higher coverings of racks and quandles.In this second article (Part II), we show that categorical Galois theory applies to theinclusion of the category of coverings into the category of surjective morphisms of racks andquandles. We characterise the induced Galois theoretic concepts of trivial covering, normalcovering and covering in this two-dimensional context. The latter is described via our definitionand study of double coverings, also called algebraically central double extensions of racks andquandles. We define a suitable and well-behaved commutator which captures the zero, oneand two-dimensional concepts of centralization in the category of quandles. We keep track ofthe links with the corresponding concepts in the category of groups. Introduction
This article (
Part II ) is the continuation of [60] which we refer to as
Part I . In the followingparagraphs we recall enough material from Part I, and the first order covering theory of racksand quandles, in order to contextualize and motivate the content of the present paper, in which,based on the firm theoretical groundings of categorical Galois theory, we identify the secondorder coverings of racks and quandles, and the relative concept of centralization , together withthe definition of a suitable commutator in this context. We refer to Part I for more details,further motivations, references, and historical remarks about the material in this introduction.1.1.
Recalls and notations.
We like to describe racks as sets equipped with a self-distributivesystem of symmetries, attached to each point. More precisely, a rack is a set A equipped with abinary operation ⊳ : A × A → A such that for each a ∈ A , the function − ⊳ a : A → A (which iscalled the symmetry attached to a ) admits an inverse (denoted − ⊳ − a : A → A ) and is compatible with the operation ⊳ (self-distributivity), i.e. for all x , a and b in A :(R1) ( x ⊳ a ) ⊳ − a = x = ( x ⊳ − a ) ⊳ a ;(R2) ( x ⊳ a ) ⊳ b = ( x ⊳ b ) ⊳ ( a ⊳ b ) .A morphism of racks is a function between racks which preserves the operation ⊳ . The thusdefined category of racks ( Rck ) is a variety of algebras in which the subvariety of quandles ( Qnd )is the full subcategory of
Rck whose objects Q are such that a ⊳ a = a for each a ∈ Q . It is thecategory of those racks whose symmetries are required to fix the point they are attached to. Wewrite F r ( free rack functor ) and F q ( free quandle functor ) for the left adjoints to the forgetful Mathematics Subject Classification.
Key words and phrases.
Two-dimensional covering theory of racks and quandles, categorical Galois theory,double central extension, commutator theory, centralization.The author is a Ph.D. student funded by
Formation à la Recherche dans l’Industrie et dans l’Agriculture ( FRIA ) as part of
Fonds de la Recherche Scientifique - FNRS . functors U :
Rck → Set and
U :
Qnd → Set to the category of sets. The units of the correspondingadjunctions are denoted by η r and η q respectively. Similarly F g : Set → Grp is the free groupfunctor and the unit of the adjunction F g ⊣ U is denoted η g .One of the earliest motivations for the study of racks and quandles arises from their link withgroup conjugation (Paragraph 1.1.4). Main applications are to be found in knot theory andsubsequently in physics or even computer science – see for instance [55, 11, 34, 18, 35, 16, 17,56, 21] and references there. Another important line of work concerns the concept of symmetricspace – see [57, 63, 2, 39]. The categories of racks and quandles are also an interesting contextfor applying and developing categorical algebra [22, 23, 24, 25, 7, 9, 10].1.1.1. The connected component adjunction.
In topology, it is convenient to view the category ofsets (
Set ) as the category of discrete topological spaces . The functor sending a topological spaceto its set of connected components is then obtained as the left adjoint to the inclusion of
Set inthe category of topological spaces (
Top ).For our purposes, we observe that
Set is isomorphic to the subvariety of trivial quandles definedby the additional axiom “ x⊳a = x ”. In other words, each set X can be viewed in a unique way as aquandle whose “system of symmetries” is obtained by attaching the identity function − ⊳ a .. = id X to each element a in X . The left adjoint of the inclusion I :
Set → Rck (or
I :
Set → Qnd )defines the connected component functor π : Rck → Set (respectively π : Qnd → Set ), with unit η : A → I( π ( A )) , where π ( A ) is the set of ( C A )-equivalence classes of elements in A . Here C A is the congruence describing when two elements of A are considered to be connected [54].Two elements x and y of a rack (or quandle) A are connected if one, e.g. y , is the result of thesuccessive action of certain symmetries (or their inverses) on the other, here x . We define (see[55]) that ( x, y ) ∈ C A if and only if there exists n ∈ N and elements a , a , . . . , a n in A suchthat y = x ⊳ δ a ⊳ δ a · · · ⊳ δ n a n , (1)for some δ i ∈ {− , } for ≤ i ≤ n ; where by convention the left-most operation is computedfirst. The data of such a formal sequence ( a δ i i ) ≤ i ≤ n is called a primitive path . Together with thedata of a head x , these describe a primitive trail , whose endpoint is then given by y . Equation (1)describes how the primitive path ( a δ i i ) ≤ i ≤ n acts on a head x in order to produce the endpoint y . Note that a same primitive path ( a δ i i ) ≤ i ≤ n can act on different heads, yielding differentendpoints.This connected component adjunction is the base of the covering theory of interest, in thesame way that, from the perspective of categorical Galois theory, the connected componentadjunction in topology is at the base of the classical covering theory of topological spaces. Thekey definitions and developments of this article (e.g. double coverings of racks and quandles, ourcommutator and the relative concept of centrality in this context) merely require a minimalistintroduction to the relevant concepts from categorical Galois theory, such as provided below, orin Part I (see for instance [46] for more details).1.1.2. Basic categorical Galois theory.
In summary, we consider a convenient particular instanceof the general theory which was developed in [41]. The axiomatic framework in which categoricalGalois theory takes place is that of a
Galois structure . For our purposes, a Galois structure,say Γ , mainly consists in the data of a category C (for instance Qnd ), a subcategory X in C (forinstance Set ), together with a reflection of C on X , i.e. the data of a left adjoint F : C → X to theinclusion
I :
X → C (e.g. π : Qnd → Set ; note that we often omit I from our notations in whatfollows). The “bigger” context C is understood to be more “sophisticated”, more difficult to study,whereas the “smaller” context X is supposedly more “primitive”, or merely better understood.In order to obtain a Galois structure from such a reflection, we also need to specify a class of IGHER COVERINGS OF RACKS AND QUANDLES – PART II 3 morphisms in C , whose “elements” will be called extensions (which are the surjective morphismsof quandles in our example).The purpose of Galois theory is then to study special classes of extensions in C which arenaturally associated to those extensions which lie in the subcategory X . In this work, we call anextension which is a morphisms in X a primitive extension . These special classes of extensions in C which are associated to primitive extensions then measure a sphere of influence of X in C (withrespect to the chosen concept of extension). In particular, the most important special class ofextensions is the class of Γ -coverings , or simply coverings , also called central extensions , definedbelow. In our example with Qnd , the induced concept of Γ -covering coincides with the conceptof covering defined by M. Eisermann in [21], as it was first proved in [22]. A covering, or centralextension of quandles (or racks), is a surjective morphism of quandles (or racks) f : A → B suchthat x ⊳ a = x ⊳ b whenever a , b and x in A are such that ( a, b ) is in the kernel pair Eq( f ) of f , i.e. such that f ( a ) = f ( b ) . A surjective morphism of racks or quandles f : A → B can be centralized , using a quotient of its domain, given by the centralization congruence C A over A generated by the pairs ( x, y ) such that there exists ( a, b ) ∈ Eq( f ) such that y = x ⊳ a ⊳ − b (see[20] and Part I).In general, given a suitable Galois structure Γ one defines three different special classes ofextensions, the simplest of which is the class of trivial Γ -coverings , or simply trivial coverings (also referred to as trivial extensions ). A trivial covering is defined (in a suitable Galois structure)as an extension t : T → E which is the pullback of a primitive extension p : X → F ( E ) in X ,along the unit morphism η E : E → F ( E ) (left-hand square in Diagram (2)). In a suitable Galoisstructure, the category of trivial coverings above an object E ∈ C is equivalent to the categoryof primitive extensions above F ( E ) in X . In topology, this Galois-theoretic definition of trivialcovering coincides with the classical definition of trivial covering [38, Section 1.3] (for a suitablechoice of Galois structure Γ T , whose reflection is the connected component functor to Set [5,Section 6.3]). Note that both in the example from topology, and in the example in
Qnd , aprimitive extension is just a surjective function in
Set . X p (cid:12) (cid:18) T t (cid:12) (cid:18) l r , A c (cid:12) (cid:18) F ( E ) E η E l r e , B (2)The class of Γ -coverings , or simply coverings is then defined (in a suitable Galois structure Γ ) asthe class of those extensions c : A → B in C , for which there exists another extension e : E → B ,which is said to split c , i.e. such that the pullback t of c along e is a trivial extension (right-handsquare in Diagram (2)). In certain contexts, coverings are also refferred to as central extensions ,such as in [46], in reference to the example from group theory described below. Again, theclassical concept of a topological covering arises as the concept of a Γ T -covering defined in thatsame suitable Galois structure for topological spaces. The remaining special class of extensionsis the class of normal Γ -coverings , or simply normal covering (or normal extension ), which arethose extensions which are split by themselves.As it is the case in topology (under some suitable assumptions) the coverings above a given ob-ject of C can be classified using data which is internal to X . Informally we say that the “behaviourof coverings is tractable using information which remains in the simpler context described by X ”.The fundamental theorem of categorical Galois theory [41, Theorem 3.7] formalises this idea insuch a way that the classification of topological coverings and other examples of so-called “Galoiscorrespondences” appear as particular instances of it [5]. One of these particular instances is theclassical Galois theory of field extensions, and both of its generalizations by A. Grothendieck andA. R. Magid. Besides the example from topology, we are also interested in the theory of centralextensions from group theory, which is another instance of categorical Galois theory [41, 46], FRANÇOIS RENAUD using the Galois structure (say Γ G ) obtained from the abelianization functor and the class ofsurjective group homomorphisms (see Paragraph 1.1.4). The Γ G -coverings are the classicallycalled central extensions of groups . As mentioned before, and generalizing from this example,the terminology central extension is used in certain contexts, such as in [46], to describe whatis more generally called a Γ -covering. Finally, the classification results for quandle coveringsobtained in [21] also derive from this fundamental theorem of categorical Galois theory, as wepartially explained in Part I.Note that in this article, we further develop the theory of quandle and rack coverings by usinghigher dimensional categorical Galois theory. In particular we identify a suitable commutator forthe study of coverings and double coverings in this context, together with a relative higher cen-trality condition which is compatible with the zero-dimensional centrality C and one-dimensionalcentrality C described before. The analogy is to be made with the corresponding developmentsin group theory as we describe in Part I and below. Because of the similarities between thecorresponding Galois structures, many aspects of the covering theory of racks and quandles canalso be interpreted using the covering theory of topological spaces (Part I and [21]).We precisely define what a “suitable” Galois structure is for our purposes in Convention 1.1.3.Again, we describe and explain the concepts at play to the extend which is necessary for ap-preciating the content of this article. As it is stated in [23] and Part I, the adjunction π ⊣ I between the categories Rck (or similarly for
Qnd ) and
Set is part of a strongly Birkhoff Galoisstructure denoted Γ .. = ( Rck , Set , π , I , η, ǫ, E ) (see Convention 1.1.3 and Section 2.3), where the class of extensions E is the class of surjective morphisms of racks. Convention 1.1.3.
For our purposes, a
Galois structure Γ .. = ( C , X , F, I , η, ǫ, E ) (see [42] ), isthe data of an inclusion I , of a full ( replete ) subcategory X in a category C , with left adjoint F : C → X , unit η , counit ǫ and a chosen class of extensions E within the morphisms of C . Theclass E is subject to the following conditions:(1) E contains all isomorphisms, and E is closed under composition;(2) the reflection (by F ) of an extension yields an extension;(3) pullbacks along extensions exist, and the pullback of an extension is an extension.For our purposes, E will always be a class of regular epimorphisms . Moreover, we require thecomponents of the unit η to be extensions, i.e. for each object X in C , η X : X → I F X is anextension. Such a category X is said to be E -reflective in C . Finally, taking pullbacks alongextensions should be a “well behaved” operation i.e. we require our extensions to be of effective E -descent in C (see Section 2.2 below).As mentioned before, we call primitive extensions , those extensions p which lie in X . A trivial Γ -covering or trivial covering is an extension t which is the pullback of a primitive extension p along a unit morphism η X (provided that Γ is admissible or strongly Birkhoff , see Section2.3). A normal Γ -covering or normal covering is such that the projections of its kernel pair aretrivial coverings. A Γ -covering or covering , sometimes called central extension , is an extension c : A → B such that there is another extension e : E → B such that the pullback t of c along e isa trivial covering. Strong connections with groups.
There is an enlightening parallel to be made between ourwork and the corresponding developments in group theory. Firstly because the algebraic struc-tures of racks and quandles are intimately related to group conjugation, and secondly becausesuch a parallel helps to navigate the possible outcomes for the development of a higher ordercovering theory of racks and quandles (see Part I).The term wrack was first used by J.C. Conway and G.C. Wraith, in an unpublished corre-spondence of 1959, to describe the “wreckage” of a group, whose multiplication operation hasbeen forgotten, and only the operation of conjugation remains. The functor
Conj :
Grp → Rck
IGHER COVERINGS OF RACKS AND QUANDLES – PART II 5 (or its restriction
Conj :
Grp → Qnd ) sends a group G to the quandle with underlying set G and operation defined by conjugation x ⊳ a .. = a − xa . Interestingly, the left adjoint of Conj plays an important role in the covering theory of racks and quandles, as it was first noticed byM. Eisermann, and further investigated in Part I. The functor
Conj is right adjoint to the functor
Pth :
Rck → Grp (first defined by D.E. Joyce as
Adconj , see also
Adj in [21]) which sends a rack A to the group Pth( A ) obtained as the following quotient in Grp : F g (U( A )) q A , Pth( A ) .. = F g (U( A )) / hh c − a − x a | a, x, c ∈ A and c = x ⊳ a ii , where hh z | [ ... ] ii stands for the normal subgroup generated by the elements z such that [ ... ] . Theunit pth of this adjunction Pth ⊣ Conj is given by pth A .. = q A η g U( A ) : A → Conj(Pth( A )) : a a .We write ~f .. = Pth( f ) for the image by Pth of a morphism of racks or quandles. We often omit
Conj , U and I from our notations.We explained in Part I how to derive the definition of Pth from looking at equivalence classesof terms in the “language of racks”. Subsequently we showed that an element g in Pth( A ) , whichwe call a path , represents an equivalence class of homotopically equivalent primitive paths – inthe sense of the covering theory of racks. We recall that the action x · g of such a path g ∈ Pth( A ) on an element x ∈ A is defined using the action of generators a ∈ Pth( A ) for which x · a .. = x ⊳ a and x · a − .. = x ⊳ − a (as we saw for primitive paths). A trail ( x, g ) in A is the data of a head x ∈ A and a path g ∈ Pth( A ) , and the endpoint of ( x, g ) is defined by x · g . For each g, h ∈ Pth( A ) and x ∈ A we have that: if e is the neutral element in Pth( A ) , then x · e = x ;moreover, x · ( gh ) = ( x · g ) · h , and ( x ⊳ y ) · g = ( x · g ) ⊳ ( y · g ) ; finally ( x · g ) .. = pth A ( x · g ) = g − xg (see augmented quandle [54]).Given a morphism of racks f : A → B , we have f ( x · g ) = f ( x ) · ~f ( g ) . If A = F r ( X ) for some set X , then Pth( A ) = F g ( X ) acts freely on A , which plays an important role in the characterizationof coverings (see Section 2.4). For the covering theory of quandles, homotopy-equivalence classesof primitive paths are represented by the elements of Pth ◦ ( Q ) , which is the normal subgroupof Pth( Q ) generated by pairs of generators a b − for a and b in Q . This divergence is easilyunderstood via the comparison of free racks and free quandles, as a consequence of the additional idempotency axiom in Qnd (see Part I). It is also the case that if Q = F q ( X ) for some set X ,then Pth ◦ ( Q ) = F g ◦ ( X ) acts freely on Q , where F g ◦ ( X ) is hh ab − | a, b ∈ F g ( X ) ii . Moreover,the foundational concepts of interest for the covering theory (relative centrality) in Rck and
Qnd coincide in the appropriate sense (and in every dimension); see Part I for more explanations.If the abelianization functor ab :
Grp → Ab , into the category of abelian groups Ab , is the leftadjoint to the inclusion of Ab in Grp , and F ab is the free abelian group functor ; then we have thefollowing square of adjunctions (Diagram (3)) describing the relationship between π ⊣ I in Rck (or
Qnd ) and the abelianization in Grp . All but one of the four possible squares of functors belowcommute – ( π , Conj , U , ab ) doesn’t. Moreover, a group is abelian if and only if its conjugationoperation is trivial. Similarly, a surjective group homomorphism ( f : G → H ) is central (in thesense of group theory, i.e., its kernel Ker( f ) ≤ Z ( G ) is in the center of G ) if and only if Conj( f ) is a covering of quandles. The image by Pth of a covering in
Rck is a central extension of groups.
Rck ⊣ π Pth (cid:18) (cid:25) ⊤ Set ⊣ I p w F ab (cid:18) (cid:25) Grp ⊤ Conj R Z ab Ab I p w U R Z (3) FRANÇOIS RENAUD
Towards higher dimensions.
In order to extend the covering theory of racks and quandlesto higher dimensions, we first look at the arrow category
ExtRck (or
ExtQnd ). Given any category C with a chosen class of extensions E (Convention 1.1.3), Ext C refers to the full subcategory ofextensions within the category of morphisms (arrow category) Arr C . A morphism α : f A → f B insuch a category of morphisms is given by a pair of morphisms in C , which we denote α = ( α ⊤ , α ⊥ ) (the top and bottom components of α ) as in the commutative Diagram (4). A ⊤ α ⊤ , p ❚❚ & - ❚❚ f A (cid:12) (cid:18) B ⊤ f B (cid:12) (cid:18) A ⊥ × B ⊥ B ⊤ π ❤❤ ❤❤ π tt u (cid:127) tt A ⊥ α ⊥ , B ⊥ (4)If all morphisms in this commutative diagram are in E , including α ’s so-called comparison map p , then α is said to be a double extension [43]. For our purposes, the class of double extensions E is the appropriate induced class of (two-dimensional) extensions in Ext C .The inclusion I :
CExtRck → ExtRck (and similarly for
I :
CExtQnd → ExtQnd ), of the fullsubcategory of coverings in the category of extensions, admits a left adjoint F : ExtRck → CExtRck . The functor F universally makes an extension into a covering (or central extension).It is said to universally centralize an extension (one-dimensional centralization) in the sameway that π universally trivializes objects (zero-dimensional centralization). The unit η of theadjunction F ⊣ I is defined for an extension f : A → B by η f .. = ( η A , id B ) , where the kernel pair Eq( η A ) of the quotient η A is denoted C f . As mentioned before, it is generated by the pairs ( x ⊳ a ⊳ − b, x ) for x , a , and b in A such that f ( a ) = f ( b ) . Then η f ∈ E is a double extensionmaking CExtRck into an E -reflective subcategory of ExtRck (Convention 1.1.3). This data thenfits into the square of adjunctions
ExtRck ⊣ F . Pth (cid:18) (cid:26) ⊤ CExtRck ⊣ I o u Pth (cid:18) (cid:26) ExtGrp ⊤ Conj S Z ab . CExtGrp I o u Conj S Z (5)where the functors Pth and Conj are the appropriate restrictions of the adjoint pairs inducedby Pth ⊣ Conj between the categories of morphisms above
Rck and
Grp . The functor ab is the centralization functor in Grp sending a surjective group homomorphism f : G → H to the centralextension of groups ab ( f ) : G/ [Ker f, G ] Grp → H obtained from the quotient of the domain of f : G/ [Ker f, G ] Grp where
Ker f is the kernel of f and for any subgroups X and Y ≤ G , the subgroup [ X, Y ] Grp .. = h xyx − y − | x ∈ X, y ∈ Y i ≤ X ∩ Y ≤ G defines the classical commutator fromgroup theory. As before, all squares of functors in Diagram (5) commute, but for the square ( F , Conj , ab , U ) which doesn’t (see Example 3.2.1).1.3. Content.
In Section 2, we first show that categorical Galois theory applies to the adjunction F ⊣ I on the top line of Diagram (5), which we fit into a strongly Birkhoff Galois structure Γ (Section 2.3). Alongside the results of Part I, this mainly consists in the recollection of classicalproperties of double extensions, including a bit of descent theory (see [52, 51] and referencestherein). The rest of the article is then aimed at the characterization and “visualization” ofthe induced notion of Γ -covering , or double central extension of racks and quandles , as it waspreviously done for groups [43], leading to the developments of [31, 27]. Note that the study of IGHER COVERINGS OF RACKS AND QUANDLES – PART II 7 the more technical categorical aspects of Section 2 is not necessary for the readers’ understandingof what follows. Section 2.4 provides a useful transition to the rest of the article, as we recall ourgeneral method for the characterization of coverings, and produce a first visual representation oftrivial Γ -coverings. We define and study the concept of double covering , also called algebraicallycentral double extension of racks and quandles in Section 3. We provide examples, and thedefinition of a meaningful and well-behaved notion of commutator, which captures (in the usualsense) the centralization congruence for objects, extensions and double extensions in the categoryof quandles. We illustrate our methods and definitions via the characterization of normal Γ -coverings, leading to a better understanding of two-dimensional centrality. The concept of doublecovering (or algebraically central double extension of racks and quandles) and the concept of Γ -covering (or double central extension of racks and quandles) are then shown to coincide in Section4. Section 4.3 is dedicated to the centralization of double extensions (i.e. the reflection of thecategory of double extensions on the category of double coverings) leading to the next step ofthe covering theory. Finally, in Section 5 we hint at further research and we adapt the concept of Galois structure with (abstract) commutators in such a way that fits to our context and remainscompatible with the developments in [44].2.
An admissible Galois structure in dimension 2
In order to apply categorical Galois theory (see Paragraph 1.1.2) to the inclusion
I :
CExt
C →
Ext C (where C stands for Rck or Qnd ) of the category of coverings of racks (or quandles) inthe category of extensions, we first fit the reflection F ⊣ I into a Galois structure Γ .. =( Ext C , CExt C , F , I , η , ǫ , E ) , satisfying the conditions of Convention 1.1.3. In order to do so,we need an appropriate class of extensions in dimension 2. In dimension 1, the base category C = Rck or C = Qnd is finitely cocomplete and
Barr-exact [1], like any variety of algebras. Inshort this means that C has finite limits and colimits, it is regular (i.e. every morphism factorsuniquely, up to isomorphism, into a regular epimorphism, followed by a monomorphism, andthese factorizations are stable under pullbacks) and, moreover, every equivalence relation is thekernel pair of its coequalizer . In such a context, a fruitful class of extensions is given by theclass of regular epimorphisms . However, for a general Barr-exact C , the category Ext C is notnecessarily Barr-exact (see comment preceding Definition 3.4 [31]); ExtRck and
ExtQnd even failto be regular categories (see below). The class of regular epimorphisms is then not appropriatefor applying Galois theory. As mentioned before, the appropriate class of extensions (in thecategory of extensions) is given by the class of double extensions ( E ).Given C and E as above, let us briefly recall some well known basic properties of the category Ext C , full subcategory of extensions within Arr C . Limits in Arr C are computed component-wise .Given a diagram D in Arr C , compute the limits L ⊤ and L ⊥ in C of the diagrams obtained as thetop component of D and the bottom component of D respectively. The limit l : L ⊤ → L ⊥ of D in Arr C is given by the induced comparison map between L ⊤ and L ⊥ . Using the regularity of C ,limits can be computed in Ext C as the regular epic part e of the regular epi-mono factorization l = me of the limit in Arr C (precompose the legs of the limit L ⊥ with the mono part m toobtain the bottom legs of the limit e in Ext C ). Pushouts in Ext C are computed component-wisein C . The initial object is the identity on the initial object of C . The coequalizer of a parallelpair of morphisms in Ext C is computed component-wise in C , and the resulting commutativesquare is a pushout square of regular epimorphisms. Given a morphism α in Ext C which is apushout-square of regular epimorphisms in C , it is the coequalizer of its kernel pair computedin Ext C . Regular epimorphisms in Ext C are thus the same as (oriented) pushout squares ofregular epimorphisms in C . Monomorphisms are morphisms for which the top component is amonomorphism in C . Regular epi-mono factorizations exist, and are unique in Ext C , howeverthese might not be pullback stable in general. FRANÇOIS RENAUD
Remark 2.0.1.
When C is Rck or Qnd , regularity of
Ext C would imply that the category ofsurjective functions ExtSet is regular (since
ExtSet is closed under regular quotients and limitsin
Ext C ). We recall that not all regular epimorphisms in ExtSet are pullback stable (see also [50,Remark 3.1] and Remark 2.2.4). Since C is Barr exact, ExtSet is equivalent to the category
ERSet of (internal) equivalence relations over
Set . Using the arguments from [49, Section 2] , a regularepimorphism in
ERSet is given by a morphism ¯ α : Eq( f A ) → Eq( f B ) and a surjective morphism α ⊤ : A ⊤ → B ⊤ that commute with the projections of the equivalence relations Eq( f A ) ⇒ A ⊤ and Eq( f B ) ⇒ B ⊤ (as in the top-right corner of the commutative Diagram (18) ) and such that ( b, b ′ ) ∈ Eq( f B ) if and only if there exists a finite sequence ( a , a ′ ) , . . . , ( a n , a ′ n ) ∈ Eq( f A ) with b = α ⊤ ( a ) , α ⊤ ( a ′ i ) = α ⊤ ( a i +1 ) for i ∈ { , . . . , n − } and α ⊤ ( a ′ n ) = b ′ [49, Proposition2.2] . Such a morphism is a pullback stable regular epimorphism if and only if it is a regularepimorphism such that ( b, b ′ ) ∈ Eq( f B ) if and only if there exists ( a, a ′ ) ∈ Eq( f A ) with b = α ⊤ ( a ) and α ⊤ ( a ′ ) = b ′ [49, Proposition 2.3(b)] . We adapt [58, Example 2.4] to this context: define A ⊤ = { (0 , a ) , (0 , b ) , (1 , a ) , (1 , b ) } , B ⊤ = { , , } and α ⊤ such that α ⊤ (0 , a ) = 0 , α ⊤ (1 , b ) = 2 and α ⊤ (0 , b ) = α ⊤ (1 , b ) = 1 ∈ B ⊤ . If Eq( f A ) is the equivalence relation generated by the pairs ((0 , a ) , (1 , a )) and ((0 , b ) , (1 , b )) ; and Eq( f B ) is B ⊤ × B ⊤ , then the pair (¯ α, α ⊤ ) defines a regularepimorphism in ERSet , but it is not pullback stable. Indeed, its pullback along the inclusion of { , } × { , } ⇒ { , } in B ⊤ × B ⊤ ⇒ B ⊤ is not a regular epimorphism. It is convenient to bring back a problem or computation in
Ext C to a couple of problems andcomputations in C , using the projections on the top and bottom components (this component-wise decrease in dimension is essential for the inductive approach to higher covering theory [27]).“From an engineering perspective”, our interest in the concept of a double extension lies in the factthat pullbacks of such, and subsequently many other constructions involving double extensions,can be computed component-wise in C .2.1. Basic properties of double extensions.
In short, we hope for the class of double ex-tensions to have as many good properties in
Ext C as the class of regular epimorphisms has inthe Barr-exact category C . Note that since double extensions are regular epimorphisms in Ext C ,a double extension which is a monomorphism in Ext C is an isomorphism. Then observe thatProposition 3.5 and Lemma 3.8 from [31] easily generalize to our context as it was observed in[30] (Example 1.11, Proposition 1.6 and Remark 1.7). For any regular [1] category C : Lemma 2.1.1. (1) If a morphism α = ( α ⊤ , α ⊥ ) in Ext C is such that α ⊤ is an extension and α ⊥ an isomorphism, then α is a double extension.(2) Double extensions are closed under composition.(3) Pullbacks along double extensions (exist in Ext C and) are computed component-wise .Moreover the pullback of a double extension is a double extension. Given a pair of morphisms α = ( α ⊤ , α ⊥ ) : f A → f B and γ = ( γ ⊤ , γ ⊥ ) : f C → f B in Ext C , their component-wise pullback is given by the following commutative diagram in C C ⊤ × B ⊤ A ⊤ π A ⊤ , π C ⊤ (cid:12) (cid:18) f z (cid:4) A ⊤ α ⊤ (cid:12) (cid:18) f A ⑧⑧⑧ z (cid:4) ⑧⑧⑧ C ⊥ × B ⊥ A ⊥ π A ⊥ , π C ⊥ (cid:12) (cid:18) A ⊥ α ⊥ (cid:12) (cid:18) C ⊤ f C z (cid:4) ⑧⑧⑧⑧⑧⑧⑧ γ ⊤ , B ⊤ f B z (cid:4) ⑧⑧⑧⑧⑧⑧⑧ C ⊥ γ ⊥ , B ⊥ IGHER COVERINGS OF RACKS AND QUANDLES – PART II 9 where the front and back faces are pullbacks – i.e. it is the pullback of α and γ in the arrowcategory Arr C . Provided that α is a double extension, Lemma 2.1.1 above says that f is anextension and the pullback of α and β in Ext C is given by f together with the projections ( π A ⊤ , π A ⊥ ) and ( π C ⊤ , π C ⊥ ) , where the latter is actually a double extension. In particular, thekernel pair of a double extension α = ( α ⊤ , α ⊥ ) exists in Ext C and is given by the kernel pairs of α ⊤ and α ⊥ in each component (together with the induced morphism between those – see Notation3.3.3). Moreover, the legs of such kernel pairs are also double extensions (see Lemma 2.2.1).Lemma 2.1.1 is important for what follows, if only because pullbacks along extensions appeareverywhere in categorical Galois theory. As we mentioned earlier, if neither α or β is knownto be a double extension, their pullback in Ext C still exists, but it is not necessarily computedcomponent-wise and thus it is badly behaved. As we move to higher dimensions, these generalpullbacks are no longer convenient for our purposes.Note that in the context of Barr-exact Mal’tsev categories [14], double extensions are thesame as pushout squares of extensions, and, as a rule, higher extensions are easier to identify –primarily using split epimorphisms. Note that the lack of such arguments is a challenge in ourmore general context where categories are not Mal’tsev categories.As we may conclude from [30, Proposition 3.3] and the fact that our categories are not
Mal’tsev ,the axiom (E4) of “right-cancellation” considered there (see also [31, Lemma 3.8]) cannot hold inour context. We have the following weaker version:
Lemma 2.1.2.
If the composite αβ is a double extension in Ext C , and β is a commutative squareof extensions in C , then α is a double extension.Proof. Since α ⊤ β ⊤ and α ⊥ β ⊥ are regular epimorphisms, α is a square of extensions in C . The restof the proof is an easy exercise. (cid:3) In particular we deduce that pullbacks along double extensions reflect double extensions.
Corollary 2.1.3.
Given a morphism α = ( α ⊤ , α ⊥ ) in Ext C , if its pullback along a double extension β yields a double extension α ′ , then α is itself a double extension. Observation 2.1.4.
Finally we note that if a composite of two double extensions αβ is a pullbacksquare, then both α and β are easily shown to be pullback squares. Beyond Barr exactness, effective descent along double extensions.
Given a Galoisstructure Γ as in Convention 1.1.3, Γ -coverings, which are the key concept of study, are defined asthose extensions c : A → B for which there is another extension e : E → B such that the pullback t of c along e is a trivial Γ -covering. In most references [5, 37, 44], e is further required to beof effective descent or effective E -descent (see [52, 51] and references therein). Such extensionsare sometimes also called monadic extensions [42, 29]. In the contexts of interest for this work,we shall always have that all our extensions are of effective E -descent, which is why we use thissimplified definition of covering. The idea is to ask that “pulling back along e is an algebraic operation”, which is necessary for the “information about coverings to be tractable in X ” in thesense of the fundamental theorem of categorical Galois theory (see for instance [42, Corollary5.4]). We didn’t insist on this requirement for Part I (see also [46]) since the class of effectivedescent morphisms in a Barr-exact C is well known to be the class of regular epimorphisms [52].Given an extension e : E → B in C , if we write Arr ( Y ) for the category of morphisms withcodomain Y , then there is an induced pair of adjoint functors: e ∗ : Arr ( E ) → Arr ( B ) , left adjointof e ∗ : Arr ( B ) → Arr ( E ) , where f ∗ ( k : X → A ) .. = f k , and f ∗ ( h : Y → B ) is given by the pullbackof h along f . This adjunction also restricts to the categories of extensions above E and B : e ∗ | : Ext ( E ) → Ext ( B ) , left adjoint of e ∗ | : Ext ( B ) → Ext ( E ) (defined similarly). We say that e is of effective (global) descent if e ∗ is monadic , and e is of effective E -descent if e ∗ | is monadic (see [53]). Let us add for the interested reader that, in order to prove the fundamental theoremof categorical Galois theory, G. Janelidze showed that (in an admissible Galois structure) if e ∗ | is monadic and we write T for the monad induced by e ∗ | ⊣ e ∗ | [59], then the category of thoseextensions which are split by e is equivalent to the category of those Eilenberg-Moore T -algebras [59] ( f : X → A, µ : T f → f ) such that f : X → A is a trivial covering (see for instance [42,Proposition 4.2,Theorem 5.3]).In this section we show that double extensions of racks and quandles are of effective globaland E -descent in the category of extensions. From Lemma 3.2 [30] and the above, we have whatcan be understood as local E -Barr exactness: Lemma 2.2.1.
Assuming that C is Barr-exact, and given a commutative square of extensionstogether with the horizontal kernel pairs and the factorization f between them; Eq( σ ⊤ ) f (cid:12) (cid:18) ( ∗ ) , , E ⊤ σ ⊤ , f E (cid:12) (cid:18) B ⊤ f B (cid:12) (cid:18) Eq( σ ⊥ ) , , E ⊥ σ ⊥ , B ⊥ (6) then, the right hand square is a double extension if and only if any of the two left hand (commu-tative) squares is an extension.If so, then σ = ( σ ⊤ , σ ⊥ ) is the coequalizer in Ext C of the parallel pair ( ∗ ) on the left, which isin turn the kernel pair of σ . Such an equivalence relation f = Eq( σ ) in Ext C is stably effective in the sense that it is the kernel pair of its coequalizer, and any pullback of its coequalizers is stilla regular epimorphism (see for instance [51, Section 2.B] ). In particular, double extensions arethe coequalizers of their kernel pairs (computed component-wise in C ).Proof. The first part is a direct consequence of Lemma 3.2 [30]. Since the component-wisecoequalizer σ of ( ∗ ) is a pushout square, it coincides with the coequalizer in Ext C . Then ( ∗ ) isthe kernel pair of σ since pullbacks along double extensions are computed component-wise. It isstably effective because everything is computed component-wise, and C is Barr-exact. (cid:3) Note that we also have the classical result, see for instance [1, Example 6.10], which is calledthe
Barr-Kock
Theorem in [8, Theorem 2.17]. From there we easily obtain (as in Remark 4.7[31], or Lemma 3.2 (2) [37]):
Lemma 2.2.2.
Double extensions are of effective (global) descent in
Ext C .Proof. Let σ : f E → f B be a double extension. The monadicity of σ ∗ in each component σ ∗ ⊤ and σ ∗ ⊥ (see [52]) easily yields the monadicity of σ ∗ itself. For instance, we use the characterizationin terms of discrete fibrations [51, Theorem 3.7].Consider a discrete fibration of equivalence relations f R : R ⊤ → R ⊥ above the kernel pair Eq( σ ) : Eq( σ ⊤ ) → Eq( σ ⊥ ) of σ = ( σ ⊤ , σ ⊥ ) as in the commutative diagram of plain arrows below.Then observe that f R is computed component-wise and consists in a comparison map betweena pair of discrete fibrations of equivalence relations R ⊤ and R ⊥ , above the pair of kernel pairs Eq( σ ⊤ ) and Eq( σ ⊥ ) with comparison map Eq( σ ) : Eq( σ ⊤ ) → Eq( σ ⊥ ) . The projections of theequivalence relation f R are also double extensions as the pullback of the projections of Eq( σ ) which are themselves double extensions by Lemma 2.2.1. We build the square ( ∗ ) on the right,first by taking the coequalizer γ = ( γ ⊤ , γ ⊥ ) of f R , which is computed component-wise (see 2.2.1again). The factorization ( ¯ β ⊤ , ¯ β ⊥ ) is then obtained by the universal property of γ . f R ˆ β (cid:12) (cid:18) , , f C ( ∗ )( γ ⊤ ,γ ⊥ ) , ❴❴❴❴ β (cid:12) (cid:18) f D ( ¯ β ⊤ , ¯ β ⊥ ) (cid:12) (cid:18) Eq( σ ) , , f E σ , f B IGHER COVERINGS OF RACKS AND QUANDLES – PART II 11
By Lemma 2.2.1, f R is the kernel pair of γ which is a double extension. Also ( ∗ ) is a pullbacksquare as it is component-wise by [8, Theorem 2.17]. Finally γ is pullback stable as a coequalizer,since everything is computed component-wise (see Lemma 2.2.1). (cid:3) What is exactly needed in our context is not effective global descent but effective E -descent.This derives from Lemma 2.2.2 because of Corollary 2.1.3, as it is explained in [52, Section 2.7]. Corollary 2.2.3.
Double extensions are of effective E -descent in Ext C . Remark 2.2.4.
It was shown in [28] that given a regular category C , Ext ( C ) is regular if and onlyif its effective global descent morphisms are the regular epimorphisms (i.e. the pushout squares ofregular epimorphisms). As far as we know, in the categories of racks and quandles, the classes ofeffective global and E -descent morphisms contain the class of double extensions and are strictlycontained in the class of regular epimorphisms. We do not need to characterize these moreprecisely for what follows. Strongly Birkhoff Galois structure.
In order for categorical Galois theory (and in par-ticular its fundamental theorem) to hold in the context of a Galois structure such as Γ fromConvention 1.1.3, Γ is further required to be admissible , in the sense of [46, 5], which impliesfor instance that pullbacks of unit morphisms along primitive extensions are unit morphisms, orsubsequently that coverings, normal coverings and trivial coverings are preserved by pullbacksalong extensions. We actually work with a stronger property for our Galois structures, whichwe require to be strongly Birkhoff in the sense of [31, Proposition 2.6], where this condition isshown to imply the admissibility condition. The Galois structure Γ is said to be strongly Birkhoff if reflection squares at extensions are double extensions. Given f : A → B in C , the reflectionsquare at f (with respect to Γ ) is the morphism ( η A , η B ) with domain f and codomain I F ( f ) in Arr ( C ) . A η A , p ❯❯❯ & - ❯❯❯ f (cid:12) (cid:18) I F ( A ) I F ( f ) (cid:12) (cid:18) B × I F ( B ) I F ( A ) π ❢❢❢ / ❢❢❢ π ♣♣♣ t | ♣♣♣ B η B , I F ( B ) (7)Our Galois structure Γ is strongly Birkhoff if the reflection squares at double extensions (asdefined in C , for C = Rck or C = Qnd ) should be double extensions in
Ext C , which defines theconcept of -fold extension (see [31, 27]). Definition 2.3.1.
Given any regular category C , define the category Ext C whose objects aredouble extensions (as in Diagram (4) ) and whose morphisms ( σ, β ) : γ → α between two doubleextensions γ and α are given by the data of the (oriented) commutative diagram in Ext C (on theleft) or equivalently in C (on the right): f C σ =( σ ⊤ ,σ ⊥ ) , π =( π ⊤ ,π ⊥ ) ) ❑❑❑❑❑ γ =( γ ⊤ ,γ ⊥ ) (cid:12) (cid:18) f Aα =( α ⊤ ,α ⊥ ) (cid:12) (cid:18) f P ❦❦❦❦❦ ❦❦❦❦❦✟✟✟✟✟ ~ (cid:8) ✟✟✟✟✟ f D β =( β ⊤ ,β ⊥ ) , f B C ⊤ σ ⊤ , γ ⊤ (cid:12) (cid:18) f C z (cid:4) ⑧⑧⑧⑧⑧⑧⑧ A ⊤ α ⊤ (cid:12) (cid:18) f A ⑧⑧⑧ z (cid:4) ⑧⑧⑧ C ⊥ σ ⊥ , γ ⊥ (cid:12) (cid:18) A ⊥ α ⊥ (cid:12) (cid:18) D ⊤ f D z (cid:4) ⑧⑧⑧⑧⑧⑧⑧ β ⊤ , B ⊤ f B z (cid:4) ⑧⑧⑧⑧⑧⑧⑧ D ⊥ β ⊥ , B ⊥ , (8) where f P is the pullback of α and β . A -fold extension ( σ, β ) in C is given by such a morphism in Ext C such that σ , β and the comparison map π are also double extensions i.e. a -fold extensionin C is the data of a double extension in Ext C . Note that most results from Sections 2.1 and 2.2generalize to -fold extensions and higher extensions in our context (see Part III). Now since all but the “very top” component of the centralization units in higher dimension(such as η above) are identities (see Corollary 5.2 [40]), we can break down the strong Birkhoffcondition in two steps: first the closure by quotients of ( -fold) coverings along double extensions,or equivalently, the fact that reflection squares at double extensions are pushout squares in Ext C (Birkhoff condition); and second, a permutability condition , in the base category C , on the kernelpair of the non-trivial component of the centralization unit η . From Section 3.4.5 in Part I, weget: Theorem 2.3.2.
The Galois structure Γ .. = ( Ext C , CExt C , F , η , ǫ , E ) , where C is either Rck or Qnd , is strongly E -Birkhoff, i.e. given a double extension of racks or quandles α =( α ⊤ , α ⊥ ) : f A → f B (as in Diagram (4) ), the reflection square at α (with respect to the reflection F ⊣ I ) is a -fold extension, i.e. the reflection square’s comparison map is a double extension,and it defines a cube of double extensions in Ext C .Proof. Since the bottom component of η is an isomorphism, it suffices to show that the topcomponent is a double extension for the whole cube to be a -fold extension. This was shown inCorollary 3.4.8 of Part I. (cid:3) In particular this justifies the study of Γ -coverings and the relative second order centrality inthe categories of racks and quandles. Remark 2.3.3.
A consequence of the strong Birkhoff condition is that if γ is a morphism of Ext C , and γ factorizes as γ = αβ , where α and β are double extensions, then if γ is a trivial Γ -covering, by Observation 2.1.4, both β and α are trivial Γ -coverings. Hence if γ is a Γ -covering(see Convention 1.1.3 or Section 2.4 below), then both α and β are Γ -coverings. From there, andby the fact that Γ -covering are reflected by pullbacks along double extensions (see Convention1.1.3), it is easy to conclude that Γ -coverings are closed under quotients along -fold extensionsin Ext C . Towards higher covering theory.
The main aim of this article is to describe what arethe double central extensions of racks and quandles as in the case of groups [43]. Following themore general terminology for coverings , this consists in characterizing the Γ -coverings of racksand quandles. These are defined abstractly in ExtRck (or
ExtQnd ) as the double extensions α : f A → f B for which there exists a double extension σ : f E → f B , such that σ splits α , i.e. thepullback of α along σ yields a trivial Γ -covering (see Convention 1.1.3).2.4.1. Projective presentations in dimension 2.
In Part I (Section 1.3.3) we reminded ourselvesthat a double extension α : f A → f B is split by some double extension σ : f E → f B if and onlyif α can be split by a projective presentation of its codomain f B – provided such a projectivepresentation exists. Hence we want to recall that given any Barr-exact category C , if we chooseextensions to be the regular epimorphisms in C , extensions in C with projective domain andprojective codomain are projective objects in Ext C (with respect to double extensions – see forinstance [27, Section 5]). Note that when C is a variety of algebras, and F : Set → C is the leftadjoint (with counit ǫ ) of the forgetful functor U :
C →
Set , the canonical projective presentationof an object B in C is given by the counit morphism ǫ B : F ( B ) → B (where we omit to write U ).Given an extension f B : B ⊤ → B ⊥ in such a C , we define the canonical projective presentation of f B to be the double extension p f B : p B → f B , below, where P .. = F ( B ⊥ ) × B ⊥ B ⊤ , is the pullback IGHER COVERINGS OF RACKS AND QUANDLES – PART II 13 of f B and ǫ B ⊥ . F ( P ) p ⊤ fB , ǫ P $ , ◗◗◗◗ p B (cid:12) (cid:18) B ⊤ f B (cid:12) (cid:18) P π ❥❥❥ ❥❥❥ π ②② w (cid:2) ②② F ( B ⊥ ) p ⊥ fB .. = ǫ B ⊥ , B ⊥ (9)2.4.2. Trivial Γ -coverings. Now we want to be able to identify when the pullback of a doubleextension α is a trivial Γ -covering in Ext C (where C stands for Rck or Qnd ). As usual, because theGalois structure Γ is strongly Birkhoff, trivial Γ -coverings are easy to characterize. Rememberthat trivial Γ -coverings are those double extensions in Ext C which “behave exactly like” theprimitive double extensions, i.e. those double extensions in CExt C – see for instance [46, Section1.3] and Example 2.4.5 below.From Part I we know that trivial ( -fold) coverings of racks (or quandles) are characterizedas those extensions that reflect loops , which are trails ( x, g ) whose endpoint y = x · g coincidewith the head x . Further remember from Paragraph 3.1.9 of Part I, that given a morphism ofracks (or quandles) f : A → B , an f -membrane M = (( a , b ) , (( a i , b i ) , δ i ) ≤ i ≤ n ) is the data ofa primitive trail in Eq( f ) , whose length is the natural number n , and whose endpoints a M and b M are the endpoints of the trails in C obtained via the projections of Eq( f ) . An f -horn is an f -membrane M = (( a , b ) , (( a i , b i ) , δ i ) ≤ i ≤ n ) such that x .. = a = b . It is said to close (into adisk) if moreover the endpoints coincide a M = b M . It is said to retract if for each ≤ k ≤ n ,the truncated horn M ≤ k .. = ( x, ( a i , b i , δ i ) ≤ i ≤ k ) closes. Finally, the associated f -symmetric pair of the membrane or horn M is given by the paths g Ma .. = a δ · · · a nδ n and g Mb .. = b δ · · · b nδ n in Pth( A ) ; in general, an f -symmetric path is a path g ∈ Pth ◦ ( A ) , such that g = g Ma ( g Mb ) − for some membrane M as above. These definitions were used in Part I to characterize a generalelement in the aforementioned centralization congruence C A of some extension f : A → B . Weshowed that ( x, y ) ∈ C A if and only if x · g = y for some f -symmetric path g . We repeat thisapproach for the two-dimensional context in Section 3. For now we observe that: Lemma 2.4.3. If α : f A → f B is a double extension in Rck (or in
Qnd ), then the followingconditions are equivalent:(1) α is a trivial Γ -covering;(2) any f A -horn which is sent by α ⊤ to a f B -disk in B ⊤ , actually closes into a disk in A ⊤ ; • } (cid:7) ✝✝✝✝ g a ✝✝✝✝ (cid:23) ! ✽✽✽✽ g b ✽✽✽✽ f A f A • f A • 7−→ α ⊤ ( • ) (cid:13) (cid:20) α ⊤ ( g a ) (cid:12) (cid:18) α ⊤ ( g b ) f B f B f B • ! = ⇒ • (cid:7) (cid:14) g a (cid:12) (cid:18) g b f A f A f A • = • (3) α ⊤ reflects f A -symmetric loops, in the sense that if the image by α ⊤ of an f A -symmetrictrail ( x, g ) loops in B ⊤ , then the trail was already a loop in A ⊤ : x · g = x .In what follows, we prefer to call a double extension α which satisfies these conditions a trivialdouble covering . This terminology will be justified by Theorem 4.2.2 where we characterize Γ -coverings to be the double coverings from Definition 3.0.1 below.Proof. Using the material from Section 2.3, we observe that our definition of trivial Γ -coveringfrom Convention 1.1.3 coincides, for an admissible or strongly Birkhoff Galois structure Gamma ,with the more common definition: the extension t is a trivial Γ -covering if and only if the reflectionsquare at t is a pullback. Hence α is a trivial Γ -covering (ore trivial double covering) if andonly if the reflection square at α is a pullback. Since pullbacks along double extensions are computed component-wise, and the bottom component is trivial, it suffices to check that thediagram below, where P .. = F i1 ( A ⊤ ) × F i1 ( B ⊤ ) B ⊤ , A ⊤ α ⊤ , p ❘❘ $ , ❘❘ η A ⊤ (cid:12) (cid:18) B ⊤ η B ⊤ (cid:12) (cid:18) P π ❤❤❤❤ ❤❤❤❤ π ②② w (cid:2) ②② F i1 ( A ⊤ ) F i1 ( α ⊤ ) , F i1 ( B ⊤ ) , is a pullback square, i.e. the comparison map p should be an isomorphism. Since, ( α ⊤ , F i1 ( α ⊤ )) isalready a double extension by Corollary 3.4.8 of Part I, it suffices to check that Eq( α ⊤ ) ∩ C ( f A ) =∆ A ⊤ (the diagonal relation on A ⊤ ). Now any element ( a, b ) ∈ C ( f A ) is either such that a and b are the endpoints of a f A -horn, or equivalently, a and b are respectively the head and endpointof an f A -symmetric trail (i.e. a trail whose path component is f A -symmetric). (cid:3) Example 2.4.4.
As a consequence of the fact that F ⊣ I is E -reflective (using Observa-tion 2.1.4) (or simply by Lemma 2.4.3 above): if the comparison map p of a double extension α : f A → f B is an isomorphism (i.e. if α is a pullback square), then both α and ( f A , f B ) aretrivial double coverings (i.e. trivial Γ -coverings). Example 2.4.5.
Since any primitive double extension (i.e. a double extensions whose domainand codomain are ( -fold) coverings) is a trivial double covering and coverings are closed underquotients along double extensions [60] , if α : f A → f B is a double extension and f A is a covering,then α : f A → f B is a trivial double covering (i.e. a trivial Γ -covering). Note that the concept of trivial double covering is not symmetric in the role of ( α ⊤ , α ⊥ ) and ( f A , f B ) . It is not true that in general ( α ⊤ , α ⊥ ) is a trivial double covering if and only if thedouble extension ( f A , f B ) is one. Example 2.4.6.
Consider the sets Q = {• , ⋆ } , Q = {• , ⋆ , ⋆ } and Q .. = { ⋆ , ⋆ , • , • } as well as the morphisms t ⋆ : Q → Q and t : Q → Q , which identify the bullets with • andthe stars with ⋆ . We write Q .. = { ⋆ , ⋆ , ⋆ , ⋆ , • , • } for the pullback of t ⋆ and t suchthat the first projection π : Q → Q identifies ⋆ with ⋆ and ⋆ with ⋆ , and symmetricallyfor the second projection, which moreover identifies • with • . Q π , , π (cid:12) (cid:18) (cid:12) (cid:18) Q t ⋆ (cid:12) (cid:18) (cid:12) (cid:18) Q t , , Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ⋆ π π ⋆ π ⋆ π ⋆ • π • Define the involutive ( ⊳ − = ⊳ ) quandle Q .. = { ⋆ , ⋆ , ⋆ , ⋆ , • , • ′ • } such that • ⊳ ⋆ = • ⊳ ⋆ = • ′ , • ′ ⊳ ⋆ = • ′ ⊳ ⋆ = • and x ⊳ y = x for any other choice of x and y in Q . Thefunction p : Q → Q which identifies • ′ with • is a surjective morphism of quandles. Then notethat the double extension ( π p, t ⋆ ) is a trivial double covering since π p is a covering. However,the double extension ( π p, t ) is not a double trivial covering since • ⊳ ⋆ = • ⊳ ⋆ even thoughtheir images by π p coincide. Finally, we give an example of trivial double covering which doesn’t arise as an instance ofExamples 2.4.4 and 2.4.5.
Example 2.4.7.
As it is explained in [21, Example 1.14] , a dihedral quandle D n is the involutivequandle obtained from the (additive) cyclic group Z n .. = Z /n Z = { , . . . , n } by x ⊳ y .. = 2 x − y ,for x and y in D n . Note that D and D are the trivial quandles (i.e. sets) with one and two IGHER COVERINGS OF RACKS AND QUANDLES – PART II 15 elements respectively. For n > , D n is the subquandle Z n ⋊ { } of the conjugation quandle Z n ⋊ Z , corresponding to the n reflections of the regular n -gon. In general it injects intothe circular quandle S defined on the unit circle in R by the “central symmetries along S ”: x ⊳ y .. = 2 h x, y i y − x , for each x and y in S , such that − ⊳ y defines the unique involution whichfixes y and sends x to − x whenever x and y are orthogonal (see [21, Section 3.6] ).Now given two natural numbers n and m we have that D nm is the product of D n and D m . Weconsider the following double extension of dihedral quandles in Qnd where for j ∈ N , ¯0 : D j → D ⊥ is the terminal map to D ⊥ and for i = 0 in N , the morphism ¯ i : D ij → D i sends x ∈ D ij to x mod i in D i : D nm ¯ m , ¯ nm % , ❙❙❙ ¯ n (cid:12) (cid:18) D m ¯0 (cid:12) (cid:18) D nm ¯ m ❤❤❤ ❤❤❤ ¯ n ✇✇ w (cid:1) ✇✇ D n ¯0 , { } , (10) Note that this double extension is symmetric in the roles of m and n . By Lemma 2.4.8 below, both (¯ n, ¯0) and ( ¯ m, ¯0) are trivial double covering whenever , m and n are coprime. If m = 2 and n arecoprime, then (¯ n, ¯0) is a trivial double covering but ( ¯ m, ¯0) is not (indeed ⊳ = 2 n = 0 ⊳ n ).See also Example 3.0.3 below. Lemma 2.4.8.
Using the notations from Example 2.4.7, the double extension (¯ n, ¯0) is a trivialdouble covering if and only if x = 0 mod n whenever mx = 0 mod n .Proof. Consider an ¯ m -horn M of length i > ∈ N which is sent to a loop by ¯ n . Such a horn M is given by the data of x ∈ D nm as well as natural numbers y j < m , a j < n and b j < n for each ≤ j ≤ i such that x + X ≤ j ≤ i ( − j y j + 2 m X ≤ j ≤ i ( − j a j = x + X ≤ j ≤ i ( − j y j + 2 m X ≤ j ≤ i ( − j b j mod n ; (11)and thus also m (cid:0) P ≤ j ≤ i ( − j ( a j − b j ) (cid:1) = 0 mod n . Now if P ≤ j ≤ i ( − j ( a j − b j ) = 0 mod n ,then Equation 11 also holds modulo nm , and the horn M closes in D nm . Conversely if Equation11 holds modulo nm , we deduce that P ≤ j ≤ i ( − j ( a j − b j ) = 0 mod n . (cid:3) Double coverings
The concepts of covering or relatively the concepts of centrality induced by the Galois theoryof racks and quandles are expressed, in each dimension, via the trivial action of certain data.In dimension zero, a rack A ⊤ is actually a set if any element a ∈ A ⊤ acts trivially on A ⊤ . Indimension 1, an extension f A : A ⊤ → A ⊥ is a covering if given elements a and b ∈ A ⊤ , such that ( a, b ) ∈ Eq( f A ) (i.e. f A ( a ) = f A ( b ) ), the action of a b − is trivial: x⊳ a⊳ − b = x for all x ∈ A ⊤ . Indimension 2, we work with double extensions α = ( α ⊤ , α ⊥ ) : f A → f B . The data we are interestedin is then given by those × matrices with entries in A ⊤ , whose rows are elements in Eq( f A ) and whose columns are elements in Eq( α ⊤ ) . -dimensional : · a -dimensional : a f A b -dimensional : a f A α ⊤ b α ⊤ d f A c Such × matrices characterize the elements of Eq( f A ) (cid:3) Eq( α ⊤ ) , namely the largest doubleequivalence relation above Eq( f A ) and Eq( α ⊤ ) [15, 62, 4, 48], also called double parallelisticrelation in [6, Definition 2.1, Proposition 2.1]. We sometimes write these elements as quadruples ( a, b, c, d ) ∈ Eq( f A ) (cid:3) Eq( α ⊤ ) which encode the entries of the corresponding × matrix as above. Their “trivial action on the elements of A ⊤ ” is the condition we are interested in. We define doublecoverings of racks and quandles and later show that these coincide with the Γ -coverings. Definition 3.0.1.
A double extension of racks (or quandles) α : f A → f B (as in Diagram (4) ) issaid to be a double covering or an algebraically central double extension if any of the equivalentconditions ( i ) - ( iv ) below are satisfied: ( i ) x ⊳ a ⊳ − b ⊳ c ⊳ − d = x, ( ii ) x ⊳ − a ⊳ d ⊳ − c ⊳ b = x, ( iii ) x ⊳ − a ⊳ b ⊳ − c ⊳ d = x, ( iv ) x ⊳ a ⊳ − d ⊳ c ⊳ − b = x, for all x ∈ A ⊤ and a f A α ⊤ b α ⊤ d f A c ∈ Eq( f A ) (cid:3) Eq( α ⊤ ) . Note that, by the symmetries of quadruples ( a, b, c, d ) in Eq( f A ) (cid:3) Eq( α ⊤ ) , one could equiva-lently use any cyclic permutation of the letters a , b , c , and d in the equalities ( i ) – ( iv ) . Theequivalence between each of these ( i ) – ( iv ) , is shown in Section 3.1. Remark 3.0.2.
In Definition 3.0.1, the roles of f A and α ⊤ are symmetric. Hence ( α ⊤ , α ⊥ ) is adouble covering (or algebraically central) if and only if ( f A , f B ) is a double covering, which canbe viewed as a property of the underlying commutative square in Rck (or
Qnd ). Unlike trivial Γ -coverings (also called trivial double coverings), the Γ -coverings are indeed expected to besymmetric in the same sense (see [27, Section 3] ). Example 3.0.3.
It is easy to show that given a double extension α : f A → f B , if either α or ( f A , f B ) is a trivial double covering, then α is a double covering. Note for instance that given aquadruple ( a, b, c, d ) ∈ Eq( f A ) (cid:3) Eq( α ⊤ ) , the α ⊤ -horn M , displayed below, is sent to a disk by f A . M : x ~ (cid:8) ✟✟✟✟ ab − cd − ✟✟✟✟ (cid:22) ✻✻✻✻ d c − c d − ✻✻✻✻ α ⊤ α ⊤ α ⊤ y α ⊤ x y .. = x · ( a b − c d − ) From Example 2.4.7, when m = 2 and n are coprime, we have that ( ¯ m, ¯0) is not a trivial doublecovering. However, it still satisfies the conditions of a double covering, which can be deducedfrom the fact that (¯ n, ¯0) is a trivial double covering. Example 3.0.4.
Not all double coverings arise from double trivial coverings. Consider thefunction t : Q → Q from Example 2.4.6 and its kernel pair π , π : Q ⇒ Q where the elementsof Q = { ⋆ , ⋆ , ⋆ , ⋆ , • , • , • , • } organise as in the Diagram 12 below. We definethe involutive quandle Q with underlying set Q ∪ {• ′ } such that, for i ∈ { , } , • ⊳ ⋆ ii = • ′ , • ′ ⊳ ⋆ ii = • and x ⊳ y = x for any other choice of x and y in Q . The function p : Q → Q defined by f ( • ′ ) = • and f ( x ) = x for all x ∈ Q ⊂ Q , is a morphism of quandles such thatthe double extension below is a double covering. Q π ′ , p " * π ′ (cid:12) (cid:18) Q t (cid:12) (cid:18) Q π ❥❥ ❥❥ π ⑧⑧ z (cid:4) ⑧⑧ Q t , Q , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ⋆ π ′ π ′ ⋆ π ′ ⋆ π ′ ⋆ • π ′ π ′ • π ′ • π ′ • • ′ (12) In anticipation of the results of Section 3.4, observe that neither ( π ′ , t ) nor ( π ′ , t ) are normal Γ -coverings since • ⊳ ⋆ = • ⊳ ⋆ even though • ⊳ ⋆ = • ⊳ ⋆ ; and also • ⊳ ⋆ = • ⊳ ⋆ even though • ⊳ ⋆ = • ⊳ ⋆ . IGHER COVERINGS OF RACKS AND QUANDLES – PART II 17
Observation 3.0.5.
Finally we relate our condition (algebraic centrality of double extensions)with the existing concept of abelian quandle (or rack) defined in [54] . If α : f A → f B is a doublecovering of racks (or quandles), then we have that ( a ⊳ d ) ⊳ ( b ⊳ c ) = a ⊳ a ⊳ c = ( a ⊳ b ) ⊳ ( d ⊳ c ) , (13) for each square a bd c in Eq( f A ) (cid:3) Eq( α ⊤ ) or Eq( α ⊤ ) (cid:3) Eq( f A ) , and symmetrically in “eachcorner” of this square (i.e. replace ( a, b, c, d ) in (13) by any cyclic permutation of the quadruple).The converse is not true in general. Thinking about a commutator.
Let A be a rack (or quandle) and ER( A ) be the latticeof (internal) equivalence relations (also called congruences – see for instance [62]), over A . Wedefine the following binary operation on ER( A ) . Definition 3.1.1.
Given a rack A and a pair of congruences R and S in ER( A ) , we define [ R, S ] , element of ER( A ) , as the congruence generated by the set of pairs of elements of A : { ( x ⊳ a ⊳ − b ⊳ c ⊳ − d, x ) | x ∈ A and a RS b S d R c ∈ R (cid:3) S } . Note that [ R, S ] is in particular included in the intersection R ∩ S . Working towards theCorollaries 3.1.4 and 3.1.5 we have that: Lemma 3.1.2.
Given a rack A and a pair of congruences R and S in ER( A ) , then [ R, S ] isgenerated by the set of pairs { ( x ⊳ − a ⊳ d ⊳ − c ⊳ b, x ) | x ∈ A and a RS b S d R c ∈ R (cid:3) S } . Proof.
By definition, [ R, S ] includes the pairs ( x ⊳ − a ⊳ a ⊳ − b ⊳ c ⊳ − d, x ⊳ − a ) for all x , a , b , c and d as in the statement. Then by compatibility with the rack operations and reflexivity, [ R, S ] also includes the pairs ( x, x ⊳ − a ⊳ d ⊳ − c ⊳ b ) , for all such x , a , b , c and d . By symmetry this then induces that [ R, S ] includes the congruencerelation generated by the set of pairs from the statement. Now a similar argument shows thatsuch a congruence includes the set of pairs defining [ R, S ] as in Definition 3.1.1. (cid:3) Corollary 3.1.3.
Given a rack A and a pair of congruences R and S in ER( A ) , the congruence [ S, R ] is generated by the set of pairs { ( x ⊳ − a ⊳ b ⊳ − c ⊳ d, x ) | x ∈ A and a RS b S d R c ∈ R (cid:3) S } . Corollary 3.1.4.
Given a rack A then for any congruences R and S in ER( A ) , the congruence [ R, S ] = [
S, R ] is equivalently generated by any of the set of pairs: ( i ) ( x ⊳ a ⊳ − b ⊳ c ⊳ − d, x ) , ( ii ) ( x ⊳ − a ⊳ d ⊳ − c ⊳ b, x ) , ( iii ) ( x ⊳ − a ⊳ b ⊳ − c ⊳ d, x ) , ( iv ) ( x ⊳ a ⊳ − d ⊳ c ⊳ − b, x ) , for all x ∈ A and a bd c ∈ R (cid:3) S. Proof.
It suffices to show that [ R, S ] contains the pairs ( x ⊳ − a ⊳ b ⊳ − c ⊳ d, x ) for any x ∈ A and ( a, b, c, d ) ∈ R (cid:3) S . Given such data, we compute that b ac d ⊳ b bc c = b ⊳ b a ⊳ bc ⊳ c d ⊳ c ∈ R (cid:3) S. Then by definition [ R, S ] contains the pair ( x ⊳ ( b ⊳ b ) ⊳ − ( a ⊳ b ) ⊳ ( d ⊳ c ) ⊳ − ( c ⊳ c ) , x ) , whichreduces to ( x ⊳ b ⊳ − b ⊳ − a ⊳ b ⊳ − c ⊳ d ⊳ c ⊳ − c, x ) . This concludes the proof. (cid:3) Corollary 3.1.5.
The conditions ( i ) - ( iv ) from Definition 3.1.1 are indeed all equivalent. More-over, a double extension α : f A → f B of racks (or quandles) is a double covering (an algebraicallycentral double extension), if and only if [Eq( f A ) , Eq( α ⊤ )] = ∆ A ⊤ (the diagonal relation on A ⊤ ). Based on this result, and in anticipation of Theorems 4.2.2 and 4.3.1, we call [Eq( f A ) , Eq( α ⊤ )] the centralization congruence of the double extension α : f A → f B . Now observe the following: Lemma 3.1.6.
Given a rack A and a congruence R on A , the congruence [ R, A × A ] is thecongruence generated by the set of pairs { ( x ⊳ a ⊳ − b, x ) | x ∈ A and ( a, b ) ∈ R } .Proof. Write S for the congruence generated by the set of pairs from the statement. Observethat given x , a and b such that ( a, b ) ∈ R we have the square a RA × A b A × A a R a. ∈ R (cid:3) ( A × A ) . By definition we then have S ≤ [ R, A × A ] . Now observe that for any ( a, b ) ∈ R and ( c, d ) ∈ R : ( x ⊳ a ⊳ − b ) S x S ( x ⊳ d ⊳ − c ) are in relation by S , and thus S also contains the generators of [ R, A × A ] . (cid:3) Corollary 3.1.7.
Given a morphism f : A → B in Rck (or
Qnd ), the congruence C ( f ) can becomputed as [Eq( f ) , A × A ] , and in particular f is a covering (in the sense of [21] ) if and onlyif [Eq( f ) , A × A ] = ∆ A . Recall that in the category of groups, we have the classical commutator [ − , − ] Grp , such thata group G is abelian if and only if its commutator subgroup is trivial [ G, G ] Grp = { e } and asurjective homomorphism f : G → H is a central extension if and only if [Ker( f ) , G ] Grp = { e } .Moreover, a double extension of groups γ : f G → f H is a double central extension of groups [43]if and only if [Ker( f G ) , Ker( γ ⊤ )] = { e } and [Ker( f G ) ∩ Ker( γ ⊤ ) , G ⊤ ] = { e } are both trivial.For the zero-dimensional case in our context, the corresponding description of centrality interms of the operation [ − , − ] only works for quandles. Indeed, if x and a are in the quandle A ,then x ⊳ a = x ⊳ − x ⊳ a , which means that ( x ⊳ a, x ) ∈ [ A × A, A × A ] . If A is a rack though, thistrick does not work. In particular we compute that [F r × F r , F r × F r
1] = ∆ A = F r × F r (F r . Corollary 3.1.8.
Given a quandle A , the congruence C ( A ) can be computed as [ A × A, A × A ] ,in particular A is a trivial quandle if and only if [ A × A, A × A ] = ∆ A . Note that in the category of groups, two-dimensional centrality is expressed using two re-quirements. In our context, one of the corresponding requirements entails the other (Corollary3.1.11). First observe that our commutator is monotone . Lemma 3.1.9.
Given a rack A , as well as congruences R , S and T in ER( A ) such that S ≤ T ,then [ R, S ] ≤ [ R, T ] . IGHER COVERINGS OF RACKS AND QUANDLES – PART II 19
Proof.
This is a direct consequence of the fact that R (cid:3) S ≤ R (cid:3) T . (cid:3) Corollary 3.1.10. If R and S are congruences on A such that R ≤ S then [ R, S ] = [
R, A × A ] .Proof. It suffices to show that [ R, S ] contains T .. = h ( x ⊳ − a, x ⊳ − b ) | aRb i . As before, observethat for any aRb , we have the quadruple ( a, b, a, a ) ∈ R (cid:3) S . (cid:3) Corollary 3.1.11. If R and S are congruences on A then [ R ∩ S, A × A ] = [ R ∩ S, S ] ≤ [ R, S ] .In particular, the comparison map p of a double covering α : f A → f B is a covering. Note that the converse of Corollary 3.1.11 is not true in general. For instance, observe thatthe double extension from Diagram (10) of Example 2.4.7 is such that the comparison map ¯ mn : D nm → D nm is always a quandle covering. However when m = 3 and n = 6 , Diagram (10)is not a double covering since ⊳ ⊳ − ⊳ ⊳ − = 0 .In Section 3.2, where we further investigate the relationship with groups, we shall see that theconverse of Corollary 3.1.11 holds for “double coverings of conjugation quandles”. More commentsand results about our commutator can be found in Section 5.3.2. The case of conjugation quandles.
Recall that a conjugation quandle is any quandlewhich is obtained as the image of a group by the functor
Conj :
Grp → Rck . As we remindedourselves in the Introduction, we use the functors
Conj and its left adjoint
Pth to comparethe covering theory of racks and quandles with the theory of central extensions of groups (see[46, 43, 44]). For instance, we mentioned that a surjective group homomorphism is central ifand only if its image is a covering in
Rck (or
Qnd – [21, Examples 2.34;1.2]). However, asthe following example shows, the centralization (in the sense of F ) of a morphism betweenconjugation quandles doesn’t coincide with the (image by Conj of the) centralization (in thesense of ab ) of a group homomorphism in Grp . Example 3.2.1.
Indeed, consider the quotient map q : S → S /A = {− , } in Grp , sending thegroup of permutations of the set of elements to the (multiplicative) group {− , } by quotienting S by A = { () , (123) , (321) } , the alternating subgroup of S . The morphism q sends -cycles to − . Observe that the (classical group) commutator [ S , A ] Grp = A . Hence the centralization of q in Grp is the identity morphism on {− , } . Now observe that x ⊳ x ⊳ − y = z for any -cycles x = y = z . Hence -cycles are also identified by the centralization of q in Qnd . However, theaction of a -cycle on a -cycle always gives the other -cycle. Hence the successive action of apair of -cycles on a -cycle does nothing. Similarly since both -cycles are inverse of each-other, -cycles act trivially on each-other. One easily deduces that if Q ab⋆ is the involutive quandle with elements whose operation is defined in the table below, then the centralization of the morphismof quandles q is obtained via the quotient η S : S → ( S / C q ) = Q ab⋆ , such that η S (123) = a , η S (321) = b and all other elements of S are sent to ⋆ . Finally we obtain F ( q ) : Q ab⋆ → S /A = {− , } which takes the values F ( q )[ a ] = 1 = F ( q )[ b ] and F ( q )[ ⋆ ] = − . ⊳ a b ⋆a a a bb b b a⋆ ⋆ ⋆ ⋆ S / [ S , A ] id ( / ❳❳❳❳❳❳❳❳❳❳❳ S L R (cid:12) (cid:18) q , S /A = {− , } S / (C q ) = Q ab⋆ F ( q ) / ❢❢❢❢❢❢❢❢ In this section we further study how our concept of double covering behaves when applied to theimage of
Conj :
Grp → Rck . First recall that given a group G , and given a path g = g δ · · · g nδ n ∈ Pth(Conj( G )) , there is always another path “of length one” g , where g = g δ · · · g δ n n ∈ G , suchthat x · g = x · g for all x ∈ Conj( G ) . A primitive path in Conj( G ) always “reduces” (as aninner-automorphism, not as a homotopy class – see Paragraph 2.1.8 of Part I) to a one-step primitive path. As a consequence, our notion of double covering simplifies significantly when thequandle operations of interest are derived from the conjugation operation in groups. Note that connectedness in symmetric spaces also reduces to strong connectedness (i.e. connectedness in“one step”) – see [21, Section 3.7] and references therein. Example 3.2.2.
Consider a group G and a pair of surjective group homomorphisms f and h with domain G in Grp . Let us write R .. = Conj(Eq( f )) and S .. = Conj(Eq( f )) (note that Conj preserves limits). In
Qnd one derives easily that [ R, S ] = [ R ∩ S, G × G ] since given asquare ( a, b, c, d ) ∈ R (cid:3) S , we have ( d, ( ab − c )) ∈ ( R ∩ S ) such that moreover x ⊳ ( ab − c ) ⊳ − d = x ⊳ a ⊳ − b ⊳ c ⊳ − d . Now observe that the functor
Pth :
Rck → Grp preserves pushouts, and thus the image by
Pth of a double extension α of racks (or quandles), yields a pushout square of extensions in Grp .Since
Grp is a Mal’tsev category,
Pth( α ) is a double extension as well (see [14] and Proposition5.4 therein). Note however that the comparison map of Pth( α ) in Grp is not the image of thecomparison map of α in Rck or Qnd .In the other direction, the conjugation functor
Conj :
Grp → Rck preserves pullbacks. Hence itsends a double extension of groups γ : f G → f H to a double extension of quandles Conj( γ ) , andit sends the comparison map p of γ to the comparison map Conj( p ) of Conj( γ ) . For a generaldouble extension of racks and quandles α , the comparison map of α being a covering is necessarybut not sufficient for alpha to be a double covering. However, by the example above and thepreceding discussion we have: Proposition 3.2.3.
Given a double extension of groups γ : f G → f H , its image by Conj is adouble covering of quandles if and only if its comparison map
Conj( p ) is a covering in Qnd orequivalently if and only if the comparison map p of γ is a central extension of groups. In particular, the image by
Conj of a double central extension of groups yields a doublecovering in
Qnd . However, one cannot deduce that γ is a double central extension of groups fromthe fact that Conj( γ ) is a double covering. Finally we show that the image by Pth of a doublecovering of quandles is not necessarily a double extension of groups.
Example 3.2.4.
Consider a double extension of groups γ : f G → f H such that k k − = k − k for some k ∈ Ker( f G ) and k ∈ Ker( γ ⊤ ) , but ka = ak for all k ∈ Ker( f G ) ∩ Ker( γ ⊤ ) and a ∈ G ⊤ .For instance, define γ ⊥ : G ⊥ → H ⊥ as the surjective group homomorphism F g ( { a, c } ) → F g ( { c } ) such that γ ⊥ ( c ) = c and γ ⊥ ( a ) = e . Similarly define f H : H ⊤ → H ⊥ as F g ( { b, c } ) → F g ( { c } ) suchthat f H ( c ) = c and f H ( b ) = e . Write P .. = G ⊥ × H ⊥ H ⊤ = F g ( { a, b, c } ) for their pullback,with projections π : P → G ⊥ and π : P → H ⊤ , and take the canonical projective presentation ǫ gP : F g ( P ) → P , obtained from the counit ǫ g of free-forgetful adjunction F g ⊣ U . Compute itscentralization ab ( ǫ gP ) : F g ( P ) / [Ker( ǫ gP ) , F g ( P )] Grp → P , and define f G .. = π ab ( ǫ gP ) . Similarlydefine γ ⊤ .. = π ab ( ǫ gP ) . The resulting double extension of groups is as required.By Proposition 3.2.3, the double extension of quandles Conj( γ ) is a double covering. Howeverwe show that the double extension Pth(Conj( γ )) cannot be a double central extension of groups.First observe that the unit pth Conj( G ⊤ ) : Conj( G ⊤ ) → Conj(Pth(Conj( G ⊤ ))) is a monomorphism,since the identity morphism on Conj( G ⊤ ) factors through it. Now, if e ⊤ is the neutral ele-ment in G ⊤ , then k e ⊤ − ∈ Ker( ~f G ) and e ⊤ k − ∈ Ker( ~α ⊤ ) . Suppose by contradiction that k e ⊤ − e ⊤ k − = e ⊤ k − k e ⊤ − . We have that k k − = k e ⊤ − e ⊤ k − and, by the compatibil-ity of pth Conj( G ⊤ ) with ⊳ , we have, moreover: k − k = ( k ⊳ e ⊤ ) − k ⊳ e ⊤ = ( k ⊳ e ⊤ ) − ( k ⊳ e ⊤ ) = e ⊤ k − e ⊤ − e ⊤ k e ⊤ − = e ⊤ k − k e ⊤ − . Hence we must also have that k k − = k − k and thus k ⊳ − k = k , which implies k ⊳ − k = k . Since pth Conj( G ⊤ ) is injective, we must have k k k − = k ⊳ − k = k ∈ Conj( G ⊤ ) IGHER COVERINGS OF RACKS AND QUANDLES – PART II 21 which is in contradiction with the hypothesis k k − = k − k . Hence it must also be that k e ⊤ − e ⊤ k − = e ⊤ k − k e ⊤ − and Pth( γ ) cannot be a double central extension of groups. Remark 3.2.5.
By anticipation of Theorems 4.2.2 and 4.3.1, we cannot hope for a direct three-dimensional version of the Diagrams 3 and 5 in which the bottom adjunction’s left adjoint wouldbe the centralization of double extensions of groups.
Now in order to further study double coverings (algebraically central double extensions) forgeneral racks and quandles, and their relation to Γ -coverings, we need a characterization forgeneral elements in the centralization congruence [Eq( f A ) , Eq( α ⊤ )] of a double extension α : f A → f B . Think about the transitive closure of the set of pairs from Definition 3.1.1. In order toidentify these general pairs, we make a detour via the generalized notion of primitive trail andthe characterization of normal Γ -coverings.3.3. A concept of primitive trail in each dimension: from membranes to volumes . Similarly to what was studied in dimension zero and one, we shall further be interested in the“action of sequences of two-dimensional data”. Given a rack A in dimension zero, we have thefundamental concept of a primitive path, which is merely a sequence of elements in A × {− , } ,viewed as a formal sequence of symmetries (Part I, Paragraph 2.3.3). Given a rack A , its centralization (or set of connected components) is obtained by identifying elements which are“connected by the action of a primitive path in A ”. In dimension one, the centralization of anextension f : A → B is in some sense obtained by the study of elements which are “linked by theaction on A of a primitive path from Eq( f ) ”, leading to the concept of a membrane (see Paragraph2.4.2 or Part I). Now given a double extension of racks α , we shall be interested in the actionon A ⊤ of primitive paths from Eq( f A ) (cid:3) Eq( α ⊤ ) . We exhibit the 2-dimensional generalizations ofthe lower-dimensional concepts of primitive trail , membrane , and horn . Definition 3.3.1.
Given a pair of morphisms f : A → B and h : A → C in Rck (or
Qnd ), wedefine an h f, h i -volume as the data V = (( a , b , c , d ) , (( a i , b i , c i , d i ) , δ i ) ≤ i ≤ n ) of a primitivetrail in Eq( f ) (cid:3) Eq( h ) . The first quadruple ( a , b , c , d ) is the head of V . We call such an h f, h i -volume V an h f, h i -horn if the head reduces to a point: a = b = c = d = .. x whichwe specify as V = ( x, (( a i , b i , c i , d i ) , δ i ) ≤ i ≤ n ) . Let us define a .. = ( a i ) ≤ i ≤ n and similarly define b , c and d . The associated h f, h i -symmetric quadruple of the volume or horn V is given by thepaths g Va .. = a δ · · · a nδ n , g Vb .. = b δ · · · b nδ n , g Vc .. = c δ · · · c nδ n and g Vd .. = d δ · · · d nδ n in Pth( A ) . The endpoints of the volume or horn are given by a V = a · g Va , b V = b · g Vb , c V = c · g Vc and d V = d · g Vd . Finally we call ( a, b ) -membrane the f -membrane defined by M V ( a,b ) .. = (( a , b ) , (( a i , b i ) , δ i ) ≤ i ≤ n ) . The other f -membrane, labelled ( c, d ) , and the two h -membranes, labelled by ( a, d ) and ( b, c ) , are defined similarly. a ✶✶ g Va > a ✶✶ · ✶✶✶ · ✶✶✶ · ✶✶✶ ··· · ✶✶✶ · ✶✶✶ · ✶✶✶ a n ✶✶ a V ✶✶ b g Vb > · · · · · · · · b V d ✶✶✶ g Vd > · ✶✶✶ · ✶✶✶ · ✶✶✶ · ✶✶✶ ··· · ✶✶✶ · ✶✶✶ · ✶✶✶ · ✶✶✶ d V ✶✶✶ c g Vc > · · · · · · · · c V Note that because double parallelistic relations are symmetric, both in the “vertical” and inthe “horizontal” direction, Definition 3.3.1 is “symmetric” in the role of opposite membranes.
Remark 3.3.2.
A morphism of racks f : A → B obviously sends a primitive trail ( x, ( a i , δ i ) ≤ i ≤ n ) in A to a primitive trail in ( f ( x ) , ( f ( a i ) , δ i ) ≤ i ≤ n ) in B . Similarly, a morphism α : f A → f B in ExtRck sends a f A -membrane to a f B -membrane, and a morphism of Ext C , such as ( σ, β ) : γ → α in Definition 2.3.1 sends a h f C , γ ⊤ i -volume to a h f A , α ⊤ i -volume (via the induced morphism (cid:3) ( σ,β ) such as in Lemma 4.1.1). Notation 3.3.3.
Given a double extension of racks (or quandles) α : f A → f B , we can build itskernel pair in ExtRck component-wise, which we denote:
Eq( α ⊤ ) π , π (cid:12) (cid:18) ¯ f z (cid:4) ⑧⑧⑧⑧⑧⑧ A ⊤ α ⊤ (cid:12) (cid:18) f A ⑧⑧ z (cid:4) ⑧⑧ Eq( α ⊥ ) p , p (cid:12) (cid:18) A ⊥ α ⊥ (cid:12) (cid:18) A ⊤ f A z (cid:4) ⑧⑧⑧⑧⑧⑧ α ⊤ , B ⊤ f B z (cid:4) ⑧⑧⑧⑧⑧⑧ A ⊥ α ⊥ , B ⊥ Remark 3.3.4.
Using Notation 3.3.3, the h f A , α ⊤ i -volumes V (see Definition 3.3.1) corre-spond bijectively to the ¯ f -membranes M in Eq( α ⊤ ) , since such an M is defined as the data ((( a , d ) , ( b , c )) , ((( a i , d i ) , ( b i , c i )) , δ i ) ≤ i ≤ n ) for a certain sequence of elements ( a i , b i , c i , d i ) in Eq( f A ) (cid:3) Eq( α ⊤ ) , where ≤ i ≤ n . Under the appropriate bijective correspondence, the ( a, b ) -membrane (and ( c, d ) -membrane) of a h f A , α ⊤ i -volume are obtained from the correspond-ing ¯ f -membrane via the projections π and π respectively. A ¯ f -horn then corresponds to a h f A , α ⊤ i -volume whose head ( a , b , c , d ) is such that a = b and c = d .Similarly, h f A , α ⊤ i -volumes correspond bijectively to ¯ α -membranes in Eq( f A ) , where ¯ α is thekernel pair of ( f A , f B ) in ExtRck . A ¯ α -horn then corresponds bijectively to a h f A , α ⊤ i -volumewhose head ( a , b , c , d ) is such that a = d and b = c . Normal Γ -coverings and rigid horns. We illustrate these definitions in the character-ization of normal Γ -coverings, which we subsequently refer to as normal double coverings . Proposition 3.4.1.
Given a double extension of racks (or quandles) α : f A → f B , it is a double Γ -covering if and only if, given a h f A , α ⊤ i -volume V = (( a , b , c , d ) , (( a i , b i , c i , d i ) , δ i ) ≤ i ≤ n ) (as in Definition 3.3.1), if its f A -membranes are horns (i.e. a = b and c = d ) then the α ⊤ -membranes of V are rigid in the sense that its ( d, c ) -horn closes if and only if its ( a, b ) -horncloses. We call a double extension satisfying this condition a normal double covering . Observe that by the “symmetries” of h f A , α ⊤ i -volumes (in the role of f A -membranes), it sufficesto show that in any such volume V , a closing ( a, b ) -horn implies a closing ( c, d ) -horn in orderto deduce that in any such volume V , a closing ( c, d ) -horn implies a closing ( a, b ) -horn (andconversely the latter implies the former). We relate this to the fact that ( π , p ) (from Notation3.3.3) is a trivial double covering if and only if ( π , p ) is one. Proof of Proposition 3.4.1.
By definition, α is a normal Γ -covering if and only if, in Notation3.3.3, the left face ( π , p ) (or equivalently the top face ( π , p ) ) is a trivial double covering.Then by Lemma 2.4.3, the double extension ( π , p ) is a trivial double covering if and only ifgiven any ¯ f -horn M , such that π ( M ) closes in A ⊤ , then M closes in Eq( α ⊤ ) , i.e. π ( M ) alsohas to close. By Remark 3.3.4 the preceding translates into the statement: ( π , p ) is a trivialdouble covering if and only if given any h f A , α ⊤ i -volume V such that a = b and c = d , if the ( a, b ) -horn of V closes then the ( c, d ) -horn of V has to close. Similarly ( π , p ) is a trivial doublecovering if and only if given any volume V such that a = b and c = d , a closing ( c, d ) -hornimplies a closing ( a, b ) -horn. (cid:3) IGHER COVERINGS OF RACKS AND QUANDLES – PART II 23
Of course trivial Γ -coverings (i.e. trivial double coverings) are examples of normal Γ -coverings(i.e. normal double coverings). However, these two concepts do not coincide. Example 3.4.2.
Consider the set { ⋆, •} , seen as a trivial quandle, as well as two copies f : Q ⋄ → { ⋆, •} and f : Q ⋄ → { ⋆, •} of the same morphism where Q ⋄ .. = { ⋆ ⋄ , ⋆, • , • } , and Q ⋄ .. = { ⋆, ⋆ ⋄ , • , • } are such that ⋆ ⋄ (respectively ⋆ ⋄ ) acts on • and • (respectively • and • )by interchanging and , and all the other actions are trivial (see also Example 2.3.14 in PartI). We then denote the kernel pair of f by Q ⋄⋄ with underlying set { ⋆ ⋄ , ⋆, ⋆ ⋄⋄ , ⋆ ⋄ , • , • , • , • } ,such that the element ⋆ ⋄ acts on bullets by interchanging the exponents and and similarlywith ⋆ ⋄ for the indices. Then ⋆ ⋄⋄ interchanges both indices and exponents of the bullets, whereas x ⊳ y = x for any other choice of x and y in Q ⋄⋄ . Q ⋄⋄ π ⋄ , , π ⋄ (cid:12) (cid:18) (cid:12) (cid:18) Q ⋄ f ⋄ (cid:12) (cid:18) (cid:12) (cid:18) Q ⋄ f ⋄ , , { ⋆, •} (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) • π ⋄ π ⋄ • π ⋄ • π ⋄ • ⋆ ⋄⋄ π ⋄ π ⋄ ⋆ ⋄ π ⋄ ⋆ ⋄ π ⋄ ⋆ (14) The projection π ⋄ identifies all the elements that have the same indices (including blanks), andsimilarly π ⋄ identifies elements with the same exponents.Observe that none of the morphisms above are quandle coverings. Moreover, both doubleextensions ( π ⋄ , f ⋄ ) and ( π ⋄ , f ⋄ ) are such that the conditions of Lemma 2.4.3 are not satisfied.However, the conditions of Proposition 3.4.1 are easily seen to be satisfied by both ( π ⋄ , f ⋄ ) and ( π ⋄ , f ⋄ ) . In order to check this, observe that the only “non-trivial” element in Eq( π ⋄ ) (cid:3) Eq( π ⋄ ) isthe square on the right of (14) (or any symmetric equivalent) and for any g , h ∈ Pth( Q ⋄⋄ ) andfor any i , j , k , l ∈ { , } , we have that • ij · g = • ij · h if and only if • kl · g = • kl · h . Even if 3.4.2 is symmetric in the sense that both ( π ⋄ , f ⋄ ) and ( π ⋄ , f ⋄ ) are double normalcoverings, Proposition 3.4.1, does not seem to be symmetric in the role of ( α ⊤ , α ⊥ ) and ( f A , f B ) .Observe that in Example 2.4.6, the double extension ( π p, t ⋆ ) is a trivial double covering and thusalso a normal double covering. However, the double extension ( π p, t ) is neither a trivial doublecovering nor a normal double covering since • ⊳ ⋆ = • ⊳ ⋆ even though • ⊳ ⋆ = • ⊳ ⋆ .Recall that any normal Γ -covering (normal double covering) α is in particular a Γ -covering,since α is split by α . Now unlike trivial double coverings and normal double coverings, Γ -coverings are expected to be symmetric in the same way that double coverings are (see Remark3.0.2). If we were to weaken the condition characterizing normal Γ -coverings to obtain a can-didate condition for the characterization of Γ -coverings, we would look for a way to make itsymmetric in the roles of f A and α ⊤ .Now observe that an obvious asymmetrical feature of the characterization in Proposition 3.4.1is the fact that we look at properties of ¯ f -horns in Eq( α ⊤ ) , some of which cannot be expressedas ¯ α -horns in Eq( f A ) . In the spirit of the discussions at page 15 and 21, we are looking atthe “successive action” of “two-dimensional data” on some “one-dimensional data” (in a fixedprivileged direction). What we are aiming for is the “successive action” of “two-dimensionaldata” on some “zero-dimensional data”.We get rid of the asymmetry in Proposition 3.4.1 by collapsing the one-dimensional head ofthe volumes we study. Looking at h f A , α ⊤ i -horns in A ⊤ , these can be described both as ¯ f -hornsin Eq( α ⊤ ) and as ¯ α -horns in Eq( f A ) . From Proposition 3.4.1 we produce the concept of a doubleextension with rigid horns . Definition 3.4.3.
A double extension of racks (or quandle) α is said to have rigid horns ifany h f A , α ⊤ i -horn V in A ⊤ has rigid α ⊤ -membranes in the sense of Proposition 3.4.1: if V = (( a , b , c , d ) , (( a i , b i , c i , d i ) , δ i ) ≤ i ≤ n ) , as in Definition 3.3.1, its ( d, c ) -horn closes if and onlyif its ( a, b ) -horn closes. Even though Definition 3.4.3 still seems asymmetric at first, it is actually not so anymore.Indeed we use the terminology rigid horns because we may show that given a double extension α : f A → f B , any h f A , α ⊤ i -horn V in A ⊤ has rigid α ⊤ -membranes if and only if any h f A , α ⊤ i -horn V in A ⊤ has rigid f A -membranes (its ( a, d ) -horn closes if and only if its ( b, c ) -horn closes).Observe that by Definition 3.4.3, the double extension ( f A , f B ) has rigid horns if and only if any h f A , α ⊤ i -horn has rigid f A -membranes. Again we may show that ( α ⊤ , α ⊥ ) has rigid horns (in thesense of Definition 3.4.3) if and only if the double extension ( f A , f B ) has rigid horns, as it is thecase for double coverings. We skip this (rather elementary) step as it can be deduced from thefact that the concepts of double covering and double extension with rigid horns coincide. Proposition 3.4.4.
A double extension of racks α : f A → f B is a double covering if and only if α has rigid horns (Definition 3.4.3).Proof. Suppose that α has rigid horns in the sense of Definition 3.4.3. Then given an element ( a, b, c, d ) ∈ Eq( f A ) (cid:3) Eq( α ⊤ ) and an element x ∈ X we build an h f A , α ⊤ i -horn V described bysuperposition of the two f A -membranes M and M below (the so-obtained “left-hand side” α ⊤ -membrane of V is as in Example 3.0.3). Since the α ⊤ -membranes of V are rigid and M closesinto a disk, we conclude that y .. = x · ( a b − c d − ) is equal to x . M : x ~ (cid:8) ✟✟✟✟ ab − cd − ✟✟✟✟ (cid:22) ✻✻✻✻ b b − c c − ✻✻✻✻ f A f A f A y f A x M : x } (cid:7) ✞✞✞✞ dc − cd − ✞✞✞✞ (cid:23) ! ✼✼✼✼ c c − c c − ✼✼✼✼ f A f A f A x f A x (15)Conversely suppose that α is a double covering and consider an h f A , α ⊤ i -horn V given by V = ( x, (( a i , b i , c i , d i ) , δ i ) ≤ i ≤ n ) as in Definition 3.3.1. Suppose that the ( c, d ) -membrane of V closes into a disk, we have to show that the ( a, b ) -membrane closes into a disk (the converse isthen given by symmetry of V in the role of the f A -membrane).More generally, and without assumption on the double extension α , we show that the endpoints a V and b V of such a horn V are in relation by [Eq( f A ) , Eq( α ⊤ )] , which we temporarily denoteby ≈ . Observe that for all z ∈ A ⊤ we have that z ⊳ − δ n d n ⊳ δ n a n ≈ z ⊳ − δ n c n ⊳ δ n b n (replace ≈ by = when α is a double covering). By taking z = d V .. = x ⊳ δ d · · · ⊳ δ n d n (and by reflexivity of ≈ and compatibility with the operation ⊳ ) we derive x ⊳ δ d · · · ⊳ δ n − d n − ⊳ δ n a n ≈ x ⊳ δ c · · · ⊳ δ n − c n − ⊳ δ n b n . (16)Then consider the square a n − ⊳ δ n a n d n − ⊳ δ n a n b n − ⊳ δ n b n c n − ⊳ δ n b n ∈ Eq( f A ) (cid:3) Eq( α ⊤ ) , and derive that for each z ∈ A ⊤ : z ⊳ − δ n − ( d n − ⊳ δ n a n ) ⊳ δ n − ( a n − ⊳ δ n a n ) ≈ z ⊳ − δ n − ( c n − ⊳ δ n b n ) ⊳ δ n − ( b n − ⊳ δ n b n ); z ⊳ − δ n a n ⊳ − δ n − d n − ⊳ δ n − a n − ⊳ δ n a n ≈ z ⊳ − δ n b n ⊳ − δ n − c n − ⊳ δ n − b n − ⊳ δ n b n . Applying this to Equation (16) we obtain x ⊳ δ d · · · ⊳ δ n − d n − ⊳ δ n − a n − ⊳ δ n a n ≈ x ⊳ δ c · · · ⊳ δ n − c n − ⊳ δ n − b n − ⊳ δ n b n . IGHER COVERINGS OF RACKS AND QUANDLES – PART II 25
We repeat the argument with a n − ⊳ δ n − a n − ⊳ δ n a n d n − ⊳ δ n − a n − ⊳ δ n a n b n − ⊳ δ n − b n − ⊳ δ n b n c n − ⊳ δ n − b n − ⊳ δ n b n ∈ Eq( f A ) (cid:3) Eq( α ⊤ ) , and conclude by induction that also x ⊳ δ a · · · ⊳ δ n a n ≈ x ⊳ δ b · · · ⊳ δ n b n . (cid:3) Given a double extension α : f A → f B , the rigid horns condition from Definition 3.4.3, ormore precisely Definition 3.4.5 below, make sense of what it means for two elements of A ⊤ to be“linked under the action of a primitive path from Eq( f A ) (cid:3) Eq( α ⊤ ) ” (see page 21). Definition 3.4.5.
Given a double extension α : f A → f B , we define the set X α to be the setof those pairs ( x, y ) in A ⊤ × A ⊤ such that there exists a h f A , α ⊤ i -horn V as in Definition 3.3.1such that x and y are the endpoints of one of the membranes M V of V , such that moreover themembrane M V , which is opposite to M V , closes into a disk. These pairs in X α are the pairs of elements which would be identified if α had rigid horns . Wejust saw that X α contains the generators of [Eq( f A ) , Eq( α ⊤ )] and moreover X α ⊆ [Eq( f A ) , Eq( α ⊤ )] .Hence if we can show that X α defines a congruence on A ⊤ , we can deduce that X α is the cen-tralizing congruence [Eq( f A ) , Eq( α ⊤ )] (see Corollary 3.5.5 below).Now recall from Part I that coverings are equivalently described via membranes or via sym-metric paths. Proposition 3.4.4 corresponds to the description via membranes. In the followingsection, we adapt the idea of a symmetric path to the two-dimensional context. Equipped withthis concept and that of a rigid horn, we provide a full description of a general element in [Eq( f A ) , Eq( α ⊤ )] .3.5. Symmetric paths for double extensions.
We describe symmetric paths in a slightlymore general context than expected, because of Lemma 3.5.7 below.
Definition 3.5.1.
Given a pair of morphisms f : G → H , h : G → K in Grp , and a generatingset A ⊆ G (i.e. such that G = h a | a ∈ A i G ), we define (implicitly with respect to A ):(i) four elements g a , g b , g c and g d in G to be h f, h i -symmetric (to each other) if thereexists n ∈ N and a sequence of quadruples ( a , b , c , d ) , . . . , ( a n , b n , c n , d n ) in the set A ∩ (Eq( f ) (cid:3) Eq( h )) , and finally, if for each ≤ i ≤ n , there is δ i ∈ {− , } such that: g a = a δ · · · a δ n n , g b = b δ · · · b δ n n , g c = c δ · · · c δ n n , g d = d δ · · · d δ n n . (17) We often call such g a , g b , g c and g d an h f, h i -symmetric quadruple .(ii) K h f,h i to be the set of h f, h i -symmetric paths , i.e. the elements g ∈ G such that g = g a g − b g c g − d for some h f, h i -symmetric quadruple g a , g b , g c and g d ∈ G . Lemma 3.5.2.
Given the hypotheses of Definition 3.5.1, the set of h f, h i -symmetric paths K h f,h i defines a normal subgroup of G .Proof. Let g a , g b , g c and g d be h f, h i -symmetric (to each other). Observe that g − d , g − c , g − b and g − a are also h f, h i -symmetric, and thus K h f,h i is closed under inverses. Moreover, if h a , h b , h c and h d are h f, h i -symmetric, and g = g a g − b g c g − d , h = h a h − b h c h − d , then gh = k a k − b k c k − d , with k a = h a h − b h b h − a g a , k b = h a h − a h b h − a g b , k c = h d h − d h b h − a g c and k d = h d h − c h b h − a g d which are h f, h i -symmetric. Finally since A generates G , for any k ∈ G , kg a , kg b , kg c and kg d are h f, h i -symmetric to each other, and thus kgk − = kg a g − b k − kg c g − d k − ∈ K h f,h i is an h f, h i -symmetric path. (cid:3) Notation 3.5.3.
For a double extension of racks (or quandles) α : f A → f B , we often write h f A , α ⊤ i -symmetric (quadruple or path) instead of h ~f , ~α ⊤ i -symmetric (quadruple or path – seefor instance Definition 3.3.1). An h f A , α ⊤ i -symmetric trail ( x, g ) in A ⊤ is a trail where g is an h f A , α ⊤ i -symmetric path. Lemma 3.5.4.
Given a double extension in
Rck (or
Qnd ) α : f A → f B , the set X α (Defin-tion 3.4.5) is the underlying set of the congruence ∼ K h fA,α ⊤ i induced by the action of h f A , α ⊤ i -symmetric paths on A ⊤ .Proof. Given x and y ∈ A ⊤ such that x ∼ K h fA,α ⊤ i y , i.e. such that y = x · ( g a g − b g c g − d ) for some h f A , α ⊤ i -symmetric quadruple as in Definition 3.5.1. The pair ( x, y ) is in X α as one can deducefrom the construction of V as in Equation (15) from the proof of Proposition 3.4.4, where onereplaces every occurrence of a by a δ · · · a nδ n and also for b by b δ · · · b nδ n , and similarly c by c δ · · · c nδ n and d by d δ · · · d nδ n .Conversely, and without loss of generality, consider an h f A , α ⊤ i -horn V given by the data V = ( x, (( a i , b i , c i , d i ) , δ i ) ≤ i ≤ n ) as in Definition 3.3.1, such that moreover the endpoints c V = d V .Observe that the endpoint b V = a V · (cid:0) ( g Va ) − g Vd ( g Vc ) − g Vb (cid:1) is obtained from the endpoint a V by the action of an h f A , α ⊤ i -symmetric path. (cid:3) As a conclusion to the discussion below Definition 3.4.5, we give a characterization of a generalelement in [Eq( f A ) , Eq( α ⊤ )] which we show to be an orbit congruence (see Part I and referencetherein). Corollary 3.5.5.
Given a double extension of racks (or quandles) α : f A → f B , the centralizationcongruence [Eq( f A ) , Eq( α ⊤ )] coincides with the congruence ∼ K h fA,α ⊤ i generated by the action of h f A , α ⊤ i -symmetric paths, also described by the set of pairs in X α (Definition 3.4.5, i.e. thosepairs of elements of A ⊤ which would be identified if α had rigid horns). Describing symmetric paths differently ?
Given a morphism f in Rck (or
Qnd ), f -symmetricpaths are described as the elements in the kernel Ker( ~f ) of ~f (which is our notation for Pth( f ) ).It is unclear to us whether this result generalizes in higher dimensions. Our understanding is thatthe question should be: given a double extension α , do the normal subgroups Ker( ~f A ) ∩ Ker( ~α ⊤ ) and K h f A ,α ⊤ i coincide ? Whether the answer is negative or positive, this would help to specifymore precisely how to understand these h f A , α ⊤ i -symmetric paths algebraically. Following thestrategy from Section 2.4.11 of Part I, we were able to show that: Lemma 3.5.7.
Given two surjective functions f : A → B and h : A → C such that Eq( f ) ◦ Eq( h ) = Eq( h ) ◦ Eq( f ) , the intersection Ker(F g ( f )) ∩ Ker(F g ( h )) of the kernels of the inducedgroup homomorphisms F g ( f ) : F g ( A ) → F g ( B ) and F g ( h ) : F g ( A ) → F g ( C ) is K h F g ( f ) , F g ( h ) i (withrespect to A ) as in Definition 3.5.1. Our proof is rather combinatorial and can be found in a separate publication [61]. Now givena double extension of racks (or quandles) α , it is easy to obtain K h f A ,α ⊤ i ≤ Ker( ~f A ) ∩ Ker( ~α ⊤ ) as the image of K h F g ( f A ) , F g ( α ⊤ ) i = Ker(F g ( f A )) ∩ Ker(F g ( α ⊤ )) by q A ⊤ : F g (U( A ⊤ )) → Pth( A ⊤ ) (defined as in Paragraph 1.1.4 above). Hence Ker( ~f A ) ∩ Ker( ~α ⊤ ) = K h ~f A , ~α ⊤ i if and only if theinduced morphism ¯ q : Ker(F g ( f A )) ∩ Ker(F g ( α ⊤ )) → Ker( ~f A ) ∩ Ker( ~α ⊤ ) is a surjection. We wereunfortunately not able to identify a reason why this should be true in general (see Observation3.5.8 for alternative descriptions).Besides, we note that even if the two groups do not coincide, it might still be that the actionof K h f A ,α ⊤ i on A ⊤ and the action of Ker( ~f A ) ∩ Ker( ~α ⊤ ) on A ⊤ define the same congruence in Rck (or
Qnd ). Finally, we ask the “even weaker” question: is
Ker( ~f A ) ∩ Ker( ~α ⊤ ) in the center of IGHER COVERINGS OF RACKS AND QUANDLES – PART II 27
Pth( A ⊤ ) ? This would imply that the image by Conj Pth of a double covering is still a doublecovering (see Section 3.2).
Observation 3.5.8.
More precisely, observe that a double extension of racks or quandles α issent to a double extension α ′ = Pth( α ) in groups since Pth preserves pushouts of surjectionsand
Grp is a Mal’tsev category. Call c ′ : Pth( A ⊤ ) ։ P ′ the surjective comparison map of α ′ .The double extension α is also sent to a double extension in Set by the forgetful functor, aspullbacks and surjections are preserved. This double extension in
Set is then sent by F g to adouble extension α ′′ in Grp , since pushouts of surjections are preserved by left-adjoints. Write c ′′ : F g ( A ⊤ ) ։ P ′′ for the comparison map of α ′′ . Finally α ′′ is sent by Conj to a double extensionin
Rck again, which is sent to a double extension α ′′′ by Pth . Write c ′′′ : F g ( A ⊤ × F g ( A ⊤ )) ։ P ′′′ for the comparison map of α ′′′ . We have thus three layers α ′′′ , α ′′ and α ′ of double extensions in Grp fitting into a fork α ′′′ ⇒ α ′′ → α ′ of -dimensional arrows, such that each arrow is a squareof double extensions, and the top pair is a reflexive graph whose legs are -fold extensions. F g ( A ⊤ × F g ( A ⊤ )) F g ( f A × F g ( f A )) , t (cid:12) (cid:18) t (cid:12) (cid:18) R L ✤✤✤✤✤✤ F g ( α ⊤ × F g ( α ⊤ )) s { ♥♥♥♥♥♥♥♥♥♥ F g ( A ⊥ × F g ( A ⊥ )) t (cid:12) (cid:18) t (cid:12) (cid:18) R L ✤✤✤✤✤✤♥♥♥♥♥ s { ♥♥♥♥♥ F g ( B ⊤ × F g ( B ⊤ )) t (cid:12) (cid:18) t (cid:12) (cid:18) R L ✤✤✤✤✤✤ , F g ( B ⊥ × F g ( B ⊥ )) t (cid:12) (cid:18) t (cid:12) (cid:18) R L ✤✤✤✤✤✤ F g ( A ⊤ ) F g ( f A ) , q A ⊤ (cid:12) (cid:18) F g ( α ⊤ ) ♥♥♥ s { ♥♥♥ F g ( A ⊥ ) q A ⊥ (cid:12) (cid:18) ♥♥♥♥♥ s { ♥♥♥♥♥ F g ( B ⊤ ) , q B ⊤ (cid:12) (cid:18) F g ( B ⊥ ) q B ⊥ (cid:12) (cid:18) Pth( A ⊤ ) ~α ⊤ ♥♥♥♥ s { ♥♥♥♥ ~f A , Pth( A ⊥ ) s { ♥♥♥♥♥♥♥♥♥♥ Pth( B ⊤ ) , Pth( B ⊥ ) F g ( A ⊤ × F g ( A ⊤ )) p A ⊤ (cid:12) (cid:18)R L ✤✤✤✤ p A ⊤ (cid:12) (cid:18) c ′′′ , , P ′′′ p (cid:12) (cid:18) p (cid:12) (cid:18) R L ✤✤✤✤ F g ( A ⊤ ) ( ∗ ) q A ⊤ (cid:12) (cid:18) (cid:12) (cid:18) c ′′ , , P ′′ q (cid:12) (cid:18) (cid:12) (cid:18) Pth( A ⊤ ) c ′ , , ❴❴❴❴❴ P ′ Figure 1.
The fork α ′′′ ⇒ α ′′ → α ′ By the universal property of the pullbacks, P ′′′ , P ′′ and P ′ , there is an induced reflexive graph p , p : P ′′′ ⇒ P ′′ , as well as a surjection q : P ′′ → P ′ which coequalises p and p , such thatthe whole fork fits into the commutative diagrams of Figure 1. By Lemma 1.2 in [6] , ( ∗ ) is adouble extension if and only if q is the coequalizer of p and p , which is also equivalent to thefork being a double extension. These three equivalent conditions are satisfied if and only if theaforementioned morphism ¯ q : Ker(F g ( f A )) ∩ Ker(F g ( α ⊤ )) → Ker( ~f A ) ∩ Ker( ~α ⊤ ) is a surjection. The Γ -coverings (or double central extensions of racks and quandles) In this section, we show that the concept of double covering of racks and quandles (or alge-braically central double extension) and the concept of Γ -covering (or double central extensionof racks and quandles) coincide. In order to do so, we first show that double coverings are re-flected and preserved by pullbacks along double extensions. Since trivial Γ -coverings are doublecoverings, this implies that Γ -coverings are also double coverings.4.1. Double coverings are reflected and preserved by pullbacks.
We first show a generalresult about morphisms induced by -fold extensions (see Definition 2.3.1). Observe that giventhe hypothesis of Lemma 2.2.1, we deduce from [6, Lemma 2.1] that if the right hand square of Diagram (6) is a double extension, then f is an extension even if C is merely regular (and notBarr-exact). Lemma 4.1.1.
Consider a -fold extension ( σ, β ) : γ → α in a regular category C . The mor-phism (cid:3) ( σ,β ) : Eq( f C ) (cid:3) Eq( γ ⊤ ) → Eq( f A ) (cid:3) Eq( α ⊤ ) induced by ( σ, β ) between the parallelistic dou-ble equivalence relations is a regular epimorphism.Proof. First we recall how to build the double parallelistic relations of interest. By taking kernelpairs horizontally and then vertically, we build the Diagrams (18), where the induced pairs ( p , p ) : R γ ⇒ Eq( f C ) and ( π , π ) : R α ⇒ Eq( f A ) on the top rows, are the kernel pairs of ¯ γ and ¯ α by a local version of the denormalised × Lemma (see [6] and Lemma 4.1.2 below). As aconsequence, all the rows and columns of Diagrams (18) are exact forks.
Eq( f γ ) (cid:12) (cid:18) (cid:12) (cid:18) p , p , Eq( f C ) (cid:12) (cid:18) (cid:12) (cid:18) ¯ γ , Eq( f D ) (cid:12) (cid:18) (cid:12) (cid:18) Eq( γ ⊤ ) f γ (cid:12) (cid:18) , , C ⊤ γ ⊤ , f C (cid:12) (cid:18) D ⊤ f D (cid:12) (cid:18) Eq( γ ⊥ ) , , C ⊥ γ ⊥ , D ⊥ Eq( f α ) (cid:12) (cid:18) (cid:12) (cid:18) π , π , Eq( f A ) (cid:12) (cid:18) (cid:12) (cid:18) ¯ α , Eq( f B ) (cid:12) (cid:18) (cid:12) (cid:18) Eq( α ⊤ ) f α (cid:12) (cid:18) , , A ⊤ α ⊤ , f A (cid:12) (cid:18) B ⊤ f B (cid:12) (cid:18) Eq( α ⊥ ) , , A ⊥ α ⊥ , B ⊥ (18)Then by Proposition 2.1 from [6], Eq( f γ ) = Eq( f C ) (cid:3) Eq( γ ⊤ ) and Eq( f α ) = Eq( f A ) (cid:3) Eq( α ⊤ ) arethe double parallelistic relations of interest.Now the -fold extension ( σ, β ) induces morphisms between the left-hand and right-handDiagrams (18), such that on the top row we have Eq( f C ) (cid:3) Eq( γ ⊤ ) (cid:3) ( σ,β ) (cid:12) (cid:18) p , p , Eq( f C ) ¯ γ , ¯ σ (cid:12) (cid:18) Eq( f D ) ¯ β (cid:12) (cid:18) Eq( f A ) (cid:3) Eq( α ⊤ ) π , π , Eq( f A ) ¯ α , Eq( f B ) . Hence, by [6, Lemma 2.1], it suffices to prove that the right hand commutative square (¯ σ, ¯ β ) isa double extension. This can be deduced from the fact that ( σ, β ) is a -fold extension. When C is Barr-exact category, we may use [30, Lemma 3.2]. However, for a general regular category C ,we consider the “fork of comparison maps”: Eq( f C ) p (cid:12) (cid:18) , , C ⊤ f C , (cid:12) (cid:18) C ⊥ (cid:12) (cid:18) Eq( f A ) × Eq( f B ) Eq( f D ) , , A ⊤ × B ⊤ D ⊤ f P , A ⊥ × B ⊥ D ⊥ . (19)where the bottom row is exact by Lemma 4.1.2 (as for the top rows in Diagrams (18) above).Moreover, the right hand square ( f C , f P ) is a double extension since ( σ, β ) is a -fold extension,and thus the morphism p is a regular epimorphism. Since p is also the comparison map of (¯ σ, ¯ β ) ,this concludes the proof. (cid:3) Using the study of the denormalised × Lemma from [6], we obtain the following result,where, as usual, we locally use double extensions instead of working globally in a Mal’tsevcategory.
Lemma 4.1.2.
Given a regular category C as well as a × diagram such as any of the twoDiagrams (18) , where all columns are exact, the middle row and the bottom row are exact, andthe bottom right-hand square is a double extension, then the top row is also exact. IGHER COVERINGS OF RACKS AND QUANDLES – PART II 29
Proof.
The top row is left-exact by [6, Theorem 2.2]. Then the top right morphism is a regularepimorphism by [6, Lemma 2.1]. We conclude by the fact that in any category with pullbacks,regular epimorphisms are the coequalizers of their kernel pairs. (cid:3)
Working in the categories
Rck and
Qnd again we obtain the following.
Corollary 4.1.3.
Double coverings are stable by pullbacks along double extensions and reflectedalong -fold extensions. In particular, Γ -coverings are double coverings.Proof. Consider a -fold extension ( σ, β ) : γ → α in Rck (or
Qnd ) such as in Definition 2.3.1.Assume that γ is a double covering. Given x ∈ A ⊤ and ( a, b, c, d ) ∈ Eq( f A ) (cid:3) Eq( α ⊤ ) , the surjec-tivity of σ ⊤ and (cid:3) ( σ,β ) (from Lemma 4.1.1) yields x ′ ∈ C ⊤ and ( a ′ , b ′ , c ′ , d ′ ) ∈ Eq( f C ) (cid:3) Eq( γ ⊤ ) suchthat σ ⊤ ( x ′ ) = x , σ ⊤ ( a ′ ) = a , σ ⊤ ( b ′ ) = b , σ ⊤ ( c ′ ) = c , and σ ⊤ ( d ′ ) = d . Since x ′ ⊳a ′ ⊳ − b ′ ⊳c ′ ⊳ − d ′ = x ′ in C ⊤ , the image of this equation by σ ⊤ yields x ⊳ a ⊳ − b ⊳ c ⊳ − d = x in A ⊤ . Hence α is a doublecovering.Conversely assume that α is a double covering and suppose that ( σ, β ) describes the pullbackof α and β , i.e. suppose that the comparison map π of ( σ, β ) is an isomorphism (see Definition2.3.1). Then we consider x ∈ C ⊤ and ( a, b, c, d ) ∈ Eq( f C ) (cid:3) Eq( γ ⊤ ) , and we have to show that y .. = x ⊳ a ⊳ − b ⊳ c ⊳ − d is equal to x . It suffices to check the equality in both components ofthe pullback C ⊤ , via the projections γ ⊤ and σ ⊤ . We have indeed γ ⊤ ( y ) = γ ⊤ ( x ) and since α is adouble covering, we have also σ ⊤ ( y ) = σ ⊤ ( x ) . Hence γ is a double covering. (cid:3) Double coverings are Γ -coverings. As described in Section 2.4, given a double covering α : f A → f B , we build the canonical double projective presentation of its codomain f B : p f B .. = ( p f B , p f B ) : p B → f B (see Diagram (9)) . We then consider the pullback of our double covering α along our projective presentation p f B .This yields a double covering γ : f C → p B with projective codomain p B : F r ( P ) → F r ( B ⊥ ) (or p B : F q ( P ) → F q ( B ⊥ ) if we work in Qnd ). We show that such double coverings are always trivialdouble coverings, which implies that the double covering α is a Γ -covering. Proposition 4.2.1.
If a double covering of racks γ = ( γ ⊤ , γ ⊥ ) has a projective codomain of theform p : F r ( P ) → F r ( B ⊥ ) for some sets P and B ⊥ : C ⊤ f C , h γ ⊤ ,f C i $ , ❘❘❘❘❘❘ γ ⊤ (cid:12) (cid:18) C ⊥ γ ⊥ (cid:12) (cid:18) Q π ❣❣❣❣❣ ❣❣❣❣❣ π ③③③ x (cid:2) ③③③ F r ( P ) p , F r ( B ⊥ ) , where Q .. = F r ( P ) × F r ( B ⊥ ) C ⊥ , then γ is a trivial double covering. The result holds similarly in Qnd , for a double covering γ with codomain of the form p : F q ( P ) → F q ( B ⊥ ) .Proof. Consider a f C -membrane M = ( x, ( a i , b i , δ i ) ≤ i ≤ n ) in C ⊤ such that the image of M by γ ⊤ closes into a disk in F r ( P ) , i.e. γ ⊤ ( x ) · (cid:0) ~γ ⊤ ( a δ · · · a nδ n b n − δ n · · · b − δ ) (cid:1) = γ ⊤ ( x ) . x w (cid:1) ✇✇✇✇✇✇ a δ ...a nδn ✇✇✇✇✇✇ (cid:29) ' ●●●●●● b δ ... b nδn ●●●●● f C f C a M .. = x · ( g Ma ) f C b M .. = x · ( g Mb ) γ ( x ) (cid:12) (cid:18) ~γ ( g Ma ) (cid:12) (cid:18) ~γ ⊤ ( g Mb ) f C f C f C γ ⊤ ( a M ) Let us write y i .. = γ ⊤ ( a i ) and x i .. = γ ⊤ ( b i ) for each ≤ i ≤ n . Then h .. = ~γ ⊤ ( a δ · · · a δ n b n − δ n · · · b − δ ) = y δ · · · y nδ n x n − δ n · · · x − δ ∈ Pth(F r ( C ⊤ )) , yields the neutral element e since the action of Pth(F r ( C ⊤ )) on F r ( C ⊤ ) is free – note that in thecontext of Qnd , this path h is in the group Pth ◦ (F q ( C ⊤ )) which acts freely on F q ( C ⊤ ) .Since p is projective (with respect to double extensions), there is a splitting s .. = ( s ⊤ , s ⊥ ) of γ such that γs = (1 F r ( P ) , F r ( B ⊥ ) ) is the identity morphism. If we define d i .. = s ⊤ ( γ ⊤ ( a i )) and c i .. = s ⊤ ( γ ⊤ ( b i )) for each ≤ i ≤ n , then we have that • c δ · · · c nδ n d n − δ n · · · d − δ = ~s ⊤ ( h − ) = e ∈ Pth( C ⊤ ) is trivial; • moreover γ ⊤ ( d i ) = γ ⊤ ( a i ) and γ ⊤ ( c i ) = γ ⊤ ( b i ) for each ≤ i ≤ n ; • and thus the product g Ma ( g Mb ) − defines an h f C , γ ⊤ i -symmetric path in Pth( C ⊤ ) since g Ma ( g Mb ) − = g Ma ( g Mb ) − ~s ⊤ ( h − ) = a δ · · · a nδ n b n − δ n · · · b − δ c δ · · · c nδ n d n − δ n · · · d − δ .Since γ is a double covering, we conclude that x = x · ( g Ma ( g Mb ) − ) = x · ( a δ · · · a nδ n b n − δ n · · · b − δ ) , which shows that γ is a trivial double covering. (cid:3) Theorem 4.2.2.
A double extension of racks (or quandles) is a Γ -covering (also called doublecentral extension of racks and quandles), if and only if it is a double covering (also called alge-braically central double extension of racks and quandles). The category of double coverings andthe category of Γ -coverings above an extension of racks (or quandles) are isomorphic. Centralizing double extensions.
Consider a double extension of racks (or quandles) α : f A → f B . We may universally centralize it (i.e. make it into a double covering) by a quotientof its initial object A ⊤ . We studied the reflection of CExtRck in ExtRck . Now in
Ext Rck (fromDefinition 2.3.1), we may identify the full subcategory
CExt Rck whose objects are the doublecoverings (or equivalently the Γ -coverings, also called double central extensions). The followingresult is the -dimensional equivalent of Theorem 3.4.1 from Part I. We define E as the class of -fold extensions from Definition 2.3.1. Theorem 4.3.1.
The category
CExt Rck is a ( E )-reflective subcategory of the category Ext Rck with left adjoint F and unit η defined for an object α : f A → f B in Ext Rck by η α .. = ( η α ⊤ , η α ⊥ ) .. = (( η A ⊤ , id A ⊥ ) , (id B ⊤ , id B ⊥ )) : α −→ F ( α ) , where η A ⊤ : A ⊤ → F i2 ( A ⊤ ) is defined as the quotient of A ⊤ by the centralizing relation C ( α ) .. =[Eq( f A ) , Eq( α ⊤ )] , and its equivalent descriptions from Corollary 3.5.5. Observing that C ( α ) ≤ Eq( f A ) ∩ Eq( α ⊤ ) , the image F ( α ) .. = (F ˆ2 ( α ⊤ ) , α ⊥ ) : F ˆ2 ( f A ) → f B is defined via the unique fac-torization of the comparison map p : A ⊤ → A ⊥ × B ⊥ B ⊤ through the quotient η A ⊤ : A ⊤ η A ⊤ ) ❑❑❑❑❑ p , A ⊥ × B ⊥ B ⊤ F i2 ( A ⊤ ) p ′ : ♠♠♠♠♠ A ⊤ η A ⊤ , α ⊤ (cid:12) (cid:18) f A z (cid:4) ⑧⑧⑧⑧⑧⑧⑧ F i2 ( A ⊤ ) F ˆ2 ( α ⊤ ) .. = π p ′ (cid:12) (cid:18) F ˆ2 ( f A ) .. = π p ′ ⑧⑧ z (cid:4) ⑧⑧ A ⊥ α ⊥ (cid:12) (cid:18) A ⊥ α ⊥ (cid:12) (cid:18) B ⊤ f B z (cid:4) ⑧⑧⑧⑧⑧⑧⑧ B ⊤ f B z (cid:4) ⑧⑧⑧⑧⑧⑧⑧ B ⊥ B ⊥ where π and π are the projections of A ⊥ × B ⊥ B ⊤ , as in Equation 4.The image by F of a morphism ( σ, β ) : γ → α is then given by the identity in all but the initialcomponent: F ( σ, β ) = ((F i2 ( σ ⊤ ) , σ ⊥ ) , ( β ⊤ , β ⊥ )) , where F i2 ( σ ⊤ ) is defined by the unique factorization IGHER COVERINGS OF RACKS AND QUANDLES – PART II 31 of η A ⊤ σ ⊤ through η C ⊤ , as displayed below, where P .. = F i2 ( C ⊤ ) × F i2 ( A ⊤ ) A ⊤ : C ⊤ σ ⊤ , p ❘❘ $ , ❘❘ η C ⊤ (cid:12) (cid:18) A ⊤ η A ⊤ (cid:12) (cid:18) P π ❤❤❤ ❤❤❤ π ③③ x (cid:2) ③③ F i2 ( C ⊤ ) F i2 ( σ ⊤ ) , ❴❴❴❴❴❴ F i2 ( A ⊤ ) (20) Proof.
First observe that the double extension F ( α ) is indeed a double covering by construc-tion. As was already mentioned in the proof of Theorem 2.3.2, η is easily seen to be a -fold extension since its bottom component is an isomorphism. Hence given ( a, b, c, d ) ∈ Eq(F ˆ2 ( f A )) (cid:3) Eq(F ˆ2 ( α ⊤ )) , it is the quotient of some ( a ′ , b ′ , c ′ , d ′ ) ∈ Eq( f A ) (cid:3) Eq( α ⊤ ) by Lemma4.1.1, and thus for any x ∈ F i2 ( A ⊤ ) , the elements x and x · ( a b − c d − ) have η A ⊤ -pre-images x ′ and x ′ · ( a ′ ( b ′ ) − c ′ ( d ′ ) − ) which are in relation by C ( α ) .Then we show that η α has the right universal property. We first show the universality of η α inthe subcategory Ext ( f B ) of double extensions over f B . Consider a double covering γ : f C → f B and a morphism τ ∈ Ext ( f B ) between α and γ , yielding the commutative diagram of plainarrows in ExtRck on the left, whose top and bottom components in C are given on the right. F ˆ2 ( f A ) ϑ .. =( ϑ ⊤ ,θ ⊥ ) (cid:19) (cid:26) F ( α ) + ❖❖❖❖❖❖❖❖❖ f Aη α ⊤ L R θ (cid:12) (cid:18) α , f B f C γ ; ♥♥♥♥♥♥♥♥♥♥♥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F i2 ( A ⊤ ) ϑ ⊤ (cid:19) (cid:26) F ˆ2 ( α ⊤ ) + PPPPPPPPP A ⊤ η A ⊤ L R θ ⊤ (cid:12) (cid:18) α ⊤ , B ⊤ C ⊤ γ ⊤ ; ♥♥♥♥♥♥♥♥♥♥♥ A ⊥ θ ⊥ (cid:19) (cid:26) α ⊥ " * ◆◆◆◆◆◆◆◆◆ A ⊥ θ ⊥ (cid:12) (cid:18) α ⊥ , B ⊥ C ⊥ γ ⊥ < ♣♣♣♣♣♣♣♣♣ Given any pair ( x · g, x ) ∈ C ( α ) , where g is some h f A , α ⊤ i -symmetric path, ~θ ( g ) is a h f C , γ ⊤ i -symmetric path and thus θ ( x ) = θ ( x ) · ~θ ( g ) since γ is a double covering. As a consequence, Eq( θ ⊤ ) ≤ C ( α ) and thus there exists a unique factorization ϑ ⊤ of θ ⊤ through η A ⊤ . Since f C ϑ ⊤ η A ⊤ = θ ⊥ F ˆ2 ( f A ) η A ⊤ , and η A ⊤ is an epimorphism, we may define the morphism ϑ .. =( ϑ ⊤ , θ ⊥ ) : F ˆ2 ( f A ) → f C , which is moreover a double extension by Lemma 2.1.2. This shows theexistence of a factorization of θ through η α ⊤ . The uniqueness of ϑ is easily deduced from theuniqueness in each component.Now working in the category Ext Rck , we consider a double covering γ : f C → f D and amorphism ( τ, ι ) : α → γ in Ext Rck . We compute the pullback ρ of γ along ι and the inducedcomparison map π of the underlying square of ( τ, ι ) in ExtRck : F ˆ2 ( f A ) F ( α ) * ❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬ f Aη α ⊤ c k ❖❖❖ α , π + τ (cid:12) (cid:18) f Bι (cid:12) (cid:18) f C × f D f B ρ ❣❣❣❣❣❣❣❣ ̺ z (cid:5) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) f C γ , f D , Since ρ is a double covering (by pullback-preservation), we obtain ϑ : F ˆ2 ( f A ) → f C × f D f B bythe preceding discussion. Then the morphism ( ̺ϑ, ι ) is a factorization of ( τ, ι ) through η ( α ) which is easily shown to be unique. (cid:3) Note that, as usual, the monadicity of I implies that CExt C , is closed under limits computedin Ext C . Also since η has regular epimorphic components, double coverings are closed under subobjects in Ext C (see for instance [46, Section 3.1]; note that the same comments hold for theadjunction F ⊣ I ). Closure by quotients along -fold extensions was discussed in Remark 2.3.3.We conclude the proof that F ⊣ I fits into a strongly Birkhoff Galois structure Γ in the nextarticle, Part III , where we study higher coverings of arbitrary dimensions.5.
Further developments
Besides the following theoretical developments, more explicit examples of double coveringsshould be studied, for instance in the known contexts of application cited in Part I. Now from theperspective of categorical Galois theory, future developments should also include the descriptionof a weakly universal double covering above an extension, and subsequently the characterizationof the fundamental double groupoid of an extension (see for instance [13]). From there, the fundamental theorem of categorical Galois theory should be applied in order to “classify” thedouble coverings above an extension.Another obvious line of work concerns the commutator defined in Section 3.1. A review ofthe links between commutators and Galois theory can be found in [45]. For instance, it shouldbe checked whether or not our commutator coincides with (or compares to) what was definedfor general varieties in terms of internal pregroupoids [48], or other theories such as the classicalapproach of [36] which was already applied in this context [3]. In the last paragraphs below, wesuggest to apply the developments of [44] to our context, with the objective to investigate thelinks between categorical Galois theory and homology [35, 17, 19] within racks and quandles (seealso [21, Section 9]).5.1.
Galois structure with abstract commutator.
As it was suggested in Part I, one of theimportant outcomes for the application of higher categorical Galois theory in groups was theelegant generalization of the higher Hopf formulae from [12] to semi-abelian categories [47, 31,27, 19]. In [44], G. Janelidze shares his perspective on how to understand the mechanics behindthe Hopf formulae from the perspective of categorical Galois theory, and in particular via thedescription and understanding of what an abstract Galois group is. He introduces the definitionof a
Galois structure with (abstract) commutators , which is suggested as another starting point(more general than that from [31]) for applying the methodology that he illustrates in the contextof groups. In this section, we adapt this definition in order to include the covering theory ofquandles as an example, in such a way which is compatible with the aims and developmentsfrom [44]. Further details about the application of the ideas from [44] to the covering theory ofquandles is left for future work.Our definition of Galois structure with (abstract) commutators is not aimed at being the mostgeneral possible. Our main point is the use of higher extensions to clarify the conditions whichare displayed in [44].
Definition 5.1.1.
A Galois structure with commutators is a system
Γ = ( C , X , F, I , η, ǫ, E , [ − , − ]) ,in which:(1) Γ = ( C , X , F, I , η, ǫ, E ) is an admissible Galois structure (see Convention 1.1.3 as well as [46, 5] for admissibility );(2) If f = pq in C and f and q are in E , then p is in E ;(3) [ − , − ] is a family of binary operations ER E ( A ) × ER E ( A ) → ER E ( A ) defined for each A in C and all written as ( S, T ) [ S, T ] ; here ER E ( A ) denotes the class of E -congruenceson C , i.e. the class of subobjects of A × A that are kernel pairs of morphisms from E ;(4) For S and T in ER E ( A ) , we always have [ S, T ] ≤ S ∩ T ;(5) If ( σ, β ) : γ → α is a morphism between the double extensions γ and β , then ( σ, β ) inducesa morphism [ σ ⊤ ] : [Eq( f C ) , Eq( γ ⊤ )] → [Eq( f A ) , Eq( α ⊤ )] ; IGHER COVERINGS OF RACKS AND QUANDLES – PART II 33 (6) if ( σ, β ) : γ → α above is a -fold extension, then [ σ ⊤ ] : [Eq( f C ) , Eq( γ ⊤ )] → [Eq( f A ) , Eq( α ⊤ )] is in E ;(7) For each A in C , F ( A ) = A/ [ A × A, A × A ] and η A is the canonical morphism A → A/ [ A × A, A × A ] ;(8) For a morphism p : E → B from E , p is a Γ -covering if and only if [ E × E, Eq( p )] = ∆ E ,i.e. [ E × E, Eq( p )] is the smallest congruence on E . Observe that conditions (5) and (6) are the two conditions which differ from G. Janelidze’spresentation (see . i ) and ( j ) in [44]). We explain how to translate from his context to ours. Inorder to avoid confusion, let us point out a small typo in [44]: the conclusions of Condition (g)in Definitions 4.1 and Condition (f) in Definition 4.4 should be that p is in F (rather than in C – see Condition (2) below). Also in Condition (i) of Definition 4.4 we should read [ E × E, Eq( p )] instead of [ E × E, Ker( p )] .Now using our notations, G. Janelidze merely considers the data of σ ⊤ : C ⊤ → A ⊤ , S .. =Eq( f C ) , T .. = Eq( γ ⊤ ) , S ′ .. = Eq( f A ) and T ′ .. = Eq( α ⊤ ) as well as induced morphisms s : S → S ′ and t : T → T ′ . From this data, we easily build the entire morphism ( σ, β ) with no furtherassumptions. The only difference is then the assumption that α and γ are not merely pushoutsquares of extensions, but also double extensions. Whenever C is a Mal’tsev category, which isthe case in the examples considered by G. Janelidze and others [26, 31, 32, 33, 28], α and γ areautomatically double extensions. In our context, this is the “natural” extra requirement to workwith (we work locally with congruences which commute since in our context, commutativity ofcongruences does not hold everywhere). Now when s and t are further required to be extensions(such as in . j ) ), by the same reasoning, the natural generalization from Mal’tsev categoriesconsists in requesting ( σ, β ) to be a square of double extensions. Finally observe that . j ) was already challenged in Remark . of [44]. Observe that under the restrictions suggested byT. Everaert or G. Janelidze, our square of double extensions ( σ, β ) becomes a -fold extension.Hence our choice of presentation is arguably an adequate and elegant variation from [44], whichis coherent with the example we are interested in, as well as the examples considered in [44] andrelated works. Example 5.1.2.
From the results of Section 3.1, and Lemma 4.1.1, we deduce that the Galoisstructure from Theorem 2.3.2 together with the operation [ − , − ] from Definition 3.1.1 satisfiesthe conditions of Definition 5.1.1. Since compatibility with unions is understood as an important property for commutators, weshow the following result which may be used to study Example 5.1.2 from the perspective of [44].Note that our hypotheses might not be optimal; we deduce the modular law locally from the lessgeneral permutability conditions on our congruences. These are arguably more suitable for thiscontext in which (repeatedly) we have been using, locally, some properties which are globallysatisfied in Mal’tsev categories.
Lemma 5.1.3.
Let A be a quandle, R , S and T congruences on A such that S ≤ R . If R , S and T commute two by two, and moreover R ∩ T commutes with S (for instance when S commuteswith all congruences), then [ R, S ∪ T ] = [ R, S ] ∪ [ R, T ] = [
S, A × A ] ∪ [ R, T ] .Proof. First observe that [ R, S ] ∪ [ R, T ] ≤ [ R, S ∪ T ] is an easy consequence of Lemma 3.1.9.Then consider a generator ( x ⊳ a ⊳ − b ⊳ c ⊳ − d, x ) ∈ [ R, S ∪ T ] for some x ∈ A and a RS ∪ T b S ∪ T d R c ∈ R (cid:3) ( S ∪ T ) . Since S and T commute, there is b ∈ A such that ( b, b ) ∈ S and ( b , c ) ∈ T . Moreover since R commutes with S and T , there are a and respectively d such that ( a, a ) ∈ S , ( a , b ) ∈ R , ( d, d ) ∈ T and ( d , b ) ∈ R . Hence ( a , d ) ∈ R ∩ ( S ∪ T ) . Using the modular law ( a , d ) ∈ S ∪ ( R ∩ T ) and thus there is a ∈ A such that ( a , a ) ∈ S and ( a , d ) ∈ ( R ∩ T ) . From thereobserve that ( a , b ) ∈ R and ( a , d ) ∈ T such that we obtain: a R S a R S a R T d R b S b S b T c Considering each of these three squares separately, by definition of [ R, S ] and [ R, T ] we derivethat x is in relation by [ R, S ] ∪ [ R, T ] with the element x ⊳ a ⊳ − b ⊳ b ⊳ − a ⊳ a ⊳ − b ⊳ b ⊳ − a ⊳ a ⊳ − b ⊳ c ⊳ − d, which reduces to x ⊳ a ⊳ − b ⊳ c ⊳ − d . (cid:3) Aknowledgments
The main aims of this paper were reached during a research visit at the
University of CapeTown , which was funded by the
Concours des bourses de voyage 2018 awarded by the
Fédéra-tion Wallonie-Bruxelles . I am very grateful to George Janelidze for our weekly conversationsand lunches on the
UCT campus. His request for the definition of a commutator and his en-couragements and advice for the completion of the main results of my Ph.D. project during mystay in South-Africa had a significant influence on the outcomes of my research. Let me addmany thanks to my supervisors Tim Van der Linden and Marino Gran for their support, advice,contributions and careful proofreading of this work.
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