HHIGHER COVERINGS OF RACKS AND QUANDLES – PART I
FRANÇOIS RENAUD
Abstract.
This article is the first part of a series of three articles, in which we develop ahigher covering theory of racks and quandles. This project is rooted in M. Eisermann’s workon quandle coverings, and the categorical perspective brought by V. Even, who characterizescoverings as those surjections which are central, relatively to trivial quandles. We extend thiswork by application of the techniques from higher categorical Galois theory (G. Janelidze), andin particular identify meaningful higher-dimensional centrality conditions defining our highercoverings of racks and quandles.In this first article (Part I), we revisit and clarify the foundations of the covering theory ofinterest, we extend it to the more general context of racks and mathematically describe howto navigate between racks and quandles. We explain the algebraic ingredients at play, andreinforce the homotopical and topological interpretations of these ingredients. In particularwe justify and insist on the crucial role of the left adjoint of the conjugation functor
Conj between groups and racks (or quandles). We rename this functor
Pth , and explain in whichsense it sends a rack to its group of homotopy classes of paths. We characterize coverings andrelative centrality using
Pth , but also develop a more visual “geometrical” understanding ofthese conditions. We use alternative generalizable and visual proofs for the characterization ofcentral extensions. We complete the recovery of M. Eisermann’s ad hoc constructions (weaklyuniversal cover, and fundamental groupoid) from a Galois-theoretic perspective. We sketchhow to deduce M. Eisermann’s detailed classification results from the fundamental theoremof categorical Galois theory. As we develop this refined understanding of the subject, we laydown all the ideas and results which will articulate the higher-dimensional theory developedin Part II and III. Introduction
Context.
We like to describe racks as sets equipped with a self-distributive system ofsymmetries, attached to each point. The term wrack was introduced by J.C. Conway andG.C. Wraith, in an unpublished correspondence of 1959. Their curiosity was driven towardsthe algebraic structure obtained from a group, when only the operations defined by conjugationare kept, and one forgets about the multiplication of elements. Sending a group to its so defined“wreckage” defines the conjugation functor from the category of groups to the category of wracks.We use the more common spelling rack as in [34] and [62]. Other names in the literature are automorphic sets by E. Brieskorn [8], crossed G-sets by P.J. Freyd and D.N. Yetter [35], and crystals by L.H. Kauffman in [54]. The former is (as far as we know) the first detailed publishedstudy of these structures.The image of the conjugation functor actually lands in the category of those racks whosesymmetries are required to fix the point they are attached to. Such structures were introducedand extensively studied by D.E. Joyce in his PhD thesis [52], under the name of quandles . Aroundthe same time (1980’s), S. Matveev was studying the same structures independently, under thename of distributive groupoids [60]. D.E. Joyce describes the theory of quandles as the algebraictheory of group conjugation, and uses them to produce a complete knot invariant for orientedknots.
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 .. a r X i v : . [ m a t h . C T ] J u l FRANÇOIS RENAUD
Over the last decades, racks and quandles have been applied to knot theory and physics invarious works – see for instance [53, 8, 34, 23, 54, 25] and references there. More historicalremarks are made in [34], including references to applications in computer science. In geometry,the earlier notion of symmetric space , as studied by O. Loos in [56], gives yet another context forapplications – see [4, 39] for up-to-date introductions to the field. This line of work goes back to1943 with M. Takasaki’s abstraction of a
Riemannian symmetric space : a kei [63], which wouldnow be called involutive quandle .More recently (2007) M. Eisermann worked on a covering theory for quandles (published in[25]). He defines quandle coverings, and studies them in analogy with topological coverings. Inparticular, he derives several classification results for coverings, in the form of Galois correspon-dences as in topology (or Galois theory). In order to do so, he works with ad hoc constructionssuch as a (weakly) universal covering or a fundamental group(oid) of a quandle. Even though thelink with coverings is unclear a priori, these constructions use the left adjoint of the conjugationfunctor, which is justified a posteriori by the fact that the theory produces the aforementionedclassification results.In his PhD thesis [26], V. Even applies categorical Galois theory, in the sense of G. Janelidze[41], to the context of quandles. By doing so, he establishes that M. Eisermann’s coverings arisefrom the admissible adjunction between trivial quandles (i.e. sets) and quandles, in the sameway that topological coverings arise from the admissible adjunction between discrete topologicalspaces (i.e. sets) and locally connected topological spaces (see Section 6.3 in [6]). He also derivesthat M. Eisermann’s notion of fundamental group of a connected, pointed quandle coincideswith the corresponding notion from categorical Galois theory. This, in turn, makes the bridgewith the fundamental group of a pointed, connected topological space. By doing so, V. Evenclarifies the analogy with topology even though his results still rely on ad hoc constructions suchas M. Eisermann’s weakly universal covers.1.2.
In this article.
We explicitly extend M. Eisermann and V. Even’s work to the more generalcontext of racks, as it was already suggested in their articles. We then clarify and justify theuse of the different algebraic and topological ingredients of their study with the perspective ofdeveloping a higher-dimensional covering theory.In Section 1.3, we describe enough of categorical Galois theory to motivate the overall projectand explain the results we seek. In particular, we specify which Galois structures we are interestedin, which conditions on these Galois structures we need in order to achieve our higher-dimensionalgoals, and finally we comment on the use of projective presentations, and a global strategy tocharacterize central extensions.In Section 2, we introduce the fundamentals of the theory of racks and quandles, with thebiased perspective of the covering theory which follows. We start (Section 2.1) with a short studyof the axioms, our first comments relating groups, racks and quandles, and the basic concepts ofsymmetry, inner automorphisms, and their actions. Next (Section 2.2), we develop some intuitionabout the geometrical features of a rack. We illustrate our comments on the construction of thefree rack, and recall the construction of the canonical projective presentation of a rack, whichpresents the elements in a rack with the geometrical features of those in the appropriate free rack.We then introduce the connected components adjunction (Section 2.3), from which the coveringtheory of interest arises. The concepts of trivializing relation, connectedness, primitive path, orbitcongruence, etc. are recalled. We propose to derive the trivializing relation from the geometricalunderstanding of free objects via projective presentations. We recall the admissibility results forthe connected components adjunction and comment on the non-local character of connectedness.We illustrate our visual approach of coverings on the characterization of trivial extensions.
IGHER COVERINGS OF RACKS AND QUANDLES – PART I 3
Section 2.4 follows with a description of the links between the construction of
Pth (the groupof paths functor), left adjoint of the conjugation functor, and the equivalence classes of tails of formal terms in the language of racks. Again, we propose to look at the simple descriptionof
Pth on free objects, and extend this description to all objects, via the canonical projectivepresentations. We describe the action of the group of paths and how it relates to inner automor-phisms and equivalence classes of primitive paths in general. The free action of this group onfree objects is recalled. We emphasize the functoriality of
Pth on all morphisms by contrast withthe non-functorial construction of inner automorphisms. We describe the kernels of the inducedmaps between groups of paths, in preparation for the characterizations of centrality. We insiston the fact that the role of
Pth (as left adjoint of the conjugation functor) is the same in racksand in quandles, although it is more intimately related to racks in design. We conclude thissurvey with a study of the adjunction between racks and quandles (Section 2.5). We derive itsadmissibility and deduce that all extensions are central with respect to this adjunction. We buildthe free quandle F q ( A ) on a set A in a way that illustrates best the journey from one contextto the other. By doing so, we explain interest for pairs of generators with opposite exponents(the transvection group, understood via the functor Pth ◦ ). We show that the normal subgroup Pth ◦ (F q ( A )) ≤ Pth(F q ( A )) of the group of paths acts freely on F q ( A ) as expected.In Section 3, we give a comprehensive Galois-theoretic account of the low-dimensional coveringtheory of quandles, which we extend to the suitable context of racks. Coverings are described,as well as their different characterizations, using the kernels of induced maps Pth( f ) betweengroups of paths, but also via the concept of closing horns . We recall that primitive extensions arecoverings, and coverings are preserved and reflected by pullbacks along surjections (i.e. centralextensions are coverings). We find counterexamples for Theorem 4.2 in [15], and finally illustrateour “geometrical” approach to centrality on the characterization of normal extensions. In Section3.2, we give proof(s) – generalizable to higher dimensions – for the characterization of centralextensions of racks and quandles. We then theoretically understand how the concepts of centralityin racks and in quandles relate (Section 3.3), using the factorization of the connected componentsadjunction through the adjunction between racks and quandles. Amongst other results, we derivethat the centralizing relations, if they exist, should be the same in both contexts. We then prove(Section 3.4) several characterizations of these centralizing relations, and extend the resultsfrom [24] on the reflectivity of coverings in extensions. In preparation for the admissibility indimension 2, we show that coverings are closed under quotients along double extensions (towards“Birkhoff”) and we show the commutativity property of the kernel pair of the centralization unit(towards “strongly Birkhoff”). We then move to Section 3.5, and the construction of weaklyuniversal covers from the centralization of canonical projective presentations. From there, webuild the fundamental Galois groupoid of a rack and of a quandle, establishing the homotopicalinterpretations of Pth and
Pth ◦ . In Section 3.6 we illustrate the use of the fundamental theoremof categorical Galois theory in this context. We conclude the article with a comment on therelationship between centrality in racks and in groups (Section 3.7).1.3. The point of view of categorical Galois theory.
Categorical Galois theory (in thesense of [41], see also [46]) is a very general (and thus abstract) theory with rich and variousinterpretations depending on the numerous contexts of application. On a theoretical level, Ga-lois theory exhibits strong links with, for example, factorization systems, commutator theory,homology and homotopy theory (see for instance [48, 17]). Looking at applications, it unifies, inparticular, the theory of field extensions from classical Galois theory, the theory of coverings oflocally connected topological spaces, and the theory of central extensions of groups. The coveringtheory of racks and quandles [25] is yet another example [26], which combines intuitive interpre-tations inspired by the topological example with features of the group theoretic case. A detailed
FRANÇOIS RENAUD historical account of the developments of Galois theory is given in [6]. In this introduction weavoid the technical details of the general theory, but hint at the very essentials needed by us.Categorical Galois theory always arises from an adjunction (say “relationship”) between twocategories (think “contexts”). For our purposes, there shall be a “primitive context”, say X , whichsits inside a “sophisticated context”, say C ; such that moreover C reflects back on X – e.g. sets,considered as discrete topological spaces, sit inside locally connected topological spaces whichreflect back on sets via the connected components functor π [6, Section 6.3]. Under certainhypotheses on these contexts and their relationship, categorical Galois theory studies a (specific) sphere of influence of the context X in the context C , with respect to this relationship – the ideaof a relative notion of centrality [36, 57]. This influence (centrality) is discussed in terms of achosen class of morphisms in these categories which we call extensions (e.g. the class of surjectiveétale maps ). Figure 1.
A kid’s drawing of categorical Galois theory
Convention 1.3.1.
For our purposes, a
Galois structure Γ .. = ( C , X , F, I , η, (cid:15), E ) (see [42] ), is thedata of such an inclusion I , of a full ( replete ) subcategory X in C , with left adjoint F : C → X ,unit η , counit (cid:15) and a chosen class of extensions E in C , such that E contains all isomorphisms, E is closed under composition and “ F ( E ) ⊂ E ” in the appropriate sense. Since the fundamentaltool at play in the theory is that of taking pullbacks [58] , pullbacks along extensions need to existand the pullback of an extension should be an extension. For our purposes E shall always be aclass of regular epimorphisms and the components of the unit will be extensions: we say that X is E -reflective in C . Given such a structure, the idea is that extensions “which live in X ”, which we call primitiveextensions , induce, in two steps, two other notions of extensions in C , which are somehow relatedto primitive extensions “in a tractable way”. The first step influence: trivial extensions , are thoseextensions t of C that are directly constructed from a primitive extension p in X , by pullback alonga unit morphism η (cid:5) (this gives topological trivial coverings in our example). Then the secondstep influence: central extensions (which are topological coverings in our example), are thoseextensions which “are locally trivial extensions”, i.e. extensions which can be split by anotherextension, where an extension e splits an extension c when the pullback t of c along e is trivial(see Figure 1). IGHER COVERINGS OF RACKS AND QUANDLES – PART I 5
For these definitions to be meaningful, we work with Galois structures such that: pullbacksof unit morphisms are unit morphisms (admissibility – see [46] for a precise definition), andmoreover: extensions are of ( relative ) effective descent (i.e. pulling back along extensions is an“algebraic operation” – see [50, 49]). Under such conditions the central extensions above a givenobject can be classified using data which is internal to X – in a form which is classically called a Galois correspondence , as in the theory of coverings in topology [38, Theorem 1.38]. We do notgive further details about admissibility (or effective descent), since it is enough to understand itas the condition for Galois theory to be applicable. We actually work with a stronger propertyfor Galois structures described in Section 1.3.2.More precisely, there is actually a third class of extensions, called normal extensions , whichare those central extensions that are split by themselves (the projections of their kernel pair aretrivial). Now if G is the image by the reflector F of the groupoid induced by the kernel pair of anormal extension n : A → B , then G is a groupoid in X . Internal groupoids and internal actionsare well explained in [51], read more about the use of groupoids in [10]. The fundamental theoremof categorical Galois theory then says that internal presheaves over that groupoid G (think“groupoid actions in X ”) yield a category which is equivalent to the category of those extensionsabove B which are split by n . If n splits all extensions above B , for instance when it is a weaklyuniversal central extension above B (see Section 1.3.3), then G is the fundamental groupoid of B , which thus classifies all extensions above B . A weakly universal central extension above B isa central extension with codomain B , which factors through any other central extension above B . Note that the conditions – connectedness, local path-connectedness and semi-local simply-connectedness – on the space X in [38, Theorem 1.38] are there to guarantee the existence of aweakly universal covering above X .In the case of groups, the adjunction of interest is ab (cid:97) I , the abelianization adjunction , wherethe left adjoint ab : Grp → Ab sends a group G to the abelian group G/ [ G, G ] , constructed byquotienting out the commutator subgroup [ G, G ] of G . In this context, the extensions are chosento be the regular epimorphisms , which are merely the surjective group homomorphisms. Giventhis Galois structure, the induced notion of central extension coincides with the definition of central extensions from group theory. The fundamental theorem can for instance be used toshow that given a perfect group G , the second integral homology group of G can be presented as a“Galois group” (see [37, 45]), which implies for instance that G has a universal central extension.Note that in Part I, the adjunction which gives rise to the covering theory of racks andquandles is related to ab (cid:97) I and is also such that X is a subvariety of algebras in C .. = Rck / Qnd .Such data always gives a Galois structure, by defining extensions to be the surjective maps [46].Moreover, X = Set is here equivalent to the category of sets, such that the left adjoint F .. = π : Rck / Qnd = C → X can be interpreted as a connected components functor like in topology.Now from the example of groups, and the aforementioned observation about links with homol-ogy, the development of Galois theory lead for instance to a generalisation [32] of the Hopf for-mulae for the (integral) homology of groups [11] to other non-abelian settings, leading to a wholenew approach to non-abelian homology, by the means of higher central extensions [43, 32, 30].This approach is compatible with settings such as the cotriple homology of Barr and Beck [2, 33],including, for instance, group homology with coefficients in the cyclic groups Z n . In order toaccess the relevant higher-dimensional information, as in [32], one actually “iterates” categoricalGalois theory. The increase in dimension consists in shifting from the context of C to the categoryof extensions of C : Ext C defined as the full subcategory of the arrow category Arr C with objectsbeing extensions. A morphism α : f A → f B in such a category of morphisms is given by a pairof morphisms in C , which we denote α = ( α , α ) (the top and bottom components of α ), such FRANÇOIS RENAUD that these form an (oriented) commutative square (on the left). A α (cid:44) (cid:50) f A (cid:12) (cid:18) ( → ) B f B (cid:12) (cid:18) A α (cid:44) (cid:50) B A α (cid:44) (cid:50) p (cid:34) (cid:42) f A (cid:12) (cid:18) B f B (cid:12) (cid:18) P π (cid:49) (cid:56) π (cid:122) (cid:5) A α (cid:44) (cid:50) B We call the comparison map of such a morphism (or commutative square) the unique map p : A → P induced by the universal property of P .. = A × B B , the pullback of α and f B .Now from the study of the admissible adjunction F (cid:97) I , Galois theory produces the concept of acentral extension, and thus we may look at the full subcategory CExt C of Ext C whose objects arecentral extensions. Central extensions are not reflective, even less so admissible, in extensionsin general (see [47]). In groups one can universally centralize an extension, along a quotientof its domain, and there CExt C is actually a full replete (regular epi)-reflective subcategory of Ext C . When such a reflection exists, one may further wonder whether there is a Galois structurebehind it, and whether it is admissible. What is the sphere of influence of central extensions inextensions, and with respect to which class of extensions of extensions , i.e. can we re-instantiateGalois theory in this induced (two-dimensional) context?An appropriate class of morphisms to work with, in order to obtain an admissible Galoisstructure in such a two-dimensional setting, is the class of double extensions [43, 18, 7, 31]. A double extension is a morphsim α = ( α , α ) in Ext C such that both α and α are extensions andthe comparison map of α is also an extension. Note that double extensions are indeed a subclassof regular epimorphisms in Ext C , provided C is a regular category (see [3]). Double centralextensions of groups were described in [43], and higher-dimensional Galois theory developedfurther [44, 32], leading to the aforementioned results in homology.Similarly in topology, higher homotopical information of spaces can be studied via the higherfundamental groupoids in the higher-dimensional Galois theory of locally connected topologicalspaces. A detailed survey about the study of higher-dimensional homotopy group(oid)s can befound in [9], see also [12]. Some insights are given in [14] where higher Galois theory is used tobuild a homotopy double groupoid for maps of spaces (see also [13]).In this article we consolidate the understanding of the one-dimensional covering theory ofracks and quandles, and introduce all the necessary ideas to start a higher-dimensional Galoistheory in this context. In Part II we obtain an admissible Galois structure for the inclusionof central extensions in extensions; we define and study double coverings, which are shown todescribe the double central extensions of racks and quandles. In Part III we generalize this toarbitrary dimensions.1.3.2.
Admissibility via the strongly Birkhoff condition, in two steps.
Note that in the literature,most instantiations of higher categorical Galois theory arise in the context of
Mal’tsev categories(see [19, 20, 18]), and this property (being Mal’tsev) is preserved in all dimensions. Admissibilityconditions as well as computations with higher extensions are easier to handle in such a context.The categories we are interested in are not Mal’tsev. Showing how higher categorical Galoistheory can apply in this more general setting thus requires some refinements on the argumentswhich are used in the existing examples.The difficulty is in the induction for higher dimensions: the study of a given Galois structureis one thing, the study of which properties of a Galois structure induce good properties of thesubsequent Galois structures in higher dimensions, is another. These subtleties will be discussedin Part III, see also [27] for the most general example known to the author. In Part I, we laydown the necessary foundations for what comes in Part II and III. Let us sketch here, withouttechnical details, which ingredients to focus on.
IGHER COVERINGS OF RACKS AND QUANDLES – PART I 7
In Part I, our context is that of [46] which we refer to for more details. We look at the inclu-sion
I :
X → C of X , a full , (regular epi)-reflective subcategory of a finitely cocomplete Barr-exact category C , such that X is closed under isomorphisms and quotients . In short Barr exactness means that C has finite limits, and every morphism admits pullback-stable regular-image fac-torizations , moreover, every equivalence relation is the kernel pair of its coequalizer [1]. Here (regular epi)-reflectiveness refers to the fact that the unit η of the adjunction F (cid:97) I (with leftadjoint F : C → X ) is a regular epimorphism (surjection), which implies (more generally) that X is also closed under subobjects. Finally, observe that (in general) monadicity of I implies that X is also closed under limits in C .The fact that X is closed under quotients is then the remaining condition for X to be called a Birkhoff subcategory of C [16, 46]. Given a more general Galois structure Γ = ( C , X , F, I , η, (cid:15), E ) such as in Remark 1.3.1, we say that Γ is Birkhoff if X is closed in C under quotients alongextensions. In the Galois structures of interest (see for instance in [46]), this condition is shown tobe equivalent to the fact that the reflection squares of extensions are pushouts . Given f : A → B in C , the reflection square at f (with respect to Γ ) is the morphism ρ f .. = ( η A , η B ) with domain f and codomain I F ( f ) in Arr ( C ) . Finally, X is said to be strongly Birkhoff in C if moreover thesereflection squares of extensions are themselves double extensions. A η A (cid:44) (cid:50) p (cid:32) (cid:41) f (cid:12) (cid:18) I F ( A ) I F ( f ) (cid:12) (cid:18) P π (cid:48) (cid:55) π (cid:123) (cid:6) B η B (cid:44) (cid:50) I F ( B ) (1)Proposition 2.6 in [32] implies that if Γ is strongly Birkhoff , then it is in particular admissible.Now observe that in the Barr-exact context from above, Proposition 5.4 in [18] implies thatif Γ is Birkhoff, it is strongly Birkhoff if and only if, for any object A in C , the kernel pair of η A commutes with any other equivalence relation on A (in the sense of [59, 18]). For instance, in thecategory of groups, any two equivalence relations commute with each other (see Mal’tsev [18]).Hence since Ab is a Birkhoff subcategory of Grp , it is actually strongly Birkhoff in
Grp , whichimplies the admissibility of ab (cid:97) I (see [46, Theorem 3.4]). However, the Mal’tsev property is toostrong, as it was noticed by V. Even in [26] and [27], where he uses the local permutability prop-erty of the kernel pairs of unit morphisms to conclude the admissibility of his Galois structure.In Part I, we briefly re-discuss these results and illustrate the argument on a new adjunction. Inhigher dimensions, we shall also aim to obtain strongly Birkhoff Galois structures by splittingthe work in two steps: (1) closure by quotients along higher extensions and (2) the permutabilitycondition on the kernel pairs of (the non-trivial component of) the unit morphisms.1.3.3. Splitting along projective presentations and weakly universal covers.
Remember that inany category, an object E is projective – with respect to a given class of morphisms, which wealways take to be our extensions – if for any extension f : A (cid:16) B and any morphism p : E → B ,there exits a factorization of p through f i.e. g : E → A such that f ◦ g = p . A projectivepresentation of an object B is then given by an extension p : E (cid:16) B such that E is projective(with respect to extensions). For instance, in varieties of algebras (in the sense of universalalgebra), there are enough projectives , i.e. each object has a canonical projective presentationgiven by the counit of the “free-forgetful” monadic adjunction with sets [58].In the Galois structures we consider (as it is the case in groups or in Part II-III) the “sophis-ticated context” C has enough projectives . Then any central extension f is in particular split by FRANÇOIS RENAUD any projective presentation p of its codomain. We have E × B A p A (cid:44) (cid:50) t (cid:37) (cid:44) p E (cid:12) (cid:18) A f (cid:12) (cid:18) T × B A p T (cid:12) (cid:18) (cid:51) (cid:59) E p (cid:44) (cid:50) p (cid:48) (cid:38) (cid:45) BT (cid:51) (cid:59) where p (cid:48) is induced by E being projective, t is induced by the universal property of T × B A and p T is a trivial extension by assumption. Then with no assumptions on C , the left hand face isa pullback since the back face and the right hand face are. Assuming that the Galois structurewe consider is admissible; trivial, central and normal extensions are then pullback stable (seefor instance [46]), and thus p E is trivial as it is the pullback of a trivial extension. Hence if C has enough projectives, then for any object B in C the category of central extensions CExt ( B ) above B is the same as the category of those extensions which are split by one given morphismsuch as the foregoing projective presentation p of B .Now when central extensions are reflective in extensions, a weakly universal central extensioncan always be obtained from the centralisation of a projective presentation. One can for exam-ple recover this idea from [61]. Consider an extension f : A → B , and the centralisation of aprojective presentation of B : E p (cid:12) (cid:18) centralisation (cid:119) (cid:1) a (cid:28) (cid:38) E (cid:48) p (cid:48) (cid:23) (cid:33) b (cid:52) (cid:60) A f (cid:126) (cid:8) B We get a since E is projective and b by the universal property of p (cid:48) . By the same argumentas above, any central extension is necessarily split by each weakly universal central extensionof its codomain. Such weakly universal central extensions above an object B are then split bythemselves which makes them normal extensions. The reflection of the kernel pair of such is thenthe fundamental Galois groupoid of B , which classifies central extensions above B .1.3.4. General strategy for characterizing central extensions.
Finally we describe our generalstrategy, suggested by G. Janelidze, when it comes to identifying a property which characterizescentral extensions. One observes that if a central extension f is split by a split epimorphism p ,then it is a trivial extension. Indeed, if the pullback f of f along p is trivial by assumption, thenthe pullback of f along the splitting s of p is again trivial and isomorphic to f : A (cid:36) (cid:44) ¯ s (cid:36) (cid:44) f (cid:12) (cid:18) A f (cid:12) (cid:18) E × B A f (cid:12) (cid:18) ¯ p (cid:51) (cid:58) B (cid:36) (cid:44) s (cid:36) (cid:44) BE p (cid:50) (cid:58) As a consequence, split epimorphic normal extensions are trivial. Also central extensions whichhave projective codomains are trivial. Now suppose one has identified a special class of extensions,called coverings , such that coverings are preserved and reflected by pullbacks along extensions.Provided primitive extensions are coverings, then all trivial extensions are coverings and alsocentral extensions are. Moreover, given a covering f : A → B , pulling back f along a projectivepresentation p of B yields a covering with projective codomain. Since f is central if and only IGHER COVERINGS OF RACKS AND QUANDLES – PART I 9 if it is split by such a p , we see that coverings are central extensions if and only if all coveringswith projective codomains are actually trivial extensions, which is usually easier to check.2. An introduction to racks and quandles
We introduce all the fundamental ingredients of the theory of racks and quandles, which wedescribe and develop from the perspective inspired by the covering theory of interest.2.1.
Axioms and basic concepts.
Racks and quandles as a system of symmetries.
Symmetry is classically modeled/studiedusing groups. Informally speaking: given a space X , one studies the group of automorphisms Aut( X ) of X . In his PhD thesis [52], D.E. Joyce describes quandles as another algebraic approachto symmetry such that, locally, each point x in a space X would be equipped with a globalsymmetry S x of the space X . Groups themselves always come with such a system of symmetriesgiven by conjugation and the definition of inner automorphisms. Quandles, and more primitivelyracks, can be seen as an algebraic generalisation of such.2.1.2. Describing the algebraic axioms.
Consider a set X that comes equipped with two functions X S (cid:44) (cid:50) S − (cid:44) (cid:50) X X , which assign functions S x and S − x in X X (the set of functions from X to X ) to each element x in X . Each element x then acts on any other y in X via those functions S x and S − x . Byconvention we shall always write actions on the right: y · S x .. = S x ( y ) y · S − x .. = S − x ( y ) The functions S x and S − x at a given point x ∈ X are required to be inverses of one anotherin the sense that for all y in X we have ( y · S − x ) · S x = y = ( y · S x ) · S − x ; note that, under this assumption, S − and S determine each other. Now we want to call suchbijections S x symmetries (or inner automorphisms ) of X . But observe that the set X is nowequipped with two binary operations X × X (cid:47) (cid:44) (cid:50) (cid:47) − (cid:44) (cid:50) X, defined by x(cid:47)y .. = x · S y and x(cid:47) − y .. = x · S − y for each x and y in X . Read “ y acts on x (positivelyor negatively)”. Automorphisms of X should then preserve these operations. In particular wethus require that for each x , y and z in X : ( x (cid:47) y ) (cid:47) z = ( x (cid:47) y ) · S z = ( x · S z ) (cid:47) ( y · S z ) = ( x (cid:47) z ) (cid:47) ( y (cid:47) z ) . Defining a rack.
Any set X equipped with such structure, i.e. two binary operations (cid:47) and (cid:47) − on X such that for all x , y and z in X :(R1) ( x (cid:47) y ) (cid:47) − y = x = ( x (cid:47) − y ) (cid:47) y ;(R2) ( x (cid:47) y ) (cid:47) z = ( x (cid:47) z ) (cid:47) ( y (cid:47) z ) ; is called a rack . We write Rck for the category of racks with rack homomorphisms defined asusual (functions preserving the operations).We refer to the axiom (R2) as self-distributivity . For each x in X , the positive (resp. negative ) symmetry at x is the automorphism S x (resp. S − x ) defined before. A symmetry , also called right-translation , of X is S x or S − x for some x in X . The symmetries of X refers to the set ofthose.2.1.4. Racks from group conjugation.
One crucial class of examples is given by group conjugation.D.E. Joyce describes quandles as “the algebraic theory of conjugation” [52]. We have the functor:
Grp
Conj (cid:44) (cid:50)
Rck , which sends a group G to the rack Conj( G ) with same underlying set, and whose rack operationsare defined by conjugation: x (cid:47) a .. = a − xa and x (cid:47) − a .. = axa − , for a and x in G . Group homomorphisms are sent to rack homomorphisms by just keeping thesame underlying function. The forgetful functor U :
Grp → Set thus factors through
U :
Rck → Set via
Conj . However the functor
Conj is not full , since given groups G and H , there are more rackhomomorphisms between Conj( G ) and Conj( H ) than there are group homomorphisms between G and H .This peculiar “inclusion” functor consists in “forgetting an operation” in comparison withsubvarieties which are about “adding an equation”. When forgetting an operation, an obviousquestion is to ask: what equations should the remaining operations satisfy? Racks is one candi-date theory. We will see that quandles (Subsection 2.1.10) give another option. In which senseis one different/better than the other? Can we characterize (as a subcategory) those racks whicharise from groups? An important ingredient for answering those questions and understanding therelationship between groups, racks and quandles is the left adjoint of Conj (Subsection 2.4). Thethorough study and understanding of this left adjoint (first defined by D.E. Joyce as
Adconj ,see also
Adj in [25]) is central to this piece of work, also with respect to its crucial role in thecovering theory of racks and quandles.In what follows, we often consider groups as racks without necessarily mentioning the functor
Conj .2.1.5.
Other identities.
Note that for the symmetries S x to define automorphisms of racks, oneneeds distributivity of (cid:47) on (cid:47) − , distributivity of (cid:47) − on (cid:47) , and self-distributivity of (cid:47) − . Allthese identities are induced by the chosen axioms. Besides, it suffices for a function f to preserveone of the operations in order for it to preserve the other. We do not give a detailed surveyof rack identities here. Bear in mind that in the theory of racks, the roles of (cid:47) and (cid:47) − areinterchangeable. Swapping them in a given equation, gives again a valid equation. Finally wefocus on an important characterization of (R2) using (R1):2.1.6. Self-distributivity.
Lemma 2.1.7.
Under the axiom (R1) , the axiom (R2) is equivalent to (R2’) x (cid:47) ( y (cid:47) z ) = (( x (cid:47) − z ) (cid:47) y ) (cid:47) z .Proof. Given (R1), we formally show that(R2) ⇒ (R2’): x (cid:47) ( y (cid:47) z ) = (( x (cid:47) − z ) (cid:47) z ) (cid:47) ( y (cid:47) z ) (by (R1)) = (( x (cid:47) − z ) (cid:47) y ) (cid:47) z (by (R2)) IGHER COVERINGS OF RACKS AND QUANDLES – PART I 11 (R2’) ⇒ (R2): ( x (cid:47) z ) (cid:47) ( y (cid:47) z ) = ((( x (cid:47) z ) (cid:47) − z ) (cid:47) y ) (cid:47) z (by (R2’)) = ( x (cid:47) y ) (cid:47) z (by (R1)) (cid:3) Similarly (R2) is also equivalent to (R2”): x(cid:47) ( y (cid:47) − z ) = (( x(cid:47)z ) (cid:47)y ) (cid:47) − z . From the precedingdiscussion we also have x (cid:47) − ( y (cid:47) − z ) = (( x (cid:47) z ) (cid:47) − y ) (cid:47) − z, and finally x (cid:47) − ( y (cid:47) z ) = (( x (cid:47) z ) (cid:47) y ) (cid:47) − z. Considering these as identities between formal terms in the language of racks (see for instanceChapter II, Section 10 in [16]), we say that the term on the right-hand side is unfolded , whereasthe term on the left hand side isn’t.2.1.8.
Composing symmetries – inner automorphisms.
By construction (see Paragraph 2.1.5),given a rack X , the images of S and S − (defined as above) are in the group of automorphismsof X . Define the group of inner automorphisms as the subgroup Inn( X ) of Aut( X ) generatedby the image of S . For each rack X , we then restrict S to the morphism X S (cid:44) (cid:50) Inn( X ) . An inner automorphism is thus a composite of symmetries. Remember that we use action onthe right, hence we use the notation z · (S x ◦ S y ) .. = S y (S x ( z )) for x , y , and z in X . We use thesame notation S for different racks X and Y . Note that the construction of the group of innerautomorphisms Inn does not define a functor from
Rck to Grp . It does so when restricted tosurjective maps (see for instance [15]).Observe that if z = x (cid:47) y in X , then S z = S − y ◦ S x ◦ S y by self-distributivity (R2’). Thefunction S is actually a rack homomorphism from X to Conj(Inn( X )) . Again this describes anatural transformation in the restricted context of surjective homomorphisms.Of course inner automorphisms of a group coincide with the inner automorphisms of theassociated conjugation rack. However, observe that for a group G , a composite of symmetries isalways a symmetry, whereas in a general rack, the composite of a sequence of symmetries doesnot always reduce to a one-step symmetry. Indeed, given a and b in a group G , then for all x ∈ G : ( x (cid:47) a ) (cid:47) b = b − a − xab = x (cid:47) ( ab ) and, moreover, x (cid:47) − a = x (cid:47) a − . So, given a group G , the morphism G S (cid:44) (cid:50) Conj(Inn( G )) = Inn( G ) is surjective.2.1.9. Acting with inner automorphisms – representing sequences of symmetries.
Given a rack X ,we have of course an action of Inn( X ) on X given by evaluation. Elements of the group of innerautomorphisms Inn( X ) allow for a “representation” of successive applications of symmetries, seenas a composite of the automorphisms S x .More explicitly, any g ∈ Inn( X ) decomposes as a product g = S δ n x ◦ · · · ◦ S δ x n for some elements x , . . . , x n in X and exponents δ , . . . , δ n in {− , } . Such a decomposition is not unique,but for any x in X the action of g on x is well defined by x · g .. = x · (S δ n x ◦ · · · ◦ S δ x n ) = x (cid:47) δ x (cid:47) δ x · · · (cid:47) δ n x n , where we omit parentheses using the convention that one should always compute the left-mostoperation first. x (cid:54) = y in a rack X , two symmetries S x and S y are identified in Inn( X ) if theydefine the same automorphism. As we move away from systematically looking at symmetrieswithin the group of inner automorphisms: from now on, and informally speaking, we considerthe data of the base-point x to be part of the data which we refer to as a symmetry denoted S x or S − x .In what follows we study different ways and motivations to organize the set of symmetriesinto a group acting on X . Note that we may understand the definition of augmented quandles (or racks) [52], see Paragraph 2.4.5, as a tool to abstract away from “representing” sequences ofsymmetries via composites of such (in the sense of the group of inner automorphisms).2.1.10. Quandles, the idempotency axiom.
As explained by D.E. Joyce, it is reasonable (in ref-erence to applications) to require that a symmetry at a given point fixes that point. If for each x in a rack X we have moreover that(Q1) x (cid:47) x = x ;then X is called a quandle . We have the category of quandles Qnd defined as before. Again,(Q1) is equivalent to (Q1’): x (cid:47) − x = x , under the axiom (R1).For the purpose of this article, we shall mainly be working in the more general context ofracks since these exhibit all the necessary features for the covering theory of interest. Actuallyall concepts of centrality and coverings shall coincide whether one works with the category ofracks or of quandles. Directions for a systematic conceptual understanding of these facts will beprovided. The addition of the idempotency axiom still has certain consequences on ingredientsof the theory such as the fundamental groupoid or the homotopy classes of paths . We shallalways make explicit these differences and similarities, also using the enlightening study of the“free-forgetful” adjunction between racks and quandles.2.1.11. Idempotency in racks.
An essential observation to make is that, even though (Q1) doesn’thold in all racks, a weaker version of the idempotency axiom still holds in a general rack as aconsequence of self-distributivity. Indeed, racks and quandles are very close – which we shallillustrate throughout this article. The axiom (Q1) requires the (cid:47) operations to be idempotent: x (cid:47) x = x . Now observe that in a rack X , such identities can be deduced by self-distributivity in“the tail of a term”: given any y and x ∈ X , we have x (cid:47) ( y (cid:47) y ) = x (cid:47) − y (cid:47) y (cid:47) y = x (cid:47) y. The symmetries S y and S ( y(cid:47)y ) , at y and y(cid:47)y are always identified in Inn( X ) , even when y (cid:54) = ( y(cid:47)y ) in X . Similarly, for x and y in X any chain y (cid:47) k y (for k ∈ Z , the action of y on y , repeated k times) is such that x (cid:47) ( y (cid:47) k y ) = x (cid:47) y . For more details, the left adjoint r F q : Rck → Qnd to theinclusion
I :
Qnd → Rck will be described in Section 2.5.1. In what follows, the present commenttranslates in several different ways, such as in Example 3.1.6 for instance.2.2.
From axioms to geometrical features.
We informally highlight two additional elementary features of the axioms which play an importantrole in what follows. We then illustrate them in the characterization of the free rack on a set A .2.2.1. Heads and tails – detachable tails.
Observe that on either side of the identities definingracks, the head x of each term is the same and does not play any role in the described identifi-cations. (R1) x (cid:47) y (cid:47) − y = x = x (cid:47) − y (cid:47) y (R2’) x (cid:47) ( y (cid:47) z ) = x (cid:47) − z (cid:47) y (cid:47) z IGHER COVERINGS OF RACKS AND QUANDLES – PART I 13
Now consider any formal term in the language of racks (built inductively from atomic variablesand the rack operations – see Chapter II Section 10 in [16]), such as for instance ( x (cid:47) y ) (cid:47) − ( · · · (( a (cid:47) b ) (cid:47) − c ) (cid:47) d ) · · · (cid:47) z. (2)Remember that roughly speaking, the elements of the free rack on a set A can be constructed asequivalence classes of such formal terms, built inductively from the atomic variables in A , wheretwo terms are identified if one can be obtained from the other by replacing subterms accordingto the axioms, or according to any provable equations derived from the axioms.Given any term such as above, we shall distinguish the head x of the term from the rest ofit which is called the tail of the term. The informal idea is that the “behaviour” of the tailis independent from the head it is attached to. It thus makes sense to consider the tails (orequivalence classes of such) separately from the heads these tails might act upon.Observe that the idempotency axiom plays a slightly different role in that respect since,although the heads of terms are left unchanged under the use of (Q1), the identifications in thetails of terms might depend on the heads these are attached to. We shall however see that thediscussion about racks still lays a clear foundation for understanding the case of quandles whichwe discuss in Section 2.5.2.2.2. Tails as sequences of symmetries.
By Paragraph 2.1.6, acting with a symmetry of the form S ( x(cid:47)y ) translates into successive applications of S − y , S x , S y from left to right. • (cid:44) (cid:50) S − y (cid:12) (cid:18) S x(cid:47)y • (cid:12) (cid:18) S x • • (cid:108) (cid:114) S y Now consider any formal term such as in Equation (2) for instance. Using (R2’) repeatedly,we may unfold the tail of a term into a string of successive actions of the form x (cid:47) y (cid:47) − c (cid:47) c (cid:47) − b (cid:47) − a (cid:47) b (cid:47) − c (cid:47) c (cid:47) d · · · (cid:47) z. We can then interpret the tail as a path of successive actions of the symmetries which are appliedto the head x . Using (R1) repeatedly again, we may also discards all possible occurrences of thesuccessive application of a symmetry and its inverse x (cid:47) y (cid:47) − b (cid:47) − a (cid:47) b (cid:47) d · · · (cid:47) z. It is then possible to show that such unfolded and reduced terms provide normal forms (uniquerepresentatives) for elements in the free rack. The elements of a free rack on a set A are thusdescribed with this architectural feature of having a head in A and an independent tail, suchthat the tail is a sequence of “representatives” of the symmetries which organize themselves asthe elements of the free group on A .2.2.3. The free rack.
The following construction can be found in [34]. It was also studied in [55].Given a set A the free rack on A is given by F r ( A ) .. = A (cid:111) F g ( A ) .. = { ( a, g ) | g ∈ F g ( A ); a ∈ A } , where F g ( A ) is the free group on A and the operations on F r ( A ) are defined for ( a, g ) and ( b, h ) in A (cid:111) F g ( A ) by ( a, g ) (cid:47) ( b, h ) .. = ( a, gh − bh ) and ( a, g ) (cid:47) − ( b, h ) .. = ( a, gh − b − h ) . In order to distinguish elements x in A from their images under the injection η gA : A → F g ( A ) ,we shall use the convention to write a .. = η gA ( a ) . Looking for the unit of the adjunction, we then have the injective function which sends anelement in A to the trivial path starting at that element, i.e. η rA : A → F r ( A ) : a (cid:31) (cid:44) (cid:50) ( a, e ) ,where e is the empty word (neutral element) in F g ( A ) .Note that since any element g ∈ F g ( A ) decomposes as a product g = g δ · · · g nδ n ∈ F g ( A ) for some g i ∈ A and exponents δ i = 1 or − , with ≤ i ≤ n , we have, for any ( a, g ) ∈ F r ( A ) , adecomposition as ( a, g ) = ( a, g δ · · · g nδ n ) = ( a, e ) (cid:47) δ ( g , e ) (cid:47) δ ( g , e ) · · · (cid:47) δ n ( g n , e ) . As we discussed before, if we have moreover that g i = g i +1 and δ i = − δ i +1 for some ≤ i ≤ n ,then ( a, e ) (cid:47) δ ( g , e ) · · · (cid:47) δ i − ( g i − , e ) (cid:47) δ i ( g i , e ) (cid:47) δ i +1 ( g i +1 , e ) (cid:47) δ i +2 ( g i +2 , e ) · · · (cid:47) δ n ( g n , e ) == ( a, g δ · · · g i − δ i − g iδ i g i +1 δ i +1 g i +2 δ i +2 · · · g nδ n )= ( a, g δ · · · g i − δ i − g i +2 δ i +2 · · · g nδ n )= ( a, e ) (cid:47) δ ( g , e ) · · · (cid:47) δ i − ( g i − , e ) (cid:47) δ i +2 ( g i +2 , e ) · · · (cid:47) δ n ( g n , e ) which expresses the first axiom of racks, using group cancellation.From there we derive the universal property of the unit: given a function f : A → X for somerack X , we show that f factors uniquely through η rA . Given an element ( a, g ) ∈ F r ( A ) , we havethat for any decomposition g = g δ · · · g nδ n as above, we must have f ( a, g ) = f ( a, g δ · · · g nδ n ) = f (cid:0) ( a, e ) (cid:47) δ ( g , e ) · · · (cid:47) δ n ( g n , e ) (cid:1) = f ( a ) (cid:47) δ f ( g ) · · · (cid:47) δ n f ( g n ) which uniquely defines the extension of f along η rA to a rack homomorphism f : F r ( A ) → X .This extension is well defined since two equivalent decompositions in F r ( A ) are equivalent after f by the first axiom of racks as displayed in Paragraph 2.2.3.The left adjoint F r : Set → Rck of the forgetful functor
U :
Rck → Set with unit η r is thendefined on functions f : A → B by F r ( f ) .. = f × F g ( f ) : A (cid:111) F g ( A ) → B (cid:111) F g ( B ) . This is easily seen to define a rack homomorphism. Functoriality of F r and naturality of η r areimmediate.2.2.3.1. Terminology and visual representation. In order to emphasise its visual representation,we call an element ( a, g ) ∈ F r ( A ) a trail . We call g the path (or tail ) component and a the head component of the trail ( a, g ) . It is understood that the path g formally acts on a to producean endpoint of the trail (see Paragraph 2.2.3). Formally ( a, g ) stands for both the trail and itsendpoint: a (cid:44) (cid:50) g ( a, g ) . The action of a trail ( b, h ) on another trail ( a, g ) consists in adding, at the end of the path g , the contribution of the symmetry associated to the endpoint of ( b, h ) (see Subsection 2.2.4and further). We say that a trail acts on another by endpoint , as in the diagram below, wherecomposition of arrows is computed by multiplication in the path component: a (cid:12) (cid:18) g (cid:47) b (cid:12) (cid:18) h = a (cid:12) (cid:18) g ( a, g ) (cid:12) (cid:18) h − bh ( a, g ) ( b, h ) ( a, gh − bh ) (3) IGHER COVERINGS OF RACKS AND QUANDLES – PART I 15
Canonical projective presentations.
Since
Rck is a variety of algebras, any object X canbe canonically presented as the quotient F r F r X F r (cid:15) rX (cid:44) (cid:50) (cid:15) r Fr X (cid:44) (cid:50) F r X F r η rX (cid:108) (cid:114) (cid:15) rX (cid:44) (cid:50) X where we have omitted to write the forgetful functor U :
Rck → Set (understand X alternativelyas a rack or a set), and (cid:15) rX is the counit of the “free-forgetful” adjunction F r (cid:97) U . This counit (cid:15) rX is the coequalizer of the reflexive graph on the left. This canonical presentation of racksallows us to capture a sense in which the geometrical features of free objects are carried throughto any general rack. We shall illustrate this on the important functorial constructions of theGalois theory of interest. Let us make explicit these objects and morphisms to exhibit someof the mechanics at play. Think of what this right-exact fork represents for groups, where theoperation is associative.First of all we may exhibit heads and tails and rewrite this right-exact fork as ( X (cid:111) F g ( X )) (cid:111) F g ( X (cid:111) F g ( X )) (cid:15) rX × F g [ (cid:15) rX ] (cid:44) (cid:50) (cid:15) r Fr X (cid:44) (cid:50) X (cid:111) F g X F r η rX (cid:108) (cid:114) (cid:15) rX (cid:44) (cid:50) X Then it is immediate from Paragraph 2.2.3 that the counit (cid:15) rX should send a pair ( x, g ) =( x, g δ · · · g nδ n ) for g i ∈ X to the element in the rack X given by: (cid:15) rX ( x, g ) = x · g .. = x (cid:47) δ g · · · (cid:47) δ n g n . Hence the canonical projective presentation (cid:15) rX of a rack X covers each element x ∈ X by allpossible formal decomposition ( x , g ) of that element x , such that x is the endpoint of the trail ( x , g ) , i.e. the result of the action of a path on a head : x = x · g . Now this head x andeach “representative of a symmetry” g iδ i in the path component g = g δ · · · g nδ n may itself beexpressed as the endpoint of some trail (i.e. x = x · h , and g i = y i · k i for h and k i in F g X ).This is what is captured by the object F r F r ( X ) on the left of the fork.Then from the definition of the counit, we may derive the two projections. These may beunderstood as expressing two things:First observe that an element t = [( a, g ); e ] in F r F r ( X ) (i.e. an element which has a trivialpath component, but an interesting head) is sent to (( a · g ) , e ) by the first projection and to ( a, g ) by the second projection. The two projections thus allow us to move part of the tail of a trailtowards the head of that trail and part of the head towards the tail.Then an element [( a, e ); ( b, h )] – i.e. an element with a trivial head component and a non trivial(but simple) tail – is sent by the first projection to ( a, ( b · h )) , and by the second projection to ( a, h − bh ) . Coequalizing these two projections expresses self-distributivity (see Paragraphs 2.1.6and 2.2.2). In other words it illustrates how to compute the representative of the symmetryassociated to the endpoint of a trail. This is already part of the definition of the rack operationin the free rack. We have the rack homomorphism on the left X (cid:111) F g ( X ) i X (cid:44) (cid:50) F g ( X )( x, g ) (cid:31) i X (cid:44) (cid:50) g − xg X η gX (cid:44) (cid:50) η rX (cid:28) (cid:39) Conj(F g ( X ))F r ( X ) i X (cid:51) (cid:59) which sends a path to the symmetry associated to its endpoint. It is actually induced by theuniversal property of free racks as displayed in the diagram on the right.2.3. The connected components adjunction.
Trivial racks and trivializing congruence.
Another important theoretical example of racksis given by the so-called trivial racks (or trivial quandles) for which each symmetry at a givenpoint is chosen to be the identity. Each point acts trivially on the rest of the rack. This may beexpressed as an additional axiom:(Triv) x (cid:47) y = x .Since each set comes with a unique structure of trivial rack and each function between trivialracks is a homomorphism, we get an isomorphism between the category of sets ( Set ) and thecategory of trivial racks. The category of sets is thus a subvariety of algebras within racks.The inclusion functor
I :
Set → Rck sends a set to the trivial rack on that set. Now thisinclusion functor should have a left adjoint which sends a rack to the freely trivialized rack.Since trivial racks are those which satisfy (Triv), a good candidate for the trivialization of a rack X is thus by quotienting out the congruence C X generated by the pairs ( x, x (cid:47) y ) . Using the comments of Section 2.2, it is not too hard to show that it actually suffices to considerthe transitive closure of the set of pairs { ( x, x ) , ( x, x (cid:47) y ) , ( x, x (cid:47) − y ) | x, y ∈ X } which givesthe congruence C X when endowed with the rack structure of the cartesian product. Symmetryand compatibility with rack operations are obtained for free. This further yields the concepts of connectedness and primitive path of Paragraph 2.3.3. Convention 2.3.2.
For the purpose of this work, understand sets, or trivial racks, to be the of the covering theory of racks (and quandles), in the same way thatabelian groups and central extensions of groups are respectively the 0-dimensional coverings and1-dimensional coverings in groups. Similarly C is the centralizing relation in dimension 0 . InSection 3 we study the subsequent 1-dimensional covering theory of racks and quandles. Connectedness and primitive paths.
Given two elements x and y in a rack A , we say that x and y are connected ( [ x ] = [ y ] ) if there exists n ∈ N and elements a , a , . . . , a n in A such that y = x (cid:47) δ a (cid:47) δ a · · · (cid:47) δ n a n , for some coefficients δ i ∈ {− , } for ≤ i ≤ n .Such a sequence of elements together with the choice of coefficients is viewed as a formalsequence of symmetries (see Paragraph 2.1.9.1). Bearing in mind Paragraphs 2.2.1 and 2.2.2,we call such a formal sequence of symmetries ( a i , δ i ) ≤ i ≤ n a primitive path of the rack A . Inparticular this specific primitive path connects x to y but may be applied to different elementsin the rack. We call the data of such a pair T = ( x, ( a i , δ i ) ≤ i ≤ n ) a primitive trail in X , where x is the head of T and y the endpoint of T .We have that ( x, y ) is in C A if and only if there exists a primitive path which connects x to y . For the sake of precision, and following the point of view from [52], let us take this asdefinition for C A .2.3.4. Left adjoint π . Then any rack homomorphism f : A → X for some trivial rack X is suchthat C A ≤ Eq( f ) since given y = x (cid:47) δ a · · · (cid:47) δ n a n in A we must have in X : f ( y ) = f ( x ) (cid:47) δ f ( a ) · · · (cid:47) δ n f ( a n ) = f ( x ) . Hence we define the functor π : Rck → Set such that π ( A ) .. = A/ (C A ) is the set of connectedcomponents of A (i.e. the set of C A -equivalence classes) and π (cid:97) I with unit A η A (cid:44) (cid:50) π ( A ) , IGHER COVERINGS OF RACKS AND QUANDLES – PART I 17 sending an element a ∈ A to its connected component η A ( a ) (also denoted [ a ] ) in π ( A ) . For any f : A → X as before, there is a unique function f (cid:48) : π ( A ) → X defined on a connected componentby the image under f of any of its representatives.2.3.5. From free objects to all – definition as a colimit.
Assume that we hadn’t explicitly con-structed π yet, and observe that the composite Set I (cid:44) (cid:50) Rck U (cid:44) (cid:50) Set gives the identity functor. As a consequence, the composite of left adjoints π F q also gives theidentity functor. More precisely we may deduce from the composite of adjunctions that, given aset X , the unit η F r ( X ) : X (cid:111) F g ( X ) → X is “projection on X ”, i.e. the connected component of atrail ( x, g ) ∈ F r ( X ) is given by projection on its head x .Since π is a left adjoint, it should preserve colimits, hence π ( X ) should be the coequalizer,in Set , of the pair: π (( X (cid:111) F g ( X )) (cid:111) F g ( X (cid:111) F g ( X ))) π ( (cid:15) rX × F g [ (cid:15) rX ]) (cid:44) (cid:50) π ( (cid:15) r Fr U X ) (cid:44) (cid:50) π ( X (cid:111) F g X ) , which indeed reduces to being the coequalizer of X × F g ( X ) p (cid:44) (cid:50) p (cid:44) (cid:50) X where p ( x, g δ · · · g nδ n ) = x (cid:47) δ g · · · (cid:47) δ n g n ; p ( x, g δ · · · g nδ n ) = x. Equivalence classes of primitive paths.
The term primitive path is used to express the ideathat it is the most unrefined way we shall use to acknowledge that two elements are connected.Literally it is just a formal sequence of symmetries.As explained in Paragraph 2.1.9, inner automorphisms also “represent” sequences of sym-metries. Again, each primitive path naturally reduces to an inner automorphism simply bycomposing all the symmetries in the sequence. We also have that ( x, y ) is in C A if and only ifthere exists g ∈ Inn( A ) such that x · g = y . In other words, C A is the congruence generatedby the action of Inn( A ) . We call it the orbit congruence of Inn( A ) (see Paragraph 2.3.9). Inwhat follows, we like to view inner automorphisms as equivalence classes of primitive paths. Asmentioned earlier we shall consider other such equivalence classes of primitive paths which lie inbetween formal sequences of symmetries and composites of such. Each of these represent differ-ent witnesses of how to connect elements in a rack. All of these generate the same trivializingcongruence C .2.3.7. Conjugacy classes.
Observe that for a group
Conj( G ) , its set of connected components isgiven by the set of conjugacy classes in G . In this case the congruence C (Conj( G )) is charac-terised as follows: ( a, b ) ∈ C (Conj( G )) if and only if there exists c ∈ G such that b = c − ac .Again, any primitive path, or sequence of symmetries, can be described via a single symmetryobtained as the symmetry of the product of the elements in the sequence.Note that if H is an abelian group, then Conj( H ) is the trivial rack on the underlying set of H . More precisely the restriction to Ab of the functor Conj yields the forgetful functor to
Set : Ab Conj restricts to U (cid:44) (cid:50) Set . Racks and quandles have the same connected components.
The functor π may be re-stricted to the domain Qnd and is then left adjoint to the inclusion functor
I :
Set → Qnd by thesame arguments as above. More precisely we have for any rack X that π F q ( X ) = π ( X ) ,where r F q ( X ) is the free quandle on the rack X . Orbit congruences permute.
In order to obtain the admissibility of
Set in Qnd , V. Evenshows that certain classes of congruences commute with all congruences. As for quandles, wedefine orbit congruences [15] as the congruences induced by the action of a normal subgroup ofthe inner automorphisms. More precisely, if X is a rack, and N a normal subgroup of Inn( X ) weshall write ∼ N for the N -orbit congruence defined for elements x and y in X by: x ∼ N y if andonly if there exists g ∈ N such that x · g = y . As it is explained in [27] (see Proposition 2.3.9),this is well defined and yields a congruence (also in Rck ).We then have the following – see [28] and [27, Lemma 3.1.2] for the proof, which also holds in
Rck . Lemma 2.3.10.
Let X be a rack, R a reflexive (internal) relation on X and N a normal subgroupof Inn( X ) , then the relations ∼ N and R permute: ∼ N ◦ R = R ◦ ∼ N . Admissibility for Galois theory.
Of course the kernel pair of the unit η X : X → π ( X ) isan orbit congruence, since by Paragraph 2.3.6, two elements are in the same connected componentif and only if they are in the same orbit under the action of Inn( X ) .As it was shortly recalled in Section 1.3.2 (see also [46]), this yields Theorem 1 of [26]: Corollary 2.3.12.
The subvariety
Set is strongly Birkhoff and thus admissible in
Rck . Similarlyfor
Set in Qnd . The Galois structure Γ .. = ( Rck , Set , π , I , η, (cid:15), E ) (respectively Γ q .. = ( Qnd , Set , π , I , η, (cid:15), E ) )(see [46]) where E is the class of surjective morphisms of racks (respectively quandles), is thusadmissible, i.e. the study of Galois theory is relevant in this context and gives rise, in principle,to a meaningful notion of relative centrality.2.3.13. Connected components are not connected.
Given an element a in a rack A , we may con-sider its connected component C a , i.e. the elements of A which are connected to a . The set C a is actually a subrack of A as it is closed under the operations in A . We may construct the rack C a as a pullback in Rck : C a (cid:44) (cid:50) (cid:12) (cid:18) [ a ] (cid:12) (cid:18) A η A (cid:44) (cid:50) π ( A ) , (4)where {∗} is the one element set, which is the terminal object in Rck and also the freequandle on the one element set. Note that if A is connected, then by definition π ( A ) = {∗} and thus C a = A . However if C a ⊂ A , then C a might have more than one connected componentitself (i.e. π (C a ) has cardinality | π (C a ) | > ), since the existence of a primitive path betweensome c and b in C a , might depend on elements which are not connected to a . Example 2.3.14.
A rack A is called involutive if the two operations (cid:47) and (cid:47) − coincide. Thesubvariety of involutive racks is thus obtained by adding the axiom (Inv) x (cid:47) y (cid:47) y = x .We define the involutive quandle Q ab(cid:63) with three elements a , b and (cid:63) such that the operation (cid:47) is defined by the following table (see Q (2 , from [25, Example 1.3] ). (cid:47) a b (cid:63)a a a bb b b a(cid:63) (cid:63) (cid:63) (cid:63) IGHER COVERINGS OF RACKS AND QUANDLES – PART I 19
The connected component of a is the trivial rack C a = { a, b } which has itself two connectedcomponents { a } and { b } . We like to say that, for racks (and quandles) the notion of connectedness is not local. Incategorical terms, we may say that the functor π is not semi-left-exact [21, 17]. This propertyis indeed characterised, in this context, by the preservation of pullbacks such as in Equation (4)above, i.e. π is semi-left-exact if and only if any such connected component ( C a ) is connected( π (C a ) = {∗} ) (see for instance [6]). This is an important difference with the case of topo-logical spaces for instance, for which the connected components are connected and thus thecorresponding π functor is semi-left-exact. See also [28] for further insights on connectedness.Finally note that the same comments apply to the context of Qnd . Looking at [22, Corollary2.5], we compute that π (F r (1) × F r (1)) = Z and thus that π : Rck → Set does not preserve finiteproducts; wheareas π : Qnd → Set does, as was shown in [26, Lemma 3.6.5].2.3.15.
Towards covering theory.
Knowing that Γ is admissible, we may now wonder what is the“sphere of influence” of Set in Rck , with respect to surjective maps, and start to develop thecovering theory. Since
Set is strongly Birkhoff in
Rck , trivial extensions (first step influence) areeasy to characterize as those surjections which are “injective on connected components”:
Corollary 2.3.16. (See also [26, 27] ) Given a surjective morphism of racks t : X → Y , TFAE:(i) t is a trivial extension;(ii) Eq( t ) ∩ C X = ∆ X ;(iii) if a and b in X are connected, then t ( a ) = t ( b ) implies a = b . Recall that the construction of inner automorphisms (
Inn ) induces a functor on surjectivemorphisms: given a surjective morphism t : X → Y , we write ˆ t or Inn( t ) : Inn( X ) → Inn( Y ) forthe induced homomorphism between the inner automorphism groups (see first two sections of[15]).We may then also describe a trivial extension as an extension which reflects loops : trivialextensions are those extensions such that for any a in A , if g in Inn( A ) is such that t ( a ) · ˆ t ( g ) = t ( a ) ,then a · g = a . ( a (cid:44) (cid:50) g a · g ) (cid:31) t (cid:44) (cid:50) t ( a ) = t ( a · g ) (cid:44) (cid:50) ˆ t ( g ) ⇒ a = a · g (cid:44) (cid:50) g In what follows, we shall use such geometrical interpretations to make sense of the algebraicconditions of interest for the covering theory. However, the non-functoriality of
Inn on generalmorphisms appears as a serious weakness (see for instance the need for Remark 2.4.7 in the proofof Proposition 3.2.1). It will become clear from what follows that a more suitable way to representsequences of symmetries is needed. This is achieved by the group of paths which we motivateand describe in the next section. It is not a new concept, but our name for the left adjoint ofthe conjugation functor, which was described by D.E. Joyce and then used by M. Eisermann toconstruct weakly universal covers and an ad hoc fundamental groupoid for quandles. However,we provide a hopefully enlightening description of the construction and the role of this functor,which naturally arises from the geometrical features described in Section 2.2.2.4.
The group of paths.
Definition.
Consider a rack X and two elements x and y in X which are connected by aprimitive path S δ x , . . . , S δ n x n : x · (S δ x , . . . , S δ n x n ) .. = x (cid:47) δ x · · · (cid:47) δ n x n = y. Because of (R1), we discussed that it makes sense to identify such formal sequences so as toobtain elements of the free group on X . Now in the same way that we used Paragraph 2.1.6 tounfold formal terms, we still have that whenever x i = b (cid:47) c for ≤ i ≤ n and b , c in X , actingwith S x i amounts to successively acting with S − c , S b and S c . From a rack X we may thus buildthe quotient: F g ( X ) q X (cid:44) (cid:50) Pth( X ) .. = F g ( X ) / (cid:104) c − a − x a | a, x, c ∈ X and c = x (cid:47) a (cid:105) , which is understood as a group of equivalence classes of primitive paths. Two primitive pathsare identified in the group of paths if and only if one can be formally obtained from the other,using the identities induced by the graph of the rack operations (such as c = x (cid:47) a ), as well as theaxioms of racks (or more precisely the axiom-induced identities between tails of formal terms).2.4.2. Unit and universal property.
The function η g : X → F g ( X ) composed with this quotient q X : F g ( X ) → Pth( X ) yields a rack homomorphism X pth X (cid:44) (cid:50) Conj(Pth( X )) which sends each element x of X to pth X ( x ) in Pth( X ) , such that pth X ( x ) “represents” thepositive symmetry at x in the same way S x does in Inn( X ) (see Paragraph 2.4.5). As for theinclusion in the free group, we shall use the convention x .. = pth X ( x ) . Now given a rack homomorphism f : X → Conj( G ) for some group G , there is a unique grouphomomorphism f (cid:48) induced by the universal property of the free group, which, moreover, factorsuniquely through the quotient q X : F g ( X ) → F g ( X ) / (cid:104) ( x (cid:47) a ) − a − x a | a, x ∈ X (cid:105) , since for any a and x in X , f ( x (cid:47) a ) = f ( a ) − f ( x ) f ( a ) in G : X η gX (cid:44) (cid:50) f (cid:29) (cid:39) F g ( X ) ∃ ! f (cid:48) (cid:12) (cid:18) q X (cid:44) (cid:50) Pth( X ) ∃ ! ¯ f (cid:117) (cid:126) G Hence, the construction
Pth uniquely defines a functor which is the left adjoint of
Conj with unit pth : 1
Rck → Conj Pth . As usual, given f : X → Y in Rck , there is a unique morphism
Pth( f ) ,such that X pth X (cid:44) (cid:50) f (cid:12) (cid:18) Conj(Pth( X )) ∃ ! Conj(Pth( f )) (cid:12) (cid:18) Y pth Y (cid:44) (cid:50) Conj(Pth( Y )) , which defines the functor Pth on morphisms.
Notation 2.4.3.
In what follows, we write (cid:126)f for the image
Pth( f ) of a morphism f from Rck . From free objects to all – construction as a colimit.
Again, observe that the composite
Pth F r is left adjoint to the forgetful functor U :
Grp → Set , i.e.
Pth(F r ( X )) = F g ( X ) . Moreprecisely, we may interpret pth as the extension to all objects of the functorial construction onfree objects i X : X (cid:111) F g ( X ) → F g ( X ) : ( x, g ) (cid:55)→ g − xg IGHER COVERINGS OF RACKS AND QUANDLES – PART I 21 which sends a trail to the “representative of the symmetry” associated to its endpoint (Subsection2.2.4). Indeed, by the composition of adjunctions, as before, this i is easily seen to define therestriction to free objects of the unit pth of the Pth (cid:97)
Conj adjunction: X η gX (cid:44) (cid:50) η rX (cid:36) (cid:44) Conj(F g ( X )) ∃ ! Conj( f (cid:48)(cid:48) ) (cid:12) (cid:18) F r ( X ) ∀ f (cid:36) (cid:44) i X =pth Fr( X ) (cid:50) (cid:58) Conj( G ) where i X ( x, e ) = i X η rX ( x ) = η gX ( x ) = x. (5)Then since Pth is a left adjoint, q X : F g ( X ) → Pth( X ) should be the coequalizer of the pair Pth(( X (cid:111) F g ( X )) (cid:111) F g ( X (cid:111) F g ( X ))) Pth( (cid:15) rX × F g [ (cid:15) rX ]) (cid:44) (cid:50) Pth( (cid:15) r Fr U X ) (cid:44) (cid:50) Pth( X (cid:111) F g X ) which, using i above, we compute to be F g ( X × F g ( X )) p (cid:44) (cid:50) p (cid:44) (cid:50) F g ( X ) where p and p are defined by p ( x, g ) = i X ( x · g, e ) = η gX ( x · g ) = x · g and p ( x, g ) = i X ( x, g ) = g − xg. The universal property of the unit and definition on morphisms then follows easily as before.We insist on the tight relationship between the left adjoint
Pth of the conjugation functor
Conj ,and the geometrical features of the free racks as described in Subsection 2.2.We also use this detailed construction of
Pth as a colimit, in the proof of Proposition 2.4.16.Finally, note that this pair p , p is reflexive and thus from the coequalizer q X we also getthe pushout q X , q X : F g ( X ) ⇒ Pth( X ) of p and p . Even though the original fork in Rck is notnecessarily a double extension , the resulting fork in
Grp is a double extension (because
Grp is anexact Mal’tsev category [18]) i.e. the comparison map p : F g ( X × F g ( X )) → Eq( q X ) to the kernelpair of the coequalizer q X , is a surjection.2.4.5. Action by inner automorphisms.
It is already clear from the construction of
Pth that thegroup of paths
Pth( X ) acts on the rack X “via representatives of the symmetries”. For any x and y in X we have x · ( y ) = x (cid:47) y, which uniquely defines the action of any element in Pth( X ) .Compare this action with the action by inner automorphisms: for each rack X , the universalproperty of pth X on S : X → Inn( X ) (defined in Subsection 2.1.8) gives X pth X (cid:44) (cid:50) S (cid:33) (cid:41) Pth( X ) s (cid:12) (cid:18) Inn( X ) , where we have omitted to write Conj , and s is the group homomorphism which relates therepresentatives of symmetries in Pth( X ) to those in Inn( X ) . Then the action of g ∈ Pth( X ) on X is also uniquely described by the action of the inner automorphism s ( g ) . If preferred, thereader may use this as the definition of action by the group of paths. The morphism s is calledthe excess of X in [34]. It is shown to be a central extension of groups in [25, Proposition 2.26].Note that if N (cid:47)
Pth( X ) is a normal subgroup of Pth( X ) , then s ( N ) is a normal subgroup of Inn( X ) . Hence the congruence ∼ N induced by the action of N on X always defines an orbitcongruence ( ∼ N = ∼ s ( N ) ) in the sense of Paragraph 2.3.9.We extend the concept of a trail from Paragraph 2.2.3.1. Definition 2.4.6.
Given a rack X , a trail (in X ) is the data of a pair ( x, g ) given by a head x ∈ X and a path g ∈ Pth( X ) . The endpoint of such a trail is then the element obtained by theaction x · g , of g on x . In some sense,
Pth( X ) is the initial such group containing representatives of the symmetriesof X and acting via those symmetries on X – whereas Inn( X ) is the terminal such. This canbe described via the notion of an augmented rack . Those are given by a group G and a rackhomomorphism ι : X → G together with a right action of G on X such that for g , h in G and x , y in X ,(1) if e is the neutral element in G , then x · e = x ;(2) x · ( gh ) = ( x · g ) · h ;(3) ( x (cid:47) y ) · g = ( x · g ) (cid:47) ( y · g ) ;(4) ι ( x · g ) = g − ι ( x ) g .Morphisms of such are as s above. Now if we fix X , then pth X : X → Pth( X ) is initial amongstaugmented racks (above X ) whereas S : X → Inn( X ) is terminal. This describes why Inn can beused as the reference to define such actions by representatives of the symmetries , described as actions by inner automorphisms . On the other hand, it also illustrates that
Pth( A ) is the freestway to produce an augmented rack. Note the resemblence between the concept of augmentedracks and the concept of a crossed module (see for instance [58]). Remark 2.4.7.
As mentioned before,
Pth has the crucial advantage of functoriality, i.e. forany morphism of racks f : X → Y (including non-surjective ones), and for any x ∈ Y , g = g δ · · · g nδ n ∈ Pth( X ) , we have that x · ( (cid:126)f ( g )) = x · ( (cid:126)f ( g δ · · · g nδ n )) = x · ( f ( g ) δ · · · f ( g n ) δ n ) = x (cid:47) δ f ( g ) · · · (cid:47) δ n f ( g n ) . In the next paragraph, we observe that in the case of free objects F r ( X ) , these two construc-tions coincide ( Pth(F r ( X )) = Inn(F r ( X )) is F g ( X ) ) and, most importantly for what follows,they act freely on F r ( X ) (also see [34, 55], where these results are first discussed). Our hope isthat, in view of the preceding discussion, these results do not take the reader by surprise anymore.2.4.8. Free action on free objects.
By Paragraph 2.4.4, and for any set X , the group of paths Pth(F r ( X )) ∼ = F g ( X ) is freely generated by the elements pth F r ( X ) [ η rX ( x )] = pth F r ( X ) [( x, e )] = ( x, e ) for x ∈ X . Using the identification ( x, e ) ↔ x , for any element ( x, g ) of F r ( X ) and any word h = h δ · · · h nδ n in Pth(F r ( X )) = F g ( X ) , with h i ∈ X and δ i ∈ {− , } for each ≤ i ≤ n , wehave that ( x, g ) · h = ( x, g ) · ( h δ · · · h nδ n ) = ( x, g ) (cid:47) δ ( h , e ) · · · (cid:47) δ n ( h n , e ) = ( x, gh ) . Corollary 2.4.9.
The action of F g ( X ) = Pth(F r ( X )) on F r ( X ) = X (cid:111) F g ( X ) corresponds tothe usual F g ( X ) right action in Set ( X × F g ( X )) × F g ( X ) (cid:44) (cid:50) X × F g ( X ) : (( a, g ) , h ) (cid:31) (cid:44) (cid:50) ( a, g ) · h = ( a, gh ) , given by multiplication in F g ( X ) . Such an action is free, since if ( a, hg ) = ( a, g ) , then hg = g and thus h = e . IGHER COVERINGS OF RACKS AND QUANDLES – PART I 23
Observe that
Inn(F r ( X )) is generated as a group by the elements in the image of S η rX . Indeedfor each ( a, g ) = ( a, g δ · · · g δ n n ) = ( a, e ) (cid:47) δ ( g , e ) · · · (cid:47) δ n ( g n , e ) = ( a, e ) · g, in F r ( A ) , as before, we have S ( a,g ) = S − δ n ( g n ,e ) · · · S − δ ( g ,e ) S ( a,e ) S δ ( g ,e ) · · · S δ n ( g n ,e ) ; see identity (4) from page 22: S ( a,e ) · g = g − S ( a,e ) g .We conclude that Inn(F r ( X )) is actually freely generated. Indeed, the group homomorphism s : Pth(F r ( X )) = F g ( X ) → Inn(F r ( X )) defined in Subsection 2.4.5, is such that: • it is surjective, since the generating set s ( X ) = { S ( x,e ) | x ∈ X } ⊂ Inn(F r ( X )) is theimage of X ⊂ F g ( X ) by s ; • it is injective, since s ( h δ · · · h nδ n ) = e for some h i ∈ X and δ i ∈ {− , } for ≤ i ≤ n ,if and only if ( x, g ) = ( x, g ) · (S δ ( h ,e ) · · · S δ n ( h n ,e ) ) = ( x, g ) · ( h δ · · · h nδ n ) , for all ( x, g ) ∈ F r ( X ) , which implies that h δ · · · h nδ n = e since the action of F g ( X ) isfree. Corollary 2.4.10.
Hence we may always identify
Inn(F r ( X )) , Pth(F r ( X )) and F g ( X ) as wellas their action on F r ( X ) , which is free. We refer to them as the group of paths of F r ( X ) . The kernels of induced morphisms (cid:126)f . In this section we introduce the results which weuse to describe the relationship between the group of paths
Pth , and the central extensions(coverings) and centralizing relations of racks and quandles.Our Lemma 2.4.13 is only a slight generalisation of a Lemma in [5]. We further generalise to higher dimensions in Part II.
Definition 2.4.12.
Given a group homomorphism f : G → H , and a generating set A ⊆ G (i.e. such that G = (cid:104) a | a ∈ A (cid:105) G ), we define (implicitly with respect to A )(i) two elements g a and g b in G to be f -symmetric (to each other) if there exists n ∈ N anda sequence of pairs ( a , b ) , . . . , ( a n , b n ) in the set ( A × A ) ∩ Eq( f ) , such that g a = a δ · · · a δ n n , and g b = b δ · · · b δ n n , for some δ i ∈ {− , } , where ≤ i ≤ n . Alternatively say that g a and g b are an f -symmetric pair .(ii) K f to be the set of f -symmetric paths defined as the elements g ∈ G such that g = g a g − b for some g a and g b ∈ G which are f -symmetric to each other. Lemma 2.4.13.
Given the hypotheses as in Definition 2.4.12, the set of f -symmetric paths K f ⊆ G defines a normal subgroup in G . More precisely it is the normal subgroup generated bythe elements of the form ab − such that a , b ∈ A , and ( a, b ) ∈ Eq( f ) : K f = G f .. = (cid:104)(cid:104) ab − | ( a, b ) ∈ ( A × A ) ∩ Eq( f ) (cid:105)(cid:105) G . Proof.
First we show that K f is a normal subgroup of G . Let g a and g b be f -symmetric (to eachother). Observe that g − b and g − a are also f -symmetric, and thus K f is closed under inverses.Moreover, if h a and h b are f -symmetric, and g = g a g − b , h = h a h − b , then gh = k a k − b , with k a = h a h − a g a and k b = h b h − a g b which are f -symmetric. Finally since A generates G , for any k ∈ G , kg a and kg b are f -symmetric to each other, and thus kgk − ∈ K f is an f -symmetricpath. Since the generators of G f are in the normal subgroup K f , it suffices to show that K f ≤ G f .Given an f -symmetric pair g a and g b , we show that g = g a g − b ∈ G f by induction, on theminimum length n g of the sequences ( a i , b i ) ≤ i ≤ n in the set ( A × A ) ∩ Eq( f ) such that g a = a δ · · · a δ n n and g b = b δ · · · b δ n n for some δ i ∈ {− , } . If n g = 1 , then g is a generator of G f .Suppose that g = g a g − b ∈ G f for all such f -symmetric pair with n g < n for some fixed n ∈ N .Then given a pair g a = a δ · · · a δ n n and g b = b δ · · · b δ n n for some ( a , b ) , . . . , ( a n , b n ) in the set ( A × A ) ∩ Eq( f ) , and δ i ∈ {− , } , we have that h a .. = a − g a and h b .. = b − g b are such that h = h a h − b ∈ G f by assumption. Moreover, g = a ha − a b − which is a product of elements in G f . (cid:3) Observation 2.4.14.
Consider a function f : A → B , and an element g = a δ · · · a δ n n ∈ F g ( A ) with a i ∈ A and δ i ∈ {− , } , for ≤ i ≤ n . As usual, a reduction of g consists in eliminating,in the word g , an adjacent pair a δ i i a δ i +1 i +1 such that δ i = − δ i +1 and a i = a i +1 . Every word g admits a unique normal form i.e. a word g (cid:48) obtained from g after a sequence of reductions, suchthat there is no possible reduction in g (cid:48) .Suppose that g is in the kernel Ker(F g ( f )) , then the normal form of the word F g ( f )( g ) = f ( a ) δ · · · f ( a n ) δ n is the empty word e ∈ F g ( B ) , and thus there is a sequence of reductions of F g ( f )( g ) such that the end result is e . From this sequence of reductions, we may obtain that n = 2 m for some m ∈ N and the elements in the sequence ν .. = ( a δ i i ) ≤ i ≤ n organize in m pairs ( a δ i i , a δ j j ) (the pre-images of those pairs that are reduced at some point in the aforementionedsequence of reductions) such that i < j , ( a i , a j ) ∈ Eq( f ) , δ i = − δ j , each element of the sequence ν appears in only one such pair and finally given any two such pairs ( a δ i i , a δ j j ) and ( a δ l l , a δ m m ) ,then l < i (respectively l > i ) if and only m > j (respectively m < j ), i.e. drawing lines whichlink those elements of the sequence ν that are identified by the pairing, none of these lines haveto cross. a δ a δ a δ a δ a δ a δ a δ a δ a δ a δ a δ a δ a δ a δ Given such a pairing of the elements of ν , for each k ∈ { , . . . , n } we write ( a δ ik i k , a δ jk j k ) for theunique pair such that either i k = k or j k = k . Note that, conversely, any word in F g ( A ) whichadmits such a pairing of its letters is necessarily in Ker(F g ( f )) . Using this observation, we characterize the kernels of maps between free groups.
Lemma 2.4.15.
Given a function f : A → B , the kernel Ker(F g ( f )) of the induced group ho-momorphism F g ( f ) : F g ( A ) → F g ( B ) is given by the normal subgroup K F g ( f ) of F g ( f ) -symmetricpaths (as in Definition 2.4.12): Ker(F g ( f )) = K F g ( f ) .Proof. The inclusion
Ker(F g ( f )) ⊇ K F g ( f ) is obvious. Consider a reduced word g = a δ · · · a δ n n oflength n ∈ N in F g ( A ) with δ i ∈ {− , } , for ≤ i ≤ n and suppose that g ∈ Ker(F g ( f )) . Thenthe elements a δ k k of the sequence ν .. = ( a δ k k ) ≤ k ≤ n organize by pairs ( a δ ik i k , a δ jk j k ) as in Observation2.4.14. Define the sequence ( b δ k k ) ≤ k ≤ n such that for each ≤ k ≤ n , b k .. = a i k . Then byconstruction the word h = b δ · · · b δ n n reduces to the empty word in F g ( A ) , such that g = gh − .Moreover, g and h form an f -symmetric pair, which shows that g ∈ K F g ( f ) . (cid:3) Finally the same characterization holds for kernels of maps
Pth( f ) = (cid:126)f : Pth( X ) → Pth( Y ) induced by a surjective morphism of racks f : X → Y . IGHER COVERINGS OF RACKS AND QUANDLES – PART I 25
Proposition 2.4.16.
Given a surjective morphism of racks f : X (cid:16) Y , the kernel Ker( (cid:126)f ) of thegroup homomorphism (cid:126)f : Pth( X ) (cid:16) Pth( Y ) is given by the normal subgroup K (cid:126)f of (cid:126)f -symmetricpaths (as in Definition 2.4.12): Ker( (cid:126)f ) = K (cid:126)f = (cid:104)(cid:104) ab − | ( a, b ) ∈ Eq( f ) (cid:105)(cid:105) Pth( X ) . Proof.
From Subsection 2.4.4, we reconstruct the image (cid:126)f as in the following diagram, where wealso draw the kernels of F g ( f ) and (cid:126)f : F g ( X × F g ( X )) (cid:12) (cid:18) (cid:12) (cid:18) F g ( f × F g ( f )) (cid:44) (cid:50) (cid:44) (cid:50) F g ( Y × F g ( Y )) (cid:12) (cid:18) (cid:12) (cid:18) Ker(F g ( f )) k (cid:12) (cid:18) (cid:44) (cid:50) ker(F g ( f ) (cid:44) (cid:50) F g ( X ) ( ∗ ) q X (cid:12) (cid:18) (cid:12) (cid:18) F g ( f ) (cid:44) (cid:50) (cid:44) (cid:50) F g ( Y ) q Y (cid:12) (cid:18) (cid:12) (cid:18) Ker( (cid:126)f ) (cid:44) (cid:50) ker(F g ( f ) (cid:44) (cid:50) Pth( X ) (cid:126)f (cid:44) (cid:50) (cid:44) (cid:50) Pth( Y ) . Since q X and q Y are the coequalizers of the pairs above (see Subsection 2.4.4 for more details),and the map F g ( f × F g ( f )) is surjective, by Lemma 1.2 in [7], the square ( ∗ ) is a double extension(regular pushout), and thus the comparison map k is surjective. Then Ker( (cid:126)f ) coincides withthe image ker F g ( f ) along of q X , by uniqueness of (regular epi)-mono factorizations in Grp . Wemay compute this image to be K (cid:126)f . Indeed, in elementary terms, any g ∈ Pth( X ) such that (cid:126)f ( g ) = e can be “covered” by an element h ∈ F g ( X ) such that q X ( h ) = g and F g ( f )[ h ] = e aswell. Then by Lemma 2.4.15, we have that h = h a h − b for some h a and h b in F g ( X ) which are F g ( f ) -symmetric to each other. The images q X ( h a ) and q X ( h b ) are then (cid:126)f -symmetric to eachother by commutativity of ( ∗ ) , hence the quotient g = q X ( h ) = q X ( h a ) q X ( h b ) − ∈ K (cid:126)f is an (cid:126)f -symmetric path. (cid:3) Notation 2.4.17.
For a morphism of racks f , we often write f -symmetric (pair or path) insteadof (cid:126)f -symmetric (pair or path). A f -symmetric trail ( x, g ) is a trail with an f -symmetric path g . The left adjoint
Pth is not faithful.
Observe that given a set A , the morphism F r ( A ) i A =Pth Fr( A ) (cid:44) (cid:50) F g ( A ) , is not injective. Indeed the elements ( a, ag ) and ( a, g ) have the same image. We shall see thatthe kernel pair of i A yields the quotient producing the free quandle from the free rack. Thenthe free quandle F q ( A ) on the set A embeds in the group Conj(F g ( A )) , which is why Joyce callsquandles the algebraic theory of conjugation. Even then, observe that not all quandles embedin a group. Example 2.4.19.
In the involutive quandle Q ab(cid:63) defined in Example 2.3.14, the elements a and b are identified in Pth( Q ab(cid:63) ) . Indeed, a and b act trivially on Q ab(cid:63) , hence they are in thecenter of the group Pth( Q ab(cid:63) ) . Moreover, a and b are in the same connected component, andthus they are also sent to conjugates in Pth( Q ab(cid:63) ) , which yields a = b . Note that from there wehave Pth( Q ab(cid:63) ) = F g ( { a, (cid:63) } ) / (cid:104)(cid:104) a − (cid:63) − a(cid:63) (cid:105)(cid:105) F g ( { a, (cid:63) } ) = F ab ( { a, (cid:63) } ) = Z × Z , where F ab is thefree abelian group functor, and in Z × Z , we have a = b = (1 , and (cid:63) = (0 , (also see [25,Proposition 2.27] ).In particular, the unit of the adjuntion Pth (cid:97)
Conj is not injective and
Pth is not faithful (notethat the right adjoint
Conj is faithful, but not full). As a consequence Q ab(cid:63) is not a subquandle of a quandle in Conj(
Grp ) since this would imply that pth Q ab(cid:63) is injective. We may also observethat a subquandle of a conjugation quandle is such that ( x (cid:47) y = x ) ⇔ ( y (cid:47) x = y ) . Racks and quandles have the same group of paths.
Observe that we may restrict
Pth tothe domain
Qnd . By the same argument
Pth I :
Qnd → Grp (which we denote
Pth ) is then leftadjoint to
Conj :
Grp → Qnd . We may conclude by uniqueness of left adjoints that if r F q is theleft adjoint to the inclusion I :
Qnd → Rck , then
Pth r F q ∼ = Pth : Rck → Grp . The adjunctionbetween racks and groups factorizes into
Rck r F q (cid:40) (cid:47) Pth (cid:26) (cid:36) ⊥ Qnd I (cid:104) (cid:111) Pth (cid:97) (cid:6) (cid:13)
Grp
Conj (cid:70) (cid:77) (cid:97)
Conj (cid:90) (cid:101) in which all possible triangles of functors commute. Considering the comment of Paragraph2.1.11 about the idempotency axiom, we may want to rephrase this as follows: for each rack X , the quotient defining Pth( X ) always identifies generators that would be identified in the freequandle on X .More informally, considering the way Pth , the left adjoint of
Conj , is constructed from equiva-lence classes of tails in the theory of racks, we may want to wonder in which sense racks could bea better context to study group conjugation. From the perspective of their respective coveringtheories, we further describe the relationship between groups, racks and quandles in what follows(see for instance Section 3.7).2.5.
Working with quandles.
We introduce the necessary material to make the transitionfrom the context of racks to the context of quandles. See also the associated quandle in [34].2.5.1.
The free quandle on a rack.
Remember from Paragraph 2.1.11 that the idempotency axiomis a consequence of the axioms of racks “for elements in the tail of a term”. In order to turn arack into a quandle the identifications that matter are thus of the form x (cid:47) δ x x (cid:47) δ x · · · x (cid:47) δ x x (cid:47) δ a · · · (cid:47) δ k a k = x (cid:47) δ a · · · (cid:47) δ k a k , for which the use of the idempotency axiom cannot be avoided by using the axioms of racks.Now by self-distributivity of the operations, we may write y .. = ( x (cid:47) δ a · · · (cid:47) δ k a k ) , and thenrewrite these identities as y (cid:47) δ x y (cid:47) δ x · · · y (cid:47) δ x y = y. Definition 2.5.2.
Given a rack X , define Q X as the relation (in Set ) defined for ( x, y ) ∈ X × X by ( x, y ) ∈ Q X if and only if x = y (cid:47) n y for some integer n (see Paragraph 2.1.11), where y (cid:47) y .. = y . Lemma 2.5.3.
Given a rack X , the relation Q X defines a congruence on X .Proof. (1) The relation Q X is reflexive by definition.(2) As aforementioned, for x and a in some rack, any chain a (cid:47) k a for some k ∈ Z is such that x(cid:47) ( a(cid:47) k a ) = x(cid:47)a . Hence Q X is symmetric since b = a(cid:47) k a implies that b(cid:47) − k b = b(cid:47) − k a = a .(3) Now Q X is transitive by self-distributivity.(4) And finally it is internal since if a = b (cid:47) k b and c = d (cid:47) l d then a (cid:47) c = ( b (cid:47) k b ) (cid:47) ( d (cid:47) l d ) =( b (cid:47) k b ) (cid:47) d = ( b (cid:47) d ) (cid:47) k ( b (cid:47) d ) . (cid:3) IGHER COVERINGS OF RACKS AND QUANDLES – PART I 27
Lemma 2.5.4.
Given a rack X , then a pair of elements ( x, y ) ∈ X × X is in the kernel pair Eq( r η qX ) of r η qX : X → r F q ( X ) if and only if y = x (cid:47) n x for some integer n , i.e. Q X = Eq( r η qX ) .Proof. Since
Rck is a Barr-exact category [1], it suffices to show that the quotient of X by theequivalence relation Q X (on the left) is the same as the quotient of X by Eq( r η qX ) (on the right): X q (cid:44) (cid:50) X/Q X X r η qX (cid:44) (cid:50) r F q ( X ) . For this we show that
X/Q X is a quandle and that q has the same universal property as r η qX .Indeed we have that q ( a ) (cid:47) q ( a ) = q ( a (cid:47) a ) = q ( a ) since ( a, a (cid:47) a ) ∈ Q X for each a . Finally observethat if f : X → Q is a rack homomorphism such that Q is a quandle, then we necessarily havethat f coequalizes the projections π , π : Q X ⇒ X of the congruence Q X . We then concludeby the universal property of the coequalizer. (cid:3) Galois theory of quandles in racks.
We study the Galois structure r Γ q .. = ( Rck , Qnd , r F q , r η q , r (cid:15) q , E ) where E is the class of surjective morphisms (see Section 1.3 and [46]).Since Qnd is a Birkhoff subcategory of
Rck , for r Γ q to be admissible, it suffices to show thatfor each rack X the kernel pair Eq( r η qX ) of the unit permutes with other congruences on X (seeSection 1.3.2). Observe that this is not a consequence of Lemma 2.3.10. Lemma 2.5.6.
Given a rack X , then the congruence Q X = Eq( r η qX ) commutes with any otherinternal relation R on X .Proof. We prove that a pair ( a, b ) ∈ X × X is in Eq( r η qX ) R if and only if it is in R Eq( r η qX ) . Asin lemma 2.3.10, we show that if there is c ∈ X such that ( a, c ) is in one of these relations (sayfor instance Eq( r η qX ) R ) and ( c, b ) in the other one, then there is a c (cid:48) ∈ X such that ( a, c (cid:48) ) is inthe latter ( R Eq( r η qX ) ) and ( c (cid:48) , b ) in the former ( Eq( r η qX ) R ). Now observe that if ( x, y ) ∈ R ,then ( x, y ) (cid:47) n ( x, y ) = ( x (cid:47) n x, y (cid:47) n y ) is in R for any integer n . The result then follows fromreading the following diagram for any n ∈ Z , where horizontal arrows represent membership in Eq( r η qX ) and vertical arrows represent membership in R . Indeed from the top right corner belowwe construct the bottom left corner and the other way around: c (cid:47) − n c = a S na (cid:44) (cid:50) c = a (cid:47) n a S − nc (cid:108) (cid:114) b (cid:47) − n b = c nc (cid:44) (cid:50) b = c (cid:47) n c − nb (cid:108) (cid:114) where we use the fact that if x = y (cid:47) n y then S x = S y . Algebraically we read ( a, c ) ∈ Q X implies c (cid:47) − n c = a for some n ∈ Z and ( c , b ) ∈ R implies ( c (cid:47) − n c , b (cid:47) − n b ) ∈ R , thus choosing c = b (cid:47) − n b yields one of the implications. The other direction translates similarly. (cid:3) Remark 2.5.7.
Given a rack X , the congruence Q X is not an orbit congruence in general. Forinstance, observe that Q F r ( { a,b } ) contains the pairs ( a, a(cid:47)a ) and ( b, b(cid:47)b ) . Suppose by contradictionthat there is a normal subgroup N ≤ Inn(F r ( { a, b } )) = F g ( { a, b } ) for which ∼ N = Q F r ( { a,b } ) .Then since F g ( { a, b } ) acts freely on F r ( X ) , both inner automorphisms S a and S b need to be in N . This leads to a contradiction since a ∼ N ( a(cid:47)b ) but ( a, a(cid:47)b ) (cid:54)∈ Q F r ( { a,b } ) . By contrast Q F r ( {∗} ) is of course an orbit congruence. Corollary 2.5.8.
Quandles form a strongly Birkhoff (and thus admissible) subcategory of
Rck .Proof.
By Proposition 5.4 in [18], the reflection squares of surjective morphisms are doubleextensions (see Section 1.3.2). This implies the admissibility of the Galois structure r Γ q , forinstance by [32, Proposition 2.6]. (cid:3) Note that the left adjoint r F q is actually semi-left-exact as we may deduce from the fact that“connected components are connected” (see Paragraph 2.3.13). Proposition 2.5.9.
Any pullback of the form C a p (cid:44) (cid:50) p (cid:12) (cid:18) [ a ] (cid:12) (cid:18) X r η qX (cid:44) (cid:50) r F q ( X ) , in Rck , is preserved by the reflector r F q , i.e. r F q ( C a ) = 1 , and thus r F q is semi-left-exact inthe sense of [21, 17] .Proof. Observe that X × ∼ = X and thus elements of the pullback C a are merely elements x ∈ X such that that r η q ( x ) = [ a ] ∈ r F q ( X ) i.e. all elements x and y in C a are such that there is k ∈ N such that x = y (cid:47) k y . Hence by Lemma 2.5.4 the image of this pullback by r F q gives indeed ,which concludes the proof. (cid:3) As a consequence we could have used absolute Galois theory in this context [41]. In order tonot overload this article, we stick to the relative approach which we developed here.Observe that there is a limit to the exactness properties satisfied by r F q : we already sawin Paragraph 2.3.13 that r F q cannot preserve finite products, since π : Qnd → Set does but π F q : Rck → Set does not. Moreover, since
Qnd is an idempotent subvariety of
Rck , Proposition2.6 of [22] induces that r F q does not have stable units (in the sense of [21]).To conclude, we show that, besides semi-left-exactness, the r F q -covering theory is “trivial” inthe sense that all surjections are r F q -central. We use the general strategy which was stated inSection 1.3.4. Since the Galois structure is strongly Birkhoff, the “first step influence” is as usual: Lemma 2.5.10.
A surjective morphism f : X → Y , in the category of racks, is r F q -trivial ifand only if Q X ∩ Eq( f ) = ∆ X .Proof. The morphism f is trivial if and only if the reflection square at f is a pullback (seeSection 1.3.2, Diagram (1)). Since this reflection square is a double extension, it suffices for thecomparison map to be injective. Since the square is a pushout, the kernel pair of the comparisonmap is given by the intersection Q X ∩ Eq( f ) of the kernel pairs of q X and f respectively. (cid:3) Proposition 2.5.11.
All surjections in racks f : X → Y are r F q -central.Proof. In order to show this, consider the canonical projective presentation (cid:15) rY : F r ( U Y ) → Y ,and take the pullback of f along (cid:15) rY . This yields a morphism ¯ f : X × Y F r ( U Y ) → F r ( U Y ) . Now any morphism g : X → F r ( Y ) with free codomain is r F q -trivial since if x = x (cid:47) k x in X for some integer k and if, moreover, f ( x ) = f ( x ) (cid:47) k f ( x ) in F r ( Y ) , then f ( x ) k = e by thefree action of Pth(F r ( Y )) on F r ( Y ) . However this can only be if k = 0 , which implies that Q X ∩ Eq( f ) = ∆ X . (cid:3) Towards the free quandle.
Given a set A , in order to develop a good candidate descriptionfor the free quandle on A (see also [52]), we may now consider F q ( A ) as the free quandle on therack F r ( A ) . As aforementioned and roughly speaking, the following identifications between terms: x (cid:47) δ x x (cid:47) δ x · · · x (cid:47) δ x x (cid:47) δ a · · · (cid:47) δ k a k = x (cid:47) δ a · · · (cid:47) δ k a k , (6)define the relation Q F r ( A ) such that F q ( A ) = F r ( A ) /Q F r ( A ) .We want to select one representative ( a, g ) ∈ A (cid:111) F g ( A ) for each equivalence class determinedby these identifications. Thinking in terms of trails, we observe that if ( a, g ) and ( b, h ) are IGHER COVERINGS OF RACKS AND QUANDLES – PART I 29 identified, then they must have the same head a = b . We thus focus on the paths and use aclever semi-direct product decomposition of F g ( A ) .2.5.12.1. Characteristic of a path. We have the following commutative diagram in Set , A η gA (cid:44) (cid:50) Cst (cid:12) (cid:18) F g ( A ) χ .. =F g (Cst) (cid:12) (cid:18) η g (cid:44) (cid:50) Z = F g (1) , where Z is the underlying set of the additive group of integers, and the composite η g Cst is theconstant function with image ∈ Z . Given an element g ∈ F g ( A ) , there exists a decomposition g = g δ · · · g δ n n for some g i ∈ A and exponents δ i = {− , } , with ≤ i ≤ n . The characteristicfunction sums up the exponents χ ( g ) = (cid:80) ni =1 δ i (of course the result doesn’t depend on the chosendecomposition of g ). We may then classify paths in F g ( A ) in terms of their characteristic (i.e. theirimage by χ ). Looking at Equation (6), two terms with same head, and same characteristic, thatare moreover identified by Q F r ( A ) , must actually be equal. In other words, given a fixed head a each equivalence class [( a, g )] in F q ( A ) has only one representative ( a, g (cid:48) ) such that the path g (cid:48) is of a given characteristic.2.5.12.2. Characteristic zero and semi-direct product decomposition. The kernel of χ defines anormal subgroup F ◦ g ( A ) ≤ F g ( A ) which is characterized (see [52] and Lemma 2.4.15) by F ◦ g ( A ) = (cid:104) ab − | a, b ∈ A (cid:105) F g ( A ) . Then for each a ∈ A , we may identify Z with the subgroup (cid:104) a n | n ∈ Z (cid:105) ≤ F g ( A ) which may beseen as the subgroup of F g ( A ) which fixes [( a, e )] ∈ F q ( A ) .. = F r ( A ) /Q F r ( A ) . This then gives asplitting for χ , on the left, yielding the split short exact sequence on the right: ι a : Z → F g ( A ) : k (cid:55)→ a k F ◦ g ( A ) (cid:44) (cid:50) ν A (cid:44) (cid:50) F g ( A ) χ (cid:44) (cid:50) (cid:44) (cid:50) Z (cid:116) (cid:124) ι a (cid:104) (cid:111) a ∈ A , any g ∈ F g ( A ) decomposes uniquely as a χ ( g ) g , where g = a − χ ( g ) g . This defines a function sending equivalenceclasses [( a, g )] ∈ F q ( A ) , to their representatives of characteristic zero ( a, g ) . Note that, for twodifferent a and b in A , the construction of g will vary, however elements of F r ( A ) with differentheads are always sent to different equivalence classes in F q ( A ) .2.5.12.4. Transporting structure. Assuming that this function is indeed bijective, we transportthe quandle structure from the quotient F r ( A ) /Q F r ( A ) to the set of representatives A × F ◦ g ( A ) .More explicitly we compute for ( b, h ) and ( a, g ) in F r ( A ) that ( a, g ) (cid:47) ( b, h ) = ( a, g h − bh ) , where w .. = g h − bh is not of characteristic zero. We then want to take w = a − g h − bh anddefine in F q ( A ) : ( b, h ) (cid:47) ( a, g ) .. = ( a, w ) . The free quandle.
After this analysis, we may confidently build the free quandle (firstdescribed in [52]) as follows.Given a set A the free quandle on A is given by F q ( A ) .. = A (cid:111) F ◦ g ( A ) .. = { ( a, g ) | g ∈ F ◦ g ( A ); a ∈ A } , where the operations on F q ( A ) are defined for ( a, g ) and ( b, h ) in A (cid:111) F ◦ g ( A ) by ( a, g ) (cid:47) ( b, h ) .. = ( a, a − gh − bh ) and ( a, g ) (cid:47) − ( b, h ) .. = ( a, a − gh − b − h ) . As before, g is the path component and a is the head component of the so-called trail ( a, g ) ∈ F q ( A ) and we say that an element ( b, h ) acts on an element ( a, g ) by endpoint . These operationsindeed define a quandle structure.From there, we translate all main results from the construction of free racks. Looking for theunit of the adjunction, we have the injective function η qA : A → F q ( A ) : a (cid:55)→ ( a, e ) .Moreover, since any element g ∈ F ◦ g ( A ) decomposes as a product g = g δ · · · g nδ n ∈ F g ( A ) for some g i ∈ A and exponents δ i ∈ {− , } , with ≤ i ≤ n , and (cid:80) i δ i = 0 , we have, for any ( a, hg ) ∈ F q ( A ) with g and h ∈ F ◦ g ( A ) , a decomposition as ( a, hg ) = ( a, hg δ · · · g nδ n ) = ( a, a (cid:80) i − δ i hg δ · · · g nδ n ) = ( a, a − δ n · · · a − δ hg δ · · · g nδ n )= ( a, h ) (cid:47) δ ( g , e ) · · · (cid:47) δ n ( g n , e ) . Observing that if g i − δ i = g i +1 δ i +1 for some ( a, g ) = ( a, g δ · · · g nδ n ) ∈ F q ( A ) as above, then ( a, e ) (cid:47) δ ( g , e ) · · · (cid:47) δ i − ( g i − , e ) (cid:47) δ i +2 ( g i +2 , e ) · · · (cid:47) δ n ( g n , e ) == ( a, g δ · · · g i − δ i − g i +2 δ i +2 · · · g nδ n ) = ( a, g δ · · · g i − δ i − g iδ i g i +1 δ i +1 g i +2 δ i +2 · · · g nδ n )= ( a, e ) (cid:47) δ ( g , e ) · · · (cid:47) δ n ( g n , e ) , which expresses the first axiom of racks, using group cancellation, as before.From there we derive the universal property of the unit: given a function f : A → Q for somequandle Q , we show that f factors uniquely through η qA . Given an element ( a, g ) ∈ F q ( A ) , wehave that for any decomposition g = g δ · · · g nδ n as above, we must have f ( a, g ) = f ( a, g δ · · · g nδ n ) = f (( a, e ) (cid:47) δ ( g , e ) · · · (cid:47) δ n ( g n , e )) = f ( a ) (cid:47) δ f ( g ) · · · (cid:47) δ n f ( g n ) which uniquely defines the extension of f along η qA to a quandle homomorphism f : F q ( A ) → Q .This extension is well defined since equal such decompositions in F q ( A ) are equal after f by thefirst axiom of racks as displayed in Paragraph 2.5.13.Finally the left adjoint F q : Set → Rck of the forgetful functor
U :
Rck → Set with unit η q isthen defined on functions f : A → B by F q ( f ) .. = f × F ◦ g ( f ) : A (cid:111) F ◦ g ( A ) → B (cid:111) F ◦ g ( B ) , where F ◦ g ( f ) is the restriction of F g ( f ) to the normal subgroup F ◦ g ( A ) ≤ F g ( A ) , whose image isin F ◦ g ( B ) . This defines quandle homomorphisms. Also functoriality of F q and naturality of η q are immediate.2.5.13.1. Free action of F ◦ g ( A ) . Now remember the action by inner automorphisms of F g =Pth(F q ( A )) defined by the commutative diagram in Set : A η gA (cid:44) (cid:50) η qA (cid:38) (cid:45) F g ( A ) s (cid:12) (cid:18) F q ( A ) pth Fq( A ) (cid:49) (cid:56) S (cid:38) (cid:45) Inn(F q ( A )) , where s is the group homomorphism induced by the universal property of η gA or equivalently thatof pth F q ( A ) .This action is not in general given by left multiplication in F ◦ g ( A ) , since in particular any h in F g ( A ) is of course not always of characteristic zero... However, from Paragraph 2.5.13 we deducethat whenever h ∈ F ◦ g ( A ) , the action of h on an element ( a, g ) ∈ F q ( A ) gives ( a, gh ) as before. IGHER COVERINGS OF RACKS AND QUANDLES – PART I 31
Corollary 2.5.14.
The action of F ◦ g ( A ) on F q ( A ) given via the restriction F ◦ g ( A ) s ◦ (cid:44) (cid:50) Inn ◦ (F q ( A )) , of s thus corresponds to the usual left-action of F ◦ g ( A ) in Set : A × F ◦ g ( A ) × F ◦ g ( A )) → A × F ◦ g ( A ) ,given by multiplication in F ◦ g ( A ) . Such an action is free since if ( a, gh ) = ( a, g ) , then gh = g and thus h = e . The group of paths of a quandle.
Observe that the construction of χ for the free group F g ( A ) = Pth(F r ( A )) generalizes to any rack X . The function Cst : X → is actually a rackhomomorphism to the trivial rack . It thus induces a group homomorphism χ = Pth(Cst) : X pth X (cid:44) (cid:50) Cst (cid:12) (cid:18)
Pth( X ) χ =Pth(Cst) (cid:12) (cid:18) pth (cid:44) (cid:50) Z = Pth(1) . As in the case of the free rack, we have the short exact sequence of groups:
Pth ◦ ( X ) (cid:44) (cid:50) ν X (cid:44) (cid:50) Pth( X ) χ (cid:44) (cid:50) (cid:44) (cid:50) Z = Pth(1) , where ν X : Pth ◦ ( X ) → Pth( X ) is the kernel of χ . This construction defines a functor Pth ◦ : Rck → Grp . Most importantly it defines a functor
Pth ◦ : Qnd → Grp which can be interpreted as sendinga quandle to its group of equivalence classes of primitive paths, such that two primitive paths areidentified if one can be obtained from the other with respect to the axioms defining quandles. Inthe same way that
Pth describes homotopy classes of paths in racks,
Pth ◦ describes homotopyclasses of paths in quandles, as it was already explained in [25] and we shall rediscover in thecovering theory described below.2.5.15.1. The transvection group. As in the case of free groups, given a rack X , Proposition2.4.16 implies that the kernel Pth ◦ ( X ) of χ is characterized as the subgroup: Pth ◦ ( X ) = (cid:104) a b − | a, b ∈ X (cid:105) Pth( X ) , (7)which is the definition that was used by D.E. Joyce in [52]. Then the restriction of the quotient s : Pth( X ) → Inn( X ) (defined in Subsection 2.1.9) yields the normal subgroup Inn ◦ ( X ) .. = (cid:104) a b − | a, b ∈ X (cid:105) Inn( X ) , which was called the transvection group of X by D.E. Joyce.This transvection group plays an important role in the literature. In the context of this work,we understand that the construction Pth ◦ has better properties such as functoriality, and is ofmore significance to the theory of coverings than its image Inn ◦ within inner automorphisms.2.5.15.2. The case of free quandles. Observe that for a set X , Pth ◦ (F q ( X )) = F ◦ g ( X ) (for instanceby Equation (7)). As in the case of free racks we get that: Proposition 2.5.16.
Given a set A , we may identify the groups Inn ◦ (F q ( A )) = Pth ◦ (F q ( A )) =F ◦ g ( A ) , and their actions on F q ( A ) . We refer to them as the group of paths of F q ( A ) . This groupacts freely on F q ( A ) by Corollary 2.5.14.Proof. Given a set A , the morphism s ◦ : F ◦ g ( A ) → Inn ◦ (F q ( A )) is a group isomorphism: • it is surjective, since Inn ◦ (F q ( A )) is generated by the set s ( A ) s ( A ) − = { S ( a,e ) (S ( b,e ) ) − | a, b ∈ A } ⊂ Inn ◦ (F q ( A )) which is the image of AA − ⊂ F ◦ g ( A ) by s ; • it is injective, as before because of the free action of F ◦ g ( A ) via s ◦ . (cid:3) Inn(F q ( A )) is not isomorphic to F g ( A ) in general. However, the only counter-example is actuallythe case A = { } : F q ( { } ) = { } is the trivial quandle on one element and Inn( { } ) = { e } is thetrivial group, whereas F g ( { } ) is Z . Of course we do have F ◦ g ( { } ) = { e } . Now in all the othercases Inn(F q ( A )) ∼ = F g ( A ) . The case A = ∅ is trivial. Then whenever x (cid:47) δ x x (cid:47) δ x · · · x (cid:47) δ x x (cid:47) δ a · · · (cid:47) δ k a k = x (cid:47) δ a · · · (cid:47) δ k a k , it suffices to pick y (cid:54) = x ∈ A and then y (cid:47) δ x x (cid:47) δ x x (cid:47) δ x · · · x (cid:47) δ a · · · (cid:47) δ k a k (cid:54) = y (cid:47) δ a · · · (cid:47) δ k a k ,showing that in Inn(F q ( A )) : x δ x x δ x · · · x δ x a δ · · · a kδ k (cid:54) = a δ · · · a kδ k , just as in Inn(F r ( A )) .3. Covering theory of racks and quandles
In this section we study the relative notion of centrality induced by the sphere of influence of
Set in Rck , with respect to extensions (surjective homomorphisms). Remember that pullbacks ofprimitive extensions (surjections in
Set ) along the unit η induce the concept of trivial extensions,which we saw are those extensions which reflect loops . Central extensions in Rck are those fromwhich a trivial extension can be reconstructed by pullback along another extension. Equivalently,central extensions are those extensions whose pullback, along a projective presentation of theircodomain, is trivial. In Section 3.1 we thus look for a condition (C) such that, if a surjective rackhomomorphism f : A → B satisfies (C), then the pullback t of f along (cid:15) rB : F r ( B ) → B reflectsloops (see Section 1.3 and references there).3.1. One-dimensional coverings.
Quandle coverings are defined in [25], then they are shownto characterize Γ q -central extensions of quandles in [26]. We give the same definition for rackcoverings (already suggested in M. Eisermann’s work), which we then characterise in severalways. In Section 3.2 we further show that these are exactly the central extensions of racks.Remember that in dimension zero, a rack A is actually a set, if zero-dimensional data , i.e. anelement a ∈ A , acts trivially on any element x ∈ A : x (cid:47) a = x . We saw that this may beexpressed by the fact that Pth( A ) acts trivially on A or alternatively by the fact that any twoelements which are connected by a primitive path are actually equal.Now in dimension one , an extension f : A (cid:16) B is a covering if one-dimensional data , i.e. apair ( a, b ) in the kernel pair of f , acts trivially on any element x ∈ A : Definition 3.1.1.
A morphism of racks f : A → B is said to be a covering if it is surjective andfor each pair ( a, b ) ∈ Eq( f ) , and any x ∈ A we have x (cid:47) a (cid:47) − b = x. Of course a trivial example is given by surjective functions between sets (the primitive exten-sions). The following implies that central extensions are coverings:
Lemma 3.1.2.
Coverings are preserved and reflected by pullbacks along surjections in
Rck .Proof.
Same proof as in [27] see also [26]. (cid:3)
Coverings and the group of paths.
Observe that given data f , x , a and b , such as inDefinition 3.1.1, we have in particular that x (cid:47) − a = x (cid:47) − a (cid:47) a (cid:47) − b = x (cid:47) − b . In fact we caneasily deduce that f is a covering if and only if for all such x , a and b as before x (cid:47) − a (cid:47) b = x. This is to say that f is a covering if and only if any path of the form a b − or a − b ∈ Pth( A ) , for a and b in A , such that f ( a ) = f ( b ) , acts trivially on elements in A . But then f is a covering if andonly if the subgroup of Pth( A ) generated by those elements acts trivially on elements of A . Now,given g ∈ Pth( A ) , if z · g = z for all z in A , then also x · a − · g · a = ( x(cid:47) − a ) · g · a = ( x(cid:47) − a ) · a = x IGHER COVERINGS OF RACKS AND QUANDLES – PART I 33 for all a ∈ A . Hence we conclude that f is a covering if and only if the normal subgroup (cid:104)(cid:104) ab − | ( a, b ) ∈ Eq( f ) (cid:105)(cid:105) Pth( A ) acts trivially on elements of A . Finally by Proposition 2.4.16 weget the following result which illustrates the importance of Pth in the covering theory of racksand quandles.
Theorem 3.1.4.
Given a surjective morphism f : A → B in Rck (or in
Qnd ), the followingconditions are equivalent:(1) f is a covering;(2) the group of (cid:126)f -symmetric paths K (cid:126)f acts trivially on A (as a subgroup of Pth( A ) ) – i.e.any f -symmetric trail loops in A ;(3) Ker( (cid:126)f ) acts trivially on A (as a subgroup of Pth( A ) );(4) Ker( (cid:126)f ) is a subobject of the kernel Ker( s ) , where s : Pth( A ) → Inn( A ) is the canonicalquotient described in Paragraph 2.4.5.Proof. The statements (1) , (2) and (3) are equivalent by the previous paragraph (and thus byProposition 2.4.16). Statement (4) is merely a way to rephrase (3) using the fact that elementsof the inner automorphism groups are defined by their action. (cid:3) As it was observed by M. Eisermann in
Qnd we have:
Corollary 3.1.5.
A rack covering f : A → B induces a surjective morphism ¯ f : Pth( B ) → Inn( A ) such that (cid:126)f ¯ f = s and thus induces an action of Pth( B ) on A given for g B ∈ Pth( B ) and x ∈ A by x · g B .. = x · g A , where g A is any element in the pre-image (cid:126)f − ( g b ) . Observe that an easy way to obtain a rack covering is by constructing a quotient f : A (cid:16) B such that (cid:126)f is an isomorphism. Example 3.1.6.
The components of the unit r η q of the r F q adjunction are rack coverings.Indeed, we discussed in Paragraph 2.4.20 that Pth r F q = Pth , also see Paragraph 2.1.11. In par-ticular, we look at the one element set and consider the map f .. = r η q F r (1) : F r (1) → F q (1) = 1 .We then compute that (cid:126)f = Pth( r η q F r (1) ) and Inn( f ) = Inn( r η q F r (1) ) are respectively the morphisms Pth(F r (1)) = Z id Z (cid:44) (cid:50) Pth(F q (1)) = Z and Inn(F r (1)) = Z (cid:44) (cid:50) Inn(F q (1)) = { e } , where Z is the infinite cyclic group, Z = Z / Z is the cyclic group with elements and { e } thetrivial group. In this case (cid:126)f is an isomorphism, but Inn( f ) is not. Remark 3.1.7.
In the article [15] , Theorem 4.2 says that quandle coverings (such as in (3) ofProposition 3.1.4 above) should coincide with rigid quotients of quandles, i.e. surjective mor-phisms f : A → B which induce an isomorphism Inn( f ) : Inn( A ) → Inn( B ) . Looking at theproof on page 1150, the authors assume “by construction” that the map η (between the excess of Q and R [34] ) is surjective, which is equivalent to asking for the bottom right-hand square c R Adconj ( h ) = Inn( h ) c Q to be a pushout. This doesn’t seem to hold in the generality asked forin [15] . Note that these results are presented in such a way that they should also hold in Rck ,since the idempotency axiom is never used. Then the example above provides a counter-exampleto [15, Theorem 4.2] in Rck . We further give a counter-example in
Qnd , which shows that [15,Theorem 4.2] must be incorrect.
Example 3.1.8.
Consider the quandle Q ab(cid:63) from Example 2.3.14, which by Example 2.4.19 issuch that Pth( Q ab(cid:63) ) = Z × Z with a = b = (1 , and (cid:63) = (0 , . Moreover, observe that the trivialquandle with two elements π ( Q ab(cid:63) ) is also such that Pth( π ( Q ab(cid:63) )) = F ab ( { [ a ] , [ (cid:63) ] } ) = Z × Z where [ a ] = (1 , and [ (cid:63) ] = (0 , . Hence the morphism of quandles f .. = η Q ab(cid:63) : Q ab(cid:63) → π ( Q ab(cid:63) ) is such that (cid:126)f = id Z × Z . In particular Ker( (cid:126)f ) = { e } is the trivial group, but Inn( f ) : Z / Z → { e } is not an isomorphism.Other such examples can be built using morphisms between quandles from Example 1.3, aswell as Proposition 2.27 and Remark 2.28 in [25] . Visualizing coverings.
Coverings are characterized by the trivial action of f -symmetricpaths, which are the elements g = g a g − b ∈ Pth( A ) such that g a and g b are f -symmetric to eachother. Notice that an f -symmetric pair g a , g b is obtained from the projections of a primitivepath in Eq( f ) . We emphasize the geometrical aspect of these 2-dimensional primitive paths bydefining membranes and horns . An f -symmetric trail is a compact 1-dimensional concept whichremains so when generalized to higher dimensions. The concept of f -horn allows for a more visual,geometrical and elementary description of these ingredients as well as their higher-dimensionalgeneralizations. Definition 3.1.10.
Given a morphism f : A → B in Rck (or
Qnd ), we define an f -membrane M = (( a , b ) , (( a i , b i ) , δ i ) ≤ i ≤ n ) to be the data of a primitive trail in Eq( f ) (see Paragraph2.3.3. We call such an f -membrane M a f -horn if a = b = .. x which we denote M =( x, ( a i , b i , δ i ) ≤ i ≤ n ) . The associated f -symmetric pair of the membrane or horn M is given bythe paths g Ma .. = a δ · · · a nδ n and g Mb .. = b δ · · · b nδ n in Pth( A ) . The top trail is t a = ( a , g Ma ) and the bottom trail is t b = ( b , g Mb ) . The endpoints of the membrane or horn are given by a M = a · g Ma and b M = b · g Mb . Given an f -symmetric trail ( x, g ) for g = g a g − b ∈ Ker( (cid:126)f ) as before, there is a f -horn such thatits associated f -symmetric pair is given by g a and g b (in particular the associated f -symmetrictrail is then ( x, g ) ). Given a horn M = ( x, ( a i , b i , δ i ) ≤ i ≤ n ) , we represent it (with n = 3 and δ i = 1 for ≤ i ≤ ) as in the left-hand diagram below. Definition 3.1.11.
A horn M = ( x, ( a i , b i , δ i ) ≤ i ≤ n ) is said to close (into a disk ) if its endpointsare equal a M = x · g Ma = x · g Mb = b M . The horn M is said to retract if for each ≤ k ≤ n , thetruncated horn M ≤ k .. = ( x, ( a i , b i , δ i ) ≤ i ≤ k ) closes. x (cid:119) (cid:2) a a a (cid:28) (cid:38) b b b ff x · ( a a a ) f x · ( b b b ) x (cid:12) (cid:18) a a a (cid:12) (cid:18) b b b fff a M = b M x (cid:12) (cid:18) a a a (cid:12) (cid:18) b b b a M = b M Corollary 3.1.12.
A surjective morphism f : A (cid:16) B in Rck (or
Qnd ) is a covering if and onlyif every f -horn retracts (or equivalently, if every f -horn closes into a disk). Visualizing normal extensions.
Normal extensions of quandles are described by V. Evenin [26]. The same description works in racks. We reinterpret it using our own terminology.
Definition 3.1.14.
Given a surjective morphism f : A → B in Rck , together with an f -membrane M = ( a i , b i , δ i ) ≤ i ≤ n , we say that the membrane M forms a cylinder if both the top and the bottomtrails of M are loops. Proposition 3.1.15.
A surjective morphism f : A → B in Rck (or
Qnd ) is a normal extension ifand only if f -membranes are rigid , i.e. if and only if given any f -membrane M = ( a i , b i , δ i ) ≤ i ≤ n , M forms a cylinder as soon as either the top or the bottom trail of M is a loop.Proof. The surjection f is normal if and only if the projections π , π : Eq( f ) ⇒ A of the kernelpair of f are trivial. Such projections are trivial if and only if they reflect loops. The π IGHER COVERINGS OF RACKS AND QUANDLES – PART I 35 (resp. π ) projection of a trail t = (( a , b ) , h ) in Eq( f ) loops if and only if there is an f -membrane M = (( a , b ) , (( a i , b i ) , δ i ) ≤ i ≤ n ) such that (cid:126)π ( h ) = g Ma , (cid:126)π ( h ) = g Mb and the top(resp. bottom) trail of M loops (see also [26, Proposition 3.2.3]). (cid:3) Characterizing central extensions.
V. Even’s strategy to prove the characterization isto split coverings along the weakly universal covers constructed by M. Eisermann. These weaklyuniversal covers can be understood as the centralization of the canonical projective presentations(using free objects – see Section 3.5). Their structure and properties used to show V. Even’s resultderive from the structure and properties of the free objects we described before. Thus even thoughV. Even’s proof can be translated to the context of racks, we prefer to work directly with freeobjects in the alternative proof below. This approach then easily generalizes to higher dimensionswithout having to build the weakly universal higher-dimensional coverings from scratch.
Proposition 3.2.1.
Any rack-covering with free codomain f : A → F r ( B ) is a trivial extension.Proof. In order to test whether f is a trivial extension, consider x ∈ A and g = a δ · · · a nδ n ∈ Pth( A ) for n ∈ N , a , . . . , a n in A and δ , . . . , δ n in {− , } . Assume that f sends the trail ( x, g ) to the loop ( f ( x ) , (cid:126)f ( g )) : f ( a ) · ( f ( a ) δ · · · f ( a n ) δ n ) = f ( x ) (cid:47) δ f ( a ) · · · (cid:47) δ n f ( a n ) = f ( x (cid:47) δ a · · · (cid:47) δ n a n ) = f ( x ) , where we write f ( a i ) .. = pth F r ( B ) ( f ( a i )) (which does not mean that f ( a i ) is in B ). We have toshow that ( x, g ) was a loop in the first place: x · g = x (cid:47) δ a · · · (cid:47) δ n a n = x. ( ∗ ) Since F r ( B ) is projective (with respect to surjective morphisms) and f is surjective, thereis a morphism of racks s : F r ( B ) → A such that f s = 1 F r ( B ) . Then s induces a group homomor-phism (cid:126)s : Pth(F r ( B )) → Pth( A ) such that for each ≤ i ≤ n , (cid:126)s [ f ( a i )] = pth A ( sf ( a i )) = .. sf ( a i ) (see Paragraph 2.4.2), and thus e = (cid:126)s [ f ( a )] δ · · · (cid:126)s [ f ( a n )] δ n = sf ( a ) δ · · · sf ( a n ) δ n . Hence in particular we have x (cid:47) δ sf ( a ) · · · (cid:47) δ n sf ( a n ) = x · ( sf ( a ) δ · · · sf ( a n ) δ n ) = x · e = x. Finally since for each ≤ i ≤ n we have f ( sf ( a i )) = f ( a i ) , M = ( x, ( a i , sf ( a i ) , δ i ) ≤ i ≤ n ) isan f -horn, which has to retract since f is a covering: x (cid:47) δ a · · · (cid:47) δ n a n = x (cid:47) δ sf ( a ) · · · (cid:47) δ n sf ( a n ) = x. (cid:3) In this one-dimensional context, the characterization of coverings from Proposition 3.1.4 allowsfor a shorter version of this proof. Since a direct generalization of Proposition 3.1.4 in higherdimensions is not yet clear to us, we prefer to keep the previous, more visual version of the proofas our main reference. However, you may want to replace what follows ( ∗ ) in the previous proofby: Proof. [ ... ] ( ∗ ) Now since the action of
Pth(F r ( B )) on F r ( B ) is free, any loop in Pth(F r ( B )) mustbe trivial, and in particular f ( a ) δ · · · f ( a n ) δ n = e . Hence g ∈ Ker( (cid:126)f ) , and thus by Proposition3.1.4, x · g = x , which concludes the proof. (cid:3) Note finally that the exact same proofs work for quandle coverings, using the fact that if A isa quandle, we may then always choose a i ’s and δ i ’s such that (cid:80) i δ i = 0 . Then f ( a ) δ · · · f ( a n ) δ n is in Pth ◦ (F q ( B )) which acts freely on F q ( B ) . The rest of the proofs remain identical. Proposition 3.2.2.
If a quandle-covering f : A → F q ( B ) has a free codomain, then it is a trivialextension. By Lemma 3.1.2, and the previous propositions, the strategy of Section 1.3.4 yields Theorem2 from [26], as well as:
Theorem 3.2.3.
Rack coverings are the same as central extensions of racks.
Comparing admissible adjunctions by factorization.
The notions of trivial object andconnectedness, or trivialising relation C , coincide in racks and quandles. These are understoodas the zero-dimensional central extensions and centralizing relation. In dimension 1, the notionsof central extensions in racks and quandles also coincide. Further we also have coincidence ofthe centralizing relations and the corresponding notions in dimension 2. Before we move on, weshow how these results are no coincidence and can be studied systematically as a consequence ofthe tight relationship between the π -admissible adjunctions of interest.Expanding on Paragraph 2.3.8 we get a factorization as in 2.4.20, where all triangles commuteand all the adjunctions are admissible: Rck r F q (cid:40) (cid:47) π (cid:26) (cid:37) ⊥ Qnd I (cid:104) (cid:111) π (cid:5) (cid:12) Set . I (cid:70) (cid:77) I (cid:90) (cid:101) (cid:97)(cid:97) Since we are dealing here with several different Galois structures: Γ from Rck to Set , r Γ q from Rck to Qnd and say Γ q .. = ( Qnd , Set , π , I , η , (cid:15) , E ) ; we specify the Galois structure with respectto which the concepts of interest are discussed. Lemma 3.3.1. If f : A → B is a Γ -trivial extension, then f is also r Γ q -trivial, and the image r F q ( f ) of f is a Γ q -trivial extension in Qnd .Proof.
The Γ -canonical square of f in Rck is given on the left, and factorizes into the compositeof double extensions on the right: A f (cid:12) (cid:18) η A (cid:44) (cid:50) π ( A ) π ( f ) (cid:12) (cid:18) B η A (cid:44) (cid:50) π ( B ) , A f (cid:12) (cid:18) r η qA (cid:44) (cid:50) r F q ( A ) r F q ( f ) (cid:12) (cid:18) η rFq( A ) (cid:44) (cid:50) π ( A ) = π ( r F q ( A )) π ( f ) (cid:12) (cid:18) B r η qB (cid:44) (cid:50) r F q ( B ) η rFq( B ) (cid:44) (cid:50) π ( B ) = π ( r F q ( B )) . Hence if f is a trivial extension, then this composite is a pullback square. The composite of twodouble extensions is a pullback if and only if both double extensions are pullbacks themselves. (cid:3) Lemma 3.3.2.
An extension f : A → B in Qnd is(i) Γ q -trivial in Qnd if and only if I( f ) is Γ -trivial in Rck ;(ii) Γ q -central in Qnd if and only if I( f ) is Γ -central in Rck .Proof.
The first point ( i ) is immediate by the previous lemma, and the fact that the π -canonicalsquares of I( f ) in Rck is the same as the image by I of the Γ q -canonical square of f in Qnd . Notealso that I preserves and reflects pullbacks.For the second statement ( ii ) , if f is Γ q -central, then there is an extension p : E → B suchthat the pullback of f along p is Γ q -trivial. We may conclude by taking the image by I of thispullback square. Now if I( f ) is Γ -central in Rck , there exists p : E → B in Rck such that the
IGHER COVERINGS OF RACKS AND QUANDLES – PART I 37 pullback t of I( f ) along p is Γ -trivial in Rck . Taking the quotient along r η q of this pullbacksquare (1) yields a factorization of (1): E × B A t (cid:12) (cid:18) r η qP (cid:44) (cid:50) r F q ( E × B A ) r F q ( t ) (cid:12) (cid:18) (cid:44) (cid:50) A f (cid:12) (cid:18) E r η qE (cid:44) (cid:50) r F q ( E ) r F q ( p ) (cid:44) (cid:50) B. Again, since the left hand square is a double extension, and the composite is a pullback, bothsquares are actually pullbacks and thus f is Γ q -central. (cid:3) Now since the π -adjunction is strongly Birkhoff (both in Rck and
Qnd ), central extensions areclosed by quotients along double extensions in
ExtRck (or
ExtQnd – see also Proposition 3.4.7).
Corollary 3.3.3.
The image by r F q of a Γ -central extension f : A → B in Rck is a Γ q -centralextension in Qnd .Proof.
The image r F q ( f ) is Γ q -central extension if and only if I( r F q ( f )) is Γ -central. Since Set isstrongly Birkhoff in
Rck , I( r F q ( f )) is the quotient of a Γ -central extension in Rck along a doubleextension and thus is still Γ -central in Rck . (cid:3) Proposition 3.3.4.
If the image by r F q of an r Γ q -trivial extension f : A → B in Rck is a Γ q -central extension in Qnd , then f is Γ -central in Rck .Proof.
Consider the following commutative cube in
Rck where we omit to write the inclusion
I :
Qnd → Rck . The back face is a pullback by construction. The right hand face is a pullbackby assumption, and the left hand face is a pullback by Proposition 2.5.11. We deduce that thefront face is a pullback as well. P (cid:44) (cid:50) t (cid:12) (cid:18) r η qP (cid:122) (cid:4) A f (cid:12) (cid:18) r η qA (cid:122) (cid:4) r F q ( P ) (cid:44) (cid:50) r F q ( t ) (cid:12) (cid:18) r F q ( A ) r F q ( f ) (cid:12) (cid:18) F r ( B ) r η q Fr( B ) (cid:122) (cid:4) (cid:15) rB (cid:44) (cid:50) B r η qB (cid:122) (cid:4) F q ( B ) r F q ( (cid:15) rB ) (cid:44) (cid:50) r F q ( B ) Since r F q ( f ) is Γ q -central by assumption, and since r F q ( (cid:15) rB ) : F q ( B ) = r F q (F r ( B )) → r F q ( B ) factorizes as F q ( B ) F q ( r η qB ) (cid:44) (cid:50) F q ( r F q ( B )) F q ( (cid:15) q rFq( B ) ) (cid:44) (cid:50) r F q ( B ) , both r F q ( t ) and t are Γ -trivial as the pullback of a trivial extension. (cid:3) Example 3.3.5.
Some extensions of racks which are not central, still have central images under r F q . Define the involutive rack with underlying set { a, a , b, b , , } , and an operation (cid:47) suchthat a , a , b and b have the same action and, moreover, (cid:47) a a b b a a a b b b b a a b b a a We may check by hand that the axioms (R1) and (R2) are satisfied. Then define the morphismof racks f , with codomain the trivial rack { x, } and which sends letters to x and numbers to (cid:63) .We have that a (cid:47) b (cid:54) = b = a (cid:47) , and thus f is not central. However we compute the morphism r F q ( f ) : Q ab(cid:63)(cid:63) → { x, (cid:63) } , where Q ab(cid:63)(cid:63) is as in Example 2.3.14 but with two distinct (cid:63) ’s which actin the same way. This morphism merely identifies the letters and the stars and thus it is central.Of course some rack homomorphisms which are not r Γ q -trivial are still Γ -central: we alreadymentioned the important example of r η qA for any rack A . Before even studying the next steps of the covering theory, we can predict that what happensin
Qnd directly follows from what happens in
Rck . Corollary 3.3.6.
If the full subcategory
CExtRck of central extensions of racks is reflective withinthe category of extensions
ExtRck (see Theorem 3.4.1 for details), then also
CExtQnd is reflectivein
ExtQnd and the reflection is computed as in
ExtRck , via the inclusion
I :
Qnd → Rck .Proof.
Since
Qnd is closed under quotients in
Rck , the centralization of an extension in
Qnd (cid:22)
Rck yields an extension in
Qnd which is moreover central by Lemma 3.3.2. The universality in
CExtQnd directly derives from the universality in
CExtRck by the same arguments. (cid:3)
Centralizing extensions.
We adapt the result from [24], showing the reflectivity of quan-dle coverings in the category of extensions, to the context of racks. We put the emphasis on ournew characterizations of the centralizing relation which works the same for racks and for quan-dles. We also prepare the ingredients to show the admissibility of coverings within extensions,and the forthcoming covering theory in dimension 2.Let us define E to be the the class of double extensions in ExtRck . Theorem 3.4.1.
The category
CExtRck is a ( E )-reflective subcategory of the category ExtRck with left adjoint F and unit η defined for an object f : A → B in ExtRck by η f .. = ( η A , id B ) ,where η A : A → F i1 ( A ) is the quotient of A by the centralizing congruence C ( f ) , which can bedefined in the following equivalent ways:(i) C ( f ) is the equivalence relation on A generated by the pairs ( x (cid:47) a (cid:47) − b, x ) for x , a , and b in A such that f ( a ) = f ( b ) ,(ii) a pair ( a, b ) of elements from A is in the equivalence relation C ( f ) if and only if a and b are the endpoints of a horn, i.e. there exists a horn M = ( x, ( a i , b i , δ i ) ≤ i ≤ n ) such that x · g Ma = a and x · g Mb = b ,(iii) C ( f ) is the orbit relation ∼ Ker( (cid:126)f ) (or equivalently ∼ K (cid:126)f ) induced by the action of thekernel of (cid:126)f (i.e. the group of f -symmetric paths).Observing that C ( f ) ≤ Eq( f ) , the image of f by F is defined as the unique factorization of f through this quotient: A η A (cid:34) (cid:42) f (cid:44) (cid:50) B F i1 ( A ) F ( f ) (cid:52) (cid:60) The definition of F on morphisms α = ( α , α ) : f A → f B decomposes into the top (initial)component F i1 ( α ) defined by the universal property of the quotients η A for f A : A → A ; andthe bottom component F ( α ) = α which is the identity. IGHER COVERINGS OF RACKS AND QUANDLES – PART I 39
Proof.
Using definition ( i ) for the centralizing relation, the proof of Theorem 5.5 in [24] easilytranslates to the context of racks. Then given an extension f : A → B , the unit η f = ( η A , id B ) is indeed a double extension since its bottom component is an isomorphism. It remains to showthat the definitions ( ii ) and ( i ) are equivalent, since ( iii ) is equivalent to ( ii ) by Proposition2.4.16.First we show by induction on n ∈ N that C ( f ) , defined as in ( i ) , contains all pairs thatare endpoints of a horn. Then we show that the collection of such pairs defines a congruencecontaining the generators of C ( f ) . This then concludes the proof.Step is satisfied by reflexivity of C ( f ) . Now assume that if ( a, b ) is a pair of elements in A ,which are endpoints of a horn M = ( x, ( a i , b i , δ i ) ≤ i ≤ n ) of length n ≤ k , for some fixed naturalnumber k , then ( a, b ) ∈ C ( f ) . We show that the endpoints a .. = x · g Ma and b .. = x · g Mb of anygiven horn M = ( x, ( a i , b i , δ i ) ≤ i ≤ k +1 ) of length k + 1 are in relation by C ( f ) . Indeed, define a (cid:48) = a (cid:47) − δ k +1 a k +1 and b (cid:48) = b (cid:47) − δ k +1 b k +1 . Then we have that ( a (cid:48) , b (cid:48) ) ∈ C ( f ) by assumptionand, moreover, a = a (cid:48) (cid:47) δ k +1 a k +1 C ( f ) b (cid:48) (cid:47) δ k +1 a k +1 C ( f ) b (cid:48) (cid:47) δ k +1 b k +1 = b, by compatibility of C ( f ) with the rack operation, together with reflexivity, and further bydefinition ( i ) of C ( f ) . We may conclude by transitivity of C ( f ) .Now define the symmetric set relation S as the subset of A × A , given by pairs of endpoints of f -horns. Looking at horns of length and , S defines a reflexive relation containing the generatorsof C ( f ) . It is also easy to observe that it is compatible with the rack operation. Thus it remainsto show transitivity. In order to do so, for k and n in N , consider a horn M = ( x, ( a i , b i , δ i ) ≤ i ≤ k ) ,and its endpoints a and b as before, as well as a horn N = ( z, ( c i , d i , γ i ) ≤ i ≤ n ) with endpoints c = z · g Na and d = z · g Nb . If b = c then also ( a, d ) is in S since: a = x (cid:47) δ a · · · (cid:47) δ k a k (cid:47) − γ n c n · · · (cid:47) − γ c (cid:47) γ c · · · (cid:47) γ n c n ,d = x (cid:47) δ b · · · (cid:47) δ k b k (cid:47) − γ n c n · · · (cid:47) − γ c (cid:47) γ d · · · (cid:47) γ n d n . (cid:3) By Corollary 3.3.6, what we deduced about the functor F restricts to the domain CExtQnd ,and so also describes the left adjoint to the inclusion in
ExtQnd from Theorem 5.5. in [24]. Inaddition to Corollary 3.3.6, we further describe how centralization behaves with respect to r F q .3.4.2. Navigating between racks and quandles.
Observe that the adjunction r F q : Rck (cid:29)
Qnd : I induces (in the obvious way) an adjunction r F q1 : ExtRck (cid:29)
ExtQnd : I with unit given by r η q = ( r η q , r η q ) . Then by Corollary 3.3.3 this adjunction restricts to the full subcategories r F q1 : CExtRck (cid:29)
CExtQnd : I . Proposition 3.4.3.
We have the following square of adjunctions, in which all possible squaresof functors commute:
ExtRck (cid:97) r F q1 (cid:47) (cid:53) F (cid:18) (cid:26) (cid:62) ExtQnd (cid:97) I (cid:111) (cid:117) F (cid:18) (cid:26) CExtRck (cid:62) I (cid:83) (cid:90) r F q1 (cid:46) (cid:53) CExtQnd . I (cid:110) (cid:117) I (cid:82) (cid:90) Proof.
Corollary 3.4.4 gives commutativity of the square F I = I F from the top right to thebottom left. In the opposite direction, I r F q1 = r F q1 I by Corollary 3.3.3 again. Finally bottom-right to top-left I I = I I commutes trivially, from which we can deduce, by uniqueness of leftadjoints, that r F q1 F = F F q1 . (cid:3) In particular we have:
Corollary 3.4.4. If f : A → B is a morphism of racks, then the centralization F ( r F q ( f )) : F i1 ( r F q ( A )) → r F q ( B ) of r F q ( f ) is equal (up to isomorphism) to the reflection r F q (F ( f )) : r F q (F i1 ( A )) → r F q ( B ) ofthe centralization F ( f ) of f . Towards admissibility in dimension 2.
A reflector such as F , of a subcategory of mor-phisms containing the identities into a larger class of morphisms can always be chosen such thatthe bottom component of the unit of the adjunction is the identity [40, Corollary 5.2]. This isan important property to obtain higher order reflections and admissibility for we relate certainproblems back to the first level context (which has the advantage of being complete, cocompleteand Barr-exact). For dimension 2, we need this reflection to be strongly Birkhoff. Below we havethe results we need for the permutability condition on the kernel pair of the unit (“strongly”) andfor the closure by quotients of central extensions (“Birkhoff”). Proposition 3.4.6.
Given a rack extension f : A → B (or in particular an extension in Qnd ) asbefore, the kernel pair C ( f ) of the domain-component η A of the unit η f .. = ( η A , id B ) , commuteswith all congruences on A , in Rck (and so also in particular in
Qnd ).Proof.
By Theorem 3.4.1, the centralizing relation C ( f ) is an orbit congruence which thuscommute with any other congruence on A . (cid:3) As we shall see in Part II and III, the following property is a consequence of the fact that theGalois structure Γ , in dimension 0, is strongly Birkhoff. For now we show by hand: Proposition 3.4.7. If α is a double extension of racks (or in particular quandles) A α (cid:44) (cid:50) p (cid:38) (cid:45) f A (cid:12) (cid:18) B f B (cid:12) (cid:18) A × B B π (cid:48) (cid:55) π (cid:117) (cid:127) A α (cid:44) (cid:50) B then the morphism ¯ α induced between the centralizing relations C ( f A ) and C ( f B ) is a regularepimorphism. Moreover, if f A is a central extension then f B is a central extension.Proof. Certainly if we show that ¯ α is a regular epimorphism, then assuming that f A is central,then its centralizing relation is trivial, hence the centralizing relation of f B is trivial, showingthat f B is central (note that in this context, it is enough to have preservation of centrality byquotients along double extensions in order to have surjectivity of ¯ α , see Part II and III).We pick a pair ( x (cid:47) y, x (cid:47) z ) amongst the generators of C ( f B ) (i.e. with f B ( y ) = f B ( z ) ). Since α is surjective we get a ∈ A such that α ( a ) = f B ( y ) . Now both pairs ( a, y ) and ( a, z ) arein the pullback A × B B hence there exist t and s in A such that α ( t ) = y , α ( s ) = z and f A ( t ) = f A ( s ) = a , by surjectivity of p . Now there is also u ∈ A such that t ( u ) = x and thepair ( u (cid:47) t, u (cid:47) s ) is a generator of C ( f A ) by definition. It is also sent to ( x (cid:47) y, x (cid:47) z ) ∈ C ( f B ) by ¯ α by construction. All generators of C ( f B ) are thus in the image of ¯ α , and this concludesthe proof. (cid:3) Corollary 3.4.8.
Given a morphism α = ( α , α ) : f A → f B in ExtRck such that α and α aresurjections, then the square below (where P .. = F i1 ( A ) × F i1 ( B ) B ) is a double extension of racks. IGHER COVERINGS OF RACKS AND QUANDLES – PART I 41
Similarly in
ExtQnd . A α (cid:44) (cid:50) p (cid:36) (cid:44) η A (cid:12) (cid:18) B η B (cid:12) (cid:18) P π (cid:48) (cid:55) π (cid:119) (cid:2) F i1 ( A ) F i1 ( α ) (cid:44) (cid:50) F i1 ( B ) Proof.
By Lemma 1.2 in [7], this square is a pushout as a consequence of Proposition 3.4.7. Thenby Proposition 5.4 in [18], p is a surjection as well, making α into a double extension. (cid:3) In Part II we complete the proof that Γ = ( ExtRck , CExtRck , F , I , η , (cid:15) , E ) forms an admis-sible Galois structure such that morphisms in E are of effective E -descent [50, 49].3.5. Weakly universal covers and the fundamental groupoid.
We insist on the impor-tance of the new results of this Section and the following, in achieving a precise theoreticalunderstanding and expansion of M. Eisermann’s covering theory of quandles (as a continuationof V. Even’s contributions).3.5.1.
Centralizing the canonical presentations.
Weakly universal covers (w.u.c.) for quandleswere described by M. Eisermann. He also indicated how to adapt his theory to the case of racks.In this section, we recover his constructions from the centralization of the canonical projectivepresentations as explained in the introduction. Note that the difference between the w.u.c. inracks and in quandles is then due to the difference between the canonical projective presentationsrather than the centralizations which are the same.Given the canonical projective presentation of a rack (cid:15) rX : F r ( X ) → X , we saw in Paragraph2.4.4 that the induced morphism (cid:126)(cid:15) rX is actually the quotient map (cid:126)(cid:15) rX = q X : F g ( X ) → Pth( X ) from Subsection 2.4. Hence the kernel of (cid:126)(cid:15) rX is given by Ker( (cid:126)(cid:15) rX ) = (cid:104)(cid:104) c − a − x a | a, x, c ∈ X and c = x (cid:47) a (cid:105)(cid:105) F g ( X ) . Since the action of
Pth(F r ( X )) = F g ( X ) is by right multiplication, two elements ( a, g ) and ( b, h ) in F r ( X ) are identified by the centralizing relation C ( (cid:15) rX ) if and only if a = b and there is k ∈ Ker( (cid:126)(cid:15) rX ) such that g = hk . In other words, the domain component η r ( X ) of the centralizationunit is given by the product X (cid:111) F g ( X ) id X × q X (cid:44) (cid:50) X (cid:111) Pth( X ) , where the operation in ˜ X .. = X (cid:111) Pth( X ) is defined as in Paragraph 2.2.3.1, Equation (3). Definition 3.5.2.
Given a rack X , we define the associated weakly universal cover of X to bethe centralised map ω X .. = F ( (cid:15) rX ) ˜ X .. = X (cid:111) Pth( X ) ω X (cid:44) (cid:50) X, where ω X sends a trail ( a, g ) ∈ ˜ X to its endpoint a · g , and trails in ˜ X “act by endpoint” asin F r ( X ) . Note that this construction is functorial in X , yielding a functor ˜ − : Rck → Rck which sends a morphism of racks f : A → B to the morphism ˜ f .. = f × (cid:126)f : ˜ A → ˜ B ; and a naturaltransformation ω : ˜ − → id Rck , whose component at X is ω X . Then the action of
Pth( X ) induced by the covering ω X on ˜ X = X (cid:111) Pth( X ) is by rightmultiplication, and is thus free. By construction, ω X splits any central extension, and is thus anormal covering itself. Given any other covering f : B → X , together with a splitting function s : X → B in Set such that f s = id X , a factorization ω f : ˜ X → B of ω X through f is given by ω f ( a, e ) .. = s ( a ) and compatibility with the action of Pth( X ) on ˜ X and B (see Corollary 3.1.5). Starting with the canonical projective presentation of a quandle (cid:15) qX : X (cid:111) Pth ◦ ( X ) → X , thesame reasoning yields a w.u.c. with the same properties ˜ X ◦ .. = X (cid:111) Pth ◦ ( X ) ω qX (cid:44) (cid:50) X, such that the quandle structure on X (cid:111) Pth ◦ ( X ) is as for F q ( X ) (Paragraph 2.5.13). As in thecase of racks, this describes a functor as well as a natural transformation whose component atany quandle A is ω qA . Observe that Corollary 3.4.4 implies that ˜ X ◦ .. = X (cid:111) Pth ◦ ( X ) is actuallythe free quandle on the rack ˜ X .. = X (cid:111) Pth( X ) and thus if X is a quandle, then ω qX is merelythe image of ω X by r F q .Since we have a normal covering ω X (respectively ω qX ) over each X in Rck (respectively in
Qnd ), their kernel pairs are sent to a groupoid by the reflection π (see [6, Lemma 5.1.22]) andthus we can construct the fundamental groupoids (see Galois groupoid of a weakly universalcentral extension as in [6]) yielding factors π r : Rck → Grpd and π q : Qnd → Grpd , with codomainthe category of ordinary groupoids
Grpd (i.e. the category of internal groupoids in
Set ). Definition 3.5.3.
The functor π : Rck → Grpd is defined on objects by sending a rack X to π r ( X ) , the image by π of the groupoid induced by taking the kernel pair of ω X . Functoriality isinduced by functoriality of ω .Similarly the functor π q : Qnd → Grpd is defined by sending a quandle X to π q ( X ) , the imageby π of the groupoid induced by taking the kernel pair of ω qX . Since every covering of X is split by ω X (respectively ω qX in Qnd ), the Galois theorem yieldsan equivalence of categories between the category of coverings of X and the category of internalcovariant presheaves over π ( X ) (and similarly for Qnd , see Section 1.3 and references).3.5.4.
The fundamental groupoid.
We show that the fundamental groupoid π ( X ) (respectively π q ( X ) ) for an object X in the category Rck (respectively
Qnd ) is indeed the groupoid inducedby the action of
Pth( X ) (respectively Pth ◦ ( X ) ) on X , as suggested in M. Eisermann’s work(see [25, Section 8]). As was mentioned in the introduction, these results, and categorical Galoistheory, give a positive answer to M. Eisermann’s questions about the relevance of his analogieswith topology. Results about the fundamental group of a connected pointed quandle were givenby V. Even in [26]. We generalize these results to the non-connected, non-pointed contextin both categories Rck and
Qnd . Exploiting the analogy with the covering theory of locallyconnected topological spaces, this result fixes the intuition that the elements of the group
Pth( X ) (respectively Pth ◦ ( X ) ) are representatives of the classes of homotopically equivalent paths whichconnect elements in the rack (respectively quandle) X . Definition 3.5.5.
Given a set X and a group G together with an action of G on X , we buildthe ordinary groupoid G ( X,G ) (in Set ) X p (cid:44) (cid:50) p (cid:44) (cid:50) m (cid:44) (cid:50) X − (cid:20) (cid:28) c (cid:44) (cid:50) d (cid:44) (cid:50) X i (cid:108) (cid:114) where X .. = X , X .. = X × G and for a ∈ X , ( a, g ) ∈ X , d ( a, g ) .. = a ; c ( a, g ) .. = a · g ; i ( a ) .. = ( a, e ); ( a, g ) − .. = ( a · g, g − ); p , p : X ⇒ X form the pullback of c and d ; and m is the composition morphism defined for (cid:104) ( a, g ) , ( b, h ) (cid:105) in X by m (cid:104) ( a, g ) , ( b, h ) (cid:105) .. = ( a, g ) · ( b, h ) .. = ( a, gh ) .Note that this construction actually defines a functor from the category of group actions to thecategory of ordinary groupoids. IGHER COVERINGS OF RACKS AND QUANDLES – PART I 43
Theorem 3.5.6.
Given an object X in Rck (respectively
Qnd ), the fundamental groupoid π ( X ) (resp. π q ( X ) ) is given by the set groupoid G ( X, Pth( X )) (resp. G ( X, Pth ◦ ( X )) ). Moreover, the groupoidmorphisms induced by f : X → Y via Pth (resp.
Pth ◦ ) and G correspond to π ( f ) (resp. π q ( f ) ).Proof. Given the kernel pair d , d : X (cid:48) ⇒ X (cid:48) of the weakly universal cover ω X : ˜ X → X (resp. ω qX : ˜ X ◦ → X ),we define the groupoid G as: X (cid:48) p (cid:48) (cid:44) (cid:50) p (cid:48) (cid:44) (cid:50) m (cid:48) (cid:44) (cid:50) X (cid:48) − (cid:19) (cid:27) d (cid:44) (cid:50) d (cid:44) (cid:50) X (cid:48) u (cid:108) (cid:114) where X (cid:48) is the pullback of d and d , and m (cid:48) is the composition morphism defined by the uniquefactorization of d ◦ p (cid:48) , d ◦ p (cid:48) : X (cid:48) ⇒ X (cid:48) through d , d : X (cid:48) ⇒ X (cid:48) .Remember that a trail ( a, g ) ∈ X (cid:48) is represented as an arrow g : a > a · g ; and the actionof a trail on another is as in Paragraph 2.2.3.1, Equation (3), where the composition of arrowsis understood by multiplication in Pth( X ) (resp. Pth ◦ ( X ) ).By definition, the elements in X (cid:48) are then pairs of trails with same endpoint (diagram on theleft), and the rack (resp. quandle) operation is defined component-wise such that we have theequality on the right: a · g = b · ha (cid:55) (cid:65) g b (cid:93) (cid:103) h ; a (cid:48) · h (cid:48) = b (cid:48) · g (cid:48)(cid:53) a · ( hk ) = b · ( gk ) a (cid:48) (cid:54) (cid:63) h (cid:48) a · h = b · g b (cid:48) (cid:95) (cid:104) g (cid:48) = a · h = b · g (cid:76) (cid:82) k a (cid:53) (cid:63) h b (cid:95) (cid:105) g a (cid:52) (cid:61) h b (cid:97) (cid:106) g (8)where k .. = ( h (cid:48) ) − a (cid:48) h (cid:48) (resp. k .. = ( a · h ) − ( h (cid:48) ) − a (cid:48) h (cid:48) ). Finally observe that X (cid:48) is composed ofpairs of elements in X (cid:48) with one matching leg (such as represented on the left), which images by m (cid:48) are given as in the right-hand diagram: a · g = b · h = a (cid:48) · g (cid:48) a (cid:51) (cid:59) g b (cid:76) (cid:82) h a (cid:48) (cid:100) (cid:108) g (cid:48) (cid:31) m (cid:48) (cid:44) (cid:50) a · g = a (cid:48) · g (cid:48) a (cid:53) (cid:62) g a (cid:48) (cid:97) (cid:105) g (cid:48) Again the operation in X (cid:48) is defined component-wise and behaves as in X (cid:48) .We compute the image π ( G ) which is π ( X ) (resp. π q ( X ) ) by definition. Working on eachobject separately, first observe that as for F r ( X ) (resp. F q ), the unit η X (cid:48) : X (cid:48) → π ( X (cid:48) ) = X sends a trail ( a, g ) ∈ X (cid:111) Pth( X ) (resp. in X (cid:111) Pth ◦ ( X ) ) to its head a ∈ X , i.e. η X (cid:48) is given bythe product projection on X . Now for each pair of trails α = (cid:104) ( a, g ) , ( b, h ) (cid:105) in X (cid:48) , we define thetrail µ ( α ) .. = ( a, gh − ) in X (cid:48) : α = a · g = b · ha (cid:53) (cid:63) g b (cid:95) (cid:105) h (cid:55)→ a · g = b · h (cid:31) (cid:41) h − a (cid:53) (cid:63) g b = .. µ ( α ) . Observe that this trail µ ( α ) is invariant under the action on α , of other pairs β = (cid:104) ( a (cid:48) , g (cid:48) ) , ( b (cid:48) , h (cid:48) ) (cid:105) in X (cid:48) , since µ ( α (cid:47) β ) = ( a, hkk − g − ) = µ ( α ) , where k = ( h (cid:48) ) − a (cid:48) h (cid:48) (resp. k = ( a · h ) − ( h (cid:48) ) − a (cid:48) h (cid:48) ) is the common part of both left and right legs as in Equation (8). Conversely supposethat α , α (cid:48) in X (cid:48) have the same image by µ , we show that α and α (cid:48) are connected in X (cid:48) . Indeed, α and α (cid:48) must then be of the form α = (cid:104) ( a, g ) , ( b, h ) (cid:105) and α (cid:48) = (cid:104) ( a, g (cid:48) ) , ( b, h (cid:48) ) (cid:105) , such that moreover gh − = g (cid:48) h (cid:48)− . Then the path l .. = h − h (cid:48) = g − g (cid:48) ∈ Pth( X ) (resp. in Pth ◦ ( X ) ) decomposes as aproduct l = x δ · · · x nδ n , such that all the pairs (cid:104) ( x i , e ) , ( x i , e ) (cid:105) are in X (cid:48) (and we have moreover (cid:80) ni =0 δ i = 0 in the context of Qnd ). By acting with these pairs “ − (cid:47) δ i (cid:104) ( x i , e ) , ( x i , e ) (cid:105) ” on α , wemay obtain α (cid:48) as in the diagram on the right: α .. = a · g = b · ha (cid:54) (cid:64) g b (cid:93) (cid:103) h and α (cid:48) .. = a · g (cid:48) = b · h (cid:48) a (cid:54) (cid:63) g (cid:48) b (cid:95) (cid:104) h (cid:48) = a · ( gl ) = a · ( hl ) a · g = b · h (cid:76) (cid:82) l a (cid:51) (cid:59) g b (cid:99) (cid:107) h Hence we have the unit morphism η X (cid:48) = µ : X (cid:48) → π ( X (cid:48) ) where π ( X (cid:48) ) is π (Eq( ω X )) = X × Pth( X ) (resp. π (Eq( ω qX )) = X × Pth ◦ ( X ) ). We may then compute π ( d ) = c , π ( d ) = d , π ( i ) = u and π ( −
1) = − , as displayed in the commutative diagram of plain arrows, X (cid:48) η X (cid:48) = µ × µ (cid:12) (cid:18) p (cid:48) (cid:44) (cid:50) p (cid:48) (cid:44) (cid:50) m (cid:48) (cid:44) (cid:50) X (cid:48) η X (cid:48) = µ (cid:12) (cid:18) − (cid:19) (cid:27) d (cid:44) (cid:50) d (cid:44) (cid:50) X (cid:48) η X (cid:48) = d (cid:12) (cid:18) u (cid:108) (cid:114) ω X (resp. ω qX ) (cid:44) (cid:50) XX p (cid:44) (cid:50) p (cid:44) (cid:50) m (cid:44) (cid:50) X − (cid:65) (cid:74) c (cid:44) (cid:50) d (cid:44) (cid:50) X i (cid:108) (cid:114) where the bottom groupoid is the inclusion in Rck (resp.
Qnd ) of the groupoid G ( X, Pth( X )) (resp. G ( X, Pth ◦ ( X )) ) from Set . Hence X = X × Pth( X ) (resp. X = X × Pth ◦ ( X ) ) has thesame underlying set as X (cid:48) , and the underlying functions of η X (cid:48) and d are both given by “projec-tion on X ”.Then since ω X (resp. ω qX ) is a normal covering, d and d are trivial extensions, such thatthe commutative squares dd = dµ and dd = cµ are actually pullback squares. Hence thepullback p (cid:48) , p (cid:48) : X (cid:48) ⇒ X (cid:48) of d and d and the pullback p , p : X ⇒ X of c and d , inducea morphism f : X (cid:48) → X which is thus the pullback of η X (cid:48) = µ and computed component-wise as f = µ × µ . By admissibility of the Galois structure Γ (see Paragraph 2.3.11 and [46]),this morphism is also the unit component f = η X (cid:48) . Finally the commutativity of the square µm (cid:48) = mη X (cid:48) is given by construction (and easy to check by hand), which concludes the proofthat π ( X ) = π ( G ) = G ( X, Pth( X )) (resp. π q ( X ) = G ( X, Pth ◦ ( X )) in Qnd ). (cid:3) regularity of the fundamental groupoid of a rack, whose domain mapis the projection map of a cartesian product: given a rack A , the set of homotopy classes of pathsof a given domain a ∈ A is always Pth( A ) and thus independent of the domain a . Since everypath is invertible, the same is true for the homotopy classes of paths of a given endpoint.One of D.E. Joyce’s main results is to show that the knot quandle is a complete invariant fororiented knots. Now the knot group of an oriented knot, which is the fundamental group of theambient space of the knot, is also computed as the group of paths of the knot quandle. In otherwords, the knot group is the fundamental group of the knot quandle, in the sense of the coveringtheory of racks (not in the sense of the covering theory of quandles).Finally observe that π ( X ) (resp. π q ( X ) ) can be equipped with a non-trivial ad-hoc structureof rack (resp. quandle) making it into an internal groupoid in Rck (resp.
Qnd ) with internalobject of objects the rack (resp. quandle) X . Given two trails ( a, g ) and ( b, h ) in X , define ( a, g ) (cid:47) ( b, h ) .. = ( a (cid:47) b, b − gh − bh ) (note that if g , h ∈ Pth ◦ ( X ) , then b − gh − bh ∈ Pth ◦ ( X ) ).Unlike in ˆ X (resp. ˆ X ◦ ), trails act on each-other with both their heads and end-points, whichmeans that both projections to X are morphisms in Rck (resp.
Qnd ). The rest of the structureis easy to derive.
IGHER COVERINGS OF RACKS AND QUANDLES – PART I 45 A , we thus also describe a skeleton S of π ( A ) (in the sense of [58, Section IV.4]). The resulting groupoid S is not regular like π ( A ) , itis totally disconnected and its vertices are the connected components of A . With the objectiveto interpret the fundamental theorem of Galois theory, the homotopical information containedin π ( A ) can be made more explicit using its skeleton. Definition 3.5.7.
Given an object A in Rck (respectively in
Qnd ), we call a pointing of A anychoice of representatives I .. = { a i } i ∈ π ( A ) ⊆ A such that η A ( a i ) = [ a i ] = i for each equivalenceclass i ∈ π ( A ) . Then for any element a ∈ A , define Loop a as the group of loops l ∈ Pth( A ) (resp. l ∈ Pth ◦ ( A ) ) such that a · l = a . Observe that if [ a ] = [ b ] , for some a and b in A , thenthere is g ∈ Pth( A ) (resp. g ∈ Pth ◦ ( A ) ) such that a = b · g and thus the subgroups Loop a and Loop b are isomorphic, via the automorphism of Pth( A ) (resp. Pth ◦ ) given by conjugation with g . Let us fix a pointing I .. = { a i } i ∈ π ( A ) ⊆ A of A , then we define the groupoid π ( A, I ) (resp. π q ( A, I ) ) as A p (cid:44) (cid:50) p (cid:44) (cid:50) m (cid:44) (cid:50) A − (cid:2) (cid:10) c (cid:44) (cid:50) d (cid:44) (cid:50) π ( A ) , i (cid:108) (cid:114) where A .. = (cid:96) i ∈ π ( A ) Loop a i is defined as the disjoint union, of the underlying sets of Loop a i ’sindexed by i ∈ π ( A ) . The domain and codomain maps send a loop l ∈ Loop a i to the index i ∈ π ( A ) . The set A is then the disjoint union of products A .. = (cid:96) i ∈ π ( A ) (Loop a i × Loop a i ) and m is defined by multiplication in Loop a i ≤ Pth( A ) (resp. Loop a i ≤ Pth ◦ ( A ) ). From the describtion of the skeleton of a groupoid obtained as in Definition 3.5.5, we deduce:
Lemma 3.5.8.
For each I pointing of A object of Rck (respectively of
Qnd ), π ( A, I ) (resp. π q ( A, I ) )is a skeleton of the fundamental groupoid π ( A ) (resp. π q ( A ) ). The fundamental theorem of categorical Galois theory.
In sections 5, 6 and 7 of[25], M. Eisermann studies in details different classification results for quandle coverings. Wewill not go into so much depth ourselves, however we show how to recover and extend the maintheorems from these sections using categorical Galois theory.Given an object A in Rck (respectively
Qnd ), the category of internal covariant presheaves over π .. = π ( A ) (resp. π .. = π q ( A ) ) are externally described as the category of functors from π to Set and thus as the category of π -groupoid actions on sets Set π . Given a pointing I of A , define π ( I ) .. = π ( A, I ) (resp. π ( I ) .. = π q ( A, I ) and deduce from π ( I ) ∼ = π that Set π ∼ = Set π ( I ) . Now π ( I ) is totally disconnected, thus the category of π ( I ) -actions is equivalent to thecategory (cid:96) i ∈ π ( A ) Set
Loop ai whose objects are sequences of Loop a i -group actions (see Definition3.5.7), indexed by i ∈ π ( A ) , and morphisms between these are π -indexed sums of group-actionmorphisms. From the fundamental theorem of categorical Galois theory (see for instance [46,Theorem 6.2]), classifying central extensions above an object we deduce in particular: Theorem 3.6.1.
Given an object A in Rck and a pointing I .. = { a i } i ∈ π ( A ) ⊆ A of A , thereis a natural equivalence of categories between the category CExt ( A ) of central extensions above A and the category Set π ( A ) . The latter category is then also equivalent (but not naturally) to Set π ( A,I ) ∼ = (cid:96) i ∈ π ( A ) Set
Loop ai . The same theorem holds in Qnd , using the appropriate definitionof
Loop a i and using π q ( A ) and π q ( A, I ) instead of π ( A ) and π ( A, I ) . Corollary 3.6.2.
The category of central extensions above a connected rack A is equivalent tothe category of Loop a -actions (from Definition 3.5.7), for any given element a ∈ A . The sameis true in Qnd . Example 3.6.3.
We illustrate this result on a trivial example, to show the difference betweenthe context of
Rck and that of
Qnd . Consider the one element set . The coverings above in Qnd should all be surjective maps to in Set , whereas the coverings above in Rck include forinstance the unit morphism r η q F r (1) = η F r1 : F r (1) → F q (1) = 1 , whose domain is not a set. Thenobserve that Pth(1) = Z and thus Pth ◦ (1) = { e } and since there is only one element ∗ ∈ , Loop ∗ is the former in Rck and the latter in
Qnd . Hence the category of coverings above in Qnd is Set { e } which is indeed equivalent to Set . The category of coverings above in Rck is givenby
Set Z , the category of Z -actions on sets, where Z is the additive group of integers. Relationship to groups and abelianization.
The following relationship between π (cid:97) I in Rck (or
Qnd ) and the abelianization in groups has played an important role in the study of thepresent paper, and in the identification of the relevant centrality conditions in higher dimensionsdescribed in Part II and III.Let us comment first of all that the subvariety of sets is absolutely not a
Mal’tsev category ,and the adjunction π (cid:97) I does not arise from an abelianization adjunction like, for instance,in the case of abelian sym quandles studied in [29] (a quandle is sym if (cid:47) is commutative). Forinstance, the distinction with the study in [29] is clear since the only connected sym quandle is {∗} , also the only group whose conjugation is sym is the trivial group { e } . The relation betweencentrality in racks/quandles and the classical notions of centrality induced by Mal’tsev or partialMal’tsev contexts, appears to us as more subtle than: one being merely an example of the other.The following comments also apply to the context of Qnd , however we like to work in themore “primitive” context of
Rck considering the role of
Pth in the comparison with groups, andits tight relationship with the axioms of racks.We study which squares of functors commute in the following square of adjunctions.
Rck (cid:97) π (cid:48) (cid:55) Pth (cid:17) (cid:24) (cid:62)
Set (cid:97) I (cid:112) (cid:119) F ab (cid:17) (cid:24) Grp (cid:62)
Conj (cid:82) (cid:89) ab (cid:49) (cid:56) Ab I (cid:112) (cid:119) U (cid:82) (cid:89) Starting with an abelian group G , conjugation in G is trivial hence Conj(I( G )) is the trivialquandle on the underlying set of G . Since also both composites send a morphism to the underlyingfunction we have Conj I = I U and thus the restriction of
Conj to abelian groups gives the forgetfulfunctor to
Set . By uniqueness of left adjoints we must also have F ab π = ab Pth . A direct proofeasily follows from the corresponding group presentations.Now starting with a set X in Set we may consider it as a trivial quandle by application of I .Then we compute Pth(I( X )) .. = F g ( X ) / (cid:104) ( x (cid:47) a ) − a − xa | a, x ∈ X (cid:105) = F g ( X ) / (cid:104) x − a − xa | a, x ∈ X (cid:105) , which shows that for each set X we have Pth(I( X )) = I F ab ( X ) , which then easily gives Pth I =I F ab , i.e. the restriction of Pth to trivial racks gives the free abelian group functor.Observe that we cannot use uniqueness of adjoints to derive that π Conj is the same as
U ab . Indeed we compute that ( π , Conj , U , ab ) is the only square of functors that doesn’tcommute. Given a group G , the image π (Conj( G )) is given by the set of conjugacy classes. Thecorresponding congruence in Qnd is given by a ∼ b ⇔ ( ∃ c ∈ G )( c − ac = b ) . (9) IGHER COVERINGS OF RACKS AND QUANDLES – PART I 47
Then the abelianization ab( G ) is the quotient of G by the congruence generated in Grp by theidentities { c − ac = a | a, c ∈ G } . We may show that in general the equivalence relation defined in(9) does not define a group congruence. A counter-example is given by the group of permutations S . It has three conjugacy classes given by cycles, two permutations and the unit. The derivedsubgroup is the alternating group A which is of order 2. This shows that there are less elementsin the abelianization of S than there are conjugacy classes in S .Understand that an “image” of the covering theory in Rck , arising from the adjunction π (cid:97) I can be studied in groups through the functor Pth and its restriction to sets F ab . Note that Pth is neither full, faithful or essentially surjective. The functor F ab is full and faithful. We will notstudy what information to extract from this image. Yet, again, we have been using ingredientsof this image to describe centrality in Rck such as in Theorem 3.4.1. Observe moreover that anycovering in racks induces a central extension between the groups of paths [25, Proposition 2.39].However, certain morphisms, such as f : Q ab(cid:63) → {∗} , which are not central in Rck (or
Qnd ) aresent by
Pth to central extensions of groups, e.g. (cid:126)f : Pth( Q ab(cid:63) ) = Z × Z → Z = Pth( {∗} ) : ( k, l ) (cid:55)→ k + l. In the other direction an “image” of the theory of central extensions of groups can be studiedin
Rck via the “inclusion”
Conj and its restriction U on abelian groups. Both Conj and U are notfull but faithful, U is moreover surjective. Again we shall not develop the full potential of thisstudy. Observe nonetheless that a morphism of groups is central if and only if it gives a coveringin racks [25, Example 2.34], see also [25, Example 1.2] and comments below. We give some moreresults about this relationship in Part II. Acknowledgements
Many thanks to my supervisors Tim Van der Linden and Marino Gran for their support,advice and careful proofreading of this work.
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