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1. Introduction Pré-Publicações do Departamento de Matemática Universidade de Coimbra Preprint Number 14–18 A CRITERION FOR REFLECTIVENESS OF NORMAL EXTENSIONS WITH AN APPLICATION TO MONOIDS ANDREA MONTOLI, DIANA RODELO AND TIM VAN DER LINDEN Dedicated to Manuela Sobral on the occasion of her seventieth birthday Abstract: We prove that the so-called special homogeneous surjections are reflec- tive amongst surjective homomorphisms of monoids. To do so, we use the recent result that these special homogeneous surjections are the normal (= central) extensi- ons with respect to the admissible Galois structure ΓMon determined by the Grothen- dieck group adjunction together with the classes of surjective homomorphisms. It is well known that such a reflection exists when the left adjoint functor of an admissible Galois structure preserves all pullbacks of fibrations along split epimorphic fibrati- ons, a property which we show to fail for ΓMon. We give a new sufficient condition for the normal extensions in an admissible Galois structure to be reflective, and we then show that this condition is indeed fulfilled by ΓMon. Keywords: categorical Galois theory; admissible Galois structure; central, nor- mal, trivial extension; Grothendieck group; group completion; homogeneous split epimorphism, special homogeneous surjection of monoids. AMS Subject Classification (2010): 20M32, 20M50, 11R32, 19C09, 18F30. 1. Introduction The original aim of our present work was to answer the following question: Is the category of special homogeneous surjections of monoids [3, 4] a reflective subcategory of the category of surjective monoid homomorphisms? Since we recently showed [17] that these special homogeneous surjections are the normal extensions in an admissible Galois structure [10, 11], we were at first convinced that this would be an immediate consequence of some known abstract Galois- theoretical result such as the ones in [13, 12]. At a second glance, we could not find any result to apply in the given situation. Received June 16, 2014. The first and the second author were supported by the Centro de Matemática da Universidade de Coimbra (CMUC), funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT–Fundação para a Ciência e a Tecnologia under the projects PEst-C/MAT/UI0324/2013 and PTDC/MAT/120222/2010 and grant number SFRH/BPD/69661/2010. The third author is a Research Associate of the Fonds de la Recherche Scientifique–FNRS. 1 2 ANDREA MONTOLI, DIANA RODELO AND TIM VAN DER LINDEN The case of commutative monoids is after all not too difficult. Here we can just use Theorem 7.1 of [13], since the reflector from commutative mo- noids to abelian groups preserves pullbacks of surjective homomorphisms along split epimorphisms (Proposition 4.2). The problem is that, for general (non- commutative) monoids, the Grothendieck group construction need not preserve such pullbacks—see Example 5.1. It turns out, however, that the special homogeneous surjections of monoids do indeed form a reflective subcategory. The process of proving this result leads naturally to a new criterion for reflectiveness of normal extensions amongst the fibrations of an admissible Galois structure (our Theorem 2.12, closely related to the subject of the recent article [7]): if the categories under consideration are Barr-exact and all fibrations in the Galois structure are regular epimorphisms, then it suffices that the category of normal extensions is closed under coequa- lisers of reflexive graphs in the category of fibrations. This sufficient condition holds for monoids (Theorem 5.5). Thus a problem in the category of monoids lead to a general result in catego- rical Galois theory. Conversely, Galois theory helped us to better understand a non-obvious result on monoids, which we believe is interesting for its own sake: it allowed us to extended Proposition 2.3.5 in [4] twice, first to Proposi- tion 5.4, then to Theorem 5.5 which says that the category of special homoge- neous surjections is closed under coequalisers of reflexive graphs of surjective homomorphisms of monoids. Before specialising to monoids, we first focus on reflectiveness results in a general Galois-theoretic setting (Section 2). In Subsection 2.8 we use the cons- truction of the normalisation functor proposed in [8] to prove our main result (Theorem 2.12). We then, in Section 3, recall the definitions and main results concerning special homogenous surjections from [3, 4] as well as the main re- sults from [17], giving the link between special homogeneous surjections and normal extensions, which are needed throughout the subsequent sections. In Section 4 we treat the commutative case, which easily leads to Theorem 4.3, the reflectiveness of special homogeneous surjections of commutative monoids. The non-commutative case is treated in the final Section 5, where we first give a counterexample against the preservation of pullbacks of split epimorphisms along spit epimorphisms by the Grothendieck group functor (Example 5.1), to prove then that the technique of Section 2 is applicable (Theorem 5.5) and to obtain Theorem 5.6—thus answering our original question. REFLECTIVENESS OF NORMAL EXTENSIONS 3 2. Reflectiveness of normal extensions In this section we work towards a general result on reflectiveness of normal extensions in an admissible Galois structure: Theorem 2.12 which says that, if the fibrations in the Galois structure are regular epimorphisms, and normal extensions are closed under coequalisers of reflexive graphs, then the normal extensions are reflective amongst the fibrations. 2.1. Galois structures. We start by recalling the definition of (admissible) Galois structure as well as the concepts of trivial, normal and central extension arising from it [10, 11, 12]. We consider the context of Barr-exact categories [1] which is general enough for our purposes and allows us to avoid some technical difficulties. Ô Õ Definition 2.2. A Galois structure Γ C , X ,H,I,η,ǫ, E , F consists of an adjunction I ,2 C à X lr H ñ with unit η : 1C ñ HI and counit ǫ: IH 1X between Barr-exact categories C and X , as well as classes of morphisms E in C and F in X such that: (G1) E and F contain all isomorphisms; (G2) E and F are pullback-stable; (G3) E and F are closed under composition; Õ (G4) H ÔF E ; Õ (G5) I ÔE F . We call the morphisms in E and F fibrations [11]. The following definitions are given with respect to a Galois structure Γ. Definition 2.3. A trivial extension is a fibration f : A Ñ B in C such that the square ηA Õ A ,2 HI ÔA Õ f HI Ôf Õ ,2 HI ÔB B ηB ¦ Õ is a pullback. A central extension is a fibration f whose pullback p Ôf along some fibration p is a trivial extension. A normal extension is a fibration such that its kernel pair projections are trivial extensions. 4 ANDREA MONTOLI, DIANA RODELO AND TIM VAN DER LINDEN It is well known and easy to see that trivial extensions are always central extensions and that any normal extension is automatically central. For any object B in C , there is an induced adjunction IB ,2 Ã Ô Ó Õ Ó Ô ÕÕ ÔE F B lr I B , HB Ó Õ Ô Ó Õ where we write ÔE B for the full subcategory of the slice category C B B Ó Ô ÕÕ determined by morphisms in E ; similarly for ÔF I B . The functor I is B Ô Õ the restriction of I, and H sends a fibration g : X Ñ I B to the pullback Õ A ,2 H ÔX B Ô Õ Õ H Ôg H g Õ ,2 HI ÔB B ηB Õ of H Ôg along ηB. Ô Õ Definition 2.4. A Galois structure Γ C , X ,H,I,η,ǫ, E , F is said to be admissible when, for every object B in C , the functor HB is full and faithful. The admissibility condition amounts to reflectiveness of trivial extensions amongst fibrations. More precisely, we have that: (1) by Proposition 2.5 below, the replete image of the functor HB is precisely Õ the category Triv ÔB of trivial extensions over B; Õ Ô Ó Õ (2) Triv ÔB is a reflective subcategory of E B ; B B Ó Õ Ñ Ô Õ (3) H I : ÔE B Triv B is its reflector. By Proposition 5.8 in [9], we obtain a left adjoint, called the trivialisation functor Õ Ñ Ô Õ Triv: FibÔC TExt C , Õ to the inclusion of the category TExtÔC of trivial extensions in C into the full Õ subcategory FibÔC of the category of arrows in C determined by the fibrations. Proposition 2.5. [13, Proposition 2.4] If Γ is an admissible Galois structure, then I : C Ñ X preserves pullbacks along trivial extensions. Hence a fibration is a trivial extension if and only if it is a pullback of some fibration in X . In particular, the trivial extensions are pullback-stable, so that every trivial extension is a normal extension. REFLECTIVENESS OF NORMAL EXTENSIONS 5 2.6. A result on reflectiveness of central extensions. Consider an ad- missible Galois structure Γ as in Definition 2.4. Given an object B in C , we Õ Ô Ó Õ let Centr ÔB denote the full subcategory of E B determined by the central extensions over B. When it exists, the left adjoint to the inclusion functor Õ ãÑ Ô Ó Õ Ô Ó Õ Ñ Ô Õ Centr ÔB E B is written Centr: E B Centr B and called the centralisation functor. In [8], it is explained how Theorem 7.1 in [13] gives us: Ô Õ Theorem 2.7.
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