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Protomodular Aspect of the Dual of a Topos

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View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ARTICLE IN PRESS Advances in Mathematics 187 (2004) 240–255 http://www.elsevier.com/locate/aim Protomodular aspect of the dual of a topos Dominique Bourn Centre Universitaire de la Mi-voix, Lab. d’Analyse, Ge´ome´trie et Alge`bre, Universite´ du Littoral, 50 rue Ferdinand Buisson, BP699, 62228 Calais Cedex, France Received 3 March 2003; accepted 15 September 2003 Communicated by Ross Street Abstract The structure of the dual of a topos is investigated under its aspect of a Barr exact and protomodular category. In particular the normal monomorphisms in the fibres of the fibration of pointed objects are characterized, and the change of base functors with respect to this same fibration are shown to reflect those normal monomorphisms. r 2003 Elsevier Inc. All rights reserved. MSC: 18B25; 18D30; 08B10 Keywords: Topos; Fibrations; Mal’cev and protomodular categories 0. Introduction It is well known that, given a topos E; the subobject classifier O is endowed with an internal structure of Heyting Algebra, and that consequently the contravariant ‘‘power-set’’ functor OðÞ : Eop-E takes its values in the category HeytE of internal Heyting algebras in E: But the category Heyt of Heyting Algebras shares with the categories Gp of groups or Rg of rings the property of being protomodular. This property [3] says that the fibration of pointed objects p : PtC-C; whose fibre at X is the category PtX C of split epimorphisms with codomain X; has its change of base functors conservative (i.e. reflecting the isomorphisms). This is not a minor property, since a protomodular category retains many aspects of the category Gp. In particular it involves an intrinsic notion of normal subobject [5] and, provided the basic category C is regular E-mail address: [email protected]. 0001-8708/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.aim.2003.09.004 ARTICLE IN PRESS D. Bourn / Advances in Mathematics 187 (2004) 240–255 241 and pointed, it implies a 3  3 lemma and a snake lemma. Moreover, when C is not pointed, the 3  3 lemma can be ‘‘denormalized’’ [8]. Now the previous property of the contravariant ‘‘power-set’’ functor transmits to the dual Eop the property of being protomodular, and the aim of this work will be to investigate Eop under this protomodular aspect. If the characterization of the normal monomorphisms in Eop appears to be a bit disappointing since, indeed, the only normal monomorphisms are the isomorphisms, the translation of the ‘‘denorma- lized’’ 3  3 lemma produces significant commutations of limits, see Propositions 2.6 and 2.7. On the contrary the normal monomorphisms in the pointed proto- op op modular fibres ðPtX EÞ will be far from trivial: in the fibre ðPt1EÞ they are characterized, in the basic topos E; as those epimorphisms which have a special kind of kernel equivalence relation, namely a disconnectedly generated equi- valence relation, i.e. such that all but one of its equivalence classes are reduced to a single element. More specifically the change of base functors of the fibration p of pointed objects will appear to reflect these normal monomorphisms (i.e. the dual of a topos is strongly protomodular, again in the same way as the category Gp). Finally the category Eop is also arithmetical [17]. This implies [6] that the normal op monomorphisms in ðPt1EÞ have at most one retraction, and consequently that the previous kind of epimorphisms in Pt1E has at most one section. This article is organized along the following line: (1) The dual of a topos is Barr exact. (2) The dual of a topos is protomodular. (3) Some right exact properties of protomodular categories. op (4) The normal monomorphisms in ðPt1EÞ : (5) The dual of a topos is strongly protomodular. (6) The dual of a topos is arithmetical. 1. The dual of a topos is Barr exact Let us recall some basic facts concerning the right exact properties of an elementary topos E; about which we shall refer to the classic book [15] and the more recent one [16]. First, it has necessarily at least the same type of right limits as of left limits. In particular, it is finitely cocomplete. This is a consequence of one of the most striking result on elementary topos: Theorem 1.1. The contravariant ‘‘power-set’’ functor P : Eop-E is monadic. It is, moreover, Barr exact [1]. Secondly, there are some very elementary results among which the following ones [15]: ARTICLE IN PRESS 242 D. Bourn / Advances in Mathematics 187 (2004) 240–255 Lemma 1.1. (1) Suppose we are given a pushout square: where the map m is a monomorphism, then the map n is also a monomorphism, and the square in question is also a pullback. (2) Let a : X1-Y and b : X2-Y be two subobjects. Take their intersection ð¼ pullbackÞ: then their union is given by the following pushout: Third, from the conjunction of these two results, comes also the following known consequence [11] (see also [2]), we shall prove here in an elementary way: Proposition 1.1. The dual Eop of an elementary topos is Barr exact and Mal’cev. Proof. The epimorphisms of Eop are the monomorphisms of E; which are all equilizers. Consequently, in Eop; all epimorphisms are regular, and the dual of part 1 in the previous lemma, means that regular epimorphisms are stable by pullbacks. The category Eop is then regular, since E is finitely cocomplete. We must show that any equivalence relation in Eop is effective. Actually we shall show that any reflexive relation is effective, which will imply that any reflexive relation is an equivalence relation, and therefore that Eop is Mal’cev [12]. So consider any reflexive relation in Eop on an object X (we shall draw the picture in E all over this article): which means that the factorization j : X þ X-Z is an epimorphism in E; and that, consequently, Z is the union of the two subobjects determined by f0 and f1: Now take ARTICLE IN PRESS D. Bourn / Advances in Mathematics 187 (2004) 240–255 243 their intersection: then the presence of the map q implies that: c ¼ q Á f0 Á c ¼ q Á f1 Á d ¼ d; and that c is the kernel of f0 and f1: Part 2 of the previous lemma implies that the following square is a pushout: Accordingly, the given reflexive relation in Eop is effective. & Remark 1.1. As a consequence, the reflexive relations in Eop are easily characterized as the coreflexive cographs in E such that the pair of monomorphisms ð f0; f1Þ is jointly strongly epic: 2. The dual of a topos is protomodular 2.1. Protomodularity and fibration of pointed objects Let us consider any finitely complete category C: We denote by PtC the category whose objects are the split epimorphisms in C with a given splitting and morphisms the commutative squares between these data. We denote by p : PtC-C the functor associating its codomain with any split epimorphism. Since the category C has pullbacks, the functor p is a fibration which is called the fibration of pointed objects. Indeed any map f : X-Y induces, by pullbacks, a change of base functor denoted à f : PtY C-PtX C; between the fibres above Y and X: Let us recall that [3] a left exact category C is said protomodular when the fibration p has its change of base functors conservative, i.e. reflecting the isomorphisms. A protomodular category is necessarily Mal’cev [4]. There are three general types of example of protomodular categories: (1) varieties of classical algebraic structures, such as the category of groups, the category of rings, the category of associative or Lie algebras over a given ring A, the category of Heyting algebras, the varieties of O-groups. Actually the protomodular varieties are completely characterized in [10]. ARTICLE IN PRESS 244 D. Bourn / Advances in Mathematics 187 (2004) 240–255 (2) categories of internal classical algebraic structures of the previous kind in a left exact category C: (3) constructions which inherit the property of being protomodular, such as the slice categories C=Z and the fibres PtZC of the fibration p of pointed objects for instance, or more generally the domain C of any pullback preserving and conservative functor U : C-D; when its codomain D is protomodular. Here is now a nonsyntactical example. Theorem 2.1. The dual of a topos is protomodular. Proof. We noticed that the ‘‘contravariant power-set functor’’ P : Eop-E takes its values in the category HeytE of internal Heyting algebras in E; i.e. we have P : Eop-HeytE: But the category Heyt of Heyting Algebras is protomodular as Johnstone showed it [4], therefore the category HeytE is protomodular. Now the functor P is left exact and conservative, thus the category Eop is itself protomodular. & One of the earlier results about protomodularity was the following [3]: Proposition 2.1. In any regular protomodular category, a pullback of regular epimorphisms is always a pushout. This is exactly the dual of part 1 in Lemma 1.1. Part 2 is also the dual of a known result in the same area [11,13]: Proposition 2.2. In any Barr exact Mal’cev category (and therefore in any Barr exact protomodular category), if you take the pushout of two regular epimorphisms f and g: then the factorization j : X-P; where P is the domain of the pullback of f 0 along g0; is a regular epimorphism.
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