Protomodular Aspect of the Dual of a Topos

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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].

D. Bourn / Advances in Mathematics 187 (2004) 240–255 241 and pointed, it implies a 3 Â 3 lemma and a snake lemma. Moreover, when C is not pointed, the 3 Â 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 ‘‘denormalized’’ 3 Â 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 equivalence 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

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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 Ã 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

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(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.

This emphazises the way how some right exact properties in a topos reﬂect the protomodularity of its dual. The protomodularity of the dual Eop of a topos asserts the following:

Theorem 2.2. Let E be an elementary topos, then the ﬁbration p : PtE-E of pointed objects is such that the co-change of base functors f! : PtX C-PtY C are conservative.

On the other hand, E being a topos, the pullback functors f Ã : E=Y-E=X; between Ã the slice categories, admit right adjoints Pf : But the functors f preserve the terminal ARTICLE IN PRESS

D. Bourn / Advances in Mathematics 187 (2004) 240–255 245 objects, and it is straightforward to show that the functors Pf : E=X-E=Y extend to Ã functors f% : PtX C-PtY C right adjoint to the functors f : PtY C-PtX C: As a Ã consequence, the change of base functors f : PtY C-PtX C are right exact.

2.2. The protomodular category Eop

So the category Eop has a very rich structure, being ﬁnitely cocomplete, Barr exact, and protomodular such that any co-change of base functor f! is left exact. As a ﬁnitely cocomplete Barr exact protomodular category, it is a category with semi- direct products in the sense of [9], i.e. such that:

Proposition 2.3. Suppose is E a topos. In Eop; any change of base functor f Ã is monadic. Or equivalently, in E; any co-change of base functor f! is comonadic.

But the main fact is that, in a protomodular category, there is an intrinsic notion of normal subobject, see [5]. Let us recall, indeed, that in any left exact category C a map m : I-X is normal to À1 an equivalence relation R when m ðRÞ is the coarse relation rI on I (i.e. the kernel relation of the terminal map tI : I-1) and when the induced map rI -R in RelC is ﬁbrant, i.e. when the commutative square with the d0 (or equivalently with the d1)in the following diagram is a pullback:

This implies that m is necessarily a monomorphism. This deﬁnition gives an intrinsic way to express that I is an equivalence class of R: But when C is moreover protomodular, the map m is normal to at most one equivalence relation and consequently the fact to be normal, in this kind of category, becomes a property.

Remark 2.1. When the category C is Barr exact, protomodular and such that any object X has a global support (i.e. the terminal map tX : X-1 is a regular epimorphism, this is the case, for instance when C is pointed), then any normal monomorphism m : I-X comes from a pullback of the following form, where we can suppose that h is a regular epi:

This observation gives us that:

Proposition 2.4. The only normal monomorphisms in Eop are the isomorphisms. ARTICLE IN PRESS

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Proof. The terminal object of Eop is the initial object of E which is strict (i.e. any morphism X-0 is an isomorphism). As a consequence, any object in Eop has a global support, and the only maps with domain the terminal object are the isomorphisms. On the other hand, the category Eop is Barr exact and protomodular, then, according to the previous remark, the only normal monomorphisms are the isomorphisms. &

2.3. The denormalized 3 Â 3 lemma

As in any regular Mal’cev category, the ‘‘denormalized’’ 3 Â 3 lemma holds in Eop: Among the many variations about that result, we shall emphasize the following translation in E of theorem 16 in [8]:

Proposition 2.5. Given a topos E; consider the following diagram where the vertical arrows are monomorphisms, where the horizontal left-hand side parts are cographs such that the pairs ðf0; f1Þ and ðg0; g1Þ are jointly strongly epic, and where the maps k and h are their respective equalizers:

then in the following extension by pushouts, the right-hand side map is the equalizer of the left-hand cograph:

And also the following translation of corollary 18 in [8], which requires the Barr exactness of Eop:

Proposition 2.6. Given a topos E; consider the following diagram where the maps k; h and b are monomorphisms:

then the right-hand square is a pullback if and only if the map g is a monomorphism. ARTICLE IN PRESS

D. Bourn / Advances in Mathematics 187 (2004) 240–255 247

2.4. The disconnectedly generated equivalence relations

We saw that the characterization of normal monomorphisms in Eop was a bit op disappointing. It will not be the same for the dual ðPtZEÞ of the fibres, which are pointed, Barr exact and protomodular, i.e. very near the leading and guiding example of protomodular category, namely the category Gp of groups. Due to the fact that the slice categories E=Z are again toposes, and that Pt1ðE=ZÞ¼PtZE ,it will be sufficient to investigate what are normal monomorphisms in the dual of the fibre Pt1E above the terminal object 1. We shall get to the heart of this matter thanks to the following observation: consider any monomorphism m : I-X in the topos E; and the following pushout:

Then the map m is a monomorphism. Consider now the following commutative square which is a pullback since m is a mono:

There is a factorization ½d0; d1 : RI -X Â X which, according to part 2 of Lemma 1.1 or to the dual of Proposition 2.3, is a monomorphism. Consequently the map ½d0; d1 : RI -X Â X determines a reﬂexive relation. Before going any further, note that the following square being again a pushout and the map m being a monomorphism, this square is also a pullback:

and thus, provided RI is an equivalence relation, the map m will be normal to RI :

Deﬁnition 2.1. In this case, the relation RI will be called the equivalence relation disconnectedly generated by I (or by m).

When E is the topos Set of sets, and I a subset of X , then the relation RI on the set 0 0 0 X is the following one: xRI x if and only if both x and x belong to I; or x ¼ x : It is clearly an equivalence relation. Moreover this proof, being entirely constructive, yields the result in every topos E: However this result is underlying a more general one, namely that the co-change of base functor m! is left exact. This is the meaning of the next section. ARTICLE IN PRESS

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3. Some right exact properties of protomodular categories

We shall consider now a Barr exact protomodular category C: First we are going to show that, when a map f : X-Y is a split epimorphism, the change of base Ã functor f : PtY C-PtX C preserves the pushout whose one edge is a regular epimorphism. For that it will be sufﬁcient, localizing the problem in the ﬁbre PtY C; to prove the following proposition:

Proposition 3.1. Given any pointed ﬁnitely cocomplete regular protomodular category C; and any object X in C; then: (1) when the map h is a regular epimorphism, then the following square is a pushout:

(2) the functor X ÂÀ preserves those pushouts having one edge a regular epimorphism.

Proof. (1) The category C; being pointed, produces canonical injections - jU : U X Â U; which themselves determine a canonical comparison map gU : X þ U-X Â U: Suppose now the map h is a regular epimorphism, then consider the following diagram:

the total rectangle is a pullback, but h is a regular epi, so this rectangle is a pushout. This is also the case of the left-hand side square, consequently the right-hand side square is a pushout. (2) Consider now any pushout:

then the map h0 is a regular epi, and the following total rectangle is a pushout as a pasting of two pushouts: ARTICLE IN PRESS

D. Bourn / Advances in Mathematics 187 (2004) 240–255 249 but it is equal to the following one:

and consequently the right-hand side square is a pushout. &

We have now the following theorem:

Theorem 3.1. Given any pointed ﬁnitely cocomplete Barr exact protomodular category C (i.e. any semi-abelian category in the terminology of [14]) such that any functor ÀþB preserves pullbacks, then, for any object X; the change of base functor Ã - - tX : Pt1C ¼ C PtX C preserves pushouts, where tX : X 1 is the terminal map.

Ã Proof. The functor tX preserves those pushouts having one edge a regular epimorphism since, according to the previous proposition, the functor X ÂÀdoes. Ã It is now sufﬁcient to prove that the functor tX preserves sums. So let ðA; BÞ be any pair of objects in C and consider the following diagram:

where the map j is the factorization induced by the map pA þ B: Since the functor ÀþB preserves pullbacks, the rectangle determined by the pasting of the two squares is a pullback. Therefore the left-hand side square is a pullback. Note that, according to Proposition 2.2, the maps gB are regular epis. Therefore the map j a regular epi. Now consider the following diagram:

The right-hand side square is a pullback, as a section of the previous left-hand side one, and, its two horizontal edges being regular epis, it is also a pushout. Now the left hand side square is always a pushout, and the total rectangle is a pushout, and therefore a sum in PtX C: &

Corollary 3.1. Given any ﬁnitely cocomplete Barr exact protomodular category C; such that, for any split epimorphism ðf ; sÞ; f : X-Y; the functor s! preserves pullbacks, Ã then the change of base functor f : PtY C-PtX C preserves pushouts. ARTICLE IN PRESS

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The dual of a topos E satisﬁes all theses conditions, and consequently:

Corollary 3.2. Given any split monomorphism m : I-X in a topos E; then the co- change of base functor m! : PtI E-PtX E is left exact.

The aim now will be to extend Corollary 3.1 to any regular epimorphism. Then Corollary 3.2 will hold for any monomorphism m in the topos E: This will be a consequence of the following general result:

Theorem 3.2. Given any ﬁnitely cocomplete and Barr exact category C such that the change of base functor (with respect to p) along any split epimorphism preserves Ã pushouts, then the change of base functor f : PtY C-PtX C along any regular epimorphism f preserves pushouts.

Proof. Localizing the question in the slice category C=Y; it will be sufﬁcient to prove the result for any regular epic terminal map tX : X-1 in a category C satisfying the hypotheses. So consider any pushout in Pt1C:

then construct the following one, and denote by j : S-X Â V 0 the induced factorization:

But the projection p1 : X Â X-X is a split epimorphism. According to the Ã hypotheses, the change of base functor p1 preserves pushouts, and consequently the following square is again a pushout:

Now, from the following three exact sequences (the terminal map tX being a regular epimorphism, the projections pI : X Â I-I are also regular epimorphisms) with successively I ¼ U; I ¼ U 0 and I ¼ V: ARTICLE IN PRESS

D. Bourn / Advances in Mathematics 187 (2004) 240–255 251 the three previous pushouts produce the following coequalizer:

Consider the following diagram:

The lower left-hand square is clearly a pullback. Consequently, the lower pair being an equivalence relation, the upper pair is underlying an equivalence relation too. But the category C is Barr exact, and consequently the upper row is not only a coequalizer but also an exact sequence (i.e. an equivalence relation with its quotient). The lower left-hand square is a pullback, the rows are exact sequences, then the right hand square is itself a pullback, and consequently the map j is Ã an isomorphism. Accordingly, the change of base functor tX preserves the pushouts. &

As a consequence we get the following:

Theorem 3.3. In any topos E; given any monomorphism m : I-X; then the co-change of base functor m! : PtI E-PtX E is left exact.

And as a corollary:

Corollary 3.3. In any topos E; given any monomorphism m : I-X and any internal category I1 with I as object of objects, then the following pushout determines an internal category X1 with X as object of objects, and an internal functor m1 : I1-X1 which is both a discrete ﬁbration and a discrete coﬁbration:

By construction this functor m1 : I1-X1 is necessarily cocartesian with respect to the - forgetful functor ðÞ0 : CatE E associating to each internal category its object of objects. When moreover I1 is respectively a groupoid, a poset, an equivalence relation, then X1 is of the same type. In particular the relation RI disconnectedly generated by I (see Section 2.4) is an equivalence relation, to which the monomorphism m is normal. ARTICLE IN PRESS

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op 4. The normal monomorphisms in ðPt1EÞ

The two previous results have an interesting consequence:

Proposition 4.1. In a topos E; consider the following square in which g and h are regular epimorphisms and m a monomorphism:

It is a pushout if and only if the following square is a pushout, where R½g and R½h are the kernel relation of g and h:

Proof. Let RelE denote the category whose objects are the pair ðX; RÞ of an object X in E endowed with an equivalence relation R and arrows are induced by the arrows in E: In any Barr exact category E and without any assumption about the map m; the ﬁrst square is a pushout if and only if the induced map m : R½g-R½h in RelE is cocartesian with respect to the forgetful functor W : RelE-E associating X to ðX; RÞ; see [7]. But m is a mono in the topos E; then, according to the previous corollary, this map m in RelE is cocartesian if and only the second square is a pushout. &

We can now characterize the normal monomorphisms in the dual of the ﬁbre Pt1E: Of course a monomorphism in this dual is an epimorphism in Pt1E; and then in E:

Theorem 4.1. Given any topos E; a regular epi f : X-Y in the ﬁbre Pt1E is normal in the dual if and only if the kernel relation R½f is disconnectedly generated in E:

Proof. Suppose f : X-Y normal in the dual of Pt1E; then, according to Remark 2.1, the map f is obtained from a pushout of the following type where m is a monomorphism:

and according to the theorem above, R½f is the disconnectedly generated equivalence relation associated with m: ARTICLE IN PRESS

D. Bourn / Advances in Mathematics 187 (2004) 240–255 253

Conversely, suppose the kernel relation R½f is disconnectedly generated in E by a monomorphism m : I-X: Then consider the following diagram, where the map q is the quotient of the equivalence relation rI :

Since the left-hand side square is a pushout, the right-hand side one is a pushout. Now Q is necessarily equal to 1, since I; being pointed by a map 1-I; has a global support. Therefore the map f is normal in the dual. &

We shall call disconnecting a regular epimorphism having a disconnectedly generated kernel relation. In the category SetsÃ of pointed sets, the disconnecting epimorphisms are those epimorphisms whose inverse images of any element but the distinguished one are reduced to a single element.

5. The dual of a topos is strongly protomodular

A category C is strongly protomodular, when it is protomodular and such that any change of base functor f Ã is a normal functor, i.e. a left exact conservative functor which reﬂects the normal monomorphisms. The main examples are the categories Gp of groups and Rg of rings. We are now going to show that Eop is strongly protomodular. First let us observe the following:

Proposition 5.1. Let ðT; l; mÞ be a left exact monad on a protomodular category C; then the category of algebras AlgT is protomodular and the forgetful functor U : AlgT-C is normal.

Proof. The forgetful functor U : AlgT-C being always left exact and conservative, then, when C is protomodular, so is AlgT: Now suppose given any monomorphism m : ðI; gÞ-ðX; aÞ in AlgT; such that m : I-X is normal in C: Let R be the equivalence relation on X to which m is normal. Let us show that the object R has an algebra structure, which will prove that U is a normal functor. But T is left exact, and consequently Tm is again a monomorphism which is normal to TR in C: The induced morphism of equivalence relations Tm : rTI -TR is thus ﬁbrant. Now C being protomodular, this ﬁbrant map is cocartesian with respect to the forgetful functor W : RelC-C [8]. Consequently the pair constituted by the map a : TX-X in C and the map m:g : rTI -R in RelC determines a map TR-R in RelC above a : TX-X: Whence a map r : TR-R in C; which is straightforwardly an algebra structure on the object R: & ARTICLE IN PRESS

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Now, as we recalled, any Barr exact protomodular category admits semi-direct products, i.e. is such that any change of base functor f Ã is monadic.

Deﬁnition 5.1. A category C will be called exactly protomodular when it is Barr exact protomodular, admits pushout of split monos and is such that any co-change of base functor f! is left exact.

Thus, according to the beginning of Section 2.2, when E is a topos, the category Eop is exactly protomodular. Now thanks to the previous proposition, we have:

Theorem 5.1. Any exactly protomodular category C is strongly protomodular. In particular, the dual Eop of any topos is strongly protomodular.

6. The dual of a topos is arithmetical

The category Heyt of Heyting algebras is not only protomodular, but also arithmetical, i.e., according to Pedicchio [17], Barr exact Mal’cev, and such that the congruence distributive property holds.

Theorem 6.1. The dual Eop of a topos is arithmetical.

Proof. The factorization of the internal power-set functor P : Eop-HeytE being left exact and conservative, and HeytE being arithmetical, so is the category Eop: &

The basic reason which can turn a Mal’cev category C in an arithmetical one is that there is no nontrivial internal group structure in the ﬁbres of p : PtC-C; see [6]. Consequently the arithmetical property is concentrated in the following proposition which says that in the dual the associated internal group object is always trivial (see Theorem 3.8 and Deﬁnition 1.10 in [6]):

Proposition 6.1. In any topos E; given any split monomorphism s : Y-X; the following square is always a pullback, where the map Y denotes the co-diagonal:

Now let us recall the following theorem, see [6]:

Theorem 6.2. In any arithmetical protomodular category C; a normal monomorphism has at most one retraction.

Now any slice category C=Z or any ﬁbre PtZC of any arithmetical protomodular category C is again an arithmetical protomodular category. Accordingly, any ARTICLE IN PRESS

D. Bourn / Advances in Mathematics 187 (2004) 240–255 255 disconnecting epimorphism in Pt1E has at most one section, as it is easily checked in the category SetsÃ of pointed sets for instance.

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