A CLASSIFICATION THEOREM FOR NORMAL EXTENSIONS MATHIEU DUCKERTS-ANTOINE AND TOMAS EVERAERT

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1 Pré-Publicações do Departamento de Matemática Universidade de Coimbra Preprint Number A CLASSIFICATION THEOREM FOR NORMAL EXTENSIONS MATHIEU DUCKERTS-ANTOINE AND TOMAS EVERAERT Abstract: For a particular class o Galois structures, we prove that the normal extensions are precisely those extensions that are locally split epic and trivial, and we use this to prove a Galois theorem or normal extensions. Furthermore, we interpret the normalisation unctor as a Kan extension o the trivialisation unctor. Keywords: Galois theory, normal extension, normalisation unctor. AMS Subject Classiication (2010): 13B05, 18A20, 18A22, 18A Introduction For an admissible Galois structure Γ = (C,X,I,H,η,ǫ,E,F), the Fundamental Theorem [6] provides, or every monadic extension p: E B, an equivalence Spl Γ (E,p) X FGal Γ (E,p) between the category o central extensions(= coverings) o B that are split by (E,p)andthecategoryodiscreteibrationsG Gal Γ (E,p)o(pre)groupoids in X over the Galois(pre)groupoidGal Γ (E,p), with componentsin the class F. When, moreover, p: E B is such that it actors through every other monadic extension o B (i.e. when it is weakly universal), then every central extension o B is split by (E,p), and the above equivalence becomes CExt Γ (B) X FGal Γ (E,p). Now, as ollows rom Lemma 2.1 below, this restricts to an equivalence CExt Γ (B) MExt E (B) X Split(F)Gal Γ (E,p) Received April 1, Research partially supported by the Université catholique de Louvain, by the Centro de Matemática da Universidade de Coimbra (CMUC), unded 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 project PEst-C/MAT/UI0324/2013 and the grant number SFRH/BPD/98155/

2 2 MATHIEU DUCKERTS-ANTOINE AND TOMAS EVERAERT between the categoryo all monadiccentralextensionso B and thato those discrete ibrations G Gal Γ (E,p) whose components are not only in F, but are also split epimorphisms. In particular, i Γ is such that every monadic central extension is normal, the latter equivalence becomes NExt Γ (B) X Split(F)Gal Γ (E,p). (1) Examples o admissible Galois structures Γ or which every monadic central extension is normal, are given by any Birkho subcategory (= a relective subcategory closed under subobjects and regular quotients) X o an exact Mal tsev category C, or E and F the classes o regular epimorphisms in C and X, respectively (see [8]). Hence, in this case, the equivalence (1) holds or every weakly universal monadic extension p: E B. The observationwe wish to make here is that there is a much largerclass o Galois structures Γ or which the equivalence (1) holds or every weakly universal monadic extension, and that such a Γ need neither be admissible nor satisy the condition that every monadic central extension is normal, in general. Among such Galois structures, there is every Γ = (C,X,I,H,η,ǫ,E,F) such that C is an additive category, X is an arbitrary ull relective subcategory o C, and E and F are the classes o all morphisms in C and X, respectively; more generally, C is a pointed protomodular category, X is a relective subcategory o C with a protoadditive [3] relector I, and E and F are the classes o all morphisms in C and X, respectively; C is an exact Mal tsev category, X is a Birkho subcategory o C, and E and F are the classes o regular epimorphisms in C and X, respectively this is the case mentioned above. In each o these cases, the ollowing two conditions are satisied, and we will show that under these two assumptions the equivalence (1) is always valid the let-adjoint unctor I: C X preserves those pullback-squares D A C g B

3 A CLASSIFICATION THEOREM FOR NORMAL EXTENSIONS 3 or which is a split epimorphism and and g are in E. the induced Galois structure Γ Split = (C,X,I,H,η,ǫ,Split(E),Split(F)) is admissible, where the classes Split(E) and Split(F) consist o those morphisms in E and F, respectively, that are also split epimorphisms. In act, in each o these cases the equivalence (1) not only holds or every weakly universal monadic extension, but or any weakly universal normal extensionp: E B as well. The existence, or every B, o such a p is related to that o a let adjoint to the inclusion unctor NExt Γ (C) Ext E (C) o the category o normal extensions into that o extensions, and we conclude the article with a closer look at this let adjoint. In particular, we explain how it can be viewed as a Kan extension o the trivialisation unctor, and we give a criterion or its existence based on this idea. 2. A characterisation o normal extensions Recall that a Galois structure [6, 7] Γ = (C,X,I,H,η,ǫ,E,F) consists o an adjunction C I X H with unit and counit η: 1 C HI and ǫ: IH 1 X, and two classes E and F o morphisms o C and X, respectively. E and F are required to be closed under pullback and composition, and to contain all isomorphisms, and one asks that I(E) F and H(F) E. Throughout, we shallcall the morphisms: A B in the class E extensions (o B) and write (C E B) and (X F Y) or the ull subcategories o the comma categories (C B) and (X Y) determined by E and F, respectively (or B C and Y X ). With respect to Γ, an extension : A B is said to be

4 4 MATHIEU DUCKERTS-ANTOINE AND TOMAS EVERAERT trivial i the naturality square A ηa HI(A) HI() B ηb HI(B) is a pullback; monadic i the change-o-base unctor : (C E B) (C E A) is monadic; central (or a covering) i it is locally trivial: there exists a monadic extension p: E B such that p () is a trivial extension; in this case one says that is split by p; normal i it is a monadic extension and i it is split by itsel, i.e. () is trivial. We denote by TExt Γ (C), MExt E (C), CExt Γ (C) and NExt Γ (C) the ull subcategories o Ext E (C) given by the trivial-, the monadic-, the central-, and the normal extensions, respectively, and by TExt Γ (B), etc., the corresponding ull subcategories o the comma category (C B) (or B C). For a given monadic extension p: E B, the ull subcategory o (C E B) whose objects are split by p will be denoted Spl Γ (E,p). By deinition, an extension is central when it is locally trivial. As it turns out, it is monadic precisely when it is locally split epic (but this would not make sense as a deinition, o course!). Lemma 2.1. An extension : A B is monadic i and only i there exists a monadic extension p: E B such that p () is a split epimorphism. Proo: For the only i part, it suices to take p =. The other implication ollows easily by applying Beck s monadicity theorem (see, or instance, [13, Theorem 2.4]). In the present article, we are particularly interested in those Galois structures Γ = (C,X,I,H,η,ǫ,E,F) or which the let-adjoint unctor I: C X preserves all pullback-squares D A C g B (2)

5 A CLASSIFICATION THEOREM FOR NORMAL EXTENSIONS 5 or which is a split epimorphism and and g are in E. For such a Γ, we have that a normal extension is the same as a morphism which is locally a split epic trivial extension. To see this, irst o all notice that a simple pullback-cancellation/composition argument yields Lemma 2.2. I I: C X preserves those pullbacks (2) or which is a split epimorphism and and g are in E, then trivial extensions which are also split epimorphisms are stable under pullback. Next, recall (or instance, rom [10, Proposition 1.6]) the ollowing Lemma 2.3. Consider a commutative diagram A a B b C A B C with a,b,c E, and assume that : (C E B ) (C E A ) relects isomorphisms. The right-hand square is a pullback as soon as both the let-hand square and the outer rectangle are pullbacks. We are now in a position to prove Proposition 2.4. Assume that I: C X preserves those pullbacks (2) or which is a split epimorphism and and g are in E. Then, or any : A B in E, the ollowing are equivalent (1) is a normal extension; (2) there exists a monadic extension p: E B such that p () is both a trivial extension (i.e. Spl Γ (E,p)) and a split epimorphism. Proo: To see that 1 implies 2, it suices to take p =. Conversely, let and p be as in 2, and consider the commutative diagram c E B A B A η A B A A B A HI(A B A) p 1 E B A p A p 1 η A HI(p 1 ) HI(A) E p B

6 6 MATHIEU DUCKERTS-ANTOINE AND TOMAS EVERAERT where the two squares on the let are pullbacks, and where p 1 and p 1 are kernel pair projections o and = p (), respectively. We must prove that the remaining square is a pullback as well. By Lemma 2.3, it will suice i we show the upper rectangle to be a pullback: indeed, since p is a monadic extension, p : (C E B) (C E E) relects isomorphisms, and this implies that the same must be true or p : (C E A) (C E (E B A)). To see that the upper rectangle is indeed a pullback, note that it coincides with the outer rectangle o the commutative diagram E B A B A HI(E B A B A) HI(A B A) p 1 E B A ηe B A HI( p 1 ) HI(E B A) HI( p) HI(p 1 ) HI(A). Here, the right-hand square is the image under HI o the let-hand upper square in the previous diagram, which is a pullback preserved by I, hence by HI; the let-hand square is induced by the unit η and is a pullback, since p 1 is a trivial extension by Lemma 2.2. Note, inally, that is a monadic extension by Lemma 2.1. I we write SSpl Γ (E,p) or the ull subcategory o Spl Γ (E,p) consisting o those (A,) Spl Γ (E,p) or which p () is a split epimorphism, then Proposition 2.4 may be expressed as an equality NExt Γ (B) = p SSpl(E, p) where p runs through all monadic extensions p: E B o B. I there is a single p: E B with SSpl(E,p ) SSpl(E,p) or every monadic extension p : E B o B, this equality moreover simpliies to NExt Γ (B) = SSpl(E,p). Such a p oten exists (assuming we are in the situation o Proposition 2.4): since split epic trivial extensions are stable under pullback (by Lemma 2.2), examples are given by any p: E B which actors through every normal extension o B. For instance, p could be a weakly universal monadic extension o B (= a weakly initial object o MExt E (B)) or a weakly universal normal extension (= a weakly initial object o NExt Γ (B)).

7 A CLASSIFICATION THEOREM FOR NORMAL EXTENSIONS 7 3. The classiication theorem Recall that an internal groupoid G in a category C is a diagram o the orm σ p 1 G 2 m G 1 p 2 d e c (3) G 0 with p 2 G 2 G 1 p 1 G 1 c d (4) G 0 a pullback and such that de = 1 = ce, dm = dp 1, cm = cp 2, m(1,ec) = 1 = m(ed,1), m(1 m) = m(m 1), dσ = c, cσ = d, m(1,σ) = ed, and m(σ,1) = ec. An internal unctor : G G between groupoids G and G in C is a triple ( 0 : G 0 G 0, 1 : G 1 G 1, 2 : G 2 G 2) o morphisms such that the evident squares in the diagram G 2 p 1 m p 2 G 1 2 p 1 G 2 m G 1 p 2 1 d e c d e c G 0 G 0 0 commute (rom which it ollows immediately that also σ 1 = 1σ). is a discrete ibration when those commutative squares are moreover pullbacks. (Note that it suices or this that the square 0 c = c 1 is a pullback.) The category o groupoids and unctors in C will be denoted by Gpd(C) and, or a ixed groupoid G, the ull subcategory o the comma category (Gpd(C) G) given by the discrete ibrations G G, by C G. When E is a class o morphisms in C, then C EG will denote the ull subcategory o C G o those ( 0, 1, 2 ): G G or which 0, 1 and 2 are in E. Any internal equivalence relation is a groupoid, which means in particular that every morphism p: E B determines, via its kernel pair (π p 1,πp 2 ), an

8 8 MATHIEU DUCKERTS-ANTOINE AND TOMAS EVERAERT internal groupoid Eq(p), as in the diagram Eq(p) E Eq(p) p p 1 τ σ Eq(p) π p 1 δ E. p p 2 π p 2 Notice, or any discrete ibration o groupoids : G G, that G is an equivalence relation as soon as G is. Letus,romnowon,consideraGaloisstructureΓ = (C,X,I,H,η,ǫ,E,F) such that the let-adjoint unctor I: C X preserves those pullback-squares (2) or which is a split epimorphism and and g are in E. I p: E B is an extension, then so are its kernel pair projectionsπ p 1 and pp 2, hence I preserves the pullback (4) or G = Eq(p). Consequently, I(Eq(p)) is again a groupoid, in X, called the Galois groupoid o p. We denote it Gal Γ (E,p). What we wish to prove now is that there is, under the additional condition 2. below, or every monadic extension p: E B an equivalence SSpl Γ (E,p) X Split(F)Gal Γ (E,p) between the category o extensions (A,) o B or which p () is a split epic trivial extension, and the category o discrete ibrations ( 0, 1, 2 ): G Gal Γ (E,p) in X whose components 0, 1 and 2 are split epimorphisms and are in F. When p is such that SSpl(E,p) = NExt Γ (B) (or instance, i p is a weakly universal monadic extension, or a weakly universal normal extension see the end o the previous section) we then obtain NExt Γ (B) X Split(F)Gal Γ (E,p) Fix an extension p: E B. By sending any extension (A,) o B to the discrete ibration induced by the right-hand pullback square in, and displayed as the let-hand side o, the diagram (5) Eq( p) P Eq( p) p p 1 Eq( p) π p 1 P p= (p) A Eq(p) E Eq(p) p p 2 p p 1 Eq(p) π p 2 π p 1 E p () p B, p p 2 π p 2

9 A CLASSIFICATION THEOREM FOR NORMAL EXTENSIONS 9 we obtain a unctor K p : (C E B) C EEq(p). It turns out (see, or instance, [11, 12]) that C EEq(p) is equivalent to the category (C E E) Tp o (Eilenberg-Moore) algebras or the monad T p = p Σ p, and that K p : (C E B) C EEq(p) corresponds, via this equivalence, to the comparison unctor K Tp : (C E B) (C E E) Tp, whence Lemma 3.1. [11, 12] An extension p: E B is monadic i and only i the unctor K p : (C E B) C EEq(p) is an equivalence o categories. K p restricts to an equivalence SSpl Γ (E,p) C TExt ΓSplit (C)Eq(p) (6) between the category o extensions (A,) o B or which p () is a split epic trivial extension, and the category o discrete ibrations ( 0, 1, 2 ): G Eq(p) in C whose components 0, 1 and 2 are split epic trivial extensions (= Γ Split -trivial extensions see the introduction, or below, or the notation Γ Split ), and we are already halway to proving (5). In order to ind an equivalence C TExt ΓSplit (C)Eq(p) X Split(F) Gal Γ (E,p), (7) irst o all notice that, since I(E) F, the relector I: C X extends, or any B C, to a unctor I B : (C E B) (X F I(B)) in an obvious way. BecauseH(F) E andsincee isstableunderpullback,i B hasarightadjoint H B : (X F I(B)) (C E B), which sends an (X,ϕ) (X F I(B)) to the extension (A,) (C E B) deined via the pullback A H(X) H(ϕ) B ηb HI(B). This gives us, or every B in C, an adjunction (C E B) which restricts to an equivalence I B (X F I(B)) H B (8) TExt Γ (B) (X F I(B))

10 10 MATHIEU DUCKERTS-ANTOINE AND TOMAS EVERAERT whenever H B is ully aithul a situation which is o interest: Deinition 3.2. [7] A Galois structure Γ is called admissible when each unctor H B : (X F I(B)) (C E B) is ully aithul. Now since I: C X preserves those pullback-squares (2) or which is a split epimorphism and and g are in E, the adjunction (8) induces an adjunction C EG X FI(G) or every groupoid G (as in (3)) in C with d and c (hence, also p 1, m and p 2 ) in E, and this, in its turn, would restrict to an equivalence C TExt Γ (C)G X FI(G). i Γ were admissible. However, instead o requiring this or Γ, we only ask that the induced Galois structure Γ Split = (C,X,I,H,η,ǫ,Split(E),Split(F)) is admissible, where the classes Split(E) and Split(F) consist o those morphisms in E and F, respectively, that are also split epimorphisms. In this case, we instead obtain an equivalence C TExt ΓSplit (C)G X Split(F) I(G) or every groupoid G in C with d and c in E. In particular, i G = Eq(p) or some monadic extension p: E B, we ind the sought-ater (7). Combining (6) and (7), we obtain: Theorem 3.3. Assume that Γ = (C,X,I,H,η,ǫ,E,F)is a Galois structure such that (1) the let adjoint I: C X preserves those pullbacks (2) or which is a split epimorphism and and g are in E. (2) the induced Galois structure is admissible. Γ Split = (C,X,I,H,η,ǫ,Split(E),Split(F))

11 A CLASSIFICATION THEOREM FOR NORMAL EXTENSIONS 11 Then, or any monadic extension p: E B, there is an equivalence o categories SSpl Γ (E,p) X Split(F)Gal Γ (E,p). Hence, i p is such that SSpl(E,p) = NExt Γ (B) (or instance, i it is a weakly universal monadic extension, or a weakly universal normal extension), there is a category equivalence NExt Γ (B) X Split(F)Gal Γ (E,p). Weakly universal monadic extensions oten exist: or instance, i C is a Barr exact category [1] with enough (regular) projectives, and E is either the class o regular epimorphisms or the class o all morphisms (in either case the monadic extensions are precisely the regular epimorphisms), then clearly every B admits a weakly universal monadic extension. It turns out that the existence o a weakly universal monadic extension p: E B at once implies that o a weakly universal normal extension o B, i we are in the situation o Theorem 3.3. Indeed, we have Proposition 3.4. I Γ = (C,X,I,H,η,ǫ,E,F) satisies the assumptions o Theorem 3.3, and i there exists, or a given object B o C, a weakly universal monadic extension p: E B, then the inclusion unctor NExt Γ (B) MExt E (B) admits a let adjoint. I there is such a p or every B, and i monadic extensions are stable under pullback, then also the inclusion unctor NExt Γ (C) MExt E (C) has a let adjoint. Proo: Since Γ Split is admissible, or every B in C the adjunction (8) induces a relection (C Split(E) B) TExt ΓSplit (B). BecauseI: C X preservesthosepullback-squares(2)orwhich isasplit epimorphism and and g are in E, these relections, in their turn, induce a relection C Split(E)G C TExt ΓSplit (C)G or every groupoid G (as in (3)) in C, with d and c (hence, also p 1, m and p 2 ) in E. In particular, i G = Eq(p) or some extension p: E B, we have a relection C Split(E)Eq(p) C TExt ΓSplit (C)Eq(p). (9)

12 12 MATHIEU DUCKERTS-ANTOINE AND TOMAS EVERAERT To prove our irst claim, it suices now to observe that the inclusion unctor in (9) coincides, up to equivalence, with the inclusion unctor NExt Γ (B) MExt E (B) whenever p is a weakly universal monadic extension: indeed, in this case, an extension : A B is monadic i and only i p () is a split epimorphism (by Lemma 2.1), and is a normal extension i and only i p () is, moreover, a trivial extension (by Proposition 2.4 and Lemma 2.2). Thus, the equivalence K p : (C E B) C EEq(p) restricts to equivalences MExt E (C) C Split(E)Eq(p) and NExt Γ (B) C TExt ΓSplit (C)Eq(p), and the inclusion unctor in (9) to the inclusion unctor NExt Γ (B) MExt E (B). The second claim ollows rom the irst by Proposition 5.8 in [5], since the stability under pullback o monadic extensions implies that o normal extensions, by Lemma 2.2. Notice that, whenever p: E B is a weakly universal monadic extension, its normalisation (=its relection in NExt Γ (B)) must be weakly universal too. The relector into NExt Γ (C) is our object o study in the next section. Here, we want to add that i we drop the assumption that a weakly universal monadic extension p: E B exists or every B, but instead require every extension to be monadic, we still have a relector MExt E (C) = Ext E (C) NExt Γ (C). Beore proving this, we note that Lemma 2.2 remains valid in this situation (see, or instance, [9, Proposition 2.1]). Lemma 3.5. I Γ Split is admissible, then trivial extensions which are also split epimorphisms are stable under pullback. Consequently, i a pullback o a normal extension is monadic, it is a normal extension as well. Proposition 3.6. Assume that Γ = (C,X,I,H,η,ǫ,E,F) satisies the assumptions o Theorem 3.3 and that every extension is monadic. The inclusion unctor NExt Γ (C) Ext E (C) admits a let adjoint. Proo: Let : A B be an extension. As in the proo o Proposition 3.4, we have a relection C Split(E)Eq() C TExt ΓSplit (C)Eq() since Γ Split is admissible, and because I: C X preserves those pullbacksquares (2) or which is a split epimorphism and and g are in E. Furthermore, the equivalence K : (C E B) C EEq() induces an equivalence ( (C E B)) (K () (C EEq() )) = (K () (C Split(E)Eq() ))

13 A CLASSIFICATION THEOREM FOR NORMAL EXTENSIONS 13 which, by Proposition 2.4 and Lemma 2.2, restricts to an equivalence ( NExt Γ (B)) (K () C TExt ΓSplit (C)Eq() ). Since K (), as an object o C Split(E)Eq(), has a relection in C TExt ΓSplit (C)Eq(), both categories (K () C TExt ΓSplit (C)Eq() ) and ( NExtΓ (B)) have an initial object. Consequently, has a relection in NExt Γ (B). Finally, since normal extensions are stable under pullback by Lemma 3.5, we can apply Proposition 5.8 in [5] and conclude that the inclusion NExt Γ (C) Ext E (C) has a let adjoint. In concluding this section, let us return to what we wrote in the introduction. The examples o Galois structures given there do indeed satisy conditions 1 and 2 o Theorem 3.3: the ormer is well known to hold in the case o an additive I: C X (between additive categories C and X ), and remains valid or a protoadditive I: C X (between pointed protomodular C and X ), by Proposition 2.2 in [4]. By Proposition 3 and Example 1 in [2], it also holds i I: C X is the relector into a Birkho subcategory X o an exact Mal tsev category C. Moreover, condition 1 implies condition 2, in each o these cases: when the adjunction I H is a relection, the admissibility o Γ Split can equivalently be described as the preservation by I: C X o every pullback (2) or which is in Split(E) and g = η C : C HI(C) is a relection unit (see Proposition 2.1 in [9]). when, moreover, η B : B HI(B) is an extension or every B, we thus have that the irst condition o Theorem 3.3 implies the second. 4. The normalisation unctor as a Kan extension Let Γ = (C,X,I,H,η,ǫ,E,F)be a Galoisstructure such that the induced Galois structure Γ Split = (C,X,I,H,η,ǫ,Split(E),Split(F)) is admissible. For every B C, I H induces an adjunction (C Split(E) B) (X Split(F) I(B)) which, by admissibility, decomposes into a relection ollowed by an equivalence (C Split(E) B) TExt ΓSplit (B) (X Split(F) I(B)).

14 14 MATHIEU DUCKERTS-ANTOINE AND TOMAS EVERAERT Inparticular,wehavethattheinclusionunctorTExt ΓSplit (B) Ext Split(E) (B) admits a let adjoint, or every B C. Since, by admissibility o Γ Split, split epic trivial extensions are stable under pullback (Lemma 3.5), we can apply Proposition 5.8 in [5] and conclude that also the inclusion unctor TExt ΓSplit (C) Ext Split(E) (C), which we denote by H 1, has a let adjoint. We call it T 1, and we write η 1 or the unit o the adjunction T 1 H 1. When also the inclusion unctor H 1 : NExt Γ (C) Ext E (C) has a let adjoint I 1 (as, or instance, in Propositions 3.4 and 3.6), we obtain a square o unctors Ext Split(E) (C) T 1 TExt ΓSplit (C) K K Ext E (C) I 1 NExt Γ (C) (10) in which K and K are the inclusion unctors. This square commutes, up to natural isomorphism. Indeed, irst o all, we have, or any split epic extension p: E B, that its relection T 1 () in TExt ΓSplit (B) is also its relection in NExt Γ (B), since (p NExt Γ (B)) = (p TExt ΓSplit (B)) by the right-cancellation property o split epimorphisms and by Lemma 4.1. I Γ Split is admissible, then a split epic extension is normal i and only i it is trivial. Proo: Every split epic extension is monadic (see [12]) and, by Lemma 3.5, split epic trivial extensions are stable under pullback, which implies they are normal. For the converse, it suices to consider, or any split epic normal extension : A B, with section s: B A, the diagram A s,1 B s Eq() π 2 π 1 A A B in which each square is a pullback, and to use, again, Lemma 3.5. In order to conclude rom this that T 1 () is also the relection in NExt Γ (C) o, or a split epic trivial extension, we would like to apply, once more,

15 A CLASSIFICATION THEOREM FOR NORMAL EXTENSIONS 15 Proposition 5.8 in [5]. While we have no reason to assume that arbitrary normal extensions are stable under pullback, the pullback stability given in Lemma 3.5 is easily seen to suice, here. Whence Lemma 4.2. Assume that Γ Split is admissible, and let p: A B be in Ext Split(E) (C). The relection T 1 (p) o p in the category TExt ΓSplit (C) is also its relection in NExt Γ (C) (irrespective o the existence o I 1 ). In particular, the square(10) indeed commutes, up to natural isomorphism, whenever I 1 exists. What we want to prove now is that under the additional condition that every extension is a regular epimorphism, the normalisationunctor I 1, when it exists, coincides with the pointwise let Kan extension o K T 1 along K. We shall irst show that the unctor K is dense, i.e. the unctor (K ) P Ext Split(E) (C) K ExtE (C) where P is the obvious orgetul unctor admits as colimit. For this, let us consider the ull subcategory J o (K ) determined by p 1 r π q 1 π q 2 where q = (π 2,), πq 1 = (τ,π 1 ), πq 2 = (p 2,π 2 ) and r = q πq 1 = q πq 2. One can easily prove that the inclusion unctor L : J (K ) is inal, i.e. or any object P = (p,(p 1,p 0 ): p ) in (K ), the category (P L ) is non-empty and connected. Let us prove here that (P L ) is connected. Any three objects q π 1 p (p (p 1,p 0 ) 1,p 0 ),(p 1,p 0 ) p 1 π (p r 1,p 1 0) (p 1,p 0 ) q

16 16 MATHIEU DUCKERTS-ANTOINE AND TOMAS EVERAERT in (P L ) are connected as shown in the commutative diagram p p p p (p 1,p 0 ),(p 1,p 0 ) (p 1,p 0 ) (p 1,p 0 ),(p 1,p 0 ) (p 1,p 0 ) p 1 π q 2 r π 1 q π q 1 r p 1 q π q 2 π 1. The last case to be considered can easily be deduced rom this. Now, rom the assumption that is a regular epimorphism (as is every morphism in E), we conclude that q = (π 2,): π 1 is the coequaliser o its kernel pair p 1 π q1 π q 2 π 1. This precisely means that the unctor J L (K ) P Ext Split(E) (C) K ExtE (C) has as colimit. Theorem 4.3. Assume that Γ = (C,X,I,H,η,ǫ,E,F)is a Galois structure such that (1) the induced Galois structure Γ Split = (C,X,I,H,η,ǫ,Split(E),Split(F)) is admissible. (2) every morphism in E is a regular epimorphism. Then the inclusion unctor H 1 : NExt Γ (C) Ext E (C) has a let adjoint I 1 i and only i the pointwise let Kan extension o K T1 along K exists and,

17 A CLASSIFICATION THEOREM FOR NORMAL EXTENSIONS 17 in this case, I 1 = Lan K ( K T 1 ). Ext E (C) K Ext Split(E) (C) = I 1 =Lan K ( K T 1 ) K T 1 NExt Γ (C). Proo: H 1 has a let adjoint i and only i or all in Ext E (C), ( H 1 ) has an initial object. We are going to show that one has an isomorphism ( H 1 ) = Cocone( K T 1 P ) or any in Ext E (C). This allows us to assert that H 1 has a let adjoint i and only i the pointwise let Kan extension Lan K ( K T 1 ) exists. Let be in Ext E (C) and λ = (λ p: p ) p ExtSplit(E) (C) be the cocone deined by the comma square (K ) P Ext Split(E) (C) λ 1 K Ext E (C). The density o K implies that one has an isomorphism ( Ext E (C)) Cocone(K P ) deined on an objet G = ((g 1,g 0 ): g,g) by λ G = (λ G p = (g 1,g 0 ) λ p: p g) p ExtSplit(E) (C). Now, when g is in NExt Γ (C), one can associate with λ G a cocone λ G = ( λ G p : KT1 (p) g) p ExtSplit(E) (C) on K T 1 P where λ G p is the (unique) actorisation o λ G p : p g through η 1 p: p T 1 (p) = KT 1 (p) (see Lemma 4.2). The assignment G λ G extends to the desired isomorphism. Now let us suppose that the normalisation unctor I 1 exists. Since K is dense, one has a let Kan extension Ext E (C) K 1 K Ext Split(E) (C) 1 ExtE (C) K Ext E (C)

18 18 MATHIEU DUCKERTS-ANTOINE AND TOMAS EVERAERT and it is preserved by I 1 (see [14]), that is I 1 is the let pointwise Kan extension o the unctor I 1 K = K T 1 along K: Ext E (C) K 1 K Ext Split(E) (C) 1 ExtE (C) K I 1 =I 1 1 ExtE (C) 1 I 1 1 K Ext E (C) = I 1 NExtΓ(C). K T 1 Corollary 4.4. I Γ = (C,X,I,H,η,ǫ,E,F) satisies the assumptions o Theorem 4.3 and i the coequaliser (let us write or its codomain) o T 1 (p 1 ) T 1 (τ,π 1 ) T 1 (p 2,π 2 ) T 1 (π 1 ) exists in NExt Γ (C) or every in Ext E (C), then the normalisation unctor I 1 exists and I 1 () =. Acknowledgements We are grateul to Tim Van der Linden or helpul comments on a preliminary version o the paper. Reerences [1] M. Barr, Exact categories, Exact categories and categories o sheaves, Lecture Notes in Math., vol. 236, Springer, 1971, pp [2] T. Everaert, Higher central extensions in Mal tsev categories, Appl. Categ. Structures 22 (2014), [3] T. Everaert and M. Gran, Homology o n-old groupoids, Theory Appl. Categ. 23 (2010), no. 2, [4], Protoadditive unctors, derived torsion theories and homology, J. Pure Appl. Algebra, in press, [5] G. B. Im and G. M. Kelly, On classes o morphisms closed under limits, J. Korean Math. Soc. 23 (1986), [6] G. Janelidze, Pure Galois theory in categories, J. Algebra 132 (1990), no. 2, [7], Categorical Galois theory: revision and some recent developments, Galois connections and applications, Math. Appl., vol. 565, Kluwer Acad. Publ., 2004, pp [8] G. Janelidze and G. M. Kelly, Galois theory and a general notion o central extension, J. Pure Appl. Algebra 97 (1994), no. 2,

19 A CLASSIFICATION THEOREM FOR NORMAL EXTENSIONS 19 [9], The relectiveness o covering morphisms in algebra and geometry, Theory Appl. Categ. 3 (1997), no. 6, [10] G. Janelidze, M. Sobral, and W. Tholen, Beyond Barr exactness: Eective descent morphisms, in Pedicchio and Tholen [15], pp [11] G. Janelidze and W. Tholen, Facets o descent I, Appl. Categ. Structures 2 (1994), [12], Facets o descent II, Appl. Categ. Structures 5 (1997), [13] J. MacDonald and M. Sobral, Aspects o monads, in Pedicchio and Tholen [15], pp [14] S. Mac Lane, Categories or the working mathematician, second ed., Grad. Texts in Math., vol. 5, Springer, [15] M. C. Pedicchio and W. Tholen (eds.), Categorical oundations: Special topics in order, topology, algebra and shea theory, Encyclopedia o Math. Appl., vol. 97, Cambridge Univ. Press, Mathieu Duckerts-Antoine CMUC, Department o Mathematics, University o Coimbra, Coimbra, Portugal address: mathieud@mat.uc.pt Tomas Everaert Vrije Universiteit Brussel, Department o Mathematics, Pleinlaan 2, 1050 Brussel, Belgium address: tomas.everaert@vub.ac.be