Aspects of algebraic exponentiation

Asp ects of algebraic exp onen tiation Domini que Bourn and James R.A. Gray Abstract W e analyse some asp ects of the notion of algebr aic exp onentiation in tro- duced b y the second auth o r [16] and satisfied b y the category Gp of group s . W e sho w ho w this notion pro vides a new approac h to the categ orical-alg ebraic question of the cen tralization. W e explore, in the category Gp , the unusual unive rsal prop erties and c onstructions determined by this notion, and we sho w ho w it is the origin of v arious prop erties of this category . In tro duction In [16], the second author observ ed, b y means of v ery straigh tforward Kan ex - tension arguments , that, in the category Gp of groups, the c hange of base f unc tor with resp e ct to the fibration of p oin ts along an y g roup homomorphism h : X → Y : h ∗ : P t Y Gp → P t X Gp has a righ t adjoin t, rev ealing, for the category Gp , a prop ert y ha ving a certain analogy with the prop e rt y , for a category E , of being lo c al ly c artesian c l o se d , na me ly the prop ert y that the following change of ba s e functor: h ∗ : E / Y → E /X has a righ t adjoint for a n y map h . On the other hand, he show ed moreo v er that , in the alg e braic con text of unital categories C , the fact that the c hange of base functor along the terminal map: τ ∗ X : C = P t 1 C → P t X C has righ t adj oin t is related to the existence of some generalized notion of cen traliza- tion. No w, the prop ert y of lo cal cartesian closednes s is ve ry p o werful and w ell kno wn to b e shared, f or instance, b y any elemen tary top os. It is not worth insisting o n its s ignificance. W e s hall dev elop some aspects of this new concept of algebraic exp onen tiation. 1 In Section 1), w e shall mor e de eply analyse the parallelism with the cartesian closedness and we shall strictly elucidate the relationship with the classical no tion of cen tra liz er, in suc h a wa y tha t, whe n a unital c ate go ry C is r e gular, any change of b ase functor τ ∗ X , as ab ove, has a right adjoint if and only if an y sub obje ct has a c en t r alizer , rev ealing that, b ehind the notions of cen tre a nd ce n tralizer, there w a s an unexpected wider-ranging phenome non of functorial nature. In Section 2) w e shall sho w that, in the Mal’cev con text, algebraic exp onen tiation along split epimorphisms allo ws us to extend the ex istence of cen tralizers f rom sub ob jects to equiv alence relations; accordingly , when the category C is moreov er exact, we get a Sc hreier-Mac Lane extens ion theorem, according to [11]. In Section 3) w e shall in v estigate the stabilit y prop erties of a lgebraic exp onen tiatio n and in part ic ular w e shall ho w, in the efficien tly regular con text, the existenc e of a righ t adjoin t t o: h ∗ : P t Y C → P t X C can b e extended from split epimorphims t o regular epimorphisms. In Section 4), in the stricter con text of protomo dular categories, w e giv e a detailed description of some constructions determined b y the algebraic exp onen tiation of all morphisms, and in particular w e shall inv estigate t w o main consequences, namely str ong pr o- tomo dularity (whic h guar an tees, a m ong o ther things, that the comm utation of t w o equiv alence relations ( R , S ) is characterized b y the comm utat ion of their asso ciated normal sub ob jects ( I R , I S ) [9]) and p eri-ab elianness (whic h is strongly related to the cohomolo gy of gr oups [10]). These last p oin ts sho w us ho w some w ell iden t ifie d particular prop erties of t he categor y Gp of groups originate from this algebraic ex- p onen tiation prop ert y . In this same category Gp , we shall explore in detail the very un usual univ ersal prop erties and constructions inv olv ed in algebraic exp onen tia tion. On the other hand w e shall enlarge the lis t of examples (Section 2.1) to some categories of topolo gical mo dels, suc h as to pological groups and top ological rings, and to non- pointed categories suc h as the fibres Gr d X ab o ve the set X of the fibration ( ) 0 : Gr d → S et from group oids to sets whose fibre Gr d 1 ab o ve 1 is nothing but the category Gp itself. 1 Cartesian clo s edness and algebraic cartes i a n closedn ess 1.1 Slice c at egories and c a rtesian closedness Let E b e an y finitely comple te c ategory and Y an o b ject in E . An y ob ject f : X → Y in the slice category E / Y has a sp ecific presen tation as the domain of an equalizer 2 of a split pair in E / Y : X / / ( f , 1) / / f # # ● ● ● ● ● ● ● ● ● ● ● ● Y × X s 0 × X / / Y × ( f , 1) / / p Y   Y × ( Y × X ) p Y w w ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ o o Y (where the common splitting o f the parallel pair is t he map p 0 × X ) whic h actually comes from a mona d on the slice category E / Y . This presen tation can b e extended to the catego ry P t Y E of p oints ab o v e Y , namely to the split epimorphisms, in the follo wing w ay: X / / ( f , 1) / / f # # ● ● ● ● ● ● ● ● ● ● ● ● Y × X s 0 × X / / Y × ( f , 1) / / p Y   Y × ( Y × X ) p Y w w ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ o o Y s c c ● ● ● ● ● ● ● ● ● ● ● ● (1 ,s ) O O ( s 0 ,s ) 7 7 ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ W e shall need later on the follo wing collateral consequence: when the category E is pointed, w e get the k ernel of f , from the previous diagram, by the follo wing equalizer: K er f / / k f / / X ι X / / ( f , 1) / / Y × X o o No w consider the c hange of base along the terminal map τ Y : Y → 1: τ ∗ Y : E − → E / Y According to o ur initial r e mark and b ecause of the left exactness of right adjoints, the question of the existence of a right adjoint to τ ∗ Y is reduced to the existence of cofree structures for the pro jections p Y : Y × X → Y in E / Y ; and t his cof ree structure is nothing but the exp onen tial X Y . In other w ords: The functor τ ∗ Y : E → E / Y has a right adjoin t if a nd only if the functor Y × − : E → E has a right adjoint. The c ate gory E is c artesia n close d if and only if a ny functor τ ∗ Y has a right adjoin t. 1.2 Fibration of p oin ts a nd a lgebraic cartesian close dness In a n algebraic con text, no suc h exp onen tial do es exist in general, among other things b ecaus e of the existence of a zero o b ject in the main instances, and consequen tly no suc h rig h t adjoin t functor to τ ∗ Y . Ho w ev er w e hav e the p ossibilit y to consider the existence of a right adjoin t to the “ c hange of base” functor τ ∗ Y : P t 1 E → P t Y E , i.e, 3 here, “only” with resp ect to the p oin ts of E / Y and E ; and ev en more generally the existence of a righ t adjoint to the change of ba s e functor f ∗ : P t E ( Y ) → P t E ( X ) for a an y map f : X → Y . This idea w a s first in tro duced b y the second aut hor [16] who show ed these right adjoin ts to f ∗ do exist in the categor ie s Gp of groups and R - Li e of Lie R -algebras, for an y commm utative ring R [17]. The previous observ atio n ab o ve concerning the equalizer presen tation of an y split epimorphism applies no w for the functor: τ ∗ Y : P t 1 E − → P t Y E The ques tion of the exis tence of a righ t adjoin t to τ ∗ Y is then red uced to the existence of cofree s tructures for the split epimorphisms ( p Y , (1 , u )) : Y × X → Y in P t Y E where u : Y ֌ X is can b e c hosen to b e a monomorphism. W e shall work now more s p ecifically in the algebraic con text of a unital [4], or ev en we alky unital cat e gory [22] C . Re call: Definition 1.1. A c ate gory C is unital (r esp. we akly unital) wh e n it is p ointe d, is finitety c omplete, and is such that any p air of maps of the fol lowin g form: X (1 X , 0) ֌ X × Y (0 , 1 Y ) ֋ Y is join tly str ongly epic (r esp. jointly epic). Accordingly a finitely complete p oin ted category C is unital if and only if the suprem um of the tw o previous sub ob jects is 1 X × Y . In these c on texts, the func- tor τ ∗ Y b ecomes f ully faithful. Recall also that there is then an in trinsic notion of comm uta tion fo r an y pair of maps with s ame co domain. W e sa y that a pair ( f , g ) comm utes: X / / (1 , 0) / / f % % ❏ ❏ ❏ ❏ ❏ ❏ ❏ X × Y φ   Y o o (0 , 1) o o g z z t t t t t t t Z when there ex its a factorization φ (called the c o op er ator of this pair), the uniqueness of φ making this existence a prop ert y of the pair ( f , g ). In these algebraic contexts , the meaning of the existence of a righ t adjoin t t o the functor τ ∗ Y ab o ve can b e made m uc h more algebraically civilized: Prop osition 1.2. Supp ose C is a unital (r esp. we akly unital) c ate gory. The functor τ ∗ Y : P t 1 E − → P t Y E admits a right adjoi n t Φ Y if and only if any sub obje ct u : Y ֌ X with domain Y admits a universal m ap ζ u : Z [ u ] → X c ommuting with it. This universal m ap ζ u is ne c essarily a mono m orphism. 4 Pr o of. The univ ersal prop ert y of ζ u translates exactly the univ ersal prop ert y of the cofree structure of the split epimorphism ( p Y , (1 , u )) : Y × X ⇄ Y with resp ect to τ ∗ Y . Indeed, the natural transformation ε : τ ∗ Y . Φ Y ⇒ I d is pro duced b y a map in P t Y C : Y × Z [ u ] ( p Y ,φ ) / / p Y % % ❑ ❑ ❑ ❑ ❑ ❑ ❑ Y × X p Y { { ✈ ✈ ✈ ✈ ✈ ✈ ✈ Y ι Y e e ❑ ❑ ❑ ❑ ❑ ❑ ❑ (1 ,u ) ; ; ✈ ✈ ✈ ✈ ✈ ✈ ✈ whic h mak es φ : Y × Z [ u ] → X the c o o p er ator of the comm uting pair ( u, ζ u ), with ζ u = φ.ι Z [ u ] . Consider no w the k ernel equiv a le nce o f the map ζ u : R [ ζ u ] p 0 / / p 1 / / Z [ u ] ζ u / / X The map ζ u .p i comm utes with u b y means of the co opera tor φ. ( Y × p i ). Its fac- torization through ζ u b eing unique, w e get p 0 = p 1 ; and the map ζ u is a monomor- phism. No w, starting fro m an y split epimorphism ( f , s ), the cofree structure Φ Y [ f , s ] is the equalizer of the follow ing upper horizon tal parallel pair induced b y the pair ( s 0 × X , Y × ( f , 1)): Φ Y [ f , s ] / / / /   Z [ s ] / / / / ζ s   Z [(1 , s ) ] ζ (1 ,s )   K er f / / k f / / X ι X / / ( f , 1) / / Y × X Since the lo w er line is a ls o an equalizer and the maps ( ζ s , ζ (1 ,s ) ) are monomorphisms, the left-hand side square is a pullbac k and Φ Y [ f , s ] = K er f ∩ Z [ s ]. According to the previous prop osition and t he parallelism with cartesian closedne ss, we shall in tro duce the following: Definition 1.3. A c ate gory C with pr o ducts is said to b e algebr aic al ly c artesian close d (a.c.c.) when any functor τ ∗ Y : P t 1 E − → P t Y E has a righ t adjoin t . On the other hand, w e ha v e the quite classical: Definition 1.4. Supp o se C is a unital (r es p . we akly unital) c ate gory. The c e ntr al- izer of a sub obje ct u : Y ֌ X is the lar gest sub obje ct c ommuting with it, i.e. the universal m onomorphism c ommuting with u . So, when C is a unital c a t e gory which is algebr aic al ly c artesian close d, any sub- obje ct u h as a c entr alizer ζ u . When C is regular (as it is the case fo r an y v ariet y of univ ersal algebras) the con v erse is true: 5 Prop osition 1.5. Supp ose C is a r e gular unital c ate gory. Then it is algebr aic al ly c artesian close d i f and only if any sub o bje c t u : Y ֌ X h as a c entr alizer. Pr o of. Let h : T → X b e any map c omm uting with u . Consider the canonical decomp osition of h throug h a regular epimorphism T ρ ։ V ¯ h ֌ X . Then, since ρ is a regular epimorphism the pair ( u, ¯ h ) of sub ob jects do es c omm ute in C ; so ¯ h , and th us h , factorizes through the cen tralizer ζ u : Z [ u ] ֌ X of u whic h therefore becomes also the univ ersal map comm uting with u . The main conseque nce of the a lgebraic cartesian closedness is a sp e cific commu- tation of limits: the functor τ ∗ Y , hav ing a right adjoin t, preserv es the colimits which exist in C . In particular, when a direct sum exits in C : U ι U → U + V ι V ← V the follo wing square is a pushout in C / Y and th us in C : Y × U Y × ι U / / Y × ( U + V ) Y (1 Y , 0) O O (1 Y , 0) / / Y × V Y × ι V O O 1.3 Examples The unital category M on of monoids is unital and, as a v ariet y of algebras, is exact and therefore regular. It has cen tralizers and th us is algebraically cartesian closed. More generally an y unital v ariety of algebras with cen tralizers (as the categories Gp of groups or C Rg of comm utativ e rings) is algebraically cartes ian closed. In the category Gp of groups an y split epimorphism ( f , s ) ab o v e Y is of the kind Y ⋉ ψ K ⇄ Y where ψ is the asso ciated action of the g roup Y on the group K . Then Φ Y [ f , s ] is nothing but the subgroup { k ∈ K/ ∀ y ∈ Y , y k = k } of K of the inv aria n t elemen ts under the action ψ . 1.4 Reflection of comm u ting pairs There is, a t the leve l of unital categories, a v ery simple res ult whic h will b e o f consequenc e later on. Prop osition 1.6. L et C b e a unital (r es p. we akly unital) c ate gory and (Γ , ǫ, ν ) a left exact c omo nad on it. Then the c ate gory C oal g Γ of Γ -c o algebr as is unital (r esp. we akly unital) and the left exact for getful functor U : C oal g Γ → C r efle cts the c ommuting p airs. Pr o of. Since the comonad (Γ , ǫ, ν ) is left exact and C is finitely complete, so is the category C oal g Γ, whic h is moro ve r p oin ted since so is C . The forgetful func tor 6 U : C oalg Γ → C is left exact. The category C oal g Γ is unital (resp. we akly unital) since the functor U is conserv ativ e (resp. f aithful). No w, suppo s e that w e ha v e a pair of morphisms in C oalg Γ: ( X , ξ ) f − → ( Z , ζ ) g ← − ( Y , υ ) whose image b y U is endo w ed with a co op erator φ . W e hav e to sho w tha t this map φ is actually a map o f coa lgebras, namely that the following quadrangle comm utes: X / / (1 , 0) / / X × Y φ   Γ ξ × Γ υ ' ' P P P P P P P Y o o (0 , 1) o o Z ζ ' ' P P P P P P P P P P Γ X × Γ Y Γ φ   Γ Z whic h can b e done b y comp osition with the t wo upp er horizontal maps. 1.5 Strongly unita l categorie s A unital category C is strongly unital [8], when in a dd ition, for an y ob ject Y , the c ha nge of base f unc tor τ ∗ Y : C → P t Y C is satur ate d on sub obje cts , na me ly suc h that a n y subo b ject R ֌ τ ∗ Y ( Z ) is, up to isomorphism, the image by τ ∗ Y of some sub ob ject S ֌ Z . In this con t e xt, the idemp oten t comonad asso ciated with the algebraic cartesian closedness has a sp e cific prop ert y: Prop osition 1.7. Supp ose C is a str ongly unital c ate gory. Supp os e the f unc tor τ ∗ Y : P t 1 C − → P t Y C admits a right adjoint Φ Y . Then the natur al tr ansform ation of the induc e d idemp otent left exact c omon a d ε Y : τ ∗ Y . Φ Y ⇒ I d is m o nomorphic. Mor e over any s ub obje ct j : • ֌ • in P t Y C pr o duc es a pul lb ack in P t Y C : τ ∗ Y . Φ Y ( • ) / / τ ∗ Y . Φ Y ( j ) / / ε Y •   τ ∗ Y . Φ Y ( • ) ε Y •   • / / j / / • Pr o of. The comonad is idempoten t ev en if C is only unital since the functor τ ∗ Y is fully faithful. F rom the construction of Φ Y [ f , s ], it is sufficien t to prov e the assertion for Z [ u ] = Φ Y [ p Y , (1 , u )]. W e observ ed that the natural map ε Y ( p Y , (1 , u )) is nothing but the map Y × Z [ u ] ( p Y ,φ ) → Y × X , where the map φ is the co op erator o f u and ζ u . 7 No w consider the follo wing diagram in C : Y ι Y / / (1 ,u )   ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ Y × Z [ u ] ( p Y ,φ )   Z [ u ] ι Z [ u ] o o (0 ,ζ u ) ~ ~ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ Y × X p Y ^ ^ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ Then, according t o Lemm a 1.8.1 8 in [4] the v ertical cen tral map is a monomorphism if and only if so is the map (0 , ζ u ), whic h is the case here. Th e end of the pro of is a consequenc e of t he following ve ry general lemma: Lemma 1.8. L et U : E → F b e a left exact ful ly faithful functor b etwe en finitely c omplete c ate gories. Supp ose mor e over that U has a right adjoint G such that the natur al tr ans formation of the induc e d idemp o t ent le f t exact c omo nad ε : U.G ⇒ I d is monomorphic. Then the functor U is satur ate d on sub obje cts if and only if, given any sub obje ct j : X ′ ֌ X in F , the fol lowing squar e is a pul lb ac k in F : U.G ( X ′ ) / / U.G ( j ) / / ε X ′   U.G ( X ) ε X   X ′ / / j / / X Pr o of. Suppose the previous condition is satisfied. Giv en any sub ob ject j , when ε X is an isomorphism, so is ε X ′ , a nd U is saturated on subob jects. Con v ersely supp ose U saturated on subo b jects, a nd consider the pullbac k of ε X along j and t he induce d monomorphic factorization η : U.G ( X ′ ) * * U.G ( j ) * * ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ε X ′   ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ $ $ η $ $ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ P / / ¯ j / / ǫ   U.G ( X ) ε X   X ′ / / j / / X Since U is saturated o n subob jects w e can choose an ob ject P = U ( T ). Accordingly there is a map ¯ η : U ( T ) = P → U.G ( X ′ ) suc h that ε X ′ . ¯ η = ǫ . Since ε X ′ and ǫ are monomorphisms, this ¯ η is the in v erse of η . Therefore t he square in question is a pullbac k. 8 2 Mal’cev context W e shall work now in the algebraic con text of Mal’cev catego ries in the sense of [13] and [14]. One w a y of saying that a category C is a Mal’cev categor y is to say that an y p oin ted fibre P t Y C is unita l or, equiv alen tly that an y p oin ted fibre P t Y C is strongly unital, see [8]. Let us introduce the following: Definition 2.1. A finitely c om plete c ate gory C is said to b e fib erwise algeb r aic al ly c artesian close d (f.a.c .c.) when every fibr e P t Y C is alge b r aic al ly c artesian close d. A morphism h : X → Y in C is said to b e algebr aic al ly exp onentiable when the change of b ase f unc tor h ∗ : P t Y C → P t X C along h admits a ri g ht adjoint. The c ate gory C is said to b e lo c al ly algebr aic al ly c a rtesia n clo s e d (l.a.c.c) when any morph i s m h is algebr aic al ly exp onentiable. So, a categor y C is fiberwise alg eb raically cartesian closed if and only if, giv en an y split e pimorphism ( f , s ) : X ⇄ Y the c hange of base functor f ∗ : P t Y C → P t X E has a r igh t adjoin t Φ f . When the categor y C is a regular Malcev catego ry , it is equiv alen t, according to Prop osition 1.5, to saying that in any fibr e P t Y C ther e exists c e ntr alizers of sub obje cts . W e get immediately: Prop osition 2.2. Supp ose C is an fib erwise algebr aic al ly c artesian close d Mal’c ev c ate gory. L et ( f , s ) : X ⇄ Y b e any split epimorphism i n C . The n the change of b ase functor f ∗ : P t Y C → P t X E is s u ch that the natur al tr ansf ormation ε f : f ∗ . Φ f ⇒ I d of the induc e d idemp otent left exact c omonad is m onomorphic. Mor e over any sub obje ct j : • ֌ • in P t X C pr o duc es a pul lb ack in P t X C : f ∗ . Φ f ( • ) / / f ∗ . Φ f ( j ) / / ε f •   f ∗ . Φ f ( • ) ε f •   • / / j / / • Pr o of. As w e recalled ab o ve, the catego ry C b eing Mal’cev, an y fibre P t Y C is not only unital but also strongly unital. Accordingly , just apply Prop osition 1.7. 2.1 Examples In [1 6 ] a nd[17], it was show n that: the catego ry C Rg of commutativ e rings is fib er- wise algebraically cartesian closed but not lo cally algebraically cartesian closed; the categories Gp of groups and R - Lie of Lie R -algebras, for any commm utativ e ring R , are lo cally algebraically car te sian closed; when a category E is a cartesian closed category with pullbac ks, the category Gp E of in ternal groups in E is lo cally alge- braically cartesian closed. On the other hand a category A w as defined as ess en tia lly affine [7] when a n y c hange of base functor h ∗ : P t Y A → P t X A is an equiv alence of 9 categories; accordingly an y essen tia lly affine category is lo cally algebraically carte- sian closed. In particular any additiv e category is locally algebraically cart e sian closed. Non-p oin ted examp les 1: slice and coslice categories Lemma 2.3. L et U : C → D b e a pul lb ack pr eserving functor. Supp ose mor e over that it is a discr ete fibr ation (r esp. disc r ete c ofibr ation). When f : X → Y is algebr aic al ly exp onentiable in C , so is U ( f ) in D . Pr o of. Straigh t forw ard. Let C b e an y category . Then, for an y ob ject Y in C , the domain functor C / Y → C is a discrete fibration whic h preserv es pullbacks . Still, for any ob ject Y in C , the co domain f unc tor Y / C → C is a left exact discrete cofibration. Ac- cordingly fib erwise algebraic cartesian c losedness (resp. lo cally algebraic cartesian closedness ) is stable unde r slicing and coslicing, g iv ing rise to non-p oin ted examples. As a consequence, when C is fiberwise algebraically cartesian close d (resp. lo cally algebraically cartesian closed), so is an y fibre P t Y C , whic h is the coslice category on the terminal ob ject of the slice category C / Y . Non-p oin ted examp les 2: the fib res of the fibration Gr d → S et Let us denote by Gr d the category of group oids and by ( ) 0 : Gr d → S et the forgetful functor associating with an y group oid Y 1 the set Y 0 of it s ob jects; it is a fibration whose cartesian maps in Gr d are t he fully faithful functors. The fibre ab o ve 1 is clearly the p oin ted catego ry Gp of groups. W e shall denote b y G rd X the fibre ab o v e t he set X : its ob jects a re the group oids whose set of o b jects is X and its arro ws are those functors b et w een suc h group oids wh ic h are bijectiv e o n ob jects . W e kno w that these fibres Gr d X are protomo dular [7] and th us Mal’cev catego ries , a nd they are no longer p oin t e d. The aim of this section is to show tha t any fibre Gr d X is lo cally algebraically cartesian clos ed; the pro of will be a sligh t gene ralization of the pro of for Gp . Lemma 2.4. L et b e gi v e n a gr oup oid Y 1 . The fibr e P t Y 1 ( Gr d Y 0 ) is in b i j e ction wi th the functor c ate gory F ( Y 1 , Gp ) . Supp ose F 1 : Y 1 → Z 1 b e any functor. Then the change of b ase functor F ∗ 1 : P t Z 1 ( Gr d Z 0 ) → P t Y 1 ( Gr d Y 0 ) is na t ur al ly isomorphic to the functor F ( F 1 , Gp ) : F ( Z 1 , Gp ) → F ( Y 1 , Gp ) . Pr o of. The category Gp can b e considered a s the full sub category of the category C at (of categories) whos e ob jects are the group oids with only one ob ject. The lemma is a s p ecific ation of the Grothendiec k construction. F r om an y functor H : Y 1 → Gp w e get a bijective on ob j e cts split cofibration H 1 : X 1 → Y 1 where a map y → y ′ in X 1 is a pair ( f , γ ) with f : y → y ′ is a map in Y 1 and γ ∈ H ( y ′ ). The composition is defined b y: ( f ′ , γ ′ ) . ( f , γ ) = ( f ′ .f , γ ′ .H ( f ′ )( γ )). The functor 10 H 1 , defined b y H 1 ( f , γ ) = f , has a splitting T 1 defined b y T 1 ( f ) = ( f , 1 H ( y ′ ) ). Con verse ly an y s plit bijectiv e on ob jects functor H 1 : X 1 → Y 1 is necessarily a split c ofibration and dete rmines a functor H : Y 1 → Gp . The end of the pro of is straigh tforw ard. Theorem 2.5. Consider the fibr ation ( ) 0 : Grd → S et ; an y of its fibr es Gr d X is lo c al ly algebr aic al ly c artesian close d. Pr o of. Giv en an y functor F 1 : Y 1 → Z 1 b et w een tw o gro upoids, the functor F ( F 1 , Gp ) : F ( Z 1 , Gp ) → F ( Y 1 , Gp ) admits a right a djoin t, giv en b y the right Kan extension along the functor F 1 . Then, according to the previous lemma, an y c ha nge of base functor F ∗ 1 : P t Z 1 ( Gr d Z 0 ) → P t Y 1 ( Gr d Y 0 ) has a rig h t adjoin t. The theorem holds b y taking F 1 a bijectiv e on ob jects functor with Z 0 = X = Y 0 . T op ologica l models In this section w e shall mak e explicit some top ological examples. Let T b e a Mal’cev theory , V ( T ) the corresponding v ar ie t y of T -algebras and T op ( T ) the category of top ological T -algebras. Recall that T op ( T ) is then a regular Mal’cev category , see[19], whose regular epimorphism s a re the op en surjectiv e maps and recall also the follo wing: Lemma 2.6. L et T b e a Mal’c ev the ory. Then the for getful le f t exact functor U : T op ( T ) → V ( T ) r efle cts the p u l lb a c k of split epimorp h isms along r e gular epi- morphisms. F rom this, w e get: Prop osition 2.7. L et T b e a Mal’c ev the ory such that V ( T ) i s fib erwise alge b r aic al ly c artesian close d. T hen t he c ate gory T o p ( T ) of t op olo gic al T -algebr as is fib erwise algebr aic al ly c artesian close d. Pr o of. Let ( f , s ) : X ⇄ Y b e a split epimorphism in T op ( T ) and ( g , t ) : V ⇄ X an ob ject in P t X T op ( T ). First conside r the f ollo wing diagram giv en b y the alg eb raic exp onen tiation in V ( T ): U ( f ) ∗ Φ U ( f ) [ U ( g ) , U ( t )] s s ε d 1 ( U ( g ) ,U ( t )) s s ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ φ / / / / x x ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ Φ U ( f ) [ U ( g ) , U ( t )] o o σ o o γ v v ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ U ( V ) U ( g )   U ( X ) U ( t ) O O U ( f ) / / / / β 8 8 ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ U ( Y ) o o U ( s ) o o τ 6 6 ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ 11 and then the follo wing one in T op ( T ) where V ′ is the algebra U ( f ) ∗ Φ U ( f ) [ U ( g ) , U ( t )] equipped with the top ology induced by the one on V : V ′ w w ¯ ε w w ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ¯ φ / / / /                     W o o ¯ σ o o ¯ γ   V g   X t O O f / / / / ¯ β ? ? Y o o s o o ¯ τ ? ? The map t in T op ( T ) and the factorization β in V ( T ) pro duce the f actorization ¯ β . Then put the quotien t top ology on Φ U ( f ) [ U ( g ) , U ( t )] to pro duce the ob ject W in T op ( T ). Then w e get the dott e d maps ab o v e the quadrangle pullbac k of our initial diagram. The previous lemma ass erts that it is a pullbac k. F rom t his situation, it is straigforward to chec k that the split epimorphism ( ¯ γ , ¯ τ ) has the desired univ ersal prop ert y with resp ect to the c hange of base functor f ∗ . Accordingly the categories T opGp and T opC Rg of top ological groups and to po- logical comm utative rings are fib erwis e a lgebraically cartesian closed. 2.2 Ab elian spli t extension A split epimorphism ( f , s ) : X ⇄ Y is said to be ab elian in a Mal’cev category C when it is an ab elian ob ject in the fibre P t Y C . Since an y righ t adjoint functor is left exact, an y algebraically exp onen tiable map h : Y → Y ′ is suc h that the restriction of Φ h : P t Y C → P t Y ′ C to the ab elian ob jects determines a functor: Φ h : AbP t Y C → AbP t Y ′ C In pa rticular, when C is p oin ted and fib erwis e algebraically cartesian closed, when ( f , s ) is ab elian, so is the ob ject Φ Y [ f , s ]. Recall that, when C is the category Gp of groups, a split epimorphism is ab elian if a nd only if it has an ab elian k ernel A : 1 / / A / / / / Y ⋉ ψ A π / / / / Y / / o o σ o o 1 The (eviden tly ab elian) subgroup Φ Y [ π , σ ] of the in v arian t elemen ts of A under the action ψ w as denoted A Y in [20] and sho wn to b e the 0- dime nsional cohomology group H 0 ψ ( Y , A ). This fa c t was used to in tro duce in the Mal’cev con text a notion of in ternal cohomology in [15]. 12 2.3 Cen tralizer of equiv alence rel ations In the Mal’cev context, there exits also an in trinsic notion of comm utation at the lev el of equiv a le nce relations, see [12]. First, the sub ob jects of the ob ject ( p 0 , s 0 ) : X × X ⇄ X in the fibre P t X C coincide exactly with the reflexiv e relations o n X , hence, in the Mal’cev con text, with the equiv alence relations on X . Recall that tw o equiv alence relations R and S on an ob ject X comm ute in C if and only if the tw o follo wing sub ob jects in the fibre P t X C do comm ute in P t X C , see Prop osition 2.6.12 in [4 ]: S d 0 ~ ~ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ( d 0 ,d 1 ) v v ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ R / / ( d 1 ,d 0 ) / / d 1 # # ● ● ● ● ● ● ● ● ● ● ● X × X p 0   X s 0 c c ● ● ● ● ● ● ● ● ● ● ● s 0 O O s 0 > > ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ the choice of this pr e sen tat ion ( R op rather t han R ) b eing made for tec hnical reasons related to the classical pr e sen tat ion of the axioms o f a Mal’cev op eration. So, in the fib erwis e a lgebraically cartesian closed con text, t he existence of centralizers can b e immediately transfered to the lev el of equiv alence relations. Prop osition 2.8. Supp os e C is a Mal’ c ev c ate gory which is fib erwise algebr aic al ly c artesian close d. L et R b e any e q u ivalenc e r elation o n the obje ct X . T hen the c entr alizer Z ( R ) of the e quivalenc e r elation R do es exist in C and is given by the domain of Φ d 1 [ p R , (1 , d 0 )] : • v v ε d 1 ( p R , (1 ,d 0 )) v v ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ / / | | ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② Z ( R ) o o o o | | ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② v v ( d 0 ,d 1 ) v v ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ R × X p R   X × X o o o o p 0   R (1 ,d 0 ) O O d 1 / / < < ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② X o o s 0 o o < < ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② s 0 O O Pr o of. Since the category C is fib erwise algebraically cartesian c losed, the unital fibre P t X C has cen tra liz ers of subob jects, and according to the previous recall about equiv alence relations and their commu tations, the cen tralizer of R in C is nothing but the cen tra liz er of the following sub ob ject in the fibre P t X C : R / / ( d 1 ,d 0 ) / / d 1 # # ● ● ● ● ● ● ● ● ● ● ● X × X p 0   X s 0 c c ● ● ● ● ● ● ● ● ● ● ● s 0 O O 13 According to Prop osition 1.2 , it is giv en b y Φ d 1 [ p R , (1 , d 0 )]. So, when the category C is exact, Mal’ce v and fib erwise alg e braically cartesian closed, the existence of cen t ralize rs mak es the Sc hreier- Mac Lane extensions classi- fication theorem hold, see [11]. 3 Some prop ertie s of th e alge b raic exp on e n tiable morphisms 3.1 Stabilit y under pullbac k al ong split epimorphisms W e show first tha t the a lgebraically exp onen tiable morphisms are stable under pull- bac k along split epimorphisms. It is a consequenc e of the following v ery general lemma: Lemma 3.1. L et E b e a c ate gory with pul lb acks and U : E → F a functor which admits a right adjoin t G . T h en, for any o bje c t X ∈ E , the induc e d functor: U X : P t X E / / P t U X F has a right adjoint G X . When mor e over the c ate gory F has pul lb acks, any map f : X → X ′ makes the fol lowing leftwar d diagr am c om mut e up to a natur al isomor- phism: P t X ′ E f ∗   U X ′ / / P t U X ′ F G X ′ o o U f ∗   P t X E U X / / P t U X F G X o o When, i n addition, U is left exact the pr evious diagr am also c ommutes at the l e vel of dote d arr ows. Pr o of. Let ( τ , σ ) : T ⇄ U X b e an o b ject of P t U X F . It is straighforw ard to c heck that G X ( τ , σ ) is g iv en by the follow ing pullbac k in E , where η X is the unit of the adjunction: •   / / GT Gτ   X η X / / O O GU X Gσ O O The second p oin t is a conseq uence of the na turalit y of the unit η and of the fact that the righ t adjo in t functor G preserv es pullback s. F r om that, the last point is straigh tforw ard. Whence the follo wing: 14 Prop osition 3.2. L et C b e a finitely c omplete c ate gory. Then the a l g e br aic al ly exp onentiable morphisms in C ar e stable under p ul lb ack along split epimorph isms. Mor e over any p ul lb ack in the c ate gory C with y algebr aic al ly exp onentiable: X f   x / / X ′ f ′   Y y / / s O O Y ′ s ′ O O satisfies the fol lowing Be ck-Cheval ley c o nditions, i.e. makes the fo l lowing dia gr am s c ommute up to natur al isomorphis m s: P t X C Φ x / / P t X ′ C x ∗ o o P t X C Φ x / / s ∗   P t X ′ C s ′∗   x ∗ o o P t Y C f ∗ O O Φ y / / P t Y ′ C y ∗ o o f ′∗ O O P t Y C Φ y / / P t Y ′ C y ∗ o o Pr o of. Apply the previous lemma to the functor y ∗ : P t Y ′ C → P t Y C and notice that w e ha ve P t ( f ′ ,s ′ ) ( P t Y ′ C ) = P t X ′ C and P t ( f ,s ) ( P t Y C ) = P t X C . Then consider the follo wing morphisms in P t Y ′ C : X ′ f ′ ! ! ❉ ❉ ❉ ❉ f ′ / / Y ′ 1 } } ④ ④ ④ ④ Y ′ 1 ! ! ❈ ❈ ❈ ❈ s ′ / / X ′ f ′ } } ④ ④ ④ ④ Y ′ 1 = = ④ ④ ④ ④ s ′ a a ❉ ❉ ❉ ❉ Y ′ s ′ = = ④ ④ ④ ④ 1 a a ❈ ❈ ❈ ❈ Then w e g et immediately the following: Corollary 3.3. Whe n a split epimorphism ( f , s ) : X ⇄ Y is algebr aic al ly exp onen- tiable, the induc e d endofunctor f ∗ . Φ f on P t X C is (up to a natur al isomorphism) e qual to the endofunctor Φ p 0 .p ∗ 1 , wher e p 0 and p 1 ar e given by the kernel e quivale n c e r elation: R [ f ] p 1 / / p 0 / / X s 0 o o f / / / / Y This endofunctor Φ p 0 .p ∗ 1 on P t X C inherits the left exact c om onad structur e induc e d by the adj o int p air ( f ∗ , Φ f ) . 3.2 The efficien tly regular con text A regular category C is said to b e efficien tly regular, when, in addition, an y equiv- alence relation S , on an ob ject X , whic h is included in an effectiv e equiv alence 15 relation S m ֌ R [ f ] b y an effec tiv e monomorphism m , is itse lf effectiv e. The main examples a re the catego ries T opGp and T opAb o f top ological gr oups and a belian groups. An y exact category is efficien tly r e gular. When the category C is moreov er efficien tly regular , w e can extend algebraic ex p onen tiability from split epimorphisms to regular epimorphisms and ha v e a kind of con vers e to Prop osition 3.2. F or that, let us b egin by the following: Prop osition 3.4. L et C b e efficie n t ly r e gular. C onsider an interna l dis c r ete c ofi- br ation: f 1 : X 1 → Y 1 b etwe en two gr oup oids: R [ d 0 ] d 1 2 / / d 1 1 / / d 1 0 / / R ( f 1 )   X 1 d 1 / / d 0 / / f 1   X 0 f 0   s 0 o o R [ d 0 ] d 1 2 / / d 1 1 / / d 1 0 / / Y 1 d 1 / / d 0 / / Y 0 s 0 o o Supp ose the morphism f 0 is algebr aic al ly exp onentiable. Then the functor f ∗ 1 : S C of Y 1 → S C of X 1 fr om t he split discr ete c o fibr ation s ab ove Y 1 to t he s p lit dis- cr ete c ofibr a t ions ab ove X 1 define d by pu l ling b ac k along the funct or f 1 admits a right adjoint whic h is c onstructe d levelwise. Pr o of. According to Prop osition 3.2, since the v ertical square with the d 0 is a pull- bac k (the f un ctor f 1 b eing a discrete cofibration), the maps f 1 and R ( f 1 ) are also algebraically exp onen tiable. Let ( α 1 , β 1 ) : T 1 ⇄ X 1 b e a split discrete fibration ab o ve X 1 . W e are going t o sho w that the split epimorphisms Φ f 0 ( α 0 , β 0 ) = ( ¯ α 0 , ¯ β 0 ) : W 0 ⇄ Y 0 and Φ f 1 ( α 1 , β 1 ) = ( ¯ α 1 , ¯ β 1 ) : W 1 ⇄ Y 1 are actually underlying a discre te cofibration ab o ve Y 1 , whic h will determine the construction of the lev elwise right adjoin t in question. F or that, let us consider the following diag ram: T 1 / / d 0 / / α 1 t t ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ T 0 α 0 t t ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ o o X 1 d 1 / / d 0 / / f 1   X 0 f 0   o o W 1 / / d 0 / / ¯ α 1 t t ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ W 0 ¯ α 0 t t ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ o o Y 1 d 1 / / d 0 / / Y 0 o o Since the square with the d 0 in the statemen t is a pullbac k underlying a pullback o f split epimorphisms, then, according to Prop osition 3.2, the Bec k-Chev alley condition 16 holds for this square. Conseq uen t ly the lo w er quadrangle with d 0 ab o ve is underlying a pullbac k of split epimorphisms. But a discrete cofibration b e t w een group oids is also a discrete fibration and the square with the d 1 in the statemen t is also a pullbac k. Moreov er, the Bec k-Chev alley condition not only says that the co-fr e e ob ject are pres erv ed b y pullback s, but also the unive rsal natural transformation f ∗ i . Φ f i ⇒ 1 P t Y i C , i ∈ { 0 , 1 } . This determin es an arrow d 1 : W 1 → Y 1 whic h mak es also the lo w er quadrangle with d 1 a pullback , and pro duces a reflexiv e gr aph W 1 ⇒ W 0 . The same Bec k-Chev a lle y c ondition mak es this reflexiv e graph underlying a group oid structure and α 1 a discrete fibration which is, by construction, a lev elwise cofree structure with resp ec t to the pulling back a long the functor f 1 . Whence the follo wing “con v erse” to Prop osition 3.2: Prop osition 3.5. L et C b e an efficiently r e gular. Consid e r the fol lowi n g pul lb ack with f ′ a r e gular epimorphism: X f / / / / x   Y y   X ′ f ′ / / / / Y ′ Then, when the morphism x is al g e br aic al ly exp onentiable, so is the morphism y , and we have the Be ck-Cheval ley c ommutation: P t X Φ x   P t Y f ∗ o o Φ y   P t X ′ P t Y ′ f ′∗ o o Pr o of. Complete the previous pullbac k b y the follo wing diagram: R 2 [ f ] p 1 2 / / p 1 1 / / p 1 0 / / R 2 ( x )   R [ f ] p 1 / / p 0 / / R ( x )   X 0 x   s 0 o o f / / / / Y y   R 2 [ f ′ ] p 1 2 / / p 1 1 / / p 1 0 / / R [ f ′ ] p 1 / / p 0 / / X ′ s 0 o o f ′ / / / / Y ′ whic h determines a discrete cofibration R 1 ( x ) : R 1 [ f ] : R 1 [ f ′ ] betw een the left hand side induced horizon tal group oids. According to the previous prop osition the c ha nge of base functor R 1 ( x ) ∗ : S C of R 1 [ f ′ ] → S C of R 1 [ f ] admits a right adjoin t. No w 17 consider the follow ing comm utative square: P t Y ′ C y ∗ / / F Y ′   P t Y C F Y   S C of R 1 [ f ′ ] R 1 ( x ) ∗ / / S C of R 1 [ f ] where t he functors F Y and F Y ′ are the canonical straighforw ard functors which a re fully faithful s ince f and f ′ are regular e pimorphisms. They are also e ssen tially surjectiv e, since, in an efficien tly regular c ategory , an y equiv alence fibration which is discretely cofibered ab o v e an effectiv e eq uiv alence relation is itself effectiv e. Ac- cordingly the functors F Y and F Y ′ are equiv alences of categor ie s, and the c ha nge of base functor y ∗ admits a righ t adjoint. This construction of the righ t adjo in t to y ∗ imp oses the Beck -Chev alley condition. Corollary 3.6. L et C b e efficiently r e gular and f : X ։ Y a r e gular epimorphism such that the map p 0 b elow: R [ f ] p 1 / / p 0 / / X s 0 o o f / / / / Y is algebr aic al ly exp on e ntiable, then f is itself algebr a i c al ly exp onentiable and we have : f ∗ . Φ f ≃ Φ p 0 .p ∗ 1 . When C is e fficiently r e gular and fi b erw i s e alg e br aic al ly c artesian clos e d, then any r e g ular epim orphism f : X ։ Y is algebr aic al ly ex p on e n t iable. Pr o of. Consider the follo wing pullbac k: R [ f ] p 1 / / / / p 0   x f   X f / / / / Y and apply the previous prop osition. 4 Protomo dular con te xt W e shall no w w ork in the stronger con text of a protomo dular categor y C [7], whic h means that an y (left exact) c hange o f base functor: h ∗ : P t Y C → P t X C is conserv a tiv e. W e get immediately the following: 18 Prop osition 4.1. Supp ose C is pr otomo dular, then any any change of b ase functor h ∗ along a n algebr aic exp onentiable map h : X → Y r e fl e cts c ommuting p airs. Pr o of. In the proto modular con text, any c hang e of base functor b eing left e xact and conserv ativ e, any algebraic exp onen tiable map h : X → Y make s this f unc tor h ∗ immediately comonadic [16]. Accordingly , the assertion in question is a direct consequenc e of Pro position 1.6. 4.1 Lo cally algebraically cartesian closed p oin ted protomo d- ular cat egories On the one hand, in [5] and [6], the notion of action r epr esen t ative category was in tro duced, i.e. a p oin ted proto modular category C in whic h eac h ob ject X admits a univ ersal split extension with k ernel X (=split extension classifier): X / / γ / / D 1 ( X ) d 0 / / / / D ( X ) o o s 0 o o in t he sense that any other split extension with k ernel X determines a unique mor- phism χ : G → D ( X ) suc h that the following diagram commutes and the right hand side squares are pullbac ks: X / / k / / 1 X   H χ 1   f / / / / G χ   o o s o o X / / γ / / D 1 ( X ) d 0 / / / / D ( X ) o o s 0 o o On the other hand, in [16], the second author sho w ed that when the c ate g o ry C is p ointe d pr otomo dular, it is lo c al ly algebr aic al ly c artesian close d if and only if the change of b ase functors along the i n it ial m aps have a righ t adjoin t . It is w orth translating in detail what this means, and, r ather surprisingly , we shall observ e that this means a kind of extended dual of t he actio n represen ta tivit y . So let C b e a lo cally algebraically cartesian closed p oin ted proto modular category . Let Y b e a n ob ject of C and α Y : 1 ֌ Y its asso ciated initia l map. W e shall denote ß Y b y the righ t adjoin t of α ∗ Y . Starting with an y o b ject T in C , the ob ject ß Y ( T ) is a split epimorphism ab ov e Y which is equipped with a (univ ersal) map from its k ernel to w ards T . In ot he r w ords it pro duces a univ ersal split exact sequence we shall denote this w ay: L( Y , T ) / / κ Y T / / l Y T   Y ⋉ L( Y , T ) ψ Y T / / / / Y o o ζ Y T o o T 19 whic h from an y giv en similar situation, i.e. a split exact sequence with co domain Y and a comparison map h : K / / k / / h * * ¯ h   X f / / / / Y ⋉ ¯ h   Y o o s o o L( Y , T ) / / κ Y T / / l Y T   Y ⋉ L( Y , T ) ψ Y T / / / / Y o o ζ Y T o o T pro duces a unique dotted factorization ¯ h . In particular the follo wing upper c anonical split exact seque nce: T / / ι T / / 1 T * * § Y T   Y × T p Y / / / / Y ⋉ § Y T   Y o o ι Y o o L( Y , T ) / / κ Y T / / l Y T   Y ⋉ L( Y , T ) ψ Y T / / / / Y o o ζ Y T o o T determines a factorization whic h will b e denoted b y § Y T . Starting from the more sp ec ific one with the dia gonal s 0 as section, w e ha v e also the f ollo wing factorization: Y / / ι 1 / / 1 Y * *  Y   Y × Y p 0 / / / / Y ⋉  Y   Y o o s 0 o o L( Y , Y ) / / κ Y Y / / l Y Y   Y ⋉ L( Y , Y ) ψ Y Y / / / / Y o o ζ Y Y o o Y whic h w e shall analyse b elo w more precisely in the category Gp . 4.2 The c ategory Gp of group s W e shall explore in detail here the v ery unusual constructions in v olved in the lo cal ex- p onen tiation prop ert y of the category Gp . In this case, w e ha v e L( Y , T ) = F ( Y , T ), namely L( Y , T ) is the set of applicatio ns from the underlying set of the group Y to the underlying set of the group T eq uipp ed with the group structure determined b y the gr oup structure on T . The action of the group Y on this group F ( Y , T ) asso ciates with the pair ( y , φ ) the application: φ ◦ τ y : Y → T ; z 7→ φ ( z .y ) 20 where τ y is t he translation on the rig h t in the group Y (in o ther w ords we get: ( y φ )( z ) = φ ( z .y )). So, in the category Gp , the parallelism b et w een cartesian closed- ness and algebraic cartesian closedness is not only simply formal, but a kind of strong memory of the underlying exp onen tiation in S et . The homomorphism l Y T : F ( Y , T ) → T is the ev aluat ion at the unit eleme n t of Y . Giv en any split extension with a map h : K / / / / h   Y ⋉ α K / / / / Y o o o o T the group homomor phis m ¯ h : K → F ( Y , T ) is then defined by ¯ h ( k )( y ) = h ( y k ). In particular w e get the group homomorphism § Y T : T → L( Y , T ) defined by § Y T ( t )( y ) = t , in o ther w ords § Y T ( t ) is the application constan t on t . And a ls o w e get  Y : Y → L( Y , Y ) defined b y  Y ( y )( z ) = z .y .z − 1 whic h is a v ery awkw ard w a y to in tegrate the “inner action” of Y inside the categor y Gp . If w e star t from: K / / / / h   X f / / / / Y o o s o o T w e get: ( Y ⋉ ¯ h )( x ) = ( f ( x ) , ¯ h ( x.s ◦ f ( x − 1 ))). 4.3 First consequences of lo cal algebraic cartesian closed- ness In this se ction, we shall in v estigat e t wo important consequences of lo cal algebraic cartesian closedne ss, namely strong protomo dularit y and p eri-ab elianness. These w ell iden tified prop erties in the category Gp no w clearly app ear to hav e originated from lo cally alg e braic cartesian closedness. 4.4 Normal functor and strong proto mo dularit y Recall that a left exact functor U : C → D is called normal when it is conserv ativ e and it refle cts the no rmal monomorphisms. A protomodular category C is said to b e str ongly pr otomo dular [4] when an y c hange of base functor h ∗ : P t Y C → P t X C with r e sp ect to the fibration of p oin ts is not only conserv ative but also normal. The categories Gp of groups and R - Lie of Lie R -algebras, for any commmutativ e ring R , are strongly protomo dular. In this section w e shall sho w that , when a protomo dular category C is lo cally algebraically cartesian closed, it is necessarily strongly protomo dular. Let us b egin b y the following o bs erv ation: 21 Lemma 4.2. L et U : C → D b e a left exac t c onserva t ive functor. Supp ose mor e ove r that D is pr otomo dular. If it has a right adjoint G , then U is norm a l. Pr o of. The right adjoin t G is left exact and consequen tly preserv es the monomor- phims and the equiv alence relatio ns . No w let m : X ′ ֌ X b e a monomorphism in C suc h that the mono morphis m U ( m ) is normal to the equiv alence relation R in D , namely suc h t hat we hav e a discrete fibration in D : U ( X ′ ) × U ( X ′ ) / / µ / / p 1   p 0   R d 1   d 0   U ( X ′ ) / / U ( m ) / / O O U ( X ) O O Since U ( X ′ ) × U ( X ′ ) = U ( X ′ × X ′ ), by a djunc tion we get a map ¯ µ in C suc h tha t ǫ R .U ( ¯ µ ) = µ : X ′ × X ′ / / β / / / / ¯ µ / / p 1   p 0   T ¯ η / / d 1   d 0   G ( R ) G ( d 1 )   G ( d 0 )   X ′ / / U ( m ) / / O O X η X / / O O G.U ( X ) O O whic h determines a mor ph ism b et w een the equiv alence relat ions ∇ X ′ and G ( R ). W e shall set T = η − 1 X ( G ( R )) a nd denote by β the induced factorization. W e are going to sho w that m is normal to the equiv alence relation T and that U ( T ) ≃ R . F or that, consider the follo wing diagram in D : U ( X ′ ) × U ( X ′ ) / / U ( β ) / / / / U ( ¯ µ ) / / p 1   p 0   µ U ( T ) U ( ¯ η ) / / U ( d 1 )   U ( d 0 )   U.G ( R ) ǫ R / /     R d 1   d 0   U ( X ′ ) / / U ( m ) / / O O U ( X ) U ( η X ) / / O O 1 U ( X ) C C U.G.U ( X ) O O ǫ U ( X ) / / U ( X ) O O The map γ = ǫ R .U ( ¯ η ) pro duce s an inclusion U ( T ) ⊂ R of equiv alence relations. Since the whole diagram is a discrete fibration and γ is a monomorphism, then the left ha nd side part of the diagram is a dis crete fibration. Accordingly U ( m ) is normal to the equiv alence relation U ( T ). No w, when D is pro tomodular, a monomor phis m is normal to at most one equiv alence relation, and γ is necessarily an isomorphism. On the o the r hand, since U is left exact and conserv ativ e it reflects the pullbac ks, so tha t m is normal to T in C . 22 Whence the follo wing: Theorem 4.3. L et C b e a pr otomo d u lar c ate g ory which is lo c al ly algebr aic al ly c arte- sian cl o se d. Then C is str ongly pr otomo dular. Pr o of. Since C is protomo dular and lo cally a lgebraically car tes ian closed, so is any fibre P t Y C . Moreov er any ch ange of base functor: h ∗ : P t Y C → P t X C is left exact and conserv ativ e since C is protomo dular; it has a righ t adjoin t since C is lo cally algebraically cartesian closed. By the previous lemma it is normal, and C is strongly protomo dular. No w suppose, in addition, that C is p oin ted. Being p oin ted and s trongly pro- tomo dular, it is suc h that t wo equiv alence relations ( R, S ) cen tralize if and only if their asso ciated normal sub ob jects ( I R , I S ) comm ute, see [9]; in other w ords the category C is suc h tha t we ha v e the so-called equation “Smith=Huq”. The assertion of the theorem ab o ve w as me n tioned by G. Janelidze during the CT 2010 c onference in Geno v a, but in the m uc h stricter context of semi-ab elian categories, as an immediate conseq uence of Prop osition 9 in [3], whic h deals w ith preserv a tion of colimits a nd cannot b e used in our context. 4.5 P eri-ab elian categories When C is a regular strongly unital category with finite colimits, the inclusion functor Ab C ֌ C from the full subcategor y of ab elian o b jects in C admits a left adjoin t whic h is giv en b y the cok ernel of the diagonal s 0 : X ֌ X × X , or by the co equalize r of the pair ( ι 0 , ι 1 ); X ⇒ X × X . Recall now the following [10]: Definition 4.4. We shal l say that a fin itely c o c omplete, r e gular Mal’c ev c ate gory D is p eri-ab e l i a n when the change of b a s e functor along any map h : Y → Y ′ with r esp e ct to the fibr a t ion o f p oints pr eserve s the asso ciate d ab elian obje ct. If ( AbP t ) D denotes the sub category of the ab elian ob jects in the fibres of the fi- bration of p oints , it is e quiv alen t to sa ying that the reg-epi re flection A () is cartesian, i.e. it preserv es the cartesian maps: ( AbP t ) D ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ / / / / P t D   ✞ ✞ ✞ ✞ ✞ ✞ ✞ A ()   D 23 The categories Gp of groups, Rg of non unitary comm utat iv e rings and K - Lie of K -Lie algebras are p eri-ab e lian. The previous definition w as in tr oduced in [10] as a to ol to pro duce some cohomolo gy isomorphisms whic h hold in the Eilen b erg-Mac Lane coho mology of groups and in the cohomolo gy of K -Lie algebras. Theorem 4.5. L et C b e a finitely c o c omplete r e gular Mal’c ev c ate g o ry which is lo c al ly algebr aic al ly c artesian close d. Then C is p eri-ab elian. Pr o of. This is a straigh tforw ard consequence of t he fact that the c hange of base- functors h ∗ , ha ving a righ t adjoin t, preserv e cok ernels or co equalize rs. 4.6 The n on - p oin ted protomo dular case W e shall supp ose here that the category is still protomo dular, but no longer p oin ted. W e shall sho w that the algebraic expo ne n tiabilit y of any split monomorphism implies the algebraic exp onen tiabilit y of an y of its retractions, and from that, in the effi- cien tly regular con text, of a large class of morphisms. This will b e the consequence of a very general result: Prop osition 4.6. L et C b e a pr otomo dular c ate gory. F or morphisms f : X → Y and g : Y → Z in C , if g .f and f ar e alg e br aic al ly exp one ntiable, then g is algebr aic al ly exp onentiable. Pr o of. Recall that for a n y morphism p : E → B in C , p ∗ preserv es all limits and reflects isomorphisms. Therefore if p ∗ has a righ t adjo in t, then by the dual of the W eak T ripleabilit y Theorem [21], p ∗ is comonadic. The result follow s from the w ell-known ad joint f unctor lifting the or e m (see e.g. [2]) applied to the diagra m of functors: P t Z ( C ) g ∗ / / ( g. f ) ∗ $ $ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ P t Y ( C ) f ∗ z z t t t t t t t t t t t P t X C Φ f : : t t t t t t t t t t t Φ gf d d ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ in whic h Φ g f and Φ f are the righ t adjoints to the functors ( g .f ) ∗ and f ∗ resp e ctiv ely . Whence the follo wing theorem: Theorem 4.7. L et C b e a pr otomo dular c ate gory. Supp ose that a ny split monom or- phism s is algebr aic a l ly exp onentiable. Th en any of its r etr actions f is a lgebr a ic al ly exp onentiable. When mor e over C is efficiently r e gular, any m a p h : X → Y is algebr aic al ly exp onentiable pr ovide d that i ts doma in X has a glob al supp ort. 24 Pr o of. The first p oin t is a straighforw ard consequenc e of the previous pro position. Moreo v er, giv en an y map h in C , w e get h = p Y . (1 X , h ), whe re the map ( 1 X , h ) is a monomorphism split by p X , a nd consequen tly the c hang e of base functor along it admits a r igh t adjoin t b y a s sumption. When C is regular and X has global supp ort, the map p Y is a regula r epimorphism. No w when C is efficien tly regular, the change of base functor along p Y admits a righ t adjoin t b y Prop osition 3.6. 4.7 Bac k to the p oin ted case The previous construction of the righ t adj oin t o b viously applies in the p oin ted case and giv es a n alternative description of the cen tralizers. F irs t, let ( f , s ) : X ⇄ Y b e an y split epimorphism; w e get a classifying map γ ( f ,s ) : K / / k / / 1 K ) ) γ ( f ,s )   X f / / / / Y ⋉ γ ( f ,s )   Y o o s o o L( Y , K ) / / κ Y K / / l Y K   Y ⋉ L( Y , K ) ψ Y K / / / / Y o o ζ Y K o o K Then Φ Y [ f , s ] is giv en b y the follo wing equalizer: Φ Y [ f , s ] / / / / K § Y K / / γ ( f ,s ) / / L( Y , K ) In particular the cen tralizer of a subo b ject u : Y ֌ X is giv en b y the fo llo wing equalizer: Z [ u ] / / ζ u / / X § Y X / / γ ( p Y , (1 ,u )) / / L( Y , X ) Let us mak e explicit these constructions in the category Gp . First w e hav e: γ ( f ,s ) ( k )( y ) = s ( y ) .k .s ( y − 1 ), and consequen tly: Φ Y [ f , s ] = { k ∈ K / ∀ y ∈ Y , k = s ( y ) .k .s ( y − 1 ) } , and th us, of course, Z [ u ] = { x ∈ X/ ∀ y ∈ Y , x = y .x.y − 1 } . References [1] M. Ba rr, Exact c ate gories , Springer L.N. in Math., 236 , 1 971, 1-120. [2] M. Barr and C. W ells, T op oses, triples and the ories Reprin ts in Theory and Applications of Categories, (12) , 2005, 1-288. 25 [3] F. Bo rc eux, Non-p oin ted strongly protomo dular theories, Applied Categorical Structures 12, 2004, 319-3 38. [4] F. Borceux and D. Bourn, Mal’c ev, pr otomo dular, homol o gic al and se mi-ab elian c ate gories , Klu w er, Mathematics and its app lic a tion s , vol. 566 , 2004. [5] F. Borceux, G. Janelidze and G.M. Kelly , In ternal ob j e ct actions, Commen ta- tiones Mathematicae Univ ersitatis Carolinae, 46 , 20 05, 235- 255. [6] F. Borceux, G. Janelidze and G.M. Kelly , On the represen tabilit y of actions in a semi-ab e lian categor y , Theory Appl. Categ., 14 , 2005, 244- 286. [7] D. Bourn, Norma lization e quivalen c e, kernel e quivalenc e and affine c ate gories , Springer LN 1488, 1991, 43-62. [8] D. Bourn, Mal’cev categories and fibration of p oin t e d ob j e cts, Applied Cate- gorical Structures, 4 , 1996, 307- 327. [9] D. Bourn, Comm utator theory in strong ly protomo dular categories, Theory Appl. Categ., 13, 2004, 2740. [10] D . Bourn, The cohomological comparison arising from the asso ciated ab elian ob ject, a rXiv :1001.090 5 (2010), 24pp. [11] D . Bourn Inte rnal profunctors a nd comm utator theory; applications to exten- sions classification and categorical Ga lois theory , Theory Appl. Categ., 24 , 2010, 441-48 8. [12] D . Bourn and M. Gran, Cen t ralit y and connectors in Maltsev categories, Alge- bra Univ ersalis, 48 , 2002, 309-33 1. [13] A. Carb oni, J. Lam b ek and M.C. P edicc hio, Diag ram c hasing in Mal’cev cate- gories, J. Pure Appl. Algebra, 69 , 1991, 271- 284. [14] A. Carb oni, M.C. P edicc hio and N. Piro v ano, In ternal g raphs and internal group oids in Mal’cev categories, CMS Conference Pro cee dings, 13 , 1 992, 97- 109. [15] J.R.A. G ra y , Algebraic exp onen tiation in general categories, Ph.D. thesis, Uni- v ersity of Cap e T own, 2010. [16] J.R.A. Gra y , Algebraic exp onen tia tion in general categories, to app ear in Ap- plied Categorical Structures, in Press. [17] J.R.A. Gray , Alg e braic exp onen tiatio n for categories of Lie algebras, to a ppear in J. Pure Appl. Algebra. 26 [18] S.A. Huq, Comm utator, nilp otenc y a nd solv ability in categories, Quart. J. Ox- ford, 19 , 1968, 363-38 9. [19] P .T. Johnstone and M.C. P edicc hio, Remarks on con tinuous Mal’cev algebras, Rend. Univ. T rieste, 1995, 277- 297. [20] S. Mac La ne , Homolo gy , Springer, 1963. [21] S. Mac Lane, Cate gories for the Working Mathematician , Springe r Science, Ne w Y ork, (2nd Edition), 1997 . [22] N. Martins-F erreira, L ow-dimen s ional internal c ate goric al structur e in we akly Mal’c ev sesquic ate gorie s , Ph.D Thesis, 2 008. [23] M.C. Pe dicc hio, A categorical a pproac h to comm utator theory , Journal of Al- gebra, 177 , 1995, 647-657 . [24] J.D .H. Smith, Mal’c ev varieties , Springer L.N. in Math., 554 , 197 6. 27