Admissibility, stable units and connected components

Unsp ecified Journal V olume 00, Number 0, Pag es 000–000 S ????- ????(XX)0000-0 ADMISSIBILITY, ST ABLE UNITS AN D CONNECTED COMPONENTS JO ˜ AO J. XAREZ Abstract. Consider a reflection from a finitely-complete category C into its full subcategory M , with unit η : 1 C → H I . Suppose there is a left- exact functor U int o the category of sets, such that U H reflects isomorphis ms and U ( η C ) i s a surjection, for eve ry C ∈ C . If, in addition, all the maps M ( T , M ) → Set (1 , U ( M )) i nduced by the f uncto r U H are surjections, where T and 1 are resp ectiv ely terminal ob j ects in C and Set , for every ob ject M i n the full sub category M , then it is true that: the reflection H ⊢ I is semi-left- exact (admissible in the sense of cate gorical Galois theory) if and only i f its connect ed comp onen ts are “connect ed”; i t has stable units if and only i f an y finite pro duct of connected comp onen ts is “connected”. Where the meaning of “connecte d” i s the usual i n catego rical Galois theory , and the definition of connect ed comp onen t with resp ect to the ground structure will b e given. Note that b oth algebraic and top ological i nstance s of Galois structures are unified in this common setting, wi th resp ect to categorical Galois theory . 1. Introduction A reflection H ⊢ I fro m a categor y C int o its full sub category M can b e seen as a Galois structure, o ne in whic h all mo rphisms are taken into account. Hence, such a reflection is semi-left-exact (in the sense of [2]) if and o nly if it is an admis- sible Galo is structur e (in the sens e of categor ical Galois theory). The fundamental theorem of catego r ical Galo is theo ry states that, for an admissible Galois structure as ab ov e, that is , a semi-left-exact reflection into a full s ub catego ry , there is an equiv alence S pl ( E , p ) ≃ M Gal ( E ,p ) , for every effective descent mor phism p : E → B in C , b et ween the full subca tegory S pl ( E , p ) of the co mma category ( C ↓ B ), deter- mined by the morphisms split by p : E → B , and the categor y M Gal ( E ,p ) of actions of the Galois pregro upoid Gal ( E , p ) in M (see [1]). T o establish the existence of such equiv alences, tha t is , in order to prov e that the re fle c tio n is semi-left-exa ct, it is necess ary to show, for e v ery B ∈ C and every ( M , g ) ∈ ( M ↓ I ( B )), that the counit mo rphism ε B ( M ,g ) : I B H B ( M , g ) → ( M , g ) is an isomor phism, wher e H B ⊢ I B : ( C ↓ B ) → ( M ↓ I ( B )) is the induced adjunction. In the current pap er, 2000 Mathematics Subje ct Classific ation. 18A40, 54B30, 18B30 , 18B40, 20M07, 20M50, 18E35. Key wor ds and phr ases. C onnected comp onen t, se mi-left-exactness, stable units, left- exactne ss, sim ple reflection, admis s ible reflection, localization, Galois theory . The author wo uld l ik e to ac knowledge the financial supp ort of Unidade de Invest iga¸ c˜ ao Matem´ atic a e Aplic a¸ c˜ oes of Universidade de Av eiro, through Pr o gr ama O p er acional Ciˆ encia e Inova¸ c˜ ao 2010 (POCI 2010 ) of the F und a¸ c˜ ao p ar a a Ciˆ encia e a T e cnolo gia (FCT), cofinanced b y the Europ ean Communit y fund FEDER. c  0000 (copyrigh t holder) 1 2 JO ˜ AO J. XAREZ we prov e it is enough to show that every ε B ( T , g ) is an isomor phism when T is a ter- minal ob ject, in order to gua r antee semi-left-exactness, provided there is a (“forget- ful”) functor U from C in to sets, satisfying cer tain conditions. Such is the case of the t wo reflections Com pHaus → Stone , compact Hausdo rff spaces into Stone spa ces, and SGr → SLat , semigroups into se mila ttices, where “co nnec ted comp onents ar e connected” (meaning that the counit morphis ms ε B ( T , g ) are all isomo rphisms, whic h amounts to the preserv a tion by the reflec tor of the “connected comp onent” pull- back diagr ams). F urthermore, these tw o examples are kno wn to satisfy a stronger condition tha n semi-left-exa ctness. In fact, b oth reflections Co mpHaus → Stone and S Gr → SLat hav e s table units (see [1 ] and [3 ], resp ectively). W e will a lso s tate that such a Galois structure with such a “for getful” functor do es hav e stable units if and only if “finite pro ducts o f connected comp onents are co nnected”. A c o nnected comp onent is simply the pullback C × ( η C ,µ ) T of a mo r phism µ : T → H I ( C ) from a ter minal o b ject T alo ng a unit mor phism η C : C → H I ( C ). Therefore, in our setting, semi- left-exactness and the stable units prop erty are simplified and the Ga- lois structures can b e classified acco rding to the reflection of connected co mponents and its pro ducts, res pectively . Besides semi-left-exactness and the stable units prop erty , there is a w eaker pro p- erty and also a stronger one. When the former ho lds, a reflection is calle d simple. A reflection where the latter ho lds is called a lo calization, meaning that the reflector is left-exa ct, that is, it preserves finite limits. In our setting, a s ufficien t co ndition, for a reflection to b e a lo calization, will b e g iv en on the connected comp onents. Also, semi-left-exa ct and s imple reflections are shown to coincide, provided a fur- ther condition holds for the left a djoin t I . Finally , the author w o uld like to mention that the results in this pap er had their origin in genera lizing the pro o f of Theorem 3 in [3], wher e it is shown that the reflection of semigr oups into semilattices has s table units. 1 2. Ground Str ucture In this s e ction 2, it is g iven the setting in which all the prop ositio ns o f the current pap er hold. Consider an adjunction H ⊢ I : C → M , with unit η : 1 C → H I , such that the category C has finite limits and the r ight a djoin t H is a full inclus ion of M in C , i.e., the adjunction is a r eflection of the catego ry C int o its full s ubcateg ory M . Consider as w ell a functor U : C → Set from C in to the c ategory of sets, with the following prop erties: ( a ) U is left exact (i.e., U preserves finite limits); ( b ) U H reflects isomo r phisms; ( c ) every map U ( η C ) : U ( C ) → U H I ( C ) is a sur jection, for every unit mor - phism η C of the r eflection ab ov e, C ∈ C ; ( d ) every map C ( T , M ) → Set ( U ( T ) , U ( M )), which is the restric tion of the functor U to the hom-s et C ( T , M ), is a sur jection, for any o b ject M ∈ M , with T a ter minal ob ject in C . Remark 2.1. It is conv enient, without no loss of gener alit y , to chosen the unit η : 1 C → H I so that the counit is an identit y I H = 1 M . 1 The prop ert y that “connecte d comp onents are connected”, i.e., s emi-left-exactness in our setting, was called attainabilit y in [ 5] , in the particular case of semigroups. ADMISSIBILITY, ST ABLE UNITS AND CONNECTED COMPONENTS 3 Remark 2.2 . It is also conv enient to assume, without no lo ss of g enerality , that T is a terminal ob ject chosen to be in M . In such ca se, C ( T , M ) = M ( T , M ) in ( d ). 2 Remark 2.3. S upp o se U H has a left adjoint F , b e ing the co unit mo rphism of such an adjunction δ : F ( U H ) → 1 M . If the counit morphism of a terminal ob ject δ T : F ( U H )( T ) → T is a split monomorphis m then condition ( d ) necessa rily holds. Notice that all functors U H , consider ed in any instance o f the gro und structure presented in last section 8, hav e a left adjoint, and the res pective counit morphisms δ T of terminal ob jects are iso morphisms, i.e., F pr e serves the terminal ob jects in Set . 3 3. Proper ties of the Reflection It is to b e defined when the r eflection I ⊣ H is 1. simple , 2. semi-left-ex act or 3. to have stable u nits (no tio ns int ro duced in [2]). One ea s ily c hecks from the definitions b elow that if I is a left-exact functor, in which case the reflection is called a lo c alization , then 1., 2. and 3. hold, and that 3 . is s tronger than 2 ., which in turn is strong e r than 1. ( I is left exact ⇒ I ⊣ H has stable units ⇒ I ⊣ H is semi- left-exact ⇒ I ⊣ H is simple). The semi-left-exactness is a lso called admissibility in categorica l Galois theo ry (see [1]). Definition 3.1. The reflec tion I ⊣ H is called simple if the morphism I ( w ) : I ( A ) → I ( C ) is an isomorphism in every diagr am of the for m (1) B C H I ( B ) , H I ( A ) H I ( f ) η B ✲ ✲ ❄ ❄ A f η A w ❅ ❅ ❅ ❅ ❘ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❯ P P P P P P P P P P P P P P q where the rectangula r part o f the diagram is a pullback sq uare, η A and η B are unit morphisms, and w is the unique morphism which makes the diagra m commute. Remark 3.1 . The functor betw een comma categor ies I B : ( C ↓ B ) → ( M ↓ I ( B )), sending f : A → B to I ( f ), has a r ight adjoint H B sending g : M → I ( B ) to its pullback along η B : B → H I ( B ), for each B ∈ C . Hence, I ⊣ H is simple if and only if I B η B is an is omorphism for ev ery B ∈ C , where η B is the unit of the adjunction I B ⊣ H B (equiv alently , ε B I B is an isomo rphism for e very B ∈ C , where ε B is the c o unit of I B ⊣ H B ). Definition 3.2 . The reflec tio n I ⊣ H is called semi-left-exact, or admissible, if the left adjoint I preserves all pullbac k s quares of the form 2 Recall that a full reflective sub category M of C i s closed for li mits in C . 3 Notice that any counit morphism is an isomorphism if it is a monomorphism, pro vided the right adjoint reflects isomorphisms . 4 JO ˜ AO J. XAREZ (2) C C × H I ( C ) M H I ( C ) , M π 1 g η C π 2 ✲ ✲ ❄ ❄ where the b ottom a rrow η C is a unit morphism, and the ob ject M , in the upper corner to the r igh t, is in the sub catego ry M . Remark 3.2. Th e refle c tion I ⊣ H is semi-left-exact if and only if the functor I pr eserves all pullback squar es in which the a rrow in the right edge is in the sub c ategory M , as it is eas y to prov e. Eq uiv a len tly , I ⊣ H is s e mi- left-exact if a nd only if the right adjoint H B is fully faithful ( ε B is an isomorphism) for every B ∈ C . Therefore, the reflec tion is s imple if it is semi-left-exact (cf. r emark 3.1). Definition 3. 3. The re flection I ⊣ H has stable units if the left adjoint I preserves all pullback squa res o f the form (3) C C × H I ( C ) D H I ( C ) , D π 1 g η C π 2 ✲ ✲ ❄ ❄ in which the b ottom arrow η C is a unit morphism. Remark 3.3. O ne could als o show that the r eflection I ⊣ H has stable units if and only if the left a djoin t I preser v es all pullbac k squares in which the ob ject at the right corner in the b ottom belongs to the sub category M . 4. Admissibility and Connected Components Definition 4.1 . Consider any morphism µ : T → H I ( C ) from a termina l ob ject T into H I ( C ), for s ome C ∈ C . The connec ted comp onen t of the mo rphism µ , with r espe c t to the gro und str uc- ture of section 2, is the pullback C µ = C × H I ( C ) T in the following pullback squa re (4) C C µ H I ( C ) . T π µ 1 µ η C π µ 2 ✲ ✲ ❄ ❄ ADMISSIBILITY, ST ABLE UNITS AND CONNECTED COMPONENTS 5 The following Theo r em 4 .1 s ta tes that, under the assumptions given in section 2, in o rder to prov e the semi-left-exactness of the full reflection I ⊣ H , one has only to esta blish the pre s erv ation by I of the pullback squar es like those in diagram (2) in which the ob ject M is termina l. So , in our con text, s emi-left-exactness r educes to connected co mponents b eing “co nnec ted” , in the sense H I ( C µ ) ∼ = T . Notice that H I ( C µ ) ∼ = T if and only if I ( π µ 2 ) is an isomorphism in diagra m (4), since H I ( T ) ∼ = T . The following Lemma 4 .1, which states a trivial res ult in sets, will b e neede d in the pro ofs of the “ if par ts” of Theorems 4 .1 and 5.1. Lemma 4. 1. L et g f b e the c omp osite of a p air f : A → B , g : B → C of s urje c- tions in the c ate gory of set s. Consider the pul lb ack pr 1 : f − 1 g − 1 ( { c } ) → A of t he function ˆ c : {∗ } → C , ˆ c ( ∗ ) = c , along the function g f : A → C , for any element c ∈ C (se e diagr am (5) b elow). Then, the function g is an inje ction if and only if, for every element c ∈ C , f w = ˆ b ! for some function ˆ b : {∗} → B (i.e., f w factorises thr ough a one p oint set), wher e ! denotes the unique function into {∗ } . (5) A f − 1 g − 1 ( { c } ) B C {∗} pr 1 ˆ c f g pr 2 ✲ ✲ ✲ ❄ ❄ {∗} ❄ ˆ b ❳ ❳ ❳ ❳ ③ Theorem 4 .1. Under the assumptions of s e ction 2, t he ful l r efle ction I ⊣ H is semi-left-exact if and only if H I ( C µ ) ∼ = T , for every c onne cte d c omp onent C µ , wher e T is any terminal obje ct. Pr o of. If I ⊣ H is semi-left-ex a ct then, b y Definition 3.2, I ( C × H I ( C ) M ) m ust b e isomorphic to I ( M ) in diag ram (2), since I ( η C ) is an isomo rphism. 4 In particula r , I ( C × H I ( C ) M ) ∼ = I ( T ) if M ∼ = T . Suppo se now that ev ery c onnected co mponent is connected, that is, I ( C µ ) ∼ = T for every µ : T → H I ( C ), C ∈ C , and consider the dia g ram: (6) pr 2 C × H I ( C ) M C gµ H I ( C × H I ( C ) M ) M T pr 1 µ η C × H I ( C ) M H I ( π 2 ) ✲ ✲ ✲ ❄ ❄ H I ( C gµ ) ❍ ❍ ❍ ❍ ❥ H I ( pr 1 ) ❳ ❳ ❳ ❳ ❳ ③ η C gµ C H I ( C ) H I ( C ) . π 1 g η C 1 H I ( C ) ✲ ✲ ❄ ❄ H I ( π 1 ) ❄ 4 ε I ( C ) I ( η C ) = 1 I ( C ) , where ε : I H → 1 M is the counit of the full reflection and therefore an isomorphism. 6 JO ˜ AO J. XAREZ The b ottom re c ta ngle in diagr am (6) is a pullback squar e of the form (2 ), since H I ( π 2 ) η C × H I ( C ) M = η M π 2 and η M is an identit y , beca use M ∈ M (cf. remark 2 .1). According to ( a ), ( b ) and ( c ) in section 2, the reflection I ⊣ H is s emi-left-exact if a nd only if U H I ( π 2 ) is a n injection in Set , in every diagr am (6). The upp er rectangle in diagram (6) (asso ciated to the equation µpr 2 = H I ( π 2 ) η C × H I ( C ) M pr 1 ) is a pullback square, ther efore the outer rec ta ngle in diagra m (6) is in fact a pullback square o f the form (4), and C gµ is the connected compo nen t asso cia ted to g µ : T → H I ( C ). Then, as ( d ) in section 2 ho lds, by Lemma 4.1, U H I ( π 2 ) is an injection since every connected comp onent is connected, in particular H I ( C gµ ) ∼ = T , for any morphisms g : M → H I ( C ), with M ∈ M , and µ : T → M , with T terminal.  5. St able Units Proper ty and Product of Connected Components Theorem 5.1. Under the assumptions of se ction 2, t he ful l r efle ct ion I ⊣ H has stable u nits if and only if H I ( C µ × D ν ) ∼ = T , for every p air of c onne cte d c omp onents C µ , D ν , wher e T is any terminal obje ct. Pr o of. If I ⊣ H has s table units then the functor I prese rves finite pro ducts, since a pro duct diag r am is a pullback square in which the rig h t cor ner in the b ottom is a terminal ob ject T ∈ M (cf. remark 3.3). Therefore, H I ( C µ × D ν ) ∼ = T since H I ( C µ ) ∼ = T ∼ = H I ( D ν ), b y Theorem 4.1, for every pair of connected comp onents C µ , D ν . Suppo se now that every pro duct of t wo connected comp o nen ts is c o nnected, i.e., H I ( C µ × D ν ) ∼ = T for every pair o f mo rphisms µ : T → H I ( C ) and ν : T → H I ( D ), C, D ∈ C , a nd consider the diagram: (7) π 2 C × H I ( C ) D H I ( C × H I ( C ) D ) H I ( D ) η D H I ( π 2 ) D ✲ ✲ ❄ ❄ C H I ( C ) H I ( C ) . π 1 H I ( g ) η C 1 H I ( C ) ✲ ✲ ❄ H I ( π 1 ) ❄ ❍ ❍ ❍ ❍ ❍ ❥ η C × H I ( C ) D C H I ( g ) ν ✲ π H I ( g ) ν 1 C H I ( g ) ν × D ν ❄ p 1 D ν ❄ π ν 1 ✲ p 2 ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❥ w The inside rectang le in diagr am (7) is a pullba c k square of the form (3), since H I ( g ) η D = η H I ( C ) g and η H I ( C ) is an identit y , b ecause H I ( C ) ∈ M (cf. remark 2.1). According to ( a ), ( b ) and ( c ) in section 2, the r eflection I ⊣ H has stable units if and only if U H I ( π 2 ) is an injection in Set , for every diagr am of the form (3). In fact, U H I ( π 2 ) is obviously a surjection, since U H I ( π 2 ) U ( η C × H I ( C ) D ) = U ( η D ) U ( π 2 ) and ADMISSIBILITY, ST ABLE UNITS AND CONNECTED COMPONENTS 7 U ( η C × H I ( C ) D ), U ( η D ) and U ( π 2 ) are all surjections by the assumptions in section 2. The morphisms p 1 and p 2 in diagr am (7) are the pro duct pro jections of the pro duct o f the c o nnected co mponents C H I ( g ) ν and D ν . The morphis m w is the unique morphism which makes diagr am (7) commute; it is well defined since H I ( g ) η D π ν 1 p 2 = H I ( g ) ν π ν 2 p 2 = = H I ( g ) ν π H I ( g ) ν 2 p 1 (beca use bo th π ν 2 p 2 and π H I ( g ) ν 2 p 1 hav e the same doma in a nd co doma in, the latter b eing the terminal ob ject T ) = η C π H I ( g ) ν 1 p 1 . Then, as ( d ) in section 2 holds, by Lemma 4.1, U H I ( π 2 ) is an injection if the outer rectangle in the following diag r am is a pullback squar e, for every mo rphism ν : T → H I ( D ) fro m the terminal ob ject into H I ( D ) (cf. dia g ram (5 )): (8) C × H I ( C ) D C H I ( g ) ν × D ν ✲ ✲ H I ( π 2 ) H I ( D ) . ❄ T ν p 2 ✲ D ν π ν 2 ✲ ❄ H I ( w ) η C H I ( g ) ν × D ν H I ( C × H I ( C ) D ) w η C × H I ( C ) D ❄ H I ( C H I ( g ) ν × D ν ) ❳ ❳ ❳ ❳ ③ In order to show that the outer rectangle in dia gram (8) is a pullback squa r e, consider a mor phis m l : A → C × H I ( C ) D such tha t H I ( π 2 ) η C × H I ( C ) D l = ν !. Let ¯ l = h l 1 , l 2 i : A → C H I ( g ) ν × D ν be the mor phis m into the pr oduct of the t wo connected comp onents, in which l 1 : A → C H I ( g ) ν and l 2 : A → D ν are the morphisms deter mined in the pullback squares of the connec ted comp onents by π C H I ( g ) ν 1 l 1 = π 1 l a nd π D ν 1 l 2 = π 2 l , resp ectively . It is then a routine calculation to verify that w is a mono morphism and w ¯ l = l .  Remark 5.1. It is an immediate co nsequence of Theorems 5.1 and 4.1 that, pr o- vided the pres erv ation of finite pro ducts by the left adjoint I is added to the as - sumptions of sectio n 2, the reflection I ⊣ H has s table units if a nd o nly if it is semi-left-exact. 6. Left-Exactness and Pullbacks of Connected Components The following Theorem 6.1 gives a sufficient condition for the r eflection I ⊣ H to b e a lo calization, that is , fo r the le ft adjoint I to be left exact (see section 3). Theorem 6 .1. Under the assumptions of s e ction 2, t he ful l r efle ction I ⊣ H is a lo c alization if H I ( A µ × C B ν ) ∼ = T , for every pul lb ack A µ × C B ν of any p air of c onne cte d c omp onents A µ , B ν , wher e T is any terminal obje ct. That is, t he left adjoint is left exact if every pul lb ack of c onne cte d c omp onents is c onne cte d. Pr o of. Conside r the diagram 8 JO ˜ AO J. XAREZ (9) A µ ✲ π µ 1 A    ✒ η A ✲ f C , ❅ ❅ ❅ ■ η C ❄ g B   ✠ η B ❄ π ν 1 B ν A µ × C B ν ❄ p 1 ✲ p 2 ❅ ❅ ❅ ❘ j A × C B ✲ π 2 ❄ π 1 ❅ ❅ ❘ η A × C B H I ( A ) H I ( f ) ✲ H I ( C ) H I ( B ) ❄ H I ( g ) H I ( A × C B ) ❄ H I ( π 1 ) H I ( π 2 ) ✲ ❅ ❅ ❘ w H I ( A ) × H I ( C ) H I ( B )    ✒ ✁ ✁ ✁ ✁ ✁ ✁ ☛ wherein A µ × C B ν = A µ × ( f π µ 1 ,gπ ν 1 ) B ν and H I ( A ) × H I ( C ) H I ( B ) = H I ( A ) × ( H I ( f ) ,H I ( g )) H I ( B ) ar e pullba cks, and j and w are the unique morphisms making the dia gram commute. One has to prov e tha t U ( w ) is alw ays a bijection. It follows from I ( A µ × C B ν ) ∼ = T that U ( A µ × C B ν ) 6 = ∅ , fo r a ll connected comp onents A µ , B ν , whic h implies that U ( w ) is a surjection, under the assumptions of section 2. Note that U ( η A × C B ) − 1 U ( w ) − 1 ( U ( A µ ) , U ( B ν )) = U ( A µ × C B ν ) in Set , which implies that U ( w ) is an injectio n, since U H I ( j ) U ( η A µ × C B ν ) = U ( η A × C B ) U ( j ) and U H I ( A µ × C B ν ) = {∗} .  7. Admissibility of a Simple Reflection Theorem 7.1. L et the fol lowing c ondition and al l assumptions of se ction 2 hold: ( e ) every map I T , C : C ( T , C ) → M ( T , I ( C )) , the r estriction of the r efle ctor I to the hom-set C ( T , C ) , is a surje ction, for every obje ct C ∈ C , with T = H I ( T ) a terminal obje ct in C . Then, the r efle ction I ⊣ H is semi-left-exact if and only if it is simple. Pr o of. Supp ose that I ⊣ H is a simple reflection, that is, I ( w ) is an iso morphism in ev ery diagr am of the form (1) in Definition 3.1, and co nsider the pullback sq uare (4) in Definition 4.1. Let w : T → C µ be the unique morphism such that π µ 1 w = ν and π µ 2 w = 1 T , where ν is s uc h tha t H I ( ν ) = µ ( ν exists by ( e ) in the sta tement). ADMISSIBILITY, ST ABLE UNITS AND CONNECTED COMPONENTS 9 Note that the comp osite I ( π µ 2 ) I ( w ) is the is omorphism 1 T . Ther e fo re, I ( π µ 2 ) is an isomorphism, since I ( w ) is an is omorphism by ass umption.  8. Examples 1. Consider the full r eflection o f compact Hausdorff spaces in to Stone spaces H ⊢ I : CompH aus → Stone , where ea c h unit map η X : X → H I ( X ) is the canonical pro jectio n of X in to the set of its comp onents, this s et being given the quotient topolog y with resp ect to η X . Hence, co ndition ( c ) in section 2 ho lds for the functor U which forgets the to polo gy . Conditions ( a ) and ( b ) of section 2 hold as well since U : CompHaus → Set is mo nadic, a nd condition ( d ) holds trivia lly . This reflection is known to hav e stable units, there fore finite pro ducts of connected comp onent s are connected. Let ˆ 0 : T → [0 , 1] a nd ˆ 1 : T → [0 , 1] b e the tw o o b vious inclusions of the one po in t top ologica l space into the closed interv al o f real num ber s [0 , 1], with the usua l top ology . Then, the pullback T × ( ˆ 0 , ˆ 1) T = ∅ is the empt y space, not connected in our sense, b e ing clea r that the reflector I is not left exact, since it do es not pre serve the pullback diagram of ˆ 0 a nd ˆ 1, and also that the sufficient co ndition of Theor e m 6.1 do es not hold. 2. With the e xception of ( d ), every assumption of s ection 2 hold for a n y reflection from a v ariet y of universal algebras into one o f its subv arieties, provided with the forgetful functor into Set . Notice that, for these r eflections, condition ( d ) of section 2 is equiv alent to idemp o tency of the algebras in the sub v ariety , meaning that every element of an a lg ebra in the subv a riety is a subalg e bra. In particula r , it is easy to chec k that condition ( d ) in section 2 ho lds for the reflection H ⊢ I : SGr → SLat of semigroups into semilattices, which is k no wn to hav e stable units (see [3]). T her efore, all finite pro ducts o f connected components are connected. The additive s e mig roup N of non-nega tiv e int egers ha s tw o co nnected comp o- nent s, { 0 } and { 1 , 2 , 3 , ... } , with r espec t to the r e flection SGr → SLat . The pull- back of the inclusio ns { 0 } → Z and { 1 , 2 , 3 , ... } → Z into the integers is the empty semigroup ∅ , which is not connected ( I ( ∅ ) = ∅ is not terminal). Hence, this r e flec- tion is not a lo calization, and a lso the sufficient condition of Theo r em 6.1 do es not hold. The re fle c tion H ⊢ I : SGr → Band of se migroups int o bands 5 is not a s emi- left-exact r eflection (cf. [3]). No twithstanding, all as sumptions in section 2 hold for this reflec tio n; therefore not every connected co mponent is connected, by Theorem 4.1 (see Exa mple 7 in [3]). Note that Theo rem 7.1 ho lds for the r eflection H ⊢ I : Band → SLat from bands into semilattices (a subr eflection of SGr → SLat ). 6 5 A semigroup is called a band i f every one of its element s is idemp oten t. 6 Remark that in algebraic instances 2., condition ( d ) i n the ground structure i s crucial, whil e condition ( b ) is the crucial one in the f ormer top ological instances 1. 10 JO ˜ AO J. XAREZ 3. Finally , we would like to remar k that the joining of new ge ometric al examples, to the algebr aic and top olo gic al well-kno w n exa mples above, has b een made p ossible by a generaliz a tion of the assumptions in the ground structure, done in [6], where a new class of instances is presented. References [1] Carb oni, A., Janelidze, G., Kelly , G. M ., Par ´ e, R., On lo ca lization and stabilization for factorization systems , App. Cat. Struct. 5 (1997) 1–58. [2] Cassidy , C., H ´ eb ert, M. , Kelly , G. M., R efle ctiv e sub c atego ries, lo ca lizations and factorization systems , J. Austral. Math. So c. 38 A (1985) 287–329. [3] Janelidze, G., Laan, V., M´ ar ki, L., Limit pr e se rvation pr op ertie s of the gre atest semilattic e image functor , In ternat. J. Algebra Comput. 5 (2008) 853–867. [4] Mac Lane, S., Categories for the Working Mathematician , 2nd ed., Springer, 1998. [5] T am ura, T., Attainability of systems of identities on semigr oups , J. Algebra 3 (1966) 261–276 . [6] Xarez, J. J. Gener alising co nne cte d c omp onents , J. Pure Appl. Algebra, acce pted. Dep ar t a mento de Ma tem ´ atica, Universidade de A veiro. Campus Universit ´ ario de San- tiago. 3810-19 3 A veiro. Por tugal. E-mail addr ess : xarez@ua.pt