A general method for building reflections

A general metho d for building reflections Olivia Caramello DPMMS, Univ ersit y of Cam bridge , Wilb erforce Road, Cam b ridge CB3 0WB, U.K. O.Caramello @dpmms.cam.ac.uk ∗ December 15, 2011 Abstract W e establish a general method for generating reflections b etw een categories. W e then apply our tec hnique to generate adjunctions s tart- ing from geometric morphisms b et w een Grothendiec k top oses; as par- ticular cases, w e recov er v arious wel l-kno wn Stone-t yp e adjunctions and establish sev eral new ones. 1 In tro duction A djunction is a fundamen tal relationship b et w een pairs of categories. An illuminating p oint of view on this not io n is pro vided b y the concept of comma category , originally in tro duced by F. W. La wv ere in his Ph.D. thesis [6]; indeed, the fa ct that t w o functors F : A → B and G : B → E b et wee n a giv en pair of categories are a djo in t to eac h o t her can b e expressed as the existence of an equiv alence b etw een the tw o comma categor ies ( F ↓ 1 B ) and (1 A ↓ G ) , and an adjunction F ⊣ G can b e obtained b y comp osing a coreflection b et w een A and ( F ↓ 1 B ) with the equiv alence ( F ↓ 1 B ) ≃ (1 A ↓ G ) and then with a reflection b etw een (1 A ↓ G ) and B . The comma category thus acts as a ‘bridge’ ob ject whic h condens es the infor matio n ab out the adjunction (via its t wo differen t represen tations) and connects the tw o categories with eac h other. ∗ The author g ratefully ackno wledges the suppor t of a Research F ello wship from Jes us College, Cambridge ( U.K.) 1 The asp ect of the comma category construction whic h constitutes the inspiration for the presen t pap er is the fact that this construction sho ws that the relationship b et w een t w o categories A and B related by a pair of adjoint functors F and G may w ell b e b est understo o d from the p oint of view of a third category , namely the comma category ( F ↓ 1 B ) ≃ (1 A ↓ G ) , in t o whic h the tw o categories A and B (canonically) em b ed. This naturally leads to the idea of a general metho d for building a djunc- tions b etw een a giv en pair of categories starting from a pair of functors from eac h of the tw o categor ies in to a third one, together with some relationships b et wee n them. In this pap er, w e sho w that this in tuition can b e materialized in a precise tec hnical sense, by pro viding in section 2 a general method for building reflections b etw een categories starting f r o m data of that kind. As sho wn in section 2.3, our metho d is c omplete , in the sense that any reflection b et wee n categories can b e obtained as an application of it. The inte rest o f our metho d lies in its inheren t tech nical flexibilit y; in- deed, it happ ens very often in practice that t w o differen t categories are b est understoo d in relationship with each other from the p oin t of view of a third category to whic h b oth are related (cf. for example [2] for an explanation of the sense in whic h Grothendiec k top oses can act a s ‘bridges’ connecting differen t mathematical theories with each other), and our metho d allo ws us to establish reflections betw een a giv en pair of categories starting from re- lations b et w een the ‘realizations’ of the tw o categories at the lev el of the ‘bridge category’ to whic h b oth categories map. In [3] we sho w ed that many ‘Stone-t yp e’ dualities or equiv alences b et w een categories of preorders and categories o f p osets, locales and top ological spaces can b e nat ura lly in terpreted as arising from the pro cess of appropriately ‘functorializing’ categorical equiv a lences b etw een top oses; in that pap er, w e also established v arious adjunctions b et w een categories of these kinds whic h extend t he given dualities or equiv alences. Amongst o t her things, in section 3 of this pap er, we sho w as applications of our general metho d that all the ‘Stone-t yp e’ reflections obtained in that pap er can b e seen as applications of our metho d for building reflection in the con text where the ‘bridge category’ is some category of top oses o r of top o ses paired with p oints (as defined in [3]). 2 2 The general metho d In this section w e desc rib e our general method for generating reflections b et wee n categories. The set of data w e shall work with consists of t w o categories H and K , a category U , t w o functors I : H → U , J : K → U and t w o binary relations R and S on O b ( H ) × O b ( K ) . In addition to this, w e supp ose to ha v e, for ev ery ( C , D ) ∈ R , an arrow ξ ( C , D ) : I ( C ) → J ( D ) in U and for ev ery ( C , D ) ∈ S an arrow χ ( C , D ) : J ( D ) → I ( C ) in U , and tw o functions Z : R → S and W : S → R suc h that Z k eeps the second comp onen t fixed and W k eeps the first comp onent fixed. Let us denote b y π H R and π K R respectiv ely the canonical pro j ections R ֒ → H × K → H and R ֒ → H × K → K ; similarly , w e denote b y π H S and π K S respectiv ely the canonical pro jections S ֒ → H ×K → H and S ֒ → H ×K → K . W e imp ose the following h yp otheses. 1. F or an y ( C , D ) ∈ R , Z (( C , D )) = ( π H S ( Z (( C , D )) ) , D ) ; we require the comp osite χ Z ( C , D ) ◦ ξ ( C , D ) : I ( C ) → J ( π H S ( Z (( C , D )) )) to b e induced by a (canonically c hosen) isomorphism in H ǫ ′ ( C , D ) − 1 : C → π H S ( Z (( C , D )) ) , as in the f ollo wing diagra m: I ( C ) ξ ( C , D ) / / I ( ǫ ′ ( C , D ) − 1 ) ' ' ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ J ( D ) χ Z ( C , D )   I ( π H S ( Z (( C , D )) )) 2. F or any ( C , D ) ∈ S , W (( C , D )) = ( C , π K R ( W (( C , D ))) ) ; we require the comp osite ξ W ( C , D ) ◦ χ ( C , D ) : J ( D ) → J ( π K R ( W (( C , D ))) ) to b e induced by a (canonically c hosen) morphism η ′ ( C , D ) : D → π K R ( W (( C , D ))) in K , as in the follo wing diagram: J ( D , K D ) χ ( C , D ) / / I ( η ′ ( C , D ) ) ( ( ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ I ( C ) ξ W ( C , D )   J ( π K R ( W (( C , D ))) ) 3 3. for any ( C , D ) ∈ R , the arrow ξ ( C , D ) : I ( C ) → J ( D ) is an isomorphism in U . [Note that, since ǫ ′ ( C , D ) is an isomorphism , this implies that χ Z ( C , D ) : J ( D ) → I ( π H S ( Z (( C , D )) )) is a n isomorphis m as w ell.] 4. F or any ( C , D ) ∈ R , η ′ Z (( C , D )) is a n isomorphis m. Let us define tw o categories ˜ R and ˜ S , as follows . The ob jects of ˜ R are the elemen ts of R while the arro ws ( C , D ) → ( C ′ , D ′ ) are the pairs ( u, v ) where u : C → C ′ , v : D → D ′ are arrow s resp ectiv ely in the categories H , K a nd such tha t the f o llo wing square comm utes: I ( C ′ ) ξ ( C ′ , D ′ )   I ( u ) / / I ( C ) ξ ( C , D )   J ( D ′ ) J ( v ) / / J ( D ) W e will o ccasionally write ( u, v , z ) for ( u, v ) , where z is the arrow π H S ( Z (( C , D )) ) → π H S ( Z (( C ′ , D ′ ))) in H g iv en b y the factorization of u across the isomorphisms ǫ ′ ( C , D ) and ǫ ′ ( C ′ , D ′ ) . The comp osition of arrows in ˜ R is defined as the comp osition of the functors fo rming the v arious comp onen ts. Similarly , w e define the category ˜ S . The ob jects of ˜ S are the elemen ts of S while the arrows ( C , D ) → ( C ′ , D ′ ) are the triples ( z , v , w ) , where v : D → D ′ , z : C → C ′ and w : π K R ( W (( C , D ))) → π K R ( W (( C ′ , D ′ ))) are morphisms resp ectiv ely in the categories K , H and K suc h that the tw o squares in the following diagram comm ute: 4 J ( D ′ ) χ ( C ′ , D ′ )   J ( v ) / / J ( D ) χ ( C , D )   I ( C ′ ) ξ ( C ′ , D ′ )   I ( z ) / / I ( C ) ξ ( C , D )   J ( π K R ( W (( C ′ , D ′ )))) J ( w ) / / J ( π K R ( W (( C , D ))) ) . Comp osition of arrows in ˜ S is defined comp onen t wise as the comp osition of the functors forming the v arious comp onen ts. Let us no w define functors ˜ Z : ˜ R → ˜ S and ˜ W : ˜ S → ˜ R , whic h extend the functions Z : R → S and W : S → R . F or an y ob ject ( C , D ) of ˜ R , w e set ˜ Z (( C , D )) = Z (( C , D )) and for a ny arro w ( u, v , z ) : ( C , D ) → ( C ′ , D ′ ) in ˜ R we set ˜ Z (( u, v , z )) equal to the t riple ( z , v , w ) , where w : π K R ( W ( Z ( C , D )) → π K R ( W ( Z ( C ′ , D ) ′ ) is the only arrow in K making the follow ing diagr a m commute (recall that b y our h yp otheses η ′ Z ( C , D ) and η ′ Z ( C ′ , D ′ ) are isomorphisms). D η ′ Z ( C , D )   v / / D ′ η ′ Z ( C ′ , D ′ )   π K R ( W ( Z ( C , D ))) w / / π K R ( W ( Z ( C ′ , D ) ′ )) T o sho w that ( z , v , w ) is an arrow in ˜ S w e ha v e to che c k that the diagram J ( D ′ ) χ ′ D   J ( v ) / / J ( D ) χ ( C , D )   I ( π H S ( Z (( C ′ , D ′ ))) ′ ) ξ ( π H S ( Z (( C , D ))))   I ( z ) / / I (( π H S ( Z (( C , D )) ))) ξ ( π H S ( Z (( C , D ))))   J ( π K R ( W ( Z (( C ′ , D ′ ))))) J ( w ) / / J ( π K R ( W ( Z (( C , D ))))) . comm utes. The top square commu tes since ( u , v , z ) is an arro w ( C , D ) → ( C ′ , D ′ ) in ˜ R . It remains to prov e the comm utativit y of the bot t o m square; but this is equiv alen t to the comm utativit y of the outer rectangle, since χ ′ D and χ D are isomorphisms (cf. our assumptions ab o v e), and to show this it suffices to in vok e the comm utativit y of the square defining w . This completes the pro of that o ur assignme n t defines a functor ˜ Z : ˜ R → ˜ S . Let us no w turn to the definition of the functor ˜ W : ˜ S → ˜ R . F or 5 an y ( C , D ) in ˜ S , we set ˜ W (( C , D )) equal to W (( C , D )) and for any arrow ( z , v , w ) : ( C , D ) → ( C ′ , D ′ ) in ˜ S , w e set ˜ W (( z , v , w )) = ( z , w ) . T his is clearly an arrow in ˜ R and hence this assignmen t actually defines a f unctor, as required. W e are now ready to state our main result. Theorem 2.1. Under the hyp otheses sp e cifie d ab ove, the functors ˜ Z : ˜ R → ˜ S and ˜ W : ˜ S → ˜ R a r e adjoint to e ach other; in fact, they yield a r efle ction in which ˜ Z is the right adjoint and ˜ W is the left adjoint. Pro of W e shall establish this adjunction b y giving its unit and counit and c hec king that they are natural transformations satisfying the triangu- lar iden tities. W e tak e as counit the transformation ǫ sending each ob- ject ( C , D ) of ˜ R the pair ǫ ( C , D ) := ( ǫ ′ ( C , D ) , η ′ Z ( C , D ) − 1 ) , regarded as an arrow ˜ W ( ˜ Z (( C , D ))) = ( π H S ( Z (( C , D )) ) , π K R ( W ( Z ( C , D )))) → ( C , D ) in ˜ R . This is indeed an arro w in ˜ R b y the comm utativity of the fo llo wing diagram. I ( C ) ξ ( C , D )   I ( π H S ( Z (( C , D )) )) I ( ǫ ′ ( C , D ) ) o o ξ W ( Z (( C , D )))   J ( D ) χ Z ( C , D ) 3 3 ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ J ( π K R ( W ( Z ( C , D )))) J ( η ′ Z ( C , D ) − 1 ) o o Note that ǫ ( C , D ) = ( ǫ ′ Z ( C , D ) , η ′ ( C , D ) − 1 , z ) where z is the unique morphism making the fo llowing diagram comm ute: C π H S ( Z ( C , D )) ǫ ′ ( C , D ) o o π H S ( Z ( C , D )) ǫ ′ ( C , D ) O O π H S ( Z ( W ( Z (( C , D ))))) z o o ǫ ′ W ( Z ( C , D )) O O that is, z = ǫ ′ W ( Z ( C , D )) . So we hav e ǫ ( C , D ) = ( ǫ ′ ( C , D ) , η ′ Z ( C , D ) − 1 , ǫ ′ W ( Z ( C , D )) ) . The transformation ǫ is natural b ecause eac h o f its components is arise from comp ositions of (morphisms whic h are naturally isomorphic t o ) the morphisms ξ and χ or their inv erses and these morphisms are natural b y the v ery definition of the arrows in the categories ˜ R and ˜ S . Let us no w define a natural transformation η : 1 ˜ S → ˜ Z ◦ ˜ W , which will serv e as the unit of our adjunction. First, recall that if X is a n ob ject o f ˜ S then the a rro w η ′ X : X = ( π H S ( X ) , π K S ( X ) ) → ˜ Z ( ˜ W ( X ) ) = ( π H S ( ˜ Z ( ˜ W ( X ))) , π K h ( ˜ Z ( ˜ W ( X )))) 6 has as second comp onen t π K S ( X ) → π K S ( ˜ Z ( ˜ W ( X ))) the in ve rse o f the isomor- phism ǫ ′ Z ( X ) : π K R ( ˜ Z ( ˜ W ( X ) )) → π K S ( X ) . F or any ob ject ( C , D ) in ˜ S , w e define η ( C , D ) to b e the triple η ( C , D ) := ( ǫ ′ W ( C , D ) − 1 , η ′ ( C , D ) , η ′ Z ( W ( C , D )) ) . Let us chec k that this triple actually defines an a r r ow ( C , D ) → ˜ Z ◦ ˜ W ( C , D ) = ( π H S ( Z ( W ( C , D ))) , π K R ( W ( C , D ))) in the category ˜ S . This amounts to v erifying that the both the sq uares in the follo wing diagram a r e comm utativ e. Note t ha t, since Z preserv es the second comp onen t, W ( π H S ( Z ( W ( C , D ))) , π K R ( W ( C , D ))) = W ( π H S ( Z ( W ( C , D )) , π K S ( Z ( W ( C , D ))) = W ( Z )( C , D ) and hence in the diag r a m b elo w w e ha ve π K R ( W ( π H S ( Z ( W ( C , D ))) , π K R ( W ( C , D )))) = π K R ( W ( Z ( W ( C , D ))) ) . J ( D ) χ D   J ( η ′ ( C , D ) ) / / J ( π K R ( W ( C , D ))) χ π K R ( W ( C , D ))   I ( C ) ξ W ( C , D )   I ( ǫ ′ W ( C , D ) − 1 ) / / I ( π H S ( Z ( W ( C , D )))) ξ π H S ( Z ( W ( C , D )))   J ( π K R ( W (( C , D ))) ) J ( η ′ Z ( W ( C , D )) ) / / J ( π K R ( W ( Z ( W ( C , D ))) )) . No w, the b ott o m square o b viously comm utes (cf. the diag ram ab ov e made of tw o tria ngles, whic h w e observ ed to comm ute), while the comm utativit y of the top square can b e prov ed b y observing that in the follow ing diagra m all the in ternal tr a p ezoids are comm utativ e whence the outer square is. 7 J ( D ) χ ( C , D ) ) ) ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ χ ( C , D )   J ( η ′ ( C , D ) ) / / J ( π K R ( W ( C , D ))) χ π K R ( W ( C , D ))   I ( C ) ξ W ( C , D )   ξ W ( C , D ) 4 4 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ J ( π K R ( W ( C , D ))) χ π K R ( W ( C , D )) * * ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ I ( π K R ( W (( C , D ))) ) ξ W ( C , D ) 5 5 ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ I ( ǫ ′ W ( C , D ) − 1 ) / / I ( π K R ( W ( Z ( W ( C , D ))) )) . The naturalit y of η is immediate from the fa ct that eac h of its componen ts is a composition of (morphisms naturally isomorphic to) the morphisms ξ and χ or their in v erses, and these morphisms a r e natural b y the ve ry definition of the arr ows in the categories ˜ R and ˜ S . So far, w e ha ve sho wn that η and ǫ are na t ura l transformations resp ec- tiv ely 1 ˜ S → ˜ Z ◦ ˜ W and ˜ W ◦ ˜ Z → 1 ˜ R . T o conclude that they are the unit and counit of an adjunction b etw een the functors ˜ Z and ˜ W , it remains to show that they satisfy the triangular iden tities. First, let us prov e that for ev ery ( C , D ) ∈ ˜ R ˜ Z ( ǫ ( C , D ) ) ◦ η ˜ Z ( C , D ) = 1 Z (( C , D )) . By definition of the functor ˜ Z , ˜ Z ( ǫ ( C , D ) ) = ( ǫ ′ W ( Z (( C , D ))) , η ′ Z ( C , D ) − 1 , w ) , where w is the unique (iso)morphism making the follow ing diagram comm ute; π K S ( Z ( W ( Z (( C , D ))))) η ′ Z ( W ( Z ( C , D )))   η ′ Z ( C , D ) − 1 / / D ′ η ′ Z ( C , D )   π K R ( W ( Z ( W ( Z (( C , D )))))) w / / π K S ( Z ( W ( Z (( C , D ))))) That is, w = η ′ Z ( W ( Z ( C , D ))) − 1 . Summarizing, ˜ Z ( ǫ ( C , D ) ) = ( ǫ ′ W ( Z (( C , D ))) , η ′ Z ( C , D ) − 1 , η ′ Z ( W ( Z ( C , D ))) − 1 ) . On the other hand, w e ha v e η Z (( C , D )) := ( ǫ ′ W ( Z (( C , D ))) − 1 , η ′ Z (( C , D )) , η ′ Z ( W ( Z (( C , D )))) ) . F rom these t w o expressions w e thus conclude that for an y ( C , D ) ∈ ˜ R , ˜ Z ( ǫ ( C , D ) ) ◦ η ˜ Z ( C , D ) = 1 Z (( C , D )) , as required. It remains to prov e that the other triangular iden tit y holds, i.e. that for an y ob ject ( C , D ) of ˜ S , ǫ ˜ W ( C , D ) ◦ ˜ W ( η ( C , D ) ) = 1 W ( C , D ) . Now, w e hav e that 8 ǫ ˜ W ( C , D ) = ( ǫ ′ W (( C , D )) , η ′ Z ( W ( C , D )) − 1 , ǫ ′ W ( Z ( W ( C , D ))) ) On the other hand, η ( C , D ) = ( ǫ ′ W ( C , D ) − 1 , η ′ ( C , D ) , η ′ Z ( W ( C , D )) ) , whence ˜ W ( η ( C , D ) ) = ( ǫ ′ W ( C , D ) − 1 , η ′ Z ( W ( C , D )) ) . F rom this it is immediate to conclude that ǫ ˜ W ( C , D ) ◦ ˜ W ( η ( C , D ) ) = 1 W ( C , D ) , as required. T o finish the pro of of the theorem, it remains to observ e that the adjunc- tion just generated is a reflection b ecause η ( C , D ) is an isomorphism in ˜ R for ev ery ob ject ( C , D ) of ˜ R , since b oth of its comp onents are isomorphisms.  One migh t naturally w onder whether our metho d for building reflections (or coreflections) can b e g eneralized to a metho d for building a rbitrary ad- junctions; the answ er to this question is p ositiv e, but w e shall not address this issue here since the details of suc h generalizations w ould bring us to o far from the scope o f this pap er, a nd w ould b e considerably more tec hnical than the treatmen t just giv en for reflections, to the exten t that the ratio b e- t w een tec hnical sophistication and conc eptual unde rstanding could b e judged rather un balanced b y most readers. On the other hand, considering that ev ery adjunction can b e naturally obta ined as a comp osite of a reflection with a coreflec tion (through the comma category construction), the c hoice of concen trating the ana lysis on reflections do es not app ear as a particularly restrictiv e one. 2.1 Dualit y principles W e note that, giv en a set of data ( H , K , U , I , J, R, S, Z , W ) satisfying the conditions of our metho d, the set of data ( K , H , U , J , I , R op , Z op , S op , W op , Z op ) also satisfies them. There is another duality principle implicit in the con text of our main result, whic h can b e fruitfully exploited as a w ay of building reflections as instances of the theorem of the last section. W e can illustrate this as follows . 9 Supp ose hav ing t w o categories H and K , a category U , t w o functors I : H → U , J : K → U and t w o binary relations R and S on O b ( H ) × O b ( K ) . In addition to this, suppo se havi ng, fo r eve ry ( C , D ) ∈ R , a n arrow α ( C , D ) : J ( C , D ) → I ( C , D ) in U , for ev ery ( C , D ) ∈ S a n arro w β ( C , D ) : I ( C , D ) → J ( C , D ) in U , and hav ing furthermore tw o functions Z : R → S and W : S → R suc h that Z k eeps the second comp onen t fixed and W k eeps the first comp onent fixed, whic h satisfy the follo wing conditions: 1. F or an y ( C , D ) ∈ R , Z (( C , D )) = ( π H S ( Z (( C , D )) ) , D ) ; we require the comp osite α ( C , D ) ◦ β W R ( C , D ) : J ( π H S ( W R (( C , D )))) → I ( C ) to b e induced by a (canonically c hosen) morphism in H λ ′ ( C , D ) − 1 : C → π H S ( Z (( C , D )) )) . 2. F or any ( C , D ) ∈ S , W (( C , D )) = ( C , π K R ( W (( C , D ))) ) ; we require the comp osite β ( C , D ) ◦ α W ( C , D ) : I ( π K R ( W S (( C , D )))) → J ( D ) to b e induced b y a ( canonically c ho sen) morphism in K µ ′ ( C , D ) : D → π K R ( W S (( C , D ))) ; 3. for any ( C , D ) ∈ R R , the arrow α ( C , D ) : J ( D ) → I ( C ) is a n isomorphis m. [Note that, since λ ′ ( C , D ) is a n isomorphis m, β Z ( C , D ) : I ( π H S ( Z (( C , D )) )) → J ( D ) is a n isomorphis m as w ell.] 4. F or any ( C , D ) ∈ R , µ ′ Z (( C , D )) is a n isomorphism. Then the set of data ( H op , K op , U op , I op , J op , R, S, Z, W ) satisfies the h yp otheses of our main theorem. The theorem th us yields a coreflection b et we en the category ˜ R and the category ˜ S , g iv en b y functors ˜ Z : ˜ R → ˜ S and ˜ W : ˜ S → ˜ R , equiv alen tly a 10 reflection b et w een the category ˜ R op and the category ˜ S op , giv en b y functors ˜ Z op : ˜ R op → ˜ S op and ˜ W op : ˜ S op → ˜ R op . The categories and functors in volv ed in this adjunction can b e describ ed explicitly as follo ws. The o b jects o f ˜ R op are the elemen ts of R while the a rro ws ( C , D ) → ( C ′ , D ′ ) are the pairs ( u, v ) , where u : C → C ′ and v : D → D ′ are arro ws resp ectiv ely in the categories H and K suc h that the follo wing diagram comm utes: I ( C ′ ) I ( u ) / / I ( C ) J ( D ′ ) α ( C ′ , D ′ ) O O J ( v ) / / J ( D ) α ( C , D ) O O W e shall o ccasionally write ( u, v , z ) for ( u, v ) , where z is the arro w π H S ( Z (( C , D )) ) → π H S ( Z (( C ′ , D ′ ))) in H giv en by the factorization of u across the isomorphisms µ ′ ( C , D ) and µ ′ ( C ′ , D ′ ) . The composition of arro ws in ˜ R op is defined as the composition of the functors fo rming the v arious comp onen ts. Similarly , w e define the category ˜ S op . The ob jects of ˜ S op are the elemen ts of S while the arro ws ( C , D ) → ( C ′ → D ′ ) are the triples ( z , v , w ) , where v : D → D ′ , z : C → C ′ and w : π K R ( W (( C , D ))) → π K R ( W (( C ′ , D ′ ))) are morphisms resp ectiv ely in the categories K , H and K suc h that the tw o squares in the following diagram comm ute: 11 J ( D ′ ) J ( v ) / / J ( D ) I ( C ′ ) β ( C ′ , D ′ ) O O I ( z ) / / I ( C ) β ( C , D ) O O J ( π K R ( W (( C ′ , D ′ )))) J ( w ) / / α ( C ′ , D ′ ) O O J ( π K R ( W (( C , D ))) ) α ( C , D ) O O . The comp osition of arrows in ˜ S op is defined as the comp osition of the functors fo rming the v arious comp onen ts. The functors ˜ Z op : ˜ R op → ˜ S op and ˜ W op : ˜ S op → ˜ R op can b e describ ed as follo ws. F or any ob ject ( C , D ) of ˜ R , ˜ Z op (( C , D )) = Z (( C , D )) and for an y a r r ow ( u, v , z ) : ( C , D ) → ( C ′ , D ′ ) in ˜ R op w e set ˜ Z op (( u, v , z )) equal to the triple ( z , v , w ) , where w : π K R ( W ( Z ( C , D )) → π K R ( W ( Z ( C ′ , D ) ′ ) is the only arrow in K making the followi ng diagram comm ute. π K R ( W ( Z ( C , D ))) w / / π K R ( W ( Z ( C ′ , D ) ′ )) D λ ′ Z ( C , D ) O O v / / D ′ λ ′ Z ( C ′ , D ′ ) O O F or a n y ( C , D ) in ˜ S , ˜ W op (( C , D )) = W (( C , D )) and for an y arrow ( z , v , w ) : ( C , D ) → ( C ′ , D ′ ) in ˜ S , ˜ W (( z , v , w )) = ( z , w ) . 2.2 F rom the relational to the functional con text In this section w e sp ecialize our general metho d for generating reflections to the con text of functional relations R and S . In fact, the main reason b ehind our choic e of a relational contex t for formulating our theorem in section 2 is the f a ct that this more general con text provide s us with dualit y principl es (cf. section 2.1 abov e) whic h do not hold in the restricte d functional con text. Supp ose ha ving tw o categories H a nd K , a category U , t wo functors I : H → U , J : K → U , tw o functions f : O b ( H ) → O b ( K ) , g : O b ( K ) → O b ( H ) , for eve ry ob ject C ∈ H an a rro w ξ C : I ( C ) → J ( f ( C )) in U , and for ev ery elemen t D ∈ K an arrow χ D : J ( D ) → I ( g ( D )) 12 in U . Assume that the comp osite χ f ( C ) ◦ ξ C is of the form I ( η C ) for some mor- phism η C : C → g ( f ( C )) , while the comp osite ξ g ( D ) ◦ χ D : J ( D ) → I ( f ( g ( D ))) is of the form J ( ǫ D − 1 ) for an isomorphism ǫ D : f ( g ( D )) → D , as in the follo wing diagrams: I ( C ) ξ C / / I ( η ′ C ) % % ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ J ( f ( C )) χ f ( C )   I ( g ( f ( C ))) J ( D ) χ D / / J ( ǫ ′ D − 1 ) & & ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ I ( g ( D )) ξ g ( D )   Sh ( f ( g ( D ))) Supp ose moreo v er that ξ C is an isomorphism for ev ery C ; note that this implies, b y definition of η C , that χ f ( C ) is a n isomorphis m for eac h C in H . Out of these data , we can construct tw o sets R R and R S and t w o f unctions Z R : R R → R S and Z S : R S → R R satisfying the h yp otheses of our main theorem, as follows. 1. W e define the relation R as the graph R f of a f unction f : O b ( H ) → O b ( K ) , i.e. the set of pairs of the fo rm ( C , f ( C )) fo r C ∈ H , and S as the in v erse o f the graph R g of a function g : O b ( K ) → O b ( H ) , i.e. the set of pairs of the form ( g ( D ) , D ) for D ∈ K . 2. W e define Z : R f → R g as the function sending a pair ( C , D ) in R f to the pair ( g ( D ) , D ) . 3. W e define W : R g → R f as the function sending a pair ( C , D ) in R g to the pair ( C , f ( C )) . These data satisfy the h yp otheses of the theorem with resp ect to the morphisms ξ ( C , D ) and χ ( C , D ) defined by: 1. F or any ( C , D ) ∈ R f , ξ ( C , D ) = ξ C ; 2. F or any ( C , D ) ∈ R g , χ ( C , D ) = χ D . Our general theorem of section 2 th us yields a coreflection b et we en the category ˜ R f and the category ˜ R g . Under the following additional assump- tions, the description of the categories ˜ R f and ˜ R g and o f the functors ˜ Z f : ˜ R f → ˜ R g radically simplifies. W e assume the follow ing conditions: 13 1. F or a n y morphism x : C → C ′ in H there is at most one morphism y : f ( C ) → f ( C ′ ) in K suc h that the diagram I ( C ) ξ C   I ( x ) / / I ( C ′ ) ξ C ′   J ( f ( C )) J ( y ) / / J ( f ( C )) comm utes; 2. F or an y morphism y : D → D ′ in K there is at most one morphism x : g ( D ) → g ( D ′ ) in H suc h that the diagram J ( D ) χ D   J ( y ) / / J ( D ′ ) χ D ′   I ( g ( D )) I ( x ) / / I ( g ( D ′ )) comm utes. Under these hy p otheses, the category ˜ R is equiv alen t to the category ˜ H whose ob jects are the ob jects of H and whose arro ws C → C ′ are the arrows s : C → C ′ in H suc h that there is a unique arr ow t s : f ( C ) → f ( C ′ ) in K suc h that the diagra m I ( C ) ξ ( C , D )   I ( s ) / / I ( C ′ ) ξ ( C ′ , D ′ )   J ( f ( C )) J ( t s ) / / J ( f ( C ′ )) comm utes. Under the same h yp otheses, the category ˜ S can b e iden tified with the category ˜ K ha ving as ob jects the ob jects of K and as arro ws D → D ′ the arro ws t : D → D ′ in K suc h that there is a (unique) arrow r t : g ( D ) → g ( D ′ ) making the diagra m J ( D ) χ ( C , D )   J ( t ) / / J ( D ′ ) χ ( C ′ , D ′ )   I ( g ( D )) I ( r t ) / / I ( g ( D ′ )) 14 comm ute, and the morphism r t defines an arrow g ( D ) → g ( D ′ ) in ˜ H , equiv alen tly there exists a unique arrow y : f ( g ( D )) → f ( g ( D ′ )) suc h that the diagra m I ( g ( D )) ξ g ( D )   I ( r t ) / / I ( g ( D ′ )) ξ g ( D ′ )   J ( f ( g ( D ))) J ( y ) / / J ( f ( g ( D ′ ))) comm utes. The functor ˜ Z f : H → K can b e c haracterized as the functor sending an y arro w s in H to t he arro w t s defined ab ov e, while the functor ˜ Z g : K → H sends an y arro w r in K to the arrow r t defined a b o v e. Notice that, b y the remarks of section 2.1, a dual v ersion o f this corollary yielding reflections (in the functional contex t) also holds. 2.3 Completeness In this section w e sho w that ev ery reflection b etw een categories can b e ob- tained a s an application of our general metho d. Let A and B categories and R : A → B , L : B → A b e tw o adjoin t functors ( L left adjoin t and R righ t adjoin t) with unit η : 1 B → R ◦ L a nd counit ǫ : L ◦ R → 1 A . If this adjunction is a coreflection then w e can obtain it as the result of applying our main theorem to the follo wing set of data. W e define H = A , K = B , U = B , I as the functor R : H = A → B = U , J as the functor 1 B : K = B → B = U , f as the underlying function on ob jects of the functor R , g as the underlying function on ob jects of the functor L , ξ a (for a ∈ A ) as the iden tity arrow 1 R ( a ) : I ( a ) = R ( a ) → R ( a ) = J ( f ( a )) , χ b (for b ∈ B ) as the arrow η b : J ( b ) = b → R ( L ( b )) = I ( g ( b )) . It is immediate to v erify that this set of da ta satisfies the h yp otheses of our metho d (rewritten in the functional con text as in section 2.2). The resulting category ˜ R f is clearly isomorphic to A (since the ξ are all iden tities), while the category ˜ R g is isomorphic to B , since for any arro w u : b → b ′ in B there exists exactly one arrow v : g ( b ) → g ( b ′ ) suc h t ha t the fo llo wing diagram comm utes, namely L ( u ) . 15 b χ b   J ( u ) / / b ′ χ b ′   I ( g ( b )) I ( v ) / / I ( g ( b ′ )) Indeed, this diagram is precisely the naturality square for the unit η with respect to the arrow u , if v = L ( u ) . Another w a y of recov ering this adjunction as an application of our metho d consists in selecting a differen t set of data leading to the same reflection: one can alternativ ely choose H = A , K = B , U = A , I as the functor 1 A : H = A → A = U , J as the functor L : K = B → A = U , f as the underlying function o n ob jects of the functor R , g as the underlying function on ob jects of the functor L , ξ a (for a ∈ A ) as the arrow ǫ a − 1 : I ( a ) = a → L ( R ( a )) = J ( f ( a )) , χ b (for b ∈ B ) as the iden tit y arro w arro w 1 L ( b ) : J ( b ) = L ( b ) → L ( b ) = I ( g ( b )) . 3 Reflections fro m geometric morphisms In this section w e apply our general metho d f o r generating reflections to a sp ecific con text, namely the top os-t heoretic inte rpretation of Stone-ty p e dualities established in [3]. W e shall b e able to naturally reco ver as sp ecial applications of our metho d all the Stone-t yp e reflections or coreflections that w e discussed in [3], pro viding at t he same time a uniform w ay for building suc h adjunctions. 1. Supp ose that H is a category of p oset structures C , whose arrow s are precisely the monotone maps C → C ′ , and that K is a category of p oset structures D , each of whic h eq uipp ed with a sub canonical Grothendiec k top ology K D , whose morphisms D → D ′ are precisely t he morphisms of sites ( D , K D ) → ( D ′ , K D ′ ) . If we take U to b e the (sk eleton of the) category of G rothendiec k top oses then w e ha v e t w o functors I : H → U and J : K op → U , defined a s follows. (a) The functor I : K → U sends a p oset C in K t o the top o s [ C , Set ] and an arrow v : C → C ′ in K to the geometric morphism E ( v ) : [ C , Set ] → [ C ′ , Set ] 16 canonically induced by v . (b) The functor J : K op → U sends a category D in K to the to p os Sh ( D , K D ) and a morphisms of sites u : ( D , K D ) → ( D ′ , K D ′ ) to the induced geometric morphism Sh ( u ) : Sh ( D ′ , K D ′ ) → Sh ( D , K D ) . If one has tw o functions f : O b ( H ) → O b ( K ) and g : O b ( K ) → O b ( H ) and geometric morphisms α C : Sh ( f ( C ) , J f ( C ) ) → [ C , Set ] (for C in H ) and β D : [ g ( D ) , Set ] → Sh ( D , J D ) (for D in K ) satisfying the h yp otheses of theorem of section 2.2 then there is a coreflection b etw een H and K op giv en b y functors ˜ Z : ˜ R f → ˜ R g and ˜ W : ˜ R g → ˜ R f . As sho wn in the previ ous section, under the follo wing h yp otheses the categories ˜ R f and ˜ R g admit simpler descrip- tions: (a) F or an y monotone map y : C → C ′ there is at most one morphism of sites x : ( f ( C ) ′ , K f ( C ′ ) ) → ( f ( C , K f ( C ) )) suc h that the diagram Sh ( f ( C ) , K f ( C ) ) α C   Sh ( x ) / / Sh ( f ( C ′ ) , K f ( C ′ ) ) α C ′   [ C , Set ] E ( y ) / / [ C ′ , Set ] comm utes; (b) F or an y mor phism of sites x : ( D , K D ) → ( D ′ , K D ′ ) there is at most one monotone map y : g ( D ) ′ → g ( D ) suc h that the diagram [ g ( D ′ ) , Set ] β D ′   E ( y ) / / [ g ( D ) , Set ] β D   Sh ( D ′ , K D ′ ) Sh ( x ) / / Sh ( D , K D ) comm utes; 17 W e note that condition ( a ) is alwa ys satisfied, the α are b eing equiv- alences. Indeed, if A and B a r e Cauc h y-complete categories (e.g. p osets), a functor l : A → B can b e recov ered up to isomorphism from the a sso ciated geometric morphism E ( l ) : [ A , Set ] → [ B → Set ] as the restriction to the full subcategories spanned b y the repres en ta bles of the left adjoin t to the in v erse image functor of E ( l ) . W e can iden tify a large class of naturally a r ising contex ts in whic h condition ( b ) is satisfied. Supp ose that for ev ery D in K , g ( D ) is a subset of D suc h that the map D → I d ( g ( D )) sending any elemen t d ∈ D to the ideal { d ′ ∈ g ( D ) | d ′ ≤ d } is a flat J g ( D ) -con tin uous flat functor F D correspo nding via Diaconesc u’s equiv alence to the geometric morphism β D : [ D , Set ] → Sh ( g ( D ) , J g ( D ) ) ; then condition ( b ) is satisfied. Indeed the comm utativit y of the square in condition ( b ) is equiv a len t to the comm utativity of the following diagram g ( D ′ ) y / / F D ′   g ( D ) F D   I d ( D ′ ) I d ( x ) / / I d ( D ) and I d ( x ) , b eing a frame homomorphism, is f orced to b e the function sending an ideal I in D ′ to the union of the ideals in D of the form F D ( x ( s )) for s suc h that F D ′ ( s ) ⊆ I . As w e already remark ed ab ov e, x is uniquely determin ed by I d ( x ) . Under these h yp otheses, if the Grothendiec k top ologies K D are C - induced for an in v ariant C satisfying the conditions in the theorems of [3] then ˜ R f can b e describ ed as the catego r y whose ob jects are the p osets in H and whose morphisms C → C ′ are the morphisms in H whose corresp onding fra me homomorphisms I d ( C ′ ) → I d ( C ) send C - compact elemen ts to C -compact elemen ts, while ˜ R g can b e desc rib ed as the category whose ob jects are the p osets in K and whose arrow s are the arrow s D → D ′ in K suc h that the map I d ( g ( D )) → I d ( g ( C ′ )) sending a n ideal I on g ( D ) to the union of the ideals of the form I s := { x ∈ g ( D ′ ) | x ≤ g ( s ) } for s ∈ I is a complete frame homo- morphism. The t wo adjunctions f or atomic frames a nd lo cally connected frames obtained in [3], as w ell as the Linden baum-T arski adjunction b et wee n sets and complete Bo olean algebras, are particular instances of this kind of adjunctions. 18 2. Supp ose that H is a category of p oset structures C , eac h of whic h equipped with a sub canonical Grothendiec k to p ology J C , whose mor- phisms C → C ′ are precisely the morphisms of sites ( C , J C ) → ( C ′ , J C ′ ) , and that K is a categor y of p oset structures D , eac h of whic h equipped with a sub canonical Gr o thendiec k top ology K D , whose morphisms D → D ′ are precisely t he morphisms of sites ( D , K D ) → ( D ′ , K D ′ ) . If w e tak e U to b e the (sk eleton of the) category of Gro t hendiec k t o p oses then we ha v e t wo functors I : H op → U and J : K op → U , defined a s follows. (a) The functor I : H op → U sends a category C in H to the top os Sh ( C , J C ) and a morphisms of sites s : ( C , J C ) → ( C ′ , J C ′ ) to the induced g eometric morphism Sh ( s ) : Sh ( C ′ , J C ′ ) → Sh ( C , K C ) . (b) The functor J : K op → U sends a category D in K to the to p os Sh ( D , K D ) and a morphisms of sites u : ( D , K D ) → ( D ′ , K D ′ ) to the induced geometric morphism Sh ( u ) : Sh ( D ′ , K D ′ ) → Sh ( D , K D ) . Supp ose that o ne has t w o functions f : O b ( H ) → O b ( K ) and g : O b ( K ) → O b ( H ) a nd geometric morphisms ξ C : Sh ( C ) , J C ) → Sh ( f ( C ) , K ( C ) ) (for C in H ) and χ D : Sh ( D , K D ) → Sh ( g ( D ) , J g ( D ) ) (for D in K ) satisfying the conditions of our metho d (sp ecialized to the functional con text as in sec tion 2 .2). Then there is a reflection b et wee n H op and K op (equiv alently , a coreflection b etw een the categories H and K ) giv en b y functors ˜ Z : ˜ R f → ˜ R g and ˜ W : ˜ R g → ˜ R f . As sho wn ab o v e, under the follo wing h yp otheses the categories ˜ R f and ˜ R g admit simple r descriptions: 19 (a) F or any morphism of sites y : ( C , J C ) → ( C ′ , J C ′ ) there is at most one morphism of sites x : ( f ( C ) , K f ( C ′ ) ) → ( f ( C ′ ) , K f ( C ) ) suc h that the diag ram Sh ( C ′ , J C ′ ) ξ C ′   Sh ( y ) / / Sh ( C , J C )) ξ C ′   Sh ( f ( C ′ ) , K f ( C ′ ) ) Sh ( x ) / / Sh ( f ( C ′ ) , K f ( C ′ ) comm utes; (b) F or an y mor phism of sites x : ( D , K D ) → ( D ′ , K D ′ ) there is at most o ne morphism of sites y : ( g ( D ) , J g ( D ) ) → ( g ( D ′ , J g ( D ′ ) ) suc h that the diagram Sh ( D ′ , K D ′ ) χ D ′   Sh ( x ) / / Sh ( D , K D ) χ D   Sh ( g ( D ′ ) , J g ( D ′ ) ) Sh ( y ) / / Sh ( g ( D ) , J g ( D ) ) comm utes. W e note that condition ( a ) is alw ays satisfied, the ξ are b eing equiv a- lences. Indeed, it is wel l-kno wn that a morphism of sub canonical sites can b e recov ered up to isomorphism (in particular, uniquely , in case of morphisms betw een p o sets) from the corresp onding geometric mor- phism a s the restriction of its inv erse image functor of the morphism to the full sub categories spanned by the represen tables. W e can iden tify a large class of naturally a r ising contex ts in whic h condition ( b ) is satisfied. Suppose that for ev ery D in K , g ( D ) is a subset of D such that the inclusion g ( D ) ֒ → D yields a morphism of sites ( g ( D ) , J g ( D ) ) → ( D , K D ) (note that, for example, this condition alwa ys holds if J g ( D ) is the Grothendiec k top o lo gy induced b y K D on g ( D ) ) whic h induce s the geometric morphism χ D ; then the commutativi t y of the diag ra m fo r ces y to b e equal to the restriction of x to g ( D ) and g ( D ′ ) and hence condition ( b ) is satisfied. Under these h yp otheses, if the Grothendiec k top ologies K D are C - induced for an in v ariant C satisfying the conditions in the theorems of [3] then ˜ R f can b e describ ed as the catego r y whose ob jects are the p osets in H and whose morphisms C → C ′ are the morphisms in H whose corr esp onding frame homomorphisms I d J C ( C ) → I d J C ′ ( C ′ ) send 20 C -compact eleme n ts to C -compact elem en ts, while ˜ R g can b e desc rib ed as the catego r y whose ob jects are the po sets in K and whose arro ws D → D ′ in K are the morphism of sites ( D , K D ) → ( D ′ , K D ′ ) whic h restrict to a morphism of sites ( g ( D ) , J g ( D ) ) → ( g ( D ′ ) , J g ( D ′ ) ) (note that, in case for ev ery D J g ( D ) is the Grothendiec k top ology in- duced b y K D on g ( D ) , for this condition to hold it suffices to require that the underly ing function D → D ′ of the morphism restricts to a function g ( D ) → g ( D ′ ) ). whic h is an arrow in H . Of course, another case in whic h condition ( b ) is satisfied is when the ξ D are equiv alences (cf. the example b elow). As a useful illustration o f this kind of adjunctions, w e describ e the fol- lo wing contex t. Let H b e a full sub category of the category of f r a mes and K be a category of posets, eac h of whic h equipped with a sub canon- ical to p ology , whose arro ws ar e the morphisms of the asso ciated sites; w e denote b y K D the Grothendiec k top olog y asso ciated to a p oset D in K . Give n a frame C in H , w e denote b y J C the canonical top o logy on C . Supp ose that w e ha v e a function f : O b ( H ) → O b ( K ) with the prop- ert y that for an y C in H , f ( C ) is a basis of C suc h that K f ( C ) = J C | f ( C ) , and denote b y g : O b ( K ) → O b ( H ) the function sending a p oset D in K to the frame g ( D ) := I d K C ( D ) . Suppo se that for ev ery D in K the canonical morphism D → I d K C ( D ) factors through the inclu- sion f ( I d K C ( D )) ֒ → I d K C ( D ) . Then the equiv alences ξ C : Sh ( C , J C ) → Sh ( f ( C ) , K f ( C ) ) and Sh ( D , K D ) ≃ Sh ( I d K C ( D ) , J I d K D ( D ) ) induced b y the Comparison Lemma satisfy the h yp otheses of our metho d. Since conditions ( a ) and ( b ) ab ov e are trivially satisfied, w e obtain a reflec- tion b et w een the categories ˜ R f and ˜ R g giv en b y the functors ˜ Z f and ˜ Z g . The category ˜ R f has as ob jects the frames in H and as arrows the frame homomorphisms C → C ′ whic h factor through the inclu- sions f ( C ) ֒ → C and f ( C ′ ) ֒ → C ′ , while the category ˜ R g has as ob jects the p osets in K and a s arrows D → D ′ the morphisms suc h that the correspo nding morphism I d K D ( D ) → I d K ′ D ( D ′ ) factors thro ug h the in- clusions f ( I d K D ( D )) ֒ → I d K D ( D ) a nd f ( I d K ′ D ( D ′ )) ֒ → I d K ′ D ( D ′ ) . The functor ˜ Z f : ˜ R f → ˜ R g sends a f rame C to the p oset f ( C ) and a frame homomorphism C → C ′ to its restriction f ( C ) → f ( D ) , while the func- tor ˜ Z g : ˜ R g → ˜ R f sends any arro w in K to the corresponding frame homomorphism. An example of a reflection of this kind is the reflection b et w een the category o f frames and the category of Bo olean a lgebras providin g a 21 lo calic v ersion of t he Stone adjunction b etw een the category of Bo olean algebras and the o pp osite o f the category of top ological spaces (cf. [3]); indeed, for a ny frame F , denoted by F c the Bo olean algebra of its complemen ted elemen ts, w e ha ve a geometric morphism Sh ( F J f ) → Sh ( F c , J coh F c ) induced b y the inclusion ( F c , J coh F c ) ֒ → ( F , J F ) (where J F is the canonical top o lo g y on F and J coh F c is the coheren t top ology on F c ) and for an y Bo olean algebra B w e ha v e a n equiv alence Sh ( F B , J F b ) ≃ Sh ( B , J coh B ) . 3. Our general metho d for building reflections can b e profitably used a lso for establishing reflections with categories of top ological spaces. F or these purp oses, it is o f ten useful to select as category U the category of top oses paired with p o in ts defined in [3]. Let us giv e a couple of examples of these kind of adjunctions. (a) The w ell-known Stone adjunction b et w een the category of Bo olean algebras a nd the opp osite of the catego ry of top o logical spaces can b e obtained as follo ws. T ak e H to b e the opp osite of the category of Bo olean algebras a nd K to b e the category of top olo gical spaces. Define U to b e the category of top oses paired with p oin ts (as de- fined in [3]). W e ha ve t w o functors I : H → U and J : K → U ; the functor I sends a Bo olean alg ebra B to the pair ( Sh ( B , J B ) , p B ) where p B is the set of all the p oints of the top os Sh ( B , J B ) and acts o n the arrows accordingly , while the functor J sends a top o- logical space X t o the pair ( Sh ( X ) , P X ) where P X is the set of p oin ts of Sh ( X ) indexed by the eleme n ts of X (as in [3]). The functions f : O b ( H ) → O b ( K ) and g : O b ( K ) → O b ( H ) can b e defined as follo ws; f sends a Bo olean algebra B to its Stone sp ec- trum X B , while g sends a top ological space to the Bo olean algebra X cl of its clop en subsets. F or each B in H w e ha v e an isomorphism ξ B : I ( B ) = ( Sh ( B , J B ) , p B )) → ( Sh ( X B ) , P X B ) = J ( f ( B )) in U , while for any X in K we hav e an arrow J ( X ) = ( Sh ( X ) , P X ) → ( Sh ( X cl , J X cl ) , p X cl ) = I ( g ( X )) in U . It is immediate to see that this se t of data satis fies the h yp otheses of our method, from whic h w e conclude that w e ha v e a reflection b et w een H and K , as re- quired. (b) The Alexandr ov adjunction b et w een the category o f preorders and the category of topolog ical spaces can b e obtained as follows . T a k e H to b e the category of preorders, and K to b e the category of top ological spaces. Define U to b e the category of top oses paired with p oints (as defined in [3]). W e ha ve tw o functors I : H → U 22 and J : K → U defined as follo ws. The functor I sends a preorder P to the pair ([ P , Set ] , E P ) , where E P is the indexing of p oin ts of [ P , Set ] by elemen ts of P (as in [3]), acting on the arrow s in the ob vious w a y . The functor J sends a top ological space X to the pair ( Sh ( X ) , P X ) where P X is the set of p oin ts of Sh ( X ) indexed b y the elemen ts of X (as in [3]). The functions f : O b ( H ) → O b ( K ) and g : O b ( K ) → O b ( H ) are defined a s follo ws; f sends a preorder P to the Alexandro v space A P based on P (i.e. the preorder P endo w ed with the Alexandro v to p ology), while the function g sends a top ological space X to its sp ecialization preorder X ≤ . F or each P in H w e ha v e an isomorphis m α P : ( Sh ( A P ) , P A P ) = J ( f ( P )) → I ( P ) = [ P , Set ] in U , and for eac h X in K w e ha ve an arro w β X : I ( g ( X )) = ([ X ≤ , Set ] , E X ≤ ) → ( Sh ( X ) , P X ) = J ( X ) in U . It is easy to v erify that these arrow s satisfy the hy p otheses of our metho d, whic h yields in t his case precisely t he Alexandro v coreflection b et wee n preorders a nd top ological spaces. References [1] M. Artin, A. Grothendiec k and J. L. V erdier, Thé orie des top os et c o - homolo gie é tale d e s schémas , Séminaire de Géométrie Algébrique du Bois-Marie, a nnée 1963-64; second edition published as Lecture No tes in Math., vols 269, 270 and 3 0 5 (Springer-V erlag, 1972 ) . [2] O. Caramello, The unification of Mathematics via T op os Theory , arXiv:math.CT/1006 . 3 930v1 . [3] O. Caramello, A top os-theoretic approach to Stone-t yp e dualities, arXiv:math.CT/1103 . 3 493v1 [4] P . T. Johnstone, S k e tches of an Elephan t: a top o s the ory c omp endium. V ols. 1-2 , v ols. 43-4 4 of Oxfor d L o gic Guides (O xford Univ ersit y Press, 2002). [5] P . T. Johnstone, Stone sp ac e s , Cam bridge Studies in A dv anced Math. No. 3 (Cambridge Univ. Press, 1982). [6] F. W. Lawv ere, F unctorial Semantics of Algebr a i c The ories and Some A lgebr aic Pr obl e ms in the c ontext of F unctorial Sem antics of Alge- br aic T he ories , Ph.D. thesis, Colum bia Univ ersit y , 1963, republis hed in R eprints in The o ry an d Appl i c ations of Cate gories , No. 5 (2004) pp. 1-121. 23