Yoneda representations of flat functors and classifying toposes

Y oneda represen tations of at funtors and lassifying top oses Olivia Caramello DPMMS, Univ ersit y of Cam bridge, Wilb erfore Road, Cam bridge CB3 0WB, UK O.Caramellodpmms.am.a.uk Otob er 23, 2018 Abstrat In this pap er, w e rst in tro due a te hnique that w e all Y oneda represen tation of at funtors, based on ideas from indexed ategory theory; then w e pro vide appliations of this te hnique to the theory of lassifying top oses. Sp eially , w e obtain results  haraterizing the mo dels of a theory lassied b y a top os of the form Sh ( C , J ) in terms of the mo dels of a theory lassied b y the top os [ C op , Set ] . 1 Preliminary fats In this setion w e in tro due the terminology and reall the fats from the theory of indexed ategories that will b e useful for our analysis. W e refer the reader to [1 ℄ (esp eially setions B1.2, B2.3 and B3.1) and to [4℄ for the ba kground. By a top os (dened) o v er Set w e mean an elemen tary top os E su h that there exists a (neessarily unique up to isomorphism) geometri morphism γ E : E → Set ; w e denote b y γ ∗ E the in v erse image funtor and b y Γ E its righ t adjoin t, that is the global setions funtor. A top os is dened o v er Set if and only if it is lo ally small and has arbitrary set-indexed op o w ers of 1 ; in partiular ev ery lo ally small o omplete top os (and hene ev ery Grothendie k top os) is dened o v er Set . 1 Giv en a ategory C and a top os E dened o v er Set , w e an alw a ys in ternalize C in to E b y means of γ ∗ E ; the resulting in ternal ategory in E will b e denoted b y C . Ev ery top os E (o v er Set ) giv es rise to a E -indexed ategory E obtained b y indexing E o v er itself; the in v erse image funtor γ ∗ E then indues an indexing of E o v er Set , whi h oinides with the anonial indexing of E pro vided that E is o omplete and lo ally small. W e will generally denote indexed ategories b y underlined letters, to distinguish them from their underlying ategories whi h will b e denoted b y the orresp onding simple letters; so for example the underlying ategory of an indexed ategory D will b e denoted b y D ; an exeption to this rule will b e the notation for indexed ategories arising as the indexing of a artesian ategory o v er itself: in this ase the indexed ategory orresp onding to a artesian ategory S will b e simply denoted b y S . Also, in ternal ategories will b e denoted b y letters C , D , et. and w e will not mo dify their notation when they are onsidered as indexed ategories. F or a top os E and an in ternal ategory C in E , w e ha v e a E -indexed ategory [ C , E ] , whose underlying ategory is the ategory [ C , E ] of diagrams of shap e C in E and morphisms b et w een them. [ C , E ] is equiv alen t (naturally in E ) to the ategory of E -indexed funtors C → E and indexed natural transformations b et w een them (b y Lemma B2.3.13 in [1℄) and also, if E is o omplete and lo ally small, to the ategory [ C , E ] (b y Corollary B2.3.14 in [1℄). F or this reason, w e will restrit our atten tion to lo ally small o omplete top oses; w e will o asionally lo osely refer to them simply as top oses. The equiv alene b et w een [ C , E ] and [ C , E ] restrits to an equiv alene b et w een the full sub ategories T ors ( C , E ) of C -torsors in E (as in setion B3.2 of [1 ℄) and Flat ( C , E ) of at funtors C → E (as in  hapter VI I of [ 3 ℄). Giv en a funtor F ∈ [ C , E ] , the in ternal diagram that orresp onds to it via the natural equiv alene [ C , E ] ≃ [ C , E ] will b e alled the in ternalization of F and denoted b y F i ; of ourse, this is dened only up to isomorphism. The E -indexed ategory [ C , E ] is lo ally small (b y Lemma B2.3.15 in [1℄); from this it follo ws that there exists a E -indexed hom funtor H o m E [ C , E ] : [ C , E ] op × [ C , E ] → E whose underlying funtor H o m E [ C , E ] : [ C , E ] op × [ C , E ] → E assigns to a pair of diagrams F and G in [ C , E ] an ob jet H o m E [ C , E ] ( F , G ) of E , whi h w e all the ob jet of morphisms from F to G in [ C , E ] . Also, there is a Y oneda E -indexed funtor Y : C → [ C op , E ] , whi h pla ys in this on text the same role as that of a Y oneda funtor in ordinary ategory theory . W e denote b y E ∗ : E → E /E the pullba k funtor along the unique arro w 2 E → 1 , that is the (logial) in v erse image funtor of the lo al homeomorphism E /E → E . Then, from the equiv alene [ C , E ] ≃ [ C , E ] w e dedue the existene of a hom funtor H o m E [ C , E ] : [ C , E ] op × [ C , E ] → E , whi h assigns to ea h pair of funtors F and G in [ C , E ] an ob jet H o m E [ C , E ] ( F , G ) of E su h that for ea h E ∈ E the morphisms E → H om E [ C , E ] ( F , G ) in E are in natural bijetion with the morphisms in [ C , E /E ] from E ∗ ◦ F to E ∗ ◦ G , that is with the natural transformations E ∗ ◦ F ⇒ E ∗ ◦ G . W e remark that, sine Flat ( C , E ) is a full sub ategory of [ C , E ] , w e ma y use the ob jets H o m E [ C , E ] ( F , G ) for F , G ∈ Flat ( C , E ) as the ob jets of morphisms from F to G in Flat ( C , E ) . Giv en a S -indexed ategory D and an ob jet I ∈ S , w e ha v e a S /I -indexed ategory D /I (dened in the ob vious w a y), whi h is alled the lo alization of D at I . If D and E are t w o S -indexed ategories, w e denote b y [ D , E ] the ategory of S -indexed funtors from D to E and indexed natural transformations b et w een them. The assignmen t I → [ D /I , E /I ] is pseudofuntorial in I ∈ S and mak es [ D , E ] in to a S -indexed ategory . 2 Y oneda represen tations It is w ell kno wn that, b y Y oneda, for ea h F ∈ [ C op , Set ] there is a natural isomorphisms of funtors F ∼ = H o m Set [ C op , Set ] ( Y ( − ) , F ) , where H o m Set [ C op , Set ] ( Y ( − ) , F ) is the funtor giv en b y the omp osite C op Y op × ∆ F − → [ C op , Set ] op × [ C op , Set ] H om Set [ C op , Set ] − → Set . Thanks to the remarks in the last setion w e are able to generalize this result to the ase of funtors with v alues in an arbitary top os. In fat, the follo wing result holds. Theorem 2.1. L et C a smal l  ate gory and E b e a lo  al ly smal l  o  omplete top os. Then for every funtor F : C op → E , ther e is a natur al isomorphism of funtors F ∼ = H o m E [ C op , E ] ( Y ( − ) , F ) , wher e Y : C → [ C op , E ] is the funtor given by the  omp osite C Y − → [ C op , Set ] γ ∗ E ◦− − → [ C op , E ] 3 and H o m E [ C op , E ] ( Y ( − ) , F ) is the funtor given by the  omp osite C op Y op × ∆ F − → [ C op , E ] op × [ C op , E ] H om E [ C op , E ] − → E . Mor e over, the isomorphism ab ove is natur al in F . Pro of One an observ e that the in ternal C op -diagram in E giv en b y the omp osite C op Y op × ∆ F i − → [ C op , E ] op × [ C op , E ] H om E [ C op , E ] − → E is on one hand equal to F i (b y an in ternal v ersion of the Y oneda's lemma) and on the other hand equal to the in ternalization of the funtor H o m E [ C op , E ] ( Y ( − ) , F ) . The v eriations are easy and left to the reader. Alternativ ely , one ma y pro eed as follo ws. W e w an t to pro v e that F ( c ) ∼ = H o m E [ C op , E ] ( Y ( c ) , F ) , naturally in c ∈ C (and in F ). It sues to observ e that w e ha v e the follo wing sequene of natural bijetions: E − → H om E [ C op , E ] ( Y ( c ) , F ) E ∗ ◦ Y ( c ) = ⇒ E ∗ ◦ F γ ∗ E /E ◦ Y ( c ) = ⇒ E ∗ ◦ F Y ( c ) = ⇒ Γ E /E ◦ E ∗ ◦ F elemen t of (Γ E /E ◦ E ∗ ◦ F )( c ) E − → F ( c ) .  In the ase of at funtors, the theorem sp eializes to the follo wing result. Corollary 2.2. L et C b e a smal l  ate gory and E b e a lo  al ly smal l  o  omplete top os. Then for every at funtor F : C op → E , ther e is a natur al isomorphism of funtors F ∼ = H o m E Flat ( C op , E ) ( Y ( − ) , F ) , wher e Y : C → Flat ( C op , E ) is the funtor given by the  omp osite C Y − → Flat ( C op , Set ) γ ∗ E ◦− − → Flat ( C op , E ) 4 and H o m E Flat ( C , E ) ( Y ( − ) , F ) is the funtor given by the  omp osite C op Y op × ∆ F − → Flat ( C op , E ) op × Flat ( C op , E ) H om E Flat ( C op , E ) − → E . Pro of This immediately follo ws from the theorem and the remarks in the rst setion.  F rom no w on w e will refer to this result as to the Y oneda represen tation of at funtors. 3 Represen tation problems In this setion w e in tro due the notion of represen tation problem in the general on text of lo ally small indexed ategories. This onept will lead to a univ ersal  haraterization of the Y oneda em b eddings, whi h will b e emplo y ed in the next setion to deriv e a riterion for a theory to b e of presheaf t yp e. Denition 3.1. Let S a artesian ategory and D a lo ally small S -indexed ategory . A S -indexed funtor F : D op → S is said to b e S -represen table if there exists an ob jet A ∈ D su h that F is isomorphi in [ D op , S ] to the omp osite D op 1 D × ∆ A − → D op × D H om S D − → S . W e denote this omp osite b y H o m S D ( − , A ) . If D is the underlying ategory of a lo ally small S -indexed ategory D then w e sa y that a funtor F : D op → S is S -represen table if it is the underlying funtor of an indexed funtor of the form H o m S D ( − , A ) . Denition 3.2. Let S a artesian ategory , D a lo ally small S -indexed ategory and K a S -indexed full sub ategory of [ D op , S ] . A lo ally small S -indexed ategory F together with S -indexed funtors i : D → F and r : K → F is said to b e a solution to the 1 -represen tation problem for K if H o m S F ( − , r ( F )) ◦ i op ∼ = F anonially in F ∈ K . D op i op   F / / S F op H om S F ( − ,r ( F )) > > ~ ~ ~ ~ 5 ( F , i : D → F , r : K → F ) is said to b e a solution to the represen tation problem for K if for ea h I ∈ S the triple ( F /I , i/I : D /I → F /I , r /I : K /I → F /I ) is a solution to the 1 -represen tation problem for the S /I -indexed ategory K /I . A solution ( F , i : D → F , r : K → F ) to the represen tation problem for K is said to b e univ ersal if for an y other solution ( F ′ , i ′ : D → F ′ , r ′ : K → F ′ ) to the same problem there exixts a unique (up to anonial isomorphism) S -indexed funtor z : F → F ′ su h that z ◦ r ∼ = r ′ and z ◦ i ∼ = i ′ anonially . Of ourse, if su h a solution exists, it is unique up to anonial isomorphism b y the univ ersal prop ert y . Prop osition 3.3. L et S a  artesian  ate gory, D a lo  al ly smal l S -indexe d  ate gory and K a S -indexe d ful l sub  ate gory of [ D op , S ] . If Y : D → [ D op , S ] fators as Y ′ : D → K thr ough the ful l emb e dding K ֒ → [ D op , S ] , then the triple ( K , Y ′ : D → K , 1 K : K → K ) is the universal solution to the r epr esentation pr oblem for the S -indexe d  ate gory K . Pro of This is an immediate onsequene of the indexed v ersion of the Y oneda lemma.  So, if C is an in ternal ategory in S , the em b edding Y : C → [ C op , S ] an b e  haraterized not only , as it is w ell kno wn, as the free S -o ompletion of C , but also as the univ ersal solution to the represen tation problem for the S -indexed ategory [ C op , S ] . Corollary 3.4. L et C b e an internal  ate gory in a top os E . Then the fatorization Y ′ : C → T ors ( C op , E ) of the Y one da indexe d funtor Y : C → [ C op , E ] thr ough the ful l emb e dding T ors ( C op , E ) ֒ → [ C op , E ] is the universal solution to the r epr esentation pr oblem for the E -indexe d  ate gory T ors ( C op , E ) .  4 Classifying top oses As promised, w e giv e a  haraterization of the (geometri) theories of presheaf t yp e based on the ideas in the last setion. W e observ e that, if T is a geometri theory , w e an regard it informally as a ategory T -mo d indexed b y the (meta)ategory of Grothendie k top oses via the pseudofuntor T -mo d (whi h assigns to ev ery top os E the ategory of T -mo dels in E ); in partiular, for ea h Grothendie k top os E , b y 6 restriting this pseudofuntor to the slies of E , w e obtain a E -indexed ategory T -mo d E , whi h is lo ally small as a E -indexed ategory . Indeed, it is w ell kno wn that T is Morita-equiv alen t (that is, has the same ategory of mo dels - up to natural equiv alene - in to ev ery Grothendie k top os E naturally in E , equiv alen tly has the same lassifying top os) to the theory of at funtors on a ategory C whi h are on tin uous with resp et to a Grothendie k top ology J on C , and the ategories of su h funtors are all full sub ategories of the orresp onding ategories of funtors on C (fr. the remarks in the rst setion). W e reall that a geometri theory T is said to b e of presheaf t yp e if its lassifying top os is a presheaf top os (equiv alen tly , the top os [ C , Set ] , where C := ( f.p. T -mo d ( Set )) is the ategory of nitely presen table T -mo dels in Set ). If T is of presheaf t yp e, then it is Morita-equiv alen t to the theory of at funtors on the ategory C op . Via this equiv alene, the Y oneda em b edding Y : C → Flat ( C op , E ) orresp onds to the em b edding of ( f.p. T -mo d ( Set )) in to T -mo d ( E ) giv en b y the in v erse image funtor γ ∗ E . Notie that the image in T -mo d ( E ) of this em b edding an b e though t as the sub ategory of onstan t T -mo dels whi h are nitely presen table in Set . As w e ha v e remark ed, for ea h Grothendie k top os E T -mo d E is lo ally small, so it do es mak e sense to ask if γ ∗ E ( − ) : f.p. T -mo d ( Set ) → T -mo d ( E ) (regarded here as a E -indexed funtor to T -mo d E ) is the univ ersal solution to the represen tation problem for the E -indexed ategory Flat (( f.p. T -mo d ( Set )) op , E ) . If this holds for ev ery E naturally in E then w e ma y onlude b y Corollary 3.4 that T is of presheaf t yp e. More onretely , w e ha v e the follo wing riterion for a theory to b e of presheaf t yp e. Theorem 4.1. L et T a ge ometri the ory. Then T is of pr eshe af typ e if and only if for e ah Gr othendie k top os E , every at funtor F : ( f.p. T -mo d ( Set )) op → E  an b e extende d to a E -r epr esentable along γ ∗ E ( − ) op : ( f.p. T -mo d ( Set )) op → T -mo d ( E ) op and  onversely every E -r epr esentable T -mo d ( E ) op → E arises up to isomorphism in this way, natur al ly in F and E . Pro of This is immediate from the disussion ab o v e.  Supp ose no w y ou ha v e a geometri theory T lassied b y the top os [ C op , Set ] and w an t to understand what the theory T ′ lassied b y the top os Sh ( C , J ) (where J is a Grothendie k top ology on C ) lo oks lik e, in terms of T , and without an y referene to at funtors. The te hnique of the Y oneda represen tation for at funtors pro vides us with a means for solving 7 this problem. Sp eially , w e are able to desrib e in terms of the T -mo dels and of the Grothendie k top ology J the T ′ -mo dels in an y Grothendie k top os E , in other w ords w e are able to iden tify T ′ up to Morita-equiv alene en tirely in terms of T and of J . W e denote b y ˇ C the Cau h y ompletion of the ategory C . Reall that ˇ C an alternativ ely b e  haraterized as the full sub ategory of Ind - C onsisting of the nitely presen table ob jets and also as the losure of C under retrats in Ind - C . It is w ell kno wn that the funtor ategories [ C op , Set ] and [ ˇ C op , Set ] are naturally equiv alen t. Sine Sh ( C , J ) is a subtop os of [ ˇ C op , Set ] , it follo ws (from the theory of elemen tary top oses) that there exists a unique Grothendie k top ology ˇ J on ˇ C su h that the top oses Sh ( C , J ) and Sh ( ˇ C , ˇ J ) are naturally equiv alen t. W e desrib e it expliitly in the theorem b elo w. W e adopt the follo wing on v en tions: if S is a siev e in C , w e denote b y S the siev e in ˇ C generated b y the mem b ers of S ; if R is a siev e in ˇ C , w e denote b y R ∩ ar r ( C ) the siev e in C formed b y the elemen ts of R whi h are arro ws in C . Moreo v er, giv en an arro w g : d → c in C and siev es S and R on c resp etiv ely in C and ˇ C , w e denote b y g ∗ C ( S ) and g ∗ ˇ C ( R ) the siev es obtained b y pulling ba k S and R along g resp etiv ely in the ategories C and ˇ C . Theorem 4.2. L et C b e a  ate gory and ˇ C its Cauhy  ompletion. Given a Gr othendie k top olo gy J on C , ther e exists a unique Gr othendie k top olo gy ˇ J on ˇ C that indu es J on C , whih is dene d by: for e ah sieve R on d ∈ ˇ C , R ∈ ˇ J ( d ) if and only if ther e exists a r etr at d i ֒ → a r → d with a ∈ C and a sieve S ∈ J ( a ) suh that R = i ∗ ( S ) . F urhermor e, if d ∈ C then R ∈ ˇ J ( d ) if and only if ther e exists a sieve S in C on d suh that R = S . Pro of Sine the full em b edding C ֒ → ˇ C is (trivially) dense with resp et to ev ery Grothendie k top ology on ˇ C , it follo ws from the Comparison Lemma (Theorem C2.2.3 in [2℄) and the remarks ab o v e that there is at most one Grothendie k top ology on ˇ C that indues J on C . Therefore, it will b e enough to pro v e that the o v erage ˇ J in the statemen t of the theorem is a Grothendie k top ology that indues J on C . This, as w ell as the seond part of the thesis, an b e easily pro v ed b y using the follo wing easy fat (whose pro of is left to the reader): Giv en an ob jet c ∈ C , the assignmen ts R → R ∩ ar r ( C ) and S → S are in v erse to ea h other and dene a bijetion b et w een the set of siev es in C on c and the set of siev es in ˇ C on c . Moreo v er, these bijetions are natural with resp et to the op erations of pullba k of siev es along an arro w in C . By w a y of example, w e pro vide the details of the pro of that ˇ J satises the stabilit y axiom for Grothendie k top ologies. 8 Giv en R ∈ ˇ J ( d ) and g : e → d in ˇ C , w e w an t to pro v e that g ∗ ( R ) ∈ ˇ J ( e ) . Sine R ∈ ˇ J ( d ) , there exists a retrat d i ֒ → a r → d with a ∈ C and a siev e S ∈ J ( a ) su h that R = i ∗ ( S ) . There exists a retrat e j ֒ → b z → e with b ∈ C . No w, g ∗ ( R ) = g ∗ ( i ∗ ( T )) = ( i ◦ g ) ∗ ( T ) = (( i ◦ g ◦ z ) ◦ j ) ∗ ( T ) = j ∗ (( i ◦ g ◦ z ) ∗ ( T )) = j ∗ (( i ◦ g ◦ z ) ∗ ˇ C ( S )) = j ∗ (( i ◦ g ◦ z ) ∗ C ( S )) . Our thesis then follo ws at one from the stabilit y axiom for J .  Theorem 4.3. L et C b e a  ate gory and J a Gr othendie k top olo gy on C . If J is the trivial top olo gy then ˇ J is the trivial top olo gy. If J is the dense (r esp e tively, the atomi) top olo gy on C , then ˇ J is the dense (r esp e tively, the atomi) top olo gy on ˇ C . Pro of All an b e easily pro v ed b y using the retrat te hnique emplo y ed in the pro of of the previous theorem. W e omit the details.  Coming ba k to our original problem, w e ha v e seen that it is natural to replae the top os Sh ( C , J ) with Sh ( ˇ C , ˇ J ) . The adv an tage for us of this replaemen t is that the ategory ˇ C , b eing Cau h y omplete, an b e reo v ered from Flat ( ˇ C op , Set ) as the full sub ategory of nitely presen table ob jets. Hene, if T is a theory lassied b y [ ˇ C , Set ] , then the natural equiv alene Flat ( ˇ C op , Set ) ≃ T -mo d ( Set ) restrits to a natural equiv alene ˇ C ≃ f.p. T -mo d ( Set ) , as in the follo wing diagram: ˇ C Y   f.p. T -mo d ( Set ) ∼ τ o o i   Flat ( ˇ C op , Set ) ∼ / / T -mo d ( Set ) No w w e w an t to rewrite the Y oneda represen tation F ∼ = H o m E [ ˇ C op , E ] ( Y ( − ) , F ) , of a at funtor F : ˇ C → E (giv en b y Corollary 2.2) in terms of T , regarded here as a E -indexed ategory . W e reall that T -mo d E is lo ally small, with H o m E T -mo d ( E ) ( M , N ) ob jet of morphisms in T -mo d E from M to N b elonging to T -mo d ( E ) . The naturalit y in E of the Morita-equiv alene b et w een T and the theory of at funtors on ˇ C op implies the omm utativit y of the follo wing diagram: Flat ( ˇ C op , Set ) ∼ / / γ ∗ E ◦−   T -mo d ( Set ) γ ∗ E ( − )   Flat ( ˇ C op , E ) ∼ / / T -mo d ( E ) 9 F rom the omm utativit y of the t w o diagrams ab o v e w e dedue the follo wing represen tation for F ◦ τ : F ◦ τ ∼ = H o m E T -mo d ( E ) ( γ ∗ E ( i ( − )) , M F ) , where M F is the T -mo del in E orresp onding to F ∈ Flat ( C op , E ) via the Morita-equiv alene. This motiv ates the follo wing denition. Denition 4.4. Let E b e a lo ally small o omplete top os and T a theory of presheaf t yp e. Giv en a Grothendie k otop ology J on C := f.p. T -mo d ( Set ) , a mo del M ∈ T -mo d ( E ) is said to b e J -homo gene ous if for ea h osiev e S ∈ J ( c ) the family of all the arro ws H o m E T -mo d ( E ) ( γ ∗ E ( i ( f )) , M ) : H om E T -mo d ( E ) ( γ ∗ E ( i ( cod ( f ))) , M ) − → H om E T -mo d ( E ) ( γ ∗ E ( i ( c )) , M ) for f ∈ S , is epimorphi in E . Remark 4.5. It is lear (from the denition of atomi top ology) that if J is the atomi otop ology on C then a mo del M ∈ T -mo d ( E ) is J -homogeneous if and only if for ea h arro w f : c → d in C , the arro w H o m E T -mo d ( E ) ( γ ∗ E ( i ( f )) , M ) : H om E T -mo d ( E ) ( γ ∗ E ( i ( d )) , M ) − → H om E T -mo d ( E ) ( γ ∗ E ( i ( c )) , M ) is an epimorphism in E . In this ase w e will simply sa y `homogeneous' instead of ` J -homogeneous'. W e observ e that M F is J -homogeneous if and only if F ◦ τ is J -on tin uous. W e th us obtain the follo wing theorem. Theorem 4.6. L et ( C , J ) b e a site and T a the ory lassie d by the top os [ C op , Set ] . Then the top os Sh ( C , J ) lassies the T -mo dels whih ar e ˇ J -homo gene ous; that is, given a ge ometri the ory T ′ to gether with a ful l and faithful indexe d funtor i : T ′ -mo d ֒ → T -mo d , then the T ′ -mo dels ar e identie d by i with the ˇ J -homo gene ous T -mo dels if and only if  T ′ is lassie d by the top os Sh ( C , J ) and  the emb e dding i is indu e d via the universal pr op erty of the lassifying top oses by the inlusion Sh ( C , J ) ֒ → [ C op , Set ] .  10 Sp eializing the theorem to the ase of the atomi top ology giv es the follo wing result. Corollary 4.7. L et ( C , J ) b e an atomi site and T a the ory lassie d by the top os [ C op , Set ] . Then the top os Sh ( C , J ) lassies the homo gene ous T -mo dels. Pro of This is immediate from the theorem and Theorem 4.3.  No w w e w an t to rephrase in more expliit terms what it means for a mo del to b e J -homogeneous; this will b e partiularly imp ortan t for the appliations. T o this end, w e rst express the ondition that a giv en family of arro ws as in Denition 4.4 is epimorphi as a logial sen tene in the in ternal language of the top os, then w e use the Kripk e-Jo y al seman tis to sp ell out what it means for that sen tene to b e v alid in the top os. Reall that if E is a o omplete top os and ( f i : C i → C | i ∈ I ) is a family of arro ws in it indexed b y a set I , then this family is epimorphi if and only if the logial form ula ( ∀ y ∈ C )( ∨ i ∈ I ( ∃ x ∈ C i ( f i x = y ))) holds in E . Giv en a lass of generators G for E , the v alidit y in E of this sen tene is in turn equiv alen t, b y the Kripk e-Jo y al seman tis, to the follo wing statemen t: for ea h E ∈ G and y : E → C there exists an epimorphi family ( r i : E i → E | i ∈ I ) and generalized elemen ts ( x i : E i → C i | i ∈ I ) su h that y ◦ r i = f i ◦ x i for ea h i ∈ I . By applying this to the families of arro ws in Denition 4.4 and b y realling that the ob jets H o m E T -mo d ( E ) ( γ ∗ E ( i ( d )) , M ) are the ob jets of morphisms from γ ∗ E ( i ( d )) to M in T -mo d E , w e obtain the follo wing  haraterization. Theorem 4.8. L et E b e a lo  al ly smal l  o  omplete top os with a lass of gener ators G and T b e a the ory of pr eshe af typ e. Given a Gr othendie k  otop olo gy J on C := f.p. T -mo d ( Set ) , a mo del M ∈ T -mo d ( E ) is J -homo gene ous if and only if for e ah  osieve S ∈ J ( c ) , obje t E ∈ G and arr ow y : E ∗ ( γ ∗ E ( i ( c ))) → E ∗ ( M ) in T -mo d ( E /E ) ther e exists an epimorphi family ( p f : E f → E , f ∈ S ) and for e ah arr ow f : c → d in S an arr ow u f : E ∗ f ( γ ∗ E ( i ( d ))) → E ∗ f ( M ) in T -mo d ( E /E ) suh that p ∗ f ( y ) = u f ◦ E ∗ f ( γ ∗ E ( i ( f ))) .  11 Notie that if E is the top os Set then b y taking as lass of generators of Set the lass ha ving as its unique elemen t the singleton 1 Set w e obtain the follo wing result. Corollary 4.9. L et T b e a the ory of pr eshe af typ e. Given a Gr othendie k  otop olo gy J on C := f.p. T -mo d ( Set ) , a mo del M ∈ T -mo d ( Set ) is J -homo gene ous if and only if for e ah  osieve S ∈ J ( c ) and arr ow y : i ( c ) → M in T -mo d ( Set ) ther e exists an arr ow f : c → d in S and an arr ow u f : i ( d ) → M in T -mo d ( Set ) suh that y = u f ◦ i ( f ) .  By sp eializing the theorem and the orollary to the ase of the atomi top ology one immediately obtains the follo wing results. Corollary 4.10. L et E b e a lo  al ly smal l  o  omplete top os with a lass of gener ators G and T b e a the ory of pr eshe af typ e. If C := f.p. T -mo d ( Set ) op satises the right Or e  ondition then a mo del M ∈ T -mo d ( E ) is homo gene ous if and only if for e ah arr ow f : c → d in C op , obje t E ∈ G and arr ow y : E ∗ ( γ ∗ E ( i ( c ))) → E ∗ ( M ) in T -mo d ( E /E ) , ther e exists an obje t E f ∈ E , an epimorphism p f : E f ։ E and an arr ow u f : E ∗ f ( γ ∗ E ( i ( d ))) → E ∗ f ( M ) in T -mo d ( E /E ) suh that p ∗ f ( y ) = u f ◦ E ∗ f ( γ ∗ E ( i ( f ))) . Corollary 4.11. L et T b e a the ory of pr eshe af typ e. If C := f.p. T -mo d ( Set ) op satises the right Or e  ondition then a mo del M ∈ T -mo d ( Set ) is homo gene ous if and only if for e ah arr ow f : c → d in f.p. T -mo d ( Set ) and arr ow y : i ( c ) → M in T -mo d ( Set ) ther e exists an arr ow u f : i ( d ) → M in T -mo d ( Set ) suh that y = u f ◦ i ( f ) : i ( c ) i ( f )   y / / M i ( d ) u f > > | | | |  Remark 4.12. W e observ e that under the h yp otheses of Denition 4.4 for ea h top os E and ob jet E ∈ E there is an isomorphism E ∗ ( H om E T -mo d ( E ) ( γ ∗ E ( i ( c )) , M )) ∼ = H o m E /E T -mo d ( E /E ) ( γ ∗ E /E ( i ( c )) , E ∗ ( M )) , whi h is natural in c ∈ C . Hene, if M ∈ T -mo d ( E ) is J -homogeneous then E ∗ ( M ) ∈ T -mo d ( E /E ) is also J -homogeneous. This implies that, while dealing with theories T ′ that one w an ts to pro v e to satisfy the onditions of Theorem 4.8, one an restrit to argue with generalized elemen ts dened on 1 , b y the lo alizing priniple. This is illustrated in the follo wing example. 12 5 An example As an appliation of Corollaries 4.7 and 4.10 , w e pro v e that the lassifying top os for the theory of dense linearly ordered ob jets without endp oin ts is giv en b y atomi top os Sh ( Ord op f m , J ) , where Ord f m is the ategory of nite ordinals and order-preserving injetions b et w een them and J is the atomi otop ology on it. The theory L ′ of dense linearly ordered ob jets without endp oin ts is dened o v er a one-sorted signature ha ving one relation sym b ol < apart from equalit y , and has the follo wing axioms: ((( x < y ) ∧ ( y < x )) ⊢ x,y ⊥ ) , ( ⊤ ⊢ x,y (( x = y ) ∨ ( x < y ) ∨ ( y < x ))) , ( ⊤ ⊢ [ ] ( ∃ x ) ⊤ ) , (( x < y ) ⊢ x,y ( ∃ z )(( x < z ) ∧ ( z < y ))) and ( x ⊢ x ( ∃ y , z )(( y < x ) ∧ ( x < z ))) . The rst t w o axioms giv e the theory L of (deidably) linearly ordered ob jets; it is w ell-kno wn that this theory is of presheaf t yp e, hene, b eing Ord f m the ategory of nitely presen table L ′ -mo dels in Set , its lassifying top os is equiv alen t to the funtor ategory [ Ord f m , Set ] . Notie also that the ategory Ord op f m satises the righ t Ore ondition, so w e an equip it with the atomi top ology J . A mo del M ∈ L -mo d ( E ) is giv en b y a pair ( I , R ) where I is an ob jet of E and R is a relation on I satisfying the diagrammati forms of the rst t w o axioms ab o v e. W e will pro v e that for ea h top os E , a mo del M = ( I , R ) ∈ L -mo d ( E ) is homogeneous if and only if it is a mo del of L ′ , that is if ( I , R ) is non-empt y , dense and without endp oin ts; this will imply (b y the orollaries) our thesis. In one diretion, let us pro v e that if M is homogeneous then ( I , R ) is dense. F or ea h ob jet E ∈ E , w e denote b y < E is the order indued b y R on H o m E ( E , I ) . By the lo alizing priniple (fr. Remark 4.12), it is enough to pro v e that if x, y : 1 → I are t w o generalized elemen ts of I with x < 1 y then there exists an ob jet E ∈ E , an epimorphism p : E ։ 1 and an arro w z : E → I su h that x ◦ p < E z < E y ◦ p . Consider the arro w f : 2 → 3 in Ord f m dened b y f (0) = 0 and f (1) = 2 ; the arro ws x and y indue, via the assignmen t (0 → x , 1 → y ) and the univ ersal prop ert y of the opro dut γ ∗ E (2) , an arro w ψ : γ ∗ E (2) → I in L -mo d ( E ) . F rom the homogeneit y of M w e obtain the existene of an ob jet E ∈ E , an epimorphism p : E ։ 1 and 13 an arro w χ : E ∗ ( γ ∗ E (3)) → E ∗ ( I ) in L -mo d ( E /E ) su h that χ ◦ E ∗ ( γ ∗ E ( f )) = E ∗ ( ψ ) . Then the omp osite arro w E ∼ = E ∗ ( γ ∗ E (1)) E ∗ ( γ ∗ E ( u )) − → E ∗ ( γ ∗ E (3)) χ → E ∗ ( I ) π I → I , where u : 1 → 3 is the arro w in Ord f m whi h pi ks out the elemen t 1 ∈ 3 , giv es an arro w z : E → I with the required prop erties. The v eriations that ( I , R ) is non-empt y and without endp oin ts are similar and left to the reader. Con v ersely , w e pro v e that if M ∈ L ′ -mo d ( E ) then M is homogeneous. Again, b y the lo alizing priniple, this amoun ts to pro ving that giv en an arro w f : n → m in Ord f m and an arro w ψ : γ ∗ E ( n ) → I in L -mo d ( E ) , there exists an ob jet E ∈ E , an epimorphism p : E ։ 1 and an arro w χ : E ∗ ( γ ∗ E ( m )) → E ∗ ( I ) in L -mo d ( E /E ) su h that χ ◦ E ∗ ( γ ∗ E ( f )) = E ∗ ( ψ ) . The arro w ψ an b e iden tied, via the univ ersal prop ert y of the opro dut γ ∗ E ( n ) , with a family ( h i : 1 → I | i ∈ n ) of generalized elemen ts of I . T o nd the required arro w χ , w e indutiv ely use the fourth or the fth axioms to obtain, starting from the h i , an ob jet E ∈ E , an epimorphism p : E ։ 1 and m generalized elemen ts ( z j : E → I | j ∈ m ) su h that for ea h i ∈ n z f ( i ) = h i ◦ p and for ea h j, j ′ ∈ m (( j < j ′ ) ⇒ ( z j < E z j ′ )) . The family ( z j : E → I | j ∈ m ) then giv es rise to an arro w χ : E ∗ ( γ ∗ E ( m )) → E ∗ ( I ) in L -mo d ( E /E ) with the required prop ert y .  A  kno wledgemen ts: I am v ery grateful to m y Ph.D. sup ervisor P eter Johnstone for his supp ort and enouragemen t. 14 Referenes [1℄ P . T. Johnstone, Skethes of an Elephant: a top os the ory  omp endium. V ol.1 , v ol. 43 of Oxfor d L o gi Guides (Oxford Univ ersit y Press, 2002). [2℄ P . T. Johnstone, Skethes of an Elephant: a top os the ory  omp endium. V ol.2 , v ol. 44 of Oxfor d L o gi Guides (Oxford Univ ersit y Press, 2002). [3℄ S. Ma Lane and I. Mo erdijk, She aves in ge ometry and lo gi: a rst intr o dution to top os the ory (Springer-V erlag, 1992). [4℄ R. P aré and D. S h uma her, Abstrat families and the A djoin t F untor Theorems, in Indexe d  ate gories and their appli ations , Leture Notes in Math. v ol. 661 (Springer-V erlag, 1978), 1-125. 15