W-types in sheaves

W-t yp es in shea v es Benno v an den Berg & Iek e Mo erdijk Septem b er 25 , 2008 Abstract In this small note w e give a concrete description of W- types in categories of sheav es. It can b e shown that an y top os with a natura l num b ers ob ject has all W- t yp es. A lthoug h there is this general res ult, it can b e useful to hav e a co nc r ete description of W-t yp es in v arious topos es. F or example, a concrete description of W-t yp es in the effective top os can b e found in [2, 3], and a co ncrete descr iptio n of W- type s in categ ories of presheav es w as given in [5]. It was claimed in [5] that W-t yp es in categor ies of sheaves are computed a s in pres he aves (Prop ositio n 5.7 in lo c.cit. ) and can ther efore be describ ed in the same wa y . Unfortunately , this claim is incor r ect, as the following (easy) counterexample shows. Let f : 1 → 1 be the identit y ma p on t he terminal ob ject. The W-t yp e asso ciated to f is the initial ob ject, which, in gene r al, is different in catego ries of presheaves and sheav es. This means that w e still lack a concrete description o f W-types in categorie s of sheaves. This note aims to fill this g ap. W e w ould like to warn rea ders who are se ns itive to such issues that o ur metatheory is ZF C . In p articular, we fr e ely use the axiom of choic e . W e leav e the issue of how to descr ibe W-types in categories of sheav es when the metathe- ory is more dema nding (i.e. w eaker) to another o ccasio n. Categorie s o f sheav es a re describ ed using (Grothendieck) sites. There are different formulations of the notion o f a site, all esse n tially eq uiv alen t ([4] pro- vides an excellent dis c ussion of this p oint), but for our purp oses we find the following (“sifted”) formulation the most useful. Definition 0.1 L et C b e a categ ory . A sieve S on an o b ject a ∈ C consists of a set of ar r ows in C all having co do main a and clos ed under precomp osition (i.e., if f : b → a and g : c → b are a r rows in C and f belongs to S , then so do es f g ). W e call the set M a of all arrows into a the m aximal sieve on a . If S is a siev e on a and f : b → a is any map in C , we write f ∗ S for the sieve { g : c → b : f g ∈ S } on b . In ca se f belong s to S , we have f ∗ S = M b . A (Gr othendie ck) top olo gy Cov on C is g iven b y assigning to every ob ject a ∈ C a collec tio n of s ie ves Cov( a ) suc h tha t the following axioms ar e satisfied: 1 (Maximal siev e) The maximal sieve M a belo ngs to Cov( a ); (Stabilit y) If f : b → a is any map and S belong s to Cov( a ), then f ∗ S b elong s to Cov( b ); (Lo cal c haracter) If S is a sieve on a for which there can b e found a sieve R ∈ Co v( a ) such that for all f : b → a ∈ R the sieve f ∗ S belo ngs to Cov( b ), then S b elongs to Cov( a ). A pair ( C , Cov) consisting o f a ca tegory C and a topo lo gy Cov on it is called a site . If a site ( C , Cov) has be e n fixed, we call the sieves belo ng ing to some Cov( a ) c overing sieves . If S b elongs to Cov( a ) we sa y that S is a sieve c overing a , o r that a is c over e d by S . With our notion of site in place, we are ready to describ e W-types in sheav es. First, we fix a map F : Y → X of sheav es. Let V b e the set of all well-founded trees, in which • no des are lab elled with triples ( a, x, S ) with a an ob ject in C , x an element of X ( a ) and S a sieve cov ering a , • edges a re labelle d with pairs ( f , y ) with f : b → a a map in C and y an element of Y ( b ), in such a w ay that • if a no de is lab elled with ( a, x, S ) and an edge in to this no de is lab elled with ( f , y ), then f b elongs to S and F ( y ) = x · f , • and if a no de is la belle d with ( a, x, S ) ther e is, for every f ∈ S and y such that F ( y ) = x · f , a unique edge in to this no de labelled with ( f , y ). In fac t, V is the W-type (in the categor y of sets) asso cia ted to the pro jectio n P ( a,x,S ) { ( f , y ) : f : b → a ∈ S, y ∈ F ( b ) , F ( y ) = x · f }   { ( a, x, S ) : a ∈ C , x ∈ X ( a ) , S ∈ Cov( a ) } . If v deno tes a well-founded tree in V , we will also use the letter v for the function that a ssigns to lab els of edges in to the ro o t of v the tree attac hed to this edge. So if ( f , y ) is a lab el of one of the edges in to the ro ot of v , we will write v ( f , y ) for the tree that is attach ed to this edge; this is a gain a n elemen t of V . Note that an element of V is uniquely determined by the la be l of its ro ot and the function we just describ ed. 2 W e say that a tree v ∈ V is r o ote d at a n o b ject a in C , if its ro ot has a lab el whose firs t comp one nt is a . W e will denote the collection of trees r o oted at a by V ( a ). This g ives the set V the structure of a presheaf, a s one can define the following restr iction op eration. Let v ∈ V ( a ) and f : b → a b e a map in C . If v has r o ot ( a, x, S ) the tre e v · f has r o ot ( b, x · f , f ∗ S ) and ( v · f )( g , y ) = v ( f g , y ) . One easily verifies that this is well-defined, and gives V the structure of a presheaf. Next, we define by tra nsfinite r ecursion an eq uiv alence re la tion ∼ o n the presheaf V : v ∼ v ′ ⇔ If the ro ot o f v is la b elled with ( a, x, S ) and the ro o t of v ′ with ( a ′ , x ′ , S ′ ), then a = a ′ , x = x ′ and ther e is a cov ering s ieve R ⊆ S ∩ S ′ such that for every f : b → a ∈ R a nd y ∈ Y ( b ) such that F ( y ) = x · f w e hav e v ( f , y ) ∼ v ′ ( f , y ). By transfinite inductio n one verifies that ∼ is an e quiv alence relation. F ur ther- more, one verifies directly that ∼ is a preshea f (i.e. v ∼ v ′ implies that v and v ′ are r o oted a t the same ob ject a and that v · f ∼ v ′ · f for all f : b → a ). Next, we define c omp osabi lity and natur ality of trees (the terminology is taken fro m [5]): • A tree v whose ro ot is lab elled with ( a, x, S ) is c omp osable , if for an y ( f , y ) with f : b → a ∈ S and y ∈ Y ( b ) suc h tha t F ( y ) = x · f , the tree v ( f , y ) is ro oted at b . • A tree v whose ro o t is lab elled with ( a, x, S ) is natur al , if for any ( f , y ) with f : b → a ∈ S , g : c → b and y ∈ Y ( b ) such that F ( y ) = x · f , w e hav e v ( f , y ) · g ∼ v ( f g , y · g ) . It is clear that natural and comp osa ble trees are stable under re striction, so that also these for m pres heav es. The sa me a pplies to the presheaf W of trees that are her e ditarily comp osa ble and natura l (i.e. not only ar e they themselves both comp osable and natural, but the same is true for all their subtrees). The relation ∼ is also an equiv alence relation on W and we let W b e its quotient in pr eshe aves (so the quotient is co mputed p o int wise). W e s how that W is a sheaf and, indeed, the W-t yp e asso ciated to F in sheav es. Lemma 0.2 I f T i s a sieve c overing a and w, w ′ ∈ W ( a ) ar e such that w · f ∼ w ′ · f f or al l f ∈ T , then w ∼ w ′ . In other wor ds, W is s ep ar ate d. 3 Pro of. If the label of the r o ot o f w is of the fo rm ( a, x, S ) and that of w ′ is of the form ( a, x ′ , S ′ ), then w · f ∼ w ′ · f im plies that x · f = x ′ · f for all f ∈ T . As X is s e parated, it follows that x = x ′ . Consider R = { g : b → a ∈ ( S ∩ S ′ ) : ∀ h : c → b, y ∈ Y ( c ) [ F ( y ) = x · g h ⇒ w ( g h , y ) ∼ w ′ ( g h, y ) ] } . R is a sieve, and the statement of the lemma will follow once we show that it is cov ering. Fix a n element f ∈ T . Tha t w · f ∼ w ′ · f holds means that there is a covering sieve R f ⊆ f ∗ S ∩ f ∗ S ′ such tha t for ev er y k : c → b ∈ R f and y ∈ Y ( c ) such that F ( y ) = x · f k we hav e w ( f k , y ) = ( w · f )( k , y ) ∼ ( w ′ · k )( g , y ) = w ′ ( f k , y ). In o ther words, R f ⊆ f ∗ R . So R is a cov ering sieve b y loc a l character.  Lemma 0.3 W is a she af. Pro of. Let S b e a cov ering sieve on a and suppo se we hav e a compatible family of elemen ts ( w f ∈ W ) f ∈ S . F or every element f ∈ S ch o ose a representativ e ( w f ∈ W ) f ∈ S such that [ w f ] = w f . F o r every f : b → a ∈ S a repr e sentativ e w f has a ro o t lab elled by something of form ( b, x f , R f ). The x f form a compatible family and, since X is a sheaf, c a n b e glued tog ether to obtain an element x ∈ X ( a ). F ur thermore, R : = { f g : f ∈ S, g ∈ R f } ⊆ S is a co vering sieve, b y lo cal character. Before we pro ce e d, w e prov e the fo llowing claim: Claim. Assume f : b → a ∈ S and g : c → b ∈ R f and f ′ : b ′ → a ∈ S and g ′ : c → b ′ ∈ R f ′ are such that f g = f ′ g ′ . If y ∈ Y ( c ) is such that F ( y ) = x · f g , then w f ( g , y ) ∼ w f ′ ( g ′ , y ) . Pro of. By co mpatibilit y of the family ( w f ∈ W ) f ∈ S we know that w f · g ∼ w f ′ · g ′ ∈ W ( c ). This means that there is a cov ering sie ve T ⊆ g ∗ R f ∩ ( g ′ ) ∗ R f ′ such that for all h : d → c ∈ T and z ∈ Y ( d ) such that F ( z ) = x · f g h , w e hav e ( w f · g )( h, z ) ∼ ( w f ′ · g ′ )( h, z ). So if h : d → c ∈ T , then w f ( g , y ) · h ∼ w f ( g h, y · h ) = ( w f · g )( h, y · h ) ∼ ( w f ′ · g ′ )( h, y · h ) = w f ′ ( g ′ h, y · h ) ∼ w f ′ ( g ′ , y ) · h. Because W is s e parated (a s w as sho wn in Lemma 0.2), it follo ws that w f ( g , y ) ∼ w f ′ ( g ′ , y ). This completes the pro of of the claim.  4 W e c o nstruct a tr ee w ∈ V s uch that [ w ] is the de s ired glueing of all the w f . It will have a ro ot lab elled with ( a, x, R ). F or any h : c → a ∈ R a nd y ∈ Y ( c ) such that F ( y ) = x · h , we cho ose f ∈ S and g ∈ R f such that h = f g a nd s et w ( h, y ) = w f ( g , y ) . This do es e ssentially depend on the choice of f and g , but any t wo c hoices yield e quivalent results: that is precisely the c o nten t of the claim w e prov ed ab ov e. It is eas y to see that the tre e that we hav e constructed is comp os able. It is also natural, since if h : c → a ∈ R and y ∈ Y ( c ) are such that F ( y ) = x · h , and we hav e chosen f ∈ S and g ∈ R f such that h = f g and we hav e set w ( h, y ) = w f ( g , y ) , and k : d → c is any other map, and we hav e chosen f ′ ∈ S a nd g ′ ∈ R f ′ such that hk = f ′ g ′ and we hav e se t w ( hk , y · k ) = w f ′ ( g ′ , y · k ) , then it follows that w ( h, y ) · k = w f ( g , y ) · k ∼ w f ( g k , y · k ) ∼ w f ′ ( g ′ , y · k ) (using the claim) = w ( hk , y · k ) , as desir ed. It remains to show that [ w ] is a glueing of all the w f , i.e. that w · f ∼ w f for a ll f ∈ S . So let f : b → a ∈ S . First of all, x · f = x f , by construction. F ur thermore, for every g : c → b ∈ R f = R f ∩ f ∗ R and y ∈ Y ( c ) such that F ( y ) = x · f g , let f ′ ∈ S and g ′ ∈ R f ′ be such that f g = f ′ g ′ and w ( f g , y ) = w f ′ ( g ′ , y ). Then ( w · f )( g , y ) = w ( f g , y ) = w f ′ ( g ′ , y ) ∼ w f ( g , y ) (using the claim) . This completes the proo f.  Lemma 0.4 W is a P F -algebr a. Pro of. W e hav e to descr ib e a natural tra ns formation sup: P F W → W . An element o f P F W ( a ) is a pair ( x, t ) co nsisting of an elemen t x ∈ X ( a ) together with a natural tr a nsformation t : Y x → W , where Y x is the presheaf defined by Y x ( b ) = { ( f : b → a, y ) : F ( y ) = x · f } . 5 W e define sup x ( t ) to b e [ w ], where tree w is the tr e e whos e ro ot ha s lab el ( a, x, M a ) and for which, for every ( f , y ) ∈ Y x , the v alue of w ( f , y ) is chosen such that [ w ( f , y )] = t ( f , y ) (this is ano ther a pplication of choice). One quickly verifies that w is compo sable a nd natural. The other v erifica tion (that o f the naturality of the sup-op eratio n) is ea s y and also left to the reader.  Lemma 0.5 W is t he initial P F -algebr a. Pro of. W e follow the usual str a tegy: we show that sup: P F W → W is monic and that W has no proper P F -subalgebra s (i.e., w e a pply Theor em 26 of [1]). W e firs t show that sup is monic. So let ( x, t ) , ( x ′ , t ′ ) ∈ P F X ( a ) b e such that sup x ( t ) = sup x ′ ( t ′ ). It follo ws that x = x ′ and that there is a cov ering siev e S on a such that for a ll h : b → a ∈ S and y ∈ Y ( b ) such that F ( y ) = x · h , we ha ve t ( h, y ) = t ′ ( h, y ). W e need to s how that t = t ′ , so let ( f , y ) ∈ Y x be arbitra r y . F or e very g ∈ f ∗ S , w e ha ve: t ( f , y ) · g = t ( f g , y · g ) = t ′ ( f g , y · g ) = t ′ ( f , y ) · g . Since f ∗ S is cov ering , it follows that t ( f , y ) = t ′ ( f , y ), as desired. The fa c t that W has no prop er P F -subalgebra s is a consequence of the inductive prop erties o f V (we recall that V is a W-type). Let I b e a shea f and P F - subalgebra o f W . W e claim that J = { v ∈ V : if v is b oth hereditarily comp osa ble and natural, then [ v ] ∈ I } is a subalg ebra of V . P r o of: Suppose v is a tree that is b oth hereditar ily comp osable and natur al. Assume moreover that ( a, x, S ) is the label of its ro ot and that for all ( f , y ) with f : b → a ∈ S and F ( y ) = x · f , we k now that [ v ( f , y )] ∈ I . O ur aim is to show that [ v ] ∈ I . F or the mo men t fix an element f : b → a ∈ S . The tree v · f has ( b, x · f , M b ) as the lab el of its ro ot and for any ( g , y ) with g : c → b ∈ M b and F ( y ) = x · f g the tree ( v · f )( g , y ) is given b y v ( f g , y ). This means that [ v ] · f = sup x · f ( t ), where t ( g , y ) = [ v ( f g , y )] ∈ I . Since I is a P F -subalgebra o f W this implies that [ v ] · f ∈ I . F rom this it follows that [ v ] ∈ I , since I is a ls o a sheaf a nd f was a n arbitrar y element of the covering sieve S . W e c o nclude that J = V and I = W . This completes the pro of.  This co mpletes the pr o of of the correctnes s of our description of W-types in categorie s of sheaves. 6 References [1] B. v an den Berg. Inductive t yp es and exact co mpletion. Ann. Pur e Appl. L o gic , 13 4:95–1 21, 2005 . [2] B. v an den Ber g . Pr e dic ative top os t he ory and mo dels for c onst ructive set the ory . P hD thesis, Universit y of Utrech t, 2 006. Av ailable from the author ’s homepage. [3] M. Hofmann, J. v an Oosten, and T. Streicher. W ell-foundedness in r ealiz- ability . Ar ch. Math. L o gic , 45(7):795 –805 , 2006. [4] P .T. Johns tone. Sketches of an elephant: a top os the ory c omp en dium. V ol- ume 2 , volume 44 of Oxf. L o gic Gu ides . Oxford Univ ersity P r ess, Oxford, 2002. [5] I. Mo erdijk and E . Palmgren. W ellfounded trees in categories. Ann. Pur e Appl. L o gic , 1 04(1-3 ):1 89–21 8, 2000. 7