Aspects of Predicative Algebraic Set Theory III: Sheaves

Asp ects of Predicativ e Algebraic Set Theory I I I: Shea v es Benno v an den Berg ∗ & Iek e Mo erdijk † Septem b er 29 , 2011 Abstract This is the third installmen t in a series of p apers on algebraic set theory . In it, w e develop a uniform approach to s heaf models of constru ct ive set theories b as ed on ideas from categorical logic. The key notion is th at of a “pred ica tive category with small maps” which ax io matises the idea of a category of classes and class m orph isms , together with a selected class of maps whose ﬁbres are sets (in some axiomatic set theory). The main result of the present pap er is that such predicative categories with small maps are stable u n der internal sheav es. W e discuss the sheaf mo dels of constructive s et theory this leads to, as w ell as ideas for futu re w ork. 1 1 In tro duc ti on This is the third in a se ries of papers on algebra ic set theory , the aim of which is to develop a categ o rical sema n tics for cons tructiv e s et theo ries, including pred- icative o nes , based on the no tio n of a “ predicativ e c a tegory with small maps” . 2 In the ﬁrst pap er in this series [9] we discus s ed how these pr e dic a tiv e categorie s with small maps provide a s ound and complete semantics fo r constructive set theory . In the seco nd one [12], we explained how rea lizabilit y extens ions of such predicative categorie s with s mall maps can b e constr ucted. The pur p ose of the present pap er is to do the same fo r shea f-theoretic extensions. This progr am was summarised in [11], where we a nnounced the r e s ults that we will pr esen t and pr o v e here. ∗ Mathematisc h Instituut, Universiteit Utrec h t, PO. Box 80010, 3508 T A Utrec h t, the Netherlands. Email address: B.v anden Berg1@uu.nl. Supp orted by Netherlands Or ganisa- tion for Scient iﬁc Research (NW O pro ject “The M o del Theory of Constructive Pr oofs”). † Corresp ond ing author. Institute for Mathematics, Astroph ysics and Particle Physics, Radboud Unive rsity , Hey endaalsew eg 135, 6525 AJ N ijmegen, the Netherlands. Emai l ad dress: i.mo erdijk@math.ru.nl. 1 MCS: 18F20; 03F50; 03E70. 2 Accessible and well-written i ntroductions to algebraic set theory are [5, 6, 32]. 1 F or the co nvenience of the reader, a nd a lso to a llo w a compar ison with the work b y other researchers, we outline the main features of our a pproach . As said, the central concept in our theo ry is tha t of a pr e dic a tiv e categor y with small maps. It axiomatises the idea of a catego ry whose o b jects ar e classes and whose mo rphisms ar e functions betw een cla sses, and which is mor eo v er equipp ed with a designa ted class of maps. The maps in the designated class ar e called small, and the intuitiv e idea is that the ﬁbres of these maps are sets (in a certa in axiomatic set theor y). Such categories are in many wa ys like top oses, and to a large extent the purp ose of our s eries of pap ers is to develop a top os theory for these ca tegories. Indeed, like top oses, predicative categ ories with s ma ll maps turn out to b e clos ed under r ealizabilit y and sheaves. On the other hand, where top o ses ca n be seen as mo dels of a typed version of (constructive) higher- o rder arithmetic, predicative categor ie s with small maps provide mo dels of (constructiv e) set the ories . F urthermore, the no tion of a predicative ca tegory with small maps is pro of-theoretically rather weak: this allows us to mo del set theories which are pro of-theoretically weak er tha n higher- order arithmetic, such as Aczel’s set theory CZF (see [1]). But at the s ame time, the notion of a predicative ca teg ory with small maps can also be strengthened, so tha t it lea ds to mo dels of set theories pro of-theoretically stronger tha n higher- order a rithmetic, lik e IZF . The reas on for this is that o ne can impo se additiona l axioms o n the class o f small maps. This a dded ﬂexibility is an imp ortant feature of algebra ic set theory . A central r esult in alg ebraic set theo ry says that the semantics pr ovided by predicative categ o ries with sma ll maps is complete. Mo re precisely , every pred- icative categ ory with small maps contains an ob ject (“the initial Z F-algebra” in the terminology of [24], or “the initial P s -algebra ” in the terminolog y o f [9] 3 ) which ca rries the structure of a model of set theor y . Whic h set-theo retic a x - ioms hold in this mo del dep ends on the pro perties of the c lass of sma ll maps and on the logic of the underlying categor y : in diﬀerent situations, this initial ZF-algebra can b e a mo del of CZF , of IZF , or of o rdinary ZF . (The axio ms of the constr uctiv e set theores CZF and IZF a re r e called in Section 2 below.) The co mpletenes s referred to a bov e results fro m the fact that fro m the s y n tax of CZF (or (I)ZF ), we c a n build a predicative categor y with sma ll maps with the prop ert y that in the initial ZF- a lgebra in this catego ry , precisely those sen tences are v alid whic h are deriv able from the axiom of CZF (see [9]). (Completeness theorems of this k ind go back to [3 1, 6]. One should also men tion that one can o btain a pr e dic a tiv e categor y with small maps from the syntax of Martin- L¨ of type theory : Aczel’s interpretation of CZF in Martin-L¨ of type theory g oes precisely via the initia l ZF-algebr a in this categ ory . In fact, our pro of of the existence of the initial ZF-a lgebra in any predica tive catego ry with s mall maps in [9] was mo delled on Aczel’s interpretation, a s it was in [29].) In algebraic se t theory w e approach the c o nstruction of realizability cate- 3 Appendix A in [24] con tains a pro of of the fact that b oth these terms refer to the same ob ject. In the sequel we wil l use these terms interc hanga bly . 2 gories and of catego ries of sheaves in a to pos-theor etic spirit; that is, we reg ard these r e a lizabilit y and sheaf cons tr uctions a s closur e pr op erties of pr e dicativ e categorie s with sma ll maps. F or realiz a bilit y this mea ns that sta rting fr om any predicative c ategory with s ma ll maps ( E , S ) one can build a predicative real- izability ca tegory with sma ll maps ( E f f E , E f f S ) ov er it. Inside b oth of these categorie s, we hav e mo dels of constructive set theory ( CZF say), a s s ho wn in the following picture. Her e , the vertical arrows are t wo insta nces of the sa me construction of the initial ZF-alg e br a, applied to diﬀere nt predicative categ ories with small maps: ( E , S )   / / ( E f f E , E f f S )   mo del of CZF / / realizability mo del of CZF T ra ditional treatments of realiz abilit y either regard it as a mo del-theoretic construction (which would corr espond to the low er edge o f the diagra m), o r as a pro of-theoretic interpretation (deﬁning a realiza bilit y mo del o f CZF inside CZF , as in [30], for instance): the latter would corresp ond to the left-hand vertical a rro w in the s pecial case wher e E is the syntactic category asso ciated to CZF . So in a wa y our treatment ca ptures bo th constructions in a uniform wa y . That r ealizabilit y is indeed a closure prop ert y o f predicative c a tegories with small maps was the pr incipa l r esult of [9]. The main result of the present pap er is that the s ame is true for sheav es, leading to a n analogo us diagra m: ( E , S )   / / (Sh E , Sh S )   mo del of CZF / / sheaf mo de l of CZF The main technical diﬃculty in s ho wing that predica tiv e categories with small maps are closed under sheaves lies in showing that the a xioms concer ning in- ductive t yp e s (W-types) and a n axiom ca lled “fullness” (needed to mode l the subset collection axio m o f CZF ) a r e inherited by shea f mo dels. The pro ofs o f these facts ar e q uite long and inv olved, and take up a la r ge part of this pa per (the situation for rea liz a bilit y was very similar). T o summarise, in our appro a c h there is one uniform co nstruction of a mo del out of a predicative catego ry with small ma ps ( E , S ), w hich one can apply to diﬀerent kinds o f such categ ories, constructed using s y n tax , us ing rea lizabilit y , using sheav es, or any iteratio n or combination of these techniques. W e pro ceed to compare our re sults with those of other authors. Ear ly work on ca tegorical semantics of se t theory (for exa mple, [1 6 ] and [15]) was concerned with sheaf and r ealizabilit y top oses deﬁned ov er S ets . The same applies to the bo ok which introduce d algebr aic set theory [24]. In par ticular, to the b est of 3 our knowledge, b efore our work a systematic ac c oun t was lacking of itera tions and co m binatio ns of r ealizabilit y and she a f interpretations. In addition these earlier paper s were co nce rned exclusively with impredicative set theories, s uch as ZF o r IZF : the only ex ception seems to hav e b een an early pap er [21] by Grayson, treating mo dels of predica tiv e s e t theory in the co ntext of wha t would now be called formal top ology . The ﬁr st pa per extending the metho ds of alg ebraic s et theory to pr edicativ e systems w as [29]. The authors of this pap er showed how catego rical mo dels of Martin-L¨ of type theory (with universes) le ad to mo dels of CZF ex tended with a choice pr inciple, which they dubbed the Axiom of Multiple Choice (AMC) . They establishe d how such ca teg orical mo dels of type theo r y ar e closed under sheav es, hence lea ding to sheaf models of a strengthening of CZF . They did not develop a sema ntics for CZF p er se and relied on a technical notion of a collection site, which we manag e to av oid here (moreov er, there was a mistake in their treatment of W-types of sheav es; we correct this in Section 4.4 below, see also [10]). Two accounts of preshe a f mo dels in the context o f algebraic set theory hav e bee n written by Ga m bino [18] and W arr e n [34]. In [18] Gambino shows how an ear lier (unpublished) construction of a mo del of constr uctiv e s e t theor y by Dana Scott can be regar ded a s an initial ZF-algebr a in a c a tegory of presheaves, and that one can p erform the construction in a predicative metatheory as well. W ar r en shows in [3 4 ] that many of the axioms that we will discus s a re inher ited by ca teg ories of coalg ebras for a Cartesian como nad, a co nstruction which in- cludes presheaf mo dels as a s pecial case. But note that neither of these authors discusses the technically complicated axio ms concerning W-t ypes a nd fullness, as we w ill do in Se c tio ns 3 and 4 b elo w. In his PhD thesis [17], Ga m bino gav e a sy s tematic account of Heyting-v a lued mo dels for CZF (see a lso [19]). This work was in the context of formal to pology (essentially , s ites whose under lying categor ies are p osets). He ha s subsequently work ed o n generalis ing this to ar bitrary sites a nd on putting this in the context of algebraic set theor y . In [2 0 ], he to ok the ﬁrst step in constr uc ting the sheaﬁﬁ- cation functor and in [7 ], written together with Awodey , Lumsdaine and W arren, he chec ks that the basic axioms for small maps are inherited by categor ies of sheav es in the g eneral setting of sheav es for a L awvere-Tierney top ology . W e will extend these results by pr o ving that for sites which hav e a pr esen tation (for a deﬁnition, see Deﬁnition 4.1 b elo w), the axioms for W-types and fo r fullness are sta ble under taking shea f extensions. Note that for pro of-theoretic rea sons, fullness cannot be stable under taking more genera l kinds of sheav es such as those for a site which do es not hav e a presentation, or for a L awvere-Tierney top ology . The po in t is that CZF extended with the Law of Excluded Middle gives ZF , a muc h stronger system pr oof-theor etically , and therefor e a double- negation interpretation of CZF in itself must fa il. The culprit turns out to be the fullness axiom, which can therefore not b e stable under taking sheaves fo r the double- negation topo logy o r sheav es for an ar bitrary site (see [19] a nd [21]). 4 W e conclude this introduction by outlining the org anisation of our paper . In Section 2 we recall the main deﬁnitions from [11, 9]. W e will introduce the axioms for a class of small ma ps necessary to obtain mo dels o f CZF and IZF . Among these necess ary axioms , we will dis c us s the fullness axiom, the ax ioms concerning W-types and the ax io m of multiple choice in detail, as these are the most complicated technically and our main r esults, which we for m ula te precisely in Section 2.5, are concerne d with these axio ms . In Section 3 we show that pre dic a tiv e ca tegories with small maps ar e closed under presheav es and that all the axioms that w e hav e listed in Section 2 a re inherited by such pres heaf mo dels. An impo rtan t part of our tre a tmen t is that we distinguish betw een tw o cla sses of small maps: the “ point wis e” and “lo cally” small ones . It turns out that for certain ax ioms it is easier to show tha t they are inherited by p oint wise sma ll maps while for other axioms it is easier to show that they are inher ited by lo cally small maps, and therefore it is an imp ortant result that these classe s of ma ps coincide. W e follow a similar strategy in Section 4, where we discuss sheaves: we again distinguish b et w een t wo classes of maps, where for some axio ms it is easier to use one deﬁnition, while for other ax ioms it turns out to be easier to use the other. T o show that these tw o cla sses coincide we use the fullness axiom and assume that the site has a pres en ta tio n. 4 This section also contains o ur main techn ical results: that sheaf mo dels inherit the fullness axiom, as well as the axioms concerning W-types. 5 Strictly sp eaking our re s ults for pr eshea v es in Section 3 are sp ecial cas e s of our res ults in Section 4. W e b eliev e, how ev er, that it is useful to give dir ect pro ofs of the results for pr eshea ves, and in ma n y cases it is helpful to see how the pro of g oes in the (easier ) presheaf cas e be fo re emb arking on the more inv o lv ed pr oofs in the sheaf case. Finally , in Section 5 we give explicit descr iptions of the shea f mo dels of constructive set theo ry our results lead to . W e also p oin t out the connection to forcing for classica l set theo r ies. This will co mplete o ur progra m for developing an abs tract semantics of c on- structive set theo ry , in particular o f Acze l’s CZF , as outlined [11]. As a res ult top os-theoretic ins ig h ts and ca teg orical metho ds can now b e used in the study of constructive set theories. F or instance, one can obtain consistency and in- depe ndence results using sheaf and r ealizabilit y mo dels or b y a combination o f these in terpretations. In future work, we will use sheaf-theoretic metho ds to show tha t the fan r ule a s well a s certain contin uity rules a re derived rules for CZF and r elated theories [13]. The main results of this pap er were pre s en ted b y the second a uthor in a 4 In [ 11 ] we claimed that (instead of fullness) the exp onen tiation axiom w ould suﬃce to establish this result, but that might not b e corr ec t. 5 One subtlety ar ises when we try to show that an axiom saying that certain inductiv es t ypes are small (axiom (WS) to be pr ec ise) is inherited by sheaf models: we show this using the axiom of multiple cho ice. In fact, we s usp ect that something of this sort is unav oidable and one has to go beyond C ZF prop er to show that its v alidity i s inherited by sheaf m odels. 5 tutorial on ca tegorical logic at the Logic Collo quium 20 06 in Nijmegen. W e a re grateful to the org a nisers of the Lo gic Collo quium for giving one of the a utho r s this opp o rtunit y . The ﬁnal dra ft of this pap er was completed during a stay of the ﬁrst author a t the Mittag- Leﬄer Institute in Stockholm. W e would like to thank the Institute and the orga nisers of the pro gram in Mathematical Log ic in F all 2009 for aw arding him a g ran t which enabled him to complete this pap er in such ex cellen t working conditions. In a ddition, we would like to a c kno wledge the helpful discussio ns we had with Steve Aw o dey , Nicola Ga mbino, Jaa p v an Oosten, Erik Palmgren, Thoma s Streicher, Michael W arren, and esp ecially Peter LeF anu Lumsdaine (see Rema r k 4.14 b elo w). 2 Preliminaries 2.1 Review of Algebraic Set Theory In this sectio n w e rec all the ma in features o f our a pproach to Algebraic Set Theory from [11, 9]. W e will a lw a ys as sume that our ambien t category E is a p ositive Heyting c ate gory . That means that E is (i) Ca rtesian, i.e., it has ﬁnite limits. (ii) r egular, i.e., morphisms fa c to r in a stable fashio n as a cov er followed by a monomorphism. 6 (iii) p ositive, i.e., it has ﬁnite sums, which ar e disjoint a nd stable. (iv) Heyting, i.e., for any mor phism f : Y / / X the induced pullback functor f ∗ : Sub ( X ) / / Sub( Y ) has a rig h t adjoint ∀ f . This means that E is rich enough to interpret ﬁr st-order intuitionistic logic. Such a ca tegory E will be calle d a c ate gory with smal l maps , if it comes equipped with a class of maps S sa tisfying a list o f axioms . T o formulate thes e , we use the notion of a cov ering square. Deﬁnition 2.1 A diagram in E of the form D f   / / C g   B p / / A 6 Recall that a map f : B → A is a co v er, if the only sub ob ject of A through which it factors, is the m ax imal one; and that f i s a r eg ular epimorphism if it is the co equa lizer of i ts k ernel pair. These tw o classes coincide in r eg ular categories (see [22, Pr oposition A1.3.4]). 6 is ca lled a quasi-pul lb ack , when the cano nic a l map D / / B × A C is a cover. If p is also a cov er, the diagram will be called a c overing squar e . When f and g ﬁt into a cov ering squar e as s hown, we say that f c overs g , or that g is c over e d by f . Deﬁnition 2.2 A cla ss o f ma ps in E satisfying the following axioms (A1-9) will b e called a class of smal l maps : (A1) (Pullback stability) In a n y pullback sq ua re D g   / / B f   C p / / A where f ∈ S , a ls o g ∈ S . (A2) (Descent) If in a pullback square as a bov e p is a cov er and g ∈ S , then also f ∈ S . (A3) (Sums) Whenever X / / Y a nd X ′ / / Y ′ belo ng to S , so do es X + X ′ / / Y + Y ′ . (A4) (Finiteness) The maps 0 / / 1 , 1 / / 1 a nd 1 + 1 / / 1 b elong to S . (A5) (Comp osition) S is closed under comp osition. (A6) (Quotients) In a commuting triang le Z h @ @ @ @ @ @ @ f / / / / Y g ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ X , if f is a cover and h b elongs to S , then so do es g . (A7) (Collection) Any tw o arrows p : Y / / X and f : X / / A where p is a cover and f b elongs to S ﬁt int o a covering square Z g   / / Y p / / / / X f   B h / / / / A, where g b elongs to S . 7 (A8) (Heyting) F or any morphism f : Y / / X belo nging to S , the r igh t adjoint to pullba ck ∀ f : Sub( Y ) / / Sub( X ) sends small mo nos to small monos . (A9) (Diagona ls) All diag onals ∆ X : X / / X × X b elong to S . F or further discuss ion of these axioms we refer to [9]. A pair ( E , S ) in which S is a class of small maps in E will b e c a lled a c ate gory with smal l maps . In such categor ies with s mall maps, ob jects A will be called smal l , if the unique map from A to the terminal o b ject is s mall. A sub ob ject A ⊆ X will b e called a smal l sub obje ct if A is a small o b ject. I f any of its representing monomo rphisms m : A → X is small, they all ar e and in this case the sub ob ject will b e called b ounde d . Remark 2 .3 In the sequel we will often implicitly use that catego ries with small maps are stable under slicing. By this we mean that for any ca teg ory with small maps ( E , S ) and ob ject X in E , the pair ( E /X , S /X ), with S /X being deﬁned by f ∈ S /X ⇔ Σ X f ∈ S , is aga in a categ o ry with small maps (here Σ X is the forgetful functor E / X → E sending a n o b ject p : A → X in E / X to A and morphisms to themselves). Moreov er, any of the further axio ms for classes o f small maps to b e intro duced below are stable under slicing, in the sense that their v alidity in the s lic e over 1 implies their v alidity in every slic e . Remark 2 .4 A very useful feature o f ca teg ories of small ma ps , a nd one we will frequently exploit, is that they satisfy an in ternal form of b ounded sepa ration. A precise s tatemen t is the following: if φ ( x ) is a formula in the internal logic of E with fre e v ar iable x ∈ X , all whose basic pr edicates are int erpreted as bo unded sub ob jects (note that this includes all e q ualities, by (A9) ), and which contains existential a nd universal quantiﬁ cations ∃ f and ∀ f along small maps f only , then A = { x ∈ X : φ ( x ) } ⊆ X deﬁnes a b ounded sub ob ject of X . In pa rticular, smallness of X implies small- ness of A . Deﬁnition 2.5 A c a tegory with small maps ( E , S ) will b e called a pr e dic ative c ate gory with smal l maps , if the following a xioms hold: ( Π E) All morphisms f ∈ S a re exp onen tiable. (WE) F o r all f : X / / Y ∈ S , the W-type W f asso ciated to f e xists. 8 (NE) E has a natural num b ers ob ject N . (NS) Mor eo v er, N / / 1 ∈ S . (Representa bility) There is a small map π : E / / U (the “universal small map”) such tha t any f : Y / / X ∈ S ﬁts into a diagram of the form Y f   B   / / o o o o E π   X A / / o o o o U, where the left hand square is cov ering a nd the right ha nd squar e is a pullback. (Bounded exactness) F or a ny eq uiv a lence relation R / / / / X × X given by a small mo no, a stable quotient X/R exists in E . (F or a detailed discussion of thes e axioms w e refer ag ain to [9 ]; W-t yp es and the axiom (WE) will also b e discussed in Section 2.3 b elow.) In predica tiv e catego ries with small maps o ne can derive the existence of a power c la ss functor, cla s sifying s ma ll sub ob jects: Deﬁnition 2.6 By a D -indexe d family of sub obje cts o f C , we mean a subo b ject R ⊆ C × D . It will b e called a D -indexe d family of smal l sub obje cts , whenever the comp osite R ⊆ C × D / / D belo ngs to S . If it exists, the p ower class obje ct P s X is the classifying ob ject for the families of small sub ob jects o f X . This means that it comes equipp ed with a P s X -indexed family of small sub ob jects of X , denoted by ∈ X ⊆ X × P s X (or simply ∈ , whenever X is understo od), with the pro p erty that for a n y Y - indexed family of small sub ob jects of X , R ⊆ X × Y say , there exists a unique map ρ : Y / / P s X such that the s q uare R     / / ∈ X     X × Y id × ρ / / X × P s X is a pullback. Prop osition 2.7 [9, Cor ollary 6.11] In a pr e dic ative c ate gory with sm a l l maps al l p ower class obje cts exist. 9 Moreov er, one can show that the a s signmen t X 7→ P s X is functor ia l and that this functor has an initial alg ebra. Theorem 2. 8 [9, Theor em 7.4] In a pr e dic ative c ate gory with smal l maps the P s -functor has an initial algebr a. The imp ortance o f this result r e sides in the fact that this initial algebra ca n b e used to mo del a w eak in tuitionistic set theory: if V is the initial algebra and E : V → P s V is the inv erse o f the P s -algebra map o n V (which is an isomorphis m, since V is an initia l alg ebra), then one can deﬁne a binary predica te ǫ o n V b y setting xǫy ⇔ x ∈ V E ( y ) , where ∈ V ⊆ V × P s V deriv es from the p o w er class structure on P s V . The resulting structure ( V , ǫ ) mo dels a weak intuitionistic set theory , which we hav e called RST (for rudimentary set theory), co nsisting o f the following axioms : Extensionality: ∀ x ( xǫa ↔ xǫb ) → a = b . Empt y set: ∃ x ∀ y ¬ y ǫx . P airing: ∃ x ∀ y ( y ǫx ↔ y = a ∨ y = b ). Union: ∃ x ∀ y ( y ǫx ↔ ∃ z ǫa y ǫz ). Set i nduction: ∀ x ( ∀ y ǫx φ ( y ) → φ ( x )) → ∀ x φ ( x ). Bounded se paration: ∃ x ∀ y ( y ǫx ↔ y ǫa ∧ φ ( y ) ), for any b o unded fo rm ula φ in which a do es not o ccur. Strong col lection: ∀ xǫa ∃ y φ ( x, y ) → ∃ b B( xǫa, y ǫb ) φ , where B( xǫa, y ǫb ) φ ab- breviates ∀ xǫa ∃ y ǫb φ ∧ ∀ y ǫb ∃ xǫa φ. Inﬁnit y: ∃ a ( ∃ x xǫa ) ∧ ( ∀ xǫa ∃ y ǫa xǫy ). In fact, as shown in [9 ], the initial P s -algebra s in predicative categor ies with small maps form a complete semantics for the set theory R ST . T o obta in com- plete sema n tics for b etter known intuitionistic set theories , like IZF and CZF , one needs further requirements on the clas s of s mall maps S . F or ex a mple, the set theory IZF is obtained from RST by adding the ax ioms F ull separation: ∃ x ∀ y ( y ǫx ↔ y ǫa ∧ φ ( y ) ), for any formula φ in which a do es not o ccur. P o wer se t: ∃ x ∀ y ( y ǫx ↔ y ⊆ a ), wher e y ⊆ a abbrev iates ∀ z ( z ǫy → z ǫa ). And to obtain a sound and complete semantics for IZF one require s of ones predicative catego ry of sma ll maps tha t it satisﬁes: 10 (M) All monomorphis ms belo ng to S . (PS) F o r any map f : Y / / X ∈ S , the p ow er class ob ject P X s ( f ) / / X in E /X belo ngs to S . The set theory CZF , intro duced by Aczel in [1], is obta ine d by adding to RST a weakening of the p o w er set a x iom ca lle d subset co llection: Subset coll ection: ∃ c ∀ z ( ∀ xǫa ∃ y ǫb φ ( x, y , z ) → ∃ dǫc B( xǫa, y ǫd ) φ ( x, y , z )). F or a suitable categorica l analog ue , see Section 2.3 b elo w. F or the sa k e of co mpletenes s we also lis t the following tw o ax ioms, saying that certain Π-types a nd W-types a re small. (The ﬁr st ther efore co rrespo nds to the exp onentiation axiom in set theory; w e will say more a bout the second in Section 2.2 below.) ( Π S) F o r any map f : Y / / X ∈ S , a functor Π f : E / Y / / E /X right adjo int to pullba c k exists and pr eserv es morphisms in S . (WS) F or all f : X / / Y ∈ S with Y small, the W-t yp e W f asso ciated to f is small. 2.2 W-t yp es In a predica tiv e categor y with sma ll maps ( E , S ) the axiom ( Π E) holds and therefore any small map f : B / / A is exp onentiable. It therefor e induces an endofunctor on E , which will b e called the p olynomia l fun ctor P f asso ciated to f . The quickest wa y to deﬁne it is as the following c o mposition: C ∼ = C / 1 B ∗ / / C /B Π f / / C / A Σ A / / C / 1 ∼ = C . In more set-theore tic terms it could b e deﬁned a s: P f ( X ) = X a ∈ A X B a . Whenever it exists, the initial algebra for the p olynomial functor P f will b e called the W-typ e asso ciate d to f . Int uitively , elements o f a W-t yp e are well-founded trees. In the categor y of sets, a ll W-types ex ist, and the W-types ha ve as elements well-founded trees, with an appropriate lab elling of its edges and no des. What is an appropr iate 11 lab elling is determined by the branching type f : B / / A : no des should be la- belle d by elements a ∈ A , edges by elements b ∈ B , in such a wa y that the edge s int o a no de lab elled by a ar e uniquely enumerated by f − 1 ( a ). The follo wing picture hop efully co n veys the idea: . . . . . . . . . . . . • u   4 4 4 4 4 4 a v         • x   7 7 7 7 7 7 7 • y   • z          f − 1 ( a ) = ∅ f − 1 ( b ) = { u, v } f − 1 ( c ) = { x, y , z } . . . a x # # G G G G G G G G G b y   c z w w o o o o o o o o o o o o o o c This set has the structur e of a P f -algebra : when a n e lemen t a ∈ A is given, together with a ma p t : B a / / W f , one can build a new element s up a t ∈ W f , as follows. First take a fresh no de, lab el it by a and draw edg es into this no de, one for every b ∈ B a , lab elling them a ccordingly . Then on the edge la belled by b ∈ B a , stick the tr ee tb . Clearly , this sup op eration is a bijective map. Moreov er, since every tree in the W-type is well-founded, it can b e thoug h t of as having b een g enerated by a p ossibly transﬁnite num b er o f itera tions of this sup o peration. That is precisely what makes this alge bra initial. The trees that can be thought of as having b een used in the gener ation of a certain element w ∈ W f are called its subtrees. O ne could call the trees tb ∈ W f the imme diate subtr e es o f sup a t , and w ′ ∈ W f a subtr e e of w ∈ W f if it is an immediate subtree, or an immediate subtree of a n immediate subtr e e, or. . . , etc. Note that with this use o f the word subtree , a tr ee is never a subtre e of itself (so pro per subtree might hav e b een a b etter ter minology). W e r e call that ther e are tw o a xioms c o ncerning W-types: (WE) F o r all f : X / / Y ∈ S , the W-type W f asso ciated to f e xists. (WS) Mor eo v er, if Y is small, als o W f is small. Maybe it is no t to o late to p oin t out the following fa c t, which ex pla ins w hy these axioms play no essential role in the impre dic a tiv e s e tt ing: Theorem 2. 9 L et ( E , S ) b e a c ate gory with smal l maps satisfying (N S) and (M) . 1. If S satisﬁes (PE) , then it also satisﬁes (WE) . 2. If S satisﬁes (PS) , then it also satisﬁes (WS) . Pro of. Note that in a catego ry with small maps satisfying (M) and (PE) the ob ject P s (1) is a sub ob ject cla s siﬁer. Therefor e the ﬁrst r e s ult can b e shown 12 along the lines of Chapter 3 in [24]. F or showing the sec ond res ult, one simply copies the argument why top oses with nno have all W-types fro m [29].  In the s e quel we will need the following res ult. W e will write P + s X for the ob ject of small inhabited sub ob jects of X : P + s X = { A ∈ P s X : ∃ x ∈ X ( x ∈ A ) } . Theorem 2. 1 0 F or any smal l map f : B → A in a pr e dic ative c ate gory with smal l maps ( E , S ) , t h e endofunctors on E deﬁne d by Φ = P f ◦ P s and Ψ = P f ◦ P + s have initial algebr as. Remark 2 .11 Befor e we sketc h the pro of of Theorem 2.10, it might b e go o d to explain the intuitiv e meaning of thes e initial algebr as. In fact, they are v ariations on the W-types explained a bov e: they ar e also classes of well-founded trees, but the conditions on the labelling s of the no des a nd edges are slight ly diﬀerent . It is still the cas e that no des ar e lab e lle d by elements a ∈ A and edges with elements b ∈ B , in such a w ay that if b ∈ B decorates a cer tain edg e, then f ( b ) deco rates the no de it po in ts to. But wherea s in a W-type , every no de in a well-founded tree lab elled with a ∈ A ha s for every b ∈ f − 1 ( a ) pr e cisely one edge into it lab elled with b , in the initial algebra s for Φ there ar e set-many, and p ossibly none , and in the initial alg ebra for Ψ ther e a re set- many, but at le ast one . Pro of. The pro of of Theo r em 2.10 is a v aria tion on that of Theo rem 7.4 in [9] and ther efore we will only sketc h the a r gumen t. Fix a universal small map π : E → U , a nd write S = { ( a ∈ A, u ∈ U, φ : E u → B a ) } . Let K b e the W-type in E as sociated to the map g ﬁtting into the pullback square R g   / / E π   S pro j / / U. An element k ∈ K is therefor e of the for m sup ( a,u,φ ) t , wher e ( a, u, φ ) ∈ S is the lab el of the ro ot of k and t : E u → K is the function that ass igns to every element e ∈ E u the tree that is attached to the ro ot of k with the edge lab elled with e . Deﬁne the following equiv a lence rela tion on K by recursion: sup ( a,u,φ ) t ∼ sup ( a ′ ,u ′ ,φ ′ ) t ′ , if a = a ′ and 13 for all e ∈ E u there is an e ′ ∈ E u ′ such that φ ( e ) = φ ′ ( e ′ ) and t ( e ) ∼ t ′ ( e ′ ), and for all e ′ ∈ E u ′ there is an e ∈ E u such that φ ( e ) = φ ′ ( e ′ ) and t ( e ) ∼ t ′ ( e ′ ). (The ex is tence o f this r elation ∼ can b e justiﬁed using the metho ds o f [8] or [9]. See Theo rem 7.4 in [9], for instance.) The equiv a le nce rela tion is bo unded (one prov es this by induction) and its quotient is the initial a lgebra for Φ. The initial algebra for Ψ is constructed in the same wa y , but with S deﬁned as S = { ( a ∈ A, u ∈ U, φ : E u → B a ) : φ is a cover } .  2.3 F ullness In order to expres s the subset co lle ction axiom, introduced by Peter Aczel in [1], in diag r ammatic terms, it is helpful to consider a n axio m which is equiv alent to it called ful lness (see [4]). In the language o f set theory one can formulate fullness using the notion of a multi-value d se ction : a m ulti-v alued section (or mvs ) o f a function φ : b / / a is a multi-v alued function s fro m a to b such that φs = id a (as relations). Identifying s with its image, this is the same as a subse t p of b such that p ⊆ b / / a is s urjectiv e. F o r us, fullness states that for any such φ ther e is a sma ll family o f mvs s such that any mvs co n tains o ne in this family . W r itten out forma lly: F ullnes s : ∃ z ( z ⊆ mvs ( φ ) ∧ ∀ x ǫ m vs ( φ ) ∃ c ǫ z ( c ⊆ x )). Here, mvs ( φ ) is an abbrev ia tion for the class of a ll multi-v alued sections of a function φ : b / / a , i.e., subsets p of b such that ∀ xǫa ∃ y ǫp φ ( y ) = x . In or der to r eform ulate this diagrammatically , we s ay that a m ulti-v alued section ( mvs ) for a small map φ : B / / A , ov er some o b ject X , is a subo b ject P ⊆ B such that the co mposite P / / A is a smal l cov er. (Smallne s s of this map is equiv alent to P being a b ounded sub ob ject of B .) W e write m vs X ( φ ) for the set of all mvs s of a map φ . This se t obviously inherits the structure of a partial o rder from Sub( B ). Note that any mo rphism f : Y / / X induces an order-pr eserving map f ∗ : mvs X ( φ ) / / m vs Y ( f ∗ φ ) , obtained by pulling back along f . T o av oid ov erburdening the notation, we will frequently talk a bout the map φ ov er Y , when we a c tually mean the map f ∗ φ ov er Y , the map f a lw a ys b eing understo o d. The ca tegorical fullness axiom now rea ds: 14 (F) F or a n y φ : B / / A ∈ S ov e r some X with A / / X ∈ S , there is a cover q : X ′ / / X and a ma p y : Y / / X ′ belo nging to S , together w ith an mvs P of φ over Y , with the following “ generic” prop ert y: if z : Z / / X ′ is any map a nd Q any mvs of φ over Z , then there is a map k : U / / Y and a cov er l : U / / Z with y k = z l s uc h that k ∗ P ≤ l ∗ Q a s mvs s o f φ over U . It is ea sy to see that in a set-theo r etic c on text fullness is a c onsequence o f the powerset axiom (be c a use then the co llection of al l m ulti-v alued sections of a map φ : b → a for ms a set) a nd implies the ex ponentiation axiom (b ecause if z is a set o f m vs s of the pr o jection p : a × b → a such that a n y mvs is r e ﬁned by one is this set, then the set of functions from a to b can b e constructed from z by selecting the univ alued elements, i.e., tho s e elements that ar e rea lly functions). Showing that in a catego rical co ntext (F) follows from (PS) and implies ( Π S) is no t muc h harder and we will therefor e not write out a formal pro of. In the sequel we will use the following tw o lemmas co ncerning the fullness axiom: Lemma 2.12 Supp ose we have the fol lowing diagr am Y 2 f 2   β / / Y 1 f 1   X 2 / / ! ! ! ! B B B B B B B B X 1     X , in which the squar e is a quasi-pul lb ack and f 1 and f 2 ar e smal l. When P is a “generic” mvs for a map φ : B / / A over X living over Y 1 (“generic” as in the statement of the ful lness axiom), t hen β ∗ P is also a generic mvs for φ , living over Y 2 . Pro of. A simple diagra m chase.  Lemma 2.13 Supp ose we ar e given a diagr am of the form B 0 / / / / ψ   B φ   A 0 / / / / i   A j   X 0 p / / / / X , 15 in which b oth squar es ar e c overing and al l t he vertic al arr ows ar e smal l. If a generic mvs for ψ exists over X 0 , t he n also a generic mvs for φ exist s over X . Pro of. This was L e mma 6.23 in [9].  2.4 Axiom of m ult iple choice The axio m of multiple choice was intro duced by Mo e rdijk and Palmgren in [29]. Their motiv ation was to hav e a choice principle which is implied by the existence of enough pro jectives (“the presentation axiom” in Aczel’s termino lo gy) and is stable under tak ing sheaves (unlike the existence of enough pro jectives). W e will use it in Section 4.4 to show that the axio m (WS) is stable under ta king sheav es. One can give a succinct formulation of the axiom of mu ltiple choice us ing the notion of a collection span (see [9, Deﬁnition 6.1 4]). 7 Deﬁnition 2.14 A span ( g , h ) in E C D h / / g o o B is called a c ol le ction sp an , when, in the internal logic, it holds tha t for any map f : E / / D c cov ering some ﬁbre of g , there is a ﬁbre D c ′ of g and a map p : D c ′ / / E such that f p is a cov er ov er B . A co llection s pa n is E / A will b e called a c ol le ction sp an over A . Diagrammatica lly , we ca n express this by a s king that for a n y map e : E / / C and a n y epi F / / E × C D there is a diagra m of the form B D g   h 2 2 E ′ × C D / / o o   F / / / / E × C D / /   D g   h l l C E ′ / / / / o o E c / / C where the middle square is a c o v ering squa r e, in volving the given map F / / D × C E , while the o ther tw o squares are pullbacks. (AMC) (Axiom of m ultiple choice) F or a n y sma ll map f : Y / / X , ther e is a 7 The wa y we formulate the Axiom of Multiple Choice here is s ligh tly diﬀeren t from ho w it wa s stated in [29]. Both formulations are equiv alen t, how ev er; see [14]. 16 cov er q : A / / X and a dia g ram D g   h / / / / q ∗ Y q ∗ f   / / / / Y f   C r / / / / A q / / / / X , in w hich the rig h t square is a pullback and the left squar e a covering sq uare in which a ll maps a r e small and in which ( g , h ) is a collection s pan over A . In the internal logic (AMC) is often applied in the following for m: Lemma 2.15 In a pr e dic ative c ate gory with smal l maps in which (AMC) holds, the fol lowi ng principle holds in the int ernal lo gic: any smal l m a p f : B / / A b etwe en smal l obje cts ﬁt s into a c overing squ ar e D q / / g   B f   C p / / A in which al l maps and obje cts ar e smal l and ( g , q ) is a c ol le ction sp an over A . Pro of. This is prov ed exactly as Pro p osition 4 .6 in [2 9].  The following res ult was prov ed in [29] as well. Recall fro m [2, 4] that the existence of many inductiv ely deﬁned sets within CZF can b e guara nteed, in a predicatively acceptable wa y , by extending CZF with Aczel’s Regular Extension Axiom. Prop osition 2.16 If ( E , S ) is a pr e dic ative c ate gory with sm a l l maps satisfying the axioms (AM C) , ( Π S) and (WS) , then Ac zel’s R e gular Extension A xiom holds in t h e initial P s -algebr a in this c ate gory. In additio n, w e will need: Prop osition 2.17 L et ( E , S ) b e a pr e dic ative c ate gory with smal l maps. If S satisﬁes the axioms (AMC) and ( Π S) , then it satisﬁes t h e axiom (F) as wel l. Pro of. W e ar g ue internally a nd us e Lemma 2.15. So supp ose that (AMC) holds and f : B → A is a small map b et ween sma ll o b jects. W e need to ﬁnd a small co llection of mvs s { P y : y ∈ Y } such that a n y mvs of f is reﬁned by o ne in this family . 17 W e apply Lemma 2.15 to A → 1 to obtain a covering squa re of the form D h / /   A   C / / 1 , such tha t fo r a n y cov er p : E → D c we ﬁnd a c ′ ∈ C and a map t : D c ′ → E such that pt is a cov er ov e r A . Let Y b e the collection o f a ll pairs ( c, s ) with c in C and s a map D c → B such that f s = h c , and let P y be the image of the map s : D c → B . Then P y is a n mvs , b ecause the h c are epi, and Y is small, b ecause ( Π S) holds. Now supp o se n : Q → B is a n y mono s uc h that f n : Q → B → A is a cov er. Pick a c ∈ C and pull back f n along h c to o btain a cov er q : E → D c , as in: E p / / q     Q f n     D c h c / / A. It follows that ther e exists a n element c ′ ∈ C and a ma p g : D c ′ → E such that q g is a cov er ov er A . Set s = npg and y = ( c ′ , s ). Then P y = Im( g ) is contained in Q .  2.5 Main results After a ll these deﬁnitions, we can for m ula te o ur main result. Let A be either { ( F ) } , or { ( AMC ) , ( ΠS ) , ( WS ) } , or { ( M ) , ( PS ) } . Theorem 2. 1 8 L et ( E , S ) b e a pr e dic ative c ate gory with smal l maps for which al l the axioms in A hold and let ( C , Cov ) b e an internal Gr othendie ck site in E , such that the c o domain map C 1 → C 0 is smal l and a pr esentation for the t o p olo gy exists. Then in the c ate gory of internal she aves Sh E ( C ) one c an identify a class of maps making it into a pr e dic ative c ate gory with smal l m a ps for which t h e axioms in A holds as wel l. In combination with Theo rem 2 .8 this r esult can b e used to prov e the exis- tence of sheaf mo dels of v arious cons tructiv e set theories: Corollary 2.19 Supp ose that ( E , S ) is a pr e dic ative c ate gory with smal l maps satisfying the axiom ( F ) and supp ose t ha t ( C , Cov) is an internal Gr othendi e ck site in E , such that the c o domain map C 1 → C 0 is smal l and a pr esentation for the top olo gy exists. Then the initial P s -algebr a in Sh E ( C ) exists and is a mo del of CZF . If, mor e over, 18 1. the axioms ( AMC ) and ( WS ) hold in E , then the initial P s -algebr a in Sh E ( C ) also mo dels A czel’s R e gular Extension Axiom. 2. the axioms ( M ) and ( PS ) hold in E , then the initial P s -algebr a in Sh E ( C ) is a mo del of IZF . 3 Preshea v es In this s ection we show that predicative categor ie s with small maps are closed under presheav es. Mo re pr ecisely , we show that if ( E , S ) is a predicative ca tegory with small maps and C is an internal ca tegory in E , then inside the category Psh E ( C ) of internal presheaves one c a n identify a class of maps such that P sh E ( C ) bec omes a predica tiv e category with small maps. Our arg umen t pr oceeds in tw o steps. First, we need to identify a suitable class of ma ps in a catego ry of internal presheav es. W e take what we will ca ll the po in twise small maps of presheaves. T o prov e that these p oin t wise small maps sa tisfy axio ms (A1-9) , we need to assume that the co domain map of C is small (note that the same assumption was made in [3 4]) . Subsequently , we show that the v alidit y in the categor y with small maps ( E , S ) of an y of the axioms int ro duced in the pr evious section implies its v alidity in any categ o ry of internal pr eshea ves ov er ( E , S ). T o av o id r epeating the convoluted e x pression “the v alidity of axiom (X) in a predica tiv e categ ory with small maps implies its v alidit y in any categ ory of internal pres hea v es over it”, we will wr ite “ (X) is inher ited by presheaf mo dels” or “ (X) is stable under presheaf extensions” to expres s this. The main r esult of this section is that the fullness ax iom (F) is stable under presheaf ex tensions. Most of the o ther stability r e s ults in this section are not really new and can in o ne form or another alrea dy be found in [24, 28, 2 9, 18, 34]. Nevertheless, for several reaso ns, we have decided to include their pro ofs here. First of all, none of the references we mentioned uses conditions on the ambien t category which a re exa ctly the sa me as o urs (in par ticular, w e assume only bo unded exactness). Secondly , these pap ers use diﬀerent deﬁnitions of the class of sma ll maps in presheaves, which we will compare in Section 3.2 b elow. And, thirdly , including them will make o ur presentation se lf- c o n ta ined. 3.1 P oin twise small maps in preshea ves Throughout this section, we w ork in a pr e dicativ e ca tegory with small maps ( E , S ) in which we are given an in ternal categor y C , whose co domain map co d: C 1 / / C 0 is small. Here we hav e written C 0 for the ob ject of o b jects of C a nd C 1 for its ob ject of ar ro ws. In addition, we will write Psh E ( C ) for the categor y of internal 19 presheav es, and π ∗ for the forgetful functor: π ∗ : P s h E ( C ) / / E / C 0 . In the sequel, w e will use capital letters for presheav es and morphisms of presheav es, and low er case letters for ob jects a nd morphisms in C . W e will also employ the following piece of notation. F o r any map of presheaves F : Y → X and element x ∈ X ( a ), we set Y M x : = { ( f ∈ C 1 , y ∈ Y (dom f )) : co d( f ) = a, F ( y ) = x · f } . (The ca pital letter M stands for the maxima l sieve on b : for this rea s on, this piece of notation is c o nsisten t with the one to be intro duced in Section 4.4.) Occasiona lly , we will reg ard Y M x as a presheaf: in that ca se, its ﬁbre at b ∈ C 0 is Y M x ( b ) = { ( f : b → a ∈ C 1 , y ∈ Y ( b )) : F b ( y ) = x · f } , and the r estriction of a n element ( f , y ) ∈ Y M x ( b ) alo ng g : c → b is given by ( f , y ) · g = ( f g , y · g ) . A map of pres hea v es F : Y → X will b e called p ointwise s mall , if π ∗ F belo ngs to S / C 0 in E / C 0 . Note that for any such p oin t wise sma ll map of pres he aves a nd for an y x ∈ X ( a ) with a ∈ C 0 the ob ject Y M x will b e s mall. This is a n immediate consequence of the fact that the co domain map is assumed to b e small. Theorem 3. 1 The p ointwise s mall maps make Psh E ( C ) into a c ate gory with smal l maps. Pro of. O bserv e that ﬁnite limits, ima ges a nd sums of pre shea v es a re computed “p oin t wise”, that is, as in E / C 0 . The universal quantiﬁcation of A ⊆ Y along F : Y / / X is given by the following for m ula: for any a ∈ C 0 , ∀ F ( A )( a ) = { x ∈ X ( a ) : ∀ ( f , y ) ∈ Y M x ( y ∈ A ) } (1) This s ho ws that Psh E ( C ) is a p ositive Heyting categor y . T o c omplete the pro of, we nee d to chec k that the p oin t wise small maps in presheaves satisfy axioms (A1-9) . W e p ostpone the pro of of the collection axiom (A7) (it will be Pr oposi- tion 3.9). The remaining axioms follow ea sily , as all we need to do is verify them po in twise. F o r verifying axiom (A8) , o ne observes that the universal quantiﬁer in (1) range s over a small ob ject.  F or most of the axioms that we intro duced in Section 2, it is r elativ ely straightforward to chec k that they are inherited by presheaf mo dels. The ex- ceptions ar e the represe n tability , collection and fullness axioms: verifying these requires an alter nativ e characterisation of the sma ll maps in presheav es and they will therefore b e discusse d in a separate section. 20 Prop osition 3.2 The fol lowing axioms ar e inherite d by pr eshe af mo dels: (M) , b ounde d exactness, (NE) and (NS) , as wel l as ( Π E) , ( Π S) and (PS) . Pro of. The mono morphisms in pr eshea ves a re precise ly thos e maps whic h are p oint wise monic and therefore the axio m (M) will b e inherited by preshea f mo dels. Similarly , pr e sheaf mo dels inherit b ounded exactness , b ecause quotients of equiv alence relations are computed p oin t wise. Since the natura l n um ber s ob jects in pr e s hea v es has that o f the base catego ry E in every ﬁbre, bo th (NE) and (N S) are inherited b y presheaf mo dels. Finally , consider the following diagra m in preshe aves, in which F is small: B G   Y F / / X . The ob ject P = Π F ( G ) ov er an element x ∈ X ( a ) is given by the formula: P x : = { s ∈ Π ( f ,y ) ∈ Y M x G − 1 ( y ) : s is natural } . This shows that ( Π E) is inherited by preshea f extensions. It also shows that ( Π S) is inherited, b ecause the fo r m ula ∀ ( f , y ) ∈ Y M x ( b ) ∀ g : c → b ( s ( f , y ) · g = s ( f g , y · g )) expressing the naturality of s is b ounded. T o see that (PS) is inherited, we ﬁr st need a descr iption o f the P s -functor in the categor y of internal pr eshea ves. This was ﬁrst given by Gam bino in [1 8] and works as follows. If X is a preshea f and y c is the r epresen table preshea f on c ∈ C 0 , then P s ( X )( c ) = { A ⊆ y c × X : A is a small subpreshe a f } , with restriction along f : d → c on a n element A ∈ P s ( X )( c ) deﬁned by ( A · f )( e ) = { ( g : e → d, x ∈ X ( e )) : ( f g , x ) ∈ A } . The members hip r elation ∈ X ⊆ X × P s X is deﬁned o n a n ob ject c ∈ C b y: fo r all x ∈ X ( c ) and A ∈ P s ( X )( c ), x ∈ X A ⇐ ⇒ ( id c , x ) ∈ A. This shows that the axiom (PS) is inherited, b ecause the fo rm ula ∀ ( f : b → c, x ) ∈ A ∀ g : a → b [ ( f g, x · g ) ∈ A ] expressing that A is a subpreshe a f is b ounded.  21 Theorem 3. 3 The axioms (WE) and (WS) ar e inherite d by pr eshe af exten- sions. Pro of. F or this pro of we need to recall the constructio n of p olynomial functors and W-type s in pr eshea v es fro m [28]. F or a mor phism of presheaves F : Y → X and a presheaf Z , the v alue of P F ( Z ) = X x ∈ X Z Y x on a n ob ject a of C 0 is g iv en by P F ( Z )( a ) = { ( x ∈ X ( a ) , t : Y M x → Z ) } , where t is suppos ed to be a morphism o f pre shea v es. The restr ic tio n of a n element ( x, t ) a lo ng a map f : b → a is g iv en by ( x · f , f ∗ t ), where ( f ∗ t )( g , y ) = t ( f g, y ) . The presheaf morphism F induces a ma p φ : X a ∈C 0 X x ∈ X ( a ) Y M x / / X a ∈C 0 X ( a ) in E whose ﬁbre ov er x ∈ X ( a ) is Y M x and which is there fore small. The W-type in presheav es will b e cons tr ucted from the W-t yp e V asso ciated to φ in E . A t ypical element v ∈ V is a tree of the form v = sup x t where x is an element of some X ( a ) and t is a function Y M x → V . F or any such v , one deﬁnes its ro ot ρ ( v ) to b e a . If one writes V ( a ) for the se t of tree s v such that ρ ( v ) = a , the ob ject V will ca r ry the structure o f a presheaf, with the restriction of an element v ∈ V ( a ) a long a ma p f : b / / a g iv e n by v · f = sup x · f f ∗ t. The W-t ype asso ciated to F in presheav es is obtained by selec ting the right trees from V , the right tr ees b eing those all whose s ubtr ees ar e (in the ter minology of [28]) comp osable and na tur al. A tree v = sup x ( t ) is called c omp osable if for all ( f , y ) ∈ Y M x , ρ ( t ( f , y )) = dom( f ) . A tree v = sup x ( t ) is natu r al , if it is compo s able and for any ( f , y ) ∈ Y M x ( a ) and a n y g : b → a , we have t ( f , y ) · g = t ( f g , y · g ) 22 (so t is actually a natural transfor ma tion). A tree will be called her e ditarily natur al , if all its subtrees (including the tree itse lf ) are na tural. In [28, Lemma 5 .5] it was shown that for any hereditarily na tural tr ee v ro oted in a a nd map f : b → a in C , the tree v · f is also hereditar ily natur al. So when W ( a ) ⊆ V ( a ) is the collection of her editarily natura l trees r ooted in a , W is a subpresheaf of V . A pr oof that W is the W-t ype for F can b e found in the sourc es mentioned ab o v e. Pres en tly , the cruc ia l po in t is tha t the construction can b e imita ted in our setting, so that (WE) is stable under presheaves. The same applies to (WS) , es sen tially b ecause W was obtained from V using b ounded separa tion (in this connection it is essential that the ob ject of all subtree s of a par ticular tree v is small, see [9, Theorem 6.13 ]).  3.2 Lo cally small maps in preshea ves F or showing that the re pr esen tabilit y , co llection and fullness axioms are inher- ited by pres hea f mo dels, we use a diﬀerent characterisa tion of the small maps in presheav es: we introduce the lo c al ly smal l maps a nd show that thes e co incide with the po int wise small maps. T o deﬁne these lo cally small maps, we hav e to set up some notation. Remark 3 .4 The functor π ∗ : P s h E ( C ) / / E / C 0 has a left adjoint, which is com- puted as follows: to any ob ject ( X , σ X : X → C 0 ) and a ∈ C 0 one asso ciates π ! ( X )( a ) = { ( x ∈ X , f : a → b ) : σ X ( x ) = b } , which is a presheaf with restrictio n g iv en by ( x, f ) · g = ( x, f g ) . This means that π ∗ π ! X ﬁts into the pullback squar e π ∗ π ! X   / / C 1 co d   X σ X / / C 0 . F ro m this one immediately sees tha t π ! preserves smallness. F ur thermore, the comp onen t maps o f the c o unit π ! π ∗ → 1 ar e small cov ers (they a re covers, bec ause under π ∗ they beco me split epis in E / C 0 ; that they are also small is another co nsequence of the fact that the co domain map is assumed to b e sma ll). In what follows, natural tr ansformations of the for m π ! B → π ! A 23 will play a cr ucial rˆ ole and there fore it will b e worthwhile to analyse them more closely . First, due to the adjunction, they corre s pond to maps in E / C 0 of the form B → π ∗ π ! A. Such a ma p is determined by tw o pieces of data: a map r : B → A in E , and, for any b ∈ B , a mo rphism s b : σ B ( b ) → σ A ( rb ) in C , as de pic ted in the following diagram: B r   s / / σ B ' ' C 1 co d   dom / / C 0 A σ A / / C 0 . (2) (Note that we do not hav e σ A r = σ B in ge ne r al, s o that it is b est to consider r as a map in E .) W e will use the expr ession ( r, s ) for the map B → π ∗ π ! A and ( r , s ) ! for the na tural transformatio n π ! B → π ! A determined by a dia gram as in (2). In the following lemma, we collect the imp ortan t prop erties of the op eration ( − , − ) ! . Lemma 3.5 1. Assu me r and s ar e as in diagr am (2). Then ( r, s ) ! : π ! B → π ! A is a p ointwise s mall map of pr eshe aves iﬀ r : B → A is smal l in E . 2. Assu me r : B → A is a c over and σ A : A → C 0 is an arbitr ary map. If we set σ B = σ A r and s b = id σ B b for every b ∈ B , then ( r , s ) ! : π ! B → π ! A is a c over. 3. If ( r, s ): B → π ∗ π ! A is a c over and σ B ( b ) = do m( s b ) for al l b ∈ B , then also ( r, s ) ! : π ! B → π ! A is a c over. 4. If ( r , s ) ! : π ! B → π ! A is a natur al t r ansformation determine d by a diagr am as in (2) and we ar e given a c ommuting diagr am V h   p / / B r   W q / / A in E , then t h ese data induc e a c ommuting squar e of pr eshe aves π ! V π ! p / / ( h,sp ) !   π ! B ( r,s ) !   π ! W π ! q / / π ! A, 24 with σ V = σ B p and σ W = σ A q . Mor e over, if the original diagr am is a pul lb ack (r esp. a quasi-pul lb ack or a c overing squar e), then so is t he induc e d diagr am. 5. If ( r, s ) ! : π ! A → π ! X and ( u, v ) ! : π ! B → π ! X ar e natur al t r ansformations with the same c o domain and for every x ∈ X and every p air ( a, b ) ∈ A × X B with x = r a = u b ther e is a pul lb ack squar e k ( a,b ) p ( a,b )   q ( a,b ) / / σ B ( b ) v b   σ A ( a ) s a / / σ X ( x ) in C , then π ! applie d to the obje ct σ A × X B : A × X B → C 0 in E / C 0 obtaine d by sending ( a, b ) ∈ A × X B to k ( a,b ) is the pu llb ack of ( r, s ) ! along ( u, v ) ! in Psh E ( C ) : π ! ( A × X B ) ( π 2 ,q ) ! / / ( π 1 ,p ) !   π ! B ( u,v ) !   π ! A ( r,s ) ! / / π ! X . Pro of. By direct insp ection.  Using the notation we hav e set up, we can list the t wo notions of a small map o f presheaves. 1. The p oin t wise deﬁnition (as in the pr e vious sec tio n): a map F : B / / A of presheav es is p ointwise smal l , when π ∗ F is a small ma p in E / C 0 . 2. The lo cal deﬁnition (as in [2 4]) : a map F : B / / A of pr eshea ves is lo c al ly smal l , when F is covered by a map of the form ( r, s ) ! in which r is small in E . W e show that these t wo clas ses of maps co incide, so that hence fo rth we can use the phrase “small map” without any dange r of ambiguit y . Prop osition 3.6 A map is p ointwise smal l iﬀ it is lo c al ly smal l. Pro of. W e have alre ady obse rv ed that maps of the form ( r , s ) ! with r small a re po in twise small, so all maps cov ered by o ne of this for m are p oin twise small as well. This shows that lo cally small maps a re point wise small. That all p oint wise small maps are also lo cally small follows from the next lemma and the fact that the counit maps π ! π ∗ Y → Y are covers.  25 Lemma 3.7 F or any p ointwise smal l map F : Z / / Y and any map L : π ! B / / Y ther e is a quasi-pul lb ack squ a r e of pr eshe aves of t he form π ! C ( k,l ) !   / / Z F   π ! B L / / Y , with k sm a l l in E . Pro of. Let S b e the pullba c k of F along L and cover S using the counit as in: π ! π ∗ S / / / / S / /   π ! B L   Z F / / Y . W e know the comp osite along the top is of the form ( k , l ) ! . Because k is the comp osite along the middle of the following diagram and bo th squares in this diagram are pullbacks, k is the comp osite of tw o s ma ll maps and hence small. C 1 co d / / C 0 π ∗ S / /   π ∗ π ! B O O π ∗ L   / / B O O π ∗ Z π ∗ F / / π ∗ Y  Corollary 3.8 Every p ointwise sm a l l map is c over e d by one of the form ( r , s ) ! in which r is s mall. In fact, every c omp osable p air ( G, F ) of p ointwise smal l maps of pr eshe aves ﬁts into a double c overing squ ar e of the form π ! C ( k,l ) !   / / / / Z G   π ! B ( r,s ) !   / / / / Y F   π ! A / / / / X , in which k and r ar e smal l in E . 26 Pro of. W e have just shown that every p oin t wise small map is cov ered by o ne of the for m ( r, s ) ! in which r is small, whic h is the ﬁrst statement. The s econd statement follows immediately from this and the pr evious lemma.  Using this a lter nativ e characterisation, we can q uickly show that the co lle c- tion axiom is inherited by preshea f mo dels, as promised. Prop osition 3.9 The c ol le ction axiom (A7) is inherite d by pr eshe af mo dels. Pro of. Let F : M / / N b e a small map and Q : E / / M b e a cov er. Without loss of genera lit y , we may assume that F is of the form ( k , l ) ! for some s mall map k : X / / Y in E . Let n be the map obtained by pullback in E / C 0 : T n / / / /   X   π ∗ E π ∗ Q / / / / π ∗ π ! X . Then use collection in E to obtain a cov ering square as follows: B m / / d   T n / / / / X k   A p / / / / Y . Using Lemma 3.5.4 this leads to a cov ering square in the categor y of presheaves π ! B π ! m / / ( d,lnm ) !   π ! T / / π ! n ( ( E Q / / / / π ! X ( k,l ) !   π ! A π ! p / / / / π ! Y , th us completing the pro of.  Prop osition 3.10 The r epr esentability axiom is inherite d by pr eshe af mo dels. 27 Pro of. Let π : E / / U b e a universal small map in E , and deﬁne the following t wo ob jects in E / C 0 : U ′ = { ( u ∈ U, c ∈ C 0 , p : E u → C 1 ) : ∀ e ∈ E u (co d( pe ) = c ) } , σ U ′ ( u, c, p ) = c, E ′ = { ( u, c, p, e ) : ( u, c, p ) ∈ U ′ , e ∈ E u } , σ E ′ ( u, c, p, e ) = dom( pe ) . If r : E ′ → U ′ is the ob vious pro jection and s : E ′ → C 1 is the map sending ( u, c, p, e ) to pe , then r and s ﬁt into a co mm uting squar e as shown: E ′ r   s / / σ E ′ ( ( C 1 co d   dom / / C 0 U ′ σ U ′ / / C 0 . W e cla im that the induced map ( r , s ) ! in the categor y of presheaves is a universal small map. T o s ho w this, we need to prov e that any small map F can b e cov ered b y a pullba c k of ( r , s ) ! . Without loss of generality , we may assume that F = ( k , l ) ! for some small map k : X / / Y in E . Since π is a universal small map, there exis ts a diagram of the form E π   V m o o h   i / / X k   l / / C 1 co d   U W n o o j / / Y σ Y / / C 0 , in which the left square is a pullback and the middle one a covering squar e. F ro m this, w e o btain a commuting diagr am o f the form C 0 V h   m ′ / / li ' ' σ V > > } } } } } } } } E ′ r   s / / C 1 co d   dom ` ` A A A A A A A W n ′ / / σ W 7 7 U ′ σ U / / C 0 28 by putting σ W = σ Y j, n ′ w = ( nw, σ W w, λe ∈ E nw .l im − 1 e ) , σ V = σ X i, m ′ v = ( n ′ hv , mv ) . T og ether these t wo commuting diag rams determine a diagr am in the categ ory of internal presheaves π ! E ′ ( r,s ) !   π ! V π ! m ′ o o ( h,li ) !   π ! i / / π ! X ( k,l ) !   π ! U ′ π ! W π ! n ′ o o π ! j / / π ! Y , in which the left s quare is a pullback and the r igh t one a cov ering square (by Lemma 3 .5.4).  Theorem 3. 1 1 (A ssuming C has chosen pul lb acks.) The ful lness axiom (F) is inherite d by pr eshe af mo dels. Pro of. In view of Lemma 2.13 and Corolla ry 3.8, we o nly ne e d to build g eneric mvs s for maps of the form ( k , l ) ! : π ! B → π ! A in which k is small, where π ! A lies ov er some ob ject of the form π ! X via a map o f the form ( r , s ) ! in which r is small. T o cons truct this g eneric mvs , we have to apply fullness in E . F or this purp ose, consider the o b ject B 0 = { ( b ∈ B , f ∈ C 1 , g ∈ C 1 ) : σ X ( rk b ) = co d( f ) , ( f ∗ l b ) g = id } . Here f ∗ l b is understo o d to b e the map ﬁtting, for any b ∈ B and f : d → c with c = σ X ( rk b ), in the double pullback dia gram f ∗ σ B ( b ) / / f ∗ l b   σ B ( b ) l b   f ∗ σ A ( k b ) / / f ∗ s kb   σ A ( k b ) s kb   d f / / c in C . If we write k 0 : B 0 → A for the map sending ( b, f , g ) to k ( b ), then this map is s mall, so we can use fullness in E to ﬁnd a cover n : W → X and a small map 29 m 0 : Z 0 / / W , together with a ge neric mvs P 0 for k 0 ov er Z 0 , a s de pic ted in the following diagr a m. P 0 / / / / ' ' ' ' Z 0 × X B 0   / / B 0 k 0   Z 0 × X A / /   A r   Z 0 m 0 / / W n / / X Now we ma k e a num ber o f deﬁnitions: Z = { ( z 0 ∈ Z 0 , f ∈ C 1 ) : co d( f ) = nm 0 ( z 0 ) and ( ∀ a ∈ A nm 0 ( z 0 ) ) ( ∃ b ∈ B a ) ( ∃ g ∈ C 1 ) ( z 0 , b, f , g ) ∈ P 0 } , σ Z ( z 0 , f ) = dom( f ) , µ ( z 0 , f ) = f , σ W ( w ) = σ X ( nw ) . In addition, we write m 1 : Z → Z 0 for the obvious (sma ll) pro jection and m = m 0 m 1 . Then we o btain the following diagram of preshe aves, in whic h bo th rectangles are pullbacks computed using Lemma 3.5.5: π ! ( Z × X B )   / / π ! B ( k,l ) !   π ! ( Z × X A )   / / π ! A ( r,s ) !   π ! Z ( m,µ ) ! / / π ! W π ! n / / π ! X . W e wish to de ﬁne a subpreshea f o f π ! ( Z × X B ) and prov e that it is the g eneric mvs of ( k , l ) ! . W e can do this b y saying: ( z 0 , f , b, h ) ∈ P if h fa ctors through a map g with ( z 0 , b, f , g ) ∈ P 0 . The inclusio n of P in π ! ( Z × X B ) is b ounded, b ecause P is deﬁned by a bo unded formula (using that the co domain map is small). F urther more, the induced map from P to π ! ( Z × X A ) is a cov e r by deﬁnition o f Z . Th us it r e mains to verify genericity . T o verify this, let E → π ! W be a n y map a nd Q be an m vs of ( k , l ) ! ov er E . Without loss o f generality , we may a ssume that E is o f the for m π ! Y (since E 30 can alwa ys b e covered using the counit). This leads to the following diagr am o f presheav es in which the recta ng les a re pullbacks: Q / / / / ( ( ( ( •   / / π ! B ( k,l ) !   •   / / π ! A ( r,s ) !   π ! Y ( d,δ ) ! / / π ! W π ! n / / π ! X . Of co urse, we will as sume that the pullbacks are computed using Lemma 3 .5.5, so that they ar e π ! ( Y × X B ) and π ! ( Y × X A ), resp ectiv ely . It follows that in E we have an mvs Q 0 for k 0 ov er Y , as in Q 0 / / / / ' ' ' ' Y × X B 0   / / B 0 k 0   Y × X A / /   A r   Y d / / W n / / X , given by ( y , b, f , g ) ∈ Q 0 if f = δ y and ( y , b, g ) ∈ Q. Therefore, b y the genericity of P 0 , ther e is a co ver e : U / / Y a nd a map c : U / / Z 0 with de = m 0 c a nd c ∗ P 0 ⊆ e ∗ Q 0 . (3) Claim: If we put t ( u ) = ( c ( u ) , δ e ( u ) ), then t ( u ) ∈ Z for every u ∈ U . Pr o of: Suppo se a ∈ A nm 0 c ( u ) . W e know that ther e are b ∈ B a , f , g ∈ C 1 such that ( c ( u ) , b, f , g ) ∈ P 0 , b ecause P 0 → Z 0 × X A is sur jectiv e, but the questio n is: do we hav e f = δ e ( u ) ? The a nsw er is yes , b ecause if ( c ( u ) , b , f , g ) ∈ P 0 , then ( e ( u ) , b , f , g ) ∈ Q 0 by (3). So we have f = δ e ( u ) by deﬁnitio n of Q 0 .  It follows that if we put σ U ( u ): = dom( δ e ( u ) ) = σ Y ( e ( u )) = σ Z ( t ( u )) , then we have the following diagra m of pres hea v es: π ! U π ! t / / π ! e     π ! ( Z ) ( m,µ ) !   π ! Y ( d,δ ) ! / / π ! W . 31 T o see that this square co mmutes, we need to chase an element from π ! U along the tw o sides and it s uﬃces to this for an element of the form ( u, id ). ( m, µ ) ! π ! t ( u, i d ) = ( m, µ ) ! ( t ( u ) , id ) = ( m, µ ) ! ( c ( u ) , δ e ( u ) , id ) = ( m 0 c ( u ) , δ e ( u ) ) = ( de ( u ) , δ e ( u ) ) = ( d, δ ) ! ( e ( u ) , i d ) = ( d, δ ) ! π ! e ( u, id ) . Therefore the pro of will b e ﬁnis he d, once we show that ( π ! t ) ∗ P ⊆ ( π ! e ) ∗ Q . T o show this, consider an e lemen t ( u , b, h ) ∈ ( π ! t ) ∗ P . W e then have ( t ( u ) , b , h ) ∈ P , which, by deﬁnition of P , means that h factors thro ugh a map g such that ( c ( u ) , b, δ e ( u ) , g ) ∈ P 0 . F rom (3) it follows that ( e ( u ) , b, δ e ( u ) , g ) ∈ Q 0 and hence ( e ( u ) , b, g ) ∈ Q , by deﬁnition o f Q 0 . Since Q is a preshea f, we also hav e ( e ( u ) , b, h ) ∈ Q , whence ( u, b, h ) ∈ ( π ! e ) ∗ Q , a s desired.  Remark 3 .12 Diagr ammatic pro ofs as the o ne we just gav e ar e hard to read and motiv ate. One can give a more unders ta ndable pro of using the internal logic of the categ ory of preshe aves: fo r those who ar e familia r with its intricacies, we present such a pro of b elo w. Theorem 3. 1 3 (A ssuming C has chosen ﬁnite pr o ducts.) The ful lness axiom (F) is inherite d by pr eshe af mo dels. Pro of. In v ie w o f Lemma 2.13 and Co rollary 3.8 we only need to ﬁnd ge ne r ic mvs s for small map ( k , l ) ! : π ! B → π ! A , where π ! A is ﬁbred by a small ma p ( r , s ) ! : π ! A → π ! X ov er π ! X . Then, by replacing C by C / π ! X = P x ∈ X C /σ X ( x ), we may even assume that X = 1 = {∗} a nd σ X ( ∗ ) = 1. Int ernal universal qua n tiﬁcatio n ov er mvs s of π ! B in the category P sh E ( C ) amounts to E -internal qua n tiﬁcatio n ov er cer tain subpre s hea v es P of π ! ( B ) × C ( − , c ), na mely thos e which ar e mvs s ov er π ! ( A ) × C ( − , c ). Such a subpresheaf satisﬁes ( ∀ a ∈ A ) ( ∀ h : d → σ A ( a )) ( ∀ f : d → c ) ( ∃ b ∈ B ) ( ∃ g : d → σ B ( b )) ( k ( b ) , l b ◦ g ) = ( a, h ) and ( b, g , f ) ∈ P ⊆ π ! ( B ) × C ( − , c ) at d, which is equiv alent to ( ∀ a ∈ A ) ( ∃ b ∈ B ) ( ∃ g : σ A ( a ) × c → σ B ( b )) ( k ( b ) , l b ◦ g ) = ( a, π 1 ) and ( b , g , π 2 ) ∈ P ⊆ π ! ( B ) × C ( − , c ) at σ A ( a ) × c, or ( ∀ a ∈ A ) ( ∃ b ∈ B ) ( ∃ g : σ A ( a ) × c → σ B ( b )) k ( b ) = a, l b ◦ g = π 1 and ( b , g , π 2 ) ∈ P ( σ A ( a ) × c ) . (4) 32 W e us e fullness in E to obtain a small family o f sub ob jects { Q i : i ∈ I } which form a g eneric family of mvs s for { ( b ∈ B , c ∈ C 0 , g : σ A ( k b ) × c → σ B ( b )) : l b ◦ g = π 1 } → A : ( b, c, g ) 7→ k ( b ) . F ro m these Q i we now constr uct an internal small family of sma ll preshe aves of π ! ( B ). Such a family is gener ated by subpresheaves of π ! ( B ) × C ( − , c ) for v arying c ∈ C 0 . F or any such c ∈ C 0 , we take the presheaves ˆ Q ( c ) i ( d ) = { ( b ∈ B , g h, π 2 h ) : g : σ A ( k b ) × c → σ B ( b ) , h : d → σ A ( k b ) × c and ( b, c, g ) ∈ Q i } , provided i and c make the map ˆ Q ( c ) i → π ! ( A ) × C ( − , c ) surjective. Now we s how these are generic. T ake c 0 ∈ C and P a mvs o ver c 0 , as in P / / / / & & & & M M M M M M M M M M M M π ! ( B ) × C ( − , c 0 ) ( k,l ) ! × id   π ! ( A ) × C ( − , c 0 ) . This P sa tis ﬁe s (4) fo r c 0 , so there is a Q i contained in { ( b ∈ B , c 0 , g : σ A ( k b ) × c 0 → σ B ( b )) : ( b, g , π 2 ) ∈ P , l b ◦ g = π 1 } . W e claim that the map ˆ Q ( c 0 ) i → π ! ( A ) × C ( − , c 0 ) is surjective, and to show this it suﬃces to prove that elements of the form ( a, π 1 : σ A ( a ) × c 0 → c 0 , π 2 : σ A ( a ) × c 0 → σ A ( a )) are hit by this map. Since Q i is a mvs , we know that there are ( b, c, g ) ∈ Q i with k ( b ) = a and l b ◦ g = π 1 . Since Q i ⊆ P , we hav e c = c 0 and hence ( b, g , π 2 ) ∈ ˆ Q ( c 0 ) i and (( k , l ) ! × id )( − , c 0 )( b, g , π 2 ) = ( a, π 1 , π 2 ). So it remains to chec k ˆ Q ( c 0 ) i ⊆ P . But if ( b, g h, π 2 h ) ∈ Q ( c 0 ) i ( d ) for some maps h : d → σ A ( k b ) × c and g : σ A ( k b ) × c → σ B ( b ) with ( b, c 0 , g ) ∈ Q i , then ( b, g , π 2 ) ∈ P ( σ A ( k b ) × c ) and hence ( b, g h, π 2 h ) = ( b, g , π 2 ) · h ∈ P ( d ).  4 Shea v es In this section we c o n tinue to work in the setting of a predica tiv e catego ry with small maps ( E , S ) together with an internal catego ry C in E whose co domain map is small. T o deﬁne a category of internal sheav es, we hav e to assume that the categ ory C comes equipp ed with a Gro thendiec k top ology , so as to b ecome a Grothendieck site. There ar e diﬀerent formulations of the notion of a s ite, all essentially equiv a len t ([23] provides an excellent dis c us sion of this point), but for our purp oses we ﬁnd the following (“sifted”) for m ula tion the most useful. 33 Deﬁnition 4.1 Let C b e a n int ernal ca tegory whose co domain map in small. A sieve S o n an o b ject a ∈ C 0 is a smal l collection of a r ro ws in C all having co domain a a nd closed under precomp osition (i.e., if f : b → a and g : c → b are arrows in C and f belo ng s to S , then so do es f g ). Since we insist that sieves are small, there is an ob ject of sieves (a s ub ob ject of P s C 1 ). W e ca ll the s e t M a of all a rrows into a the maximal sieve o n a (it is a sieve, since we ar e assuming that the c o domain ma p is small). If S is a sieve 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 case f b elongs to S , we hav e f ∗ S = M b . A (Gr othendie ck) top olo gy Co v on C is g iv en by assigning to every ob ject a ∈ C a co llection of sieves Cov( a ) s uch that the following axioms are satisﬁed: (Maximality ) The maximal sieve M a belo ngs to Cov( a ); (Stabilit y) If f : b → a is any map and S b elongs to Cov( a ), then f ∗ S b elongs to Cov( b ); (Lo cal c harac ter) If S is a sieve on a and R ∈ Cov( a ) is such that for all f : b → a ∈ R the sie v e f ∗ S b elongs to Cov( b ), then S belo ngs to Cov( a ). A pair ( C , Co v) consis ting of a catego ry C and a topo logy Cov on it is c a lled a site . If a site ( C , Cov) ha s bee n ﬁxed, we ca ll the sieves b elonging to so me Cov( a ) c overing sieves . If S b elongs to Cov( a ) we say that S is a sieve c overing a , or that a is c over e d by S . Finally , a pr esen tatio n for a s ite ( C , Cov) is a function BCov which yields, for every a ∈ C 0 , a smal l collection of b asic c overing sieves BCov( a ) such that: S ∈ Cov( a ) ⇔ ∃ R ∈ B C ov( a ): R ⊆ S. A site for which such a presentation exists will b e c alled pr esentable . 8 Our ﬁrst goa l in this sectio n is prov e that a n y ca tegory of internal sheav es ov er a predicative catego ry with small ma ps ( E , S ) is a positive Heyting categor y . The pro of of this relies on the existence o f a she a ﬁﬁcation functor (a le ft adjoint to the inclusion of sheaves in presheav es), and s ince this functor is built b y taking a quotient, we use the bo unded exa ctness of ( E , S ). T o ensure that the equiv alence relation by which we q uotien t is b ounded, we will hav e to assume that the site is presentable. Next, we have to identify a clas s of small maps in any catego ry o f int ernal sheav es ov er ( E , S ). W e will deﬁne p oin t wise small and lo cally small maps of sheav es and w e will insist that these should a gain coincide (as happ ened in pres he aves). F or this to work out, we again seem to need the assumption that the s ite is pres en ta ble; moreov er, we will assume tha t the fullness a xiom holds in E (note that similar ass umpt ions were ma de in [21]). 8 This is supposed to b e reminiscent of Aczel’s notion of a set-presen table formal space (see [3]). Note that i n IZF ev ery si t e is presen table. 34 So, in eﬀect, we will work in a predicative catego ry with sma ll maps ( E , S ) equipp e d with a Grothendieck site ( C , Cov) such that: 1. The fullness a x iom (F) holds in E . 2. The co domain ma p co d: C 1 → C 0 is small. 3. The site is pres e ntable. After w e hav e shown that a catego ry o f sheav es can b e given the s tructure of a categ ory with small maps, we prov e that the v alidity o f a n y of the axioms int ro duced in Section 2 in ( E , S ) implies its v alidity in any catego ry of internal sheav es ov er it (Theor e ms 4.8– 4.11 and Theor em 4.17 ): we will say that the axiom is “ inherited by sheaf mo dels”. There is one exc e ptio n to this, how ever: we will not b e able to show that the axiom (WS) is inherited by sheaf mo dels. W e will discuss the pro blem and pr o vide a solution based on the axiom of m ultiple choice in Section 4.4 b elo w (see T he o rem 4.20 and Theorem 4.21). The main r esults of this s ection ar e that we establish the stability of fullness (F) under sheav es and we corr ect the treatment of W-types in [29]. In addition, we show that the tw o diﬀerent no tions o f a class of small maps that o ccur in the litera ture coincide in our setting. As far as the basic axioms ar e concerned, their sta bilit y can in o ne for m or another already b e found in the literature (see [24, 29, 20, 7]). In particular , w e should p oint out that [7] es tablishes the more general res ult that they a re stable under sheav es for a Lawvere-Tierney top ology . Nevertheless, it is not q uite true that our res ults are a specia l cas e of theirs, b ecause, to a c hieve this gener a lit y , they w ork in a setting which has full (not just b ounded) exactnes s. In a ddition, as we alr eady mentioned in the int ro duction, it is not true that the fullness axiom (F) is stable under sheav es for a Lawv ere-Tierney top ology . 4.1 Sheaﬁﬁcation Our next theorem shows the ex is tence of a shea ﬁﬁcation functor, a Cartesia n left adjoint to the inclusion of sheav es in presheaves. The pro of relies in an essential wa y o n the assumption o f b ounded exactness a nd on the fact that our site is presentable. Theorem 4. 2 The inclusion i ∗ : Sh E ( C ) / / / / Psh E ( C ) has a Cartesian left adjoint i ∗ (a “ she aﬁﬁc ation functor”). Pro of. W e verify that it is p ossible to imitate the standa rd construction (see [27, Section II I.5]). 35 Let P be a presheaf. A pair ( R , x ) will be ca lled a c omp atible family on a ∈ C 0 , if R is a covering sieve on a , a nd x sp eciﬁes for every f : b / / a ∈ R an element x f ∈ P ( b ), such that for any g : c / / b the equality ( x f ) · g = x f g holds. Because ( Π E) holds and sieves ar e small, by deﬁnition, ther e is an o b ject of compatible families. Actually , the co mpatible families for m a presheaf Comp( P ) with restriction given by ( R, x ) · f = ( f ∗ R, x · f ) , where ( x · f ) g = x f g . W e deﬁne an equiv a lence rela tion on Co mp( P ) by declaring tw o compatible families ( R, x ) and ( T , y ) on a equiv alent, when there is a covering sieve S ⊆ R ∩ T on a with x f = y f for a ll f ∈ S . Since the s ite is a ssumed to b e presentable, this quantiﬁcation ov er the (large) collection of cov ering sieves S on a , can b e replaced with a q ua n tiﬁca tion ov er the small collection o f ba sic covering sieves on a . Therefor e the equiv alence rela tion is b ounded and has a quotient P + . This ob ject P + is easily seen to car ry a preshea f str ucture in such a wa y that the quotient map Comp( P ) → P + is a morphism of pr eshea ves. First claim: P + is separated. P roo f: Supp ose tw o elements [ R , x ] a nd [ S, y ] of P + ( a ) agree on a cover T . Pick repr esen tativ es ( R , x ) and ( S, y ), and deﬁne: Q = { f : b / / a ∈ R ∩ S : x f = y f } . Once we show that Q is cov ering, we are done. But this follows immediately from the lo cal character axiom for sites: for any f ∈ T , the sieve f ∗ Q is cov er ing, by a ssumption. Second claim: when P is separa ted, P + is a shea f. Pr oof: Let R b e a covering sieve on a , and let compatible elements p f ∈ P + ( b ) b e given for every f : b / / a ∈ R . Using the collectio n axiom, w e ﬁnd for every f ∈ R a family o f r epresen tatives ( R ( f ,i ) , x ( f ,i ) ) of p f , with the v aria ble i running through some inhabited and smal l index set I f . There fo re S = { f ◦ g : f ∈ R, i ∈ I f , g ∈ R ( f ,i ) } is s mall; in fact, it is a covering s iev e, by lo cal character. W e now prov e that for any tw o tr iples ( f ∈ R, i ∈ I f , g ∈ R ( f ,i ) ) and ( f ′ ∈ R, i ′ ∈ I f ′ , g ′ ∈ R ( f ′ ,i ′ ) ) with f g = f ′ g ′ , we must hav e x ( f ,i ) g = x ( f ′ ,i ′ ) g ′ . Since the elements p f are assumed to b e compatible, the equality [ R ( f ,i ) , x ( f ,i ) ] · g = p f g = p f ′ g ′ = [ R ( f ′ ,i ′ ) , x ( f ′ ,i ′ ) ] · g ′ holds. Hence the elements x ( f ,i ) g and x ( f ′ ,i ′ ) g ′ agree on a cov ering s iev e. Since P is a ssumed to b e separ ated, this implies that the elements x ( f ,i ) g and x ( f ′ ,i ′ ) g ′ are in fact identical. 36 This a r gumen t shows that the deﬁnition z f g = x ( f ,i ) g is unambiguous for f g ∈ S , and also that ( S, z ) is a co mpatible family . As its equiv a lence class [ S, z ] is the glueing of the family p f we s tarted with, the second claim is proved. F ro m the co nstruction it is clear that for a ny presheaf P the sheaf P ++ has to b e its sheaﬁﬁcation. So we hav e shown that the construction of the sheaﬁﬁcation functor carries thr ough in the setting we are working in; that this ass ig nmen t is moreov er functorial as well as Cartesia n is pr o v ed in the usual manner.  Theorem 4. 3 Sh E ( C ) is a p ositive Heyting c ate gory. Pro of. The categor y of s hea v es has ﬁnite limits, b ecause these a re co mput ed po in twise, as in presheaves. Using the following description of ima ges and covers in catego ries of sheaves, one can e asily show these categ ories have to b e regular : the image of a map F : Y / / X o f sheaves consists of those x ∈ X ( a ) that are “lo cally” hit by F , i.e., for which there is a sie v e S covering a such that for a n y f : b / / a ∈ S there is an element y ∈ Y ( b ) with F ( y ) = x · f . Ther efore a map F : Y / / X is a cov er, if fo r every x ∈ X ( a ) there is a sieve S cov ering a and for any f : b → a ∈ S a n element y ∈ Y ( b ) such that F ( y ) = x · f (suc h maps are also called lo c al ly su rje ctive ). The Heyting structure in sheav es is the same as in preshe aves, so the universal quantiﬁcation of A ⊆ Y along F : Y / / X is given by the formula (1). In- deed, fro m this description it is readily seen that belo nging to ∀ F ( A ) is a lo cal prop ert y . The sums in sheav es ar e obtained by shea ﬁfying the sums in presheaves. They are still disjoin t and stable, b ecause the s heaﬁﬁcation functor is Cartesia n.  4.2 Small maps in shea v es W e will now deﬁne tw o classes o f maps in the ca tegories o f sheav es, tho se w hich are p oin t wise small and those which a r e lo cally small. Using that (F) holds in E and the fact that the site is presentable, we will then show that they c oincide. But b efore we deﬁne these tw o c la sses of maps, note that we hav e the following diagram of functors: E / C 0 π ! / / ρ ! $ $ I I I I I I I I I Psh E ( C ) π ∗ o o i ∗ y y s s s s s s s s s s Sh E ( C ) , ρ ∗ d d I I I I I I I I I i ∗ 9 9 s s s s s s s s s s 37 where the maps ρ ∗ and ρ ! are deﬁned as the comp osites of π and i via the diagram. So ρ ∗ is the forgetful functor, ρ ! is deﬁned as ρ ! X = i ∗ π ! X , and they ar e adjo int. It follows immediately from the maximality axiom for sites that the comp onen ts of the co unit ρ ! ρ ∗ / / 1 a re cov ers. One ﬁnal remar k b efore we give the deﬁnitions. W e hav e se e n that a n y pair of maps ( r , s ) in E mak ing B r   s / / σ B ' ' C 1 co d   dom / / C 0 A σ A / / C 0 . commute determines a map ( r, s ) ! : π ! B → π ! B of presheaves. Therefore it also determines a map i ∗ ( r , s ) ! : ρ ! B → ρ ! A of sheav es, but note that now not all maps ρ ! B → ρ ! A will be of this form, in contrast to what happ ened in the presheaf case. Finally , the tw o cla sses of maps are deﬁned as: 1. The po in twise deﬁnition: a morphism F : B / / A of sheav es is p ointwise smal l , when ρ ∗ F is a small ma p in E / C 0 . 2. The lo cal deﬁnition (as in [24]): a morphism F : B / / A o f sheav es is lo c al ly smal l in ca se it is cov ered by a map of the form i ∗ ( r , s ) ! where r is a s mall map in E . That these tw o class es of ma ps coincide will follow from the next t wo prop osi- tions, b oth whose pro ofs us e the fullness a xiom. Prop osition 4.4 The she aﬁﬁc ation functor i ∗ pr eserves p ointwise smal lness: if F is a (p ointwise) smal l map of pr eshe aves, then i ∗ F is a p ointwise smal l map of she aves. Pro of. T o prove the pr oposition, it suﬃces to show that the ( − ) + -construction preserves smallness. So let F : P / / Q b e a (p oin t wise) sma ll mor phism of presheav es and q b e an element of Q + ( a ), i.e. q = [ R , x ] wher e R is a sieve and ( x f ) f ∈ R is a family of compa tible elements. The ﬁbre of F + ov er q con- sists of e q uiv a lence cla sses of a ll those compatible families ( S, y ) o n a such tha t ( S, F ( y )) and ( R, q ) are equiv alen t (by F ( y ) we of co urse mea n the family given by F ( y ) f = F ( y f )). Beca use every such equiv a le nce clas s is repr esen ted b y a 38 compatible family ( S, y ) wher e S is a b asic cov ering sieve contained in R and F ( y f ) = x f for all f ∈ S , the ﬁbre of F over q is cov ered by the ob ject: X S ∈ BCo v( a ) ,S ⊆ R Y f ∈ S F − 1 ( x f ) . It follows from the fullness ax io m in E that this ob ject is sma ll (actually , the exp onen tiation a xiom ( Π S) would suﬃce for this purp ose) and then it follows from the quotient axio m (A6) that the ﬁbre o f F ov e r q is small as well.  Prop osition 4.5 The p ointwise smal l m a ps in she aves ar e close d u nder c over e d maps: if X F   / / A G   Y P / / / / B is a c overing squ a r e of she aves (i.e., P and the induc e d map X → Y × B A ar e lo c al ly surje ctive) and F is p ointwise smal l, then also G is p ointwise smal l. Pro of. T o make the pr o of mor e per s picuous, we will split the ar gumen t in tw o : ﬁrst we s how closure of p oin t wise small maps under quotients a nd then under descent. So s uppose ﬁrs t that we hav e a co mm uting triang le of sheaves Y G / / / / F @ @ @ @ @ @ @ X H ~ ~ } } } } } } } B , with F p oin t wise sma ll and G lo cally surjective. Fix an element b ∈ B ( c ). The fullness axiom in E implies that for any bas ic cov ering s iev e S ∈ BCov( c ) ther e is a small generic fa mily P S b of mvs s of the obvious (small) pro jection map p S b : Y S b = { ( f : d → c ∈ S, y ∈ Y ( d )) : F d ( y ) = b · f } / / S, such that a n y mvs of this map is reﬁned by one in P S b (recall that an mvs of p S b would b e a sub o b ject L ⊆ Y S b such that the c omposite L ⊆ Y S b → S is a small cov er). Strictly sp eaking, the fullness axio m says that for every S ∈ BCov( c ) such a generic mvs exists, not necessarily as a function of S . This do es follow, how ever, using the collection axiom: for this a x iom tells us that there is a sma ll family { P i : i ∈ I S b } of such mvs s for every S . So we ca n set P S b = S i ∈ I S b P i to get a generic mvs of p S b as a function of S . 39 Call an element L ∈ P S b c omp atible after G , if for a n y pair of elements ( f : d → c, y ) and ( f ′ : d ′ → c, y ′ ) in L we hav e ∀ g : e → d, g ′ : e → d ′ ( f g = f ′ g ′ ⇒ G d ( y ) · g = G d ′ ( y ′ ) · g ′ ) . Note that there is a map q : X S ∈ BCo v( c ) { L ∈ P S b : L compa tible after G } → H − 1 c ( b ) , which one obtains b y sending ( S, L ) to the glueing of the e lemen ts { G d ( y ) : ( f : d → c, y ) ∈ L } in X . The do ma in of this map q is small, s o the desired result will fol- low, once we show that this map is a cover. F or this we use the lo cal surjectivity of G . Lo cal sur jectivit y o f G mea ns that for every x ∈ X ( c ) in the ﬁbr e ov er b ∈ B ( c ), there is a basic cov ering sieve S ∈ BCov( c ) such that ∀ f : d → c ∈ S ∃ y ∈ Y ( d ) : G d ( y ) = x · f . But G d ( y ) = x · f implies that F d ( y ) = b · f , so { ( f : d → c, y ∈ Y ( d )) : G d ( y ) = x · f } is a n mvs o f p S b and there fore it is reﬁned by an ele ment of P S b . Since this element must be compatible after G , we hav e shown that q is a cover. Second, suppo se we have a pullback s quare of s hea v es X F   Q / / A G   Y P / / / / B , where F is p oin t wise sma ll and P and Q a r e lo cally surjective. Aga in, for any b ∈ B ( c ) and basic cov ering sieve S of c , let p S b be the map p S b : Y S b = { ( f : d → c ∈ S, y ∈ Y ( d )) : P d ( y ) = b · f } / / S, as ab o v e. F urthermor e, let mvs ( p S b ) be the ob ject of mvs s o f p S b and s et Y ′ ( c ) = X b ∈ B ( c ) X S ∈ BCo v( c ) m vs( p S b ) , X ′ ( c ) = X ( b,S,L ) ∈ Y ′ ( c ) { k ∈ Y ( f : d → c,y ) ∈ L F − 1 d ( y ) : k compatible after Q } , where we call k ∈ Q ( f : d → c,y ) ∈ L F − 1 d ( y ) c omp atible after Q , if for any ( f : d → c, y ) and ( f ′ : d ′ → c, y ′ ) in L we hav e ∀ g : e → d, g ′ : e → d ′ ∈ C ( f g = f ′ g ′ ⇒ Q d ( k ( f ,y ) ) · g = Q d ′ ( k ( f ′ ,y ′ ) ) · g ′ ) . 40 This leads to a commuting square in E / C 0 X ′ ( c ) Q ′ c / / / / F ′ c   A ( c ) G c   Y ′ ( c ) P ′ c / / / / B ( c ) , in which P ′ and F ′ are the obvious pr o jections and Q ′ sends ( b, S, L, k ) to the glueing of { Q d ( k ( f ,y ) ) : ( f , y ) ∈ L } . The squa re is a pullback in which the map P ′ is a cover (this uses the collection axiom) and F ′ is small, so that G c is a small map by descent (A2) in E / C 0 . This completes the pro of.  Theorem 4. 6 The p ointwise smal l maps and lo c al ly smal l maps of she aves c oincide. Pro of. That all lo cally small maps of sheav es are also p oin t wise s mall follows from the pre vious tw o pro positions. T o prov e that all p oin t wise small maps ar e also lo cally small w e use that the p oin t wise and lo cally small maps co incide in presheav es. So consider a p oin t wise small map F : B / / A of sheav es. Since i ∗ F is a p oin t- wise sma ll map of pres hea v es, there is a small map of presheav es ( k , l ) ! with k small in E such that π ! X ( k,l ) !   / / i ∗ B i ∗ F   π ! Y / / i ∗ A is a cov er ing square in presheaves. Applying sheaﬁﬁca tion i ∗ and using that i ∗ i ∗ ∼ = 1, we obtain a diagra m of the desir ed form.  Corollary 4.7 Any p ointwise smal l map is c over e d by one of the form i ∗ ( r , s ) ! with r smal l in E . In fact, every c omp osable p air ( G, F ) of p ointwise s mall maps of she aves ﬁts into a double c overing squar e of the form ρ ! C i ∗ ( k,l ) !   / / / / Z G   ρ ! B i ∗ ( r,s ) !   / / / / Y F   ρ ! A / / / / X , in which k and r ar e smal l in E . 41 Pro of. Immediate from the previo us theor em and the cor r esponding fact for presheav es (Corollar y 3.8).  Henceforth we can therefore use the term “sma ll map” without danger o f ambiguit y . The ﬁrst thing to do now is to show that the small ma ps in sheaves really satisfy the axio ms for a c la ss o f small maps. Theorem 4. 8 The smal l maps in she aves satisfy axioms (A1-9) . Pro of. Aga in, we p o stpone the pro of of the collection axiom (A7) (it will be Theorem 4 .10 ). Because limits in sheav es are computed as in pres hea v es, (A1) and (A9) are inherited from presheaves. Colimits in sheav es ar e computed by shea ﬁfying the res ult in presheav es, hence the ax ioms (A3) and (A4) follow from Pro position 4.4. That p oint wise small ma ps are clo sed under covered maps was P r opositio n 4.5: this disp oses of (A2) and (A6) . Poin twise small maps are clo s ed under co mp osition, so (A5) holds as well. Fina lly , since univ ersal quantiﬁcation in sheaves is computed as in pr eshea ves, the axio m (A8) holds in sheav es, b ecause it holds in pre shea v es.  Theorem 4. 9 The fol lo wing axioms ar e inherite d by she af mo dels: b ou nde d exactness, r epr esentability, (NE), (NS), ( Π E), ( Π S) , (M) and (PS) . Pro of. Bo unded exactnes s is inherited by sheaf mo dels, since one can sheaﬁfy the quotient in presheaves. Representabilit y is inherited fo r the same reason: one s he a ﬁﬁes the univ ersal sma ll maps in pres hea v es. Also the natura l num- ber s ob ject in sheav es is obtained by s hea ﬁfying the natur al num b ers ob ject in presheav es, so (NE) and (N S) are inherited by sheaf mo dels. Since Π-types in pres he aves ar e computed as in sheav es and ( Π E) and ( Π S) are inher ited by presheaf mo dels, they will a lso b e inherited by she a f mo dels. Finally , since monos in sheav es are p oin t wise, (M) is inherited a s well. The P s -functor in sheaves is obtained by quotienting the P s -functor in pr eshea ves (see P ropositio n 3.2) by the following equiv ale nce rela tio n (basically , bis im ula- tion understo od as in sheaves): if A, A ′ ⊆ y c × X , then A ∼ A ′ if for a ll ( f : b → c, x ) ∈ A ( c ), the sieve { g : a → b : ( f g , x · g ) ∈ A ′ } cov ers b , and for all ( f ′ : b ′ → c, x ′ ) ∈ A ′ ( c ) the sieve { g ′ : a ′ → b ′ : ( f ′ g ′ , x ′ · g ′ ) ∈ A } cov ers b ′ . One easily veriﬁes that this deﬁnes an equiv alence relation in presheaves; mor e- ov er, it is b ounded, since the site is as sumed to be pr esen table. Its quotient 42 P has the structure of a sheaf (a s we hav e seen several times, to constr uct the glueing o ne uses the co llection axiom to s elect small collections of r epresen ta- tives from ea c h equiv alence class). One deﬁnes the relatio n ∈ X ⊆ X × P on an ob ject c ∈ C by putting for any x ∈ X ( c ) a nd A ∈ P s ( X )( c ), x ∈ [ A ] ⇐ ⇒ the sie ve { f : d → c : ( f , x · f ) ∈ A } cov ers c. A str a igh tforw ard veriﬁcation establishes that this is indeed the p o wer clas s o b- ject of X in sheav es. Hence the ax io m (PS) is inherited by sheaf mo dels.  In the coming tw o s ubs ections we will dis cuss the collection and fullness axioms and W-types in sheaf categ ories. 4.3 Collection and fullness in shea v es Theorem 4. 1 0 The c ol le ction axiom (A7) is inherite d by she af mo dels. Pro of. Let F : M / / N be small map and E : Y / / M b e a cover in sheav e s (i.e. E is lo cally surjective). Without loss of g eneralit y we may assume that F is o f the form i ∗ ( k , l ) ! : ρ ! B → ρ ! A . If the map Q : X → π ! B of presheaves is obtaine d b y pulling back the map i ∗ E along the compo nen t o f the unit 1 → i ∗ i ∗ at π ! B a s in X Q   / / i ∗ Y i ∗ E   π ! B / / i ∗ ρ ! B = i ∗ i ∗ π ! B , then this ma p Q also has to b e lo cally surjective. This means tha t for the following ob ject in E C = { ( b ∈ B , S ∈ Cov( c )) : σ B ( b ) = c a nd ( ∀ f : d → c ∈ S ) ( ∃ x ∈ X ( d )) ( Q ( x ) = ( b, f )) } , the obvious pro jection s 0 : C → B is a cover. Ther efore we can apply the collec- tion axiom in E to obtain a cov ering squar e of the form: V s 1 / / l   C s 0 / / / / B k   U r 0 / / A, (5) with l sma ll in E . W e wish to apply the collection axiom again. F or this purp ose, 43 deﬁne the following tw o ob jects in E : V ′ = { ( v ∈ V , f ∈ C ) : if s 1 ( v ) = ( b, S ) , then f ∈ S } , W = { ( v ∈ V , f : d → c, x ∈ X ( d )) : if s 1 ( d ) = ( b , S ) , then f ∈ S and Q d ( x ) = ( b, f ) } , and let s 3 : W → V ′ and s 2 : V ′ → V be the obvious pro jections. s 3 is a cov er (essentially by deﬁnition of C ), and the co mp osite l ′ = l s 2 is sma ll. So we can apply collection to obtain a cov ering squa re in E J s 4 / / m   W s 3 / / / / V ′ l ′   I r 1 / / U, (6) in which m is small. W r iting r = r 0 r 1 and s = s 0 s 1 s 2 s 3 s 4 , we obtain a com- m uting square J s / / m   B k   I r / / A, with every j ∈ J determining an element b ∈ B , a sieve S on σ B ( b ), a n arrow f ∈ S and an element x ∈ X (dom f ) such that Q ( x ) = ( b, f ) ∈ π ! B . If for such an element j ∈ J w e put ρ J ( j ) = dom( f ) , t j = f , n j = l b ◦ f and for every i ∈ I we deﬁne σ I ( i ) = σ A ( ri ), then we obtain a square of presheaves: π ! J ( s,t ) ! / / ( m,n ) !   π ! B ( k,l ) !   π ! I π ! r / / π ! A. T o see that it co mm utes, we chase a n ele ment around the tw o sides of the diagram and it suﬃces to do that for an element o f the form ( j, id ). So π ! r ( m, n ) ! ( j, id ) = π ! r ( mj, l b ◦ f ) = ( rmj, l b ◦ f ) , and ( k , l ) ! ( s, t ) ! ( j, id ) = ( k , l ) ! ( sj, f ) = ( k sj, l b ◦ f ). W e claim that s heaﬁfying the squar e gives a cov ering squa r e. Since r is a cov er and ρ ! preserves these, this means that we have to show that the map from π ! J to the pullback of the ab ov e square is lo cally s urjectiv e. Lemma 3.5.4 tells us tha t we may assume that the pullback is of the for m π ! ( I × A B ) with σ I × A B ( i, b ) = σ B ( b ). The induced map K : π ! J / / π ! ( I × A B ) sends ( j, g ) to (( mj, sj ) , f ◦ g ), where f is the ele men t in C 1 determined b y j ∈ J as ab ov e . T o show that this map is lo cally surjective, it suﬃces to pr o v e that every element 44 (( i, b ) , id ) ∈ π ! ( I × A B ) is lo cally hit by K . The element i ∈ I determines an element r 1 i ∈ U , and s ince (5) is a cov ering square , we ﬁnd a v ∈ V with l v = r 1 i and s 0 s 1 v = b , hence a cov er ing s ie v e S on ρ B ( b ). Moreov er, since (6) is a covering square, w e ﬁnd for every f ∈ S a n element j ∈ J such that m ( j ) = i and s ( j ) = b . Then K ( j, id ) = (( i, b ) , f ) = (( i, b ) , i d ) · f , whic h prov es that K is lo cally sur jectiv e . T o complete the pro of, we need to show that ( s, t ) ! : π ! J / / π ! B factor s through Q : X / / π ! B . T her e is a map ( p, q ): J / / π ∗ X which sends every j ∈ J to the x ∈ X (dom f ) that it determines . Its tr anspose ( p, q ) ! sends ( j, i d ) to x ∈ X whic h in turn is sent by Q to Q ( x ) = ( sj, f ) = ( s, t ) ! ( j, id ). Therefore ( s, t ) ! = Q ( p, q ) ! .  Theorem 4. 1 1 (A ssuming C has chosen pul lb acks.) The ful lness axiom (F) is inherite d by she af mo dels. Pro of. In view of Lemma 2.1 3 and Co rollary 4 .7, it will suﬃce to show that there exists a generic mvs for any map o f the form i ∗ ( k , κ ) ! : ρ ! B / / ρ ! A , living ov er so me ob ject of the form ρ ! X via so me map i ∗ ( r , ρ ) ! : ρ ! A / / ρ ! X , with k and r small. W e ﬁrs t c o nstruct the generic mvs P . T o this end, deﬁne: S 0 = { ( a ∈ A, α : d → c, S ∈ BCov( α ∗ σ A ( a ))) : σ X ( ra ) = c } M 0 = { ( a ∈ A, α : d → c, S ∈ BCov( α ∗ σ A ( a )) , β ∈ S ) : σ X ( ra ) = c } B 0 = { ( b ∈ B , α : d → c, S ∈ BCov( α ∗ σ A ( k b )) , β ∈ S, γ ∈ C 1 : σ X ( rk b ) = c, α ∗ κ b ◦ γ = β } (In the deﬁnition of S 0 and M 0 we hav e used that any pair co nsisting o f a ma p α : d → c ∈ C and elemen t a ∈ A with σ X ( ra ) = c determines a pullback diagram α ∗ σ A ( a ) / / α ∗ ρ a   σ A ( a ) ρ a   d α / / c in C ; in the deﬁnition of B 0 we have used that a n y pair consisting of a map α : d → c ∈ C and element b ∈ B with σ X ( rk b ) = c determines a do uble pullba ck diagram α ∗ σ B ( b ) / / α ∗ κ b   σ B ( b ) κ b   α ∗ σ A ( k b ) / / α ∗ ρ kb   σ A ( k b ) ρ kb   d α / / c 45 in C .) One easily chec ks that all the pro jections in the chain B 0 / / M 0 / / S 0 / / A / / X are small. F or the cons tr uction o f P , we ﬁrst build a gener ic mvs for S 0 / / A ov er X . This means we hav e a cov er n : W / / X and a small map m 0 : Z 1 / / W , together with a g eneric mvs P 1 for S 0 / / A ov er Z 1 , as in the diagr am P 1 / / / / ' ' ' ' S 1   / / S 0   A 1   / / A   Z 1 m 0 / / W n / / / / X , where the rectangles are understo o d to b e pullbacks. Next, we pull B 0 → M 0 → S 0 back along P 1 → S 0 and obtain the diagr a m B 1 / /   B 0   M 1   / / M 0   P 1 / / S 0 . Then we build a generic mvs for B 1 → M 1 ov er Z 1 . This we obtain ov e r an ob ject Z 2 via a small map Z 2 / / W ′ and a cov er W ′ / / Z 1 . Without lo ss of generality , we may a ssume that the latter map W ′ → Z 1 is the identit y . (Pro of: apply the collection axio m to the small ma p Z 1 / / W and the cov er W ′ / / Z 1 to obta in a small map S / / R cov e ring the morphism Z 1 / / W . Lemma 2.12 tells us that there lives a gener ic mvs for S 0 / / A ov er S as well. By a nother application of Lemma 2.12, there lives a g eneric mvs for B 1 → M 1 ov er T , if T → S is the pullback of Z / / W ′ along the map S / / W ′ .) So we may assume there is a s mall map m 1 : Z 2 / / Z 1 , such that ov er Z 2 there is a g eneric mvs P 2 for B 1 → M 1 , as in the following diagra m P 2 / / / / ' ' ' ' B 2   / / B 1 / /   B 0   M 2   / / M 1 / /   M 0   Z 2 m 1 / / t 4 4 Z 1 m 0 / / W n / / / / X , 46 where all the rectangles are supp osed to b e pullbacks. F o convenience, write t = nm 0 m 1 . W e make s ome deﬁnitions. Fir st of all, let Z = { ( z 2 ∈ Z 2 , δ : d → c ) : σ X ( t ( z 2 )) = c and ( ∀ a ∈ A t ( z 2 ) ) ( ∃ S ∈ B Co v( δ ∗ σ A ( a )) ( m 1 ( z ) , a, δ, S ) ∈ P 1 } . F urther mo re, we write m 2 : Z → Z 2 for the obvious pr o jection and put m = m 0 m 1 m 2 . Finally , we let P 3 be the pullback of P 2 along m 2 . W e wish to co ns truct a diagram of presheaves of the form: P / / / / ' ' π ! ( Z × X B )   / / π ! B ( k,κ ) !   π ! ( Z × X A )   / / π ! A ( r,ρ ) !   π ! Z ( m,µ ) ! / / π ! W π ! n / / π ! X , which we can do by putting σ Z ( z 2 , δ ) = co d( δ ) and µ ( z 2 ,δ ) = δ . Note that π ! n is a cover and ( m, µ ) ! is small. In addition, P is deﬁned by s a ying that an element ( z ∈ Z , b ∈ B , η : c → d ) ∈ π ! ( Z × X B )( c ) b elongs to P ( c ) if there is a s ie v e S ∈ BCov( µ ∗ z σ A ( k b )), a map β ∈ S and a map γ ∈ C 1 such tha t ( z , b, µ z , S, β , γ ) b elongs to P 3 and η factor s thro ugh γ . By construction, the map P / / π ! ( Z × X A ) is lo cally surjective. By shea ﬁfying the whole diag ram, we therefor e obtain an mvs i ∗ P for i ∗ ( k , κ ) ! ov er ρ ! Z in the category of sheav es. The r emainder of the pro of will s ho w it is generic. T o that purpo se, let V / / ρ ! W b e a map o f sheav es and Q b e a n m vs for i ∗ ( k , κ ) ! ov er V . Let Y b e the pullback in presheaves of V along the map π ! W → ρ ! W and cov er Y using the counit π ! π ∗ Y → Y . W riting Y = π ∗ Y , this means we hav e a commuting square of presheaves π ! Y ( l,λ ) ! / /   π ! W   V / / ρ ! W , in which the vertical arrows are lo cally sur jectiv e and the to p ar r o w is of the form ( l , λ ) ! . Finally , let Q b e the pullba c k of Q along π ! Y → V . This mea ns we 47 hav e the fo llo wing diagr am o f presheaves: Q / / / / ' ' π ! ( Y × X B )   / / π ! B ( k,κ ) !   π ! ( Y × X A )   / / π ! A ( r,ρ ) !   π ! Y ( l,λ ) ! / / π ! W π ! n / / π ! X , , where the r ectangles are pullbacks, computed, as usual, using Lemma 3.5.5 (so σ Y × X A ( y , a ) = λ ∗ y σ A ( a ) and σ Y × X B ( y , b ) = λ ∗ y σ B ( b )). The map Q → π ! ( Y × X A ) is lo cally surjective, and there fo re Q 1 = { ( y , a, λ y , S ∈ BCov( λ ∗ y σ A ( a ))) : ( ∀ β ∈ S ) ( ∃ b ∈ B a ) ( ∃ γ ∈ C 1 ) ( y , b, γ ) ∈ Q and λ ∗ y κ b ◦ γ = β } = { ( y , a, λ y , S ∈ BCov( λ ∗ y σ A ( a ))) : ( ∀ β ∈ S ) ( ∃ b ∈ B a ) ( ∃ γ ∈ C 1 ) ( y , b, γ ) ∈ Q and ( b, λ y , S, β , γ ) ∈ B 0 } is an mvs of S 0 → A ov er Y . By the generic it y of P 1 this implies the e x istence of a ma p v 1 : U 1 → Z 1 and a cov er w 1 : U 1 → Y s uc h that m 0 v 1 = lw 1 and v ∗ 1 P 1 ≤ w ∗ 1 Q 1 as mvs s of S 0 → A 0 ov er U 1 . Note that this means that ( v 1 ( u 1 ) , a, α, S ) ∈ P 1 = ⇒ α = λ w 1 ( u 1 ) . (7) Next, deﬁne the sub ob ject Q 2 ⊆ v ∗ 1 B 1 by saying for any element ( u 1 ∈ U 1 , b ∈ B , S ∈ BCov( λ ∗ w 1 ( u 1 ) σ A ( k b )) , β ∈ S, γ ∈ C 1 ) ∈ v ∗ 1 B 1 : ( u 1 , b, S, β , γ ) ∈ Q 2 ⇐ ⇒ ( w 1 u 1 , b, γ ) ∈ Q (dom( γ )) . It follows from (7) a nd the deﬁnition of Q 1 that Q 2 is a s mall mvs o f B 1 → M 1 ov er U 1 . Therefore there is a ma p v 2 : U / / Z 2 and a cover w 2 : U / / U 1 such that v 1 w 2 = m 1 v 2 and v ∗ 2 P 2 ≤ w ∗ 2 Q 2 . No te that (7 ) implies tha t v 2 factors through m 2 : Z → Z 2 via a map v : U → Z given by v ( u ) = ( v 2 ( u ) , λ w 1 w 2 ( u ) ). If we put w = w 1 w 2 , then l w = l w 1 w 2 = m 0 v 1 w 2 = m 0 m 1 v 2 = m 0 m 1 m 2 v = mv . Since for each u ∈ U , σ Z ( v u ) = dom( λ wu ) = σ Y ( wu ), we may put σ U ( u ) = σ Z ( v u ) = σ Y ( wu ) and then π ! w and π ! v deﬁne maps π ! U → π ! Y and π ! U → π ! Z , resp ectiv ely , such that ( l , λ ) ! π ! w = ( m, µ ) ! π ! v . Because π ! w is a cover, the pro of will b e ﬁnished, once we show that ( π ! v ) ∗ P ≤ ( π ! w ) ∗ Q . T o show this, cons ider an element ( u ∈ U, b ∈ B , η : c → d ∈ C ) ∈ π ! ( U × X B )( c ) for which we hav e ( u , b, η ) ∈ ( π ! v ) ∗ P ( c ). This mea ns that ( v u, b, η ) ∈ P ( c ) and hence that ther e is a sieve S ∈ B Co v( µ ∗ vu σ A ( k b )), a map β ∈ S and a map γ : e → d ∈ C 1 such that ( v u, b, µ vu , S, β , γ ) ∈ P 3 and η factors thro ugh γ . The former means tha t ( v 2 u, b, µ vu , S, β , γ ) ∈ P 2 and since v ∗ 2 P 2 ≤ w ∗ 2 Q 2 , it follows 48 that ( w 2 u, b, S, β , γ ) ∈ Q 2 . B y deﬁnition this mea ns that ( wu , b, γ ) ∈ Q ( e ). Since Q is a presheaf, a lso ( w u, b, η ) ∈ Q ( c ) and hence ( u, b, η ) ∈ ( π ! w ) ∗ Q ( c ). This completes the pro of.  Remark 4 .12 Aga in, one can als o prove this result using the internal logic of categorie s of sheaves. Also to illustrate its p ow er , we g iv e o ne such pro of her e. Theorem 4. 1 3 (A ssuming C has chosen ﬁnite pr o ducts.) The ful lness axiom (F) is inherite d by she af mo dels. Pro of. In view of Lemma 2.13 and Corolla ry 3.8, we o nly ne e d to build g eneric mvs s fo r maps of the form i ∗ ( k , l ) ! : ρ ! B → ρ ! A in whic h k is sma ll, where ρ ! A lies over some ob ject of the form ρ ! X via a map of the form i ∗ ( r , s ) ! in which r is sma ll. Again, b y replacing C by C /ρ ! ( X ), we may a s sume that X = 1 = {∗ } and σ X ( ∗ ) = 1. Note that for a ﬁxed c ∈ C 0 an mvs of i ∗ ( k , l ) ! ov er i ∗ C ( − , c ) as in P / / / / & & & & M M M M M M M M M M M M ρ ! ( B ) × i ∗ C ( − , c ) i ∗ (( k,l ) ! × id )   ρ ! ( A ) × i ∗ C ( − , c ) satisﬁes ( ∀ a ∈ A ) ( ∃ S ∈ BCov( σ A ( a ) × c )) ( ∀ g : d → σ A ( a ) × c ∈ S ) ( ∃ b ∈ B ) ( ∃ f : d → σ B ( b )) ( ∃ h : d → c ) ( k ( b ) , l b ◦ f ) = ( a, π 1 ◦ g ) , π 2 ◦ g = h and ( b, f , h ) ∈ P ( d ) , or ( ∀ a ∈ A ) ( ∃ S ∈ BCov( σ A ( a )) × c ) ( ∀ g : d → σ A ( a ) × c ∈ S ) ( ∃ b ∈ B ) ( ∃ f : d → σ B ( b )) k ( b ) = a, l b ◦ f = π 1 ◦ g and ( b, f , π 2 ◦ g ) ∈ P ( d ) . (8) W e ﬁrs t a pply fullness in E to the map { ( a ∈ A, c ∈ C 0 , S ∈ BCov( σ A ( a ) × c )) } → A : ( a, c, S ) → a to o btain a gener ic small family o f mvs s ( Q j ) j ∈ J . W riting for every j ∈ J ¯ Q j : = { ( a, c, S, g ) : ( a, c, S ) ∈ Q j , g ∈ S } , ˜ Q j : = { ( a, c, S, g , b, f ) : ( a, c, S ) ∈ Q j , g ∈ S, b ∈ B , k ( b ) = a, l b f = π 1 g } , we have an obvious pro jectio n ˜ Q j → ¯ Q j . 49 Applying fullness a nd using the collection axio m we o bta in gener ic families { P ij } i ∈ I j for these maps as well. (The co llection a xiom is employ ed here to obtain these generic families as a function of j .) F or ﬁxed c ∈ C , j ∈ J a nd i ∈ I j the ob ject P ij determines a subsheaf ˆ P ( c ) ij ⊆ ρ ! ( B ) × i ∗ C ( − , c ) generated by those elements ( b, f , π 2 ◦ g ) for whic h there is a basic cov ering sieve S such tha t ( k ( b ) , c, S, g , b , f ) ∈ P ij . Again, we only take those which a re mvs s, i.e., map in a lo cally surjective manner to ρ ! ( A ) × i ∗ C ( − , c ). Now supp ose c 0 is arbitr ary and R ⊆ ρ ! ( B ) × C ( − , c 0 ) is an mvs . This means that (8) holds with c = c 0 . Hence ther e is a Q j with Q j ⊆ { ( a, c 0 , S ) : ( ∀ g : d → σ A ( a ) × c ∈ S ) ( ∃ b ∈ B ) ( ∀ f : d → σ B ( b )) k ( b ) = a, l b ◦ f = π 1 ◦ g and ( b, f , π 2 ◦ g ) ∈ R ( d ) } (9) and a n i ∈ I j with P ij ⊆ { ( a, c 0 , S, g , b, f ) : ( b, f , π 2 ◦ g ) ∈ R ( d ) } . It is c le ar that ˆ P ( c 0 ) ij ⊆ R , so it remains to verify that ˆ P ( c 0 ) ij is an mvs . W e check (8) for c = c 0 . So take a ∈ A . W e wan t to show that the gener ator ( a, π 1 , π 2 ) ∈ ρ ! ( A ) × i ∗ C ( − , c 0 ) is lo cally hit by i ∗ (( k , l ) ! × id ). Beca use Q j is an mvs , there a re e ∈ C 0 and S ∈ BC ov( σ A ( a ) × e ) such that ( a, e, S ) ∈ Q j ; morever, we must have e = c 0 , b ecause (9) holds. Since P ij is a n mvs w e know that for every g ∈ S there are b ∈ B and f ∈ C 1 with ( a, c 0 , S, g , b, f ) ∈ P ij . In particular ( a, c 0 , S, g , b, f ) ∈ ˜ Q j , s o k ( b ) = a and l b ◦ f = π 1 ◦ g . By construction ( b, f , π 2 ◦ g ) ∈ ˆ P ( c 0 ) ij and fo r this elemen t the equation i ∗ (( k , l ) ! × id )( b, f , π 2 ◦ g ) = ( k ( b ) , l b ◦ f , π 2 ◦ g ) = ( a, π 1 , π 2 ) · g holds. This co ncludes the pro of.  4.4 W-t yp es in shea ves In this ﬁnal subsectio n, we s ho w that the a xiom (WE) is inher ited by sheaf mo dels. It turns out that the construction of W-types in categor ie s of sheav es is co nsiderably mo re involv ed than in the presheaf ca se (in [10] w e s ho w ed that some of the complications c an b e avoided if the metatheory includes the ax iom of choice). W e then go on to show that the a xiom (WS) is inher ited as well, if we a ssume the axiom of multiple choice. Remark 4 .14 In [2 8 ] the authors claimed that W-t yp e s in categories of sheav es are c o mputed as in presheaves (Pr oposition 5 .7 in lo c.cit. ) and can therefo re be 50 describ ed in the same (relatively ea s y) wa y . B ut, unfortunately , this claim is incorrect, as the following counterexample shows. Let F : 1 → 1 b e the identit y map on the terminal o b ject. The W-type ass ociated to F is the initial ob ject, which, in genera l, is diﬀere n t in ca tegories o f presheaves a nd sheav es. (This was noticed by Peter Lumsdaine together with the ﬁrst a uthor.) W e ﬁx a small map F : Y → X o f she aves. If x ∈ X ( a ) and S is a cov ering sieve on a , then we put Y S x : = { ( f : b → a ∈ S, y ∈ Y ( b )) : F ( y ) = x · f } . Observe that Y S x is small and write ψ for the o b vio us pr o jection ψ : X ( S,x ) Y S x → X × C 0 Cov . Let Ψ = P ψ ◦ P + s and let V b e its initial a lgebra (see Theorem 2.1 0) . Ele ments v of V are ther efore of the for m sup ( a,x,S ) t with ( a, x, S ) ∈ X × C 0 Cov and t : Y S x → P + s V . W e will think of such an elemen t v as a lab elled well-founded tree, with a ro ot lab elled with ( a, x, S ). T o this ro ot is attached, for every ( f , y ) ∈ Y S x and w ∈ t ( f , y ), the tr e e w with a n edge lab elled with ( f , y ). T o simplify the nota tio n, we will denote by v ( f , y ) the smal l collection of a ll trees that are a ttac hed to the ro ot of v with an edge that has the lab el ( f , y ). W e now wish to deﬁne a presheaf structure on V . W e say that a tr ee v ∈ V is r o ote d at an ob ject a in C , if its r oot has a lab el who se ﬁrs t comp onen t is a . If v = sup ( a,x,S ) t is r ooted a t a and f : b → a is a ma p in C , then we can deﬁne a tr ee v · f ro oted a t b , as follows: v · f = sup ( b,x · f ,f ∗ S ) f ∗ t, with ( f ∗ t )( g , y ) = t ( f g, y ) . This clearly gives V the structure of a presheaf. Note that ( v · f )( g , y ) = v ( f g , y ) . Next, we deﬁne by transﬁnite recursio n a r elation on V : v ∼ v ′ ⇔ if the ro ot of v is lab elled with ( a, x, S ) and the ro ot of v ′ with ( a ′ , x ′ , S ′ ), then a = a ′ , x = x ′ and there is a cov er ing sie ve R ⊆ S ∩ S ′ such that for every ( f , y ) ∈ Y R x we have v ( f , y ) ∼ v ′ ( f , y ). Here, the formula v ( f , y ) ∼ v ′ ( f , y ) is s upposed to mean ∀ m ∈ v ( f , y ) , n ∈ v ′ ( f , y ) : m ∼ n. 51 In general, we will write M ∼ N for small s ubob jects M a nd N of V to mean ∀ m ∈ M , n ∈ N : m ∼ n. In a similar vein, we will write for such a sub ob ject M , M · f = { m · f : m ∈ M } . That the rela tion ∼ is indeed deﬁna ble c a n be shown by the methods o f [8 ] o r [9]. By transﬁnite induction o ne can show that ∼ is symmetric and transitive, and co mpatible with the pre s heaf structure ( v ∼ w ⇒ v · f ∼ w · f ). Next, we deﬁne c omp osability and n atur ality of tr e e s (as we did in the presheaf case, see Theorem 3.3). • A tr ee v ∈ V whose ro ot is la belled with ( a, x, S ) is c omp osable , if for any ( f : b → a, y ) ∈ Y S x and w ∈ v ( f , y ), the tre e w is ro oted at b . • A tree v ∈ V whose ro ot is la belled with ( a, x, S ) is natur al , if it is com- po sable and for any ( f : b → a, y ) ∈ Y S x and g : c → b , v ( f , y ) · g ∼ v ( f g , y · g ) . One can show that if v is natur al, a nd v ∼ w , then also w is natura l; mor eo v er, natural trees a r e stable under restriction. The same a pplies to the tr ees that are her e ditarily natura l (i.e. no t only a re they themselves natur al, but the sa me is tr ue for all their subtree s). W e sha ll write W for the ob ject consis ting of those trees tha t are hereditarily natural. The rela tio n ∼ deﬁnes an eq uiv alence on W , for if a tree v = sup ( a,x,S ) t is natural, then for all ( f , y ) ∈ Y S x one has v ( f , y ) · i d ∼ v ( f · id , y · id ), that is, v ( f , y ) ∼ v ( f , y ), and ther efore v ∼ v . By induction o ne proves that the equiv alence relation ∼ on W is b o unded and hence a quotient exists. W e deno te it by W . It follows fro m w ha t we have said that the quotient W is a presheaf, but more is tr ue: one can actually show that W is a s heaf a nd, indeed, the W-t yp e ass ociated to F in sheav es. Lemma 4.15 L et w, w ′ ∈ W b e r o ote d at a ∈ C . If T is a sieve c overing a and w · f ∼ w ′ · f for al l f ∈ T , t he n w ∼ w ′ . In other wor ds, W is sep ar ate d. Pro of. If the lab el of the r oot of w is of the form ( a, x, S ) a nd that of w ′ is of the form ( a, x ′ , S ′ ), then w · f ∼ w ′ · f implies that x · f = x ′ · f for all f ∈ T . As X is sepa rated, it follows that x = x ′ . Consider R = { g : b → a ∈ ( S ∩ S ′ ) : ∀ ( h, y ) ∈ Y M b x · g [ w ( g h, y ) ∼ w ′ ( g h, y ) ] } . 52 R is a sieve, and the statement of the lemma will follow o nce we hav e shown that it is cov ering. Fix an element f ∈ T . That w · f ∼ w ′ · f holds means that ther e is a cov ering sieve R f ⊆ f ∗ S ∩ f ∗ S ′ such that for ev ery ( k, y ) ∈ Y R f x · f we hav e w ( f k, y ) = ( w · f )( k , y ) ∼ ( w ′ · f )( k, y ) = w ′ ( f k , y ). In other words, R f ⊆ f ∗ R . So R is a cov ering sieve by lo cal character .  Lemma 4.16 W is a she af. Pro of. Let S b e a cov ering sieve o n a and supp ose we hav e a compatible family of elements ( w f ∈ W ) f ∈ S . Using the collection axio m, we know that there must be a s pan S ← J → W f j ← [ j 7→ w j with J small and [ w j ] = w f j for a ll j ∈ J . Every w j is of form s up ( a j ,x j ,R j ) t j . If f j = f j ′ , then w j ∼ w j ′ , so x j = x j ′ . Thus the x j form a compatible family and, since X is a s heaf, can b e glued together to obtain a n e le men t x ∈ X ( a ). W e c laim that the desired glueing is [ w ], where w = sup ( a,x,R ) t ∈ V is deﬁned by: R = { f j g : j ∈ J, g ∈ R j } , t ( h, y ) = [ j ∈ J { t j ( g , y ) : f j g = h } F or this to make sense, we ﬁrst need to show that w ∈ W , i.e., that w is hereditarily natura l. In order to do this, we prov e the following claim. Claim. Assume we ar e given ( h, y ) ∈ Y R x , with h = f j g for some j ∈ J . Then w ( h, y ) ∼ w j ( g , y ) . Pro of. Since w ( h, y ) = [ j ′ ∈ J { w j ′ ( g ′ , y ) : f j ′ g ′ = h } , it suﬃces to show that w j ( g , y ) ∼ w j ′ ( g ′ , y ) if h = f j ′ g ′ . By compa tibility of the fa mily ( w f ∈ W ) f ∈ S we know that w j · g ∼ w j ′ · g ′ ∈ W ( c ). This means that there is a cov ering sieve T ⊆ g ∗ R j ∩ ( g ′ ) ∗ R j ′ such tha t for all ( k , z ) ∈ Y T x · h , we hav e ( w j · g )( k , z ) ∼ ( w j ′ · g ′ )( k , z ). So if k : d → c ∈ T , then w j ( g , y ) · k ∼ w j ( g k , y · k ) = ( w j · g )( k , y · k ) ∼ ( w j ′ · g ′ )( k , y · k ) = w j ′ ( g ′ k , y · k ) ∼ w j ′ ( g ′ , y ) · k . 53 Because W is separated (a s w as s ho wn in Lemma 4.15), it follows that w j ( g , y ) ∼ w j ′ ( g ′ , y ). This pr o v es the claim.  An y subtr ee of w is a subtree of some w j and therefor e natural. Hence we only need to prove of w itself that it is c omposable and natura l. Direct insp e ction shows that the tree that we hav e constructed is co mp osable. F o r verifying that w is also na tural, let ( h : c → a, y ) ∈ Y R x and k : d → c . Since h ∈ R , there ar e j ∈ J and g ∈ R j such that h = f j g . Then w ( h, y ) · k ∼ w j ( g , y ) · k ∼ w j ( g k , y · k ) ∼ w ( hk , y · k ) , by us ing natura lit y of w j and the c laim (twice). It rema ins to show that [ w ] is a g lue ing of a ll the w f , i.e., that w · f j ∼ w j for all j ∈ J . So let j ∈ J . First of all, x · f j = x j , b y construction. Seco ndly , for every g : c → b ∈ R j = ( R j ∩ f ∗ j R ) and y ∈ Y ( c ) such tha t F ( y ) = x · f j g , we hav e ( w · f j )( g , y ) = w ( f j g , y ) ∼ w j ( g , y ) . This completes the pro of.  Lemma 4.17 W is a P F -algebr a. Pro of. W e hav e to describ e a natural trans fo rmation S : P F W → W . An element of P F W ( a ) is a pair ( x, t ) consisting of an element x ∈ X ( a ) together with a natural tra nsformation G : Y M a x → W . Using c o llection, there is a map Y M a x t / / P + s W (10) such that [ w ] = G ( y , f ), for all ( f , y ) ∈ Y M a x and w ∈ t ( f , y ). W e deﬁne S x G to be [ sup ( a,x,M a ) t ] . One now needs to chec k that w is heredita r ily natur a l. And then ano ther ver- iﬁcation is needed to check that [ w ] do es not dep end o n the choice o f the map in (10). Finally , one needs to chec k the natura lit y of S . Thes e veriﬁcations are all relatively straightforw ard and simila r to some of the earlier calculatio ns, a nd therefore we leav e all of them to the rea der.  Lemma 4.18 W is the initial P F -algebr a. Pro of. W e will show that S : P F W → W is mo nic and that W has no pr oper P F -subalgebra s; it will then follow from Theorem 26 o f [8] (or Theo rem 6.13 in [9]) that W is the W-type o f F . 54 W e ﬁrst show that S is monic. So let ( x, G ) , ( x ′ , G ′ ) ∈ P F X ( a ) b e such that S x G = S x ′ G ′ ∈ W . It follows that x = x ′ and that there is a c o v ering sieve S on a s uc h that for all ( h, y ) ∈ Y S x , we have G ( h, y ) = G ′ ( h, y ). W e need to show that G = G ′ , so let ( f , y ) ∈ Y M a x be arbitrar y . F or every g ∈ f ∗ S , we hav e: G ( f , y ) · g = G ( f g , y · g ) = G ′ ( f g , y · g ) = G ′ ( f , y ) · g . Since f ∗ S is covering, it follows tha t G ( f , y ) = G ′ ( f , y ), as desir ed. The fact that W has no pr o per P F -subalgebra s is a conseq uence of the inductive prop erties of V (reca ll that V is an initial algebra). Let A b e a sheaf and P F - subalgebra of W . W e cla im that B = { v ∈ V : if v is her editarily natural, then [ v ] ∈ A} is a subalgebr a o f V . Pr oof: Suppose v is a tree that is hereditarily natural. Assume moreover tha t v = sup ( a,x,S ) t and for all ( f , y ) ∈ Y S x and w ∈ t ( f , y ), we k no w that [ w ] ∈ A . Our aim is to show that [ v ] ∈ A . F or the moment ﬁx an element f : b → a ∈ S . Since v · f has a ro ot lab elled by ( b, x · f , M b ) and ( v · f )( g , y ) = v ( f g , y ) fo r all ( g , y ) ∈ Y M b x · f , we hav e that [ v ] · f = S x · f G , where G ( g , y ) = [ v ( f g , y )] ∈ A . Because A is a P F -subalgebra of W this implies that [ v ] · f ∈ A . Since this holds for every f ∈ S , while S is a cov ering s iev e and A is a subsheaf o f W , we obtain that [ v ] ∈ A , as desired. W e co nclude that B = V and hence A = W . This completes the pro of.  T o wra p up: Theorem 4. 1 9 The axiom (WE) is inherite d by she af mo dels. W e b elieve that one has to ma k e additional assumptions on o nes predica tiv e category with small ma ps ( E , S ) to s ho w that the axiom (WS) is inher ited by sheaf mo dels (the a rgumen t a bov e do es no t establish this, the pr oblem b eing that the initial algebra V will b e large, even when the c odomain of the map F : Y → X we hav e computed the W-t ype of is small). W e w ill now show that this problem can b e circumv en ted if we ass ume that the axiom of multiple choice (AMC) holds in E . It is quite likely that one can a lso solve this problem by using Aczel’s Regular Extension Axiom: it implies the axio m (WS) a nd is claimed to b e stable under shea f extensions (but, as far as we ar e aw a re, no pro of of that claim has b een published). Theorem 4. 2 0 The axiom (AMC) is inherite d by she af mo dels. Pro of. This was pr o v ed in Section 10 o f [2 9 ].  55 Theorem 4. 2 1 (A ssuming t ha t (AMC) holds in E .) The axiom (WS) is inherite d by she af mo dels. Pro of. W e will co n tinue to use the notation from the pro of of the previous theorem. So , aga in, w e assume we hav e a small map F : Y → X of sheaves. Moreov er, w e let ψ b e the map in E and Ψ b e the endofunctor on E deﬁned ab o v e, w e le t V b e its initial algebra and ∼ b e the symmetric and transitive relation w e deﬁned on V , and W the W-type as s ociated to F , obtained b y quotienting the hereditarily natur a l elements in V by ∼ . Assume that X is a small sheaf. Since (AMC) holds in E , it is the ca se that, int ernally in E / C 0 , the map ψ ﬁts into a covering squa re a s shown D q / / g   P ( S,x ) Y S x ψ   C p / / X × C 0 Cov , in whic h all ob jects and maps are small in E / C 0 and ( g , q ) is a co lle c tion span ov er X × C 0 Cov. The W-type U = W g in E / C 0 is small in E / C 0 , b ecause we are assuming tha t (WS) holds in E (and hence also in E / C 0 ). The idea is to us e this to show that W is s mall as well. Every element u = sup c s ∈ U determines an element in ϕ ( u ) ∈ V as follows: ﬁrst compute p ( c ) = ( a, x, S ). Then let for every ( y , f ) ∈ Y S x the element t ( y , f ) be deﬁned by t ( y , f ) = { ( ϕ ◦ s )( d ) : d ∈ q − 1 c ( y , f ) } . Then ϕ ( u ) = s up ( a,x,S ) t (so this is an inductive deﬁnition). W e claim that for every hereditar ily natura l tree v ∈ W there is an element u ∈ U such that v ∼ ϕ ( u ). The desired result follows readily from this claim. W e pr o v e the claim by induction: so let v = sup ( a,x,S ) t be a her editarily natura l element of V a nd as sume the claim holds for all subtrees o f v . Since all s ubtrees of v a re her editarily natura l as w ell, this means that for every ( y , f ) ∈ Y S x and w ∈ t ( y , f ) there is an elemen t u ∈ U suc h that ϕ ( u ) = w . F r om the fact tha t ( g , q ) is a collectio n span ov er X × C 0 Cov, it follows that there is a c ∈ C with p ( c ) = ( a, x, S ) together with tw o functions: ﬁrst o ne picking for e very d ∈ D c an e le men t r ( d ) ∈ t ( y , f ) (b ecause t ( y , f ) is non-empty) and a s e cond one pick- ing for every d ∈ D an elemen t s ( d ) ∈ U such that ϕ ( s ( d )) ∼ r ( d ). It is not hard to se e that v ∼ ϕ (sup c s ), using that v is natural a nd therefo re all elements in t ( y , f ) are equiv alen t to e a c h o ther.  This co mpletes the pro of of o ur main result, Theore m 2.18. 56 5 Sheaf mo dels of constructiv e set theory Our ma in result Theorem 2.18 in combination with Theor em 2 .8 yields the existence of sheaf mo dels for CZF and IZF (see Coro llary 2.19). F or the sake of completeness a nd in order to allow a comparison with cla ssical for cing, we describ e this mo del in concrete terms. W e will not present veriﬁcations of the correctnes s o f our descriptions, b ecause they could in principle b e o btained b y un winding the existence pro ofs, and other descr iptions which diﬀer o nly slightly from what we present here can already b e found in the liter ature. T o construct the initial P s -algebra in a catego ry of internal presheaves ov er a predicative category with s mall maps ( E , S ), let W b e the initial algebra o f the endofunctor Φ = P co d ◦ P s on E (see Theor em 2.10). Elements of w ∈ W ar e therefore o f the form sup c t , with c ∈ C 0 and t a function from { f ∈ C 1 : co d( f ) = c } to P s W . W e think o f such an ele ment w as a well-founded tree, where the ro ot is labelled with c a nd for every v ∈ t ( f ), the tree v is connected to the ro ot of w with an edge lab elled with f . The o b ject W carr ies the structure of a presheaf, with W ( c ) cons is ting of trees whose ro ot is lab elled with c , a nd with a r estriction op eration deﬁned b y putting for any w = sup c t a nd f : d → c , w · f = sup d t ( f ◦ − ) . The initial P s -algebra V in the categ ory of presheav es is constructed from W b y selec ting those tree s that a re hereditar ily comp osable and natural: • A tree w = sup c ( t ) ∈ W is c omp osable , if for any f : d → c a nd v ∈ t ( f ), the tree v has a ro ot lab elled with d . • A tree w = sup c ( t ) ∈ W is natu r al , if it is comp osable and for any f : d → c , g : e → d and v ∈ t ( f ), we have v · g ∈ t ( f g ). The P s -algebra str ucture, or, equiv alently , the membership relation on V , is given by the formula ( x, s up c t ∈ V ) x ∈ sup c t ⇐ ⇒ x ∈ t ( id c ) . The easiest w ay to prov e the correctness of the des cription we gav e is by app eal- ing to Theor em 1.1 from [25] (or Theo rem 7.3 from [9]). This mo del was ﬁrst presented in the pap er [18] by Gambino, based on unpublished work b y Dana Scott. The initial P s -algebra in categor ies of internal sheav es is obtained as a quo- tien t of this ob ject V . Roug hly sp eaking, we q uotien t by bis imulation in a wa y which reﬂects the s e ma n tics of a ca teg ory of sheav es. More prec isely , we take V as deﬁned ab o v e and we w r ite: sup c t ∼ sup c t ′ if for all f : d → c and v ∈ t ( f ), the sieve { g : e → d : ∃ v ′ ∈ t ′ ( f g ) ( v · g ∼ v ′ ) } 57 cov ers d a nd for all f ′ : d → c and v ′ ∈ t ′ ( f ′ ), the sieve { g : e → d : ∃ v ∈ t ( f ′ g ) ( v ′ · g ∼ v ) } cov ers d . O n the quotient the membership rela tio n is deﬁned b y: [ v ] ∈ [s up c t ] ⇐ ⇒ the sie ve { f : d → c : ∃ v ′ ∈ t ( f ) ( v · g ∼ v ′ ) } covers c. T o see that this is correc t, one should verify that ∼ deﬁnes a b ounded equiv a- lence r elation and the quotient is a sheaf. Then one pr o v es tha t it is the initial P s -algebra by a pp ealing to Theore m 1.1 fro m [25] (or Theor em 7.3 fro m [9]). The reader who wishes to see more details, should co nsult [33]. Remark 5 .1 T o see the a nalogy with c lassical forc ing (as in [26], for example), note tha t any p oset P determines a site, by decla ring that S cov ers p whenever S is dens e b elow p . In this case, the elements o f V a re a particular k ind of names (as they are traditionally ca lled). One could regar d c omposability and naturality as satura tion prop erties of names (so that, in eﬀect, we only cons ide r nice, satur ated na mes). It is not to o hard to show that every name (in the usual sense ) is equal in a for cing mo del to s uch a s aturated na me, so that in the case of classical ZF the mo dels that we have constructed are not diﬀerent from standard forcing mo dels. References [1] P . Aczel. The t yp e theoretic interpretation of c o nstructiv e set theor y . 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