A GENERAL CONSTRUCTION OF INTERNAL SHEAVES IN ALGEBRAIC SET THEORY

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1 A GENERAL CONSTRUCTION OF INTERNAL SHEAVES IN ALGEBRAIC SET THEORY S. AWODEY, N. GAMBINO, P. L. LUMSDAINE, AND M. A. WARREN Abstract. We present a solution to te problem o deining a counterpart in Algebraic Set Teory o te construction o internal seaves in Topos Teory. Our approac is general in tat we consider seaves as determined by Lawvere-Tierney coverages, rater tan by Grotendieck coverages, and assume only a weakening o te axioms or small maps originally introduced by Joyal and Moerdijk, tus subsuming te existing topos-teoretic results. Introduction Algebraic Set Teory provides a general ramework or te study o category-teoretic models o set teories [17]. Te undamental objects o interest are pairs (E, S) consisting o a category E equipped wit a distinguised amily o maps S, wose elements are reerred to as small maps. Te category E is tougt o as a category o classes, and S as te amily o unctions between classes wose ibers are sets. Te researc in te area as been ollowing two general directions: te irst is concerned wit isolating axioms or te pair (E, S) tat guarantee te existence in E o a model or a given set teory; te second is concerned wit te study o constructions, suc as tat o internal seaves, tat allow us to obtain new pairs (E, S) rom given ones, in analogy wit te existing development o Topos Teory [18, Capter 5]. Te combination o tese developments is intented to give general metods tat subsume te known tecniques to deine sea and realizability models or classical, intuitionistic, and constructive set teories [9, 10, 12, 15, 19, 21, 24]. Our aim ere is to contribute to te study o te construction o internal seaves in Algebraic Set Teory. Te starting point o our development is te notion o a Lawvere-Tierney coverage. I our ambient category E were an elementary topos, Lawvere-Tierney coverages would be in bijective correspondence wit Lawvere-Tierney local operators on te subobject classiier o te topos. However, since E is assumed ere to be only a Heyting pretopos, we work wit te more general Lawvere-Tierney coverages. As we will see, wen E is a category o internal preseaves, tese correspond bijectively to te Grotendieck coverages considered in [13]. Tereore, our development gives as a special case a treatment o te construction o internal seaves Date: November 9t,

2 2 S. AWODEY, N. GAMBINO, P. L. LUMSDAINE, AND M. A. WARREN relative to tose Grotendieck sites, extending te results in [13]. Given a Lawvere-Tierney coverage, we can deine an associated universal closure operator on te subobjects o E, wic allows us to deine te notion o a sea as in te standard topos-teoretic context. Our main result asserts tat te category o internal seaves or a Lawvere- Tierney coverage is a Heyting pretopos and tat it can be equipped wit a amily o small maps satisying te same axioms tat we assumed on te small maps S in te ambient category E. Te irst part o tis result involves te deinition o an associated sea unctor, a inite-limit preserving let adjoint to te inclusion o seaves into E. For tis, we adapt te toposteoretic argument due to Lawvere [18, V.3]. Since te argument involves te construction o power-objects, wic in our setting classiy indexed amilies o small subobjects, our proo requires a preliminary analysis o locally small maps, wic orm te amily o small maps between seaves. We will apply tis analysis also to prove te second part o our main result, wic involves te veriication tat locally small maps between seaves satisy te axioms or a amily o small maps. In recent years, substantial work as been devoted to isolating axioms on (E, S) tat provide a basic setting or bot directions o researc mentioned above. Let us briely consider two suc possible settings. Te irst, to wic we sall reer as te exact setting, involves assuming tat E is a Heyting pretopos, and tat S satisies a a weakening o te axioms or small maps introduced in [17]. Te second, to wic we sall reer as te bounded exact setting, involves assuming tat E is a Heyting category, tat S satisies not only te axioms or small maps o te exact setting, but also te axiom asserting tat or every object X E, te diagonal X : X X X is a small map, tat universal quantiication along small maps preserves smallness o monomorpisms, and inally tat E as quotients o bounded equivalence relations, tat is to say equivalence relations given by small monomorpisms [5]. Categories o ideals provide examples o te bounded exact setting [2]. Te exact completion and te bounded exact completion o syntactic categories o classes arising rom constructive set teories provide oter examples o te exact setting and o te bounded exact setting, respectively [5, Proposition 2.10]. Neiter setting is included in te oter, and tey are someow incompatible. Indeed, i we wis to avoid te assumption tat every equivalence relation is given by a small monomorpism, wic is necessary to include constructive set teories [1] witin te general development, it is not possible to assume bot tat E is exact and tat every object as a small diagonal. Eac setting as speciic advantages. On te one and, te assumption o exactness o E is useul to deine an internal version o te associated sea unctor [13]. On te oter and, te assumption tat diagonals are small as been applied in te coalgebra construction or cartesian comonads [26] and to establis results on W -types [5, Proposition 6.16].

3 INTERNAL SHEAVES IN ALGEBRAIC SET THEORY 3 Te coice o developing our teory witin te exact setting is motivated by te desire or te teory to be appropriately general. Even or te special case o Grotendieck sites, te assumption tat te ambient category is exact seems to be essential in order to deine te associated sea unctor witout additional assumptions on te site [13]. In te bounded exact setting, Benno van den Berg and Ieke Moerdijk ave recently announced a result concerning internal seaves on a site [6, Teorem 6.1], building on previous work o Ieke Moerdijk and Erik Palmgren [22] and Benno van den Berg [4]. Apart rom te axioms or small maps tat are part o te bounded exact setting, tis result assumes a urter axiom or small maps, te Exponentiation Axiom, and te additional ypotesis tat te Grotendieck site as a basis. We preer to avoid tese assumptions since te Grotendieck site tat provides a category-teoretic version o te double-negation translation cannot be sown to ave a basis witout assuming additional axioms or small maps, wic do not old in categories o classes arising rom constructive set teories [12, 15]. 1. Algebraic Set Teory in Heyting pretoposes 1.1. Preliminaries. We begin by stating precisely te axioms or small maps tat we are going to work wit. As in [17], we assume tat E is a Heyting pretopos, and tat S is a amily o maps in E satisying te axioms (A1)-(A7) stated below. (A1) Te amily S contains isomorpisms and is closed under composition. (A2) For every pullback square o te orm g Y k X B A (1) i : X A is in S, ten so is g : Y B. (A3) For every pullback square as (1), i : B A is an epimorpism and g : Y B is in S, ten : X A is in S. (A4) Te maps 0 1 and are in S. (A5) I : X A and g : Y B are in S, ten + g : X + Y A + B is in S. (A6) For every commutative triangle o te orm X Y g A were : X Y is an epimorpism, i : X A is in S, ten g : Y A is in S.

4 4 S. AWODEY, N. GAMBINO, P. L. LUMSDAINE, AND M. A. WARREN (A7) For map : X A in S and every epimorpism p : P X, tere exists a quasi-pullback diagram o te orm Y P p X g B A were g : Y B is in S and : B A is an epimorpism. We reer to (A2) as te Pullback Stability axiom, to (A3) as te Descent axiom, to (A6) as te Quotients axiom, to (A7) as te Collection axiom Power objects. Our basic axiomatisation o small maps involves one more axiom. In order to state it, we need some terminology. Given a amily o maps S satisying (A1)-(A7), an S-object is an object X suc tat te unique map X 1 is in S. For a ixed object A E, an A-indexed amily o S-subobjects is a subobject S A X suc tat its composite wit te irst projection A X A is in S. We abbreviate tis by saying tat te diagram S A X A is an indexed amily o S-subobjects. Recall tat, writing Γ() : X A X A or te evident indexed amily o subobjects consisting o te grap o a map : X A, it olds tat is in S i and only i Γ() is an indexed amily o S-subobjects. Axiom (P1), stated below, expresses tat indexed amilies o S-subobjects can be classiied. (P1) For eac object X o E tere exists an object P(X) o E, called te power object o X, and an indexed amily o S-subobjects o X X P(X) X P(X), called te membersip relation on X, suc tat or any indexed amily o S-subobjects S A X A o X, tere exists a unique map χ S : A P(X) itting in a double pullback diagram o te orm S A X X P(X) X A χ S P(X). Writing Sub S (X)(A) or te lattice o A-indexed amilies o S-subobjects o X, axiom (P1) can be expressed equivalently by saying tat or every map χ : A P(X), te unctions Hom(A, P(X)) Sub S (X)(A),

5 INTERNAL SHEAVES IN ALGEBRAIC SET THEORY 5 deined by pulling back X P(X) X P(X), are a amily o bijections, natural in A. In te ollowing, we oten omit te subscript in te membersip relation. Our basic axiomatisation o small maps involves only axioms (A1)-(A7) and (P1). Tereore, wen we speak o a amily o small maps witout urter speciication, we mean a amily S satisying axioms (A1)-(A7) and (P1). In tis case, elements o S will be reerred to as small maps, and we speak o small objects and indexed amilies o small subobjects rater tan S-objects and indexed amilies o S-subobjects, respectively Exponentiability and Weak Representabily. One can also consider an alternative axiomatisation o small maps by requiring, in place o (P1), te axioms o Exponentiability (S1) and Weak Representability (S2), stated below. (S1) I : X A is in S, ten te pullback unctor : E/A E/X as a rigt adjoint, wic we write Π : E/X E/A. (S2) Tere exists a map u : E U in S suc tat every map : X A in S its in a diagram o orm X A Y B were : B A is an epimorpism, te square on te let-and side is a quasi-pullback and te square on te rigt-and side is a pullback. Te axiomatisation o small maps wit (A1)-(A7) and (S1)-(S2) is a sligt variant o te one introduced in [17]. Te only dierence concerns te ormulation o te Weak Representability axiom, wic is a weakening o te Representability axiom in [17, Deinition 1.1]. Te weakening involves aving a quasi-pullback rater tan a genuine pullback in te let-and side square o te diagram in (2). Example and Example illustrate ow te weaker orm o representability in (S2) is te most appropriate to consider wen working witin exact categories witout assuming additional axioms or small maps. See [2, 25] or oter orms o representability. Tis axiomatisation is a strengtening o te one consisting o (A1)-(A7) and (P1). On te one and, te combination o (S1) and (S2) implies (P1), since te proo in [17, I.3] carries over wen Weak Representability is assumed instead o Representability [6]. On te oter and, Example sows tat tere are examples satisying (P1) but not (S2). Let us also recall tat (P1) implies (S1) by an argument similar to te usual construction o exponentials rom power objects in a topos [2, Proposition 5.17] Internal language. We will make extensive use o te internal language o Heyting pretoposes [20] Tis is a orm o many-sorted irst-order E U u (2)

6 6 S. AWODEY, N. GAMBINO, P. L. LUMSDAINE, AND M. A. WARREN intuitionistic logic wic allows us to manipulate objects and maps o E syntactically. As an illustration o te internal language, let us recall tat or any map : X Y, we ave a direct image map! : P(X) P(Y ). Assuming : X Y to be small, tere is also an inverse image map : P(Y ) P(X), wic is related to! : P(X) P(Y ) by te internal adjointness expressed in te internal language as ollows: ( s : P(X))( t : P(Y )) (! (s) t s (t) ). (3) Te internal language allow us also to give a caracterisation o small maps. Indeed, a map : X A is small i and only i te ollowing sentence is valid: ( a : A)( s : P(X))( x : X) ( (x) = a x s ). (4) Te sentence in (4) can be understood inormally as expressing tat te ibers o : X A are small. Formulation o some o te axioms or small in te internal language can be ound in [3]. For s : P(X) and a ormula φ(x) were x : X is a ree variable, we deine te restricted quantiiers by letting ( x s)φ(x) = de ( x : X) ( x s φ(x) ), ( x s)φ(x) = de ( x : X) ( x s φ(x) ). We denote anonymous variables o sort X by writing : X Examples. We end tis section wit some examples o Heyting pretoposes equipped wit amilies o small maps. Example sows tat our development o internal seaves applies to elementary toposes [18, Capter V], wile Example and Example sow tat it includes important examples or wic te ambient category E is not an elementary topos. Note tat neiter Example nor Example satisies te Representability axiom o [17, Deinition 1.1], but only te Weak Representabily Axiom, as stated in (S2) above. Furtermore, neiter o tese examples satisies te additional axiom tat every object X o E as a small diagonal map X : X X X Example. Consider an elementary topos E and let S consist o all maps in E. It is evident tat te axioms (A1)-(A7) and (P1) are veriied, wile (S2) is not Example. Consider Constructive Zermelo-Fraenkel set teory (CZF), presented in [1]. We take E to be te exact completion [7, 8] o te corresponding category o classes [3, 11], considered as a regular category. By te general teory o exact completions, te category E is a Heyting pretopos and te category o classes o CZF embeds aitully in it [7, 8]. We write te objects o E as X/r X, were X is a class and r X X X is an equivalence relation on it. A map : X/r X A/r A in E is a relation X A tat is unctional and preserves te equivalence relation, in te sense made

7 INTERNAL SHEAVES IN ALGEBRAIC SET THEORY 7 precise in [7]. We declare a map : X/r X A/r A to be small i it its into a quasi-pullback o te orm Y k X/r X g B A/r A were : B A is an epimorpism and g : Y B is a unction o classes wose ibers are sets. Tis amily satisies te axioms (A1)-(A7) and (S1)-(S2) by a combination o te results on te category o classes o CZF in [3, 11] wit tose on small maps in exact completions in [5, Section 4] Example. Consider Martin-Lö s constructive type teory wit rules or all te standard orms o dependent types and or a type universe relecting tem [23, 14]. We take E to be te corresponding category o setoids, wic as been sown to be a Heyting pretopos in [22, Teorem 12.1]. We declare a map : X A in E to be small i it its into a quasi-pullback o te orm g Y k X B A were : B A is an epimorpism and g : Y B is a map suc tat or every b B te setoid g 1 (b), as deined in [22, Section 12], is isomorpic to setoid wose carrier and equivalence relation are given by elements o te type universe. Tis amily satisies te axioms (A1)-(A7) and (S1)-(S2) by combining te results on display maps in [5, Section 4] wit tose on setoids in [22, Section 12]. 2. Lawvere-Tierney seaves 2.1. Lawvere-Tierney coverages. Let E be a Heyting pretopos equipped wit a amily o small maps S satisying axioms (A1)-(A7) and (P1). We deine te object o small trut values Ω by letting Ω = de P(1). By te universal property o Ω, it is immediate to see tat we ave a global element : 1 Ω. To simpliy notation, we write p instead o p =, or p : Ω. For example, tis allows us to write {p : Ω p} instead o {p : Ω p = }. Note, ten, tat te monomorpism {p : Ω p} Ω is te map : 1 Ω. Similarly, p φ is equivalent to ( p)φ, or every p : Ω and every ormula φ o te internal language. Te internal language is used in Deinition to speciy wat will be our starting point to introduce a notion o sea Deinition. Let (E, S) be a Heyting pretopos wit a amily o small maps. A Lawvere-Tierney coverage in E is a subobject J Ω making te ollowing sentences valid in E

8 8 S. AWODEY, N. GAMBINO, P. L. LUMSDAINE, AND M. A. WARREN (C1) J( ), (C2) ( p : Ω)( q : Ω)[ (p J(q) ) ( J(p) J(q) ) ] Remark. Our development o internal seaves relative to a Lawvere- Tierney coverage generalises te existing teory o internal seaves relative to a Lawvere-Tierney local operator in an elementary topos [18]. Indeed, wen E is an elementary topos and S consists o all maps in E, as in Example 1.5.1, Lawvere-Tierney coverages are in bijective correspondence wit Lawvere-Tierney local operators [16, A.4.4.1]. Te correspondence is given by te universal property o Ω, wic is te subobject classiier o E, via a pullback square o te orm J 1 Ω j Ω Te veriication o te correspondence between te axioms or a Lawvere- Tierney coverage and tose or a Lawvere-Tierney local operator is a simple calculation. Since in te general te monomorpism J Ω may ail to be small, we ocus on Lawvere-Tierney coverages. From now on, we will work wit a ixed Lawvere-Tierney coverage as in Deinition Our irst step towards deining seaves is to construct a universal closure operator on subobjects, tat is to say a natural amily o unctions C X : Sub(X) Sub(X), or X E, satisying te amiliar monononicity, inlationarity, and idempotency properties [16, A.4.3]. Note tat we do not need to require meetstability, since tis ollows rom te oter properties by te assumption tat te operator is natural [16, Lemma A.4.3.3]. Naturality o te operator means tat or : X Y, te diagram below commutes Sub(Y ) Sub(X) C Y C X Sub(Y ) Sub(X) We deine te universal closure operator associated to te Lawvere-Tierney coverage by letting, or S X C X (S) = de { x : X ( p : Ω) ( J(p) ( p S(x) ))}. (5) Proposition. Te amily C X : Sub(X) Sub(X), or X E, associated to a Lawvere-Tierney operator is a universal closure operator.

9 INTERNAL SHEAVES IN ALGEBRAIC SET THEORY 9 Proo. First, we veriy tat te operator is natural. Tis is immediate, since or a subobject T Y we ave C X ( T ) = { x : X ( p : Ω) ( J(p) ( p T (x) ))} = {y : Y ( p : Ω) ( J(p) ( p T (y) ))} = (C Y (T )). For inlationarity, let S X, x : X and assume tat S(x) olds. Ten, deine p : Ω by letting p = de. We ave tat J(p) olds by (C1), and tat p S(x) olds by assumption. Monotonicity is immediate by te deinition in (5). Idempotence is te only part tat is not straigtorward, since it makes use o te Collection Axiom or small maps. For S X, we need to sow tat C 2 (S) C(S). Let x : X and assume tat tere exists p : Ω suc tat J(p) and p C X (S)(x) old. For : 1 and q : Ω, let us deine φ(, q) = de J(q) ( q S(x) ). By te deinition in (5), p C X (S)(x) implies ( p)( q : Ω)φ(, q). We can apply Collection and derive te existence o u : P(Ω) suc tat ( p)( q u)φ(, q) ( q u)( p)φ(, q). Deine r : Ω by r = de u. We wis to sow tat J(r) and r S(x) old, wic will allow us to conclude CS X (x), as required. To prove tat J(r) olds, we observe tat p ( q u) J(q) ( q : Ω) ( J(q) q r ) J(r). Tereore p J(r). Since we ave J(p) by ypotesis, Axiom (C2) or a Lawvere-Tierney coverage allows to derive J(r), as required. By deinition o r : Ω, we note tat r S(x) olds i and only i or every q u we ave q S(x). But since q u implies q S(x), we obtain r S(x), as desired Remark. Given a universal closure operator on E, we can deine a Lawvere-Tierney coverage J Ω by taking J to be te closure o {p : Ω p}. Te closure operator induced by J coincides wit te given one i and only i te latter satisies te equation in (5). Tereore, we are considering ere only a special class o universal closure operators. Some restriction on te universal closure operators seems necessary since it does not seem possible to develop a treatment o seaves or arbitrary universal closure operators witout assuming additional axioms or small maps, suc as tat asserting tat every monomorpism is small. Focusing on te class o universal closure operations determined by Lawvere-Tierney coverages captures an

10 10 S. AWODEY, N. GAMBINO, P. L. LUMSDAINE, AND M. A. WARREN appropriate level o generality. First, as we will see in Section 2.3, tey correspond precisely to te Grotendieck sites considered in [13]. Secondly, as we will see in Section 4 and Section 5, tey allow us to develop a treatment o internal seaves. We sall be particularly interested in te closure o te membersip relation X P(X) X X, wic we are going to write as X P(X) X P(X). For x : X and s : P(X), te deinition in (5) implies x s ( p : Ω) ( J(p) ( p x s )), 2.2. Seaves. Having deined a universal closure operator on E, we can deine a notion o sea and introduce te amily o maps tat we will consider as small maps between seaves. In order to do tis, we deine a monomorpism m : B A to be dense i it olds tat C A (B) = A Deinition. An object X o E is said to be a separated i or every dense monomorpism m : B A te unction Hom(m, X) : Hom(A, X) Hom(B, X), (6) induced by composition wit m, is injective. Equivalently, X is a separated i and only i or every map v : B X tere exists at most one extension u : A X making te ollowing diagram commute m B v A u X (7) We say tat X is a sea i te unction in (6) is bijective. Equivalently, X is a sea i and only i every map v : B X as a unique extension u : A X as in (7). Deinition deines wat it means or a map in E to be locally small. As explained urter in Section 3, te idea underlying tis notion is tat a map is locally small i and only i eac o its ibers contains a small dense subobject. By teir very deinition, locally small maps are stable under pullback, since te properties o te deining diagrams all are Deinition. A map : X A in E is said to be locally small i its grap Γ() : X A X A its in a diagram o te orm T B X X A X π B B A π A

11 INTERNAL SHEAVES IN ALGEBRAIC SET THEORY 11 were : B A is an epimorpism, T B X B is an indexed amily o small subobjects o X, and te canonical monomorpism T B A X is dense. An indexed amily o subobjects S A X X is said to be an indexed amily o locally small subobjects i te composite map S A is locally small. We write E J or te ull subcategory o E wose objects are seaves, and S J or te amily o locally small maps in E J. Te aim o te remainder o te paper is to prove te ollowing result. As ixed in Section 1.2, a amily o small maps is required to satisy only axioms (A1)-(A7) and (P1) Teorem. Let (E, S) be a Heyting pretopos wit a amily o small maps. For every Lawvere-Tierney coverage J in E, (E J, S J ) is a Heyting pretopos equipped wit a amily o small maps. Furtermore, i S satisies te Exponentiability and Weak Representability axioms, so does S J Grotendieck seaves. Beore developing te teory required to prove Teorem 2.2.3, we explain ow tis result subsumes te treatment o seaves or a Grotendieck site. Let C be a small internal category in E. Smallness o C means tat bot its objects C 0 and its arrows C 1 are given by small objects in E. We write category Ps E (C) o internal preseaves over C. It is well-known tat Ps E (C) is a Heyting pretopos. Wen C 0 and C 1 ave small diagonals, so tat bot te equality o objects and tat o arrows is given by a small monomorpism, it is possible to equip Ps E (C) wit a amily o small maps, consisting o te internal natural transormations tat are pointwise small maps in E [22, 26]. Lawvere-Tierney coverages in Ps E (C) are in bijective correspondence wit Grotendieck coverages wit small covers on C, as deined in [13]. To explain tis correspondence, we need to recall some terminology and notation. A sieve P on a C is a subobject P y(a), were we write y(a) or te Yoneda embedding o a. Suc a sieve can be identiied wit a amily o arrows wit codomain a tat is closed under composition, in te sense tat or every pair o composable maps φ : b a and ψ : c b in C, φ P implies φ ψ P. For a sieve P on a and arrow φ : b a, we write P φ or te sieve on b deined by letting P φ = de {ψ : c b φ ψ P }. (8) Recall rom [13] tat a Grotendieck coverage wit small covers on C consists o a amily (Cov(a) a C) suc tat elements o Cov(a) are small sieves, and te conditions o Maximality (M), Local Caracter (L), and Transitivity (T) old: (M) M a Cov(a). (L) I φ : b a and S Cov(a), ten S φ Cov(b). (T) I S Cov(a), T is a small sieve on a, and or all φ : b a S we ave T φ Cov(b), ten T Cov(a).

12 12 S. AWODEY, N. GAMBINO, P. L. LUMSDAINE, AND M. A. WARREN Te object Ω in Ps E (C) is given by Ω(a) = de {S y(a) S small}. Tereore, a Grotendieck coverage wit small covers can be identiied wit a amily o subobjects Cov(a) Ω(a). Condition (L) means tat tis amily is a subpresea o Ω, wile conditions (M) and (T) or a Grotendieck coverage are te rewriting o conditions (C1) and (C2) or a Lawvere-Tierney coverage. By instanciating te general deinitions o Section 2.2 we obtain a notion o sea, wic can be sown to be equivalent to te amiliar notion o a sea or a Grotendieck coverage, and a notion o small map. Writing S E (C, Cov) or te category o internal seaves, and S(C, Cov) or te corresponding amily o small maps, Teorem implies te ollowing result Corollary. Let (E, S) be a Heyting pretopos wit a amily o small maps. Let (C, Cov) be a small category wit small diagonals equipped wit a Grotendieck coverage wit small covers. Ten ( S E (C, Cov), S(C, Cov) ) is a Heyting pretopos wit a amily o small maps. Furtermore, i S satisies te Exponentiability and Weak Representability axioms, so does S J. 3. Classiication o locally small subobjects 3.1. Locally small maps. We begin by caracterising locally small maps in te internal language, analogously to ow small maps are caracterised in (4). For eac object X, deine an equivalence relation R P(X) P(X) by letting R = de {(s, t) : P(X) P(X) ( x : X) ( x s x t ) }. Inormally, R(s, t) olds wenever s and t ave te same closure. Using te exactness o te Heyting pretopos E, we deine P J (X) as te quotient o P(X) by R, itting into an exact diagram o te orm R π 1 π 2 P(X) [ ] P J (X). (9) Te quotient map P(X) P J (X) is to be interpreted as perorming te closure o a small subobject o X Remark. Te exactness o te Heyting pretopos E is exploited ere in a crucial way to deine P J (X). In particular, witout urter assumptions on te Lawvere-Tierney coverage, te equivalence relation in (9) cannot be sown to be given by a small monomorpism. We deine a new indexed amily o subobjects o X, J P J (X) X P J (X), by letting, or x : X and p : P J (X), x J p ( s : PX) ( p = [s] x s ).

13 INTERNAL SHEAVES IN ALGEBRAIC SET THEORY 13 In particular, or x : X and s : PX, tis gives x J [s] x s, wic is to say tat te ollowing squares are pullbacks P(X) X J P J (X) X (10) P(X) [ ] P J (X) Tis deinition o J and o te relation R imply tat P J (X) satisies a orm o extensionality, in te sense tat or p, q : P J (X) it olds tat p = q ( x : X)(x J p x J q). From diagram (10), we also see tat J is closed in X P J X, since it is closed wen pulled back along an epimorpism. Given a map : X A, it is convenient to deine, or a : A, s : P(X), s 1 (a) = de ( x : X) [(x s (x) = a) ((x) = a x s)] Lemma. A map : X A is locally small i and only i te ollowing sentence is valid: ( a : A)( s : PX)s 1 (a). (11) By Deini- Proo. First, let us assume tat : X A is locally small. tion tere is a diagram T B X B X A X A were T B X B is an indexed amily o small subobjects o X, : B A is an epimorpism, and te monomorpism T B A X is dense. Tere is a classiying map χ T : B P(X) suc tat x χ T (b) T (b, x), and te commutativity o te diagram implies x χ T (b) (x) = (b), wile te density o T B A X implies (x) = (b) x χ T (b). Given a : A, tere exists b : B suc tat (b) = a and so, deining s = de χ T (b), we obtain te data required to prove te statement. For te converse

14 14 S. AWODEY, N. GAMBINO, P. L. LUMSDAINE, AND M. A. WARREN implication, assume (11). We deine and B = de {(a, s) : A P(X) s 1 (a)} T = de { ( (a, s), x ) : B X x s}. We ave tat T B X B is small by construction and tat te projection : B A is an epimorpism by ypotesis. Since B A X = { ( (a, s), x ) : B X (x) = a}, we ave tat T B A X is dense by te deinition o B Remark. We could ave considered maps : X A satisying te condition ( a : A)( s : PX)( x : X.)(x) = a x s. Tis amounts to saying tat eac iber o is te closure o a small subobject. Let us call suc maps closed-small. For maps wit codomain a separtated object, and so or maps wit codomain a sea, tese deinitions coincide, so eiter would give our desired class o small maps on E J. However, considered on te wole o E, tey generally give dierent classes o maps, eac retaining dierent properties. For instance, closed-small subobjects o X are classiied by P J (X), wile locally small subobjects may not be classiied. On te oter and, locally small maps satisy axioms (A1)-(A7) in E, wereas te closed-small maps may not, since te identity map on a non-separated object will not be closed-small. Tus, te coice o eiter as te extension o local smallness rom E J to E is a matter o convenience Universal property o P J (X). We conclude tis section by sowing ow locally small closed subobjects can be classiied. Tis is needed in Section 4 or te proo o te associated sea unctor teorem Lemma. An indexed amily o subobjects S A X A is an indexed amily o locally small subobjects i and only i tere exists a diagram o te orm T S B X π B B A X were : B A is an epimorpism, T B X S is an indexed amily o small subobjects o X, and te canonical monomorpism T B A S is dense. Proo. Assume S A X A to be an indexed amily o locally small subobjects. By Deinition 2.2.2, tis means tat te composite S A is a A π A

15 INTERNAL SHEAVES IN ALGEBRAIC SET THEORY 15 locally small map, wic in turn implies tat tere exists a diagram o te orm T S (12) B S π B B A S were : B A is an epimorpism, T B S S is an indexed amily o small subobjects o S, and te canonical monomorpism T B A S is dense. Te composite A T B S B A X B X can be sown to be a monomorpism using te commutativity o te diagram in (12). We obtain an indexed amily o small subobjects T B X B, wic clearly satisies te required property. Observe tat J P J (X) X P J (X) is a amily o locally small subobjects o X, as witnessed by te diagram P(X) X π A J P J (X) X P(X) [ ] P J (X) Te pullback o J P J (X) X P J (X) along a map χ : A P J (X) is tereore an A-indexed amily o locally small closed subobjects o X. We write Sub SJ (X)(A) or te lattice o suc subobjects Proposition. For every object X, te object P J (X) classiies indexed amilies o locally small closed subobjects o X, wic is to say tat or every suc amily S A X A tere exists a unique map φ S : A P J (X) suc tat bot squares in te diagram S J A X φ S 1 X P J (X) X π A A φ S π PJ (X) P J (X)

16 16 S. AWODEY, N. GAMBINO, P. L. LUMSDAINE, AND M. A. WARREN are pullbacks. Equivalently, te unctions Hom(A, P J (X)) Sub SJ (X)(A) given by pulling back J P J (X) X P J (X) are a amily o bijections, natural in A. Proo. Let an indexed amily as in te statement be given, togeter wit data as in Lemma By te universal property o P(X), we get a classiying map χ T : B P(X). Using te naturality o te closure operation, we obtain (χ T 1 X ) C( X ) = C ( (χ T 1 X ) ( X ) ) = C(T ) = B A S, were te last equality is a consequence o te density o T in B A S. Tereore, we ave a sequence o pullbacks o te orm J P J (X) X P J (X) [ ] P(X) X P(X) χ T B A S B X B S A X A Tis, combined wit te epimorpism : B A, allows us to sow tat ( a : A)( p : P J (X))( x : X) ( S(a, x) x J p ). But te deinition o P J (X) as a quotient ensures tat or a given a : A, tere is a unique p : P J (X) satisying S(a, x) x J p or all x : X. Hence, by unctional completeness we obtain te existence o a map φ S : A P J (X) suc tat ( a : A)( x : X) S(a, x) x J φ S (x), as required. 4. Te associated sea unctor teorem 4.1. Te associated sea unctor. We are now ready to deine te associated sea unctor a : E E J, te let adjoint to te inclusion i : E J E o te ull subcategory o seaves into E. Given X E, deine σ : X P J (X) to be te composite X { } P(X) [ ] P J (X). To deine te associated sea unctor, irst actor σ : X P J (X) as an epimorpism X X ollowed by a monomorpism X P J (X). Ten, deine a(x) to be te closure o te subobject X in P J (X) a(x) = de C(X ).

17 INTERNAL SHEAVES IN ALGEBRAIC SET THEORY 17 Te unit o te adjunction η X : X a(x) is ten deined as te composite o X X wit te inclusion X C(X ). We need to sow tat a(x) is indeed a sea and tat it satisies te appropriate universal property. Te proo o te ormer involves te veriication tat P J (X) is a sea. Tis, in turn, requires urter analysis o te notion o locally small map, wic we carry out in Section 4.2 below Caracterisation o locally small maps. We say tat a map is dense i it actors as an epimorpism ollowed by a dense monomorpism. For monomorpisms tis deinition agrees wit te deinition o dense monomorpism given in Section 2.2. It is immediate to see tat te pullback o a dense map is again dense, and a direct calculation sows tat dense maps are closed under composition. It will be convenient to introduce some additional terminology: we reer to a commutative square o te orm Y X B suc tat te canonical map Y B A X is dense as a local quasi-pullback. Note tat every dense map : B A its into a local quasi-pullback o te orm B A m A B p A were m : B B and p : B A are respectively te dense monomorpism and te epimorpism orming te actorisation o. Te diagram is a local quasi-pullback because te map B B A A is m itsel. Let us also observe tat i bot squares in a diagram Z Y 1 A X C B A are local quasi-pullbacks, ten te wole rectangle is also a local quasipullback. Tis implies tat any inite pasting o local quasi-pullbacks is again a local quasi-pullback. We establis a very useul actorisation or dense mononomorpisms Lemma. Every dense monomorpism : B A can be actored as B m A p B

18 18 S. AWODEY, N. GAMBINO, P. L. LUMSDAINE, AND M. A. WARREN were m : B B is a small dense monomorpism and p : B A is an epimorpism. Proo. Let us deine B = de {(p, a) : Ω A p B(a)}. By te deinition o closure in (5) and te assumption tat is dense, te projection π 2 : B A is an epimorpism. Furtermore, tere is a monomorpism m : B B deined by mapping b : B into (, b) : B. Diagrammatically, we ave a pullback o te orm B 1 m B π 1 J Te map : 1 J is small and dense. It is small because it is te pullback o te map : 1 Ω, wic is small by te deinition o Ω, along te inclusion J Ω. It is dense by te very deinition o closure in (5). Tereore, preservation o smallness and density along pullbacks implies tat m : B B is small and dense, as required Remark. Lemma exploits in a crucial way te act tat te closure operation, and ence te notion o density, are determined by a Lawvere- Tierney coverage. Indeed, it does not seem possible to prove an analogue o its statement or arbitary closure operations witout assuming urter axioms or small maps Lemma. A map : X A is locally small i and only i it its into a local quasi-pullback square o te orm g Y k X B A were : B A is an epimorpism and g : Y B is small. (13) Proo. Assuming tat : X A is locally small, te required diagram is already given by Deinition For te converse implication, consider a diagram as in (13). Consider te actorisation o te canonical map Y B A X as an epimorpism ollowed by a monomorpism, say Y T B A X. We ave a diagram o orm Y T X B Y B X A X π B B 1 B B π B A π A

19 INTERNAL SHEAVES IN ALGEBRAIC SET THEORY 19 By te Quotients Axiom or small maps T B X B is a amily o small subobjects. Furtermore, T B A X is dense by construction. Note tat te map k : Y X in Lemma is also dense, since it is te composition o te pullback o an epimorpism wit a dense map Proposition. A map : X A is locally small i and only i it its into a pullback diagram o te orm g Y k X B A (14) were : B A is dense and g : Y B is locally small. Proo. Assume to be given a diagram as (14). By te actorisation o dense maps as epimorpisms ollowed by dense monomorpisms and Lemma 4.2.3, it is suicient to prove te statement wen : B A is a dense monomorpism. We construct a diagram o te orm B C Z Z Y k X B (5) (3) g (1) C B A n (4) m (2) A p A p A 1 A Te given diagram is in (1). First, actor : B A using Lemma as a small dense monomorpism m : B A ollowed by an epimorpism p : A A. Te resulting commutative square in (2) is a local quasipullback since m is dense. Next, apply Lemma to te locally small map g : Y B so as to obtain te local quasi-pullback in (3) wit Z C a small map and C B an epimorpism. Next, we apply te Collection Axiom to te small map m : B A and te epimorpism C B, so as to obtain te quasi-pullback in (4) were n : B A is a small map and p : A A is an epimorpism. Finally, we construct te pullback square in (5) to obtain anoter small map B C Z B. Te wole diagram is a local quasi-pullback, and we can apply Lemma to deduce tat : X A is locally small. Te converse implication is immediate since locally small maps are stable under pullback. Corollary sows tat te assumption o being an epimorpism or te map : B A in Lemma can be weakened.

20 20 S. AWODEY, N. GAMBINO, P. L. LUMSDAINE, AND M. A. WARREN Corollary. I a map : X A its into a local quasi-pullback diagram o te orm k Y X (15) g B A were g : Y B is small and : B A is dense, ten : X A is locally small. Proo. Given te diagram in (15), we construct te ollowing one g Y B B A X 1 B B Te let-and side square is a local quasi-pullback by assumption. So, by Lemma 4.2.3, te map g : B A X B is locally small. Te rigt-and side square is a pullback by deinition. So, by Proposition 4.2.4, te map : X A is locally small, as desired Proo o te associated sea unctor teorem. We exploit our caracterisation o locally small maps in te proo o te ollowing proposition Proposition. For every object X o E, P J (X) is a sea. Proo. By Proposition 3.2.2, it suices to sow tat every dense monomorpism m : B A induces by pullback an isomorpism g X A m : Sub SJ (X)(A) Sub SJ (X)(B). Let us deine a proposed inverse m as ollows. For a amily o locally small closed subobjects T B X B, we deine m (T ) = C A X (T ). Here, we view T as a subobject o A X via composition wit te evident monomorpism B X A X. We need to sow tat te result is a locally small amily, and tat m and m are mutually inverse. First, note tat m is just intersection wit B X. Tereore, or a locally small closed amily T B X B we ave m m (T ) = m C A X (T ) = C B X (m T ) = C B X (T ) = T, were we used te naturality o te closure operation, tat T A X, and te assumption tat T is closed. Tis also sows tat m (T ) becomes locally

21 INTERNAL SHEAVES IN ALGEBRAIC SET THEORY 21 small over B wen pulled back along te dense monomorpism m : B A. By Proposition 4.2.4, it is locally small over A, as required. Finally, or a locally small closed amily S A X A, we ave as desired. m m (S) = m (S (B X)) = C A X (S (B X)) = C A X (S) C A X (B X) = S (A X) = S, Lemma. A subobject o a sea is a sea i and only i it is closed. Proo. Let i : S X be a monomorpism, and assume tat X is a sea. We begin by proving tat i S is closed, ten it is a sea. For tis, assume given a dense monomorpism m : B A and a map : B S. Deine g : B X to be i : B X. By te assumption tat X is a sea, tere exists a unique ḡ : A X making te ollowing diagram commute B S i X We ave m A ḡ Im(ḡ) = ḡ! (A) = ḡ! (C A (B)) C X (ḡ! (B)) = C X (Im(g)) = S, C X (S) were te irst inequality ollows rom adjointness. Tereore, ḡ : A X actors troug : A S, extending : B S as required. Uniqueness o tis extension ollows by te uniqueness o ḡ : A X and te assumption tat i : S X is a monomorpism. For te converse implication, we need to sow tat i S is a sea ten it is closed. Te monomorpism m : S C X (S) is dense and tereore te identity 1 S : S S as an extension n : C X (S) S wit n m = 1 S. But by te preceding part, we know tat C X (S) is a sea and tat m n m = n = 1 CX (S) m. Tereore, bot n m and 1 C(S) are extensions o m : S C X (S) to C X (S). Since S is a sea, tey must be equal. Tus m and n are mutually inverse, and so S = C X (S) as desired. Lemma and Lemma imply tat te object a(x) is a sea or every X E, since a(x) is a closed subobject o te sea P J (X). In

22 22 S. AWODEY, N. GAMBINO, P. L. LUMSDAINE, AND M. A. WARREN order to sow tat we ave indeed deined a let adjoint to te inclusion, we need te ollowing Lemma 4.3.3, concerning te map η X : X a(x), also deined in Section 4.1 using te map σ X : X P J (X) taking x : X into [{x}] : P J (X) Lemma. For every X E, we ave in Sub(X X). Proo. For x, y : X, we ave as required. Ker(η X ) = Ker(σ X ) = C X X ( X ) σ(x) = σ(y) C({x}) = C({y}) ( p : Ω) ( J(p) ( p x {y} )) (x, y) C X X ( X ), Generally, a map : X Y wit Ker() C( X ) is called codense Teorem. For every X E, a(x) E J is te associated sea o X, in te sense tat or every map : X Y into a sea Y, tere exists a unique : a(x) Y making te ollowing diagram commute X η X a(x) Te resulting let adjoint a : E E J preserves inite limits. Proo. Given : X Y, Ker() is a pullback o Y Y Y. Since Y Y is a sea, Lemma implies tat Y is closed, and ence Ker() is closed in X X. But certainly X Ker() and tereore Y Ker(σ X ) = C( X ) Ker(). Since any epimorpism is te coequaliser o its kernel pair, actors uniquely troug te codense epimorpism X Im(σ X ). Since Y is a sea, te map rom Im(σ X ) to Y extends uniquely along te dense monomorpism Im(σ X ) a(x), giving a unique actorisation o troug η X as desired. To sow tat te associated sea unctor preserves inite limits, we may proceed exactly as in [18, V.3], since te argument tere uses only te structure o a Heyting category on E and te act tat te associated sea unctor is deined by embedding eac object X by a codense map into a sea. In particular, injectivity o te sea in wic X is embedded is not required to carry over te proo. Teorem allows us to deduce Proposition 4.3.5, wic contains te irst part o Teorem Ater stating it, we discuss in some detail te structure o te category E J.

23 INTERNAL SHEAVES IN ALGEBRAIC SET THEORY Proposition. Te category E J is a Heyting pretopos. Proo. Tis is an immediate consequence o Teorem Te Heyting pretopos structure o seaves. We discuss te relationsip between te structure o E and tat o E J is some detail. Here and subsequently, operations in E, will be denoted witout subscript, suc as Im, wile teir counterparts in E J will be denoted wit subscript, suc as Im J. Limits in E J are just limits in E, since seaves are closed under limits. Colimits are te seaiications o colimits taken in E, since te associated sea unctor, being a let adjoint, preserves colimits. Te lattice Sub J (X) is te sub-lattice o closed elements o Sub(X). Here, te closure operation is a relection and tereore meets in Sub J (X) are meets in Sub(X), and joins in Sub J (X) are closures o joins in Sub(X). Moreover, or S, T Sub(X), we ave S C(S T ) C(S) C(S T ) = C(S (S T )) C(T ) = T. Tereore C(S T ) (S T ) and so S T is closed. Tus, te implication o Sub(X) restricts so as to give implication in Sub J (X). Since limits in E J agree wit limits in E, te inverse image unctors in E J are just te restrictions o te inverse image unctors o E. It is clear tat te seaiication o a dense monomorpism m : B A is an isomorpism, since or any sea X we ave E J ( a(a), X ) = E(A, X) = E(B, X) = EJ ( a(b), X ). Tereore, epimorpisms in E J are precisely te dense maps in E, since coequalisers in E J are seaiications o coequalisers in E, and dense maps are te seaiications o epimorpisms o E. We ave seen beore tat dense maps are stable under pullback. Consequently, quasi-pullbacks in E J are exactly tose squares tat are local quasi-pullbacks in E. I (r 1, r 2 ) : R X X is an equivalence relation in E J, it is also an equivalence relation in E, so as an eective quotient q : X Q in E. Its seaiication a(q) = η Q q : X a(q) is ten a coequaliser or (r 1, r 2 ) in seaves; but tis is an eective quotient or R, since Ker(η Q q) = q C( Q ) = C(Ker q) = C(R), and R is closed in X X, as it is a subsea. Images in E J are te closures o images in E. For : X Y, Im J () Y is te least subsea o Y trougt wic actors, so te least closed subobject o Y containing Im(), wic can be readily identiied wit C Y (Im ). Consequently, te corresponding orward image unctor ( J ) : Sub J (X) Sub J (Y )

24 24 S. AWODEY, N. GAMBINO, P. L. LUMSDAINE, AND M. A. WARREN is given by ( J ) (S) = C Y ( S). Dual image unctors in E J are again just te restrictions o te dual images unctors o E, since i S X is closed so is (S) Y, and tereore C( S) (CS). Tese give us an immediate translation rom te internal logic o E J into tat o E. 5. Small maps in seaves 5.1. Preliminaries. Witin tis section, we study locally small maps between seaves, completing te proo o Teorem Te associated sea unctor ollowed by te inclusion o seaves into E preserves dense maps. Indeed, avoiding explicit mention o te inclusion unctor, i : B A is dense ten we ave Im(η A ) = Im(a() η B ) Im(a()). But η A : B a(a) is a dense map, and so its image is dense in a(a). Also, i X is a sea ten η X : X a(x) is easily seen to be an isomorpism Lemma. For every : X Y in E, te naturality square is a local quasi-pullback. X η X a(x) a() Y η Y a(y ) Proo. Te canonical map X ax ay Y, wic we wis to sow to be dense, actors troug X ay Y as in te diagram below: X X a(y ) Y a(x) a(y ) Y Y 1 X X η X a(x) a() η Y a(y ) Te map X ay Y ax ay Y is a pullback o η X and tereore it is dense. We just need to sow tat X X ay Y is dense. For tis, let us consider

25 INTERNAL SHEAVES IN ALGEBRAIC SET THEORY 25 te ollowing diagram, in wic eac square is a pullback X = Γ X a(y ) Y X (,) Y C( Y ) π 1 Y = π 2 η Y Y η Y a(y ) We ten ave te ollowing cain o isomorpisms o subobjects o X Y X ay Y = X Y Ker(η Y ) = (, 1 Y ) C Y Y ( Y ) = C X Y ( (, 1Y ) Y ) = C X Y (Γ ). Tereore, te map X X ay Y can be identiied as te map Γ C(Γ ), wic is dense Corollary. (i) I : X Y is small, ten a() : a(x) a(y ) is locally small. (ii) I : X A is locally small, ten tere is some small map g : Y B and dense : a(b) A suc tat te ollowing diagram is a pullback a(y ) a(g) a(b) Proo. Since te components o te unit o te adjunction are dense maps, part (i) ollows rom Corollary and Lemma For part (ii), use te deinition o locally small map to get a dense map : B A and an indexed amily o subobjects Y B X B, togeter wit a diagram o orm Y X B X B X A A X suc tat Y B A X is dense. Ten, let g : Y B be te evident map, wic is small. Te required square can be readily obtained by recalling tat a : E E J preserves pullbacks, sends dense monomorpisms to isomorpisms, and preserves dense maps. A

26 26 S. AWODEY, N. GAMBINO, P. L. LUMSDAINE, AND M. A. WARREN Corollary allows us to regard te amily o locally small maps as te smallest amily o maps in E J containing te seaiications o te small maps in E and closed under descent along dense maps Lemma. (i) Identity maps are locally small. (ii) Composites o locally small maps are locally small. Proo. All identities are trivially locally small. For composition, suppose : X A and : X X are locally small. We construct te diagram Y C Z Y (5) Z (4) (3) Y X X (2) X C Y X g (1) B B p as ollows. We begin by applying Lemma to : X A so as to obtain te local quasi-pullback (1), were g : Y B is a small map and p : B A is an epimorpism. Ten, we construct te pullback (2) and obtain te locally small map : Y X X Y. We can apply Lemma to it so as to obtain te local quasi-pullback in (3), were : Z C is a small map and C Y is an epimorpism. Next, we apply te Collection Axiom to te small map Y B and te epimorpism C Y, so as to obtain (4), were Y B is a small map and B B is an epimorpism. Finally, (5) is a pullback and tereore Y C Z Y is small since : Z C is so. To conclude tat te composite o : X X and : X A is locally small, it is suicient to apply Lemma to te wole diagram, wic is a local quasi-pullback since it is obtained as te pasting o local quasi-pullbacks Lemma. For every commutative diagram o te orm d X X A i : X A is locally small and d : X X is dense, ten : X A is locally small. Proo. By Lemma we ave a local quasi-pullback g Y k X B A A

27 INTERNAL SHEAVES IN ALGEBRAIC SET THEORY 27 were g : Y B is a small map and : B A is an epimorpism. Since : X A is d : X A, we can expand te diagram above as ollows Y k g B A X B A X X d X B A Te map Y B A X is dense since it is te composite o te dense map Y B A X wit te pullback o te dense map d : X X. Hence, te commutative square g Y d k X B A is a local quasi-pullback, and so : X A is locally small Lemma. Let X, A, P be seaves. For every locally small map : X A and every dense map P X, tere exists a local quasi-pullback o te orm Y P X g B were g : Y B is locally small and : B A is dense. A Proo. Given suc a pair, we construct te diagram 1 Y X P Y X P P Z C Z Z Y X P Y (3) e (2) d Z (6) (4) C B Y X (5) C B A (1)

28 28 S. AWODEY, N. GAMBINO, P. L. LUMSDAINE, AND M. A. WARREN as ollows. Te commutative square (1) is obtained by applying Lemma to : X A. In particular, it is a local quasi-pullback. Diagram (2) is a pullback. To construct (3), irst we actor te dense map e : Y X P Y irst as an epimorpism Y X P Y ollowed by a dense monomorpism Y Y, and ten we actor te dense monomorpism as a small dense monomorpism Y B ollowed by an epimorpism B Y by Lemma We can ten apply te Collection Axiom in E to construct a diagram in (4) wic is a quasi-pullback. By te deinition o quasi-pullback and te act tat Y B is dense, it ollows tat Z C is a dense map since it is te composition o a dense map wit an epimorpism. We apply again te Collection Axiom to construct (5), and inally (6) is obtained by anoter pullback. Since te pasting o (1) and (5) is a local quasi-pullback and te map Z C Z Z is dense, te resulting diagram Z C Z P d X C is a local quasi-pullback as well. Note tat some o te objects in te diagram above need not be seaves, since tey ave been obtained by applying te Collection Axiom in E. To complete te proo, it suices to apply te associated sea unctor, so as to obtain te diagram A a(z C Z) P d X a(c ) A Tis provides te required diagram, since te associated sea unctor preserves dense maps and pullbacks and sends small maps into locally small maps Proposition. Te amily S J o locally small maps in E J satisies te axioms or a amily o small maps. Proo. Lemma proves Axiom (A1). Axiom (A2), asserting stability under pullbacks, olds by te very deinition o locally small map, as observed beore Deinition Axiom (A3) ollows by Proposition Axioms (A4) and (A5) ollow rom te corresponding axioms in E, using te act tat seaiication o small maps is locally small and te act tat te initial object and te coproducts o E J are te seaiication o te initial object and o coproducts in E, respectively. Lemma proves Axiom (A6), and Lemma proves Axiom (A7). Axiom (P1) ollows by Proposition and Lemma