Approximations of superstability in concrete accessible categories
aa r X i v : . [ m a t h . L O ] M a y APPROXIMATIONS OF SUPERSTABILITY INCONCRETE ACCESSIBLE CATEGORIES
M. LIEBERMAN AND J. ROSICK ´Y
Abstract.
We generalize the constructions and results of Chap-ter 10 in [2] to coherent accessible categories with concrete di-rected colimits and concrete monomorphisms. In particular, weprove that if any category of this form is categorical in a successor,directed colimits of saturated objects are themselves saturated. Introduction
A longstanding preoccupation in model theory is the problem ofdetermining, given a category of models K , the structure of the fullsubcategory Sat λ ( K ) consisting of λ -saturated models, where satura-tion is characterized variously in terms of the realization of syntac-tic types or, in more general contexts, of Galois types. In particular,there are a number of results on the conditions under which the cate-gory Sat λ ( K ) is closed under unions of increasing chains—which is tosay, closed under directed colimits —a pleasantly surprising property:in general, one could not reasonably hope for the union of a chain of λ -saturated models of small cofinality to be itself λ -saturated. Results of[15] and [7] guarantee that if K = Mod( T ) with T a superstable first-order theory, Sat λ ( K ) has precisely this property. In the literatureon abstract elementary classes, versions of this property— admitting λ -saturated unions , in the language of [2]—have typically been used asan analogue of superstability. Theorem 15.8 in [2] shows that for anyAEC K satisfying the amalgamation and joint embedding propertiesand containing arbitrarily large models, if K is κ -categorical for κ a reg-ular cardinal then it admits λ -saturated unions for any LS ( K ) < λ < κ .That is to say, in such an AEC, Sat λ ( K ) is closed under directed col-imits for any LS ( K ) < λ < κ . A recent result of [5] extends this tosuperstable tame abstract elementary classes with amalgamation, andshows, moreover, that if K is κ -categorical for κ sufficiently large, then Date : May 6, 2015.Supported by the Grant Agency of the Czech Republic under the grantP201/12/G028. in fact K admits λ -saturated unions for all sufficiently large λ . In-deed, they show that Sat λ ( K ) (which they denote by K λ − sat ) is itselfan AEC.In the context of accessible categories, λ -saturation is more usuallydefined as an injectivity condition: M ∈ K is λ -saturated if for anymorphism f : N → N ′ with N and N ′ λ -presentable (roughly, of sizeless than λ ) and for any g : N → M , there is a morphism h : N ′ → M such that hf = g . The structure of Sat λ ( K ), here interpreted as thefull subcategory of λ -saturated objects in an accessible category K ,has already been studied to some degree in [12], which gives conditionsunder which Sat λ ( K ) is itself accessible. Purely in terms of closure,it is clear that Sat λ ( K ) should be closed under λ -directed colimits,but there are not presently any results giving sufficient conditions forclosure under µ -directed colimits for µ < λ , let alone under arbitrarydirected colimits. This paper represents a first attempt at bridging thisgap. We will not work with general accessible categories, of course:our focus will be on a slight generalization of the κ -CAECs of [9],which we call weak κ -CAECs , in which we drop the assumptions ofrepleteness and iso-fullness included in the definition of κ -CAECs. Suchcategories fall at the extreme right of the following schematic diagramof generalizations: AECs = ℵ − CAECs ①①①①①①①①①①①①①①①①①①①①①① [8] ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯
Weak ℵ − CAECs [LR] mAECs ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ κ − CAECs [9] ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯
Weak κ − CAECs κ − AECs[5],[19] [10] ③③③③③③③③③③③③③③③③③③③③③③ κ − accessiblecategories PPROXIMATIONS OF SUPERSTABILITY IN CONCRETE ACCESSIBLE CATEGORIES3
We note that weak ℵ -CAECs are precisely the coherent accessi-ble categories with concrete directed colimits and concrete monomor-phisms considered in [8]. Results of [9] are stated for κ -CAECs, butare in fact valid for weak κ -CAECs. Here we deduce the existence ofuniversal extensions from stability in a general weak κ -CAEC, beforespecializing again to the case κ = ℵ . Ultimately, in Theorem 5.5, wewill prove an analogue of Theorem 10.22 in [2] in the latter context,guaranteeing the saturation of short chains (and, by extension, arbi-trary directed colimits) of saturated objects under the assumption ofcategoricity. This illustrates that the property of isomorphism fullnesswhich distinguishes AECs and weak ℵ -CAECs is not essential for thisresult. Moreover, we obtain our result by a more abstract toolkit thatshould enable further generalizations in the future.2. Weak κ -CAECs The broad framework in which we work is a twofold generalizationof the category-theoretic characterization of AECs given in [8], namelyas pairs ( K , U ) where K is a category and U : K →
Set is a faithfulfunctor such that • K is an accessible category with directed colimits all of whosemorphisms are monomorphisms. • ( K , U ) is coherent, and has concrete monomorphisms. • U preserves directed colimits. • K admits a replete, iso-full embedding into the category ofstructures in a canonical finitary signature Σ U derived from U .In [9], we note that the assumption that U preserves directed col-imits, i.e. that directed colimits are concrete, may be too limiting.Crucially, metric AECs, a generalization of AECs in which the under-lying objects of the models are not sets but complete metric spaces,do not fall within this scheme: although any mAEC K is closed underdirected colimits, the forgetful functor U : K →
Set does not preservedirected colimits. It does, however, preserve ℵ -directed colimits; thatis, ℵ -directed colimits are concrete. This leads naturally to the in-troduction of κ -concrete AECs (or κ -CAECs), which are pairs ( K , U )satisfying the axioms above with the exception that U is only requiredto preserve κ -directed colimits. M. LIEBERMAN AND J. ROSICK ´Y
In this account, we also drop the assumptions of repleteness and iso-fullness. This has already been done in some measure in [8], wherethey are explicitly dispensed with, and in [9], where they are includedin the definition of a κ -CAEC but are never used. Having dropped thisassumption, we pass from κ -CAECs to weak κ -CAECs : Definition 2.1.
We say that a pair ( K , U ) consisting of a category K and faithful functor U : K →
Set is a weak κ -concrete AEC , or weak κ -CAEC , if(1) K is accessible with directed colimits, and all of its morphismsare monomorphisms.(2) ( K , U ) is coherent, and has concrete monomorphims.(3) U preserves κ -directed colimits.When necessary, we incorporate the index of accessibility of K intoour notation: if ( K , U ) is a weak κ -concrete category with K λ -accessible,we say that ( K , U ) is a weak ( κ, λ ) -concrete AEC , or weak ( κ, λ ) -CAEC .We may occasionally abuse this notation by referring to K itself as aweak κ -CAEC, or weak ( κ, λ )-CAEC.We collect a few basic facts about weak κ -CAECs: Remark 2.2.
Let ( K , U ) be a large weak ( κ, λ )-CAEC.(1) Because K is λ -accessible and has all directed colimits, it is well accessible ; that is, K is µ -accessible for all regular µ ≥ λ ([3] 4.1). Note that this is not true for a general λ -accessiblecategory, e.g. Pos λ , the category of λ -directed posets and sub-structure embeddings, which is µ -accessible only in regular µ satisfying the sharp inequality µ D λ . (For more on the rela-tion D , see [1] 2.11, 2.12, and 2.13.)(2) The presentability rank of any object M in K is a successorcardinal, say | M | + . We call | M | the size of M . We note thatthis is a notion of size internal to the category K —the size ofan object M need not, in general, correspond to | U ( M ) | . Thatis, U need not preserve sizes. (See [3] 4.2.)(3) There exists a minimal cardinal λ U ≥ λ such that U preserves λ U -presentable objects. Indeed, one can show that U will pre-serve all sizes µ such that µ + ⊲ κ and µ + ≥ λ U . (See [9] 4.11(1) PPROXIMATIONS OF SUPERSTABILITY IN CONCRETE ACCESSIBLE CATEGORIES5 and 4.12)(4) K admits an EM-functor: there is a faithful functor E : Lin →K that is faithful and preserves directed colimits. Moreover,there is a cardinal λ E such that E preserves sizes µ for which µ + ≥ λ E . (See [9] 5.6.)We refer readers to [9] for further background, including a descrip-tion of Galois types, saturation, and stability in this context ([9] § Assumption 2.3.
All weak κ -AECs are assumed to be large (roughly,to have arbitrarily large objects), and to satisfy the amalgamation andjoint embedding properties. Moreover, we assume that there exists aproper class of cardinals λ with λ <λ = λ , to ensure the existence ofarbitrarily large monster objects, C .Note that weak ℵ -CAECs are precisely the coherent accessible cat-egories with concrete directed colimits and concrete monomorphismsthat are the principal focus of [8]. Because µ + ⊲ ℵ for any infinitecardinal µ , the statement on preservations of sizes in Remark 2.2(2)simplifies considerably: U preserves all sizes starting with λ U . Werequire one additional result for weak ℵ -CAECs, which appears asCorollary 7.9 in [8]. Theorem 2.4.
Let K be a large weak ℵ -CAEC. If K is λ -categoricalfor a regular cardinal λ ≥ λ U + λ E , then the unique object of size ν + issaturated. Universal Extensions
In our progress toward Theorem 5.5, we will essentially follow theargument of Chapter 10 in [2]. We begin by connecting stability firstto the existence of 1-special extensions, defined below, then to theexistence of universal extensions, which is of independent interest.The following is Definition 10.1 in [2], transferred to the currentcontext. We work exclusively with Galois 1-types—in the future, wewill simply refer to them as “types.”
Definition 3.1.
Given
M, N ∈ K both of size µ , we say that N is a1 -special extension of µ if there is a continuous chain of morphisms M → M → · · · → M i → . . . where M = M , M i +1 realizes all types over M i , and N = colim i<µ M i . M. LIEBERMAN AND J. ROSICK ´Y
Lemma 3.2.
Let ( K , U ) be a weak ( κ, λ ) -CAEC. If K is µ -stable for µ + ⊲κ and µ ≥ λ , then any object M of size µ has a -special extension.Proof. By µ -stability, there are at most µ types over M , say { ( f, a i ) | i <µ } with f : M → C . As C is a µ + -directed colimit of objects of size µ ,and such colimits are preserved by U , f factors through another objectof size µ , say as M → M → C , such that a i ∈ M for all i < µ .Then M realizes all types over M . We continue in this way, and takecolimits at limit stages: note that, by 1.16 in [1], the colimit of a chainof fewer than µ objects of size µ , i.e. µ + -presentable objects, mustbe µ + -presentable, hence itself of size µ . Thus we build a continuouschain h M i → M j | i < j < µ i , where M i +1 realizes all 1-types over M i for i < µ . The colimit of this chain—of size µ , by the reasoningimmediately above—is the desired 1-special extension. (cid:3) Lemma 3.3.
Let ( K , U ) be a weak ( κ, λ ) -CAEC. Given g : M → M ′ ,if M ′ realizes all types over M , then M ′ realizes all types over any N with f : N → M .Proof. By joint embedding and saturation of C , we have embeddings u : N → C , v : M → C , and w : M ′ → C . By Remark 4.3 in [8]—whichis stated there for accessible categories with concrete directed colimits,but whose proof does not in fact require any concreteness of colimitswhatsoever—there is an automorphism ¯ f of C extending f , in the sensethat the following diagram commutes: C ¯ f / / C N f / / u O O M v O O Without loss of generality, any type over N is of the form ( u, a ), with a ∈ C . We use ¯ f to transform this into a type over M : consider( v, U ¯ f ( a )). This type is realized in M ′ , meaning that there is b ∈ U ( M ′ ) and an automorphism s of C with U s ( U ¯ f ( a )) = U w ( b ), i.e. U ( s ¯ f )( a ) = U w ( b ), and such that the following diagram commutes: C s / / C M v O O g / / M ′ w O O PPROXIMATIONS OF SUPERSTABILITY IN CONCRETE ACCESSIBLE CATEGORIES7
Gluing the two commutative squares along v , we have s ¯ f u = wgf meaning that the automorphism s ¯ f witnesses the equivalence of ( u, a )and ( gf, b ), meaning that ( u, a ) is realized in M ′ . (cid:3) We wish to show that any 1-special extension m : M → ¯ M (subjectto certain size conditions) is universal in the following sense, againadapted only superficially from Definition 10.4 in [2]. Definition 3.4.
Given a subobject m : M → ¯ M , we say that ¯ M is µ -universal over M if for every h : M → N with M and N of size atmost µ there exists t : N → ¯ M such that th = f . If M and ¯ M areboth of size µ , we simply say that ¯ M is universal over M .The following generalizes a result first proven for AECs in [6]: Theorem 3.5.
Let ( K , U ) be a weak κ -CAEC. If m : M → ¯ M is a -special extension where M and ¯ M are of size µ with µ + ⊲ κ and µ ≥ λ U ,then ¯ M is universal over M .Proof. By assumption, ¯ M is the colimit of a continuous chain M = M m , / / M / / . . . / / M i m i,i +1 / / M i +1 / / . . . ¯ M with i < µ , where all Galois 1-types over M i are realized in M i +1 viathe embedding m i,i +1 : M i → M i +1 . Let h : M → N , with N also ofsize µ . We must construct an embedding t of N into ¯ M so that thefollowing triangle commutes: M m / / h ' ' ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ¯ MN t O O We employ a variant of the construction used in the proof of Proposi-tion 6.2 in [8], essentially replacing the Galois-saturated model K withthe chain M = M → M → . . . To begin, we note that N , being of size µ with µ + ⊲ κ and µ ≥ λ U , must satisfy | U ( N ) | ≤ µ (see Corollary 4.12in [9]). Hence we may enumerate U ( N ) \ ( U h )( U ( M )) as { a i | i < µ } .In fact, we define, for each i < µ , X i = ( U h )( U ( M )) ∪ { a k | k < i } To facilitate our work with types, we choose, using joint embedding,maps g : ¯ M → C and g : N → C . We build the desired mapfrom N to ¯ M element by element. In particular, we construct a chain M. LIEBERMAN AND J. ROSICK ´Y h n ij : N i → N j | i < j < µ i , and K -morphisms f i : N i → M i and v i : N i → C , together with a family of set maps t i : X i → U ( N i ). Webegin with N = M = M , f = Id M , v = g h , and t = ( U h ) − .Abusing notation slightly (the maps within and from the bottom roware set maps; the rest are in K ), we aim to build the following diagram: C M m , / / u M / / u ? ? . . . / / M i m i,i +1 / / u i Y Y ✸✸✸✸✸✸✸✸✸✸✸✸✸✸ M i +1 / / u i +1 m m . . . ¯ M g n n N n , / / f O O v : : N / / f O O v N N . . . / / N i n i,i +1 / / f i O O v i R R N i +1 / / f i +1 O O v i +1 d d . . . ¯ N ¯ f O O X t O O (cid:31) (cid:127) / / X t O O (cid:31) (cid:127) / / . . . (cid:31) (cid:127) / / X it i O O (cid:31) (cid:127) / / X i +1 t i +1 O O (cid:31) (cid:127) / / . . . U ( N ) t O O Here the maps u i : M i → C are just the compositions of the colimitmaps M i → ¯ M with g : ¯ M → C , meaning that the triangles of u and m morphisms commute automatically. We will ensure that all squarescommute and, in addition, that v i +1 n i,i +1 = v i .Suppose we have constructed up to the i th stage, i.e. we have N i , f i , v i , and t i . If a i ∈ U ( N i ), we take N i +1 = N i , and so on. Supposethat a i U ( N i ), and consider the type ( v i , U g ( a i )) over N i . Since wehave f i : N i → M i and M i +1 realizes all types over M i , Lemma 3.3implies that this type is realized in M i +1 . Hence there is b ∈ U ( M i +1 )so that ( v i , U g ( a i )) is equivalent to ( m i,i +1 f, b ). That is, there is anautomorphism s of C with su i +1 m i,i +1 f i +1 = v i and U ( su i +1 )( b ) = U ( g )( a i ). As u i +1 m i,i +1 = u i , we can simplify the first equation to su i f i , v i . Take N i +1 of size µ with c ∈ U ( N i +1 ) and morphisms n i,i +1 : N i → N i +1 and f i +1 : N i +1 → M i +1 so that f i +1 n i,i +1 = m i,i +1 f i and U f i +1 ( c ) = b . Set v i +1 = su i +1 f i +1 (this guarantees that the upper rowof squares commutes). Notice that v i +1 n i,i +1 = su i +1 f i +1 n i,i +1 = su i +1 m i,i +1 f i = su i f i = v i so the triangles commute as desired. Finally, we set t i +1 = t i ∪ { ( a i , c ) } .Notice that U v i +1 ( c ) = U ( su i +1 f i +1 )( c ) = U ( su i +1 )( b ) = U g ( a i )In fact, then, ( U v i +1 ) t i +1 = U g . PPROXIMATIONS OF SUPERSTABILITY IN CONCRETE ACCESSIBLE CATEGORIES9
While we might be inclined to simply take directed colimits in case i is a limit ordinal, there is a slight—very slight—wrinkle: such colim-its need not be concrete. We note, however, that we may take suchdirected colimits freely in the chains of N ’s and M ’s, and must only ac-count for the map t ′ i = S k
Corollary 3.6.
Let ( K , U ) be a weak ( κ, λ ) -CAEC. If K is µ -stable for µ + ⊲ κ and µ ≥ λ + λ U , then any object M of size µ has a universalextension. Or, more particularly,
Corollary 3.7.
Let K be an mAEC. If K is µ -stable for µ + ⊲ ℵ and µ ≥ LS d ( K ) + LS d ( K ) ℵ , then any model of density character µ has auniversal extension also of density character µ . We note that Theorem 2.11 in [21] establishes the latter result givenstability in any µ : our methods, which are fundamentally discreterather than metric, yield a weaker result in this case.4. Weak ℵ -CAECs: Saturation of EM-objects Recall that, by Remark 2.8 in [8], any coherent accessible categorywith directed colimits K whose morphisms are monomorphisms—andtherefore any weak ( κ, λ )-CAEC—admits an EM-functor E : Lin → K that is faithful, preserves directed colimits, and preserves sizes startingwith some cardinal λ E . In this section, we focus on the conditions on K and on a linear order I that ensure the EM-object E ( I ) is µ -saturated.The following technical result concerning the EM-functor is an ap-proximation of Lemma 10.11 in [2] in the context of weak ℵ -CAECs:for emphasis, we note again that these are equivalent to the coherentaccessible categories with concrete directed colimits considered in [8].Notice that this proof removes any reference to terms and signatures:purely category-theoretic properties of the functor E are sufficient. Proposition 4.1.
Let K be a weak ℵ -CAEC. Suppose K is λ -categoricalfor regular λ > λ U + λ E . For any linear order J with λ E < | J | < λ and the property that J contains an increasing sequence of length θ + for all λ E + λ U < θ < | J | , E ( J ) is saturated.Proof. Let J be a linear order with the property described above, andlet M = E ( J ). We wish to show that for any θ with λ E + λ U < θ < | J | , M is θ + -saturated. Take such a θ . By assumption, there is J ⊆ J oforder type θ + . Define J ′ = J + λ as the ordered sum. Then | J ′ | = λ and since λ > λ E , the object N = E ( J ′ ) is of size λ . Hence it is isomorphic to the categoricalobject—by Corollary 7.9 in [8], N is therefore saturated.Consider a subobject f : M → M , with M of size less than θ . Bycategoricity and joint embedding, there is an embedding g : M → N .Moreover, because N is saturated, any type over M is equivalent to( g, b ) for some b ∈ U ( N ). Notice that we may express J ′ as the directedcolimit of its finite suborders, J ′ = colim i ∈ I J ′ i . Since both E and U preserve directed colimits, U ( N ) = U E ( J ′ ) = U E (colim i ∈ I J ′ i ) = [ i ∈ I U E ( J ′ i )In particular, b ∈ U E ( J ′ i ) for some i ∈ I . As a finite suborder of J ′ , J ′ i = J ′ i, + J ′ i, , where J ′ i, ⊆ J and J ′ i, ⊆ λ , and both are finite. Take K with J ′ i, ⊆ K ⊆ J and | K | = θ so that U f ( U ( M )) ⊆ U E ( K ). As J is of order type θ + , there is room to choose J ′′ ⊆ J with K + J ′ i, order-isomorphic to K + J ′′ over K . This induces an isomorphism E ( K + J i, ) → E ( K + J ′′ ) over E ( K ), hence over U f ( U M ). Theimage of b under this isomorphism—call it a —lies in M and satisfies( f, a ) ∼ ( g, b ). (cid:3) By a slight generalization of this argument, we also have:
Proposition 4.2.
Let K be a weak ℵ -CAEC. Suppose K is λ -categoricalfor regular λ > λ U + λ E . For any linear order J with λ E < | J | < λ andthe property that J contains an increasing sequence of length θ + for all λ E + λ U < θ < µ for some µ ≤ | J | , E ( J ) is µ -saturated. We note that these propositions tell us a great deal about the satu-ration of EM-objects based entirely on the properties of the associatedlinear orders. The following is, essentially, Corollary 10.14 in [2]:
Corollary 4.3.
Let K be a weak ℵ -AEC, and suppose that K is λ -categorical for some regular λ > λ U + λ E . (1) For any µ with λ E + λ U < µ ≤ λ , E ( µ ) and E ( µ <ω ) are satu-rated. PPROXIMATIONS OF SUPERSTABILITY IN CONCRETE ACCESSIBLE CATEGORIES11 (2) If E ( I ) is µ -saturated with I a linear order satisfying the con-ditions of Proposition 4.2, then for any extension I of I , E ( I ) is µ -saturated as well. Weak ℵ -CAECs: Limit Objects and Saturation We now define the notion of a limit model in our context.
Definition 5.1.
Let h M i → M j | i ≤ j < δ i be a continuous chain ofobjects of size µ with δ ≤ µ + . We say that it is a ( µ, δ ) -chain if M i +1 µ -universal over M i . In case δ is a limit ordinal, we say the colimit ¯ M is a ( µ, δ ) -limit object .An analogue of saturated models, limit models were introduced in[14], and have subsequently been used in a number of contexts withinabstract model theory: in the analysis of categoricity in AECs with nomaximal models ([16] and [17]), for example. In particular, the unique-ness of limit models has been used heavily in the development of thestability theory of AECs and mAECs (e.g. in [4] and [19], respectively).This uniqueness appears as an axiom for good frames in [14], and hasbeen proven from superstability-like assumptions in AECs and mAECs(e.g. in [4] and [20], respectively). Here, we require only a very mildversion of uniqueness: Lemma 5.2.
Let ( K , U ) be a weak ℵ -CAEC. Let δ be a limit ordinalwith δ < µ + . Let ¯ M and ¯ N be ( µ, µ × δ ) -limits over M and N ,respectively, with M ∼ = N . Then ¯ M ∼ = ¯ N and, moreover, the chainsare term-by-term isomorphic on all terms indexed by α × i with α ≤ µ a limit ordinal and i < δ .Proof. The proof proceeds by a back-and-forth construction. As thecurrent context is slightly exotic, we sketch the argument. In particular,we will consider the chain up to M ω, , the first interesting case. Fornotational simplicity, we omit the second subscript in writing the chainmaps φ Mi,j : M i, → M j, and φ Mi : M i, → ¯ M , as well as the chain maps φ Ni,j and φ Nj , and the system of back and forth maps. Let f : M , → N , be an isomorphism. By universality of M , over M , , there isa morphism g : N , → M , such that g f = φ M , . By universalityof N , over N , , there is a morphism f : M , → N , such that f g = φ N , . The process continues, yielding the following diagram in which all triangles commute: M , f (cid:15) (cid:15) φ M , / / M , φ M , / / f (cid:15) (cid:15) M , . . . f (cid:15) (cid:15) M i, φ Mi,i +1 / / f i (cid:15) (cid:15) M i +1 , . . . f i +1 (cid:15) (cid:15) M ω, f ω (cid:20) (cid:20) N , φ N , / / g < < ①①①①①①①① N , φ N , / / g : : ✉✉✉✉✉✉✉✉✉ N , . . . N i, φ Ni,i +1 / / g i +1 ttttttttt N i +1 , . . . N ω, g ω T T Here f ω : M ω → N ω is the map induced by the morphisms φ Ni,ω f i : M i → N ω , and g ω : N ω → M ω is the map induced by the morphisms φ i +1 ,ω g i +1 : N i → M ω . Consequently, we have φ Ni,ω f i = f ω φ Mi,ω φ Mi +1 ,ω g i +1 = g ω φ Ni,ω as well as g i +1 f i = φ Mi,i +1 f i +1 g i +1 = φ Ni,i +1 by construction. We claim that f ω and g ω are inverses. To begin, wecompute: φ Mi +1 ,ω g i +1 = g ω φ Ni,ω φ Mi +1 ,ω g i +1 f i = g ω φ Ni,ω f i φ Mi +1 ,ω φ Mi,i +1 = g ω f ω φ Mi,ω φ Mi,ω = g ω f ω φ Mi,ω
Since the colimit maps φ Mi,ω form a jointly epimorphic family, it followsthat g ω f ω = Id M ω Similarly, f ω g ω φ Ni − ,ω = φ Ni − ,ω . Since the φ Ni − ,ω are jointly epimorphicas well, we have f ω g ω = Id N ω Beyond stage ω ×
0, we resume the back-and-forth sequence, and con-tinue this process. (cid:3)
In case K is categorical this gives a characterization of limits overEM-objects, which slightly generalizes Lemma 10.16(3) in [2]. Lemma 5.3.
Let ( K , U ) be a weak ℵ -CAEC that is λ -categorical forregular λ > λ U + λ E , and let λ E < µ < λ . Let δ < µ + be a limitordinal and let I = µ <ω . Every ( µ, δ ) -chain over E ( I ) is isomorphic to E ( I × δ ) .Proof. We first prove h E ( I × α ) | α < δ i is a ( µ, δ )-chain over E ( I ). Inparticular, we show that E ( I × ( α + 1)) is µ -universal over E ( I × α ).Consider f : E ( I × α ) → M with M of size µ . By saturation of E ( I × λ )—which follows from categoricity in λ , again by Corollary 7.9 PPROXIMATIONS OF SUPERSTABILITY IN CONCRETE ACCESSIBLE CATEGORIES13 in [8]—there is a morphism g : M → E ( I × λ ) such that the followingdiagram commutes: E ( I × α ) f / / ¯ i & & ▼▼▼▼▼▼▼▼▼▼ M g (cid:15) (cid:15) E ( I × λ )where ¯ i is the K -morphism induced by the inclusion i : I × α → I × λ .As I × λ can be expressed as a µ + -directed colimit of suborders I × α ⊆ Y ⊆ I × λ of size µ , and the functor E preserves both notions, E ( I × λ )is a µ + -directed colimit of the E ( Y ), all of which are of size µ . As M is µ + -presentable, g factors through some E ( Y ) → E ( I × λ ). So we have E ( I × α ) f / / ¯ j % % ❑❑❑❑❑❑❑❑❑❑ M g (cid:15) (cid:15) g ′ y y sssssssssss E ( Y ) % % ❑❑❑❑❑❑❑❑❑ E ( I × λ )where the upper triangle commutes. So M embeds in E ( Y ) over E ( I × α ), hence it suffices to show that E ( Y ) embeds in E ( I × ( α +1)). This issimple: Y consists of a disjoint union of Y ⊆ I × α and Y ⊆ I × λ \ I × α of size at most µ . By universality of I = µ <ω , Y embeds in the ( α +1)stcopy of I . So we have an induced morphism E ( Y ) → E ( I × ( α + 1))that fixes E ( I × α ). Hence h E ( I × α ) | α < δ i is a ( µ, δ )-chain over E ( I ), as claimed.The conclusion now follows from the uniqueness of ( µ, µ × δ )-limitsover E ( I ) proven in Lemma 5.2. (cid:3) We require one more flavor of uniqueness for limit objects, which,like Lemma 5.2 (or Lemma 10.8 in [2]), can be obtained via a simpleback-and-forth argument.
Lemma 5.4. If M is of size µ and ¯ M and ¯ N are, respectively, ( µ, δ ) -and ( µ, µ × δ ) -limits over M with δ < µ + , then ¯ M ∼ = ¯ N . Theorem 5.5.
Let ( K , U ) be a weak ℵ -CAEC that is categorical in λ + , and let λ U + λ E < χ < λ + with χ a limit cardinal. Then thecolimit of any continuous δ -chain of χ -saturated objects with δ < χ + is χ -saturated. Proof.
By Corollary 4.3, χ -saturated objects exist. Let N be the colimitof a chain h φ i,j : N i → N j | i ≤ j < δ i , where each N i is χ -saturated.Consider f : M → N with M of size κ < χ . We wish to show that anytype over M is realized in N .By concreteness of directed colimits, U ( N ) = [ i<δ U φ i ( U ( N i ))and U f ( U ( M )) is a subset. Let I = { i < δ | U φ i +1 ( U ( N i +1 )) ∩ U f (( U ( M )) \ U φ i ( U ( N i )) = ∅} Note that we can list this index set as I = { i α | α < δ ′ } where δ ′ < max( κ + , cf( δ ) + ) < χ. We can express U ( M ) as the union of an increasing δ ′ -chain of sets X α = { x ∈ U ( M ) | U f ( x ) ∈ U φ i ( U ( N i )) } For each α < δ ′ , let f α : X α → N i α such that ( U φ i α ) f α = U f . We builda continuous chain h ψ i,j : M α → M β | α ≤ β < δ ′ i such that each M i is κ + -saturated, each ψ α,α +1 : M α → M α +1 is µ -universal. Moreover, wewish to build a series of set maps t α : X α → U ( M α ) and K -morphisms g α : M α → N i α so that all cells of the following diagram commute: N i φ i ,i / / N i φ i ,i / / N i / / . . . NM ψ , / / g = Id O O M ψ , / / g O O M / / g O O . . . ¯ M g O O X / / t O O f V V X / / t O O f V V X / / t O O f V V . . . U ( M ) t O O f W W We begin with M = N i and g the identity, with t = f . By κ ++ -saturation of N , we can find ψ , : M → M and g : M → N i where M is of size κ + , is κ + -saturated, and ψ , : M → M is µ -universal. Moreover, we may do so in such a way as to ensure that f ( X ) ⊆ U g ( M ). Choose t so that ( U g ) t = f . Notice that f and f are compatible with φ , by design, meaning that the outer PPROXIMATIONS OF SUPERSTABILITY IN CONCRETE ACCESSIBLE CATEGORIES15 rectangle of the diagram below commutes, as does the upper square. N i φ i ,i / / N i M ψ , / / g = Id O O M g O O X / / t O O X t O O as all morphisms are mono, this implies that the lower square commutesas well. We proceed in the same way at each successor stage. For limit α , take colimits. The t α induce a set map t : U ( M ) → U ( ¯ M )), the g α induce a K -morphism g : ¯ M → N , and the construction ensures that( U g ) t = U f . By coherence, there is a K -morphism ¯ t : M → ¯ M so that U ¯ t = t .If κ < cf( δ ), M embeds in some M α +1 with α < δ ′ . By κ + -saturationof M α +1 , any type over M is realized in M α +1 , hence in ¯ M , hence in N . Otherwise, h M α | α < δ ′ i is a ( κ + , δ ′ )-chain over M . Without lossof generality (but possibly at the cost of deleting M ) we may replace M by E (( κ + ) <ω ). Being a ( κ + , δ ′ )-limit over E (( κ + ) <ω ),¯ M ∼ = E (( κ + ) <ω × δ ′ )by Lemma 5.3. Hence, by Corollary 4.3(2), ¯ M is κ + -saturated, and weare finished. (cid:3) As Sat χ ( K ), the full subcategory of K consisting of χ -saturated mod-els, is certainly closed under chains of length (or, rather, cofinality) atleast χ + , Theorem 5.5 implies that Sat χ ( K ) is closed under colimits ofarbitrary chains, hence, in fact, under arbitrary directed colimits: Corollary 5.6.
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