Tameness, Powerful Images, and Large Cardinals

TAMENESS, POWERFUL IMAGES, AND LARGE CARDINALS

WILL BONEY AND MICHAEL LIEBERMAN

Abstract. We provide comprehensive, level-by-level characterizations of large cardinals, in the range from weakly compact to strongly compact, by closure properties of powerful images of accessible functors. In the process, we show that these properties are also equivalent toboun various formsBT-R of tamenesslcatth for abstractLRclass elementary classes. This systematizes and extends results of [BU17], [BTR16], [Lie18], and [LR16].

1. Introduction

Recent years have seen the rapid development of a literature surrounding equivalences of large cardinal principles, locality properties of Galois types in abstract elementary classes (AECs), and properties of (powerful) images of accessible functors. This synthesis of set theory, abstract model boneylarge LRclass theory, and category theory has its roots in [Bon14] and [LR16]. The centerpiece result of the former is that if κ is a strongly compact cardinal, Galois types in any AEC essentially below κ are < κ-tame; that is, types are completely determined by their restrictions to < κ-sized submodels of their domains. The latter subsequently derived the same result, but by diﬀerent means—one can characterize equivalence of Galois types via the powerful image of an accessible functor, in which case < κ-tameness corresponds precisely to κ-accessibility (and accessible embeddability) of makkai-pare this powerful image. By an older result of [MP89], this κ-accessibility, too, follows from strong compactness of κ. BT-R This chain of implications can be tightened to form an equivalence. In [BTR16], a careful reworking makkai-pare of the argument of [MP89] reveals that accessibility of the powerful image of a λ-accessible functor F : K → L that preserves µL-presentable objects (µL a cardinal computed from λ and L, which we recall below) follows from the weaker assumption of µ -strong compactness of κ, or, if you L boun sh932 prefer, almost strong compactness of κ, µL < κ. The loop is closed in [BU17] (building on [She]), where the authors give a combinatorial construction that allows one to infer, subject to certain technical conditions, almost strong compactness of a cardinal κ from the fact that AECs below κ are < κ-tame. So we are left with an equivalence: boun Theorem 1.1 ([BU17, Corollary 4.14]). Let κ be an inﬁnite cardinal with µω < κ for all µ < κ. The following are equivalent: (1) κ is almost strongly compact. (2) The powerful image of any λ-accessible functor F : K → L, µL < κ, is κ-accessible and κ-accessibly embedded in L. (3) Every AEC K with LS(K) < κ is < κ-tame.

Date: April 2, 2019 AMS 2010 Subject Classiﬁcation: Primary: 03E55, 18C35. Secondary: 03E75, 03C20, 03C48, 03C75. Key words and phrases. almost measurable cardinals, accessible categories, abstract elementary classes, Galois types, locality. The second author was supported by the Grant Agency of the Czech Republic under the grant P201/12/G028. 1 2 W. BONEY AND M. LIEBERMAN

boneylargeboun There is a great deal more to be said, however. In conjunction with [Bon14], [BU17] gives a much broader and subtler array of equivalences between gradations of strong compactness (respectively, measurability) and gradations of tameness (respectively, locality). One can also ﬁne-tune the ar- BT-R lcatth gument of [BTR16] to match these ﬁner gradations. As noted in [Lie18], for any reasonably large cardinal κ, the powerful image of an accessible functor F : K → L with µL < κ will always be κ-preaccessible—what changes, depending what precise kind of large cardinal κ is, are the kinds of lcatth colimits under which the powerful image of F is closed in L. While [Lie18] is concerned with almost measurable cardinals κ, closure of the powerful image under κ-chains, and κ-locality of Galois types, we refocus on compactness. framework-sec After a brief review of the terminology involved, Section 3 provides a general framework for the arguments that connect large cardinals, Galois type locality, and accessibility of powerful images. awcsec In Section 4, we apply this framework to show that a cardinal κ is δ-weakly compact just in case accessible functors below κ have powerful images closed under κ+-small κ-directed colimits of κ- awc-thm ascsec presentable objects, and just in case AECs below κ are (< κ, κ)-tame (Theorem 4.1). In Section 5, nearlystrongcptequiv we prove a similar, level-by-level equivalence for (δ, θ)-strong compact cardinals (Theorem 5.3).

2. Preliminaries

We consider large cardinals around the notions of weak compactness and strong compactness. Many of the definitions that are equivalent globally become separate when viewed on a level-by-level basis. We add the adjective ‘logically’ to denote that we are picking out the definitions in terms of compactness of logics. Definition 2.1. Let κ be an uncountable cardinal.

(1) (a) κ is logically δ-weakly compact if every < κ-satisfiable theory in Lδ,δ theory of size κ is satisfiable. (b) κ is almost logically weakly compact if it is logically δ-weakly compact for all δ < κ. (c) κ is logically weakly compact if it is logically κ-weakly compact. (2) (a) κ is logically (δ, λ)-strongly compact for δ ≤ κ ≤ λ if every < κ-satisfiable theory in Lδ,δ in a language of size λ is satisfiable. (b) κ is logically (δ, ∞)-strongly compact if it is logically (δ, λ)-strongly compact for all λ ≥ κ. (c) κ is logically λ-strongly compact if it is logically (κ, λ)-strongly compact. (d) κ is almost logically strongly compact if it is logically (δ, ∞)-strongly compact for all δ < κ. (e) κ is logically strongly compact if it is logically (κ, ∞)-strongly compact. kanamori Remark 2.2. As can be seen in standard texts (e.g., [Kan03]), the logical global versions (‘logically weakly compact,’ ‘logically strong compact,’ etc.) agree with the standard global versions (‘weakly compact,’ ‘strong compact,’ etc.). On a level-by-level basis (where the relevant definitions can be boun h-partial-strong found in [BU17, Definition 2.1], things separate a bit more. Hayut [Hay] studies this behavior when δ = κ, and the same arguments work in δ < κ case. Consider the following three notions: cto-set ((∗)δ,κ,λ) κ is logically (δ, λ)-strongly compact cto-set ((∗∗)δ,κ,λ) any κ-complete filter on λ can be extended to a δ-complete ultrafilter cto-set ((∗ ∗ ∗)δ,κ,λ) there is a δ-complete, fine ultrafilter on Pκ, λ TAMENESS, POWERFUL IMAGES, AND LARGE CARDINALS 3

Then, if δ < κ ≤ λ = λ<κ, we have1

(∗ ∗ ∗)δ,κ,2λ =⇒ (∗∗)δ,κ,λ ⇐⇒ (∗)δ,κ,2λ =⇒ (∗ ∗ ∗)δ,κ,λ We make use of the logical characterizations of these cardinals, so prefer this deﬁnition.

adamek-rosicky We recall the following terminology related to accessible categories, and refer readers to [AR94] and makkai-pare [MP89] for further details: Deﬁnition 2.3. Let λ be a regular cardinal. (1) The colimit of a diagram D : I → K in a category K is λ-directed if the underlying poset I is λ-directed, i.e. any J ⊂ I with |J| < λ has upper bound in I. (2) An object M in a category K is λ-presentable if the associated hom-functor

HomK(M, −): K → Set preserves λ-directed colimits. (3) A category K is λ-preaccessible if • it contains, up to isomorphism, a set of λ-presentable objects, and • any object in K is a λ-directed colimit of λ-presentable objects. (4) A category K is λ-accessible if it is λ-preaccessible and closed under λ-directed colimits. We say K is accessible if it is λ-accessible for some λ. Remark 2.4. In an accessible category K, every object M is λ-presentable for some λ. We define the presentability rank of M to be the least such λ. remsharpclosed Remark 2.5. Recall that accessibility (and preaccessibility) do not pass upward as nicely as one might hope: in general, a λ-accessible category K is µ-accessible, µ > λ regular, if and only if µ is sharply greater than λ, denoted µ . λ. The difficulty is related to, and commensurate with, that of ensuring that a subset of cardinality µ in a λ-directed poset can be completed to a λ-directed set makkai-pare also of cardinality µ. This sharp inequality relation, defined in [MP89, 2.3.1], can be reduced to the internal-sizes-v2 following (see [LRV, 2.5]): if µ is λ-closed, i.e. θ<λ < µ for all θ < µ, then µ . λ. When µ > 2<λ, the converse holds. Remark 2.6. We recall an important parameter, µ , associated with each accessible category K BT-R K ([BTR16]). Fix λ such that K is λ-accessible—in fact, µK will depend on λ, but we do not include it explicitly in the notation: we trust that no confusion will result. Let Presλ(K) denote a full subcategory of K containing exactly one representative of each isomorphism class of λ-presentable objects, β = | Presλ(K)|. Let γK be the smallest cardinal with γK ≥ β and γK D λ. Define

<γK + µK = (γK ) .

By design, µK D γK D λ. As we will see, this parameter gives a kind of measure of K, giving an upper bound on the linguistic resources needed to describe K as a well-behaved class of structures.

defaccbelow Deﬁnition 2.7. We say that an accessible category K is below κ, κ a cardinal, if µK < κ. We say K is sharply below κ if µK / κ.

functsdef Deﬁnition 2.8. Let λ be a regular cardinal. (1) We say that a functor F : K → L is λ-accessible if K and L are λ-accessible, and F preserves λ-directed colimits.

1 h-partial-strong λ See [Hay], especially Theorem 3 there; Hayut uses ‘Lκ,κ-compactness for languages of size 2 ’ to refer to (∗∗)κ,κ,λ. 4 W. BONEY AND M. LIEBERMAN strongacc-functsdef (2) We say that F : K → L is strongly λ-accessible if it is λ-accessible and preserves λ- presentable objects.2 accemb-functsdef (3) Given a subcategory K of a category L, we say K is λ-accessibly embedded in L if it is full and K is closed under λ-directed colimits in L. Remark 2.9. If a subcategory embedding K ,→ L is strongly λ-accessible, any object in K can be expressed as a λ-directed colimit in L of λ-presentable objects of L that happen to be in K— something like saying that K is λ-preaccessible in the sense of L. Closure of K in L under λ-directed colimits, on the other hand, does not follow: this will be of critical importance in the sequel. In any case, this means that—in an unfortunate quirk of the existing notation—strong λ-accessibility of the embedding K ,→ L does not imply K is λ-accessibly embedded in L. Deﬁnition 2.10. We say that a λ-accessible functor F is below (respectively, sharply below) κ if µL < κ (respectively, µK / κ).

We will primarily be concerned with powerful images of accessible functors, but we will also make occasional reference to two related notions.

defimgs Deﬁnition 2.11. Let F : K → L be a functor. (1) The full image of F is the full subcategory of L on objects FA, A ∈ K. imgsieve-def (2) We denote by S(F ) the maximal sieve on the image of F , i.e. the full subcategory of L on objects B ∈ L that admit L-morphisms B → FA, A ∈ K. powim-def (3) The powerful image of F , denoted P(F ), is the closure of the full image of F under L- imgsieve-def subobjects: as in (2), then, but with monomorphisms in place of general morphisms. (4) The λ-pure powerful image of F , denoted P (F ), is the closure of the full image of F under adamek-rosicky λ powim-def λ-pure subobjects (see [AR94, 2.27]): as (in 3), but with λ-pure monomorphisms in place of monomorphisms.

We also review a few of the ideas we need in connection with abstract elementary classes (AECs). shelahaecs First defined in [She87], AECs are a semantic (or, if you like, category-theoretic) abstraction of the classes of models and embeddings arising in well-behaved nonelementary classes, such as those axiomatizable in Lλ,ω or, indeed, in the elementary classes from finitary first-order logic. Crucially, AECs retain, in the structure of their class of designated embeddings, certain essential properties of these more elementary cousins. For our purposes, it is enough, perhaps, to recall that an AEC K has arbitrary directed colimits of K-embeddings, and that it has an associated Löwenheim-Skolem number LS(K) such that any object in K is a LS(K)+-directed colimit of LS(K)+-presentable objects. Here the presentability rank of an object M ∈ K is precisely |M|+, i.e. the successor of the cardinality of the underlying set of M. Notation 2.12. Here (and in general) we do not distinguish notationally between an object M ∈ K and its underlying set. We denote by Kλ the set of all M ∈ K with |M| = λ, and define K<λ, K≤λ in the obvious way.

Remark 2.13. Notice that for any AEC K and λ > LS(K), the full subcategory of K on K<λ is equivalent to Presλ(K).

In AECs, the syntactic types familiar from ﬁrst-order model theory are replaced by Galois (or orbital) types. Classically, a Galois pretype over M ∈ K consists of a triple (M, a, N), with N K M and a ∈ N; equivalently, we may deﬁne pretypes to be pairs (f, a), with f : M → N a K-embedding chornros 2This handy piece of terminology has not seen much use: see e.g. [CR12, 2.7] (in the class-accessible context). TAMENESS, POWERFUL IMAGES, AND LARGE CARDINALS 5 and a ∈ N.A Galois type over M is an equivalence class of pretypes, under the transitive closure of the following relation of atomic equivalence:(M, a1,N1) ≡AT (M, a2,N2) if and only if there are K-embeddings gi : Ni → N, i = 1, 2, with g1 M = g2 M and g1(a1) = g2(a2). If we adopt the more category-theoretic formulation, pretypes (f1, a1) and (f2, a2) are equivalent if the pointed span

f1 f2 a1 ∈ N1 ← M → N2 3 a2 can be extended to a commutative square

g1 N1 / N O O

f1 g2

M / N2 f2 with g1(a1) = g2(a2). We toggle between the two viewpoints, but largely employ the standard model- theoretic formulation. Note that if K has the amalgamation property, ≡AT is already transitive, so the equivalence notions coincide. The notion of tameness of Galois types in an AEC K—essentially the requirement that equivalence of pretypes over any M is determined by restrictions to K-substructures of M of some fixed small grovantame size—was first isolated in [GV06], and has come to play an important role in the development of the classification theory of AECs. We consider several parameterizations of this notion.

Deﬁnition 2.14. Let K be an AEC, and κ ≤ λ.

(1) We say that K is (< κ, λ)-tame if for every M ∈ Kλ and types p 6= q over M, there is M0 K M of size less than κ such that p M0 6= q M0. (2) We say K is κ-tame if it is (< κ, µ)-tame for all µ ≥ κ.

We also introduce a related notion: atomic tameness, which is the tameness property for atomic equivalence ≡AT of pretypes. As noted above, this will coincide with conventional tameness in AECs with amalgamation, but we wish to avoid that assumption wherever possible. In any case, we awcsec ascsec will see in Sections 4 and 5 that global assertions of atomic tameness (i.e. “all AECs are atomically tame...”) are equivalent to global assertions of tameness.

Deﬁnition 2.15. Let K be an AEC, and κ ≤ λ.

(1) We say that K is (< κ, λ)-atomically tame if for every M ∈ Kλ and pretypes (M, a1,N1) 6≡AT (M, a2,N2) over M, there is M0 K M of size less than κ such that (M0, a1,N1) 6≡AT (M0, a2,N2). (2) We say K is κ-atomically tame if it is (< κ, µ)-atomically tame for all µ ≥ κ.

It is important to note that, in the category-theoretic formulation, everything that we have described here (and, indeed, all the arguments below) will go through in contexts much more general than mu-aec-jpaa AECs. In particular, everything generalizes in straightforward fashion to µ-AECs ([BGL+16]), which are, morally speaking, just accessible categories whose morphisms are monomorphisms: A µ-AEC K with L¨owenheim-Skolem-Tarski number LS(K) = λ is a λ+-accessible category with all morphisms monomorphisms, and a λ-accessible category K0 whose morphisms are monomorphisms mu-aec-jpaa is equivalent to a λ-AEC K with LS(K) = µ ([BGL+16]). This is the level of generality involved LRclass LiRo17 lcatth K in, e.g. [LR16], [LR17], and [Lie18]. 6 W. BONEY AND M. LIEBERMAN

3. General criteria framework-sec makkai-pare, LRclass, BT-R, lcatth We extract some general points from the work of [MP89, LR16, BTR16, Lie18] to aid in our analysis. There are two main motivations for this. The ﬁrst is to create a general framework for variations on awc-thm nearlystrongcptequiv these results (which we exploit in later sections to obtain, e.g. Theorems 4.1 and 5.3). The second is to emphasize which part of these constructions are tied to the large cardinal properties, and which are true in ZFC. Note that there is no ‘tameness to compactness’ discussion, as this was already boun done in a modular manner in [BU17].

3.1. Compactness to powerful images. Given a λ-accessible category K, one can readily (see, adamek-rosicky λ λ λ e.g. [AR94, 4.18, 5.33]) build a language τK and a theory TK ⊂ LµL,µL (τK) such that op λ Presλ(K) • Str τK is essentially the functor category Set ; λ λ • there is a full embedding EK : K → Str τK, given by the canonical embedding of K into op SetPresλ(K) ; and λ λ • EK induces an equivalence between K and Mod TK. Note that we have decorated these notions with the accessibility cardinal that we are concerned with, but one can typically omit this in practice. In our set-up, we have the following data: λ-accessible categories K and L, and a λ-accessible functor F : K → L that preserves µL-presentable objects—note that, since µL D λ, F is strongly µL-accessible. From this, we form the comma category CF = IdL ↓ F , whose objects are morphisms f : L → FK (for K ∈ K, L ∈ L) and the arrows between objects f : L → FK and f 0 : L0 → FK0 are pairs of morphisms l : L → L0 and k : K → K0 with F (k)f = f 0l. In fact, we will want to require more—namely that the objects are monomorphisms L → FK (respectively, λ-pure monomorphisms) to get the powerful image (respectively, λ-pure powerful image) of F . We postpone this discussion for the moment. adamek-rosicky The category CF is µL-accessible ([AR94, 2.43]). Consider the restriction of the domain projection → functor D : L → L to the subcategory CF , H : CF → L, that takes each f : L → FK to L. Note: (1) Since D preserves λ-directed colimits and µ -presentable objects, so does H. λ defimgsimgsieve-def (2) The full image of H is precisely the subcategory S(F ) of L (Deﬁnition 2.11(2)). So, in thinking about S(F ), we can think instead about the full image of H. In fact, we translate once more, using the syntactic characterization of accessible categories described above to replace H with a reduct functor, R, and realize the full image of H—and therefore S(F )—as a well-behaved projective class of structures, susceptible to analysis via logical compactness.

The µL-accessibility of CF gives rise to a functor ECF into an associated category of structures, as above, on general grounds. However, the key is that this functor (and the corresponding language and theory) can be written in terms of those of K and L. In particular, given f : L → FK and x : L0 → L for L0 ∈ L<λ, λ-presentability of L0 means that the arrow f ◦ x : L0 → FK factors through one of the objects in the canonical decomposition of K as a λ-directed colimit of λ-presentables. That means there is an essentially unique y : K0 → K and h : L0 → FK0 such λ λ L0 K0 that f ◦ x = F y ◦ hf . Thus, we add to τK∪˙ τL a relation Rh ⊂ S × S that holds in exactly this 3 λ λ circumstance. Then we add to the theory TK∪˙ TK the sentences (for each L0 ∈ L<λ and K0 ∈ K<λ)

L0 K0 _ (?) ∀x ∈ S ∃y ∈ S Rh(x, y)

h:L0→FK0 makkai-pareBT-R 3We use essentially the same coding as in [MP89] or [BTR16], which agree in most respects. TAMENESS, POWERFUL IMAGES, AND LARGE CARDINALS 7

λ λ Call this language τF and the theory TF . By deﬁnition of µL, this can all be done in LµL,µL . λ λ Let R be the reduct functor R : Mod TF → Mod TL , which simply forgets the interpretations of λ λ BT-R the symbols in τF \ τL. This functor is strongly µL-accessible, [BTR16, p. 9]. In model-theoretic λ λ terminology, the image of this functor is precisely the pseudo-elementary class PC(TF , τL). This λ λ means that, given M TL , we have that M ∈ imR if and only if it has an expansion realizing TF . At long last, we connect this discussion to compactness. Recall that, if N is a τ-structure, then τ(N) := τ∪{˙ cn : n ∈ N} is the language where a constant for each element of N has been added. λ + λ Given M ∈ Str τL, we set TM := CD (M) to be the Lω,ω τL(M) -theory consisting of the positive, + quantiﬁer-free sentences true in M. Then N CD (M) if and only if there is a homomorphism N from M to N, and this homomorphism is induced by taking m ∈ M to cm ∈ N. Thus, we have the following chain of equivalences:

λ λ M ∈ imR ⇐⇒ M has an expansion that models TF ⇐⇒ TF ∪ TM is satisﬁable

This ﬁnal piece is the syntactical formulation that we will use.

As noted above, we must adjust our deﬁnition of CF if we wish to obtain the powerful or λ-pure powerful image of F .

powsetup (1) (Powerful image, P(F )): We are interested in monomorphisms L → FK, which also raptisros chornros form a µL-accessible category ([RR15, Pseudopullback Theorem], [CR12, 3.1]), here denoted CF,mono. We may proceed more or less as above, again realizing P(F ) as the full image of a suitable restriction of the domain projection, and translating into structures. Note, though, that we must now ensure syntactically that there is a monomorphism from M into a model of TF . To achieve this, we expand TM to mono 0 TM = TM ∪ {cm 6= cm0 : m 6= m ∈ M}

lpowsetup (2) (λ-pure powerful image, Pλ(F )) The story is similar, with “λ-pure monomorphism” in place of “monomorphism,” and with the resulting arrow category denoted by C . BT-R F,λ−pure This category is µL-accessible, once again, by [BTR16, p. 8]. Here we replace TM by the λ-pure diagram of M, which consists of all positive-primitive and negated positive-primitive λ formulas in Lλ,λ τL(M) that are true in M (recall that a formula is positive primitive if it is of the form ∃xψ¯ (¯x), where ψ is a conjunction of atomic formulas).

λ In either case, we must also change the additional sentence in TF , namely that in the displayed equation (?) above. In particular, we simply restrict the disjunction to be over monomorphisms λ h : L0 → FK0 (respectively, λ-pure morphisms h : L0 → FK0), obtaining a new theory TF,mono λ λ (respectively, TF,λ−pure) for use in place of TF . We conclude with a useful lemma. prelim-lem Lemma 3.1. Suppose that K is the colimit of a δ-directed diagram D : I → K with |I| + λ.

λ λ (1) if X ⊂ EK(K) of size < δ, there is i∗ ∈ I such that X ⊂ EK (D(i∗)). λ (2) |EK(A)| = λ + δ.

The proof is straightforward. The following also reduces the work necessary in our later results, and follows from the syntactic characterization and strong µL-accessibility of the reduct functor, R: 8 W. BONEY AND M. LIEBERMAN strongacc-lem Lemma 3.2. For any λ-accessible functor F : K → L that preserves µL-presentable objects, the inclusion of S(F ) in L is strongly κ-accessible for all κ D µL. The same holds for P(F ) and Pλ(F ). awcsecascsec Crucially, this holds in ZFC. The large cardinal assumptions considered in Sections 4 and 5 will guar- antee partial closure under colimits of these categories in L; that is, the large cardinal properties of κ will determine how close these subcategories come to being κ-accessibly embedded (Deﬁni- functsdefaccemb-functsdef tion 2.8(3)).

3.2. Powerful images to tameness. Fix an AEC K with LS(K) < λ. In order to properly code Galois types in our framework, we deﬁne two auxiliary AECs:

< LRclass (1) K consists of all Galois pretypes over the same domain (this is L2 in the context of [LR16, < 5.2]). Structures in K consist of (M,N1,N2, a1, a2) in τ(K) ∪ {R,R1,R2, c1, c2} such that (a) M ≺K N` for ` = 1, 2 (b) a` ∈ N` ∗ ∗ ∗ ∗ ∗ Then (M,N1,N2, a1, a2) ≺< (M ,N1 ,N2 , a1, a2) if and only if ∗ ∗ (a) M ≺K M and N` ≺K N` ∗ (b) a` = a` (2) consists of all witnesses to atomic equality of pretypes (this is L in the context of KLRclass 1 [LR16, 5.2]). Structures in K consist of tuples (M,N1,N2,N+, a1, a2, f1, f2) in τ(K) ∪ {R,R1,R2,R+, c1, c2,F1,F2} such that < (a) (M,N1,N2, a1, a2) ∈ K and N+ ∈ K (b) f` : N` → N+ is a K-embedding (c) f1, f2 agree on M and f1(a1) = f2(a2) Then the following is straightforward: Proposition 3.3. Let K be an AEC. Then K< and K are AECs with LS(K<) = LS(K) = LS(K). Moreover, considering these AECs as categories in the obvious way, µ < = µ = µ . K K K

< We also build UK : K → K by forgetting the extra structure. This functor preserves directed colimits, but it is by no means clear that its image is an AEC or even closed under directed K < colimits. Indeed, write EAT for the full image of UK in K . This notation is suggestive because K (M,N1,N2, a1, a2) ∈ EAT if and only if

(a1,M,N1) ≡AT (a2,M,N2)

This category is closed under subobjects in < so is also the powerful image of U . The crucial LRclass K K K observation of [LR16] is that (under amalgamation at least) the closure of EAT under colimits of certain kinds is intimately connected to the type locality properties (tameness, locality, etc.) of . lcatth K We revisit this in later sections (building also on [Lie18]).

4. Almost weakly compact awcsec Recall that a colimit is said to be κ-small if its diagram is of cardinality less than κ. awc-thm Theorem 4.1. Suppose δ = δω < κ = κδ. The following are equivalent: (1) κ is logically δ-weakly compact. (2) If F : K → L is λ-accessible and preserves µL-presentable objects, µL / δ, then the inclusion P(F ) ,→ L is strongly κ-accessible and P(F ) is closed in L under κ-small κ-directed colimits of κ-presentable objects. TAMENESS, POWERFUL IMAGES, AND LARGE CARDINALS 9

(3) Every AEC K with LS(K) < δ is (< κ, κ)-atomically tame. (4) Every AEC K with LS(K) < δ is (< κ, κ)-tame. boun boun Proof: (3) =⇒ (1): The proof of [BU17, Theorem 4.9.(1)] uses an AEC with amalgamation ([BU17, Claim 4.7]), so atomic tameness suffices: it is equivalent to tameness in such classes. strongacc-lem (1) =⇒ (2): Strong λ-accessibility of the inclusion follows from Lemma 3.2. Fix D : I → P(F ) ⊂ L as in the hypothesis. Since I is κ-directed, it is also λ-directed, so there is a colimit A ∈ L. Then prelim-lem EL(A) is a τL-structure and |EL(A)| = κ by Lemma 3.1. Thus T mono has size κ, in which case T mono ∪ T mono is an (τ (E (A))∪˙ τ )-theory of size κ, EL(A) EL(A) F LµL,µL L L F so we are in a position to use δ-weak compactness of κ to prove its satisfiability. We need to show that every < κ-sized subset of this theory is satisfiable. mono mono mono Let T0 ⊂ T ∪ T of size < κ. Write T1 for T0 ∩ T . Then there is some X ⊂ EL(A) of EL(A) F prelim-lemEL(A) size < κ that contains every cm in T1. By Lemma 3.1, there is i ∈ I so im ≤ i for all cm ∈ X.

Pushing back through EL, this means that T1 ⊂ TEL(D(i)). But then T ⊂ T mono ∪ T mono 0 EL(D(i)) F which is satisﬁable by virtue of the fact that D maps to P(F ). Thus, by the δ-weak compactness of κ, T mono ∪ T mono is satisﬁable and E (A) ∈ PC(T , τ ). So EL(A) F L F L A ∈ P(F ). 3 (2) =⇒ (3): Fix such an AEC K.(a1,M,N1), (a2,M,N2) ∈ Kκ such that

(a1,M0,N1) ≡AT (a2,M0,N2) < for every M0 ∈ K<κ with M0 ≺ M. Note that, in K ,(a1, a2,M,N1,N2) is the κ-directed colimit of the collection of (a1, a2,M0,N1,N2) for M0 ∈ I := {M0 ∈ K<κ : M0 ≺ M} (this is its canonical cocone up to isomorphism), and each such (a1, a2,M,N1,N2) is in the image of UK. So we have a D as in (2). Thus, (a1, a2,M,N1,N2) is in the image of UK, as desired. boun (4) =⇒ (1): This is [BU17, Theorem 4.9.(3)]. boneylarge (1) =⇒ (4): This is [Bon14, Theorem 6.4]. †

prelim-lem By a similar argument, and the discussion preceding Lemma 3.1, the same holds if we replace the powerful image with Pλ(F ), the λ-pure powerful image of F , or with S(F ), the maximal sieve on the image of F .

5. Level-by-level almost strong compact ascsec nearlystrongcptequiv Theorem 5.1. Let δ be an inaccessible cardinal, and θ = θ<κ. Each of the following statements implies the next:

(1) κ is logically (δ, θ)-strong compact. (2) If F : K → L is λ-accessible, preserves µL-presentable objects, and µL < δ, then the inclusion P(F ) ,→ L is strongly λ-accessible and P(F ) is closed under θ+-small κ-directed colimits of κ-presentables. (3) Every AEC K with LS(K) < δ is (< κ, θ)-atomically tame. (4) Every AEC K with LS(K) < δ is (< κ, θ)-tame. 10 W. BONEY AND M. LIEBERMAN

boun Proof. (3) =⇒ (1): As before, atomic tameness suﬃces in the proof of [BU17, Theorem 4.9.(3)] since the AEC under consideration has amalgamation. strongacc-lem (1) =⇒ (2): Strong λ-accessibility of the inclusion follows from Lemma 3.2, as before. Fix D : I → P(F ) ⊂ L as in the hypothesis. I is κ-directed, has colimit A ∈ L. Then E (A) is a τ -structure prelim-lem L L and |E (A)| = θ by Lemma 3.1, meaning that T mono ∪ T mono is in a language of size θ. L EL(A) F mono mono mono Let T0 ⊂ T ∪ T of size < κ. Write T1 for T0 ∩ T . Then there is some X ⊂ EL(A) of EL(A) F prelim-lemEL(A) size < κ that contains every cm in T1. By Lemma 3.1, there is i ∈ I so im ≤ i for all cm ∈ X.

Pulling back through EL, this means that T1 ⊂ TEL(D(i)). But then T ⊂ T mono ∪ T mono 0 EL(D(i)) F which is satisﬁable by virtue of the fact that D maps to P(F ). By the (δ, θ)-strong compactness of κ, then, T mono ∪ T mono is satisﬁable and E (A) ∈ PC(T , τ ). EL(A) F L F L So A ∈ P(F ). 3 (2) =⇒ (3): Fix such an AEC K. Let (a1,M,N1), (a2,M,N2) ∈ K such that

(a1,M0,N1) ≡AT (a2,M0,N2) < for every M0 ∈ K<κ with M0 ≺ M. Note that, in K ,(a1, a2,M,N1,N2) is the κ-directed colimit of the collection of (a1, a2,M0,N1,N2) for M0 ∈ I := {M0 ∈ K<κ : M0 ≺ M} (this is its canonical cocone, up to isomorphism), and each such (a1, a2,M0,N1,N2) is in the image of UK. So we have a D as in (2). Thus, (a1, a2,M,N1,N2) is in the image of UK, as desired. †

boun (4) =⇒ (1): This is [BU17, Theorem 4.9.(1)]. boneylarge (1) =⇒ (4): This is [Bon14, Theorem 4.5]. awc-thm As noted after Theorem 4.1, we may replicate this argument in the λ-pure powerful case, as well. nearlystrongcptequiv Note that we do not have a perfect equivalence in Theorem 5.3. The following would close the loop, <κ but the cardinal arithmetic is just oﬀ (since δ(θ ) ≥ 2θ > θ).

boun <κ bu-str-fact Fact 5.2 ([BU17, Theorem 4.9.(3)]). If every AEC K with LS(K) = δ is (< κ, δ(θ ))-tame, then κ + is (δ , θ)-strong compact (in the sense that (∗ ∗ ∗)δ+,κ,θ holds).

This allows us to give a nice characterization at κ-closed, strong limit cardinals. Adopting the notational convention that a class is (µ, < χ)-tame if it is (µ, χ0)-tame for all µ ≤ χ0 < χ, we have: nearlystrongcptequiv Theorem 5.3. Let δ be an inaccessible cardinal, and θ be a κ-closed strong limit cardinal. The following are equivalent:

(1) κ is logically (δ, < θ)-strong compact. (2) If F : K → L is λ-accessible, µL < δ, and preserves µL-presentable objects, then the inclusion PF ,→ L is strongly κ-accessible and P(F ) is closed under < θ-small κ-directed colimits of κ-presentables. (3) Any AEC K with LS(K) < δ is (< κ, < θ)-tame. nearlystrongcptequivbu-str-fact Proof. Combine Theorem 5.3 with Fact 5.2. TAMENESS, POWERFUL IMAGES, AND LARGE CARDINALS 11

One ﬁnal time, we note that this argument can be adapted, in straightforward fashion, to the case of Pλ(F ), the λ-pure powerful image, or S(F ), the maximal sieve on the image of F .

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URL: http://math.harvard.edu/~wboney/

Department of Mathematics, Harvard University, Cambridge, Massachusetts, USA

URL: http://www.math.muni.cz/~lieberman/

Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Brno, Czech Repub- lic

Institute of Mathematics, Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic