Sizes and filtrations in accessible categories
aa r X i v : . [ m a t h . L O ] J un SIZES AND FILTRATIONS IN ACCESSIBLE CATEGORIES
MICHAEL LIEBERMAN, JIˇR´I ROSICK ´Y, AND SEBASTIEN VASEY
Abstract.
Accessible categories admit a purely category-theoretic replace-ment for cardinality: the internal size. Generalizing results and methodsfrom [LRV19b], we examine set-theoretic problems related to internal sizesand prove several L¨owenheim-Skolem theorems for accessible categories. Forexample, assuming the singular cardinal hypothesis, we show that a large ac-cessible category has an object in all internal sizes of high-enough cofinality.We also prove that accessible categories with directed colimits have filtrations:any object of sufficiently high internal size is (the retract of) a colimit of achain of strictly smaller objects.
Contents
1. Introduction 12. Preliminaries 43. Directed systems and cofinal posets 94. Presentation theorem and axiomatizability 125. On successor presentability ranks 146. The existence spectrum of a µ -AEC 167. The existence spectrum of an accessible category 198. Filtrations 22References 261. Introduction
Recent years have seen a burst of research activity connecting accessible categorieswith abstract model theory. Abstract model theory, which has always had the aimof generalizing—in a uniform way—fragments of the rich classification theory offirst order logic to encompass the broader nonelementary classes of structures that
Date : June 6, 2019AMS 2010 Subject Classification: Primary 18C35. Secondary: 03C45, 03C48, 03C52, 03C55,03C75, 03E05.
Key words and phrases. internal size, presentability rank, existence spectrum, accessibilityspectrum, filtrations, singular cardinal hypothesis.The second author is supported by the Grant agency of the Czech republic under the grant19-00902S. abound in mathematics proper, is perhaps most closely identified with abstractelementary classes (AECs, [She87]), but also encompasses metric AECs (mAECs,[HH09]), compact abstract theories (cats, [BY05]), and a host of other proposedframeworks. While accessible categories appear in many areas that model theoryfears to tread—homotopy theory, for example—they are, fundamentally, generalizedcategories of models, and the ambition to recover a portion of classification theoryin this context has been present since the very beginning, [MP89, p. 6]. That thesefields are connected has been evident for some time—the first recognition that AECsare special accessible categories came independently in [BR12] and [Lie11]—but itis only recently that a precise middle-ground has been identified: the µ -AECs of[BGL + µ -AEC below, we note that they are a nat-ural generalization of AECs in which the ambient language is allowed to be µ -ary,one assumes closure only under µ -directed unions rather than unions of arbitrarychains, and the L¨owenheim-Skolem-Tarski property is weakened accordingly. Themotivations for this definition were largely model-theoretic—in a typical AEC, forexample, the subclass of µ -saturated models is not an AEC, but does form a µ -AEC—but it turns out, remarkably, that µ -AECs are, up to equivalence, preciselythe accessible categories all of whose morphisms are monomorphisms (Fact 2.14).This provides an immediate link between model- and category-theoretic analyses ofproblems in classification theory, a middle ground in which the tools of each disci-pline can be brought to bear (and, moreover, this forms the basis of a broader collec-tion of correspondences between µ -AECs with additional properties—universality,admitting intersections—and accessible categories with added structure—locallymultipresentable, locally polypresentable [LRV19c]).Among other things, this link forces a careful consideration of how one shouldmeasure the size of an object: in µ -AECs, we can speak of the cardinality of theunderlying set, but we also have a purely category-theoretic notion of internal size ,which is defined—and more or less well-behaved—in any accessible category (seeDefinition 2.5). This is derived in straightforward fashion from the presentabilityrank of an object M , namely the least regular cardinal λ (if it exists) such thatany morphism sending M into the colimit of a λ -directed system factors through acomponent of the system. In most cases, the presentability rank is a successor, andthe internal size is then defined to be the predecessor of the presentability rank.The latter notion generalizes, e.g. cardinality in sets (and more generally in AECs),density character in complete metric spaces, cardinality of orthonormal bases inHilbert spaces, and minimal cardinality of a generator in classes of algebras (Ex-ample 2.6). In a sense, one upshot of [LRV19b] is that internal size is the moresuitable notion for classification theory, not least because eventual categoricity inpower fails miserably, while eventual categoricity in internal size is still very muchopen. A related question is that of LS-accessibility : in an accessibly category K ,is it the case that there is an object of internal size λ for every sufficiently large λ ? Under what ambient set-theoretic assumptions, or concrete category-theoreticassumptions on K , does this hold? Broadly, approximations to LS-accessibility canbe thought of as replacements for the L¨owenheim-Skolem theorem in accessiblecategories. Notice that the analogous statement for cardinality fails miserably: forexample, there are no Hilbert spaces whose cardinality has countable cofinality. IZES AND FILTRATIONS IN ACCESSIBLE CATEGORIES 3
One broad aim of the present paper is to relate classical properties of a category(phrased in terms of limits and colimits) to the good behavior of internal sizesin this category. To properly frame these classical properties, we first recall thedefinition of an accessible category (see Section 2 for more details). For λ a regularcardinal, a category is λ -accessible if it has λ -directed colimits, has only a set (upto isomorphism) of λ -presentable objects (i.e. objects with presentability rank atmost λ ), and every object can be written as a λ -directed colimit of λ -presentableobjects. A category is accessible if it is λ -accessible for some λ . We will say that acategory is large if it has a proper class of non-isomorphic objects. Thus a categoryof structures is large exactly when it has objects of arbitrarily large cardinality.Note that λ -accessible does not always imply λ ′ -accessible for λ ′ > λ (see also Fact2.11). If a given category has this property (i.e. it is accessible on a tail of regularcardinal), then we call it well accessible . In general, the class of cardinals λ suchthat a given category is λ -accessible (the accessibility spectrum ) is a key measureof the complexity of the category. For example, accessible categories with directedcolimits [BR12, 4.1] or µ -AECs with intersections [LRV19b, 5.4] are both knownto be well accessible while general µ -AECs need not be. In the present paper,we attempt to systematically relate the accessibility spectrum to the behavior ofinternal sizes. For example: • We prove that in any well accessible category, high-enough presentabilityranks have to be successors (Corollary 5.4). This holds more generallyof categories where the accessibility spectrum is unbounded below weaklyinaccessibles. In particular, we recover the known results that any acces-sible category with directed colimits [BR12, 4.2], any µ -AEC with inter-sections [LRV19b, 5.5(1)], and—assuming the singular cardinal hypothe-sis (SCH)—any accessible category [LRV19b, 3.11], has high-enough pre-sentability ranks successor. • We prove, assuming SCH, that in large accessible categories with all mor-phisms monos, for all high-enough cardinals λ , λ + -accessibility implies ex-istence of an object of internal size λ (Corollary 6.11). In this sense, theaccessibility spectrum is contained in the existence spectrum. In particular,well accessible categories with all morphisms monos are LS-accessible. • Assuming SCH, any large accessible category has objects of all internalsizes with high-enough cofinality. In particular, it is weakly LS-accessible(i.e. has objects of all high-enough regular internal sizes). This is Theorem7.13.Regarding the SCH assumption, we point out that we use a weaker version (“even-tual SCH”, see Definition 2.1(7)) which follows from the existence of a stronglycompact cardinal [Jec03, 20.8]. Thus our conclusions follow from this large cardi-nal axiom. In reality, we work primarily in ZFC, obtain some local results dependingon cardinal arithmetic, and then apply SCH to simplify the statements. Sometimesweaker assumptions than SCH suffice, but we do not yet know whether the conclu-sions above hold in ZFC itself. Unsurprisingly, dealing with successors of regular cardinals is often easier, and can sometimes be done in ZFC.
LIEBERMAN, ROSICK´Y, AND VASEY
Another contribution of the present account is the following: throughout the model-and set-theoretic literature, one finds countless constructions that rely on the ex-istence of filtrations, i.e. the fact that models can be realized as the union of acontinuous increasing chain of models of strictly smaller size. In a λ -accessiblecategory, on the other hand, one has that any object can be realized as the col-imit of a (more general) λ -directed system of λ -presentable objects, but there isno guarantee that one can extract from this system a cofinal chain consisting ofobjects that are also small. We here introduce the notion of well filtrable accessiblecategory (Definition 8.5), in which the internal size analog of this essential model-theoretic property holds, and show that certain well-behaved classes of accessiblecategories are well filtrable. We prove general results on existence of filtrations inarbitrary accessible categories (Theorem 8.8), and deduce that accessible categorieswith directed colimits are well filtrable (or really a slight technical weakening ofthis, see Corollary 8.9). This result improves on [Ros97, Lemma 1] (which estab-lished existence of filtrations only for object of regular internal sizes) and is usedin a forthcoming paper on forking independence [LRV] (a follow-up to [LRV19a]).The background required to read this paper is a familiarity with classical set theory(e.g. [Jec03, § µ -AEC, whose definition we recall below,first appears in [BGL + µ -AECsthat nevertheless have not appeared exactly in this form before. In particular, wereprove the presentation theorem for µ -AECs, fixing a mistake in [BGL + § Preliminaries
We start by recalling the necessary set-theoretic notation.
Definition 2.1.
Let λ and µ be infinite cardinals with µ regular.(1) For A a set, we write [ A ] <λ for the set of all subsets of A of cardinalitystrictly less than λ , and similarly define [ A ] λ (we will sometimes think of itas being partially ordered by containment). For B a set, we write B A forthe set of all functions from B to A , and let <λ A := S α<λ α A .(2) A partially ordered set (poset for short) I is µ -directed (for µ a regularcardinal) if any subset of I of cardinality strictly less than µ has an upperbound. When µ = ℵ , we omit it.(3) We write λ − for the predecessor of the cardinal λ , defined as follows: λ − = (cid:26) θ if λ = θ + λ if λ is limit IZES AND FILTRATIONS IN ACCESSIBLE CATEGORIES 5 (4) We say that λ is µ -closed if θ <µ < λ for all θ < λ .(5) For µ < λ regular, we write µ ⊳ λ if cf([ θ ] <µ ]) < λ for all θ < λ (see [AR94,2.12]). We extend this notion to singular cardinals λ as follows : when λ issingular, we write µ ⊳ λ if for all λ < λ , there exists a regular λ ′ ∈ ( λ , λ ]so that µ ⊳ λ ′ .(6) When we write a statement like “for all high-enough θ , ...”, we mean “thereexists a cardinal θ such that for all θ ≥ θ , ...”.(7) We say that SCH holds at λ if λ cf( λ ) = 2 cf( λ ) + λ + (SCH stands for the singular cardinal hypothesis — note that the equation is always true forregular λ ). We say that SCH holds above λ if SCH holds at θ for allcardinals θ ≥ λ . The eventual singular cardinal hypothesis (ESCH) is thestatement “SCH holds at all high-enough θ ”, or more precisely “there exists θ such that SCH holds above θ ”. Remark 2.2.
Let µ be a regular cardinal. If λ > µ is µ -closed, then µ ⊳ λ (becausecf([ θ ] <µ ) ≤ θ <µ ). Conversely, if λ > <µ and µ ⊳ λ , then λ is µ -closed (see [LRV19b,2.5]). Note also that if µ ⊳ λ and λ is weakly inaccessible, then an easy closureargument shows that there exists unboundedly-many successor cardinals λ < λ such that µ ⊳ λ . In fact, in the present paper, we can sometimes replace anassumption of the form “ λ is µ -closed” with the slightly more precise “ µ ⊳ λ ”,however for simplicity we will rarely do so.It is a result of Solovay (see [Jec03, 20.8]) that SCH holds above a strongly compactcardinal. Thus ESCH follows from this large cardinal axiom. We will assume ESCHin several results of the present paper.The facts below are well-known to set theorists. See [Jec03, § Fact 2.3. (1) If λ is a µ -closed cardinal ( µ regular), then λ = λ <µ if and only if λ hascofinality at least µ .(2) If SCH holds above an infinite cardinal θ , then for every cardinal λ andevery regular cardinal µ , λ <µ ≤ λ + + sup θ <θ θ <µ . In particular, everycardinal strictly greater than sup θ <θ θ <µ which is not the successor of acardinal of cofinality strictly less than µ is µ -closed.The following easy result will be used in the proof of Lemma 6.5 Lemma 2.4.
Let µ be a regular cardinal and let I be a µ -directed poset. Let α < µ and let h I i : i < α i be a sequence of subsets of I . If I = S i<α I i , then thereexists i < α such that I i is cofinal in I . Proof.
Suppose not. Then for each i < α there exists p i ∈ I such that no elementof I i bounds p i . Since I is µ -directed, there exists p ∈ I such that p i ≤ p for all i < α . There is i ∈ I such that p ∈ I i , contradicting the choice of p i . (cid:3) The ideas at the heart of the category-theory-enriched form of classification theoryat work here, in [LRV19b], and in [LRV19a], are the notions of presentability rank and internal size . Lurie [Lur09, A.2.6.3] writes µ ≪ λ to mean that µ < λ are both regular and λ is µ -closed. This was suggested by Mike Shulman.
LIEBERMAN, ROSICK´Y, AND VASEY
Definition 2.5.
Let λ and µ be infinite cardinals, µ regular, and let K be a category.(1) We say that a diagram D : I → K is µ -directed if I is a µ -directed poset.A µ -directed colimit is just the colimit of a µ -directed diagram.(2) We say that an object M in K is µ -presentable if the hom-functorHom K ( M, − ) : K →
Setpreserves µ -directed colimits. Equivalently, M is µ -presentable if everymorphism f : M → N , with N a µ -directed colimit with cocone h N i d i → N | i ∈ I i , the map f factors essentially uniquely through one of the d i ’s.(3) We say that M is ( < λ ) -presentable if it is θ -presentable for some regular θ < λ + ℵ .(4) The presentability rank of an object M in K , denoted r K ( M ), is the smallest µ such that M is µ -presentable. We sometimes drop K from the notationif it is clear from context.(5) The internal size of M in K is defined to be | M | K = r K ( M ) − . Again, wemay drop K from the notation if it is clear from context. Example 2.6.
We recall that internal size corresponds to the natural notion ofsize in familiar categories: • In the category of sets, the internal size of any infinite set is precisely itscardinality. In an AEC, too, the internal size of any sufficiently big modelwill be its cardinality (see [Lie11, 4.3] or Fact 2.15 here). • In the category of complete metric spaces and contractions, the internalsize of any infinite space is its density character (the minimal cardinality ofa dense subset). This is true, as well, for sufficiently big models in a generalmetric AEC, [LR17, 3.1]. • In the category of Hilbert spaces and linear contractions, the internal sizeof any infinite dimensional space is the cardinality of its orthonormal basis. • In the category of free algebras with exactly one ω -ary function, the in-ternal size is the minimal cardinality of a generator. In fact, a similarcharacterization holds in any µ -AEC with a notion of generation (i.e. withintersections), see [LRV19b, 5.7].As the above examples indicate, the relationship between internal size and car-dinality can be very delicate—particularly in a context as general as µ -AECs or,equivalently, accessible categories with monomorphisms (henceforth monos )—andseems to become tractable only under mild set- or category-theoretic assumptions.This is the substance of [LRV19b], results of which we refine in the present paper.We recall from [LRV19b, 3.1, 3.3] an essential piece of terminology: Definition 2.7.
Let λ and µ be infinite cardinals, µ regular.(1) A ( µ, < λ ) -system in a category K is a µ -directed diagram consisting of( < λ )-presentable objects. A ( µ, λ ) -system (for λ regular) is a ( µ, < λ + )-system.(2) We say that a ( µ, < λ )-system with colimit M is proper if the identity mapon M does not factor through any object in the system. IZES AND FILTRATIONS IN ACCESSIBLE CATEGORIES 7
The following two results on the relationship between presentability and directed-ness of systems are basic.
Fact 2.8.
Let λ , µ , and θ be infinite cardinals, µ regular. Let K be a categorywith µ -directed colimits.(1) [LRV19b, 3.5] For λ a regular cardinal, the colimit of a ( µ, λ )-system with θ objects is always ( θ + + λ )-presentable. In fact (for λ not necessarily regular),if cf( λ ) > θ and λ is not the successor of a singular cardinal, the colimit ofa ( µ, < λ )-system with θ objects is always ( < ( θ ++ + λ ))-presentable.(2) [LRV19b, 3.4] Let M be the colimit of a ( µ, < λ )-system.(a) If M is µ -presentable, then the system is not proper.(b) If the system is not proper, then M is ( < λ )-presentable.We hereby obtain a criterion for the existence of objects whose presentability rank isthe successor of a regular cardinal (this is already implicit in the proof of [LRV19b,3.12]): Corollary 2.9.
Let µ be a regular cardinal. In a category with µ -directed colimits,the colimit of a proper ( µ, µ + )-system containing at most µ objects has presentabil-ity rank µ + . Proof.
Let M be this colimit. By Fact 2.8(1), M is µ + -presentable. By Fact 2.8(2), M is not µ -presentable. (cid:3) The notion of a ( µ, < λ )-system also allows a more parameterized and compactrephrasing of the definition of an accessible category:
Definition 2.10.
Let K be a category and λ and µ be infinite cardinals, µ regular.(1) [LRV19b, 3.6] We say that K is ( µ, < λ )-accessible if it has the followingproperties:(a) K has µ -directed colimits.(b) K contains a set of ( < λ )-presentable objects, up to isomorphism.(c) Any object in K is the colimit of a ( µ, < λ )-system.(2) We say that K is ( µ, λ ) -accessible if λ is regular and K is ( µ, < λ + )-accessible. We say that K is µ -accessible if it is ( µ, µ )-accessible (this cor-responds to the usual definition from, e.g. [AR94, MP89]). We sometimessay finitely accessible instead of ℵ -accessible.(3) [BR12, 2.1] We say that K is well µ -accessible if it is θ -accessible for eachregular cardinal θ ≥ µ . We say that K is well accessible if it is well µ -accessible for some regular cardinal µ .We will use the following result, allowing us to change the index of accessibility ofa category (see [MP89, 2.3.10] or [LRV19b, 3.8]): Fact 2.11.
Let K be a ( µ, < λ )-accessible category. If θ is regular and µ ⊳ θ , then K is ( θ, < ( λ + θ + ))-accessible. Moreover, every object of K can be written as a θ -directed diagram, where each object is a µ -directed colimit of strictly fewer than θ -many ( < λ )-presentables. LIEBERMAN, ROSICK´Y, AND VASEY
The following two definitions describe good behavior of the existence spectrum of anaccessible category (LS-accessibility appears in [BR12, 2.4], weak LS-accessibilityis introduced in [LRV19b, A.1]):
Definition 2.12.
An accessible category K is LS-accessible if it has objects of allhigh-enough successor presentability ranks. We say that K is weakly LS-accessible if it has objects of all high-enough presentability ranks that are successors of regular cardinals.Similar to an accessible category, a µ -AEC is an abstract class of structures inwhich any model can be obtained by sufficiently highly directed colimits of smallobjects: Definition 2.13 ([BGL + § . Let µ be a regular cardinal.(1) A ( µ -ary) abstract class is a pair K = ( K, ≤ K ) such that K is a class ofstructures in a fixed µ -ary vocabulary τ = τ ( K ), and ≤ K is a partial orderon K that respects isomorphisms and extends the τ -substructure relation.For M ∈ K we write U M for the universe of M .(2) An abstract class K is a µ -abstract elementary class (or µ -AEC for short)if it satisfies the following three axioms:(a) Coherence: for any M , M , M ∈ K , if M ⊆ M ≤ K M and M ≤ K M , then M ≤ K M .(b) Chain axioms: if h M i : i ∈ I i is a µ -directed system in K , then:(i) M := S i ∈ I M i is in K .(ii) M i ≤ K M for all i ∈ I .(iii) If M i ≤ K N for all i ∈ I , then M ≤ K N .(c) L¨owenheim-Skolem-Tarski (LST) axiom: there exists a cardinal λ = λ <µ ≥ | τ ( K ) | + µ such that for any M ∈ K and any A ⊆ U M , thereexists M ∈ K with M ≤ K M , A ⊆ U M , and | U M | ≤ | A | <µ + λ .We write LS( K ) for the least such λ .When µ = ℵ , we omit it and call K an abstract elementary class (AECfor short).One can see any µ -AEC K (or, indeed, any µ -ary abstract class) as a category ina natural way: a morphism between models M and N in K is a map f : M → N which induces an isomorphism from M onto f [ M ], and such that f [ M ] ≤ K N . Weabuse notation slightly: we will still use boldface when referring to this category,i.e. we denote it by K and not K , to emphasize the concreteness of the category.As mentioned in the introduction, these classes are an ideal locus of interactionbetween abstract model theory and accessible categories: Fact 2.14 ([BGL + § . Any µ -AEC K with LS( K ) = λ is a λ + -accessiblecategory with µ -directed colimits and all morphisms monos. Conversely, any µ -accessible category K with all morphisms monos is equivalent (as a category) to a µ -AEC K with LS( K ) ≤ max( µ, ν ) <µ , where ν is the size of the full subcategoryof K on the set of (representatives) of µ -presentable objects.We finish with two facts on internal sizes in µ -AECs. The first describes the rela-tionship between presentability and cardinality: IZES AND FILTRATIONS IN ACCESSIBLE CATEGORIES 9
Fact 2.15.
Let K be a µ -AEC and let M ∈ K . Then:(1) [LRV19b, 4.5] r K ( M ) ≤ | U M | + + µ .(2) If λ > LS( K ) is a µ -closed cardinal such that M is ( < λ + )-presentable,then | U M | < λ . Proof of (2). If λ is singular, we can replace λ by a regular λ ∈ [LS( K ) + , λ ) suchthat M is λ -presentable and λ is µ -closed, so without loss of generality, λ isregular. Now apply [LRV19b, 4.8, 4.9]. (cid:3) The second gives a sufficient condition for existence of objects of presentability rankthe successor of a regular cardinal. The proof is very short given what has alreadybeen said, so we give it.
Fact 2.16 ([LRV19b, 3.12]) . Let λ and µ be infinite cardinals with µ regular. If K is a ( µ, < λ )-accessible category with all morphisms monos and K has an objectthat is not ( < λ )-presentable, then K has a proper ( µ, < λ )-system with µ objects.In particular, if in addition λ ≤ µ ++ , then K has an object of presentability rank µ + . Proof.
Let N be an object of K that is not ( < λ )-presentable. Since K is ( µ, < λ )-accessible, N is the colimit of a ( µ, < λ )-system h N i : i ∈ I i . Since N is not( < λ )-presentable, the system is proper (Fact 2.8(2)). Since I is µ -directed andall morphisms of K are monos, we can pick a strictly increasing chain h i k : k < µ i inside I such that h M i k : k < µ i is still proper. This then gives the desired proper( µ, < λ )-system with µ objects. The “in particular” part follows from Corollary2.9. (cid:3) Directed systems and cofinal posets
As observed in [BGL +
16, 4.1], if K is a µ -AEC then any M ∈ K is the LS( K ) + -directed union of all its K -substructures of cardinality at most LS( K ). In anAEC (i.e. when µ = ℵ ), it is well known that furthermore one can write M = S s ∈ [ UM ] <µ M s , where the M s ’s form a µ -directed system of objects of cardinalityat most LS( K ), and s ⊆ U M s for all s . In the proof of [BGL +
16, 3.2], it was assertedwithout proof that the corresponding statement was also true when µ > ℵ . Thiswas a key ingredient of the proof of the presentation theorem there. It was pointedout to us (by Marcos Mazari-Armida) that the proof for AECs does not generalizeto µ -AECs: since we cannot take unions, there are problems at limit steps. Thus wein fact do not know whether the statement is still true for µ -AECs. In this section,we prove a weakening and in the next two sections reprove the presentation theoremand related axiomatizability results.We will more generally develop the theory of subposets of posets that are cofinalin a generalized sense: Definition 3.1. A partially ordered set (or poset) is a binary relation ( P , ≤ ) whichis transitive, reflexive, and antisymmetric. We may not always explicitly mention ≤ . A subposet of a partially ordered set P is a poset ( P ∗ , ≤ ∗ ) with P ∗ ⊆ P and x ≤ ∗ y implying x ≤ y . Definition 3.2.
For θ a cardinal, a subposet P ∗ of a poset P is θ -cofinal if for any p ∈ P and any sequence h p i : i < θ i of elements of P ∗ with p i ≤ p for all i < θ ,there exists q ∈ P ∗ such that p ≤ q and p i ≤ ∗ q for all i < θ . We say that P ∗ is( < θ ) -cofinal if it is θ -cofinal for all θ < θ .Note that P ∗ is 0-cofinal if and only if it is cofinal in P as a set. Being 1-cofinalmeans that if p ≤ p with p ∈ P and p ∈ P ∗ , there exists q ∈ P ∗ so that p ≤ q and p ≤ ∗ q . If P ∗ is 1-cofinal, an induction shows that it is automatically n -cofinal forall n < ω (a similar argument appears in [SV, 3.9]). More generally, we can getthat it is θ -cofinal assuming chain bounds (a poset has α -chain bounds if any chainof length α has an upper bound; define ( < α )-chain bounds, ( < ≤ α )-chain bounds,in the expected way): Lemma 3.3.
Let θ be a cardinal and let P ∗ be a subposet of a poset P . If P ∗ is1-cofinal in P and P ∗ has ( ≤ θ )-chain bounds, then P ∗ is θ -cofinal in P . Proof.
Let p ∈ P and h p i : i < θ i be below p and in P ∗ . We proceed by inductionon θ . If θ ≤
1, the result is true by assumption. Assume now that θ > θ < θ . We build an increasing chain h q i : i < θ i in P ∗ such that p ≤ q and for all j < i < θ , p j ≤ ∗ q i . This is possible by successive use of theinduction hypothesis and 1-cofinality (at limits, use the chain bound assumption).Now take q ∈ P ∗ ≤ ∗ -above each q i . (cid:3) Definition 3.2 helps us understand the relationship between θ -directedness of P ∗ and P . The proof is straightforward, so we omit it. Lemma 3.4.
Let P ∗ be a 0-cofinal subposet of a poset P and let θ be an infinitecardinal.(1) If P ∗ is θ -directed, then P is θ -directed.(2) If P ∗ is ( < θ )-cofinal and P is θ -directed, then P ∗ is θ -directed.The next result extracts, from a diagram in P , a certain cofinal extension of thatdiagram in P ∗ . It will be applied to µ -AECs. Theorem 3.5.
Let θ be an infinite cardinal, let P be a θ -directed poset, and let P ∗ be a ( < θ )-cofinal subposet of P . Let ( I, ≤ ) be a partial order and let h p i : i ∈ I i be a diagram in P (that is, if i ≤ j are in I , then p i ≤ p j ). If for all i ∈ I , |{ j ∈ I | j < i }| < θ , then there exists I ⊆ I cofinal and a diagram h q i : i ∈ I i in P ∗ such that for all i ∈ I , p i ≤ q i . Proof.
We define a new poset F . Its objects are diagrams h q i : i ∈ I i in P ∗ ,with I ⊆ I (not necessarily cofinal) and with p i ≤ q i for all i ∈ I . Order F byextension. Now F is not empty (the empty diagram is in F ) and every chain in F has an upper bound (its union). By Zorn’s lemma, there is a maximal element h q i : i ∈ I i in F . We show that I is cofinal in I . Suppose not and let i ∗ ∈ I besuch that i ∗ j for any j ∈ I .Since P ∗ is 0-cofinal in P , there exists p ′ i ∗ in P ∗ such that p i ∗ ≤ p ′ i ∗ . Consider theset Q := { p ′ i ∗ } ∪ { q i | i ∈ I , i < i ∗ } . Note that | Q | < θ and by Lemma 3.4 P ∗ is θ -directed, so pick q i ∗ ∈ P ∗ an upper bound for Q . IZES AND FILTRATIONS IN ACCESSIBLE CATEGORIES 11
Consider h q i : i ∈ I ∪ { i ∗ }i . We show this is an element of F , contradicting themaximality of h q i : i ∈ I i . First, it is clear from the definition that p i ≤ q i forall i ∈ I ∪ { i ∗ } . It remains to see that h q i : i ∈ I ∪ { i ∗ }i is a diagram in P ∗ .So let i ≤ j be two elements of I ∪ { i ∗ } . If i, j ∈ I , then we are done by theassumption that h q i : i ∈ I i is a diagram. Thus at least one of i or j is equal to i ∗ . If i = j = i ∗ , then since q i ∗ ∈ P ∗ we are also done. If i ∈ I and j = i ∗ , we havemade sure in the construction that q i ≤ ∗ q j . Finally, we cannot have that i = i ∗ and j ∈ I by the choice of i ∗ (a witness to the non-cofinality of I ). (cid:3) As an application, we can study sufficiently closed objects M in an abstract class.While we may not be able to resolve such an object with a system indexed by[ U M ] <θ , we can at least get a system indexed by a cofinal subset of [ U M ] <θ : Definition 3.6.
Let K be an abstract class and let θ be an infinite cardinal. Anobject M ∈ K is θ -closed if for any A ∈ [ U M ] <θ , there exists M ∈ K with M ≤ K M , A ⊆ U M , and | U M | < θ . Corollary 3.7.
Let K be an abstract class, let θ be an infinite cardinal, and let M ∈ K be θ -closed. Let I ⊆ [ U M ] <θ be such that for any s ∈ I , |P ( s ) ∩ I | < cf( θ ).Then there exists I ⊆ I and h M s : s ∈ I i such that I is cofinal in I and for any s, t ∈ I :(1) M s ≤ M .(2) | U M s | < θ .(3) s ⊆ U M s .(4) s ⊆ t implies U M s ⊆ U M t . Proof.
Let P be the partially ordered set [ U M ] <θ , ordered by subset inclusion. Notethat P is θ -directed. Let P ∗ be the partially ordered set { U M | M ∈ K , M ≤ K M, | U M | < θ } , also ordered by subset inclusion. It is easy to check that P ∗ is asubposet of P . It is also straightforward to see that P ∗ is 1-cofinal in P (using that M is θ -closed). Similarly, P ∗ has ( < cf( θ ))-chain bounds. By Lemma 3.3, P ∗ is( < cf( θ ))-cofinal in P . Apply Theorem 3.5, with θ there being cf( θ ) here and p s = s for each s ∈ I . (cid:3) As an application of Corollary 3.7, we study what happens if we weaken theL¨owenheim-Skolem-Tarski (LST) axiom of µ -AECs to the “weak LST axiom”:there exists a cardinal λ ≥ | τ ( K ) | + µ such that λ = λ <µ and every object of K is λ + -closed (i.e. for all M ∈ K and all A ⊆ U M of cardinality at most λ , thereexists M ∈ K with M ≤ K M , | U M | ≤ λ , and A ⊆ U M ). It was shown in[BGL +
16, 4.6] that such a weakening still implies the original LST axiom, but theproof did not give that the minimal λ satisfying the weak LST axiom should be theL¨owenheim-Skolem-Tarski number. We prove this now. Corollary 3.8.
Let µ be a regular cardinal, let K be an abstract class satisfyingthe coherence and chain axioms of µ -AECs, and let λ be an infinite cardinal. If λ = λ <µ and any element of K is λ + -closed, then K is a µ -AEC with LS( K ) ≤ λ . Proof.
Let M ∈ K and let A ⊆ U M . Apply Corollary 3.7 with I := [ A ] <µ and θ := λ + (note that 2 <µ ≤ λ <µ = λ < θ , so the cardinal arithmetic condition thereis satisfied). Let h M s : s ∈ I i be as given there. This is a µ -directed system by coherence, so let N be its union. By construction, N ≤ M and A ⊆ U N . Moreover, | U N | ≤ | I | · λ ≤ | A | <µ + λ , as needed. (cid:3) Presentation theorem and axiomatizability
We reprove here the presentation theorem for µ -AECs (and more generally foraccessible categories with µ -directed colimits and all morphisms monos), in theform outlined and motivated in [LRV19c, §
6] (there, additional assumptions on theexistence of certain directed colimits had to be inserted to make the proof work).The idea is simple: any µ -accessible category is equivalent to the category of modelsof an L ∞ ,µ -sentence, and we can Skolemize such a sentence to obtain the desiredfunctor. We first state the three facts we will use. Recall that Mod( φ ) denotesthe category of models of the sentence φ , with morphisms all homomorphisms (i.e.maps preserving functions and relations). See also [LRV19c, §
4] for a summary ofwhat is known on axiomatizability of accessible categories.
Fact 4.1 ([MP89, 3.2.3, 3.3.5, 4.3.2]) . Any µ -accessible category is equivalent toMod( φ ), for φ an L ∞ ,µ -formula.Recall [LRV19c, 2.1] that (for a regular cardinal µ ) a µ -universal class is a class ofstructures in a µ -ary vocabulary that is closed under isomorphisms, substructure,and µ -directed unions. Fact 4.2 (Skolemization) . Let µ ≤ λ . If φ is an L λ + ,µ -sentence in the vocabulary τ , there exists an expansion τ + of τ with function symbols and a µ -universal class K + with vocabulary τ + such that:(1) | τ + | ≤ λ .(2) The reduct map is a faithful functor from K + into Mod( φ ) that is surjectiveon objects and preserves µ -directed colimits. Proof sketch.
Let Φ be a fragment (i.e. set of formulas in L λ + ,µ closed under sub-formulas) which contains φ and has cardinality at most λ . Add a Skolem functionfor each formula in Φ, forming a vocabulary τ + , and let K + be the set of all τ + -structures whose reduct is a model of φ and where the Skolem functions performas expected. (cid:3) The result below was stated for µ = ℵ in [LR16, 2.5], but the proof easily gener-alizes. Fact 4.3 ([LR16, 2.5]) . If K is an accessible category with µ -directed colimitsand all morphisms monos, there exists a µ -accessible category L and a faithfulessentially surjective functor F : L → K preserving µ -directed colimits. In fact,if K is λ -accessible, L is the free completion under µ -directed colimits of the fullsubcategory of K induced by its λ -presentable objects. Theorem 4.4 (The presentation theorem for µ -AECs) . If K is an accessible cat-egory with µ -directed colimits and all morphisms monos, then there exists a µ -universal class L and an essentially surjective faithful functor F : L → K preserving µ -directed colimits. IZES AND FILTRATIONS IN ACCESSIBLE CATEGORIES 13
Proof.
By Fact 4.3, there exists a µ -accessible category K and a faithful essentiallysurjective functor F : K → K . By Fact 4.1, K is equivalent to Mod( φ ), for some L ∞ ,µ -sentence φ . Since all morphisms are monos, we may assume that non-equalityis part of the vocabulary of φ . By Fact 4.2, we can find a µ -universal class K + sothat the reduct map F : K + → Mod( φ ) is an essentially surjective faithful functorpreserving µ -directed colimits. Set L := K + , F := F ◦ F . (cid:3) Note that if we apply Theorem 4.4 to a µ -AEC K , the functor is not directly givenby a reduct from some expansion of K (we first have to pass through several equiv-alence of categories). Thus Theorem 4.4 does not immediately prove (as in [Bon14]for AECs) that µ -AECs are closed under sufficiently complete ultraproducts. Forthis, we will prove that a certain functorial expansion of the µ -AEC is axiomatizableby an infinitary logic ( without passing to an equivalent category): Definition 4.5.
Let K be a µ -AEC. The substructure functorial expansion of K is the abstract class K + defined as follows:(1) τ ( K + ) = τ ( K ) ∪ { P } , where P is an LS( K )-ary predicate.(2) M + ∈ K + if and only if M + ↾ τ ( K ) ∈ K and for any ¯ a ∈ LS( K ) M + , P M + (¯ a ) holds if and only if ran(¯ a ) ≤ K M + ↾ τ ( K ), where we see ran(¯ a ) asa τ ( K )-structure.(3) For M + , N + ∈ K + , M + ≤ K + N + if and only if M + ↾ τ ( K ) ≤ K N + ↾ τ ( K ).The substructure expansion is “functorial” in the sense of [Vas16, 3.1]: the reductfunctor gives an isomorphism of concrete categories. The substructure functorialexpansion has the property of having very simple morphisms: Theorem 4.6.
Let K be a µ -AEC and let K + be its substructure functorial ex-pansion. If M + , N + ∈ K + are such that M + ⊆ N + , then M + ≤ K + N + . Proof.
For M ∈ K , write M + for the expansion of M to K + . Let M, N ∈ K and assume that M + ⊆ N + . We have to see that M ≤ K N . For this, it isenough to show that for any M ≤ K M of cardinality at most LS( K ), we alsohave that M ≤ K N (indeed, we can then take the LS( K ) + -directed union of allsuch M ’s). So let M ≤ K M have cardinality at most LS( K ): we must show that M ≤ K N . Let ¯ a be an enumeration of M . We have that M + | = P [¯ a ] (where P is the additional predicate in τ ( K ) + ), so N + | = P [¯ a ] (as M + is a substructure of N + ). This means that M ≤ K N , as desired. (cid:3) The substructure functorial expansion of a µ -AEC can be axiomatized (a morecomplication variation of this, for AECs, is due to Baldwin and Boney [BB17,3.9]). Since the ordering is trivial by the previous result, this shows that any µ -AEC is isomorphic (as a category) to the category of models of an L ∞ , ∞ sentence,where the morphisms are injective homomorphisms. Theorem 4.7.
Let K be a µ -AEC and let K + be its substructure functorial ex-pansion. There is an L ( LS( K ) ) + , LS( K ) + sentence φ such that K + is the class ofmodels of φ . Proof.
First note that for each M ∈ K ≤ LS( K ) , there is a sentence ψ M (¯ x ) of L LS( K ) + , LS( K ) + coding its isomorphism type, i.e. whenever M | = ψ M [¯ a ], then ¯ a is an enumeration of an isomorphic copy of M . Similarly, whenever M , M are in K ≤ LS( K ) with M ≤ K M , there is ψ M ,M (¯ x, ¯ y ) that codes that (¯ x, ¯ y ) is isomor-phic to ( M , M ) (so in particular ¯ x ≤ K ¯ y ). Let S be a complete set of membersof K ≤ LS( K ) (i.e. any other model is isomorphic to it) and let T be a complete setof pairs ( M , M ), with each in K ≤ LS( K ) , such that M ≤ K M . Now define thefollowing: φ = ∀ ¯ x ∃ ¯ y _ M ∈ S ψ M (¯ y ) ! ∧ ¯ x ⊆ ¯ y ∧ P (¯ y ) ! φ = ∀ ¯ x ∀ ¯ y (¯ x ⊆ ¯ y ∧ P (¯ x ) ∧ P (¯ y )) → _ ( M ,M ) ∈ T ψ M ,M (¯ x, ¯ y ) φ = φ ∧ φ Where ¯ x ⊆ ¯ y abbreviates the obvious formula. This works. First, any M + ∈ K + satisfies φ by the LST axiom and satisfies φ by the coherence axiom. Conversely,assume that M + | = φ and let M := M + ↾ τ ( K ). Consider the set: I = { M ∈ K ≤ LS( K ) | U M ⊆ U M, P M + ( ¯ M ) } Where ¯ M refers to some enumeration of M . Then by construction of φ , I is a µ -directed system in K and S I = M , so M ∈ K . Similarly, P M + (¯ a ) holds if andonly if ran(¯ a ) ≤ K M , so M + ∈ K + . (cid:3) Corollary 4.8.
Any µ -AEC K is closed under (cid:0) LS( K ) (cid:1) + -complete ultraproducts(in the sense that the appropriate generalization of [Bon14, 4.3] holds). Proof.
By Theorems 4.6 and 4.7, Lo´s’ theorem, and the fact that taking reductscommutes with ultraproducts. (cid:3) On successor presentability ranks
We start our study of the existence spectrum of an accessible category K : the setof regular cardinals λ such that K has an object of presentability rank λ . The goalis to say as much as possible by just looking at the accessibility spectrum: the setof cardinals λ such that K is λ -accessible.In this section, we consider the question, first systematically investigated in [BR12],of whether the presentability rank of an object always has to be a successor (or,said differently, whether there can be objects of weakly inaccessible presentabilityrank). Assuming the accessibility spectrum is sufficiently large, we show thereare no objects of weakly inaccessible presentability rank, and explain how thisgeneralizes previous results.The following easy lemma characterizes existence in terms of the accessibility spec-trum. It will also be used in the next section: IZES AND FILTRATIONS IN ACCESSIBLE CATEGORIES 15
Lemma 5.1.
Let λ be a regular cardinal and let K be a category. The followingare equivalent:(1) K is ( λ, < λ )-accessible.(2) K is λ -accessible and has no objects of presentability rank λ . Proof.
Assume K is ( λ, < λ )-accessible. By definition, K is clearly λ -accessible. If M is a λ -presentable object of K , then it is a λ -directed colimit of ( < λ )-presentableobjects, hence by Fact 2.8 must itself be ( < λ )-presentable.Conversely, if K is λ -accessible and has no objects of presentability rank λ , thenany ( λ, λ )-system must be a ( λ, < λ )-system, hence K is ( λ, < λ )-accessible. (cid:3) The following new result gives a criterion for ( λ, < λ )-accessibility when λ is weaklyinaccessible. Theorem 5.2. If λ is weakly inaccessible and K is ( µ, < λ )-accessible for unboundedly-many µ < λ , then K is ( λ, < λ )-accessible. Proof.
Let S be the set of all regular cardinals µ < λ such that K is ( µ, < λ )-accessible. Let M be an object of K . For each µ ∈ S , fix a µ -directed system h M µi : i ∈ I µ i , with maps h f µi,j : i ≤ j ∈ I µ i whose colimit is M (with colimit maps f µi , i ∈ I µ ). Let I := { ( i, µ ) | µ ∈ S, i ∈ I µ } . Order it by ( i, µ ) ≤ ( j, µ ) if andonly if µ ≤ µ , and there exists a unique map g : M µ i → M µ j so that f µ i = f µ j g .Observe that ( I, ≤ ) is a partial order. Also, if we fix µ ∈ S and i ∈ I µ , then M µi is( < λ )-presentable, hence µ -presentable for some regular µ ∈ S , µ ≤ µ . Thus forany µ ∈ S with µ ≤ µ , there exists j ∈ I µ such that ( i, µ ) ≤ ( j, µ ).The last paragraph quickly implies that I is λ -directed, and so the diagram inducedby I is the desired ( λ, < λ )-system with colimit M . (cid:3) Corollary 5.3. If λ is weakly inaccessible and K is µ -accessible for unboundedly-many µ < λ , then K does not have an object of presentability rank λ . Proof.
By Theorem 5.2, K is ( λ, < λ )-accessible. Now apply Lemma 5.1. (cid:3) We obtain that any high-enough presentability rank in a well-accessible category(recall Definition 2.10(3)) must be a successor, which improves on [BR12, 4.2] and[LRV19b, 5.5]:
Corollary 5.4. If K is a well µ -accessible category, then the presentability rank ofany object that is not µ -presentable must be a successor. Proof.
Immediate from Corollary 5.3. (cid:3)
We have also recovered [LRV19b, 3.11]:
Corollary 5.5.
Let µ be a regular cardinal and let λ > µ be a weakly inaccessiblecardinal. If K is a ( µ, < λ )-accessible category and λ is µ -closed, then K has noobject of presentability rank λ .In particular, assuming ESCH, high-enough presentability ranks are successors inany accessible category. Proof.
Since λ is µ -closed and limit, there are unboundedly-many regular θ ∈ [ µ, λ )that are µ -closed. By Fact 2.11, for any such θ , K is ( θ, < λ )-accessible. ByTheorem 5.2, K is ( λ, < λ )-accessible, hence by Lemma 5.1 cannot have an objectof presentability rank λ . (cid:3) The existence spectrum of a µ -AEC We now refine a few results of [LRV19b] concerning the existence spectrum of µ -AECs, especially [LRV19b, 4.13].We aim to study proper ( λ, < λ )-systems—in the sense of Definition 2.7—and showthat under certain conditions they do not exist. This will give conditions underwhich an object of presentability rank λ does exist: Lemma 6.1.
Let λ be a regular cardinal and let K be a λ -accessible category. If K has an object that is not ( < λ )-presentable and K has no proper ( λ, < λ )-systems,then K has an object of presentability rank λ . Proof.
By Lemma 5.1, it suffices to show that K is not ( λ, < λ )-accessible. Supposefor a contradiction that K is ( λ, < λ )-accessible. Let M be an object that is not( < λ )-presentable. Then M is the colimit of a ( λ, < λ )-system, which must beproper because M is not ( < λ )-presentable (see Fact 2.8), a contradiction to theassumption that there are no proper ( λ, < λ )-systems. (cid:3) To help the reader, let us consider what a ( λ + , < λ + )-system should be in an AEC K with λ > LS( K ). Since internal sizes correspond to cardinalities in that context(see Fact 2.15 or simply [Lie11, 4.3]), such a system must be a λ + -directed systemconsisting of object of cardinality strictly less than λ . Because it is “too directed,”the system cannot be proper (i.e. its colimit will just be a member of the system).We attempt here to generalize such an argument to suitable µ -AECs. We willsucceed when λ is µ -closed (Theorem 6.6 — notice that this is automatic when µ = ℵ ).We will use the following key bound on the internal size of a subobject: Lemma 6.2. If K is a µ -AEC, M ≤ K N are in K , λ > LS( K ) is a µ -closedcardinal, and N is ( < λ + )-presentable, then M is ( < λ + )-presentable. Proof.
By Fact 2.15, | U N | < λ . Of course, | U M | ≤ |
U N | , so | U M | < λ By Fact2.15 again, r K ( M ) ≤ | U M | + + µ , so r K ( M ) ≤ λ + µ = λ , so M is ( < λ + )-presentable, as desired. (cid:3) We require an additional refinement, concerning systems in which bounded subsys-tems have small colimits:
Definition 6.3.
A system h M i : i ∈ I i in a given category is boundedly ( < λ ) -presentable if whenever I ⊆ I is bounded in I , the colimit of h M i : i ∈ I i is( < λ )-presentable (whenever it exists). Lemma 6.4.
Let K be a µ -AEC and let λ > LS( K ) + be such that λ − is µ -closed. Then any system in K consisting of ( < λ )-presentable objects is boundedly( < λ )-presentable. IZES AND FILTRATIONS IN ACCESSIBLE CATEGORIES 17
Proof.
Let h M i : i ∈ I i be a system consisting of ( < λ )-presentable objects. Let I ⊆ I be bounded in I , say by i , and such that the colimit M I of the resultingsystem exists. We have that M I ≤ K M i . Since λ − is µ -closed and M i is ( < λ )-presentable, we can find λ ∈ [LS( K ) , λ ) regular and µ -closed such that M i is λ -presentable. By Lemma 6.2, M I is also λ -presentable, hence ( < λ )-presentable,as desired. (cid:3) Using the bound of Lemma 6.2 again, we now show that for most successor cardinals θ , there are no proper θ -directed boundedly ( < θ )-presentable systems: Lemma 6.5.
Let K be a µ -AEC. If λ > LS( K ) is µ -closed, then there are noproper λ + -directed boundedly ( < λ + )-presentable systems. Proof.
Assume for a contradiction that h M i : i ∈ I i is such a system, with colimit M . First, if λ is regular, then using λ + -directedness and properness we can find achain I ⊆ I of type λ such that i < j in I implies M i = M j . Then h M i : i ∈ I i is proper, so its colimit (union) M I is not λ -presentable (Fact 2.8). However, I is bounded as I is λ + -directed, a contradiction to the hypothesis of bounded( < λ + )-presentability.Assume now that λ is singular. Let δ := cf( λ ) and write λ = sup α<δ λ α , withLS( K ) < λ and each λ α regular and µ -closed. Let I α := { i ∈ I | M i is λ α -presentable } .Note that I = S α<δ I α .By Lemma 2.4, there exists α < δ such that I α is cofinal in I . By renaming, wecan assume without loss of generality that α = 0: I is already cofinal in I , hence I α is cofinal in I for all α < δ . Note that I must itself be λ + -directed.Now pick h i j : j < λ i an increasing sequence in I such that h M i j : j < λ i is strictlyincreasing (this is possible by properness of the system). For k < λ of cofinality atleast µ , let N k = S j Theorem 6.6. Let K be a µ -AEC. If λ > LS( K ) is µ -closed, then there are noproper ( λ + , < λ + )-systems in K . Proof. By Lemma 6.4 (with λ + in place of λ ) and Lemma 6.5. (cid:3) We get that, at least for successors of high-enough µ -closed cardinals (or under SCH,see below), the presentability rank spectrum contains the accessibility spectrum. Corollary 6.7. Let K be a µ -AEC. If λ > LS( K ) is a µ -closed cardinal such that K is λ + -accessible and K has an object of cardinality at least λ , then K has anobject of presentability rank λ + . Proof. Let M ∈ K have cardinality at least λ . Using Fact 2.15 together with theassumption that λ is µ -closed, M is not ( < λ + )-presentable. By Theorem 6.6, there are no proper ( λ + , < λ + )-systems in K . By Lemma 6.1, K has an object ofpresentability rank λ + . (cid:3) We have in particular recovered [LRV19b, 4.13]. This will later be further general-ized to any accessible category (Theorem 7.12). Corollary 6.8. Let K be a µ -AEC and let λ = λ <µ ≥ LS( K ). We have that K has an object of presentability rank λ + if at least one of the following conditionshold:(1) λ > LS( K ), λ is µ -closed, and K has an object of cardinality at least λ .(2) λ is regular and K has an object of cardinality at least λ + . Proof. Since λ = λ <µ , λ + is µ -closed, so by Fact 2.11, K is λ + -accessible. Now:(1) This follows from Corollary 6.7.(2) Since K has µ -directed colimits, it is also ( λ, λ + )-accessible. Since K has anobject of cardinality at least λ + and λ + is µ -closed, Fact 2.15 (with λ + inplace of λ ) implies that this object is not λ + -presentable. Now apply Fact2.16. (cid:3) Remark 6.9. The example of well-orderings ordered by initial segment given in[LRV19b, 6.2] shows that, even when µ = ℵ , we may not have an object of rankLS( K ) + if LS( K ) is singular.Under SCH, the statements simplify and we recover [LRV19b, 4.15]: Corollary 6.10. Let K be a large µ -AEC. If SCH holds above LS( K ), then forevery λ > LS( K ) of cofinality at least µ , K has an object of presentability rank λ + .In particular, K is weakly LS-accessible. Proof. By Fact 2.3, λ = λ <µ . If λ is regular, we may apply Corollary 6.8(2). If λ is singular, then by Fact 2.3 it is µ -closed so one may apply Corollary 6.8(1). (cid:3) Still under SCH, we obtain that the (successor) accessibility spectrum is eventuallycontained in the existence spectrum: Corollary 6.11. Assume ESCH. Let K be a large category with all morphismsmonos. For all high-enough successor cardinals θ , if K is θ -accessible, then K hasan object of presentability rank θ . Proof. By Fact 2.14, K is equivalent to a µ -AEC K . By replacing K by a tailsegment if necessary, we can assume without loss of generality that SCH holdsabove LS( K ). Pick θ > LS( K ) + successor such that K is θ -accessible. Write θ = λ + . If cf( λ ) ≥ µ , Corollary 6.10 gives the result, so we may assume thatcf( λ ) < µ . In this case, the SCH assumption implies that λ is µ -closed so we canapply Corollary 6.7. (cid:3) Recall (Definition 2.12) that a category is LS-accessible if it has objects of all high-enough internal sizes. IZES AND FILTRATIONS IN ACCESSIBLE CATEGORIES 19 Corollary 6.12. Assuming ESCH, any large well accessible category with all mor-phisms monos is LS-accessible.Note that Corollary 6.12 can be seen as a joint generalization of [LR16, 2.7] (LS-accessibility of large accessible categories with directed colimits and all morphismsmonos) and [LRV19b, 5.9] (LS-accessibility of large µ -AECs with intersections): inboth cases, the categories in question are well accessible with all morphisms monos(see Fact 2.11, [LRV19b, 5.4]). Note however that the proof of LS-accessibility oflarge accessible categories with directed colimits and all morphisms monos does notassume SCH.7. The existence spectrum of an accessible category A downside of the previous section was the assumption that all morphisms weremonos. In the present section, we look at what can be said for arbitrary accessiblecategories. The main tool is the fact that the inclusion functor K mono → K is (in asense we make precise) accessible, hence plays reasonably well with internal sizes.While the notion of an accessible functor appears already in [MP89, § Definition 7.1. Let λ and µ be infinite cardinals, with µ regular. A functor F : K → L is ( µ, < λ ) -accessible if it preserves µ -directed colimits and both K and L are ( µ, < λ )-accessible. We say that F is ( µ, λ ) -accessible if λ is regularand F is ( µ, < λ + )-accessible. We say that F is µ -accessible precisely when it is( µ, µ )-accessible. Fact 7.2 ([LRV19a, 6.2]) . If K is a µ -accessible category, there there exists acardinal λ ≥ µ such that K mono is ( µ, λ )-accessible and moreover the inclusionfunctor F of K mono into K is ( µ, λ )-accessible.The following properties, describing the interaction of an accessible functor withpresentability, were first systematically investigated in [BR12, § Definition 7.3. A functor F : K → L preserves λ -presentable objects if whenever M is λ -presentable in K , then F ( M ) is λ -presentable (in L ). We say that F reflects λ -presentable objects if M is λ -presentable in K whenever F ( M ) is λ -presentable L . We also say that F preserves λ -ranked objects if whenever r K ( M ) = λ , then r L ( F M ) = λ . Similarly define what it means for F to reflect λ -ranked objects .Of course, there is a simple test to determine when a functor preserves rank giveninformation as to whether it preserves and reflects presentable objects: Lemma 7.4. Let F : K → L be an accessible functor and let µ be a regular cardinal.If F preserves µ -presentable objects and reflects ( < µ )-presentable objects, then F preserves µ -ranked objects. Similarly, if F preserves ( < µ )-presentable objects andreflects µ -presentable objects, then F reflects µ -ranked objects. Proof. We prove the first statement (the proof of the second is similar). Let M be an object of K such that r ( M ) = µ . Then r ( F M ) ≤ µ because F preserves µ -presentable objects, and if r ( F M ) < µ , then r ( M ) < µ because F reflects ( < µ )-presentable objects, contradiction. Thus r ( M ) = r ( F M ). (cid:3) For a functor to reflect λ -presentable objects, it enough that it is sufficiently acces-sible and that the functor reflects split epimorphisms (i.e. if F f is a split epi, then f is a split epi). This was isolated in [BR12, 3.6]. We now proceed to mine theproof of this result to extract what can be said in our more parameterized setup: Lemma 7.5. Let λ and µ be cardinals, µ regular. Let F : K → L be a functorreflecting split epimorphisms and preserving µ -directed colimits. Let h M i : i ∈ I i be a ( µ, < λ )-system with colimit M . If h M i : i ∈ I i is proper, then h F M i : i ∈ I i is proper. If in addition F preserves ( < λ )-presentable objects, then h F M i : i ∈ I i is a ( µ, < λ )-system. Proof. Since F preserves µ -directed colimits, the colimit of h F M i : i ∈ I i is F M .Suppose that the identity map on F M factors through some F M i , via a map g : F M → F M i . That is, ( F f i ) g = id F M , where f i : M i → M is a colimit map.Then F f i is a split epimorphism, hence f i is a split epimorphism, i.e. f i g = id M , so h M i : i ∈ I i is not proper. The last sentence is immediate from the definition. (cid:3) Fact 7.6 ([BR12, 3.6]) . Let λ be an uncountable cardinal, and let F : K → L bea functor reflecting split epimorphisms. If there exists a regular cardinal λ < λ such that F is ( λ , < λ )-accessible and for all regular cardinals µ ∈ [ λ , λ ), K is( µ, < λ )-accessible, then F reflects ( < λ )-presentable objects. Proof. Assume that F M is ( < λ )-presentable. Pick a regular cardinal µ ∈ [ λ , λ )such that F M is µ -presentable. By ( µ, < λ )-accessibility, M is the colimit of a( µ, < λ )-system h M i : i ∈ I i . Assume for a contradiction that M is not ( < λ )-presentable. Then h M i : i ∈ I i must be proper by Fact 2.8(2). By Lemma 7.5, h F M i : i ∈ I i is proper, hence its colimit F M cannot be µ -presentable by Fact2.8(2), a contradiction. (cid:3) In passing, we can deduce the following powerful criterion for existence of an objectof regular internal size. Notice that this is a generalization of Fact 2.16 (which isthe special case of the identity functor). Theorem 7.7. Let λ be a regular cardinal and let F : K → L be a ( λ, λ + )-accessible functor that preserves λ + -presentable objects and reflects isomorphisms.If all morphisms in the image of F are monos and K has an object that is not λ + -presentable, then L has an object of presentability rank λ + . Proof. Since F reflects isomorphisms and all morphisms in the image of F aremonos, F reflects split epimorphisms. By Fact 2.16, K has a proper ( λ, λ + )-system h M i : i ∈ I i with λ -many objects. By Lemma 7.5, h F M i : i ∈ I i is a proper( λ, λ + )-system. By Corollary 2.9, the colimit of this system in L has presentabilityrank λ + . (cid:3) To preserve λ -presentable objects, a cardinal arithmetic assumption on λ suffices: Fact 7.8. Let F : K → L be a ( µ, λ )-accessible functor. Let λ ≥ λ be a regularcardinal such that the image of any λ -presentable object is λ -presentable. If λ > λ is such that λ − is µ -closed, then F preserves ( < λ )-presentable objects. IZES AND FILTRATIONS IN ACCESSIBLE CATEGORIES 21 Proof. This is similar to the proof of [AR94, 2.19]. We include some details forthe convenience of the reader. Let A be a ( < λ )-presentable object of K . Pick aregular θ < λ such that A is θ -presentable. If λ is a successor and λ − is regular,let θ := λ − . Otherwise, λ − is limit and we let θ := (cid:0) ( θ + λ ) <µ (cid:1) + . In eithercase, we have that λ ≤ θ < λ (because λ − is µ -closed), θ is regular, θ is µ -closed (by [AR94, 2.13(5)]), and A is θ -presentable. By the proof of [MP89, 2.3.10](see [AR94, 2.15]), A is a θ -directed colimit of a diagram consisting of µ -directedcolimits of fewer than θ -many λ -presentable objects. Since A is θ -presentable, A is a retract of such a µ -directed colimit. Since F preserves µ -directed colimits andany functor preserves retractions, F A is a retract of a µ -directed colimit of fewerthan θ -many λ -presentable objects. By Fact 2.8(1), F A is θ -presentable, hence( < λ )-presentable. (cid:3) Remark 7.9. Instead of λ − µ -closed, it suffices to assume that µ⊳λ − (see Definition2.1(5) and Remark 2.2). Conversely, if µ < λ are cardinals such that µ is regularand every µ -accessible functor which preserves µ -presentable objects also preserves( < λ )-presentable objects, then µ ⊳ λ − (consider the functor A [ A ] <µ from thecategory of sets to the category of µ -directed posets).We obtain the following cardinal arithmetic test for preservation and reflection ofranks: Lemma 7.10. Let F : K → L be a ( µ, λ )-accessible functor that preserves λ -presentable objects and reflects split epimorphisms.If θ > λ is the successor of a µ -closed cardinal of cofinality at least µ , then F preserves and reflects θ -ranked objects. Proof. Write θ = λ + , with λ a µ -closed cardinal of cofinality at least µ . Notethat λ <µ = λ , since it has cofinality at least µ . Thus both λ and θ are µ -closed.This implies that F preserves θ -presentable objects and preserves ( < θ )-presentableobjects (Fact 7.8). We also have that F reflects ( < θ )-presentable objects and F reflects θ -presentable objects (use Facts 2.11 and 7.6). Now apply Lemma 7.4. (cid:3) Using Corollary 6.8, we obtain the following existence spectrum result if the domainof the functor is a large µ -AEC: Lemma 7.11. Let K be a µ -AEC, let λ > LS( K ) be regular, and let F : K → L be a ( µ, λ )-accessible functor that preserves λ -presentable objects and reflectsisomorphisms. Let λ ≥ λ be a µ -closed cardinal of cofinality at least µ . If K hasan object of cardinality at least λ , then L has an object of presentability rank λ + . Proof. Since all morphisms of K are monos, F reflects split epimorphisms. Since λ is µ -closed and has cofinality at least µ , λ = λ <µ . By Corollary 6.8, K has anobject M of presentability rank λ + . By Lemma 7.10 (where θ there stand for λ + here), F preserves λ + -ranked objects, so F M has presentability rank λ + . (cid:3) Putting all the results together, we obtain an existence spectrum result for anylarge accessible category. This extends Corollary 6.8. Theorem 7.12. Let K be a large µ -accessible category. (1) For every high-enough regular λ such that λ = λ <µ , K has an object ofpresentability rank λ + .(2) There exists a regular cardinal µ ′ such that for every high-enough µ ′ -closedcardinal λ of cofinality at least µ ′ , K has an object of presentability rank λ + . Proof. (1) By Fact 7.2, there exists a cardinal λ such that the inclusion functor F of K mono into K is ( µ, λ )-accessible. Of course, F also reflects isomorphisms.Let λ > λ be regular such that λ = λ <µ . By Fact 7.8, F preserves λ + -presentable objects and by Fact 2.11, F is ( λ, λ + )-accessible. Since K is large, K mono is also large, so by Theorem 7.7, K has an object ofpresentability rank λ + .(2) As before, K mono is an accessible category with all morphisms monos, soby Fact 2.14, it is equivalent to a µ ′ -AEC K ∗ , for some regular cardinal µ ′ .Let F : K ∗ → K be the composition of the equivalence with the inclusionof K mono into K . Then F is ( µ ′ , λ )-accessible, for some regular cardinal λ > LS( K ), and F reflects isomorphisms. Taking λ bigger if needed(and using Fact 7.8), we can assume without loss of generality that F alsopreserves λ -presentable objects. Let λ ≥ λ be a µ ′ -closed cardinal ofcofinality at least µ ′ . Since K is large, Lemma 7.11 applies and so L hasan object of presentability rank λ + . (cid:3) We obtain the main result of this section. This extends for example [LRV19b, A.2]— weak LS-accessibility of large locally multipresentable categories — at the costof ESCH: Corollary 7.13. Assuming ESCH, any large accessible category has objects of allinternal sizes of high-enough cofinality. In particular, any large accessible categoryis weakly LS-accessible. Proof. Let K be a large µ -accessible category, and let µ ′ be as given by Theorem7.12. Let λ be a high-enough cardinal of cofinality at least µ ′ (the proof will givehow big we need to take it). If λ is a regular cardinal, then (by ESCH, see Fact 2.3) λ = λ <µ , so by Theorem 7.12, K has an object of presentability rank λ + . Assumenow that λ is a singular cardinal. Then λ is in particular a limit cardinal, so (byESCH) λ is µ ′ -closed. By Theorem 7.12, K has an object of presentability rank λ + . (cid:3) Filtrations We consider conditions under which, in a general category, we can ensure than anyobject is not merely the colimit of an appropriately directed system of objects ofstrictly smaller internal size, but rather the colimit of a chain of such objects. Theexistence of such filtrations (sometimes also called resolutions ) is crucial to a hostof model-theoretic constructions, and should be of considerable use in the furtherdevelopment of classification theory at the present level of generality. IZES AND FILTRATIONS IN ACCESSIBLE CATEGORIES 23 Definition 8.1. For µ a regular cardinal and λ an infinite cardinal, a ( µ, < λ ) -chain (in a category K ) is a diagram h M i : i < µ i indexed by µ , all of whoseobjects are ( < λ )-presentable. We call µ the length of the chain. A ( < λ ) -chain isa ( µ, < λ )-chain for some regular µ < λ . For λ a regular cardinal, a λ -chain is a( < λ + )-chain.For θ a regular cardinal, we say that a chain h M i : i < µ i is θ -smooth if for every i < µ of cofinality at least θ , M i is the colimit of h M j : j < i i .Note that ( µ, < λ )-chains are ( µ, < λ )-systems in the sense of Definition 2.7. Wewill use the terminology of systems introduced in the preliminaries. The readermay also wonder why we are looking only at chains indexed by a regular cardinal.This is because any system h M i : i ∈ I i indexed by a linear order I has a cofinalsubsystem of the form h M i j : i j < µ i , where µ is the cofinality of I .The next definition is the object of study of this section: Definition 8.2. Let K be a category. A filtration of an object M is a ( < r K ( M ))-chain with colimit M . We call M filtrable if it (has a presentability rank and) hasa filtration.Observe that the length of a filtration is determined by the presentability rank: Lemma 8.3. Let λ be a regular cardinal and let K be a category. If there existsa ( < λ )-chain whose colimit is not ( < λ )-presentable, then λ is a successor and thechain must have length cf( λ − ). In particular, any filtrable object M has successorpresentability rank and any of its filtrations will have length cf( | M | K ). Proof. Let µ < λ be a regular cardinal and let h M i : i < µ i be a ( µ, < λ )-chainin K with a colimit M that is not ( < λ )-presentable. By Fact 2.8(2), the chain isproper. Next, assume for a contradiction that λ is weakly inaccessible. By Fact2.8(1), M is ( < λ )-presentable, a contradiction to the fact that it has presentabilityrank λ . This shows that λ is a successor. Let λ = λ − . We now have to see thatcf( λ ) = µ . We consider two cases depending on whether λ is regular or singular: • If λ is regular, then h M i : i < µ i is a ( µ, λ )-system. Since µ < λ , weknow that µ ≤ λ . If µ < λ , then by Fact 2.8(1), M would be ( µ + + λ )-presentable, hence λ -presentable, contradicting that it has presentabilityrank λ . Thus µ = λ = cf( λ ). • If λ is singular, then h M i : i < µ i is a ( µ, < λ )-system, and moreover(because µ is regular) µ = λ , so µ + < λ . If cf( λ ) > µ , then there existsa regular λ < λ such that h M i : i < µ i is a ( µ, λ )-system. By Fact 2.8(1), M is ( µ + + λ )-presentable, hence ( < λ )-presentable, a contradiction.Thus cf( λ ) ≤ µ . If cf( λ ) < µ , let h θ α : α < cf( λ ) i be an increasingchain of regular cardinals cofinal in λ . For α < cf( λ ), let I α := { i <µ | M i is θ α -presentable } . By Lemma 2.4 (applied to I = µ ), there exists α < cf( λ ) so that I α is cofinal in µ . In particular, M is still the colimit of h M i : i ∈ I α i . The latter system is a ( µ, θ α )-system, hence (again by Fact2.8(1)) M is ( µ + + θ α )-presentable, so ( < λ )-presentable, a contradiction.The only remaining possibility is that cf( λ ) = µ , which is what we wantedto prove. (cid:3) It follows that if the category has directed colimits, we can take the filtration to besmooth. More generally: Lemma 8.4. Any filtration of an object M of presentability rank λ is boundedly( < λ )-presentable (Definition 6.3). In particular, if θ < λ is regular such that K has θ -directed colimits, then M has a θ -smooth filtration. Proof. Let µ < λ be regular, and let h M i : i < µ i be a ( < λ )-chain whose colimitis M . By Lemma 8.3, λ is a successor cardinal and µ = cf( λ − ). Let δ < µ be alimit ordinal. We will show that the colimit of h M i : i < δ i (assuming it exists) is( < λ )-presentable. The “in particular” part will then follow, since, when cf( δ ) ≥ θ ,it suffices to replace M δ by the colimit of h M i : i < δ i .Note that δ < µ ≤ λ − , so δ + < λ . By cofinality considerations, there exists aregular cardinal λ < λ such that for all i < δ , M i is λ -presentable. By Fact2.8(1), we get that the colimit of h M i : i < δ i is ( δ + + λ )-presentable, hence( < λ )-presentable, as desired. (cid:3) Using the definition of presentability, it is also easy to generalize the well knownfacts that, for objects of regular cardinality (that is, of presentability rank thesuccessor of a regular cardinal), any two smooth filtrations are the same on a club :a closed unbounded set of indices. See for example [BGL + 16, 6.11].We give a name to categories where every object of a given presentability rank isfiltrable: Definition 8.5. For a regular cardinal λ , we say an accessible category K is λ -filtrable if any object of presentability rank λ is filtrable. For a regular cardinal µ ,we say that K is well µ -filtrable if it is λ -filtrable for any regular λ ≥ µ . We say K is well filtrable if it is well µ -filtrable for some regular cardinal µ .Similarly, we say that K is almost λ -filtrable if any object M of presentabilityrank λ is a retract of a filtrable λ -presentable object N (i.e. there exists a splitepimorphisms from N to M ). Define almost well λ -filtrable and almost well filtrable as expected. Remark 8.6. The technical notion of being almost filtrable is included here be-cause we do not know whether retracts of filtrable objects are filtrable. Of course,in categories where all morphisms are monos, retractions are just isomorphisms andso this technical distinction is irrelevant.In the rest of this section, we give some conditions implying existence of filtrations.The next result is our main tool: Lemma 8.7. Let K be a category, let µ ≤ θ ≤ λ be given with µ regular, θ regular,and µ ⊳ λ (see Definition 2.1(5)). Let A ∈ K be an object of presentability rank λ + .If K has µ -directed colimits and A is the colimit of a ( µ, θ )-system of cardinality atmost λ , then A is filtrable. Proof. Let D : I → K be a ( µ, θ )-system with | I | ≤ λ . It is a direct consequence ofthe definition of the ⊳ relation that we can find h I i : i < λ i an increasing chain of IZES AND FILTRATIONS IN ACCESSIBLE CATEGORIES 25 µ -directed subposets of I such that S i<λ I i = I and for all i < λ , | I i | < λ i for someregular λ i ∈ [ θ, λ ]. By Fact 2.8(1), for each i < λ we have that A i := colim( D ↾ I i )is λ i -presentable, hence ( < λ + )-presentable, so the system consisting of h A i : i < λ i is the desired chain. (cid:3) We deduce the main theorem of this section. Note that the third part is an im-provement on [Ros97, Lemma 1], which assumed in addition existence of directedcolimits. Theorem 8.8. Let µ be a regular cardinal, and let λ be a cardinal such that µ ⊳ λ and µ ⊳ λ + .(1) If K is a category which is ( µ, θ )-accessible for some regular θ ≤ λ , then K is almost λ + -filtrable.(2) Any µ -accessible category is λ + -filtrable.(3) Any µ -accessible category is µ + -filtrable. Proof. (1) Fix θ ≤ λ regular such that K is ( µ, θ )-accessible. By definition of µ ⊳ λ ,we can increase θ if necessary to assume without loss of generality that µ ⊳ θ . Let A be an object of presentability rank λ + . By the moreover partof Fact 2.11, A is a λ + -directed colimit of a diagram whose objects are all µ -directed colimits of at most λ -many θ -presentables. Thus A is a retractof such a colimit, and the result follows from Lemma 8.7.(2) As before, but this time, by [MP89, 2.3.11], we actually have that A isa µ -directed colimits of at most λ -many µ -presentable objects (not just aretract of such a colimit).(3) Let A be an object of presentability rank µ + . Since µ ⊳ µ + , by [MP89,2.3.11], A is a the colimit of a µ -directed diagram D : I → K containingat most µ -many µ -presentable objects. Enumerate I as { i α : α < µ } . Webuild h j α : α < µ i an increasing sequence in I such that j α ≥ i β whenever β < α . This is possible because I is µ -directed, and in the end we get that h D j α : α < µ i is the desired chain. (cid:3) We deduce some results on accessible categories with directed colimits. Note inparticular that we have recovered the well known fact that any AEC K is wellLS( K ) ++ -filtrable. Corollary 8.9. (1) Any finitely accessible category is well ℵ -filtrable.(2) Any λ -accessible category with directed colimits is almost well λ + -filtrable.(3) Any λ -accessible category with directed colimits and all morphisms monosis well λ + -filtrable. If λ is regular this follows, but not necessarily if λ is singular (e.g. if λ is strong limit). Proof. The first two results are immediate by setting µ := ℵ in Theorem 8.8.For the third, simply observe that a retraction which is a monomorphism is anisomorphism. (cid:3) Remark 8.10. Using a fat small object argument to eliminate retracts (see [MRV14]),we can also show that any locally λ -presentable category is well λ + -filtrable. References [AHS04] Jiˇr´ı Ad´amek, Horst Herrlich, and George E. Strecker, Abstract and concrete categories ,online edition ed., 2004, Available from http://katmat.math.uni-bremen.de/acc/ .[AR94] Jiˇr´ı Ad´amek and Jiˇr´ı Rosick´y, Locally presentable and accessible categories , LondonMath. Society Lecture Notes, Cambridge University Press, 1994.[BB17] John T. Baldwin and Will Boney, Hanf numbers and presentation theorems in AECs ,Beyond first order model theory (Jos´e Iovino, ed.), CRC Press, 2017, pp. 327–352.[BGL + 16] Will Boney, Rami Grossberg, Michael J. Lieberman, Jiˇr´ı Rosick´y, and Sebastien Vasey, µ -Abstract elementary classes and other generalizations , The Journal of Pure andApplied Algebra (2016), no. 9, 3048–3066.[Bon14] Will Boney, Tameness from large cardinal axioms , The Journal of Symbolic Logic (2014), no. 4, 1092–1119.[BR12] Tibor Beke and Jiˇr´ı Rosick´y, Abstract elementary classes and accessible categories ,Annals of Pure and Applied Logic (2012), 2008–2017.[BY05] Itay Ben-Yaacov, Uncountable dense categoricity in cats , The Journal of SymbolicLogic (2005), no. 3, 829–860.[HH09] ˚Asa Hirvonen and Tapani Hyttinen, Categoricity in homogeneous complete metricspaces , Archive for Mathematical Logic (2009), 269–322.[Jec03] Thomas Jech, Set theory , 3rd ed., Springer-Verlag, 2003.[Lie11] Michael J. Lieberman, Category-theoretic aspects of abstract elementary classes , An-nals of Pure and Applied Logic (2011), no. 11, 903–915.[LR16] Michael J. Lieberman and Jiˇr´ı Rosick´y, Classification theory for accessible categories ,The Journal of Symbolic Logic (2016), no. 1, 151–165.[LR17] , Metric abstract elementary classes as accessible categories , The Journal ofSymbolic Logic (2017), no. 3, 1022–1040.[LRV] Michael J. Lieberman, Jiˇr´ı Rosick´y, and Sebastien Vasey, Weak factorization systemsand stable independence , Preprint. URL: https://arxiv.org/abs/1904.05691v2 .[LRV19a] , Forking independence from the categorical point of view , Advances in Mathe-matics (2019), 719–772.[LRV19b] , Internal sizes in µ -abstract elementary classes , Journal of Pure and AppliedAlgebra (2019), no. 10, 4560–4582.[LRV19c] , Universal abstract elementary classes and locally multipresentable categories ,Proceedings of the American Mathematical Society (2019), no. 3, 1283–1298.[Lur09] Jacob Lurie, Higher topos theory , Annals of Mathematics Studies, no. 170, PrincetonUniversity Press, 2009.[MP89] Michael Makkai and Robert Par´e, Accessible categories: The foundations of categoricalmodel theory , Contemporary Mathematics, vol. 104, American Mathematical Society,1989.[MRV14] Michael Makkai, Jiˇr´ı Rosick´y, and Luk´aˇs Vokˇr´ınek, On a fat small object argument ,Advances in Mathematics (2014), 49–68.[Ros97] Jiˇr´ı Rosick´y, Accessible categories, saturation and categoricity , The Journal of Sym-bolic Logic (1997), no. 3, 891–901.[She87] Saharon Shelah, Classification of non elementary classes II. Abstract elementaryclasses , Classification Theory (Chicago, IL, 1985) (John T. Baldwin, ed.), LectureNotes in Mathematics, vol. 1292, Springer-Verlag, 1987, pp. 419–497.[SV] Saharon Shelah and Sebastien Vasey, Categoricity and multidimensional diagrams ,Preprint. URL: http://arxiv.org/abs/1805.06291v2 .[Vas] Sebastien Vasey, Accessible categories, set theory, and model theory: an invitation ,Preprint. URL: https://arxiv.org/abs/1904.11307v1 . IZES AND FILTRATIONS IN ACCESSIBLE CATEGORIES 27 [Vas16] , Infinitary stability theory , Archive for Mathematical Logic (2016), 567–592. E-mail address : [email protected] URL : Department of Mathematics and Statistics, Faculty of Science, Masaryk University,Brno, Czech RepublicInstitute of Mathematics, Faculty of Mechanical Engineering, Brno University ofTechnology, Brno, Czech Republic E-mail address : [email protected] URL : Department of Mathematics and Statistics, Faculty of Science, Masaryk University,Brno, Czech Republic E-mail address : [email protected] URL : http://math.harvard.edu/~sebv/http://math.harvard.edu/~sebv/