Definable orthogonality classes in accessible categories are small

DEFINABLE OR THOGONALI T Y CLASSES IN A CCESSIBLE CA TEGORIES ARE SMALL JO AN BAGARIA, CARLES CASA CUBER T A, A. R. D. MA TH I AS, AND JI ˇ R ´ I ROSICK ´ Y Abstra ct. W e l ow er substantially the strength of th e assumptions needed for the v alidity of ce rtain results in category theory and ho- motopy theory whic h were kn own to follo w fro m V opˇ enk a’s principle. W e prov e that the necessary large-cardinal hyp otheses dep end on t he complexity of the form ulas deﬁning the given classes, in the sense of the L´ evy hierarc hy . F or example, the statement t h at, for a class S of mo r- phisms in a lo cally presentable category C of structures, the orth ogonal class of ob jects S ⊥ is a small -orthogonality cl ass (hence reﬂective) can b e prove d in ZFC if S is Σ 1 , while it follo ws from the existence of a prop er class of sup ercompact cardinals if S is Σ 2 , and from th e exis- tence of a prop er class of what we call C ( n )-extendible cardinals if S is Σ n +2 for n ≥ 1. These cardinals form a n ew hierarch y , and w e show that V opˇ enk a’s principle is equ iv alent to the existence of C ( n )-extend ible cardinals for all n . As a consequence, we prov e that the existence of cohomological lo- calizations of simplicial sets, a long-standing op en problem in algebraic top ology , is implied by the existence of arbitrarily large sup ercompact cardinals. This follow s from the fact that the class of E ∗ -equivalences is Σ 2 -definable, where E denotes a sp ectrum treated as a parameter. In contrast with t his fact, the class of E ∗ -equivalences is Σ 1 -definable, from which it follo ws (as is w ell k now n) t hat th e existence of h omologica l localizations is prov able in ZFC. Introduction The an s w ers to certain questions in category theory turn out to dep end on set theory . A typical examp le is whether ev ery full limit-closed su b category of a complete category C is reﬂectiv e. On the one hand, there are counterex- amples inv olving the category of top o logical spaces and con tinuous fu n c- tions [45]. On the other h and, as explained in [2], an aﬃrmativ e answe r to this qu estion for lo cally pr esen table categories is implied by a large-cardinal axiom called V op ˇ enk a’s prin ciple (stating that, for ev ery prop er class of structures of the same typ e, there exists a nont rivial elemen tary embedd ing b et we en tw o of them). Large ca rdin als were used in a similar wa y in [17] to sho w that the ex- istence of cohomolo gical lo calizations, a famous unsolv ed pr oblem, follo ws Date : Nove mber 11, 2018. 2000 M athematics Subje ct Cl assiﬁc ation. 03E55, 03C55, 18A40, 18C35, 55P60. The auth ors w ere supp orted by the Spanish Ministry of Science and Innov ation under gran ts MTM2 007-63277, MTM2008 -03389, MTM2010-1583 1 and MTM2 011-25229, by the Generalitat de Catalunya under grants 2005 SGR 606, 2005 SGR 738, 2009 S GR 119 and 2009 SGR 187, and by t he Ministry of Edu cation of the Czec h Repub lic under pro ject MSM002162 2409. This researc h was supp orted through the Researc h in P airs programme by the Mathematisc hes F orsch ungsinstitut Ob erw olfac h in 2008. 1 DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 2 from V opˇ enk a’s principle. Other relev an t consequences of V op ˇ enk a’s princi- ple in algebraic top ology w ere found in [15], [16], [19], [43]. Ho wev er, the precise consistency stren gth of m any implications of this axiom in category theory or homotop y theory is not kno wn, an d in some cases the question of whether suc h statement s are p r o v able in ZF C remains unanswered. A rele- v an t step in this direction was made in [42]. In another direction, it w as p oin ted out in [9] th at certain results ab o ut accessible categories th at follo w from V op ˇ enk a’s principle are still true un der m uc h we ak er large-ca rdin al assumptions. This claim is based on the follo w- ing ﬁnd ing, w hic h is the sub ject of the present article: the assumptions ne e de d to infer r eﬂe ctivity or smal lness of ortho gonality classes in ac c essib le c ate gories may d ep end on the c omplexity of the formulas in the language of set the ory deﬁning these classes . Here “complexit y” is meant in th e sense of the L ´ evy h ierarc h y [31, Ch . 13]. Recall that Σ n form ulas and Π n form ulas are deﬁn ed ind uctiv ely as follo ws: Π 0 form ulas are the same as Σ 0 form ulas, namely formulas in which all qu an tiﬁers are b oun ded; Σ n +1 form ulas are of the form ∃ x ϕ w here ϕ is Π n , an d Π n +1 form ulas are of the form ∀ x ϕ where ϕ is Σ n . F or example, as we pro v e in this article, if S is a f ull limit-closed sub cate- gory of a lo c ally presenta ble category C of s tructures, and S can b e deﬁn ed with a Σ 2 form ula (p o ssibly with parameters), then the existence of a prop er class of su p ercompact cardin als s uﬃces to ensur e reﬂectivit y of S . Moreo v er, remark ably , if S can b e d eﬁned with a Σ 1 form ula, then the r eﬂ ectivit y of S is prov able in ZF C. In case of a more complex deﬁnition of S , its reﬂectivit y follo ws from the existence of a prop er class of what w e call C ( n ) - extendible c ar dinals , for some n . T hese cardinals f orm a n atur al h ierarc h y ranging from extendible cardinals [31, 20.22] wh en n = 1 to V opˇ enk a’s prin ciple. In deed, as s tated in C orollary 6.9 b elow, V op ˇ enk a’s p rinciple is equiv alen t to th e claim that there exists a C ( n )-extendible cardinal for ev ery n < ω . W e denote by C ( n ) the pr op er class of cardinals α su ch that V α is a Σ n -elemen tary submo d el of the set-theoretic u niv erse V , and sa y that a cardinal κ is C ( n )-extendible if κ ∈ C ( n ) and for all λ > κ in C ( n ) there is an elemen tary em b edd ing j : V λ → V µ for some µ ∈ C ( n ) with critical p oi nt κ , su c h that j ( κ ) ∈ C ( n ) and j ( κ ) > λ . By wa y of this approac h , w e pro v e that the existence of cohomolog i- cal lo calizations of simplicial sets follo ws from the existence of a prop er class of sup ercompact cardin als. This r esu lt uses the fact, prov ed in Theo- rem 9.3 b elo w, that for ev ery (Bousﬁeld–F r iedlander) sp ectrum E the class of E ∗ -acyclic simp licial sets (wher e E ∗ denotes the r educed cohomology the- ory represen ted by E ) can b e deﬁned by means of a Σ 2 form ula with E as a parameter. Ho wev er, the class of E ∗ -acyclic simplicial s ets (where E ∗ no w denotes homology) can b e deﬁned with a Σ 1 form ula. This is consistent with the fact that the existence of homological lo calizations can b e p r o v ed in ZF C, as done in deed b y Bousﬁeld in [11]; see also [5]. The reason why classes of homology acyclics h a v e lo wer complexit y than classes of cohomology acyclics is that, for a ﬁbr an t s im p licial set Y with basep oint , the statemen t “all p ointed maps f : S n → Y are nullhomotopic”, where S n is the simp licial n -sphere, is absolute b etw een transitive m o dels of ZFC, sin ce a simplicial map S n → Y is determined by a sin gle n -simplex DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 3 of Y satisfying certain conditions expressible in terms of Y with b ounded quan tiﬁers; cf. [40, 3.6]. Ho wev er, if X and Y are simp licial sets with base- p oint s x 0 and y 0 , then the s tatemen t “all p oint ed maps f : X → Y are n ullhomotopic” inv olve s unb ounded quan tiﬁers, since it is formalized, for example, b y stating that ∀ f ( f is a map from X to Y → ∃ h ( h is a h omotopy from f to y 0 )) . Therefore, f or a sp ectrum E , th ere might exist E ∗ -acyclic simp licial sets in a transitive mo d el of ZF C con taining E that fail to b e E ∗ -acyclic in some larger mo del, while the class of E ∗ -acyclic simplicial sets is absolute. See Section 9 for a detailed d iscussion of th ese f acts. Another consequence of this article is th at the main theorem of [9] can n ow b e pro v ed for r eﬂ ections, n ot n ecessarily epireﬂections. Th us, if there are arbitrarily large su p ercompact cardin als, then ev ery r eﬂ ection L on an acces- sible category of structures is an F -reﬂection for some set of morp hisms F , pro vided that the class of L -equiv alences is Σ 2 ; see Corollary 8.5 b elo w. (Boldface t yp e s Σ n or Π n are us ed to d enote the fact that the corresp ond- ing formulas m a y con tain parameters.) W e also prov e that the F r eyd–Kelly orthogonal s ub category problem [25], asking if S ⊥ is reﬂ ective for a class of morp hisms S in a suitable category , h as an aﬃrmativ e answer in Z F C f or Σ 1 classes in lo cally pr esen table categories of s tructures. It is also true for Σ 2 classes if a prop er class of sup ercompact cardinals is assumed to exist, and for Σ n +2 classes if there is a prop er class of C ( n )-extendible cardinals for n ≥ 1. W e say that S is deﬁnable with suﬃciently low c omplexity to en compass all these cases in a single ph rase. Essen tially the same argumen ts hold in the homotop y categ ory of sim- plicial sets, hence yielding a simpler and more accurate answer than in [17] (where V opˇ enk a’s p rinciple wa s used) to F arj oun’s question in [20] of whether ev ery homotop y r eﬂection on simp licial sets is an f -localizatio n for some map f . Lo calizations with resp ect to sets of maps were constructed in [12], [21], [28], and the extension to prop er classes of maps w as carried ou t in [17] using V op ˇ enk a’s p rinciple. Here we pro v e that lo calizations with resp ect to prop er classes of maps exist wh enev er the giv en classes are deﬁnable w ith suﬃcien tly lo w complexit y . W e w arn the reader that in this article, as w ell as in [9 ], complexit y of classes of ob jects or morp h isms in an accessible category C is meant under th e assumption that C is accessibly embedd ed into a category of structures. This happ en s canonically with the catego ry of simplicial sets and with the cat- egory of Bousﬁ eld–F riedlander sp ectra, or, more generally , with categories of mo dels of basic theories in any language. T ermin ology and bac kground can b e found in [2, 5.B], where it is prov ed that every accessible category is equiv alen t to one which is accessibly embedd ed in to a catego ry of structures. Ac knowledgemen ts W e are muc h indebted to the referee for a deep and careful reading of the m an uscript and a num b er of p ertinen t corrections. 1. Ca tegories of structures Most of the results in this article refer to catego ries of structur es (p ossibly man y-sorted, in a language of any cardinalit y). F or the con v enience of the reader, w e start by recalling terminology and b ac kgrou n d ab out structures DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 4 and m o dels in this s ection. Additional d etails can b e foun d, among many other sources, in [2 , Ch. 5] and [31, Ch . 12]. F or a regular cardinal λ , a λ - ary S -sorte d signatur e Σ consists of a set S of sorts , a set Σ op of op er ation symb ols , another set Σ rel of r elation symb ols , and an arity function that assigns to eac h op eration symbol an ord inal α < λ , a sequence h s i : i ∈ α i of input sorts and an output sort s ∈ S , and to eac h relation symb ol an ord in al β < λ and a sequence of sorts h s j : j ∈ β i . An op eration symbol with α = ∅ is called a c onstant symb ol . A signature Σ is called op er ational if Σ rel = ∅ and r elational if Σ op = ∅ . Giv en an S -sorted signature Σ , a Σ - structur e is a triple X = h{ X s : s ∈ S } , { σ X : σ ∈ Σ op } , { ρ X : ρ ∈ Σ rel }i consisting of an underlying S -sorte d set or universe , denoted b y { X s : s ∈ S } or ( X s ) s ∈ S , together w ith a f u nction σ X : Y i ∈ α X s i − → X s for eac h op erat ion symb ol σ ∈ Σ op of arit y h s i : i ∈ α i → s (includ in g a distinguished elemen t of X s for eac h constan t symb ol of sort s ), and a set ρ X ⊆ Y j ∈ β X s j for eac h relation symbol ρ ∈ Σ rel of arit y h s j : j ∈ β i . A homomorphism f : X → Y b et wee n tw o Σ -str u ctures is an S -sorted function ( f s : X s → Y s ) s ∈ S preserving op erations and relatio ns. F or eac h signature Σ, the category of Σ-structures and their homomorph isms will b e denoted by Str Σ. Giv en a λ -ary S -sorted signature Σ, the language L λ (Σ) consists of sets of variables , terms , an d formulas , wh ic h are d eﬁned as follo ws. T here is a family W = { W s : s ∈ S } of sets of cardinalit y λ , the elemen ts of W s b eing variables of sort s . One d eﬁnes terms b y declaring that eac h v ariable is a term and, for eac h op erati on sym b ol σ ∈ Σ op of arit y h s i : i ∈ α i → s and eac h collection of terms τ i of s ort s i , the expression σ ( τ i ) i ∈ α is a term of sort s . Atomic formulas are expressions of the form τ 1 = τ 2 and ρ ( τ j ) j ∈ β , where ρ ∈ Σ rel is a relation s y mb ol of arity h s j : j ∈ β i and eac h τ j is a term of sort s j with j ∈ β . F ormulas are built in ﬁnitely many steps from th e atomic formulas by means of logical conn ectiv es and quant iﬁers. Th us, if { ϕ i : i ∈ I } are formulas and | I | < λ , then so are the conjunction V i ∈ I ϕ i and the disju nction W i ∈ I ϕ i . Quant iﬁcation is allo w ed o v er sets of v ariables of cardinalit y smaller than λ ; that is, ( ∀ ( x i ) i ∈ I ) ϕ and ( ∃ ( x i ) i ∈ I ) ϕ are form ulas if ϕ is a formula and | I | < λ . V ariables that app ear unquantiﬁed in a formula are called fr e e . If a f or- m ula is denoted by ϕ ( x i ) i ∈ I , it is mean t that eac h x i is a fr ee v ariable. Eac h language L λ (Σ) determines a satisfaction r elation b et w een Σ-struc- tures and formulas with an assignmen t for their fr ee v ariables. If ϕ ( x i ) i ∈ I is a form ula where eac h x i is a fr ee v ariable of sort s i and X is a Σ-stru cture, a variable assignment , denoted by x i 7→ a i , is a function a : I → ∪ s ∈ S X s suc h that a ( i ) ∈ X s i for all i . Satisfacti on of a formula ϕ in a Σ-structur e X is deﬁned inductiv ely , starting with the atomic f orm ulas and quantifying o v er subsets of ∪ s ∈ S X s of cardinalit y sm aller than λ ; see [2 , § 5.26] for d etails. DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 5 W e wr ite X | = ϕ ( a i ) i ∈ I if ϕ is satisﬁed in X und er an assignmen t x i 7→ a i for all its f ree v ariables x i . A formula w ithout free v ariables is called a sentenc e . A set of sen tences is called a the ory . A mo del of a theory T in a language L λ (Σ) is a Σ-structur e satisfying all sentences of T . F or eac h theory T , we denote b y Mo d T the full sub category of Str Σ consisting of all mo d els of T . A language L λ (Σ) is called ﬁnitary if λ = ω (the least in ﬁnite card in al); otherwise it is inﬁnitary . A n esp eci ally imp ortant ﬁ nitary language is th e language of set the ory . T his is the ﬁr st-order ﬁnitary language corresp onding to the signature with one sort, namely “sets”, and one b inary relation symb ol (“mem b ership”). Hence the atomic formulas are x = y and x ∈ y , wh ere x and y are sets. Deﬁne, r ecursiv ely on the class of ord inals, V 0 = ∅ , V α +1 = P ( V α ) for all α , wher e P denotes th e p o w er-set op eration, and V λ = S α<λ V α if λ is a limit ordin al. Th en ev ery set is an elemen t of some V α ; see [30, Lemma 9.3] or [31 , Lemma 6.3]. The r ank of a set X is the least ordinal α suc h that X ∈ V α +1 . Hence V α is the set of all s ets whose rank is less than α . The universe V of all sets is the u n ion of V α for all ord inals α . Ev erything in this article is form ulated in ZFC (Zermelo–F raenk el set theory with the axiom of c hoice). Thus, a c lass consists of all sets for w hic h a certain form ula of the language of set theory is satisﬁed, p ossibly with parameters. More p r ecisely , a class C is deﬁne d by a formula ϕ ( x, y 1 , . . . , y n ) with p ar ameters p 1 , . . . , p n if C = { x : ϕ ( x, p 1 , . . . , p n ) } , where satisfaction, if unsp eciﬁed, is meant in the unive rse V . The sets p 1 , . . . , p n are ﬁxed v alues of y 1 , . . . , y n under ev ery v ariable assignment. T o simplify the n otatio n, we often replace p 1 , . . . , p n b y a single parameter p = { p 1 , . . . , p n } . A class whic h is not a s et is called a pr op er class . E ac h set A is deﬁnable with A itself as a parameter by A = { x : x ∈ A } . In this article, a mo del of ZFC will b e a p air h M , ∈i where M is a set or a prop er class and ∈ is the restriction of th e members h ip relation to M , in whic h the formalized ZFC axioms are satisﬁed. Th us, if we neglect the fact that M can b e a p rop er class, we ma y view h M , ∈i as a Σ-stru cture w here Σ is the relational signature of the language of set theory , and in fact a mo del of the theory consisting of the formalized ZFC axioms. In particular, h V , ∈i itself is suc h a mo del. A class M is tr ansitive if ev ery elemen t of an elemen t of M is an elemen t of M . W e shall alw a ys assume that mo dels of ZF C are transitive , b ut not necessarily in ner (a mo d el is called inner if it is trans itiv e and contai ns all the ord inals). 2. The L ´ evy hierarchy In th is section we sp ec ialize to the language of set theory . Th us, giv en t w o classes M ⊆ N , w e sa y that a f orm ula ϕ ( x 1 , . . . , x k ) is absolute b etwe en M and N if, f or all a 1 , . . . , a k in M , N | = ϕ ( a 1 , . . . , a k ) if and only if M | = ϕ ( a 1 , . . . , a k ) . W e sa y th at a form ula ϕ ( x 1 , . . . , x k ) is upwar d absolute for transitiv e mo dels of some theory T if, give n any t w o suc h mo d els M ⊆ N and given DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 6 a 1 , . . . , a k ∈ M for wh ic h ϕ ( a 1 , . . . , a k ) is true in M , ϕ ( a 1 , . . . , a k ) is also true in N . And w e say that ϕ is downwar d absolute if, in the same situation, if ϕ ( a 1 , . . . , a k ) holds in N th en it holds in M . A f orm ula is absolute if it is b oth u p wa rd and do wnw ard absolute. If T is unsp ec iﬁed, th en it should b e understo o d that T is b y default the set of all form alized Z FC axioms. If it is mean t, on th e con trary , that T = ∅ , th en we sp eak of absoluteness b et we en transitiv e classes. A class C is u pwar d absolute b et w een transitiv e classes M ⊆ N if it is deﬁnable, p ossibly with a set p of parameters, by a f orm ula that is upw ard absolute b et wee n M and N . Downwar d absolute classes are deﬁned anal- ogously , an d we s a y that C is absolute b et w een M and N if it is upw ard absolute and do wnw ard absolute, hence allo wing the p ossibilit y that C = { x : ϕ ( x, p ) } = { x : ψ ( x, p ) } where ϕ is u p wa rd absolute an d ψ is do wnw ard absolute. In this situation, N | = x ∈ C if and only if M | = x ∈ C , assuming that p ∈ M . The follo wing termin ology is due to L ´ evy; see [31 , Ch. 13]. A formula of the language of set th eory is said to b e Σ 0 if all its quantiﬁers are b oun ded, that is, of the form ∃ x ∈ a or ∀ x ∈ a . Then Σ n formulas and Π n formulas are deﬁn ed ind uctiv ely as follo ws: Π 0 form ulas are the same as Σ 0 form ulas; Σ n +1 form ulas are of the form ( ∃ x 1 . . . x k ) ϕ , wh ere ϕ is Π n ; and Π n +1 form ulas are of the form ( ∀ x 1 . . . x k ) ϕ , where ϕ is Σ n . W e sa y that a formula is Σ n ∧ Π n if it is a conjunction of a Σ n form ula and a Π n form ula. Classes can b e deﬁned b y distinct formulas and, m ore generally , prop erties and mathematical statemen ts can b e formalized in the language of set theory in m any d iﬀeren t wa ys. W e s ay that a class C is Σ n -deﬁnable (or, shortly , that C is Σ n ) if there is a Σ n form ula ϕ ( x, y ) s uc h that C = { x : ϕ ( x, p ) } for a set p of parameters. Similarly , a class is Π n if it can b e deﬁn ed by some Π n form ula with parameters. A class is call ed ∆ n if it is b ot h Σ n and Π n . F or n otatio nal conv enience, if no p arameters are inv olve d, then we write that a class C is Σ n , Π n or ∆ n , using lightfac e t yp es. The same terminology is used with statement s or inform al expressions; for example, “ λ is a cardinal” is a Π 1 statemen t [31, Lemm a 13.13], w h ile “ f is a function”, “ α is an ord in al” or “ ω is the least nonzero limit ordin al” are ∆ 0 statemen ts [31, Lemma 12.10]. If a class C is Σ 1 with a set p of parameters, th en it is upw ard abs olute for transitiv e classes con taining p . In fact, given a Σ 1 form ula ∃ x ϕ ( x, y ) where ϕ is Σ 0 and given a set p of parameters, supp ose that M ⊆ N are transitiv e classes with p ∈ M . Then, if M | = ∃ x ϕ ( x, p ), we may infer that N | = ∃ x ϕ ( x, p ) as well, since if a ∈ M w itnesses that ϕ ( a, p ) holds in M , then a ∈ N and ϕ ( a, p ) also holds in N , since ϕ is absolute. Con v ersely , if a class C is upw ard absolute for tran s itiv e m o dels of some ﬁnite fragmen t Z F C ∗ of ZFC, then it is Σ 1 . T o p ro v e this claim, su pp o se that C is deﬁ ned by a formula ϕ ( x, y ) that is upw ard absolute for transitive mo dels of ZFC ∗ with a set p of parameters. Then C is also deﬁned by the follo win g Σ 1 form ula: (2.1) ∃ M [ M is transitiv e ∧ { x, p } ⊂ M ∧ M | = ( ϕ ( x, p ) ∧ ( V ZF C ∗ ))] . Indeed, if a ∈ C then ϕ ( a, p ) holds in V , and it follo ws from the Reﬂection Principle [31, Theorem 12.14] that there is an ord inal α with { a, p } ∈ V α DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 7 suc h that V α | = ϕ ( a, p ) and all the sen tences in the ﬁnite set ZF C ∗ are satisﬁed in V α , s o V α witnesses (2.1). And , if a set M witnesses (2.1 ) for some v ariable assignmen t x 7→ a , then, since ϕ ( x, y ) is upw ard absolute f or transitiv e mo d els of ZF C ∗ , we infer that ϕ ( a, p ) holds in V , that is, a ∈ C . Similarly , if a class C is deﬁned by a Π 1 form ula w ith parameters, then it is do wnw ard absolute for transitiv e classes con taining the parameters, and, if C is d o wn wa rd absolute for transitiv e mo dels of some ﬁnite fragmen t of Z F C, then it is Π 1 , analogously as in (2.1 ). W e conclude that ∆ 1 classes are absolute for transitiv e classes con taining the p arameters. The f ollo wing are examples of nonabsoluteness whic h will b e relev ant in this article. Example 2.1. The class of top ological spaces is Π 1 , sin ce the u nion of ev ery collect ion of op en s ets must b e op en. T hus, a top ology on a set X in some mo del of ZFC may fail to b e a top ology on X in a larger mo d el. Ho wev er, the class of s im p licial sets is ∆ 0 (see Section 9). Example 2.2. Let C b e the class of all ab e lian groups of the form Z κ , wh ere κ is a card in al. Then A ∈ C if and only if ∃ x ( x is a cardinal ∧ ∀ y ( y ∈ A ↔ y is a fun ction from x to Z )) , whic h is a Σ 2 form ula, since the expr ession wr itten within the outer paren- theses is Π 1 . In ev ery mo d el of Z FC with measurable cardinals, the follo wing sen tence is true: ∃ κ ∃ f ( κ is an inﬁnite cardinal ∧ f is a group homomorph ism from Z κ to Z ∧ f ( Z <κ ) = 0 ∧ f 6 = 0) , while if this holds then th e sm allest κ with th is prop ert y is measurable, according to [22]; see [23] for further details. Therefore, this sen tence is false in a mo d el of ZFC without measurable cardinals wh ile it is true in a mo del of ZF C with measurable cardinals. Example 2.3. F or a cardinal λ and a set X , we denote by P λ ( X ) the set of all subsets of X wh ose cardin ality is smaller th an λ . Note ﬁ rst that, although the statemen t “ A is a su bset of B ” is ∆ 0 , the statemen t “ A is the set of all subsets of B ” is formalized with the follo w in g Π 1 form ula: ∀ a ∈ A ( a ⊆ B ) ∧ ∀ x ( x ⊆ B → x ∈ A ) . This s tatemen t cannot b e formalized with any upw ard absolute formula, since, if we pic k a countable transitiv e mo del M of ZF C and A is th e set of all su bsets of the natural n umb er s N in M , then A cannot b e the set of all subsets of N in th e unive rse V , since A is countable. The assertion “ x is ﬁnite” is ∆ 1 , since it is equiv alen t to the statemen t that ther e exists a bijection b et w een x and a ﬁnite ordinal (whic h is Σ 1 ) and it is also equ iv alen t to the statement that every injectiv e function fr om x to itself is su r jectiv e (whic h is Π 1 ). Note also that, if a set x is ﬁn ite and eac h of its elemen ts b elo ngs to a m o del M of ZF C, then w e may infer that x ∈ M using th e pairing and un ion axioms. F rom this f act it follo ws that the statemen t A = P ω ( B ) —th at is, “ A is the set of all ﬁnite su bsets of B ”— is absolute for transitiv e mo dels of a suitable ﬁnite fragmen t of ZF C, hence ∆ 1 . Nev er th eless, if M and N are just trans itiv e classes with M ⊂ N and B ∈ M , it can h app en that the claim “ P ω ( B ) exists” is tr ue in N but not in M , as discussed in [39, Sections 5 and 6]. DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 8 F or a cardinal λ > ω , the expression A = P λ ( B ) can b e formalized by claiming that λ is a cardinal and ∀ x ( x ∈ A ↔ ( x ⊆ B ∧ | x | < λ )). The clause | x | < λ is, on one h and, equiv alent to ( ∃ α ∈ λ ) ∃ f ( f is a bijectiv e fu nction fr om x to α ) , whic h is Σ 1 , and on the other h and it is the negation of λ ≤ | x | , hence equiv alen t to the Π 1 claim that there is n o injectiv e function from λ to x . Therefore, the statemen t A = P λ ( B ) is Π 1 . 3. Complexity o f ca te gories In order to s im p lify expr essions , if C is a category we shall d enote by X ∈ C th e statement that X is an ob ject of C and b y f ∈ C ( X, Y ) the claim that X and Y are ob jects of C and f is a morph ism from X to Y . Deﬁnition 3.1. F or n ≥ 0, a catego ry C is called Σ n - deﬁnable (shortly , Σ n ) with a set p of parameters if there is a Σ n form ula ϕ of the language of set theory such that ϕ ( X , Y , Z, f , g, h, i, p ) is true if and only if f ∈ C ( X, Y ), g ∈ C ( Y , Z ), h is th e comp osite of f and g , and i is the iden tit y of X . If a categ ory C is Σ n with a set p of parameters, then th er e are Σ n form ulas ψ Ob ( x, y ) and ψ Mor ( x, y , z , t ) suc h that ψ Ob ( X, p ) is true if and only if X ∈ C and ψ Mor ( X, Y , f , p ) is true if and only if f ∈ C ( X, Y ). Sp ec iﬁcally , from a form ula ϕ as in Deﬁnition 3.1 w e can c h o o se ψ Mor ( x, y , z , t ) to b e ∃ i ϕ ( x, x, y , i, z , z , i, t ), and next choose ψ Ob ( x, y ) to b e ∃ z ψ Mor ( x, x, z , y ). If C is Σ n , then the statemen t F = C ( X , Y ) is formalized with the fol- lo w ing Σ n ∧ Π n form ula: ( ∀ f ∈ F ) f ∈ C ( X, Y ) ∧ ∀ g ( g ∈ C ( X , Y ) → g ∈ F ) . W e say that a category is Π n for n ≥ 0 if there are Π n form ulas deﬁning its ob jects, morp hisms, comp o sition and identities. A category w ill b e called ∆ n if it is b oth Σ n and Π n . A category is upwar d absolute for trans itiv e classes if its ob jects, mor- phisms, comp ositi on and iden tities can b e deﬁ ned b y formulas that are upw ard absolute for trans itiv e classes. D ownwar d absolute catego ries are deﬁned in the same w a y , and a category will b e called absolute if it is b ot h upw ard absolute and d o wn wa rd absolute. T hus, ∆ 1 catego ries are absolute for tr ansitiv e classes con taining the inv olve d parameters. If C is a su b category of the category of sets, then comp osition an d id en- tities in C are prescrib ed by those of sets. Th erefore, the complexit y of a sub category of sets is the same if d eﬁned as in Deﬁnition 3.1 or if simply treated as a class of sets together with a class of fu nctions. Man y imp ortan t categories whic h cannot b e em b edd ed into Set ha ve nev ertheless a complexit y in our sense. F or example, th e homotopy category of simplicial sets cannot b e em b edd ed in to Set according to [24], and y et it can b e deﬁned with a Σ 2 form ula, since µ is a morph ism from X to Y if and only if th ere exists a simplicial map f from X to a ﬁ b ran t replacemen t of Y suc h that µ is the set of all simplicial maps h omotopic to f , and comp osition is d eﬁned accordingly (ﬁbrant replacemen ts are discu s sed in Section 9). F or a category C and an ob ject A of C , we d enote by ( C ↓ A ) the slic e c ate gory whose ob jects are pairs h X , f i wh ere f ∈ C ( X, A ) and whose mor- phisms h X , f i → h X ′ , f ′ i are morp hisms g ∈ C ( X, X ′ ) suc h that f = f ′ ◦ g . DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 9 Dually , the ob jec ts of the c oslic e c ate gory ( A ↓ C ) are pairs h X , f i wh ere f ∈ C ( A, X ), with corresp ondin g morp h isms. Both ( C ↓ A ) and ( A ↓ C ) are deﬁnable with the same complexit y as C , with A as an add itional parame- ter. Slice and coslice catego ries are (non-full) s u b categories of the c ate gory of arr ows Arr C , wh ose ob jects are triples h A, B , f i w ith f ∈ C ( A, B ) and where a morphism f → g is a comm utativ e s quare A / / f   C g   B / / D . Lemma 3.2. If Σ is any sig natur e, then ther e is a signatur e Σ ′ such that Arr Str Σ ful ly emb e ds into St r Σ ′ , and, if A is a Σ -structur e, then ther e is a signatur e Σ ′′ such that ( A ↓ St r Σ) ful ly emb e ds into Str Σ ′′ . In b oth c ases, the emb e dding pr eserves c omplexity. Pr o of. Let S b e the set of sorts of Σ. Consider a new set of sorts S ′ with t w o elemen ts s 0 and s 1 for eac h s ∈ S , and let Σ ′ b e the S ′ -sorted s ignature with the follo wing op eration symb ols and relation symb ols. Th e set Σ ′ op has tw o sym b ols σ 0 and σ 1 of r esp ectiv e arities h ( s i ) 0 : i ∈ α i → s 0 and h ( s i ) 1 : i ∈ α i → s 1 for eac h sy mb ol σ ∈ Σ op of arit y h s i : i ∈ α i → s , and an add itional symb ol µ s of arit y s 0 → s 1 for eac h s ∈ S . Th e set Σ ′ rel has t w o sy mb ols ρ 0 and ρ 1 of resp ec tiv e arities h ( s j ) 0 : j ∈ β i and h ( s j ) 1 : j ∈ β i for eac h symbol ρ ∈ Σ rel of arit y h s j : j ∈ β i . Then a Σ ′ -structure is a p air of Σ-structures X 0 and X 1 together with an S -sorted fu nction µ : X 0 → X 1 . T herefore, Arr Str Σ is canonically isomor- phic to the full sub cat egory of St r Σ ′ whose ob jects are trip les h X 0 , X 1 , µ i for w hic h µ is a h omomorphism of Σ-structures. F or th e second claim, deﬁn e, as in [2, 1.57(2)], a signature Σ ′′ b y addin g to Σ a new relation symb ol ρ a of arit y s for eac h elemen t a ∈ A s . It then follo ws that ( A ↓ Str Σ ) is canonical ly isomorp hic to the full sub ca tegory of Str Σ ′′ whose ob jects are those Y ∈ Str Σ f or whic h ( ρ a ) Y consists of a single elemen t of Y s for eac h a ∈ A s and the fu nction ρ Y : A → Y giv en by ρ Y ( a ) = ( ρ a ) Y is a h omomorphism of Σ-structures. Both emb eddings pr eserv e complexity d ue to their canonical n atur e. In more detail, su pp ose given a Σ n class F of ob jects in Arr Str Σ. Then its image F ′ in Str Σ ′ is d eﬁned as th e class of Σ ′ -structures X = h{ X s 0 : s ∈ S } ∪ { X s 1 : s ∈ S } , { ( σ 0 ) X : σ ∈ Σ op } ∪ { ( σ 1 ) X : σ ∈ Σ op } ∪ { ( µ s ) X : s ∈ S } , { ( ρ 0 ) X : ρ ∈ Σ rel } ∪ { ( ρ 1 ) X : ρ ∈ Σ rel }i for w hic h the triple consisting of X 0 = h{ X s 0 : s ∈ S } , { ( σ 0 ) X : σ ∈ Σ op } , { ( ρ 0 ) X : ρ ∈ Σ rel }i , X 1 = h{ X s 1 : s ∈ S } , { ( σ 1 ) X : σ ∈ Σ op } , { ( ρ 1 ) X : ρ ∈ Σ rel }i , together with the S -sorted fun ction f : X 0 → X 1 giv en by f s = ( µ s ) X for all s ∈ S is in the class F . Hence, F ′ is also Σ n , and analogously with Π n . The argument for ( A ↓ Str Σ ) is s im ilar.  DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 10 Prop osition 3.3. If Σ i s a λ - ary signatur e for a r e gular c ar dinal λ , then the f ol lowing assertions hold: (a) The c ate gory Str Σ of Σ -structur es is Π 1 with p ar ameters { λ, Σ } , and it is absolute b etwe en tr ansitive classes close d under se quenc es of length less than λ and c ontaining the p ar ameters. (b) Mor e gener al ly, the c ate gory Mo d T of mo dels of a the ory T in L λ (Σ) is ∆ 2 with p ar ameters { λ, Σ , T } , and i t is absolute b etwe en tr ansitive classes close d under se quenc e s of length less than λ and c ontaining the p ar ameters. Pr o of. In order to claim th at X is a Σ-structur e, we need to formalize the follo win g s tatemen t: “ λ is a regular cardin al, and Σ = h S, Σ op , Σ rel , ar i is a λ -ary signature, and X = h{ X s : s ∈ S } , { σ X : σ ∈ Σ op } , { ρ X : ρ ∈ Σ op }i is a Σ-stru ctur e”. W r iting down that λ is a regular cardinal is Π 1 b y [31, Lemma 13.13], and add in g that Σ is a λ -ary signature do es not increase complexit y . The assertion that X is a Σ -structure includ es the Π 1 form ula ( ∀ σ ∈ Σ op ) ( ∀ α ∈ λ ) ( ∀ x ) [[ x is a function α → ∪ s ∈ S X s ∧ ar( σ ) = ( h s i : i ∈ α i → s ) ∧ ( ∀ i ∈ α ) x ( i ) ∈ X s i ] → σ X ( x ) ∈ X s ] . Hence, the whole statemen t is Π 1 . S im ilarly , the assertion that f : X → Y is a h omomorphism of Σ -structures is Π 1 , sin ce w e need to imp ose that f ( σ X ( x )) = σ Y ( f ( x )) for all fu nctions x : α → ∪ s ∈ S X s with x ( i ) ∈ X s i for all i ∈ α , for eac h op eration symb ol σ of arit y h s i : i ∈ α i → s . Stating th at f ( x ) ∈ ρ Y for ev ery x ∈ ρ X and eac h relation symbol ρ do es not require unboun ded quant iﬁers. If λ = ω , then we can omit the clause “ λ is a r egular cardinal” and there is only need to quan tify o v er ﬁnite sequ ences in ∪ s ∈ S X s , w hic h is ∆ 1 , as discussed in Ex amp le 2.3. In order to state that X is a mo del of a theory T , we n eed to assert th at “ X is a λ -ary Σ-structure, and T is a set of sen tences of th e language of Σ, and ev ery sentence of T is satisﬁed in X ”. If λ = ω , then this is again ∆ 1 , since satisfaction of sen tences of a ﬁnitary language in X on ly dep ends on ﬁnite subsets of X . F or an arb itrary regular cardinal λ , the last t w o clauses are absolute b et we en transitiv e classes that are closed u nder sequences of length less than λ . Hence, b y the Reﬂection Principle, X is a mo del of T if and only if every ϕ ∈ T is a sen tence of th e language of Σ , and X is a Σ-structure, and there is a ﬁnite fragmen t ZFC ∗ of ZFC su c h that (3.1) ∃ M ( M is tr ansitiv e and closed und er < λ -sequences ∧ { λ, Σ , T , X } ⊂ M ∧ M | = V ZF C ∗ ∧ M | = ( ∀ ϕ ∈ T ) X | = ϕ ) , whic h can b e replaced with (3.2) ∀ M (( M is transitiv e and closed un der < λ -sequen ces ∧ { λ, Σ , T , X } ⊂ M ∧ M | = V ZF C ∗ ) → M | = ( ∀ ϕ ∈ T ) X | = ϕ ) . Since (3.1) is Σ 2 and (3.2) is Π 2 , the statemen t “ X is a mo del of T ” is ∆ 2 . And a morphism b et we en mo dels of T is just a homomorphism of Σ-stru c- tures, so the p ro of is complete.  4. Suppor ting elem e nt ar y e mbeddings An elementary emb e dding of a Σ-structure X into another Σ-structur e Y (where X and Y can b e prop er classes) is a function j : X → Y that preserv es DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 11 and reﬂects tru th. That is, for ev ery form ula ϕ ( x i ) i ∈ I of the language of Σ and all { a i : i ∈ I } in X , the sentence ϕ ( a i ) i ∈ I is satisﬁed in X if and only if ϕ ( j ( a i )) i ∈ I is satisﬁed in Y . In what follo ws, we consid er element ary em b eddin gs b et we en structures of the language of s et theory . If j : V → M is a non trivial element ary em b eddin g of the unive rse V of all s ets into a transitiv e class M , then its critic al p oint (i.e., the least ordinal m o v ed b y j ) is a measurable cardinal. In fact, the existence of a n on trivial elemen tary em b edd ing of the set-theoretic unive rse into a transitiv e class is equ iv alen t to th e existence of a measurable cardinal [31, Lemm a 17.3]. Giv en a su b category C of the category of sets and an elemen tary embed - ding j : V → M , we say that j is supp orte d by C if, for eve ry ob ject X in C , the set j ( X ) is also in C and the restriction fu nction j ↾ X : X → j ( X ) is a morphism in C . Theorem 4.1. L et j : V → M b e an elementary emb e dding with critic al p oint κ . L et Σ b e a λ - ary signatur e in V κ for a r e gular c ar dinal λ < κ su c h that M is close d under se quenc es of length less than λ . If X is a Σ -structur e, then j ( X ) is also a Σ -structur e and j ↾ X : X → j ( X ) is an e lementary emb e dding of Σ -structur es. Pr o of. First, observe th at j ( λ ) = λ and h ence λ is also a regular cardinal in M . Next, j (Σ) = Σ as Σ ∈ V κ . Th erefore, since j is an elemen tary em b eddin g, if X is a Σ-structure then j ( X ) is a Σ-structure in M . It f ollo ws that j ( X ) is also a Σ-stru ctur e in V , b eca use, by Prop osit ion 3.3, b ei ng a λ -ary Σ-stru cture is absolute for transitive classes con taining λ and closed under sequences of length less than λ . W e next c h ec k, by indu ction on the complexit y of form ulas of L λ (Σ), that j ↾ X is an elemen tary embedd ing of Σ-structures. F or atomic formulas, let σ ∈ Σ op b e an op erat ion symb ol with arit y h s i : i ∈ α i → s w h ere α < λ , so j ( α ) = α . Thus, if a i ∈ X s i for all i ∈ α , and a ∈ X s , then, since j is elemen tary , X | = ( σ X ( a i ) i ∈ α = a ) if and only if M | =  j ( X ) | = ( σ j ( X ) ( j ( a i )) i ∈ α = j ( a ))  . Since the s tatemen t j ( X ) | = ( σ j ( X ) ( j ( a i )) i ∈ α = j ( a )) is absolute for transi- tiv e classes, it h olds in M if an d on ly if it holds in V , as needed. Relation sym b ols ρ ∈ Σ rel are dealt with similarly , and th e cases of negation an d conjunction are immediate. Thus, there only remains to consid er existen- tial formulas. I f X | = ∃ x ϕ ( x, a ) for s ome a ∈ X , then ther e exists b ∈ X suc h that X | = ϕ ( b, a ). By indu ction h yp othesis, j ( X ) | = ϕ ( j ( b ) , j ( a )); hence j ( X ) | = ∃ x ϕ ( x, j ( a )). F or the con v erse, observ e ﬁrs t th at, since M is transitive and closed u nder sequences of length less than λ , satisfaction in j ( X ) of form ulas of L λ (Σ) is absolute b et w een M and V . Hence, if j ( X ) | = ∃ x ϕ ( x, j ( a )) for some a ∈ X , th en M | = ( j ( X ) | = ∃ x ϕ ( x, j ( a ))), and, b y elemen tarit y of j , we conclude that X | = ∃ x ϕ ( x, a ).  Since element ary embed d ings of Σ-str u ctures are homomorph isms, Theo- rem 4.1 tells us that categories of structures sup p ort elemen tary em b edd ings with suﬃ cien tly large critical p oint . The follo wing generalization of this fact is a more accurate restatemen t of [9, Prop osit ion 4.4]. DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 12 Theorem 4.2. L e t C b e a class of Σ - structur es for some λ -ary sig natur e Σ , wher e λ is a r e gular c ar dinal. Supp ose that C is Σ 1 with a set p of p ar am- eters. L et j : V → M b e an elementary emb e dding with critic al p oint κ > λ such that M is close d under se quenc es of length less than λ and { p, Σ } ∈ V κ . If X ∈ C , then j ( X ) ∈ C and j ↾ X : X → j ( X ) i s an elementary emb e dding of Σ -structur es. Pr o of. The pro of follo w s the same steps as the pro of of Theorem 4.1, using the fact that Σ 1 form ulas are upw ard absolute to inf er that j ( X ) ∈ C for ev ery X ∈ C .  5. V op ˇ enka ’s pr inciple and sup ercomp a ct ca r dinals F or any tw o stru ctures M ⊆ N of the language of set theory an d n < ω , w e w r ite M  n N and say that M is a Σ n -elementary su bstructur e of N if, for every Σ n form ula ϕ ( x 1 , . . . , x k ) and all a 1 , . . . , a k ∈ M , N | = ϕ ( a 1 , . . . , a k ) if and only if M | = ϕ ( a 1 , . . . , a k ) . F or a cardin al λ , we den ote b y H ( λ ) the set of all s ets whose transitiv e closure has cardinalit y less than λ . Thus H ( λ ) is a transitive set conta ined in V λ , and , if λ is strongly inaccessible, th en H ( λ ) = V λ ; see [35, Lemma 6.2]. A class C of ordinals is unb ounde d if it cont ains arbitrarily large ordinals, and it is close d if, for eve ry ordinal α , if S ( C ∩ α ) = α then α ∈ C . The abbreviation club means closed and unb ounded. As a consequence of the Reﬂection Principle [31, Theorem 12.14], for ev ery n there exists a club class of cardinals λ such th at H ( λ )  n V . In addition, if λ is u ncount able, then H ( λ )  1 V . In what follo w s , structur es are m ean t to b e sets, not pr op er classes. W e sa y that X and Y are structur es of the same typ e if they are b o th Σ-structures for some signature Σ . V op ˇ enka’s principle is the follo wing as- sertion; compare with [2, Ch. 6] or [31, (20.29)]: VP: F or every pr op er class C of structur es of the same typ e, ther e exist distinct X and Y in C and an e lementary emb e dding of X into Y . This is a statemen t inv olving classes. In the language of set theory , one can also formulate VP , but as an axiom schema, that is, an inﬁnite set of axioms; namely , one axiom for eac h formula ϕ ( x, y ) of the language of s et theory with t w o free v ariables, as f ollo ws: ∀ x [( ∀ y ∀ z (( ϕ ( x, y ) ∧ ϕ ( x, z )) → y and z are structures of the same t yp e) ∧ ∀ α ( α is an ord inal → ∃ y (rank( y ) > α ∧ ϕ ( x, y )) )) → ∃ y ∃ z ( ϕ ( x, y ) ∧ ϕ ( x, z ) ∧ y 6 = z ∧ ∃ e ( e : y → z is elemen tary ))] . In this article, VP will b e un dersto o d as this axiom schema, and similarly with the v arian ts of VP deﬁn ed b el o w. In the statemen t of VP , the requirement that there is an elemen tary em- b eddin g b et wee n t w o distinct stru ctures is sometimes replaced b y the re- quirement th at there is a nontrivial elemen tary em b edding b et we en tw o p ossibly equal s tructures. It follo ws from [14] that it is consisten t with ZF C to assum e that the t w o f orm ulations are equiv alent . Equiv alence can b e pro v ed using rigid graphs , as in [2, § 6.A], although this seems to r equire the use of global choice . DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 13 The theory ZFC + VP is v ery strong. It imp lies, for instance, that the class of extendib le cardinals is statio nary , that is, eve ry club pr op er class con tains an extendible cardinal [37]. The consistency of ZF C + VP follo ws from that of ZF C plus the existence of an almost-huge cardinal; see [31] or [33]. If λ and ν are cardinals, we denote by ν <λ the u nion of ν α for all α < λ . If f : A → B is a homomorphism of s tr uctures and M is an y s et, wh en we write that f ∈ M w e mean that A, B ∈ M and { ( a, f ( a )) : a ∈ A } ∈ M . Theorem 5.1. L et C b e a ful l sub c ate gory of Σ -structur es deﬁnable by a Σ 1 formula with a set p of p ar ameters for some λ -ary signatur e Σ . L et κ b e a r e gular c ar dinal bigge r than λ such that { p, Σ } ∈ H ( κ ) and with the pr op erty that ν <λ < κ for al l ν < κ . Then the fol lowing hold: (a) F or every homomorp hism g : A → Y of Σ -structur es with A ∈ H ( κ ) and Y ∈ C ther e is a homomor phism f : A → X with X ∈ C ∩ H ( κ ) and a c ommutative triangle A f ~ ~ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ g   ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ X e / / Y wher e e is an elementary emb e dding. (b) Every obje ct Y ∈ C has a sub obje ct X ∈ C ∩ H ( κ ) . Pr o of. W e on ly hav e to p ro v e (a), since (b) then f ollo ws with A = ∅ . Note that every element ary em b edd in g of Σ-stru ctures is an injectiv e homomor- phism and, sin ce C is a full su b category , e : X → Y is in C , so X is a sub ob ject of Y , s ince, in a sub cat egory of sets, ev ery injectiv e morph ism is a monomorphism; see [1, Pr op osition 7.37]. Th us, sup p ose that C , view ed as a class, is d eﬁnable as C = { x : ϕ ( x, p ) } , where ϕ is Σ 1 and p ∈ H ( κ ). Giv en g : A → Y with A ∈ H ( κ ) and Y ∈ C , let µ b e a regular cardinal bigger than κ su c h that Y ∈ H ( µ ) and s uc h that H ( µ ) | = ϕ ( Y , p ). In this situation, th e L¨ ow enheim–Skole m Theorem imp lies the existence of an elemen tary substr u cture h N , ∈ i of h H ( µ ) , ∈i of cardinalit y sm aller than κ and closed un der sequences of length less than λ (here w e use th e assumption that ν <λ < κ for all ν < κ ) such th at g ∈ N and w ith the transitiv e closure of { p, Σ , A } con tained in N . By elementa rit y , g is a homomorphism of Σ-structures in N and N | = ϕ ( Y , p ). Let M b e th e transitive collapse of N , and let j : M → N b e th e isomor- phism giv en by the collapse; that is, j is inv erse to th e fun ction π : N → M giv en b y π ( x ) = { π ( z ) : z ∈ x } ; see [31, 6.13]. S ince N is closed un d er sequences of length less than λ , s o is M , an d the critical p oint of j is greater than or equal to λ . And since N con tains th e transitiv e closure of { p, Σ , A } , w e hav e that π ( p ) = p , π (Σ) = Σ and π ( A ) = A . Moreo ve r, the restriction j ↾ A is the identit y . No w let X ∈ M b e such that j ( X ) = Y and let f : A → X b e such that j ( f ) = g . Then X ∈ H ( κ ) since | M | < κ and M is transitiv e. S ince j is an isomorphism and j ( p ) = p , we in fer that M | = ϕ ( X , p ), and h ence, as Σ 1 form ulas are upw ard ab s olute f or transitiv e classes, we conclude that X ∈ C in V . Since j (Σ) = Σ and M and N are closed under sequences DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 14 of length less than λ , the ob ject X is a Σ -structure an d , since j is an iso- morphism, the r estriction e = j ↾ X is an element ary em b e ddin g, h ence a homomorphism of Σ-structures. Moreo ver, f is also a homomorp h ism and the triangle commutes since f has b een deﬁned so that g ( a ) = j ( f ( a )) f or all a ∈ A .  Recall th at a cardin al κ is λ -sup er c omp act if there is an element ary em- b eddin g j : V → M with M transitive and with critical p o int κ , such that j ( κ ) > λ an d M is closed und er λ -sequences. Note that it then f ollo ws that H ( λ ) ∈ M . A cardin al κ is called sup er c omp act if it is λ -sup ercompact f or all ord inals λ . The follo wing theorem is an upgraded version of [9, Theorem 4.5], w here a similar r esu lt w as p ro v ed for absolute classes. Theorem 5.2. L et C b e a ful l sub c ate gory of Σ -structur es deﬁnable by a Σ 2 formula with a set p of p ar ameters. Supp ose that ther e exists a sup er c omp act c ar dinal κ b i gger than the r ank of p and Σ . Then the fol lowing hold: (a) F or every homomorph ism g : A → Y of Σ - structur es with A ∈ V κ and Y ∈ C ther e is a homo morphism f : A → X with X ∈ C ∩ V κ and an elementary emb e dding e : X → Y with e ◦ f = g . (b) Every obje ct Y ∈ C has a sub obje ct X ∈ C ∩ V κ . Pr o of. As w ith Theorem 5.1, we only ha ve to pro v e (a), since (b) follo ws b y taking A = ∅ . S upp ose that κ is a sup ercompact cardinal for whic h { p, Σ , A } ∈ V κ . Th en , since κ is strongly inaccessible, we hav e V κ = H ( κ ) and, sin ce κ is r egular, it is bigger than the supremum of the ordinals of the arities of all th e op e ration sym b ols and relation s ym b ols of Σ, so Σ is κ -ary . Giv en a homomorphism g : A → Y with Y ∈ C , let µ b e a cardinal bigger than κ such that Y ∈ H ( µ ) and H ( µ )  2 V . Let j : V → M b e an elemen- tary embedd ing with M transitiv e and critical p oint κ , su c h th at j ( κ ) > µ and M is closed un der µ -sequences. Then j ( A ) = A since A is in H ( κ ), and g and th e restriction j ↾ Y : Y → j ( Y ) are in M b ecause A, Y ∈ M and M is closed un der µ -sequences. In addition, g : A → Y is a homomorp hism of Σ-structures in M , since, by Prop ositi on 3.3, b eing a homomorphism of κ -ary Σ-stru ctures is absolute for trans itive classes conta ining Σ and closed under sequences of length less th an κ . Moreo ver, by Theorem 4.1, since Σ ∈ V κ , th e r estriction j ↾ Y : Y → j ( Y ) is an elementa ry em b edding of Σ-structures. Since b eing a cardinal is Π 1 and hence d o w n wa rd absolute, µ is a cardinal in M , and this implies that H ( µ ) in th e sense of M coincides w ith H ( µ ). It follo ws that H ( µ )  1 M , since every Σ 1 sen tence ψ whic h h olds in M also holds in V (as Σ 1 sen tences are upw ard ab s olute) and therefore ψ holds in H ( µ ) b ecause H ( µ )  2 V . Hence, Σ 2 form ulas are up w ard absolute b et we en H ( µ ) and M . Since H ( µ )  2 V and the class C is d eﬁned by a Σ 2 form ula ϕ ( x, y ), we ha v e that H ( µ ) | = ϕ ( Y , p ) and thus M | = ϕ ( Y , p ). No w rank( Y ) < µ < j ( κ ) in V and also in M . Thus, as witnessed by g : A → Y , in M there exists a homomorphism f : A → X of Σ -str u ctures suc h that rank( X ) < j ( κ ) and ϕ ( X, p ) h olds, and there is an elementa ry em b eddin g e : X → j ( Y ) s uc h that e ◦ f = j ( g ). By element arit y of j , the corresp ondin g statemen t is true in V ; that is, th ere exists a homomorphism of Σ -structures f : A → X suc h that rank( X ) < κ and ϕ ( X, p ) holds, so DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 15 X ∈ C , and there is an elemen tary em b edd in g e : X → Y with e ◦ f = g , as w e w ant ed to pr o ve.  Theorem 5.2 tells us that the existence of arb itrarily large sup ercompact cardinals implies that VP h olds for Σ 2 prop er classes. Th e follo wing theorem yields a str ong con v erse of this fact. Theorem 5.3. Supp ose that, for every ∆ 2 pr op er class C of structur es in the language of set the ory with one add itional c onstant symb ol, ther e exist distinct X and Y in C and an elementary emb e dding of X into Y . Then ther e exists a pr op er class of sup er c omp act c ar dinals. Pr o of. Let ξ b e an y ordin al and supp ose, tow ards a con tradiction, that there are no s up ercompact cardinals bigger than ξ . Then the class function F giv en as follo ws is well deﬁn ed on ord inals ζ > ξ : F ( ζ ) equals the least car- dinal λ > ζ suc h th at no cardinal κ such that ξ < κ ≤ ζ is λ -sup ercompact. Since the assertion “ ζ is λ -sup erco mpact” is ∆ 2 in ZFC (see [33, § 22]), F is ∆ 2 -deﬁnable with ξ as a p arameter. Let C 0 = { α : α is a limit ord inal, ξ < α , and ∀ ζ ( ξ < ζ < α → F ( ζ ) < α ) } . Then C 0 is a club class ∆ 2 -deﬁnable with ξ as a parameter. Fix a rigid binary relation (i.e., a rigid graph) R on ξ + 1 (see [41]). F or eac h ordin al α , let λ α b e the least element of C 0 greater than λ . The prop er class C = {h V λ α +2 , ∈ , h α, R ii : α > ξ } is ∆ 2 -deﬁnable w ith R as a parameter. By our assumption, there exist α < β greate r than ξ and an elemen tary embedd ing j : h V λ α +2 , ∈ , h α, R ii − → h V λ β +2 , ∈ , h β , R ii . Since j must send α to β , it is not the id en tit y . Hence, by Kunen’s Theorem ([31, Theorem 17.7], [34]), we ha v e λ α < λ β . Let κ ≤ α b e the critical p oint of j . Then, as in [37, Lemma 2], it follo w s that κ is λ α -sup ercompact. But this is imp ossible, since F ( κ ) < λ α b ecause λ α ∈ C 0 .  In order to summarize w h at we hav e pro v ed so far, we introdu ce some useful notatio n. Let Γ b e one of Σ n , Π n , ∆ n , Σ n ∧ Π n or Σ n , Π n , ∆ n , Σ n ∧ Π n , for any n . F or an inﬁn ite cardinal κ and a signature Σ ∈ H ( κ ), w e write: VP Σ (Γ): F or every Γ pr op er class C of Σ -structur es, ther e exist distinct X and Y in C and an elementary emb e dding of X into Y . SVP Σ κ (Γ): F or every pr op er class C of Σ - structur es admitting a Γ deﬁnition whose p ar ameters, if any, ar e in H ( κ ) , and for every Y ∈ C , ther e exists X ∈ C ∩ H ( κ ) and an elementary emb e dding of X into Y . If Σ is omitted from the notation, we mean that the corresp onding state- men t holds for all admissible signatures. Thus, VP(Γ) means VP Σ (Γ) f or all Σ , wh ile SVP κ (Γ) means S VP Σ κ (Γ) for ev ery Σ ∈ H ( κ ). Ev en though SVP Σ κ (Γ) is an app aren tly stronger statement than VP Σ (Γ) (hence the notation SVP), in the case of Σ 2 classes of structur es they turn out to b e equiv alen t, as we next prov e. Corollary 5.4. The fol lo wing statements ar e e quivalent: (1) SVP κ ( Σ 2 ) holds f or a pr op er class of c ar dinals κ . DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 16 (2) VP( Σ 2 ) holds. (3) VP Σ ( ∆ 2 ) holds if Σ is the signatur e of the language of set the ory with one additional c onstant symb ol. (4) Ther e exists a pr op er class of sup er c omp act c ar dinals. Pr o of. In order to chec k that (1) ⇒ (2), supp ose that (1) is true, and let Σ b e any signature. Let C b e any p rop er class of Σ-structures deﬁn ed by a Σ 2 form ula with parameters, and let κ b e bigger th an the ranks of the parameters and suc h that S VP Σ κ ( Σ 2 ) holds. Sin ce C is a p rop er class, w e ma y choose Y of r an k bigger than κ , s o an y X ∈ C ∩ H ( κ ) will necessarily b e distinct from Y . Hence, there exist distinct X and Y such that X is elemen- tarily emb eddable in to Y , so VP Σ ( Σ 2 ) holds, as needed. T he implication (2) ⇒ (3) is trivial, and T heorem 5.3 imp lies that (3) ⇒ (4). Finally , to see that (4) ⇒ (1), let ξ b e any cardin al and pic k a su p ercompact cardinal κ > ξ . Since H ( κ ) = V κ , Th eorem 5.2 tells u s that S VP κ ( Σ 2 ) holds .  The follo wing is a corresp onding version w ith ou t p arameters, with the same (in f act, simpler) pro of. Corollary 5.5. The fol lo wing statements ar e e quivalent: (1) SVP κ (Σ 2 ) holds for some c ar dinal κ . (2) VP(Σ 2 ) holds. (3) VP Σ (∆ 2 ) holds if Σ is the signatur e of the language of set the ory. (4) Ther e exists a su p er c omp act c ar dinal. 6. V op ˇ enka ’s principle and extend ible cardinal s F or cardinals κ < λ , we sa y that κ is λ -extendible if th er e is an element ary em b eddin g j : V λ → V µ for s ome µ , with critical p oint κ and with j ( κ ) > λ . A cardin al κ is called extendible if it is λ -extendible for all cardinals λ > κ . As sho wn in [31, 20.24], extendible cardin als are sup ercompact. See [31] or [33] for more inf ormation ab out extendible cardinals. F or eac h n < ω , let C ( n ) denote the club prop er class of inﬁnite cardinals κ that are Σ n -c orr e ct in V , that is, V κ  n V . Since th e satisfaction relation | = n for Σ n sen tences (which is, in fact, a prop e r class) is Σ n -deﬁnable for n ≥ 1 [33, § 0.2], it follo w s th at, for n ≥ 1, the class C ( n ) is Π n . T o see this, note ﬁ rst that C (0) is the class of all inﬁnite cardinals, and therefore it is Π 1 -deﬁnable. F or κ an inﬁn ite cardinal, κ ∈ C (1) if an d only if κ is an un - coun table cardinal and V κ = H ( κ ), wh ic h implies that C (1) is Π 1 -deﬁnable. In general, for n ≥ 1 and for an y inﬁn ite cardinal κ , we h a ve V κ  n +1 V if and only if κ ∈ C ( n ) ∧ ( ∀ ϕ ( x ) ∈ Σ n +1 ) ( ∀ a ∈ V κ ) ( | = n +1 ϕ ( a ) → V κ | = ϕ ( a )) , whic h is a Π n +1 form ula sho wing that C ( n + 1) is Π n +1 -deﬁnable. W e shall use the follo w ing new strong f orm of extendibility . Deﬁnition 6.1. F or C a club prop e r class of cardinals and κ < λ in C , we sa y that κ is λ - C -extendible if th er e is an elemen tary embed ding j : V λ → V µ for s ome µ ∈ C , with critical p oin t κ , suc h that j ( κ ) > λ and j ( κ ) ∈ C . W e say that a cardinal κ in C is C -extendible if it is λ - C -extendible for all λ in C greater than κ . Note that, for all n , if κ is C ( n )-extendible, then κ is extendible. There- fore, a cardinal is C (0)-extendible if and only if it is extendible. DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 17 Prop osition 6.2. Every extendible c ar dinal is C (1) -extendible. Pr o of. Sup p ose th at κ is extendible and λ ∈ C (1) is greater than κ . Note that the existence of an extend ib le cardinal imp lies the existence of a prop er class of inaccessible cardinals, as the image of κ un d er any elementa ry em- b eddin g j : V λ → V µ , with critical p o int κ and λ a cardinal, is alwa ys an inaccessible cardinal in V . So we can pick an inaccessible cardinal λ ′ ≥ λ . Let j ′ : V λ ′ → V µ ′ b e an elemen tary em b edd ing with critical p oint κ and suc h that j ′ ( κ ) > λ ′ . S ince V λ ′ = H ( λ ′ ), it follo ws b y elemen tarity of j ′ that V µ ′ = H ( µ ′ ). Hence, µ ′ ∈ C (1). Let u s see that j = j ′ ↾ V λ : V λ → V j ′ ( λ ) witnesses the λ - C (1)-extendibilit y of κ . W e only need to chec k that µ = j ′ ( λ ) ∈ C (1). But sin ce V λ  1 V λ ′ , it follo ws by elemen tarit y of j ′ that V µ  1 V µ ′ . Hence, sin ce µ ′ ∈ C (1), also µ ∈ C (1).  Hence, a cardinal is C (1)-extendible if and only if it is extendible. Let u s also observe th at, if there exists a C ( n + 2)-extendible cardinal for n ≥ 1, then there exists a p rop er class of C ( n )-extendible cardinals; see [7]. Lemma 6.3. If κ is C ( n ) -extendible, then κ ∈ C ( n + 2) . Pr o of. By ind uction on n . F or n = 0, since κ ∈ C (1), we only need to sho w that if ∃ x ϕ ( x ) is a Σ 2 sen tence, where ϕ is Π 1 and has parameters in V κ , that holds in V , then it holds in V κ . S o s u pp o se that a is such that ϕ ( a ) holds in V . Let λ ∈ C ( n ) b e greater than κ and with a ∈ V λ , and let j : V λ → V µ b e elemen tary , with critical p oi nt κ and with j ( κ ) > λ . T hen V j ( κ ) | = ϕ ( a ), and so, b y elemen tarit y , V κ | = ∃ x ϕ ( x ). No w su pp ose that κ is C ( n )-extendible and ∃ x ϕ ( x ) is a Σ n +2 sen tence, where ϕ is Π n +1 and h as parameters in V κ . If ∃ x ϕ ( x ) holds in V κ , th en, since by the in duction hyp othesis κ ∈ C ( n + 1), w e ha v e that ∃ x ϕ ( x ) h olds in V . No w su pp ose that a is such that ϕ ( a ) h olds in V . Let λ ∈ C ( n ) b e greater than κ and such that a ∈ V λ , and let j : V λ → V µ b e elemen tary with critical p oin t κ and with j ( κ ) > λ . Th en, sin ce j ( κ ) ∈ C ( n ), we h av e V j ( κ ) | = ϕ ( a ), and so, b y elemen tarit y , V κ | = ∃ x ϕ ( x ).  Theorem 6.4. F or every n ≥ 1 , if κ is a C ( n ) -extendible c ar dinal, then SVP κ ( Σ n +2 ) holds. Pr o of. Fix a Σ n +2 form ula ∃ x ϕ ( x, y , z ), where ϕ is Π n +1 , su c h th at C = { Y : ∃ x ϕ ( x, Y , p ) } is a p rop er class of structures of the same typ e for some set p ∈ V κ . Fix Y ∈ C and let λ ∈ C ( n + 2) b e greater than κ and the r anks of p and Y . Thus, V λ | = ∃ x ϕ ( x, B , p ). Let j : V λ → V µ for some µ ∈ C ( n ) b e an elemen tary em b edd ing with critical p oin t κ , with j ( κ ) > λ and j ( κ ) ∈ C ( n ). Note that b oth Y and j ↾ Y : Y → j ( Y ) are in V µ . Since κ, λ ∈ C ( n + 2) by Lemma 6.3, and κ < λ , w e hav e V κ  n +2 V λ . It follo ws that V j ( κ )  n +2 V µ . Ind eed, the follo wing h olds : V λ | = ( ∀ x ∈ V κ ) ( ∀ θ ∈ Σ n +2 ) ( V κ | = θ ( x ) ↔ | = n +2 θ ( x )) . Hence, by elemen tarity , V µ | = ( ∀ x ∈ V j ( κ ) ) ( ∀ θ ∈ Σ n +2 ) ( V j ( κ ) | = θ ( x ) ↔ | = n +2 θ ( x )) , whic h implies that V j ( κ )  n +2 V µ . DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 18 Since j ( κ ) ∈ C ( n ), w e h a v e V λ  n +1 V j ( κ ) , and th er efore V λ  n +1 V µ . It follo ws that V µ | = ∃ x ϕ ( x, Y , b ). Th us, in V µ it is tru e that there exists X ∈ V j ( κ ) suc h that X ∈ C , namely Y , and there exists an elemen tary em b edd ing e : X → j ( Y ), namely j ↾ Y . Therefore, by elemen tarity of j , the same is tru e in V λ , that is, there exists X ∈ V κ suc h that X ∈ C , and th er e exists an element ary embedd ing e : X → Y . Since λ ∈ C ( n + 2), w e ha v e X ∈ C and we are done.  Corollary 6.5. If κ is an extendible c ar dinal, then SVP κ ( Σ 3 ) holds. Pr o of. This is the assertion of Theorem 6.4 for n = 1.  Corollary 6.6. L et C b e a ful l sub c ate gory of Σ -structur es deﬁnable by a Σ n +2 formula with a set p of p ar ameters, wher e n ≥ 1 . Supp ose that ther e exists a C ( n ) - extendible c ar dinal κ bigger than the r ank of p and Σ . Then the f ol lowing hold: (a) F or every homomorph ism g : A → Y of Σ - structur es with A ∈ V κ and Y ∈ C ther e is a homo morphism f : A → X with X ∈ C ∩ V κ and an elementary emb e dding e : X → Y with e ◦ f = g . (b) Every obje ct Y ∈ C has a sub obje ct X ∈ C ∩ V κ . Pr o of. P art (b) is a consequence of T h eorem 6.4 and part (a) is a more general v arian t pr ov ed as in Theorem 5.2.  The follo wing th eorem yields a conv erse to Theorem 6.4. Theorem 6.7. L et n ≥ 1 , and supp ose that VP Σ (Σ n +1 ∧ Π n +1 ) holds when Σ is the signatur e of the language of set the ory with ﬁnitely many additional 1 -ary r e lation symb ols. Then ther e exists a C ( n ) -extendible c ar dinal. Pr o of. Sup p ose, to the contrary , that th ere is n o C ( n )-extendible cardinal. Then the class fu nction F on ordinals giv en by deﬁning F ( ζ ) to b e the least λ > ζ su c h that λ ∈ C ( n ) and ζ is not λ - C ( n )-extendible is well d eﬁned. F or λ ∈ C ( n ), the relation “ ζ is λ - C ( n )-extendible” is Σ n +1 , for it holds if and only if ζ ∈ C ( n ) and ∃ µ ∃ j : V λ → V µ ( j is elemen tary ∧ cp( j ) = ζ ∧ j ( ζ ) > λ ∧ µ, j ( ζ ) ∈ C ( n )) , where cp( j ) denotes the critical p oin t of j . Hence F is Σ n +1 ∧ Π n +1 . Let C = { α : α is a limit ord inal and ( ∀ ζ < α ) F ( ζ ) < α } . S o, C is a Σ n +1 ∧ Π n +1 closed unb ounded prop e r class. F or eac h ordinal α , let λ α b e th e ﬁrst limit p oin t of D = C ∩ C ( n ) ab o v e α . Note that the class fun ction f on ordin als suc h that f ( α ) = λ α is (Σ n +1 ∧ Π n +1 )-deﬁnable. No w let C = {h V λ α , ∈ , α, λ α , C ∩ α + 1 i : α ∈ D } . W e claim that C is (Σ n +1 ∧ Π n +1 )-deﬁnable. Indeed, X ∈ C if an d only if X = h X 0 , X 1 , X 2 , X 3 , X 4 i , where (1) X 2 ∈ C ; (2) X 3 = λ X 2 ; (3) X 0 = V X 3 ; (4) X 1 = ∈ ↾ X 0 ; (5) X 4 = C ∩ X 2 + 1 . W e h a v e already seen th at (1) and (2) are Σ n +1 ∧ Π n +1 expressible. And so are (3) and (4). As for (5), note that X 4 = C ∩ α + 1 holds in V if and only if it holds in V X 3 . DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 19 So C is a Σ n +1 ∧ Π n +1 prop er class of structur es of the same t yp e in the language of set theory with three additional relation symbols. By ou r assumption, there are α < β in D and an elemen tary embedd ing j : h V λ α , ∈ , α, λ α , C ∩ α + 1 i − → h V λ β , ∈ , β , λ β , C ∩ β + 1 i . Since j sends α to β , it is not the id en tit y . Let κ b e the critical p o int of j . Since α ∈ C , we ha v e κ < F ( κ ) < α . T h us, j ↾ V F ( κ ) : V F ( κ ) − → V j ( F ( κ )) is elementa ry , with critical p oint κ . W e claim that κ ∈ D . Other w ise, γ = su p( D ∩ κ ) < κ . Let δ b e the least ordinal in D greater th an γ with κ < δ < λ α . Since δ is d eﬁ nable from γ in the stru cture h V λ α , ∈ , α, C ∩ α + 1 i , and since j ( γ ) = γ , we must also ha v e j ( δ ) = δ . But then j ↾ V δ +2 : V δ +2 → V δ +2 is an elemen tary em b edding, con tradicting Kunen ’s Theorem [34]. By elementa rit y , j ( κ ) ∈ C ( n ). Moreo v er, since F ( κ ) ∈ C ( n ) and λ β ∈ C ( n ), we ha v e j ( F ( κ )) ∈ C ( n ). Since κ ∈ C , by elementa rit y we also ha v e j ( κ ) ∈ C . Hence, j ( κ ) > F ( κ ). This shows that j ↾ V F ( κ ) witnesses that κ is F ( κ )- C ( n )-extendible, and th is contradict s the deﬁn ition of F .  The pr o of of Th eorem 6.7 easily generalizes to the b oldface case (see the pro of of Theorem 5.3), namely if VP( Σ n +1 ∧ Π n +1 ) holds , then there is a prop er class of C ( n )-extendible card inals. In fact it is suﬃcient to assum e that VP Σ ( Σ n +1 ∧ Π n +1 ) holds w hen Σ is the signature of the language of set theory w ith a ﬁ nite num b er of additional 1-ary r elation symb ols. The follo wing corollaries summ arize our results in this section. Corollary 6.8. The fol lo wing statements ar e e quivalent for n ≥ 1 : (1) SVP κ ( Σ n +2 ) holds f or some c ar dinal κ . (2) VP(Σ n +1 ∧ Π n +1 ) holds. (3) VP Σ (Σ n +1 ∧ Π n +1 ) holds when Σ is the signatur e of the language of set the ory with a ﬁnite numb er of additional 1 - ary r elation symb ols. (4) Ther e exists a C ( n ) -extendible c ar dinal. Corollary 6.9. The fol lo wing statements ar e e quivalent: (1) F or every n , SVP κ ( Σ n ) holds f or a pr op er class of c ar dinals κ . (2) F or every n , SVP κ ( Σ n ) holds f or some c ar dinal κ . (3) VP( Σ n ) holds f or al l n . (4) VP Σ (Σ n ) holds f or al l n when Σ is the signatur e of the language of set the ory with a ﬁnite numb er of additional 1 - ary r elation symb ols. (5) Ther e exists a C ( n ) -extendible c ar dinal for ev e ry n . (6) Ther e exists a pr op er class of C ( n ) -extendible c ar dinals for eve ry n . (7) V opˇ enka’s principle holds. 7. A cc essible ca tegor ies A category is smal l if its ob jects form a set, and essential ly smal l if the isomorphism classes of its ob jects form a set. Let λ b e a regular cardinal. A n onempt y category K is called λ - ﬁlter e d if, giv en any set of ob jects { k i } i ∈ I in K wh ere | I | < λ , there is an ob j ect k ∈ K and a morphism k i → k for eac h i ∈ I , and , moreo v er, giv en an y set of parallel arr o ws b et we en any t wo ob jects { f j : k → k ′ } j ∈ J where | J | < λ , DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 20 there is a morp hism g : k ′ → k ′′ suc h that g ◦ f j is the same morp hism for all j ∈ J . If C is any category , a fun ctor D : K → C w h ere K is a λ -ﬁltered small category is called a λ - ﬁlter e d diagr am , and, if D h as a colimit L , then L is called a λ - ﬁlter e d c olimit . F or example, ev ery set is a λ -ﬁltered colimit of its su bsets of cardin ality sm aller th an λ (partially ord ered b y inclusion). An ob ject A of a category C is λ - pr esentable if the fu nctor C ( A, − ) pre- serv es λ -ﬁltered colimits; that is, for eac h λ -ﬁltered diagram D : K → C with a colimit L , eac h m orphism A → L factors through a morph ism A → D k for some k ∈ K , and if t w o morph isms A → D k an d A → D k ′ comp ose to the same morph ism A → L , then there is some k ′′ ∈ K and morph isms k → k ′′ and k ′ → k ′′ in K such that th e t wo comp osites A → D k ′′ are equ al; s ee [26, § 6.1] or [38 , § 2.1]. F or a s mall f ull su b category A of C and an ob ject X in C , the c anonic al diagr am ( A ↓ X ) → C sen d s eac h pair h A, f i with f ∈ C ( A, X ) to A . Recall from [2, 1.23] that A is called dense in C if eac h ob ject X of C is a colimit of the canonical diagram ( A ↓ X ) → C . A category C is b ounde d if it has a dense small full sub c ategory . A category C is called λ -ac c essible if λ -ﬁltered colimits exist in C and there is a set A of λ -present able ob jects such that ev er y ob j ect of C is a λ -ﬁltered colimit of ob jects fr om A . A category C is called ac c essible if it is λ -accessible for s ome r egular cardinal λ . As sh o wn in [3, p. 226 ] or [2, p. 73], if C is λ -accessible, then the full su b category of its λ -pr esen table ob jects is essen tially sm all and, if w e denote b y C λ a set of representa tiv es of all isomorphism classes of λ -presen table ob jects of C , then C λ is dense in C . Moreo ver, f or ev ery X ∈ C , the slice category ( C λ ↓ X ) is λ -ﬁltered and X is a colimit of th e canonical diagram ( C λ ↓ X ) → C . Thus, every accessible catego ry is b ounded. An accessible category is called lo c al ly pr esentable if all colimits exist in it. It then follo ws , by [2, Corollary 1.28], that all limits exist as well . Ev ery category of structures Str Σ is lo cally presenta ble [2, 5.1(5) ], an d the forgetful fun ctor Str Σ → Set S creates limits and colimits, where S is th e set of sorts of Σ and Set S denotes the category of S -sorted sets. Theorem 7.1. L et λ b e a r e gular c ar dinal and let C b e a λ -ac c essible c at- e gory. Then ther e is a fu l l emb e dding of C into a c ate gory of r e lational structur es that pr eserves λ -ﬁlter e d c olimits. Pr o of. Let us assume, with greater generalit y , th at C is a b ounded category and let A b e a d en se small full sub category of C . Denote by Set A op the catego ry of fun ctors A op → Set , where A op is the opp osit e of A . Then there are f u ll emb ed dings (7.1) C − → Set A op − → St r Σ , deﬁned as follo ws [2, Ch. 1]: The embedd ing of C in to Set A op is of Y oneda t yp e, s ending eac h ob ject X to the restriction of C ( − , X ) to A op . T he fact that it is full and faithful is prov ed in [2, Prop ositio n 1.26]. The signature Σ is c hosen b y pic king the ob jects of A as sorts and the morp hisms of A op as relation sym b ols. The full em b edd in g of Set A op in to Str Σ sends eac h fun ctor F to the A -sorted set { F A : A ∈ A} together with a r elation { ( x, ( F f ) x ) : x ∈ F A } ⊂ F A × F B for eac h morp hism f : B → A in A . DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 21 Hence, (7.1 ) sends eac h ob ject X ∈ C to h{C ( A, X ) : A ∈ A} , {{ ( α, α ◦ f ) : α ∈ C ( A, X ) } : f ∈ A ( B , A ) }i . If C is λ -accessible and we let A b e a set of r epresen tativ es of all isomor- phism classes of λ -presen table ob jects in C , then (7.1) p r eserv es λ -ﬁltered colimits, sin ce the ﬁrst arro w p r eserv es λ -ﬁltered colimits by [2, Prop osi- tion 1.26], and the second arr ow preserve s all ﬁltered colimits; see [2, Ex- ample 1.41].  As in [2, Deﬁnition 2.35], w e s a y that a su b category C of a category D is ac c essibly emb e dde d if C is full and closed under λ -ﬁltered colimits in D for some regular cardinal λ . Hence, in particular, C is isomorphism-closed; that is, ev ery ob ject of D whic h is isomorphic to an ob ject of C is in C . Moreo v er, the inclusion C ֒ → D creates λ -ﬁltered colimits. If D is accessible and C is accessibly em b ed ded in to D , then C is itself accessible if and only if, for some regular cardinal λ , every λ -ﬁltered colimit of split sub ob jects of ob jects of C is in C ; see [2, C orollary 2.36] f or d etails. V op ˇ enk a’s principle implies that ev ery full em b eddin g b et wee n acce ssi- ble categorie s is accessible. The same conclusion can b e inferred f r om the existence of s u ﬃcien tly large C ( n )-extendible card in als [8]. A theory T in a λ -ary language is b asic if eac h of its sen tences h as the form ∀ { x i : i ∈ I } ( ϕ ( x i ) i ∈ I → ψ ( x i ) i ∈ I ) where ϕ and ψ are disj unctions of p ositiv e-primitiv e formulas and | I | < λ . A f orm ula is p ositive-primitive if it has the form ∃{ y j : j ∈ J } η (( y j ) j ∈ J , ( z k ) k ∈ K ) in which η is a conjunction of atomic form ulas and | J | , | K | < λ . It follo ws f rom Theorem 7.1 th at ev ery accessible category is equiv alen t to an accessibly emb edded su b cate gory of a category of relational stru ctures, namely to the closure of the image of (7.1 ) un der isomorph isms. Moreo ve r, the follo wing fund amen tal fact is prov ed in [2 ]: Theorem 7.2. Every ac c essibly emb e dde d ac c essible sub c ate gory of a c ate- gory of structur es is a c ate gory of mo dels for some b asic the ory, and for eve ry b asic the ory T in some language L λ (Σ) , the c ate gory Mo d T is ac c essible and ac c essibly emb e dde d i nto St r Σ . Pr o of. This is sho wn in [2, Theorem 4.17 and Theorem 5.35].  W e shall use the follo w ing terminology in ord er to simp lify s tatement s: Deﬁnition 7.3. An ac c essible c ate gory of structur es is a f ull su b cate gory of Str Σ that is accessible and accessibly em b edded, for some signature Σ. W e sa w in Prop osition 3.3 that eac h category Mo d T is ∆ 2 with param- eters { λ, Σ , T } . Hence, Theorem 7.2 implies that eve ry accessible category of structur es is at most ∆ 2 . I n many cases the complexit y will b e low er; for example, if Σ is ﬁn itary , th en , according to Prop osition 3.3, Mo d T is ∆ 1 with parameters { Σ , T } . This amends the statemen t of [9 , Prop osition 4.2]. Although, in th e rest of the article, we shall r estrict most of our discu ssion to acce ssible categories of structures, r esults in v olving only concepts that are inv arian t und er equiv alence of catego ries will remain tr u e for arb itrary accessible catego ries, by Theorem 7.1. A regular cardinal κ is said to b e sharply bigger than another regular car- dinal λ if κ > λ and, for eac h set X of cardinalit y less than κ , the set P λ ( X ) has a coﬁnal su bset of cardinalit y less than κ . T his n otion wa s introdu ced DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 22 in [38 , § 2.3], where it was pr ov ed that κ is s h arply b igger than λ if and only if every λ -accessible catego ry is κ -accessible; see also [2, Theorem 2.11]. If κ has the p rop erty that ν <λ < κ for all ν < κ (which was used in Theorem 5.1 ab ov e) and κ > λ , then κ is sh arply bigger than λ , since, for a set X of cardinalit y ν , the cardinalit y of P λ ( X ) is precisely ν <λ . Therefore, if λ ≤ µ , then (2 µ ) + is sharply bigger than λ . This w as ﬁr st observ ed in [38, Prop osition 2.3.5 ] and shows that for ev ery λ th ere are arb itrarily large regular cardinals sh arp ly bigger th an λ . Moreo v er, if κ is strongly inaccessible and κ > λ , th en κ is sh arply bigger than λ . In w hat follo ws, f or an S -sorted signature Σ and a Σ-structure A , the c ar dinality of A designates the sum Σ s ∈ S | A s | of the cardinalities of the comp onent s of its un d erlying S -sorted set. Lemma 7.4. L et Σ b e a λ -ary signatur e f or a r e gular c ar dinal λ , and let C b e a ful l λ -ac c essible su b c ate gory of Str Σ close d under λ -ﬁlter e d c olimits. L et κ b e a r e gular c ar dinal sharply bigge r than λ and bigger than the c ar dinalities of al l λ -pr esentable obje cts in C , and suc h that Σ ∈ H ( κ ) . Then an obje c t A ∈ C is κ -pr esentable if and only if its c ar dinality is smal ler than κ . Pr o of. Let S b e the set of sorts of Σ; let Σ op b e its set of op e ration sym b ols and Σ rel its s et of r elation sy mb ols. Let A b e a Σ -structure, and su pp ose ﬁrst that its cardin alit y Σ s ∈ S | A s | is smaller than κ . Let D : K → C b e a κ -ﬁltered diagram with a colimit L . Th en D is also λ -ﬁltered and therefore the inclusion of C into Str Σ p r eserv es its colimit. Su pp ose giv en a homo- morphism f : A → L . S ince ev ery set A s has cardinality less than κ and D is κ -ﬁltered, eac h function f s : A s → L s factors thr ough D ( k s ) for some k s ∈ K . Since | S | < κ , w e infer th at f factors (as a fu nction) through D k for some k ∈ K . Moreo v er, sin ce the cardin ality of the set of all α -sequences h a i : i ∈ α i with a i ∈ A s i for all i and with α < λ is less than κ , and the cardinalities of th e sets Σ op and Σ rel are also sm aller than κ , w e can ﬁn d a morphism k → l in K su ch that the comp osite A → D k → D l is a homo- morphism of Σ -structures. F or the same r eason, giv en t w o homomorphisms A → D k and A → D k ′ whic h coincide in L , there is an ob ject k ′′ ∈ K and morphisms k → k ′′ and k ′ → k ′′ suc h that the comp osit es A → D k → D k ′′ and A → D k ′ → D k ′′ are equal. Hence A is κ -presentable. F or the con v erse, b y [38, Prop osition 2.3.11], if κ is sharp ly bigger th an λ then ev ery κ -pr esen table ob ject A in C is a λ -ﬁltered colimit of λ -p resen table ob jects ind exed b y a category w ith less than κ morphisms. Therefore, since eac h λ -presentable ob ject has card inalit y smaller than κ an d th e colimit is created in Set S , it follo ws that A also has cardinalit y smaller than κ .  The follo wing is our main result in this s ection. Theorem 7.5. L et C b e an ac c essible c ate gory of structur e s and let S b e a Σ n ful l su b c ate gory of C , wher e n ≥ 1 . Supp ose that ther e is a pr op er class of sup er c omp act c ar dinals if n = 2 or that ther e is a pr op er class of C ( n − 2) -extendible c ar dinals if n ≥ 3 . Then ther e i s a dense smal l ful l sub c ate gory D ⊆ S and ther e ar e arbitr arily lar ge r e gular c ar dinals κ such that, for al l Y ∈ S , the c ate gory ( D ↓ Y ) is κ -ﬁlter e d and Y is a c olimit of the c anonic al diagr ams ( D ↓ Y ) → S and ( D ↓ Y ) → C . DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 23 Pr o of. Note ﬁrst that, if S is essenti ally small, then the result trivially holds with D a full sub cat egory of S conta ining one representat iv e of eac h isomor- phism class of ob jects in S , if κ is c hosen bigger than the cardin alit y of the set of ob jects of D . Therefore we assume from now on th at there is a prop er class of nonisomorphic ob jects in S . Cho ose a Σ n form ula deﬁning S w ith a set p of parameters. Sup p ose that C em b eds accessibly int o St r Σ for a signature Σ , and pic k a regular cardinal λ suc h that Σ is λ -ary and C is λ -accessible and closed un der λ -ﬁ ltered colimits in Str Σ. Let C λ b e a set of r epresen tativ es of all isomorphism classes of λ -presen table ob jects in C . No w let α b e an y giv en ord inal. Cho ose a regular cardinal κ bigger than α and λ , and large enou gh so that eac h ob ject in C λ is in H ( κ ) and { p, Σ } ∈ H ( κ ) as well. Moreo v er, if n = 1 then p ic k κ of th e f orm (2 µ ) + with µ ≥ λ ; if n = 2 th en c ho ose instead κ su p ercompact, and if n ≥ 3 then c ho ose it C ( n − 2)-extendible. With an y of these c hoices, κ is sharp ly b igger than λ and ther efore C is κ -accessible. Let D b e a full su b cate gory of S conta ining one r epresen tativ e of eac h isomorphism class of ob jects in the set S ∩ H ( κ ). Note that, since eac h ob ject of D is in H ( κ ), all ob jects of D are κ -presen table in C , by Lemma 7.4 . Let C κ b e a s et of rep r esen tativ es of all isomorph ism classes of κ -presen t- able ob jects of C , chose n so that D ⊆ C κ and all ob jects of C κ are in H ( κ ). The latter is p ossible since, if A ∈ C and A is κ -present able, then A has car- dinalit y smaller than κ by Lemma 7.4 and therefore A ∼ = A ′ as Σ -structures for some A ′ ∈ H ( κ ). Since C is isomorph ism-closed, A ′ is in C and we ma y pic k A ′ as a memb er of C κ . Let Y be an y ob ject of S . Sin ce C is κ -accessible, w e kno w that Y is a colimit of the canonical d iagram ( C κ ↓ Y ) → C , which is κ -ﬁltered, b y [2, p. 73]. Th erefore, if we pro v e that ( D ↓ Y ) is c oﬁnal in ( C κ ↓ Y ), it w ill then follo w that Y is a colimit of the canonical diagram ( D ↓ Y ) → C , and that ( D ↓ Y ) is κ -ﬁltered. Moreo v er, since Y is in S , we shall b e able to conclude that Y is also a colimit of the canonical diagram ( D ↓ Y ) → S , as w e w ant ed to show. Th us, to wards proving that ( D ↓ Y ) is coﬁnal in ( C κ ↓ Y ), let A b e an y ob ject of C κ and let a morphism g : A → Y be give n. If n = 1, then, since A ∈ H ( κ ), it follo ws from part (a) of Theorem 5.1 that there is an ob ject h X , f i in ( A ↓ S ) with X ∈ S ∩ H ( κ ), together with an elemen tary em b eddin g e : X → Y of Σ -structures suc h that e ◦ f = g . I f n > 1, then Theorem 5.2 if n = 2 or Theorem 6.6 if n ≥ 3 lead to the same conclusion (recall that H ( κ ) = V κ if κ is str ongly inaccessible). I n eac h case, we rep lace, if necessary , X by an isomorphic ob ject within S ∩ H ( κ ), so we ma y assume that X ∈ D . W e therefore hav e a comm utativ e triangle A g / / f ❅ ❅ ❅ ❅ ❅ ❅ ❅ Y X e > > ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ where f can also b e view ed as a morp hism from h A, g i to h X, e i in ( C κ ↓ Y ). Since ( C κ ↓ Y ) is ﬁltered, this tells us that ( D ↓ Y ) is coﬁn al in ( C κ ↓ Y ), as w e w ant ed to show.  DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 24 Corollary 7.6. If ther e is a pr op e r class of sup er c omp act c ar dinals, then every ac c essible c ate gory is c o-wel lp ower e d. Pr o of. Let C b e an accessible category . Since accessibilit y and co-w ellp o w er- edness are inv arian t und er equiv alence of categories, we can assume that C is a cate gory of m o dels of a basic theory T for some signature Σ, b y Theorem 7.1 and T h eorem 7.2. F or an ob ject A ∈ C , let E A b e the full su b category of ( A ↓ C ) whose ob jects are the epimorp hisms. T h en E A is a partially ordered class, since b et we en an y tw o of its ob jects there is at most one morp hism. Moreov er, E A is closed u nder colimits in ( A ↓ C ) and , if a d iagram D : K → E A has a colimit, then the colimit is a supremum of the set { D k : k ∈ K} , hen ce determined by this s et up to isomorphism . Therefore, in order to p ro v e that C is co-w ellp o w ered, it is enough to pr o v e that E A is b ounded for eve ry A , since this implies that E A is essential ly small. F rom the fact that C is ∆ 2 it follo ws that E A is Π 2 , s ince an ob ject of E A is a p air h Y , g i where g ∈ C ( A, Y ) and ∀ Z ∀ h ∀ h ′ [( h ∈ C ( Y , Z ) ∧ h ′ ∈ C ( Y , Z ) ∧ h ◦ g = h ′ ◦ g ) → h = h ′ ] , and a morphism h Y , g i → h Y ′ , g ′ i is a morph ism d ∈ C ( Y , Y ′ ) with g ′ = d ◦ g . Hence, T heorem 7.5 implies that E A is b oun ded und er the assum ption th at there are arb itrarily large extendible cardinals. Ho wev er, as we next show, it is enough to assume that there are arbitrarily large sup e r c omp act cardinals. F or this, we need to rep eat the argument used in the pro of of Theorem 7.5 and the one used in the p ro of of Th eorem 5.2, adapted to our cur ren t situation. If C is accessible, then ( A ↓ C ) is also accessible, by [2, Corollary 2.44]. Pic k a regular cardinal λ such that ( A ↓ C ) is λ -accessible. Ass uming that there exists a prop er class of su p ercompact cardinals, we may choose a sup er compact cardinal κ bigger than λ , suc h that Σ , T ∈ H ( κ ) and su c h that all λ -presen table ob jects of ( A ↓ C ) are in H ( κ ). Since κ is s tr ongly inaccessible, it is sharp ly b igger than λ and therefore ( A ↓ C ) is κ -accessible. Cho ose a full sub cat egory D of E A con taining one r epresen tativ e of eac h isomorphism class of ob j ects in E A ∩ H ( κ ). By Lemma 7.4, all ob jects in D are κ -presen table. Cho ose also a set ( A ↓ C ) κ of repr esentati ve s of all isomorphism classes of κ -presen table ob jects of ( A ↓ C ), con taining D and suc h that all its ob jects are in H ( κ ), which is p ossible by Lemma 7.4. No w let h Y , g i b e an y ob ject of E A , so g : A → Y is an epimorphism . W e kno w that h Y , g i is a colimit of the canonical diagram (( A ↓ C ) κ ↓ h Y , g i ) − → ( A ↓ C ) . Hence it suﬃces to pro v e that ( D ↓ h Y , g i ) is coﬁnal in (( A ↓ C ) κ ↓ h Y , g i ). F or th is, pic k any ob ject in (( A ↓ C ) κ ↓ h Y , g i ), which consists of a κ -pres- en table ob ject h B , a i of ( A ↓ C ) together w ith a morp hism d : B → Y su ch that d ◦ a = g . Pick a cardin al µ > κ such that h Y , g i ∈ H ( µ ). T hen d is also in H ( µ ) since B ∈ H ( κ ). Let j : V → M b e an elemen tary embedd ing with M tr an s itiv e and critical p oint κ , suc h that j ( κ ) > µ and M is closed und er µ -sequences. Th en g and d are in M since H ( µ ) ∈ M . Moreo ve r, C is absolute b e t w een M and V , by part (b) of Prop ositi on 3.3. Th erefore g is also an epimorp hism in M , since, DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 25 if h, h ′ ∈ C ( Y , Z ) satisfy h ◦ g = h ′ ◦ g in M , then h and h ′ also b elo ng to C ( Y , Z ) in V and therefore h = h ′ , since g is an epimorph ism in V . Since Y ∈ H ( µ ), the restriction j ↾ Y : Y → j ( Y ) is in M , and it is an elemen tary em b e ddin g of Σ-structures b y Th eorem 4.1. Since A and B are in H ( κ ), we ha ve j ( A ) = A and j ( B ) = B . Therefore, as in the pro of of Theorem 5.2, g : A → Y and d : B → Y witness that in M there exists an ob ject X (n amely , Y ) and an epimorp hism f ∈ C ( A, X ) with rank( X ) < j ( κ ), together w ith an eleme nta ry em b eddin g e : X → j ( Y ) suc h that e ◦ f = j ( g ) and a morphism c ∈ C ( B , X ) su c h that c ◦ a = f and e ◦ c = j ( d ). This implies, by elemen tarit y of j , that in V there is an epimorphism f ∈ C ( A, X ) with rank( X ) < κ , together with an elemen tary em b eddin g e : X → Y su c h that e ◦ f = g and a m orp hism c ∈ C ( B , X ) su c h that c ◦ a = f and e ◦ c = d . In other w ords, there is a comm u tativ e d iagram A a ~ ~ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ f   g ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ B d ; ; c / / X e / / Y . Here we ma y rep lace h X, f i by an isomorphic ob ject wh ic h is in D . This sho ws that ( D ↓ h Y , g i ) is coﬁnal in (( A ↓ C ) κ ↓ h Y , g i ), and consequently the category E A is b oun ded, as needed.  On the other hand, as shown in [2, A.19], if eac h accessible category is co-w ellp o w ered then there exists a pr op er class of measurab le cardin als. Therefore, the statemen t that ev ery accessible category is co-w ellp o we red is set-theoretica l. Its p recise consistency s trength is not kno wn; see [2, Op en Problem 11]. By part (i) of [38, Th eorem 6.3.8], together with the fact that catego ries of ep imorphisms can b e ske tc hed by a p ushout sketc h (as done in [2, p. 101]), the statemen t that ev ery accessible category is co-w ellp o we red is implied by the existence of a p rop er class of strongly compact cardinals, a large-ca rdin al assum ption that is not known to b e weak er, consistency-wise, than the existence of a prop er class of sup ercompact cardinals. In order to simplify the statemen ts of s everal corollaries of T heorem 7.5, w e shall use th e follo win g terminology . Deﬁnition 7.7. W e sa y that a class S is deﬁnable with suﬃciently low c omplexity if any of the follo wing conditions is satisﬁed: (1) S is Σ 1 . (2) There is a prop er class of sup ercompact cardinals and S is Σ 2 . (3) There is a prop er class of C ( n )-extendible cardin als for some n ≥ 1 and S is Σ n +2 . By Corollary 6.9, if V op ˇ enk a’s principle holds, then all classes are d eﬁ nable with suﬃcien tly lo w complexit y . 8. Small-or thogonal ity cl asses An ob ject X and a morph ism f : A → B in a categ ory C are called ortho g onal [25] if th e function C ( f , X ) : C ( B , X ) − → C ( A, X ) DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 26 is bijectiv e. That is, X and f are orthogonal if and only if f or eve ry m orphism g : A → X th ere is a uniqu e morp hism h : B → X su c h that h ◦ f = g . F or a class of ob jects X , we denote by ⊥ X the class of morph ism s that are orthogonal to all the ob jects of X . Similarly , f or a class of morphisms F , w e denote by F ⊥ the class of ob jects th at are orthogonal to all the morphism s of F . Classes of ob jects of th e form F ⊥ are called ortho gonality classes , and, if F is a set (not a pr op er class), then F ⊥ is a smal l-ortho gonality class . In wh at follo ws , we view eac h class of morphisms in C as a full sub c ategory of the category of arr o ws Arr C . Lemma 8.1. F or a r e gular c ar dinal λ , let F b e a class of morphisms in a λ - ac c essible c ate gory C , and let D ⊆ F . Supp ose that every f ∈ F is a λ -ﬁlter e d c olimit of elements of D , and supp ose that the inclusion of F into Arr C pr eserves the c olimit. Then D ⊥ = F ⊥ . Pr o of. T o prov e this claim, only the inclusion D ⊥ ⊆ F ⊥ needs to b e chec ke d. Let X ∈ D ⊥ and let f : A → B b e an y elemen t of F . By assumption, f = colim d k where d k : A k → B k is in D for all k ∈ K , and K is λ -ﬁltered. Since C is λ -accessible, the colimits colim A k and colim B k exist, and the induced arrow g : colim A k → colim B k is a colimit of the arro ws d k in Arr C . Since f is also a colimit of the same diagram, we infer that g ∼ = f . Hence, f induces bijections C ( B , X ) ∼ = C (co lim B k , X ) ∼ = lim C ( B k , X ) ∼ = lim C ( A k , X ) ∼ = C (co lim A k , X ) ∼ = C ( A, X ) , whic h means that X ∈ F ⊥ , as n eeded.  Lemma 8.2. If S is a Σ n +1 ful l sub c ate g ory of a Σ n c ate gory C , then ⊥ S is Π n +1 if n ≥ 1 , and it is Π 2 if n = 0 . Pr o of. The class of morphism s ⊥ S can b e deﬁn ed as follo w s: h A, B , f i ∈ ⊥ S if and only if (8.1) ∀ X ∀ g [( X ∈ S ∧ g ∈ C ( A, X )) → ∃ h ( h ∈ C ( B , X ) ∧ h ◦ f = g )] ∧ ∀ X ∀ h 1 ∀ h 2 [( X ∈ S ∧ h 1 ∈ C ( B , X ) ∧ h 2 ∈ C ( B , X ) ∧ h 1 ◦ f = h 2 ◦ f ) → h 1 = h 2 ] . Recall that P → Q means ¬ ( P ∧ ¬ Q ), or ¬ P ∨ Q . Th erefore, (8.1) is at least Π 2 , and it is Π n +1 if S is Σ n +1 and C is at most Σ n with n ≥ 1.  Theorem 8.3. Assu me the existenc e of a pr op er class of C ( n ) -extendible c ar dinals, wher e n ≥ 2 . Then e ach Σ n +1 ortho g onality class in an ac c essible c ate gory C of structur es is a smal l-ortho gonality class. Pr o of. Let S b e a full sub cate gory of C whose ob jects form a Σ n +1 orthog- onalit y class. Thus S = F ⊥ for s ome F , and this implies th at ( ⊥ S ) ⊥ = ( ⊥ ( F ⊥ )) ⊥ = F ⊥ = S . Since C is ∆ 2 b y P r op osition 3.3, we infer fr om Lemma 8.2 that ⊥ S is Π n +1 . No w the category of arro ws Arr C is accessible and em b eds accessibly in to a category of s tr uctures in suc h a w a y that complexit y is pr eserv ed, by Lemma 3.2. Hence, by Theorem 7.5, ⊥ S has a dense small full su b category D and there is a regular cardin al κ (whic h we ma y c h o ose so that C is κ -acce ssible) suc h that every arro w f ∈ ⊥ S is a κ -ﬁltered colimit of elemen ts DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 27 of D , b oth in ⊥ S and in Arr C . Th en D ⊥ = ( ⊥ S ) ⊥ = S b y Lemma 8.1, so S is indeed a small-orthogonalit y class.  This result can b e sharp ened as follo ws. A r eﬂe ction on a category is a left adjoin t (when it exists) of the inclusion of a f ull sub cate gory [36], whic h is then called r eﬂe ctive . F or example, in the category of group s, the ab elianization fun ctor is a reﬂection onto the r eﬂectiv e fu ll sub cate gory of ab elian group s . F or ev ery reﬂection L , the closure und er isomorph isms of its image is an orthogonalit y class, and it is in fact orth ogonal to the class of L - e quivalenc es , i.e., morp hisms f suc h th at Lf is an isomorphism. A r eﬂection L is called an F - r eﬂe ction , wh ere F is a set or a prop er class of morphisms , if the closure und er isomorphisms of the image of L is equal to F ⊥ . T his notion is particularly relev an t when F can b e chosen to b e a set (or even b et ter a single morp h ism). In the previous example, ab elianization is an f -reﬂection w here f is the canonical pro jection of a free group on tw o generators on to a free ab elian group on t wo generators, since the groups orthogonal to f are precisely the ab elian groups. Theorem 8.4. L et L b e a r eﬂe ction on an ac c essible c ate gory C of struc- tur es. Then L is an F -r e ﬂe ction for some set F of morphism s under any of the f ol lowing assumptions: (1) The class of L -e quiv alenc es is deﬁnable with suﬃciently low c om- plexity. (2) The class of obje cts isomorph ic to LX for some X is Σ n +1 for n ≥ 2 and ther e is a pr op er c lass of C ( n ) -e xtendible c ar dinals. Pr o of. T o p ro v e case (1), let S b e the fu ll sub ca tegory of L -equiv alences in the category of arrows of C . It then follo ws fr om Th eorem 7.5 that th er e is a s m all full sub ca tegory D of S whic h is dense and satisﬁes S ⊥ = D ⊥ , by Lemma 8.1, as n eeded. Case (2) follo ws as a sp ecia l case of T h eorem 8.3.  The follo wing coroll ary is a stronger v arian t of [9, Corollary 4.6]. The assumptions th at L b e an ep ir eﬂection and th at C b e balanced, whic h w ere made in [9], are not at all necessary h ere. Corollary 8.5. Supp ose that ther e is a pr op er class of sup er c omp act c ar di- nals. If L is a r eﬂe ction on an ac c essib le c ate g ory C of structur es and the class of L - e quivalenc es is Σ 2 , then L is an F -r eﬂe ction for some set F of morphism s. Pr o of. By assumption, the class of L -equiv alences is deﬁn able with suﬃ- cien tly lo w complexit y . Hence, Theorem 8.4 applies.  As already s ho wn in [17, Th eorem 6.3], the assertion th at ev ery reﬂection on an accessible category is an F -reﬂection for some set F of morp hisms cannot b e prov ed in ZF C. Sp eciﬁcally , if on e assumes that measurable car- dinals do not exist and considers reﬂection on the category of group s with resp ect to the class Z of h omomorphisms of the f orm Z κ / Z <κ → { 0 } , where κ ru ns ov er all cardinals (see Example 2.2), then there is no set F of group homomorphisms suc h that F -reﬂection coincides with Z -reﬂection. This fact was also used in [9]. Theorem 8.6. If C is a lo c al ly pr esentable c ate gory of structur es, then every ful l sub c ate gory S of C close d under limits and deﬁnable with suﬃciently low c omplexity is r eﬂe c tiv e. DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 28 Pr o of. As in the pro of of Theorem 7.5, for ev ery A ∈ C we can c h o ose a small full sub ca tegory D of S (dep ending on th e cardin alit y of A an d the parameters of C ) su c h that every arr o w f : A → Y with Y in S factors through some ob ject X ∈ D . Hence the inclus ion functor S ֒ → C satisﬁes the solution-set cond ition for eve ry A in C , as required in the F r eyd Ad join t F unctor T h eorem [36, V.6], from which the existence of a reﬂection of C on to S follo ws .  The follo wing result is a further improv ement, since it implies, among other things, that, if S is Σ 1 , then the reﬂectivit y of S ⊥ is pro v able in ZF C. This yields, in particular, a solution of the F reyd–Kelly orthogo nal sub category problem [25] in ZF C for Σ 1 classes. Theorem 8.7. L et S b e a class of morphisms deﬁnable with suﬃci e ntly low c omplexity in an ac c essible c ate gory C of structur es. Then S ⊥ is a smal l- ortho g onality c lass and, if C is c o c omplete, then S ⊥ is r eﬂe ctive. Pr o of. If w e view S as a fu ll sub category of the category of arro ws of C , then Theorem 7.5 ensu res that S has a dense small fu ll sub category D and Lemma 8.1 imp lies th at D ⊥ = S ⊥ . Hence S ⊥ is a small-orthogonalit y class, and small-orthogonalit y classes are r eﬂectiv e if colimits exist [2, 1.37].  If w e weak en the assumption that S is closed und er limits in Th eorem 8.6, b y imp osing only that it is closed under pro du cts and retracts, then we ma y infer similarly that S is wea kly r eﬂectiv e, und er the hyp otheses made in the statemen t. O n the other hand, it is sho wn in [16] that, assuming the nonexistence of measurable cardinals, there is a Σ 2 full sub category S of the cate gory of ab elian groups w h ic h is closed und er pro ducts and retracts bu t not wea kly reﬂectiv e. Sp eciﬁcall y , S is the closur e of the class of groups Z κ / Z <κ under pro d ucts and retracts, w h ere κ ru ns o v er all cardinals. Hence, the statemen t th at all Σ 2 full sub cat egories closed un der pro du cts and r etracts in locally presentable categories are w eakly r eﬂectiv e imp lies the existence of measurab le card inals, wh ile it follo ws from the existence of sup er compact cardinals. Theorem 8.8. Every ful l sub c ate gory close d under c olimits and deﬁnable with suﬃcie ntly low c omplexity in a lo c al ly pr esentable c ate gory C of struc- tur es is c or eﬂe ctive. Pr o of. Argue as in [2, Theorem 6.28].  9. Consequences in homot o py theo r y Ho vey conjectured in [29] that for ev ery cohomology theory deﬁn ed on sp ectra there is a homology theory w ith th e same acyclics. Th is conjecture remains so far unsolv ed. In a diﬀerent but closely related direction, the existence of cohomological lo calizations is also an op en problem in ZFC, al- though it is known that it follo ws from V op ˇ enk a’s pr inciple, b oth in unstable homotop y and in stable homotop y , b y [17] and [15, Theorem 1.5]. Motiv ated by these pr oblems, in th is section we compare homological acyclic classes with cohomological acyclic classes from the p oint of view of complexit y of th eir deﬁnitions. W e consider homology theories and coho- mology theories deﬁned on simplicial sets and represente d by sp ec tra. DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 29 Sp ectra will b e meant in the sense of Bousﬁeld–F riedland er [13]. Thus, a sp e ctrum E is a sequence of p ointe d simplicial sets h ( E n , p n ) : p n ∈ ( E n ) 0 , 0 ≤ n < ω i equipp ed with p ointe d simp licial maps σ n : S E n → E n +1 for all n . Here S denotes susp ension , that is, S X = S 1 ∧ X . F or k ≥ 1, we denote by S k the simplicial k -sp here, namely S k = ∆[ k ] /∂ ∆[ k ], where ∆[ k ] is th e standard k -simplex an d ∂ ∆[ k ] is its b oundary . F or p oin ted simplicial sets X and Y , the smash pr o duct X ∧ Y is th e quotien t of the pro du ct X × Y by the w edge sum X ∨ Y , and w e denote by map ∗ ( X, Y ) the p ointe d function c omplex from X to Y , whose n -simp lices are the p ointe d maps X ∧ ∆ [ n ] + → Y , where the su bscript + means that a disjoint basep oin t has b een added . A simplicial set is ﬁbr ant if it is a Kan complex [32]. F or the pu r p oses of this article, it will b e conv enient to use K an ’s Ex ∞ construction as a ﬁb ran t replacemen t fun ctor. Thus, there is a natural (injectiv e) w eak equiv alence j Y : Y ֒ → Ex ∞ Y f or all Y , wh ere Ex ∞ Y is ﬁb ran t. Let [ X, Y ] den ote th e set of morphisms f rom X to Y in the p ointe d homotop y catego ry of simp licial sets, which can b e describ ed as the s et of p oint ed h omotopy classes of maps X → Ex ∞ Y . If Y is ﬁbr an t, then this is in bijectiv e corresp ond ence, via j Y , with the set of p ointed homotop y classes of maps X → Y . A s p ectrum E is an Ω - sp e ctrum if eac h E n is ﬁbrant and the adjoin ts τ n : E n → Ω E n +1 of the str u cture maps σ n : S E n → E n +1 are wea k equiv a- lences, w here Ω d enotes the lo op sp ac e functor Ω X = map ∗ ( S 1 , X ). Eac h sp ect rum E deﬁnes a reduced h omology theory E ∗ on simplicial sets b y (9.1) E k ( X ) = colim n π n + k ( X ∧ E n ) = colim n [ S n + k , X ∧ E n ] for k ∈ Z , and, if E is an Ω-sp ectrum , then E deﬁnes a r educed cohomology theory E ∗ on sim p licial sets by (9.2) E k ( X ) = colim n π n − k (map ∗ ( X, E n )) = colim n [ S n X, E n + k ] for k ∈ Z . Note that, if k ≥ 0, then simply E k ( X ) ∼ = [ X, E k ]. Suc h h omology or cohomology theories are called r epr esentable , and we shall only consider these in this article. Although n ot ev ery generalized homology or cohomology theory in the sense of Eilenberg–Steenro d is r ep- resen table [44, Ex amp le I I.3.17], h omologic al lo calizatio ns ha ve only b een constructed and studied assuming represen tabilit y [5 ], [11]. According to Bro wn’s r epresen tabilit y theorem, every cohomology th eory which is additive (i.e., sendin g copro ducts to pro ducts) is repr esen ted by some Ω-sp ect rum. Similarly , h omology theories that p r eserv e ﬁltered colimits are repr esen table. See [4] or [44] f or fu rther d etails. In most of w hat follo ws, we assume that E is an Ω-sp ect rum . A simplicial set X is called E ∗ -acyclic if E k ( X ) = 0 f or all k ∈ Z , and, similarly , X is E ∗ -acyclic if E k ( X ) = 0 f or all k ∈ Z . Observe that, by (9.2), the statemen t that X is E ∗ -acyclic is equiv alent to the statemen t that the p ointed function complex map ∗ ( X, E n ) is w eakly contract ible (that is, connected and with v anishing homotop y groups) for all n . A map f : X → Y is an E ∗ -e quivalenc e if E k ( f ) : E k ( X ) − → E k ( Y ) DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 30 is an isomorph ism of ab elia n groups for all k ∈ Z , and similarly for coho- mology . Let C f d enote the mapping c one of f , whic h is obtained fr om the disjoin t u nion of Y and X × ∆[1] b y identifying X × { 0 } with f ( X ) ⊆ Y using f , and collapsing X × { 1 } to a p oint. Using the Ma yer–Viet oris axiom, one ﬁ nds th at f is an E ∗ -equiv alence if and only if C f is E ∗ -acyclic , and analogously for cohomology . The categ ory of simplicial sets is ∆ 0 , lo cally present able, and it has a canonical accessible em b edd ing int o a category of stru ctures with a ﬁn itary ω -sorted op erati onal s ignatur e. In fact, one can write down explicitly a f or- m ula without unb ounded quan tiﬁers expressin g that X and Y are simplicial sets and f is a simp licial map from X to Y . T his amounts to formalizing the claim that a simplicial set X is a sequence of sets h X n : 0 ≤ n < ω i (where the elemen ts of X n are called n - simplic es ), together with fun ctions d n i : X n → X n − 1 (called fac es ) for n ≥ 1 and 0 ≤ i ≤ n , and s n i : X n → X n +1 (called de gener acies ) for n ≥ 0 and 0 ≤ i ≤ n , satisfying the simplicial iden tities; see [40, Deﬁnition 1.1]. A s im p licial map f : X → Y is a sequence of fu nctions h f n : X n → Y n i 0 ≤ n<ω compatible with faces and degeneracies. Similarly , th e category of sp ectra is ∆ 0 , lo cally pr esen table, and it also has an accessible em b edding into a category of str u ctures with a ﬁ nitary ω -sorted op erational signature, s in ce a sp ectrum E consists of a sequence of p oint ed simplicial sets h ( E m , p m ) : 0 ≤ m < ω i , w h ere p m ∈ ( E m ) 0 , and a sequence of p oi nte d m aps h σ m : S E m → E m +1 i 0 ≤ m<ω , eac h of whic h can b e view ed as a map ∆[1] × E m → E m +1 sending ∂ ∆[1] × E m and ∆[1] × { p m } to the basep o int p m +1 . Giving a map f : ∆[1] × E m → E m +1 is equiv alen t to giving a collection of functions f 0 0 , f 1 0 : ( E m ) 0 → ( E m +1 ) 0 and f 0 k , f 1 k , f 01 k : ( E m ) k → ( E m +1 ) k for k ≥ 1, with comm utativit y cond itions f 0 0 ◦ d 1 0 = d 1 0 ◦ f 0 1 , f 1 0 ◦ d 1 0 = d 1 0 ◦ f 1 1 , f 0 0 ◦ d 1 0 = d 1 0 ◦ f 01 1 , f 0 0 ◦ d 1 1 = d 1 1 ◦ f 0 1 , f 1 0 ◦ d 1 1 = d 1 1 ◦ f 1 1 , f 1 0 ◦ d 1 1 = d 1 1 ◦ f 01 1 , s 0 0 ◦ f 0 0 = f 0 1 ◦ s 0 0 , s 0 0 ◦ f 1 0 = f 1 1 ◦ s 0 0 , and corresp ondin gly for k ≥ 1. Prop osition 9.1. The fol lowing ar e ∆ 1 classes: (1) Fibr ant simplicial sets. (2) We ak e qu ivalenc es of simplicial sets. (3) We akly c ontr actible sp e ctr a. (4) Ω -sp e ctr a. Pr o of. The assertion that a giv en s im p licial set X is ﬁbr an t can b e formal- ized by means of the Kan extension condition, as in [40, Deﬁnition 1.3]. Explicitly , a simplicial set X is ﬁb ran t if and only if for ev ery 1 ≤ n < ω an d ev ery k ≤ n + 1, the follo wing sente nce h olds: F or all x 0 , x 1 , . . . , x n +1 ∈ X n suc h that d n i x j = d n j − 1 x i for i < j , i 6 = k and j 6 = k , ther e exists x ∈ X n +1 suc h that d n +1 i x = x i for i 6 = k . Since quantiﬁcatio n o ver ﬁnite subsets is ∆ 1 (see E x amp le 2.3), the class of ﬁ brant simplicial sets is ∆ 1 -deﬁnable. T o wa rds (2), recall that a map of simplicial sets f : X → Y is a w eak equiv alence if and only if it induces a bijection of connected comp o nents and isomorphism s of homotop y groups for eve ry choic e of a basep oi nt. Let us assu me ﬁr st that X and Y are ﬁbrant. T h en f indu ces a bijection of DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 31 connected comp onent s if and only if, for all x 0 and x 1 of X 0 , if there exists v ∈ Y 1 with d 1 0 v = f ( x 0 ) and d 1 1 v = f ( x 1 ), th en there exists u ∈ X 1 with d 1 0 u = x 0 and d 1 1 u = x 1 , and moreo v er for eac h y ∈ Y 0 there exist x ∈ X 0 and v ∈ Y 1 suc h that d 1 0 v = y and d 1 1 v = f ( x ). Hence, the statemen t that f induces a b ijection of conn ected comp onen ts is ∆ 0 . Similarly , if a simplicial set X is ﬁb ran t, then the n th homotopy group π n ( X, p ) with basep oint p ∈ X 0 is the quotien t of th e set of all x ∈ X n suc h that d n i x = sp for all i (where s = s n − 2 n − 2 ◦ · · · ◦ s 0 0 ) by the homotop y relation, where x ∼ x ′ if d n i x = d n i x ′ for all i and there exists z ∈ X n +1 with d n +1 n +1 z = x , d n +1 n z = x ′ , and d n +1 i z = s n − 1 d n i x for 0 ≤ i < n ; compare with [40, Deﬁnition 3.1]. Therefore, if X and Y a re ﬁ brant, then f ind u ces an isomorphism π n ( X, p ) ∼ = π n ( Y , q ), where p ∈ X 0 and q = f ( p ), if and only if the follo wing sen tence holds: ∀ y ∈ Y n [ ∀ i ≤ n ( d n i y = sq ) → [ ∃ x ∈ X n ( ∀ i ≤ n ( d n i x = sp ) ∧ f n ( x ) ∼ y ∧ ∀ x ′ ∈ X n (( ∀ i ≤ n ( d n i x ′ = sp ) ∧ f n ( x ′ ) ∼ y ) → x ∼ x ′ ))]] . This sho ws that the statemen t that a map b et wee n ﬁbr ant simp licial sets is a w eak equiv alence is ∆ 1 . Next w e analyze the complexity of a ﬁb ran t replacemen t. F or a simplicial set X , the map j X : X ֒ → Ex ∞ X can b e d eﬁned as the inclusion of X into a simplicial set Ex ∞ X d eﬁned as f ollo ws. Let Ex 1 X b e the simplicial set whose set of n -simp lices is the set of all maps fr om the b arycen tric s ub d ivision of ∆[ n ] into X . The barycen tric su b division sd ∆[ n ] is the n erv e of the p oset of nondegenerate simplices of ∆[ n ] (see [27, Ch. I II , § 4]). T he last vertex map sd ∆[ n ] → ∆[ n ] yields an inclusion X ֒ → Ex 1 X . Then Ex ∞ X is the union of a s equ ence of in clusions Ex k X ֒ → Ex k +1 X for k ≥ 1, wh ere Ex k is the comp osite of Ex 1 with itself k times. Let p b e any vertex of X . Eac h element in π n (Ex ∞ Y , f ( p )) is r epresen ted b y a map S n → Ex k Y b ased at f ( p ) f or some k < ω , that is, a map fr om ∆[ n ] to Ex k Y se ndin g the b o und ary of ∆[ n ] to f ( p ). By adjoin tness, the maps ∆[ n ] → Ex k Y corresp ond bijectiv ely with the maps sd k ∆[ n ] → Y , where sd k is an iterated barycen tric sub division. Let a k ,n b e the num b er of nondegenerate n -simplices of sd k ∆[ n ] and let R k ,n b e the set of all relations among their f aces. F or example, a 2 , 1 = 4 and R 2 , 1 consists of th e equalities d 1 1 x (0 → 001) = d 1 1 x (01 → 001) , d 1 0 x (01 → 001) = d 1 0 x (01 → 011) , d 1 1 x (01 → 011) = d 1 1 x (1 → 011) . Th us, eac h map ∆ [ n ] → E x k Y is determined by a sequence of a k ,n (not necessarily distinct) elemen ts of Y n satisfying a set R k ,n of equ alities among their faces. In what follo ws , when w e write “a map β : S n → Ex k Y ” w e implicitly formalize it as an order ed sequ ence of a k ,n elemen ts of Y n satisfying a set S k ,n of sen tences, including those of R k ,n and those needed to expr ess the fact that ∂ ∆ [ n ] is sent to the basep oin t f ( p ). Homoto pies in to Ex k Y are formalized similarly . The assertion that f : X → Y induces π n (Ex ∞ X, p ) ∼ = π n (Ex ∞ Y , f ( p )) for ev ery p ∈ X 0 can therefore b e expressed by stating that f or ev ery k < ω and every map β : S n → Ex k Y based at f ( p ) there exist l < ω and a map α : S n → Ex l X based at p and a h omotop y H : S n ∧ ∆[1] + → Ex r Y from (Ex r f ) ◦ α to β , where r ≥ k and r ≥ l , and , moreo v er, if α ′ : S n → Ex m X DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 32 is based at p and there is a h omotopy from (Ex r f ) ◦ α ′ to β with r ≥ k and r ≥ m , then there is a homotop y H : S n ∧ ∆[1] + → Ex s X from α to α ′ with s ≥ l and s ≥ m . Therefore, the class of w eak equiv alences b et w een simplicial s ets is ∆ 1 -deﬁnable. Ha vin g pr o v ed (1) and (2), we next add r ess (3). A sp ectrum F is w eakly con tractible if and only if all its homotop y groups v anish, th at is, colim n [ S n + k , F n ] = 0 for all k ∈ Z . This is equiv alent to imp osing that, for all k ∈ Z and n ≥ 0 suc h that n + k ≥ 0, eac h p oin ted map β : S n + k → Ex ∞ F n b ecomes n ullhomotopic after susp ending it a ﬁnite num b er of times (sa y , m times) and comp osing with the str ucture maps σ n : S F n → F n +1 . More pr ecisely , on the one hand, w e ha v e: (9.3) S n + m + k S m β / / S m Ex ∞ F n j / / Ex ∞ S m Ex ∞ F n , and, on the other h and, there are m aps Ex ∞ S m Ex ∞ F n Ex ∞ S m F n Ex ∞ S m j o o Ex ∞ σ / / Ex ∞ F n + m , where σ is an abbreviation for σ n + m − 1 ◦ S σ n + m − 2 ◦ · · · ◦ S m − 2 σ n +1 ◦ S m − 1 σ n . The maps j and Ex ∞ S m j are natural we ak equiv alences. Hence, F is we akly contrac tible if and only if, for eac h k ∈ Z and eac h ( n + k )-simplex x ∈ Ex ∞ F n whose faces are equal to the basep o int , there is an ( n + m + k )-simplex y ∈ Ex ∞ S m F n whose faces are equal to the basep oi nt and an ( n + m + k + 1)-simplex z ∈ Ex ∞ F n + m whose top f ace is y and all its other f aces are equ al to the basep oint , and (Ex ∞ S m j ) y ∼ j ( S m x ). W e ﬁ n ally pr o v e (4). In order to formalize the f act that a sp ectrum E is an Ω-sp ectrum, we ﬁrst need th at eac h simp licial set E n b e ﬁbrant. T hen w e need to deﬁn e the adjoin t maps τ n : E n → Ω E n +1 and we need to imp ose that eac h τ n b e a weak equiv alence. T o deﬁne τ n , let x b e a k -simp lex of E n . Its image in Ω E n +1 = map ∗ ( S 1 , E n +1 ) is a map S 1 ∧ ∆ [ k ] + → E n +1 whic h is d etermined by imp osing th at ( τ n ( x ))( se 1 , e k ) = σ n ( se 1 , x ) , where e 1 is the nondegenerate 1-simplex of S 1 and e k is the n ondegenerate k -simplex of ∆[ k ], and s denotes a comp osition of d egeneracies.  In what follo ws, w e denote by sSet ∗ the category of p ointed simplicial sets and p oin ted map s . Theorem 9.2. The class of E ∗ -acyclic simplicial sets for a sp e ctrum E is ∆ 1 with E as a p ar ameter. Pr o of. If ( X , p ) and ( Y , q ) are p ointe d simplicial sets, then W = X ∨ Y is a p oint ed sim p licial set con tained in X × Y suc h that W n con tains all elemen ts of the form ( x, sq ) with x ∈ X n and all th ose of the form ( sp, y ) with y ∈ Y n , where s is a comp o sition of degeneracies, with basep oint ( p, q ). The s m ash pro du ct X ∧ Y is obtained from X × Y by collapsing X ∨ Y to a p o int . Hence, ( X ∧ Y ) n = ( X n × Y n ) \ ( W n \ { ( sp, sq ) } ) for all n , and we d eclare equal to ( sp, s q ) all faces of elemen ts of X n +1 × Y n +1 and all degeneracies of elemen ts of X n − 1 × Y n − 1 taking v alues in W n . If ( X , p ) is a p o int ed simplicial set and E is a sp ect rum with structur e maps h σ n : 0 ≤ n < ω i , then X ∧ E is a sp ect rum with ( X ∧ E ) n = X ∧ E n DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 33 and stru cture maps (id ∧ σ n ) ◦ ( τ ∧ id ) for all n , where τ : S 1 ∧ X → X ∧ S 1 is the t wist map. By part (3) of P rop osition 9.1, the statement that X ∧ E is wea kly con tractible is ∆ 1 . Ho we ve r, a formula expressing this fact has to conta in a deﬁnition of X ∧ E , wh er e E is a give n sp ectrum treated as a parameter. This can b e done in t w o equiv alen t wa ys, as follo ws: (9.4) X ∈ sSet ∗ ∧ ∃ F [ F is a sp ectrum ∧ ( ∀ n < ω )(( F n = X ∧ E n ) ∧ σ F n = (id ∧ σ E n ) ◦ ( τ ∧ id )) ∧ F is we akly contrac tible ]; (9.5) X ∈ sSet ∗ ∧ ∀ F [[ F is a sp ectrum ∧ ( ∀ n < ω )(( F n = X ∧ E n ) ∧ σ F n = (id ∧ σ E n ) ◦ ( τ ∧ id ))] → F is we akly contract ible ] . Since (9.4 ) is Σ 1 and (9.5) is Π 1 , the th eorem is prov ed.  As explained in Section 2, the fact that homological acycli c classes are ∆ 1 implies that they are absolute. This means that, if E is a sp ectrum an d M is a transitiv e mo d el of ZFC such that E ∈ M (in wh ic h case E is a sp ectrum in M as w ell, since b eing a sp ec trum is ∆ 0 ), then a simplicial set X ∈ M is E ∗ -acyclic in M if and only if it is E ∗ -acyclic . W e thank F ederico Cantero for p ertinen t r emarks ab out the argumen t giv en in the pro o f of th e next r esult. Theorem 9.3. The class of E ∗ -acyclic simplicial sets for an Ω -sp e ctrum E is ∆ 2 with E as a p ar ameter. Pr o of. Let E b e an Ω-sp ect rum, which w ill b e used as a parameter. By part (4) of Prop o sition 9.1, ev ery trans itiv e mo del of ZF C conta ining E w ill agree with the f act that E is an Ω-sp ec trum. A simplicial set X is E ∗ -acyclic if and only if, for all k ∈ Z and n ≥ 0 w ith n + k ≥ 0, ev ery map S n X → E n + k b ecomes nullhomoto pic after su sp end in g it a ﬁn ite num b er of times and comp osing with the structur e maps of E as in (9.3). Th is claim leads to a Π 2 form ula —note that a map S n X → E n + k is n o longer determined b y any ﬁ nite set of simp lices of E n + k . Next w e sho w that it is p ossible to restate it by means of a Σ 2 form ula. A p ointe d simp licial s et ( X, p ) is E ∗ -acyclic if and only if for all n < ω the simplicial set map ∗ ( X, E n ) is wea kly con tr actible, assuming that E is an Ω-sp ectrum. Thus, X is E ∗ -acyclic if and only if the f ollo wing formula holds, where w e need to deﬁne M = map ∗ ( X, E n ): X ∈ sSet ∗ ∧ ( ∀ n < ω ) ∃ M [ M ∈ sSet ∗ ∧ ( ∀ k < ω ) [( ∀ f ∈ M k ) f ∈ sSet ∗ ( X ∧ ∆[ k ] + , E n ) ∧ ∀ g ( g ∈ sSet ∗ ( X ∧ ∆[ k ] + , E n ) → g ∈ M k )] ∧ M is w eakly con tractible] . According to P rop osition 9.1, this is a Σ 2 form ula.  In order to state and prov e the n ext results, we use the term homotopy r eﬂe ction (also called homotopy lo c alization elsewhere) to designate a functor L : sSet ∗ → sSet ∗ equipp ed w ith a n atur al transform ation η : Id → L whic h preserve s w eak equiv alences and b ecome s a reﬂection when passing to the homotop y category . F or a homotopy r eﬂection L , an L -e qu ivalenc e is a m ap f : X → Y suc h th at Lf : LX → LY is an isomorphism in the homotop y catego ry , and a simplicial set X is called L -lo c al if it is ﬁbr an t and weakly equiv alen t to LX for some X . DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 34 W e also recall that, for a p oi nte d map f : A → B , a connected ﬁbr an t simplicial set X is f -lo c al if the induced map of p o int ed function complexes map ∗ ( f , X ) : map ∗ ( B , X ) − → map ∗ ( A, X ) is a w eak equ iv alence, and a nonconnected X is f -lo cal if eac h of its con- nected comp onents is f -local with any c hoice of basep oint; cf. [21, 1.A.1]. Note that, if X is f -local for a map f : A → B , then f induces a bijection [ B , X ] ∼ = [ A, X ], since [ B , X ] is in natur al bijectiv e corresp ondence with the set of connected comp on ents of m ap ∗ ( B , X ). Hence, b eing f -lo cal is a stronger cond ition th an b eing orthogonal to f in the homotop y category . The same termin ology is used for a set or a pr op er class of maps F ; that is, a simplicial set is F -lo c al if it is f -lo cal f or all f ∈ F . An F -lo c alization is a homotop y r eﬂection L such th at the class of L -lo cal spaces coincides with the class of F -lo cal spaces. Lemma 9.4. Given any class of p ointe d maps S b etwe en simplicial sets, if ther e is a sub c lass F ⊆ S such that e ach element of S is a ﬁlter e d c olimit of elements of F , then every F -lo c al sp ac e is S -lo c al. Pr o of. The argument is analogous to the one us ed in the pr o of of Lemm a 8.1. Let f : A → B b e any elemen t of S and let X b e an F -lo cal simplicial set, whic h we ma y assum e connected. W rite f = colim f k (in the cate gory of p oint ed maps b et w een simp licial sets), where f k : A k → B k is in F for all k ∈ K , and K is ﬁltered. No w we use, as in [17, Lemma 5.2], the fact that the n atural map ho colim f k − → colim f k is a w eak equiv alence, since h omotop y groups commute w ith ﬁltered colimits (here h o colim is a p ointed homotop y colimit [28, 18.8]). Hence, map ∗ ( B , X ) ≃ map ∗ (ho colim B k , X ) ≃ holim map ∗ ( B k , X ) ≃ holim map ∗ ( A k , X ) ≃ m ap ∗ (ho colim A k , X ) ≃ map ∗ ( A, X ) , from whic h it follo w s indeed that X is S -lo cal.  Theorem 9.5. A ssume the existenc e of arbitr arily lar ge sup er c omp act c ar- dinals. Then for every additive c ohomolo gy the ory E ∗ deﬁne d on si mplicial sets ther e is a homotop y r eﬂe ction L such that the L -e qu ivalenc es ar e pr e- cisely the E ∗ -e quivalenc es. Pr o of. Let S b e th e class of E ∗ -equiv alences for a giv en additive cohomology theory E ∗ , and view it as a full sub ca tegory of the category of p oin ted maps b et we en sim p licial sets, whic h is accessibly emb edded in to a category of structures, by L emma 3.2. S ince th e class of E ∗ -equiv alences coincides with the class of maps w h ose mappin g cone is E ∗ -acyclic , Theorem 9.3 tells us that S is ∆ 2 , h ence Σ 2 . Con s equen tly , it follo ws from Theorem 7.5 that there is a r egular cardinal κ and a set F of E ∗ -equiv alences such that ev ery E ∗ -equiv alence is a κ -ﬁltered colimit of elements of F in th e categ ory of p oint ed maps b etw een simp licial sets. T o conclud e the pro of, let f : A → B b e th e copro duct of all the ele- men ts of F , and let L b e f -lo calizati on, as constructed in [12], [21 ] or [28]. Since all the elemen ts of F are E ∗ -equiv alences and E ∗ is add itive, f is an E ∗ -equiv alence. DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 35 Let E b e an Ω-sp ect rum representing E ∗ . S ince f is an E ∗ -equiv alence, it induces bijections [ B , E n ] ∼ = [ A, E n ] for all n , and in fact w eak equiv alences map ∗ ( B , E n ) ≃ map ∗ ( A, E n ) for all n . In other w ords, the basep o int com- p onent of E n is f -local for all n . Sin ce E n is a lo op sp ace, all its connected comp onent s h a ve the same homotopy type and therefore E n itself is f -lo cal for all n . It follo ws that ev er y L -equiv alence g : X → Y induces a wea k equiv alence map ∗ ( Y , E n ) ≃ map ∗ ( X, E n ) for all n , and w e conclude that all L -equiv alences are E ∗ -equiv alences. Con v ersely , ev ery E ∗ -equiv alence is, as said ab o v e, a κ -ﬁltered colimit of ob jects from F . According to Lemma 9.4 , ev ery L -lo cal simplicial set is E ∗ -lo cal, and th er efore all E ∗ -equiv alences are L -equiv alences. Th is com- pletes the argument .  What w e ha v e p ro v ed is that lo calizati on with resp ect to an y ad d itiv e cohomology theory exists on the homotop y catego ry of simplicial sets if arbitrarily large sup ercompact cardinals exist. Th is is a substantial im- pro v emen t of [17, Corollary 5.4], where it was p ro v ed that the existence of cohomologi cal lo calizations follo ws fr om V op ˇ en k a’s principle. W e also emphasize that from Th eorem 9.2 it follo ws, by a similar metho d as in the pro of of Th eorem 9.5 (or using Th eorem 9.7 b elo w), that the existence of homolo gic al localizations (for representa ble homology th eories) is p ro v able in ZF C. Bousﬁeld did it in deed in [11]. The same line of argum en t pr o v id es an answer to F arj ou n ’s qu estion in [20] of whether all h omotopy reﬂections are f -localizations for some map f . It w as sho wn in [17] that the answ er is aﬃr mativ e under V op ˇ enk a’s principle, and Prze ´ zdziec ki pro v ed in [42 ] that an aﬃrmativ e an s w er is in f act equiv a- len t to V op ˇ enk a’s p rinciple. Here w e prov e an analogue of Theorem 8.4. Theorem 9.6. A homoto py r eﬂe ction L on simplicial se ts is an f -lo c aliza- tion f or some map f under any of the fol lowing assumptions: (1) The class of L -e quiv alenc es is deﬁnable with suﬃciently low c om- plexity. (2) The class of L -lo c al simplicial sets is Σ n +1 for n ≥ 2 and ther e is a pr op er class of C ( n ) -extendible c ar dinals. Pr o of. F or (1), w e m a y c ho ose, by Th eorem 7.5, a s et F of L -equiv alences suc h that every L -equiv alence is a ﬁ ltered colimit of elements of F in th e catego ry of p oin ted maps b et w een s im p licial sets. Let f b e the copro duct of all the elemen ts of F . Then f is an L -equiv alence, since the class of L -equiv alences is closed under copro d ucts. Th er efore, ev ery L -lo cal simpli- cial set is f -local, by [17, Corollary 4.4]. Con v ersely , every f -lo cal simplicial set is L -lo cal b y Lemma 9.4. In order to pro v e (2), note that, if the class of L -lo cal simplicial sets is Σ n +1 , then the class of L -equiv alences is Π n +1 , since f : A → B is an L -equiv alence if and only if the ind uced fun ction [ B , X ] → [ A, X ] is a bijection for eac h L -lo cal sp ace X , whic h can b e formalized as ∀ X ∀ g [( X is an L -lo cal simp licial set ∧ g ∈ sSet ∗ ( A, X )) → ( ∃ h ( h ∈ sSet ∗ ( B , X ) ∧ h ◦ f ≃ g ) ∧ any tw o suc h maps are homotopic)] . The statement “an y tw o such maps are h omotopic” can b e formally written as a Π 2 form ula. Hence the same argument as in p art (1) applies under DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 36 the assumption that a prop er class of C ( n )-extendible cardinals exists, by means of Theorem 7.5.  The corresp ondin g analogue of T heorem 8.7 is the n ext result. Lo caliza- tion with r esp ect to prop er classes of maps w as shown to exist in [18] under restrictiv e conditions. Theorem 9.7. L e t S b e any (p ossibly pr op er) class of maps of simplicial sets. If S is deﬁnable with suﬃciently low c omplexity, then an S -lo c alization exists. Pr o of. Theorem 7.5 implies that there is a set F ⊆ S su c h that ev ery f ∈ S is a ﬁ ltered colimit of elemen ts of F . Then F -lo calization exists s in ce F is a set, and ev ery F -lo cal simplicial set is S -lo cal by Lemma 9.4. Since F ⊆ S , all S -lo cal simplicial sets are F -lo cal, so the pr o of is complete.  10. Bergman’s qu estion If Σ is a ﬁnitary op erational signature, then Σ-stru ctures are universal algebr as . If C is a full sub category of Str Σ and n is a nonnegativ e in teger, an n -ary implicit op er ation f on C is a n atural tr an s formation f r om the n -fold p ro duct fun ctor to the identit y fun ctor; that is, a collection of maps f X : X n → X indexed b y ob jects X of C suc h that the square X n h n / / f X   Y n f Y   X h / / Y comm utes for eac h homomorphism h : X → Y . Su c h implicit op erations are very usefu l in ﬁnite u niv ersal alge bra; see [6]. If C is a prop er class with no h omomorp hisms except iden tities, th en eac h collection { f X } X ∈C is an implicit op eration. Thus, assuming the n egation of V op ˇ enk a’s prin ciple, there is a prop er class of imp licit op erations on C . In connection with [10], Bergman ask ed whether this can happ en assuming V opˇ enk a’s prin ciple. Theorem 10.1. F or a ﬁnitary op er ational sig natur e Σ , V opˇ enka’s principle implies that ther e i s only a set of implicit op er ations on e ach ful l sub c ate gory of Str Σ . Pr o of. Let C b e a full sub cate gory of Str Σ , where Σ is S -sorted. By [3 ], V op ˇ enk a’s principle imp lies that there is a regular card inal κ and a set A of ob jects in C such that eac h ob ject of C is a κ -ﬁltered colimit of ob jects of A . Since the forgetful functor Str Σ → Se t S and the n -fold pro du ct functor ( − ) n : Set S → Set S preserve colimits, eac h imp licit op eration f X with X ∈ C is u niquely determined by { f A } A ∈A . Hence there is only a set of d istin ct imp licit op erations on C .  W e improv e this r esult as follo ws. Theorem 10.2. F or a ﬁnitary op er ational signatur e Σ , every ful l sub c ate- gory S of St r Σ deﬁnable with suﬃcie ntly low c omplexity has only a set of implicit op er ations. DEFINABLE O R THOGON A LITY CLASSES ARE SMALL 37 Pr o of. As sho wn in the pro of of Theorem 7.5, f or eac h ob ject Y of S the s lice catego ry ( S ∩ H ( κ ) ↓ Y ) is coﬁnal in ( K ↓ Y ) f or some regular cardinal κ , where K is the (essen tially small) class of κ -pr esen table ob jec ts in Str Σ. Th us eac h ob ject of S is a κ -ﬁltered colimit of ob jects from the set S ∩ H ( κ ). The rest is the same as in the p ro of of Th eorem 10.1.  Referen ces [1] J. A d´ amek, H. Herrlic h, and G. 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Joan Bagari a, ICREA (Instituci´ o Catalana de Recerca i Estudis Av an¸ cats) and Dep arta- ment de L` ogica, H ist` oria i Filosoﬁa de la Ci` en cia, Universitat de Barcelona, Montalegre 6, 08001 Barcelona, Spain, joan.bagaria @icrea.cat; bagaria @ub.edu . Carles Casacub erta, Departament d’ ` Algebra i Geometria and In stitut d e Matem` atica, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, S p ain, carles.cas acuberta@ub.e du . A. R . D. Mathias, ER MIT, U niversit ´ e de la R´ eun ion, U FR Sciences et T ec hnologies, Lab o- ratoire d ’I n formatique et de Math´ ematiques, 2 rue Joseph W etzel, Bˆ atiment 2, F-97490 Sainte Clotilde, F rance outre-mer, ardm@uni v-reunion.fr; ardm@dpmm s.cam.ac.uk . Ji ˇ r ´ ı Rosic k´ y, Department of Mathematics and Statistics, Masaryk Universit y , Kotl´ a ˇ rsk´ a 2, 600 00 Brno, Czec h Rep u blic, rosicky@math.mun i.cz .

Definable orthogonality classes in accessible categories are small

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