The orthogonal subcategory problem in homotopy theory

THE ORTHOGONAL SUBCATEGORY PROBLEM IN HOMOTOPY THEORY CARLES CASACUBERTA AND BORIS CHORNY Abstract. It is known that, in a locally presentable category, localization exists with respect to every set of morphisms, while the statement that localization with respect to every (possibly proper) class of morphisms exists in locally presentable categories is equivalent to a large-cardinal axiom from set theory. One proves similarly, on one hand, that homotopy localization exists with respect to sets of maps in every cofibrantly generated, left proper, simplicial model category Mwhose underlying category is locally presentable. On the other hand, as we show in this article, the existence of localization with respect to possibly proper classes of maps in a model category M satisfying the above assumptions is implied by a large-cardinal axiom called Vopˇenka’s principle, although we do not know if the reverse implication holds. We also show that, under the same assumptions on M, every endofunctor of Mthat is idempotent up to homotopy is equivalent to localization with respect to some class Sof maps, and if Vopˇenka’s principle holds then Scan be chosen to be a set. There are examples showing that the latter need not be true if Mis not cofibrantly generated. The above assumptions on Mare satisfied by simplicial sets and symmetric spectra over simplicial sets, among many other model categories. Introduction Locally presentable categories were introduced by Gabriel and Ulmer in [18]. This concept has proved to be very useful in category theory. Among other things, the orthogonal subcategory problem (asking if localization with respect to a given class of morphisms exists) has a positive solution 2000 Mathematics Subject Classification. Primary 55U35; Secondary 55P60, 18G55. Key words and phrases. Model category, localization, large cardinal. During the preparation of this paper, the second-named author was a fellow of the Marie Curie Training Site with reference no. HPMT-CT-2000-00075 of the European Commission, hosted by the CRM, Barcelona. The first-named author was supported by the Spanish Ministry of Education and Science under MEC-FEDER grant MTM2004- 03629. 1 2 CARLES CASACUBERTA AND BORIS CHORNY in locally presentable categories if the given class of morphisms is a set; see, e.g., [1, 1.37]. Moreover, if one assumes the validity of a suitable settheoretical principle, then there is also a positive solution to this problem for every proper class of morphisms. In fact, Ad´amek, Rosick´y and Trnkov´a proved in [2] that the statement that the orthogonal subcategory problem has a solution for every class of morphisms in locally presentable categories is equivalent to the weak Vopˇenka principle, a large-cardinal principle that cannot be proved using the usual ZFC axioms (Zermelo–Fraenkel axioms with the axiom of choice). Localizing with respect to sets of maps is a common technique in homotopy theory, as well as in other areas of Mathematics. However, localizing with respect to proper classes of maps is a more delicate issue, since the standard methods may fall into set-theoretical difficulties (see for instance [9], where positive results in equivariant homotopy theory involving localization with respect to proper classes of maps were obtained). Due to difficulties of this sort, it is still unknown whether the existence of arbitrary cohomological localizations of spaces can be proved or not using the ZFC axioms. An interesting step was made in [8], based on results from [1], by showing that Vopˇenka’s principle implies the existence of localization with respect to every proper class of maps in the category of simplicial sets. The existence of cohomological localizations follows of course as a special case. Vopˇenka’s principle is equivalent to the statement that the category of ordinals cannot be fully embedded into the category of graphs (where a graph is meant to be a binary relation). This statement has a place in the hierarchy of large-cardinal principles; see [1]. In this article we contribute further to the ongoing program of extending basic results from locally presentable categories to homotopy theory, which may perhaps give answers to other open problems under large-cardinal assumptions. In order to achieve this, one has to work in suitable model categories. Specifically, our results are stated in left proper, combinatorial, simplicial model categories. The term “combinatorial” means that the model category is cofibrantly generated and the underlying category is locally presentable (see [1] and [20] for the definitions of these concepts). This notion is due to J. H. Smith, who constructed (in unpublished work) localizations of combinatorial model category structures with respect to sets of maps. In [15], Dugger proved that every combinatorial model category is equivalent to a localization of a category of diagrams of simplicial sets, hence generalizing [1, 1.46]. Among many other examples, the model category of simplicial sets and the model category of symmetric spectra based on simplicial sets are combinatorial. THE ORTHOGONAL SUBCATEGORY PROBLEM IN HOMOTOPY THEORY 3 In Section 1 we show that Vopˇenka’s principle implies the existence of homotopy localization with respect to every class of maps in left proper, combinatorial, simplicial model categories. This fact can also be deduced, with a different argument, from results obtained by Rosick´y and Tholen in [24, §2]. Furthermore, under Vopˇenka’s principle, every such localization is equivalent to localization with respect to some set of maps. Next, we address a closely related question, raised by Dror Farjoun in [11], asking if any functor Lon simplicial sets that is idempotent up to homotopy is equivalent to localization with respect to some single map f. He himself showed in [12] that, if Lis assumed to be, in addition, continuous, then it is indeed equivalent to localization with respect to a proper class of maps. This result was improved in [8] by showing that the assumption that Lbe continuous is unnecessary, and that, under Vopˇenka’s principle, the proper class of maps defining Lcan be replaced by a set (in the category of simplicial sets). Furthermore, it was shown that such a replacement of a class by a set cannot be done using only the ZFC axioms, since a counterexample was exhibited by means of another assumption (the nonexistence of measurable cardinals), which is relatively consistent with ZFC. In Section 2 we show (without resorting to large-cardinal principles) that every homotopy idempotent functor Lin a simplicial model category Mis equivalent to localization with respect to a proper class of maps, assuming either that Lis continuous or that Msatisfies suitable hypotheses allowing to approximate any homotopy functor by a continuous functor. For this, one may assume that Mis a simplicial model category that is proper, cofibrantly generated and stable (as in [23]), or left proper and either combinatorial or cellular (as in [14]). Furthermore, if one assumes that Vopˇenka’s principle is true and Mis combinatorial, then, again, the proper class of maps defining Lcan be replaced by a set. In most cases of interest, such a set of maps can further be replaced by a single map (by taking the coproduct of all maps in the set), but not always, as we show by means of an example at the end of the paper. In [10] an example was given of a homotopy idempotent functor in a locally presentable (but not cofibrantly generated) model category that fails to be a localization with respect to any set of maps. Namely, in the category of maps between simplicial sets with the model structure generated by the collection of orbits (as defined in [13]), the functor that sends every map to the final object (i.e., a map between two points) is not a localization with respect to any set of maps. Hence our results in Section 2 below are sharp. Acknowledgements: Discussions with Mark Hovey, Jiˇr´ı Rosick´y, Brooke Shipley and Jeff Smith are greatly appreciated. 4 CARLES CASACUBERTA AND BORIS CHORNY 1. Simplicial orthogonality Model categories were introduced by Quillen in [22] and have recently been discussed in the books [16], [19], [20], [21], among many other places, with slight changes in the terminology and even in the assumptions. In this article we will assume that model categories are complete, cocomplete, and equipped with functorial factorizations. See [16, §9], [20, §7] or [21, §1] for more details. Although our main results are stated for simplicial model categories (for the definition, see for example [19, II.3] or [20, 9.1.5]), several of our steps require only the use of homotopy function complexes, as introduced in [17] and discussed in [20, Ch. 17] or [21, §5]. Thus, for any given model category M, we choose functorially a fibrant simplicial set map(X, Y ) for each Xand Yin M, whose homotopy type is the same as the diagonal of the bisimplicial set M(X∗, Y∗) where X∗→Xis a cosimplicial resolution of Xand Y→Y∗ is a simplicial resolution of Y. The homotopy type of map(X, Y ) remains unchanged if Xor Yare replaced by weakly equivalent objects. If Mis a simplicial model category and Map(X, Y ) denotes the simplicial set given as part of the structure of M, then Map(QX, RY ) is a good choice of a homotopy function complex, where Qis a cofibrant approximation functor and Ris a fibrant approximation functor in M. Before discussing simplicial orthogonality in model categories by means of homotopy function complexes, we recall the following older concepts from category theory. If Cis any category, an object Xand a morphism f:A→ Bare called orthogonal (see [1] or [7] for details and motivation) if the induced function C(f, X): C(B, X)−→ C(A, X) is bijective. (We denote by C(X, Y ) the set of morphisms from Xto Y in C.) If Lis an endofunctor of Cequipped with a natural transformation η: Id →Lsuch that Lη :L→LL is an isomorphism and ηL =Lη, then L is called an idempotent functor or a localization. Then every object isomorphic to LX for some Xis orthogonal to every morphism fsuch that Lf is an isomorphism, and these two classes determine each other by the orthogonality relation; that is, an object is isomorphic to LX for some Xif and only if it is orthogonal to all morphisms fsuch that Lf is an isomorphism, and reciprocally. As a special case, this terminology applies to the homotopy category HoMassociated with any model category M. Thus, orthogonality in HoM between an object Xand a map f:A→Bamounts to the condition that (1) [f, X]: [B, X]−→ [A, X] THE ORTHOGONAL SUBCATEGORY PROBLEM IN HOMOTOPY THEORY 5 be bijective, where [X, Y ] means, as usual, HoM(X, Y ). Examples of idempotent functors in the homotopy category of simplicial sets, such as homological localizations, have been studied since several decades ago; see [4]. Throughout the extensive study of localizations undertaken since then in homotopy theory, a stronger notion of “simplicially enriched orthogonality” came to be considered. There is no widely agreed terminology for it yet. It was called simplicial orthogonality in [8] and homotopy orthogonality in [20, §17]. Thus, if Mis any model category with a choice of homotopy function complexes, an object Xand a map f:A→Bwill be called simplicially orthogonal or homotopy orthogonal (not to be confused with orthogonality in HoM) if the induced map (2) map(f, X): map(B, X)−→ map(A, X) is a weak equivalence of simplicial sets. Since there is a natural bijection between π0map(X, Y ) and [X, Y ], homotopy orthogonality implies indeed orthogonality in the homotopy category HoM. The fibrant objects that are homotopy orthogonal to a given map fare usually called f-local. More generally, if Sis any class of maps, the fibrant objects that are homotopy orthogonal to all the maps in Sare called S- local. We denote by Sh ⊥the closure under weak equivalences of the class of S-local objects, and call it the homotopy orthogonal complement of S. Similarly, for a class Dof objects, we denote by Dh ⊥the class of maps that are homotopy orthogonal to all the objects in D. In particular, the maps in (Sh ⊥)h ⊥are called S-local equivalences, or shortly S-equivalences. The homotopy-theoretical version of the orthogonal subcategory problem asks if Sh ⊥is reflective in HoMfor a model category Mand a given class of maps Sin M(that is, if a localization Lexists in HoMsuch that the closure of the image of Lunder isomorphisms is precisely the class Sh ⊥). One reason for using (2) instead of (1) as orthogonality relation is the fact that the answer to the orthogonal subcategory problem would too often be negative using (1). For instance, there is no localization in the homotopy category of simplicial sets onto the class of simply connected spaces. See [5] for a more elaborate counterexample. In order to construct localizations in HoM, one normally operates in the corresponding model category M. The following terminology is commonly used. A homotopy idempotent functor is an endofunctor L:M→M preserving weak equivalences, taking fibrant values, and equipped with a natural transformation η: Id →L(called a coaugmentation) which is idempotent up to homotopy; that is, for each object X, the morphisms LηXand ηLX from LX to LLX coincide in HoMand are weak equivalences. Thus Ldefines indeed a localization in HoM. 6 CARLES CASACUBERTA AND BORIS CHORNY If Lis a homotopy idempotent functor such that LX is S-local for all Xand ηX:X→LX is an S-equivalence for all X, where Sis any class of maps, then it is said that Lis a homotopy localization with respect to the class S, or shortly an S-localization. In this case, the class Sh ⊥is indeed the closure of the image of Lunder isomorphisms in HoM. (In order to prove that every object Yin Sh ⊥is weakly equivalent to LX for some X, consider the coaugmentation ηY:Y→LY , which induces a weak equivalence map(LY, Y )≃map(Y, Y ), hence a bijection [LY, Y ]∼ =[Y, Y ] yielding a map LY →Yin HoMwhich is inverse to ηY, so Y≃LY , as needed.) It is well known that the orthogonal subcategory problem for a class of maps Sin a model category Mhas a positive solution whenever Sis a set and Msatisfies certain assumptions, which vary slightly depending on the authors. We will call a model category combinatorial if it is cofibrantly generated and the underlying category is locally presentable. The definition of a locally presentable category can be found in [1] or [18], and the definition of a cofibrantly generated model category is contained, e.g., in [20]. The notion of properness is also discussed in [20]. Theorem 1.1. Let Mbe a left proper, combinatorial, simplicial model category. For every set of maps Sthere is a homotopy localization with respect to S. Proof. The core of the proof is in [3]. See [20] for an updated approach. ¤ As far as we know, there is no way to prove this when Sis a proper class, not even for simplicial sets, using the ordinary axioms of set theory. In [8] it was shown that the statement of Theorem 1.1 holds for a proper class Sin the category of simplicial sets using a suitable large-cardinal axiom (Vopˇenka’s principle). We now undertake a generalization of this fact to other model categories. If Mis a cofibrantly generated model category and Cis a small category, then the projective model structure on the category MCof C-indexed diagrams in Mhas objectwise weak equivalences and objectwise fibrations, while the injective model structure has objectwise weak equivalences and objectwise cofibrations. The projective model structure is discussed in [20, 11.6 and 11.7], where it is shown that, if the model category Mis simplicial, then MCwith the projective model structure is also simplicial. Lemma 1.2. Let Mbe a cofibrantly generated simplicial model category and Ca small category. Suppose that Ais a cofibrant diagram in the projective model category structure of MCand Xis a fibrant object of M. Then Map(A, X)is fibrant in the injective model structure on the category of Cop-diagrams of simplicial sets. THE ORTHOGONAL SUBCATEGORY PROBLEM IN HOMOTOPY THEORY 7 Proof. We have to show that any commutative square CÄ_ ²O i ²² //Map(A, X) ²²²² D//∗, where iis an objectwise trivial cofibration of Cop-diagrams of simplicial sets, admits a lift. By adjunction (as in [20, 18.3.9]), this problem is equivalent to finding a lift in the following commutative square in M: A⊗CC ²² //X ²²²² A⊗CD//∗. This problem is equivalent, by another adjunction, to finding a lift in the following commutative square in MC: ∅Ä_ ²² //XD ²O Xi ²²²² A//XC. In the last square a lift exists, since Ais projectively cofibrant and Xiis a projective trivial fibration. ¤ Recall that a partially ordered set Iis called λ-directed, where λis a regular cardinal, if every subset of Iof cardinality smaller than λhas an upper bound. Lemma 1.3. Let Dbe any class of objects in a combinatorial simplicial model category M, and let Sbe its homotopy orthogonal complement. Then there exists a regular cardinal λsuch that Sis closed under λ-directed colimits in the category of maps of M. Proof. Let Cbe a set of generating cofibrations for M. Choose a regular cardinal λsuch that any member of the set of domains and codomains of maps in Cis λ-presentable (such a cardinal exists since Mis locally presentable). Let Ibe any λ-directed partially ordered set, and suppose given a diagram f:I→Arr M, where Arr Mis the category of maps in M. Let us denote by X:I→Mthe domain of fand by Y:I→Mthe codomain, so fcan also be seen as a map from Xto Yin MI. Let us depict 8 CARLES CASACUBERTA AND BORIS CHORNY it, for simplicity, as a chain: (3) X0−−−−→ X1−−−−→ · · · −−−−→ Xn−−−−→ · · · f0  yf1  yfn  y Y0−−−−→ Y1−−−−→ · · · −−−−→ Yn−−−−→ · · · . Suppose that the maps fiare in Sfor each i∈I. Since Mis cocomplete, Arr Mis cocomplete as well, and we may consider the colimit of the diagram f. We need to show that the map colim f: colim X−→ colim Yis in S. Choose a cofibrant approximation ˜ f:˜ X→˜ Yto fusing the projective model structure on MI, hence obtaining the following commutative diagram in M: ˜ X0 ˜ f0²² Â_ ÂÂÂÂ ? ? ? ÂÄ//˜ X1 ˜ f1²² Â_ ÂÂÂÂ ? ? ? ÂÄ//· · ·ÂÄ//˜ Xn ˜ fn²² Â_ ÂÂÂÂ ? ? ? ÂÄ//· · · X0 f0 ²² //X1 f1 ²² //· · · //Xn// fn ²² · · · ˜ Y0Â_ ÂÂÂÂ ? ? ? ÂÄ//˜ Y1Â_ ÂÂÂÂ ? ? ? ÂÄ//· · ·ÂÄ//˜ YnÂ_ ÂÂÂÂ ? ? ? ÂÄ//· · · Y0//Y1//· · · //Yn//· · · , where ˜ fiis a cofibrant approximation to fi. For every Z∈ D, let ˆ Zbe a fibrant approximation to Z. The induced map Map(colim ˜ f, ˆ Z): Map(colim ˜ Y , ˆ Z)−→ Map(colim ˜ X, ˆ Z) can be written as a limit of a diagram of maps of simplicial sets lim Map( ˜ f, ˆ Z): lim Map( ˜ Y , ˆ Z)−→ lim Map( ˜ X, ˆ Z). The Iop-diagrams of simplicial sets Map( ˜ X, ˆ Z) and Map( ˜ Y , ˆ Z) are fibrant in the injective model structure by Lemma 1.2. Therefore, their inverse limits are homotopy inverse limits (since the constant diagram of points is cofibrant in the injective model structure). Hence, Map(colim ˜ f, ˆ Z) is a weak equivalence, as a map induced between homotopy inverse limits by levelwise weak equivalences Map( ˜ fi,ˆ Z). This shows that colim ˜ fis in S. Now trivial fibrations in Mare preserved under λ-directed colimits, since the set Cof generating cofibrations has λ-presentable domains and codomains. From the commutative diagram colim ˜ X∼//// colim ˜ f ²² colim X colim f ²² colim ˜ Y∼////colim Y THE ORTHOGONAL SUBCATEGORY PROBLEM IN HOMOTOPY THEORY 9 we conclude that the map colim ˜ fis a cofibrant approximation to the map colim f, since both colim ˜ Xand colim ˜ Yare cofibrant in M(indeed, ˜ Xand ˜ Yare cofibrant diagrams in MI, and the colimit functor MI→Mis left Quillen by [20, 11.6.8]). Hence, colim fis in S, as claimed. ¤ The statement of Vopˇenka’s principle and enough motivation for its use in this context can be found in [1], [2], [8], and [24]. Lemma 1.4. Suppose that Vopˇenka’s principle is true. Let Dbe any class of objects in a combinatorial simplicial model category M, and let S=Dh ⊥. Then there exists a set of maps Xsuch that Xh ⊥=Sh ⊥. Proof. By abuse of notation, we also denote by Sthe full subcategory of Arr Mgenerated by the class S. Since Mis locally presentable, Arr Mis also locally presentable. Then, assuming Vopˇenka’s principle, it follows from [1, Theorem 6.6] that Sis bounded, i.e., it has a small dense subcategory. We have shown in Lemma 1.3 that there exists a regular cardinal λsuch that Sis closed under λ-directed colimits in the category Arr M. Hence, by [1, Corollary 6.18], the full subcategory generated by Sin Arr Mis accessible. Thus, for a certain regular cardinal λ0≥λ, the class Scontains a set Xof λ0-presentable objects such that every object of Sis a λ0-directed colimit of objects of X. Since X ⊂ S, we have Xh ⊥⊃ Sh ⊥and (Xh ⊥)h ⊥⊂(Sh ⊥)h ⊥=S. Our aim now is to show the reverse inclusion (Xh ⊥)h ⊥⊃ S. By Lemma 1.3, (Xh ⊥)h ⊥ is closed under λ-directed colimits. Hence (Xh ⊥)h ⊥is also closed under λ0- directed colimits and every element of Sis a λ0-directed colimit of elements of X. This concludes the proof, since ((Xh ⊥)h ⊥)h ⊥=Xh ⊥.¤ Theorem 1.5. Let Mbe a left proper, combinatorial, simplicial model category. If Vopˇenka’s principle is assumed true, then for any (possibly proper) class of maps Sthere is a homotopy localization with respect to S. Proof. By Lemma 1.4, there exists a set Xof maps in Msuch that Xh ⊥= Sh ⊥. Then the homotopy localization with respect to X, which exists by Theorem 1.1, is an S-localization. ¤ Thus, the statement of Theorem 1.5 is a positive answer to the orthogonal subcategory problem for all classes of maps in sufficiently good model categories. 2. Idempotent functors and simplicial orthogonality The next theorem is motivated by results of Dror Farjoun in [12]. We consider a model category Mand assume, as in the beginning of the previous