DUALITY AND SMALL FUNCTORS

Transcription

1 DUALITY AND SMALL FUNCTORS GEORG BIEDERMANN AND BORIS CHORNY Abstract. The homotopy theory o small unctors is a useul tool or studying various questions in homotopy theory. In this paper, we develop the homotopy theory o small unctors rom spectra to spectra, and study its interplay with Spanier-Whitehead duality and enriched representability in the dual category o spectra. We note that Spanier-Whitehead duality unctor D : Sp Sp op actors through the category o small unctors rom spectra to spectra, and construct a new model structure on the category o small unctors, which is Quillen equivalent to Sp op. In this new ramework or the Spanier-Whitehead duality, Sp and Sp op are ull subcategories o the category o small unctors and dualization becomes just a ibrant replacement in our new model structure. Contents 1. Introduction 1 Acknowledgements 4 2. Yoneda embedding or large categories 4 3. Homotopy theory o small unctors 6 4. Models o spectra Homotopy unctors The Yoneda embedding as a Quillen equivalence Enriched representability in the dual category o spectra 27 Appendix A. A non-unctorial Bousield-Friedlander localization 32 Reerences Introduction In this paper we give an extension o Spanier-Whitehead duality by producing a Quillen equivalent model or the opposite category o spectra. Theorem Let Y : Sp op Sp Sp be the Yoneda embedding and Z its let adjoint unctor. There is a Quillen equivalence Z : Sp Sp Sp op : Y or a certain model structure on the category Sp Sp o small endounctors o spectra. As a consequence we prove the ollowing theorem about enriched representability o small covariant unctors rom spectra to spectra up to weak equivalence. Date: December 21,

2 2 GEORG BIEDERMANN AND BORIS CHORNY Theorem 7.4. Let F : Sp Sp be a small unctor. Assume that F takes homotopy pullbacks to homotopy pullbacks and also preserves arbitrary products up to homotopy. Then there exists a coibrant spectrum Y and a natural transormation F ( ) R Y ( ), inducing a weak equivalence F (X) R Y (X) or all ibrant X Sp. The deinitions o representable and small unctors are given at the end o the introduction, beore the description o the structural organization o the paper. Let Sp denote a closed symmetric monoidal model or the stable homotopy category that is locally presentable, with coibrant unit S, and that satisies the monoid axiom [31, De. 2.2]. We call the objects spectra. In Section 4 we prove that symmetric spectra [24] and Lydakis pointed simplicial unctors [28] with the linear model structure meet the criteria. Taking a ibrant representative Ŝ or the sphere spectrum S, the Spanier-Whitehead dual o a spectrum A is given by the enriched morphism object DA = hom Sp (A, Ŝ) in Sp. We point out that we do not insist on A to be compact. It coincides with the classical notion o Spanier-Whitehead dual i A is compact and coibrant. This unctor D : Sp op Sp is adjoint to itsel, since hom Sp (A, DB) = hom Sp (A, hom Sp (B, Ŝ)) = hom Sp (B, hom Sp (A, Ŝ)) = hom Sp op(da, B). This adjunction actors through the category Sp Sp o small unctors: Sp op D Sp. D Z W Y evŝ Sp Sp Here Y is the Yoneda embedding. Further, or all F Sp Sp we set Z(F ) = hom(f, Id) to be spectrum o natural transormation rom F to the identity unctor o Sp and evŝ(f ) = F (Ŝ) the unctor which evaluates every F at the chosen ibrant replacement Ŝ o the sphere spectrum S. For all A Sp, we set W (A) = A RŜ, where RŜ = hom Sp (Ŝ, ) is the unctor represented by Ŝ. See Section 2 or more details. The let adjoint unctors are depicted by the solid arrows. We view theorem 6.11 as another approach to the extension o Spanier-Whitehead duality to non-compact spectra as the one proposed by J. D. Christensen and D. C. Isaksen [13], where the model or Sp op was constructed on the category o pro-spectra. There is an interesting eature that distinguishes our construction: Proposition 6.12 states that every object in Sp Sp is weakly equivalent to an ℵ 0 -small

3 DUALITY AND SMALL FUNCTORS 3 representable unctor, which is ibrant and coibrant in our model structure. Since the category o small unctors contains ull subcategories equivalent to Sp and Sp op, which intersect precisely at the category o compact spectra (see Lemma 7.2), we obtain a coherent picture o (extended) Spanier-Whitehead duality or non-compact spectra. Let us move on to Theorem 7.4. How does it relate to other representability theorems? Roughly speaking, in category theory there are two main types o representability theorems: Freyd representability and Brown representability. Freyd representability theorem takes its origin in the oundational book [21] by P. Freyd on abelian categories and states that limit preserving set valued unctors deined on an arbitrary complete category and satisying the solution set condition are representable. It is intimately related to the celebrated adjoint unctor theorem. The irst Brown representability theorem was proven in a seminal article [6] by E.H. Brown on cohomology theories and states that an arbitrary semi-exact unctor deined on the homotopy category o pointed connected spaces and taking values in the category o pointed sets is representable. Both theorems have been applied many times and extended to new rameworks. The main dierence between the two representability results is that Freyd s theorem imposes the solution set condition on the unctor, while not demanding any set theoretical restrictions rom the domain category o the unctor. On the other side, Brown s theorem uses in a signiicant way the presence o a set o small generators in the domain category, while not imposing any set theoretical conditions on the unctor itsel. Enriched Freyd representability was proven by M. Kelly, [26, 4.84]. J. Lurie, [27, ] proved the analog in the ramework o (,1)-categories. The solution set condition is replaced by the accessibility condition on the unctor in both cases. Note that a covariant unctor with an accessible category in the domain is small i and only i it is accessible, but the concept o small unctor is applicable even i the domain category is not accessible. The enriched version o Brown representability theorem or contravariant unctors rom spaces to spaces was proven by the second author in [8]. J. F. Jardine [25] generalized the theorem or unctors deined on a coibrantly generated simplicial model category with a set o compact generators. The smallness assumption on the unctor classiies our theorem as a Freyd-type result up to homotopy. On the other hand, our exactness assumptions on the unctor are less restrictive than in Freyd s theorem and closer to a Brown-type theorem. Brown representability or covariant unctors rom the homotopy category o spectra to abelian groups was proven by A. Neeman [30]. An enriched version o Neeman s theorem is still not proven. In homotopy theory, there is a third kind o theorem: G.W. Whitehead s [33] representability o homological unctors where, or a covariant homological unctor F, an object C is constructed together with an objectwise weak equivalence F ( ) C ( ). Its enriched counterpart was proven by T. Goodwillie [22] as classiication o linear unctors. Whithead s representability is related to Brown s representability on inite spectra through the Spanier-Whitehead duality, as it was explained by J.F. Adams [1]. An enriched version o this connection is contained in Lemma 7.2 and is central in our proo o the representability theorem.

4 4 GEORG BIEDERMANN AND BORIS CHORNY I a category K is enriched in a closed symmetric monoidal category V, then a unctor F : K V is called (V-enriched) representable i there exists an object K K and a natural isomorphism o unctors η : F ( ) hom K (, K), where hom K (, ): K op K V is the enriched hom unctor. Our notation or representable unctors is R K ( ) = hom K (, K) and R K = hom K (K, ). A small unctor rom one large category to another is a let Kan extension o a unctor deined on a small, not necessarily ixed, subcategory o the domain. Equivalently, i the domain category is enriched over the range category, small unctors are small weighted colimits o representable unctors. The category o small unctors is a reasonable substitute or the non-locally small category o all unctors, provided that we are interested in studying global phenomena and not satisied with changing the universe as an alternative solution. Several variations o this concept or set-valued unctors were extensively studied by P. Freyd [20]. In algebraic geometry, small unctors were used by W. C. Waterhouse [32] under the name basically bounded presheaves in order to treat categories o presheaves over large sites without changing the universe, since such a change might also alter the sets o solutions o certain Diophantine equations. For enriched settings, our main reerence is the work o B. Day and S. Lack [15]. Recently, several applications o small unctors rom spaces to spaces have appeared in homotopy theory [2], [10]. The paper is organized as ollows. Section 3 is devoted to the construction o a new model category structure on small unctors, which is close to the projective model category, except that weak equivalences and ibrations are determined only on the values o the unctors on ibrant objects. Hence, it is called the ibrantprojective model structure. Its goal is to create an initial ramework in which the adjunction (Z, Y ) is a Quillen pair. In Section 4 we provide model categories or spectra that satisy the conditions given in the previous section. In Section 5 we obtain an auxiliary result 5.9. To obtain the promised new Quillen equivalent model or Sp op, where every spectrum corresponds to a representable unctor, we perorm in Section 6 a non-unctorial version o Bousield-Friedlander s Q-construction on Sp Sp. This is the crucial technical part o this paper. We localize the ibrantprojective model structure on Sp Sp with respect to the derived unit o the adjunction (Z, Y ). Our localization construction ails to be unctorial; nevertheless, it preserves enough good properties to allow us to get a let Bousield localization o Sp Sp along the lines o the Bousield-Friedlander localization theorem [5]. In the Appendix A, we provide an appropriate generalization o the Bousield-Friedlander machinery to encompass non-unctorial homotopy localizations. The represntability theorem 7.4 is derived in the last Section 7. Acknowledgements. We would like to that A. K. Bousield or helping us to prove the only i part in the classiication o Q-ibrations in Theorem A.8 and the anonymous reeree or many useul suggestions. 2. Yoneda embedding or large categories In this article, we consider enriched categories and enriched unctors. Some sources do not distinguish between the cases o small and large domain categories, although unctors rom large categories have large hom-sets, i.e., proper classes. Morphism sets and internal mapping objects only make sense ater a change o universes. Unortunately, we cannot adopt this approach, as the internal mapping objects will play a crucial role in the construction o homotopy theories on unctors.

5 DUALITY AND SMALL FUNCTORS 5 Thus, we will use small unctors and the Yoneda embedding with values in the category o small unctors. The language o enriched category theory is used throughout the paper. The basic deinitions and notations may be ound in Max Kelly s book, [26]. Deinition 2.1. Let V be a symmetric monoidal category and K a V-category. A V-unctor rom K to V is called a small unctor i it is a V-let Kan extension o a V-unctor deined on a small but not necessarily ixed subcategory o K. The category o small unctors is denoted by V K. The main example o the symmetric monoidal model category V considered in this paper is the category Sp o spectra. As explained in Section 4 we can work with either symmetric spectra, [24], or Lydakis category o linear unctors, [28]. In the uture we hope to extend the ideas o this paper to make them applicable or unctors enriched in simplicial sets S or chain complexes, so we record the basic results in bigger generality, than required or the present paper. Deinition 2.2. The enriched covariant Yoneda embedding unctor Y : K op V K is given by mapping an object K in K to the V-enriched covariant representable unctor R K : K V, L hom K (K, L) = R K (L). Remark 2.3. For all K the unctor R K is small as it is Kan extended rom the ull subcategory o K given by the object K. Deinition 2.4. We denote the V-let adjoint to Y to be the end construction Z(F ) = K K hom V(F (K), K). Note that i K = V, as we will assume rom some point in this paper, then the end in the deinition above becomes just a mapping object in the category o small unctors V V : F V V, Z(F ) = hom V V(F, Id V ). We obtain the Yoneda adjunction (2.1) Z : V K K op :Y, which we turn into a Quillen adjunction in Proposition 3.7. Let us briely veriy that the unctor Z is indeed the let adjoint o Y. F V K and X K, then hom K op(z(f ), X) = hom K (X, Z(F )) by deinition o Z(F ) = hom K (X, K F (K) ) by the universal property o an end K K = hom K (X, K F (K) ) since K is cotensored over V K K = hom V (F (K), hom K (X, K)) by deinition o the object K K = hom V K(F, Y (X)). Let o natural transormations

6 6 GEORG BIEDERMANN AND BORIS CHORNY In [8], K = S op and the Yoneda embedding Y : S S Sop was o central importance. In the current article, we take K = Sp. The Yoneda embedding Y : Sp op Sp Sp plays an analogous role as beore and we will turn the adjunction (Z, Y ) into a Quillen equivalence in Theorem Homotopy theory o small unctors We want the Yoneda adjunction (2.1) in the case V = Sp to be a Quillen pair between suitable model structures on each side. The projective model structure constructed by Chorny and Dwyer [10] on the category o small unctors, where weak equivalences and ibrations are objectwise, is not suitable: i we apply Y (v) = V(, v) to a trivial ibration in V op, aka. a trivial coibration in V, then or nonibrant v this map will not remain a weak equivalence. So Y is not right Quillen. We remedy this shortcoming with the ollowing new model structure, which is introduced ater we recall a ew standard deinitions. Deinition 3.1. Let I be a class o maps in a category C. Following standard conventions [23, ], we denote by I-inj the class o maps that have the right liting property with respect to all maps in I. We denote by I-co the class o maps that have the let liting property with respect to all maps in I-inj. We denote by I-cell the class o relative cell complexes obtained rom all maps in I as deined in [23, ]. Deinition 3.2. Let V be a closed symmetric monoidal model category and let K be a V-model category. A V-natural transormation : F G in the category o small unctors V K is a ibrant-projective weak equivalence (resp., a ibrant-projective ibration) i or all ibrant K K the map (K): F (K) G(K) o objects o V is a weak equivalence (resp., a ibration). We oten abbreviate the unctor category V K by F. The main result o this section is Theorem 3.6, where we show that the ibrantprojective weak equivalences and the ibrant-projective ibrations equip F with a model structure, which is, naturally, called ibrant-projective. Deinition 3.3. We recall the ollowing deinitions. (1) A category is class µ-locally presentable, [12], i it is complete and cocomplete and has a class A o µ-presentable objects such that every other object is a iltered colimit o the elements o A. It is class locally presentable i there is a µ or which A is class µ-locally presentable. (2) A model category is class µ-coibrantly generated, [11], i there exist classes o generating (trivial) coibrations with µ-presentable domains and codomains satisying the generalized small object argument [7]. A model category is class coibrantly generated i it is class µ-coibrantly generated or some cardinal µ. (3) A model category is class µ-combinatorial i it is class µ-locally presentable and class µ-coibrantly generated. A model category is class combinatorial i it is class µ-combinatorial or some cardinal µ. (4) A V-model category is class combinatorial, [11], i its underlying category is so. An object o a V-category is λ-presentable i it is λ-presentable in the underlying category.

7 DUALITY AND SMALL FUNCTORS 7 I in the previous deinition the various classes required to exist are in act sets one recovers the well-known concepts o µ-local presentability, coibrant generation and so orth. Deinition 3.4 ([31]). Let tco V be the class o trivial coibrations in V. Let E V be the class o relative cell complexes in V generated by the class o morphisms {j A j tco V, A obv}. The model structure on V satisies the monoid axiom i every morphism in E V is a weak equivalence. Deinition 3.5 ([18] De. 4.6). Let co V be the class o coibrations in V. Let D V be the class o relative cell complexes generated by the class o morphisms {i A i co V, A obv}. The model structure on V is strongly let proper i the cobase change o a weak equivalence along any map in D V is a weak equivalence. Now we state the main result o this section. Theorem 3.6. Let λ be a regular cardinal. Let V be a closed symmetric monoidal category equipped with a λ-combinatorial model structure such that the unit S V is a coibrant object and the monoid axiom 3.4 is satisied. Let K be a λ-combinatorial V-model category. Then the category o small unctors V K = F with the ibrantprojective weak equivalences, ibrant-projective ibrations and the coibrations given by the let liting property is a class-combinatorial V-model category. It is right proper i the model structure on V is. It is let proper i the model structure on V is strongly let proper. Proo. The category F is complete by the main result o [15] and cocomplete by [26, Prop. 5.34]. We use the usual recognition principle [23, ] due to Kan to establish the remaining axioms or a class-coibrantly generated model structure. (1) Weak equivalences are obviously closed under retracts and 2-out-o-3. (2) There are classes o generating coibrations I F and trivial coibrations J F deined in 3.8 that admit the generalized small object argument in the sense o [7] as proved in (3) A map is I F -injective i and only i it is J F -injective and a weak equivalence by Lemma (4) Every J F -coibration is a weak equivalence by The model structure is a V-model structure by Proposition Right properness can be checked by evaluating on all ibrant objects in K and then ollows rom the right properness o V. Let properness is proved in [18, 4.7,4.8]. The key observation is that any ibrant-projective coibration is objectwise a retract o maps in D V. Corollary 3.7. I we equip the category V K = F with the ibrant-projective model structure constructed in Theorem 3.6, then the adjunction (2.1) becomes a Quillen pair. Proo. In the opposite category K op consider a (trivial) ibration op, which in act is a (trivial) coibration : A B in K. The induced map Y (): R B R A is a (trivial) ibration in the ibrant-projective model structure, since hom(, W ) is

8 8 GEORG BIEDERMANN AND BORIS CHORNY a (trivial) ibration or every ibrant object W in V. Thus, the unctor Y is right Quillen. The rest o this section is devoted to the missing steps in the proo o Theorem 3.6. We assume that the closed symmetric monoidal model category V and the V-model category K satisy the conditions o Theorem 3.6. The category V K is tensored over V by applying the tensor product o V objectwise: (F V )(K) = F (K) V or a unctor F in V K and objects V in V and K in K. Deinition 3.8. Let I V and J V be sets o generating coibrations and generating trivial coibrations or V. We deine the ollowing two classes o morphisms in F: { } I F = R X A R X A B I V ; X K, B { } J F = R X C R X C D J V ; X K, D where K K is the subcategory o ibrant objects. Remark 3.9. By [17, Prop ], or any λ-combinatorial model category K there exists a suiciently large cardinal µ, such that K is µ-combinatorial and there exists a µ-accessible ibrant replacement unctor : K K sending µ-presentable objects to µ-presentable objects, i.e., or each X K there is a natural trivial coibration X ˆX such that ˆX is µ-presentable whenever X is and commutes with µ-iltered colimits. From now on and or the rest o the whole article we ix a choice o a ibrant replacement unctor as in the previous remark on the source category. Remark Every ibrant object X is a retract o ˆX. It ollows that the generating classes I F and J F can be replaced by the classes { } I F = R ˆX A R ˆX A B I V ; X K, B { } J F = R ˆX C R ˆX C D J V ; X K, D because the retract argument allows one to see that the classes o maps with the respective right liting properties coincide. Deinition We deine R to be the class o maps in V K that are trivial ibrations when evaluated on all ibrant objects. We deine T to be the class o maps that are ibrations when evaluated on all ibrant objects. Lemma We have: (1) A map is in R i and only i it has the right liting property with respect to all maps in I F : R = I F -inj. (2) A map is in T i and only i it has the right liting property with respect to all maps in J F : T = J F -inj. Proo. Straightorward.

9 DUALITY AND SMALL FUNCTORS 9 Lemma For any ibrant object X in K, the canonical map R X has the let liting property with respect to all maps in R, i.e., it is in I F -co. Proo. Because the unit S o V is coibrant, it is easy to see that the map R X S = R X has the let liting property with respect to all maps in R. Lemma Every relative J F -cell complex is a ibrant-projective weak equivalence. Proo. Because ibrant-projective weak equivalences can be detected by evaluating on ibrant objects, one easily veriies that the lemma ollows rom the monoid axiom that holds in V. Now we present the crucial technical part in the proo o the existence o the ibrant-projective model structure. The generalized small-object argument, [7], may be applied on a class o maps I satisying certain co-solution set condition (see below), so that on each step o the transinite induction we could attach one coibration, through which all other maps in I actor. Lemma The classes I F and J F admit the generalized small object argument. Proo. Since the domains and codomains o the maps in I V and J V are λ-presentable, so are the maps in I F and J F. It remains to show that I F and J F satisy the ollowing co-solution set condition: (CSSC): Every map : F G in F may be equipped with a commutative square C g F D G, so that g I F -co (resp. g J F -co) and every morphism o maps i with i I F (resp. i J F ) actors through g. We will prove this condition in the irst case, where we construct g I F -co. The second case with g J F -co will be dealt with in brackets along the way. For the proo o (CSSC) we consider a morphism o maps i or some i I F (resp. i J F ) and arbitrary in F as above. Let the diagram (3.1) R X A i R X B be this morphism. Here A B is in I V (resp. J V ) and X is a ibrant object in K. By adjunction, this square corresponds to the ollowing commutative diagram o solid arrows: (3.2) R X F G ϕ W F A A G B G A.

10 10 GEORG BIEDERMANN AND BORIS CHORNY where W = F A G A G B is the pullback and ϕ is the universal map. We claim that or such W in (3.2) there exists a map p: W W where p I F -inj, and the canonical map W is in I F -co. In other words, W is a coibrant replacement o W in the yet to be constructed ibrant-projective model structure. The proo o this claim will be postponed to Lemma We proceed with the proo that property (CSSC) holds. The map ϕ lits along the map W W by Lemma Unrolling the adjunction, we ind that the morphism i rom (3.1) actors through the map w A B : W A W B, which is in I F -co (resp. J F -co) as we are now going to prove. We choose the required map g : C D to be g = w A B A I V B and resp. w A B A J V B We need inally to show that g I F -co (resp. g J F -co). It suices to show w A B I F -co (resp. w A B J F -co) or each w A B : W A W B rom above. So, let q : M N be an arbitrary map in S = I F inj (resp. T = J F inj). Consider any commutative square as ollows: (3.3) W A w A B W B We claim this diagram admits a dotted lit. We actually construct a dotted arrow in the ollowing adjoint solid arrow diagram W M B P M N q. M A M A N B N B N A, where P = M A N A N B denotes the pullback. The induced map M B P is in S, which can be checked by evaluating on ibrant objects o K because the model structure on V is monoidal and we are in one o the ollowing cases: (1) The coibration A B is a weak equivalence in V. This is the case inside the brackets above; (2) The map q is in S = I F -inj and hence a trivial ibration when evaluated on ibrant objects. This is the case outside the brackets above.

11 DUALITY AND SMALL FUNCTORS 11 The dotted arrow exists because we get a lit to P by its universal property and then a lit to M B since W is in I F -co. This corresponds to the lit in the original square (3.3) inishing the proo o property (CSSC). Deinition The ull subcategory o K given by the µ-presentable objects will be denoted by K µ. In the previous proo, we have used the ollowing Lemma For each W in diagram (3.2) the canonical map W can be actored into a map W in I F -co ollowed by W W in S = I F -inj. Proo. By assumptions, V and K are λ-combinatorial model categories. We know, by [17, Prop. 2.3(iii)], that there exists a λ-accessible ibrant replacement unctor in K denoted by ˆ, such that or every suiciently large regular cardinal µ λ and or every µ-presentable object X, ˆX is also µ-presentable. We ix this cardinal µ. Here we have chosen µ λ so, that every λ-accessible category is also µ-accessible, [29]. The unctor W is small. In other words, it is a let Kan extension o a unctor deined on a small subcategory K W o K. Alternatively, we can write W as a weighted colimit o a diagram o representable unctors R: K op W VK, K W K R K with a unctor o weights M: K W V. The ull image o M is an essentially small subcategory o V denoted by V W. Thereore, W is a colimit o a set o unctors {R K M K K W, M V W }. Enlarging µ i necessary, we ensure that every K K W is µ-presentable. Then every R K is a µ-accessible unctor. Hence, W is a µ-accessible unctor as a colimit o µ-accessible unctors. In order to construct the stated actorization, we apply to the map W the ordinary small object argument on the ollowing set o maps: I W = { R ˆX A R ˆX B } A I V, X K µ, B where K µ is, as in Deinition 3.16, the ull subcategory o K given by the µ- presentable objects. Note, that by the choice o µ, ˆX Kµ or all X K µ. The map W is then in I W -cell I F -co. The natural transormation o unctors p: W W has the property that or all X K µ the map p( ˆX): W ( ˆX) W ( ˆX) is a trivial ibration in V. We need to show that p I F -inj, i.e. that it is a trivial ibration on all ibrant X. Since ˆX K µ or all X K µ, the unctors R ˆX are µ-accessible or all X K µ. Hence, the unctor W is also a µ-accessible unctor as a colimit o µ-accessible unctors. Since we have chosen µ λ, we obtain that K is a locally µ-presentable category. Hence, every X K is a µ-iltered colimit o X i K µ. Thereore ˆX = colim i ˆXi, since the ibrant replacement was chosen to be λ-accessible, which means that it is also µ-accessible. Then, the map p( ˆX): W ( ˆX) W ( ˆX)

12 12 GEORG BIEDERMANN AND BORIS CHORNY is a µ-iltered colimit o trivial ibrations p( ˆX i ): W ( ˆX i ) W ( ˆX i ) with X i K µ, since both W and W are µ-accessible unctors. Thereore, all the maps p( ˆX), X K are trivial ibrations. Given a ibrant object X K, it is a retract o its ibrant replacement ˆX. Thereore, the map p(x) is a retract o p( ˆX) by naturality o p, i.e. p(x) is a trivial ibration or all ibrant X. We conclude that p is in S. We have completed the proo that the ibrant-projective model structure on the category o small unctors exists. Now we show that it is equipped with an additional structure o a V-model category. Proposition The ibrant-projective model structure makes V K into a V- model category. Proo. We will show that or any coibration i: A B and or any ibration p: X Y in V K, the induced map hom (i, p): hom(b, X) hom(a, X) hom(a,y ) hom(b, Y ) is a ibration. Moreover, hom (i, p) is a weak equivalence i either i, or p is. The retract argument shows that it suices to prove the statement or cellular coibrations. We proceed by induction on the construction o the cellular (trivial) coibration i. Suppose or induction that B 0 = A and the statement is true or all cardinals smaller than d. I d is a successor cardinal, then there is a pushout square R Z K B d 1 R X j R Z L with j : K L in I V (resp., in J V ) and Z K. We apply hom (, p) to the above pushout square, obtaining the ollowing commutative diagram with the let and the right vertical aces being pullback squares. X B d X B d 1 B d X(Z) L Y (Z) L P Y B d X(Z) K Y (Z) K Q Y B d 1 Let P = X(Z) K Y (Z) K Y (Z) L and Q = X B d 1 Y B d 1 Y B d, then also Q = X B d 1 Y (Z) K Y (Z) L as a concatenation o two pullback squares. Applying [23, i d

13 DUALITY AND SMALL FUNCTORS 13 Proposition (2)] twice, we conclude, irst, that Q = X B d 1 X(Z) K P, and next, that X B d = Q P X(Z) L. Then hom (i d, p): hom(b d, X) hom(b d 1, X) hom(bd 1,Y ) hom(b d, Y ) (the dashed map in the ront ace o the diagram above) is a (trivial) ibration as a base change o the (trivial) ibration hom (R Z j, p): hom(r Z L, X) hom(r Z K, X) hom(rz K,Y )hom(r Z L, Y ) (the dotted map in the back ace o the diagram above). The latter map is a (trivial) ibration, since, by adjunction, it is equal to hom (j, p RZ ): hom(l, X(Z)) hom(k, X(Z)) hom(k,y (Z)) hom(l, Y (Z)), which is a trivial ibration by the analog o SM7(b) in the closed symmetric monoidal model category V. Now consider the ollowing commutative diagram computing hom (i d... i 2 i 1, p). X B d Y B d Q X B d 1 P Q Y B d 1 X A Y A In this diagram P = Y B d 1 Y A X A, Q = Y B d Y B d 1 X B d 1, and Q = Y B d Y A X A. We need to show that the natural map hom (i d... i 2 i 1, p): X B d Q is a (trivial) ibration. But the map hom (i d, p): X B d Q is a (trivial) ibration by the previous argument (this is the dashed map in the previous diagram), hence, it is suicient to show that the induced map Q Q is a (trivial) ibration. Applying [23, Proposition (2)] twice, we conclude, irst, that Q = Y B d Y B d 1 P, and next, that Q = Q P X B d 1. The natural map hom (i d 1, p): X B d 1 P is a (trivial) ibration by the inductive assumption, hence Q Q is a (trivial) ibration as a base change o hom (i d 1, p). Continuing this process we conclude that hom (i d... i 1, p) is a (trivial) ibration as a transinite inverse composition o (trivial) ibrations also in the case that d is a limit cardinal. Thereore, hom (i, p) is a (trivial) ibration. 4. Models o spectra We want to apply Theorem 3.6 to a model or the stable homotopy category o spectra. Thereore, we need to demonstrate that there are models that satisy all assumptions. The model category o S-modules rom [19] cannot be used here since its unit or the monoidal structure is not coibrant. Symmetric spectra over simplicial sets constructed by Hovey/Shipley/Smith [24] serve as an acceptable model or us. The monoid axiom 3.4 is proved in [24, section 5.4]. Strong let properness 3.5 is not explicitly stated. We prove it now.

14 14 GEORG BIEDERMANN AND BORIS CHORNY Lemma 4.1. The stable model structure on symmetric spectra over simplicial sets is strongly let proper. Proo. We will use reely the language o Hovey et al. in [24] and all the reerences mentioned here are taken rom their paper. Theorem 5.3.7(3) states that, i is an S-coibration and g a level coibration, their pushout product g is a level coibration. Because any stable coibration i is an S-coibration and any symmetric spectrum A is level coibrant, any map o the orm i A is a level coibration. Because level coibrations are stable under cobase change and iltered colimits, all maps in D V are level coibrations. By Lemma 5.5.3(1), the stable equivalences are stable under cobase change along level coibrations. Now we turn to Lydakis simplicial unctor model [28] or Sp. The category V is now given by the pointed simplicial unctors rom inite pointed simplicial sets S in to pointed simplicial sets S. The symmetric monoidal product is given by Day s convolution product [14]. The monoid axiom or the stable model structure on pointed simplicial unctors is proved by Dundas et. al. [18, Lemma 6.30] or more general source and target categories. Lemma 4.2. Lydakis stable model structure on pointed simplicial unctors is strongly let proper. Proo. Recall rom Deinition 3.5 that D V is the class o relative cell complexes generated by all morphisms o the orm i A, where i is a coibration and A an object in V. We claim that all maps in D V are objectwise coibrations. Since S is let proper it suices to prove that i A is an objectwise coibration or i in a generating set o coibrations and all objects A. Stable coibrations coincide with the projective ones by [28, Lemma 9.4]. A generating set or projective coibrations in V is I V = {R X (Λ n k) + R X ( n ) + n k 0, n > 0, X S in }. For i I V the map i A is isomorphic to (R X A) (Λ n k) + (R X A) ( n ) +. By [28, Lemma 5.13] or [18, Cor. 2.8], using Lydakis assembly map F G F G, this is isomorphic to (A R X ) (Λ n k) + (A R X ) ( n ) +. Ater evaluating on an arbitrary inite pointed simplicial set K this map A ( S (K, X) ) i: A ( S (K, X) ) (Λ n k) + A ( S (K, X) ) ( n ) + is clearly a coibration. We have shown that D V consists o objectwise coibrations. Given a stable weak equivalence, we actor it into a trivial stable coibration ollowed by a trivial stable ibration. Every cobase change o the irst map remains a trivial stable coibration. The second map is an objectwise weak equivalence and pushes out along an objectwise coibration to an objectwise weak equivalence by let properness o S. The composite o both cobase changes is a stable weak equivalence.

15 DUALITY AND SMALL FUNCTORS 15 In conclusion, i we take symmetric spectra on simplicial sets or Lydakis simplicial unctors as models or Sp, the ibrant-projective model structure exists on the category Sp Sp o small endounctors and is proper. Here ollow some properties o the model structure on Sp that we will use urther down the line. Remark 4.3. This property o the ibrant-projective model structure on the category Sp Sp is used in Lemma 7.2 below: For any coibrant object A in Sp the unctor A maps stable equivalences to stable equivalences. For symmetric spectra this ollows rom [24, ]. For simplicial unctors this ollows rom [28, Thm. 12.6]. The ibrant-projective model structure on Sp Sp is simplicial. both models by the ollowing reasoning. This is true or Lemma 4.4. Suppose that V is a symmetric closed monoidal model category and that F : S V is a strict symmetric monoidal let Quillen unctor rom the category o pointed simplicial sets. Then every V-category is an S-category where the simplicial tensor is given by V S K = V V F (K). Proo. The pointed simplicial structure is supplied by [3, Prop ]. The veriication o compatibility is routine. Thus it suices to exhibit a unctor F : S Sp as in the previous lemma. No surprises here; or symmetric spectra this is the symmetric suspension spectrum K Σ K [24, p. 163]. For the Lydakis model F is given by K Id K where Id is the inclusion unctor o inite pointed simplicial sets to all pointed simplicial sets and the smash is objectwise. Thus, we have Remark 4.5. For either symmetric spectra or Lydakis simplicial unctors as models or Sp the ibrant-prjective model structure on Sp Sp is simplicial. Remark 4.6. Since both models or spectra are obtained by localization o either the projective model structure [28], or the strict model structure [24], the sets o generating coibrations have initely presentable domains and codomains. 5. Homotopy unctors In this section we assume, like in Section 3, that V is a closed symmetric monoidal combinatorial model category and K is a combinatorial V-model category, so that the category o small unctors supports the ibrant-projective model structure constructed in Theorem 3.6. We assume, in addition to the previous assumptions, that V is a strongly let proper model category, so that the ibrant-projective model structure on the category V K = F o small unctors is let proper. This allows us to localize unctors in F turning them into a homotopy unctors. The whole section is subsumed in Lemma 5.9 which later enters in the proo o Proposition 7.3. Deinition 5.1. By a homotopy unctor in F we mean any unctor preserving weak equivalences between ibrant objects. Usually, a homotopy unctor is required to preserve all weak equivalences. I desired, a homotopy unctor in our sense here may be turned into a usual homotopy unctor by precomposing with a ibrant approximation unctor in K, while preserving the ibrant-projective homotopy type.

16 16 GEORG BIEDERMANN AND BORIS CHORNY Deinition 5.2. Consider the class o maps between coibrant unctors: H = {R B R A A B weak equivalence o ibrant objects in K} We use the standard notions o H-local object and H-(local) equivalence deined by Hirschhorn [23, 3.1.4]. Lemma 5.3. A unctor in V K is H-local i and only i it is ibrant in the ibrantprojective model structure and a homotopy unctor in the sense o Deinition 5.1. Proo. Let to the reader. On V K there exists the projective model structure [10] whose ibrant unctors are the objectwise ibrant ones. Obviously, every projectively ibrant unctor is ibrant-projectively ibrant. Proposition 5.4. For every small unctor X V K, there exists an H-equivalence η X : X HX such that HX is a homotopy unctor with ibrant values on ibrant objects. Proo. Similarly to the proo o Lemma 3.17, let µ be the maximal cardinal between the accessibility rank o the small (hence, accessible) unctor X and the degree o accessibility o the subcategory o weak equivalences in the combinatorial model category V; then, it suices to construct a localization o X with respect to the set H µ H o maps with µ accessible domains and codomains. Since V K is let proper, it suices to apply the small object argument with respect to the ollowing set o maps: L = Hor(H µ) J µ, where J µ J F is the subset o generating trivial coibrations with µ-accessible domains and codomains, H µ is a set o coibrations obtained rom H µ, and Hor( ) denotes the horns on a set o maps deined in [23, 1.3.2] The ollowing corollary is a standard conclusion rom the application o the (generalized) small-object argument, [7]. Corollary 5.5. For every map : X Y, where Y is a ibrant-projectively ibrant homotopy unctor, there exists a map g : HX Y, unique up to homotopy, such that gη X =. Remark 5.6. We have constructed, so ar, or every small unctor F V K a map into a homotopy unctor F HF, which is initial, up to homotopy, among the maps into arbitrary homotopy unctors. Unlike a similar localization in [2] or the projective model structure on S S, our current construction is not unctorial (since it depends on the accessibility rank o a small unctor, which we are localizing), so the corresponding let Bousield localization o the model category is more involved, [9, 3.2]. We do not use the localized model category in this paper. Deinition 5.7. Recall rom Deintion 3.16 that K µ denotes the ull subcategory o K given by the µ-presentable objects. Recall that K µ is small, since K is locally presentable. Let K c µ be the set o ibrant and coibrant objects in K µ. We deine the ollowing set o maps in V K : C µ = {R A K R A L A K c µ, K I V }. L

17 DUALITY AND SMALL FUNCTORS 17 And denote the proper class C = C µ, where the union is indexed by all ordinals. A unctor X in V K is called C-cellular i the map X is in C µ -cell or some µ. Proposition 5.8. Let X be a homotopy unctor in V K. Then there exists a ibrantprojective weak equivalence X C X where XC is C-cellular. Proo. Let µ be a regular cardinal such that the small unctor X is µ-accessible. The construction o the required coibrant approximation is the same as in Lemma 3.17, except that we will use only the coibrations in C µ. The application o the small object argument produces a map X C X, such that X C (A) X(A) is a weak equivalence or every ibrant and coibrant object A K µ. But or such A, every unctor R A is a homotopy unctor. Moreover, X C is also a homotopy unctor, as may be proved by cellular induction using the Cube Lemma [23, ]. Since X is also a homotopy unctor, the map X C X is a ibrant projective equivalence. Lemma 5.9. Every small unctor is H-equivalent to a C-cellular unctor. Proo. For every unctor X, we construct a homotopy approximation using Proposition 5.4. We obtain an H-equivalence X HX, such that HX is a homotopy unctor. Proposition 5.8 then allows the construction o a cellular approximation or HX HX. We obtain a zig-zag X HX HX o H-local equivalences. 6. The Yoneda embedding as a Quillen equivalence The important part o this section is Theorem 6.11 where we establish that the the Yoneda adjunction (2.1) (6.1) Z : Sp Sp Sp op :Y is a Quillen equivalence. One irst notes that the counit ZY (X) X is an isomorphism or all spectra X. We are done once the unit η F : F Y Z(F ) is a weak equivalence or all small unctors F. Since this is not the case or the ibrant-projective model structure on Sp Sp we perorm a localization o it which orces η F to become a local equivalence. This localization will be a generalization o the Bousield-Friedlander technique [5] where conditions on a coaugmented unctor Q are given such that Q becomes the desired localization unctor. Our generalization o it deals with the existence o a Q-local model structure even in situations where Q is not unctorial. This is necessary in categories o small unctors, since the actorizations are not unctorial or at least we do not have unctorial constructions o these actorizations. We develop this non-unctorial localization in Appendix A. Here we will apply it by exhibiting a Q that suits our purpose. Beore we proceed let us recall the simplicial mapping cylinder construction. For a map : A B in a simplicial model category we deine Cyl() as the ollowing

18 18 GEORG BIEDERMANN AND BORIS CHORNY combined pushout A ı 0 A A i0 i1 A 1 Id B B A Cyl() where ı 0 is the inclusion into the irst summand and i 0, i 1 are the inclusions on the bottom and top o the cylinder. It is a standard argument using the right hand pushout to see that the map (6.2) l 1 = i 1 : A Cyl() is a coibration as long as B is coibrant. The universal property o the pushout yields a simplicial equivalence q : Cyl() B with a section given by the lower horizontal map in the previous diagram such that = qi 0. The irst idea to take or Q the adjunction (6.1) η : F Y Z(F ) itsel does not work because Y Z does not preserve ibrant-projective weak equivalences. However, one can do the ollowing: given F, consider irst its coibrant replacement F and apply the let Quillen unctor Z, then replace Z( F ) by a ibrant (Z( F )) and the apply the right Quillen unctor Y. (We put the standard notation o the ibrant replacement ( ) on the on the righthand side when the hat becomes awkwardly large; Ẑ = Z denotes the composition o Z with the ibrant replacement unctor.) The composition Y Ẑ preserves ibrant-projective weak equivalences between ibrant-projectively coibrant unctors. Finally, this construction has to be equipped with a coaugmentation or arbitrary F. This uses the simplicial mapping cylinder as ollows: η F F Y Z( F ) Y Ẑ( F ) l 1 q Cyl() F i QF, where is a composition o the unit η F with an application o Y on the ibrant replacement Z( F ) Ẑ( F ) in Sp op (coibrant replacement in Sp), and QF = F F Cyl(). The codomain Y Ẑ F is coibrant in the ibrant-projective model structure on Sp Sp by Lemma Thus, the map l 1 : F Cyl() is a coibration and the map Cyl() Y Ẑ( F ) is a weak equivalence. Let properness o the ibrant-projective model structure implies that Cyl() QF is a weak equivalence. The advantage o using the mapping cylinder instead o the actorization into a coibration ollowed by a trivial ibration, guaranteed by the model structure, is that the mapping cylinder construction is unctorial. The construction o QF still lacks unctoriality, since the coibrant replacements are not unctorial in our model

19 DUALITY AND SMALL FUNCTORS 19 category, but the unctoriality o the middle step is essential or the veriication o various properties o QF in Proposition 6.9. To summarize, we describe the deinition stage by stage. Deinition 6.1. For every F Sp Sp we deine QF together with the coaugmentation map i F : F QF as ollows: Choose a coibrant approximation o F to obtain F ; Factor the composition o the unit o the adjunction (6.1) with the map Y (Z( F ) Ẑ( F )) into a coibration ollowed by a weak equivalence in a unctorial way: F Cyl() Y Ẑ( F ); Put QF = F F Cyl() with the induced map i F : F QF. We also deine Q on maps. Given a map between unctors, we need to choose a map on their coibrant replacements using the liting axiom o the model structure. It is unique up to simplicial homotopy. The rest o the stages in the deinition are unctorial. Thereore, once the map o coibrant replacements is chosen, Q is deined. This deinition o i F : F QF gives rise to a homotopy localization construction as in Deinition A.1. It remains to check the conditions (A.2) (A.6) o the generalized Bousield-Friedlander localization given in Theorem A.8. Proposition 6.2. The construction Q rom Deinition 6.1 is homotopy idempotent in the sense that i QF : QF QQF and Q(i F ): QF QQF are weak equivalences or all F. Proo. This is a simple diagram chase relying on Yoneda s lemma: ZY (X) = X or all spectra X. The map i QF is constructed as ollows: F F Y a Ẑ( F ) m Y ( ZY (Z F ) ) Y (Zc) c Cyl( F ) Y ( Z(Cyl( k F )) ) Y (Zb) b i F F QF QF Y ( Z QF i QF l QF ) QQF ). Cyl( QF Y ( (Z F ) ) Here ( ) replaces the hat notation or ibrant replacement. We irst conclude that the upper horizontal map m = Y (Z( F )) is a weak equivalence, since this is an application o Y on a ibrant approximation o a ibrant object. Next, we apply the 2-out-o-3 axiom to the lower horizontal maps k and concluding that they QF are weak equivalences too. Finally we can see that i QF is a trivial coibration as a cobase change o the trivial coibration l, which is a weak equivalence by the 2-out-o-3 axiom again. Now we turn to Q(i F ) which is depicted on the right in the diagram below. It suices to show that the map γ = Y (Zĩ F ) is a weak equivalence. One o the

20 20 GEORG BIEDERMANN AND BORIS CHORNY possibilities or choosing a coibrant approximation to i F is to take the composition ba rom the commutative diagram above. Consider the ollowing commutative diagram: ĩ F a F F Cyl( F ) b QF Y Ẑ F Y (Z F ) F Cyl( F ) Y Ẑ F δ Y ( ZY ( Z F ) ) Y (Za) Y ( Z(Cyl( F )) ) QF Y (Zb) Y (Z QF ) γ The three lower horizontal maps are weak equivalences, but this is irrelevant to the proo. The map δ = Y Ẑ F is a weak equivalence since it is weakly equivalent to a second ibrant replacement by Yoneda s lemma as used beore: ZY (X) = X or every spectrum X. Hence, the map Y Ẑa is a weak equivalence by the 2-out-o-3 property. Thereore, the composition Y Ẑb Y Ẑa = Y Ẑ(ba) = Y Ẑĩ F is a weak equivalence. The ollowing proposition veriies condition A.2. Proposition 6.3. Let : F G be a natural transormation o unctors in Sp Sp ; then, Q i F = i G, i.e., the ollowing square is commutative. F i F QF G ig QG Proo. Following the deinition o Q, we notice that the only non-unctorial stage o the deinition is computing the coibrant replacement o the domain and the codomain o. But we choose a map : F G, so that the square F G becomes commutative. The remaining steps in the deinition are unctorial, and hence we end up with the required commutative square. Our next goal is to veriy that Q satisies conditions A.3 and A.4. Again, the veriication would be immediate i Q were a unctorial localization construction. Our approach to this question is to show that Q induces a unctor on the level o homotopy category. Let Γ: Sp Sp Ho(Sp Sp ) be the canonical unctor. Lemma 6.4. The Q-construction is a unctor up to homotopy: The composition ΓQ: Sp Sp Ho(Sp Sp ) is a unctor too. F G Q

21 DUALITY AND SMALL FUNCTORS 21 Proo. For any commutative triangle (6.3) B g A C in Sp Sp, we have to show that the triangle h (6.4) QB Q Qg QA QC Qh is commutative up to homotopy, i.e., i we apply on it the unctor Γ, we obtain a commutative triangle in Ho(Sp Sp ). We will ollow the stages o the construction o triangle (6.4) and make sure that at each stage the commutativity is preserved up to homotopy. Recall that the ibrant-projective model structure on Sp Sp is simplicial by Remark 4.5. The irst stage is applying a coibrant replacement on the vertices o triangle (6.3) obtaining the ollowing triangle with the edges constructed using the liting axiom. (6.5) B g à C h Triangle (6.5) is commutative up to simplicial homotopy by [23, Prop ], since the maps h and g are the lits in the commutative square à The next stage in the construction o Q is the application o simplicial unctors Z,Y and the unctorial coibrant replacement in spectra (ibrant replacement in Sp op ) in between. Simplicial unctors preserve simplicial homotopies o maps. Coibrant replacement in any simplicial model category allows or the lit o simplicial homotopy: i J S is a generalized interval, the simplicial homotopy o Z h and Z Z g is a map H : Zà Z C J, such that ev 0 H = Z h and ev 1 H = Z gz, and hence H can be lited to a simplicial homotopy H J : Zà Z C C C. Zà H Zà H Z CJ Z C J,

22 22 GEORG BIEDERMANN AND BORIS CHORNY so that each o the simplicially homotopic maps ev 0 H and ev1 H is a lit to the coibrant replacements o the maps Z h and Z Z g, respectively. On the other hand, the maps ev 0 H and ev1 H are simplicially homotopic to the unctorially induced maps o coibrant replacements in Sp, i.e., the maps Z h and Z g Z are simplicially homotopic by transitivity o the simplicial homotopy relation. So ar, we have obtained two triangles commutative up to simplicial homotopy with a natural map between them: Y ẐÃ Y Ẑ B Y Ẑ Y Ẑ g Y Ẑ h Y Ẑ C Ã B h g The completion o the localization construction involves actoring the dotted maps into coibrations ollowed by a weak equivalence and then applying the cobase change. Both operations are natural and change only the commuting triangle in the homotopy category up to a natural isomorphism, preserving the commutativity. Proposition 6.5. The localization construction Q satisies conditions A.3 and A.4. Proo. It ollows immediately rom Lemma 6.4. The ollowing property is reminiscent o unctoriality and veriies A.5. Proposition 6.6. For every commutative square o small unctors (6.6) A h X there exists a commutative cube (6.7) Q A A B C B k Y h Q B k k h g X Y g Q X g Q Y or some choice o Q A QA, Q B QB, Q X QX, Q Y QY. Moreover, every edge o the cube connecting the ront ace with the back ace actors through