Are all localizing subcategories of stable homotopy categories coreflective?

ARE ALL LOCALIZING SUBCA TEGORIES OF ST ABLE HOMOTOPY CA TEGORI E S COREFLECTIVE? CARLES CASA CUBER T A, JA VIER J. GUTI ´ ERREZ AND JI ˇ R ´ I R OSICK ´ Y ∗ Abstract. W e prov e that, in a tr ia ngulated catego ry with combinatorial mo de ls , every lo calizing sub c ategory is coreﬂective and every colo ca liz ing sub categ ory is re ﬂectiv e if a certain large-c a rdinal axiom (V opˇ enk a ’s principle) is assumed true. It follows that, under the same a s sumptions, orthogona lit y sets up a bijective cor resp ondence b etw een lo calizing sub catego ries and colo calizing sub categories . The existence o f such a bijection was left as an o pen problem by Hov ey , Palmieri and Str ickland in their axiomatic s tudy of stable homotopy ca tegories and also by Neeman in the con text of w ell-genera ted triangulated categor ies. Intr oduction The main purp o se of t his article is to address a question a ske d in [37, p. 35] of whether ev ery lo calizing sub category (i.e., a full triangulated subcatego ry closed under copro ducts) of a stable homotop y category T is the ke rnel of a lo calization on T ( o r, equiv alen tly , the image of a colo calization). W e pro v e that the answ er is aﬃrmativ e if T arises from a com binatorial mo del category , assuming the truth of a larg e-cardinal axiom from set theory called V opˇ enk a’s principle [2], [38]. A mo del category (in the sense of Quillen) is called c ombinatorial if it is coﬁbran tly generated [33], [36] a nd its underly ing category is lo cally presen table [2], [2 7]. Many t r ia ngulated categories of in terest admit com binatorial mo dels, includ ing deriv ed categories o f rings and the homotopy category of sp ectra. More precisely , w e sho w that, if K is a stable combinatorial mo del category , then ev ery semilocalizing subcatego ry C of the homotop y category Ho( K ) is coreﬂectiv e under V opˇ enk a’s principle, and the coreﬂection is exact if C is lo calizing. W e call C semilo c alizin g if it is closed under copro ducts, coﬁbres and extensions, but not necessarily under ﬁbres. Examples include k ernels of nulliﬁc ations in the sens e of [11] or [1 9 ] o n the homoto py category of spectra. W e also pro v e that, under the same hypotheses, eve ry semilo calizing sub category C is singly gener ate d ; that is, there is an ob ject A suc h tha t C is the smallest semilo calizing sub category con taining A . The same result is true for lo calizing sub categories. The Date : Octob er 19, 2 018. 2010 Mathematics Subje ct Classiﬁc ation. 18E30 , 1 8G55, 55 P42, 55P6 0, 03E55 . ∗ The t wo ﬁrst-named authors w ere suppor ted b y the Spanish Ministry of Educa tio n and Science under MEC-FEDER grants MTM20 07-632 77 and MTM2010-158 31, and by the Gener alitat de Ca talun ya as mem b ers of the team 2009 SGR 119. The thir d- named author was suppo rted by the Ministr y o f Educatio n of the Czech Republic under the pro ject MSM 00216 2 2409 and by the Czech Science F oundation under grant 201/1 1 /0528 . He gratefully ac knowledges the hospitality of the Universit y of Barcelona and the Cent re de Recerca Matem` atica. 1 ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 2 question o f whether ev ery lo calizing sub cat ego ry is singly generated in a w ell-generated triangulated category w as ask ed in [48, Problem 7.2]. W e note that, a s sho wn in [50, Prop osition 6 .10], triangulated categories with combinatorial mo dels are w ell generated. In a n arbitrary triangulated category T , lo calizing sub catego r ies need neither b e singly generated nor coreﬂectiv e. Indeed, the existence o f a coreﬂection on to a lo calizing sub cat - egory C is equiv alent to the existence of a right adjoin t for the V erdier functor T → T / C ; see [46, Prop osition 9.1.18]. Hence, if a coreﬂection onto C exists, then T / C has small hom-sets. This need not happ en if no restriction is imp osed on T ; a coun terexample was giv en in [14]. Dually , w e prov e that ev ery full sub category L closed under pro ducts and ﬁbres in a triangulated category with lo cally presen t able mo dels is reﬂectiv e under V op ˇ enk a’s prin- ciple. The r eﬂection is semiexact if L is closed under extensions, and it is exact if L is colo calizing, as in the dual case. Ho w ev er, we ha v e not b een able to prov e that colo cal- izing (or sem icolo calizing) sub cat ego ries are necessarily singly generated, not ev en under large-cardinal assumptions. This apparent lack of symmetry is not entirely surprising, in view of some w ell-kno wn facts in v olving torsion t heories. In ab elian categories, a full sub category closed under colimits and extensions is called a torsion clas s , a nd one closed under limits and extens ions is called a torsion-fr e e class . These are analogues o f semilo calizing and semicolo calizing sub categories of triangulated categories. T orsion theories hav e also been considered in triangulated categories b y Beligiannis and Reiten in [6], in connection with t -structures. In w ell-p o w ered ab elian categories, to rsion classes are necessarily coreﬂectiv e and torsion- free classe s are reﬂectiv e [18]. As sho wn in [21 ] and [29], V opˇ enk a’s principle implies that ev ery torsion class of ab elian groups is singly generated. How ev er, there exist torsion-free classes that are not singly generated in ZFC; for example, the class of ab elian groups whose coun table subgroups a re free [20, Theorem 5.4 ]. ( In this article, w e do not make a distinction b et w ee n the terms “singly generated” a nd “singly cog enerated”.) Our results imply that, if T is the homotopy category of a stable com binatorial mo del category and V op ˇ enk a’s principle holds, then there is a bijection b et w een lo calizing and colo calizing sub categories of T , giv en by orthogonality . This was ask ed in [5 4, § 6] and in [48, Problem 7.3 ]. In f act, we prov e that there is a bijection b et w een semilo calizing and semicolo calizing sub categor ies as we ll, and eac h of t hose determines a t -structure in T . The lac k of symmetry b etw een reﬂections and coreﬂections a lso sho ws up in the fa ct that singly g enerated semilo calizing sub categor ies are coreﬂectiv e (in ZF C) in triangulat ed categories with combinatorial mo dels. A detailed pro of of this claim is given in Theo- rem 3.7 b elo w; the arg umen t g o es bac k to Bousﬁeld [9], [10] in the case of sp ectra, and ha s subseque n tly b een adapted to other sp ecial cases in [4 ], [6], [37], [3 9], [43] —o ur vers ion generalizes some of these. How ev er, we do not kno w if singly generated semic olo calizing sub categories can b e sho wn to b e reﬂectiv e in ZFC. A p o sitiv e answ er w ould imply the existence of cohomological lo calizations of sp ectra, whic h is so far unse ttled in ZFC . It remains of course to decide if V opˇ enk a’s principle (or an y other large-cardinal princi- ple) is really needed in order to answ e r all t hese questions . Although w e cannot ascertain ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 3 this, w e prov e that there is a full sub category of the ho mo t o p y category of sp ectra closed under retracts a nd pro ducts whic h fa ils to b e w eakly reﬂectiv e, assuming that there are no measurable cardinals. This follo ws from the existence o f a full sub category of ab elian groups with the same pro p ert y , and hence solv es a n op en problem pro p osed in [2, p. 296]. In connection with this problem, see also [4 9]. 1. Reflections and coreflections in triangula ted ca tegories In this ﬁrst section w e recall basic concepts and ﬁx our terminology , whic h is mostly standard, except for small discrepancies in the notation for orthogo na lit y and lo calization in a num b er of recen t articles and monographs ab out tr ia ngulated categories, suc h as [6], [7], [37], [41], [46] or [48]. The essen tials of triangulated categories can b e fo und in [46]. F or a category T , we denote b y T ( X , Y ) the set of morphisms from X to Y . W e tacitly assume that sub categories are isomorphism-closed, and denote indistinctly a full sub category and the class o f its ob jects. 1.1. Reﬂections and coreﬂections. A full sub category L of a catego ry T is r eﬂe ctive if the inclusion L ֒ → T ha s a left adjo in t T → L . Then the comp osite L : T → T is called a r eﬂe ction onto L . Suc h a functor L will b e called a lo c alization and ob jects in L will b e called L -lo c al . There is a natural transformation l : Id → L (namely , the unit of the adjunction) suc h that Ll : L → LL is an isomorphism, lL is equal to Ll , and, for eac h X , the morphism l X : X → LX is initial in T among morphisms from X to ob jects in L . Similarly , a full sub category C of T is c or eﬂe ctive if the inclusion C ֒ → T has a righ t adjoin t. The comp osite C : T → T is called a c or eﬂe ction or a c olo c al i z ation o nto C , and it is equipp ed with a natural transformation c : C → Id (the counit of the adjunction) suc h that C c : C C → C is an isomorphism, cC is equal to C c , and c X : C X → X is terminal in T , for eac h X , among morphisms from ob jects in C (which a re called C -c o l o c al ) in to X . A full subcategory L of a category T is called we akly r eﬂe ctive if for ev ery ob j ect X of T there is a morphism l X : X → X ∗ with X ∗ in L and suc h that t he function T ( l X , Y ) : T ( X ∗ , Y ) − → T ( X, Y ) is surjectiv e for all ob jects Y of L . Th us, ev e ry morphism from X to an ob ject of L factors through l X , not necessarily in a unique wa y . If suc h a factorization is unique for all ob jects X , then the morphisms l X : X → X ∗ for a ll X deﬁne to gether a reﬂection, so L is then reﬂectiv e. One deﬁnes we a k ly c or eﬂe ctive sub categories dually . If a w eakly reﬂectiv e sub catego r y is closed under retra cts, then it is closed under all pro ducts that exist in T ; see [2, Remark 4.5(3 ) ]. Dually , weak ly coreﬂectiv e sub categories closed under retracts are closed under copro ducts. Reﬂectiv e sub cat ego ries are closed under limits, while coreﬂectiv e subcategories are closed under colimits. If L is a reﬂection on an additive category T , then the ob jects X suc h that LX = 0 are called L -acyclic . The full subcatego r y of L -acyclic ob jects is closed under colimits. F or a coreﬂection C , the class of ob jects X such that C X = 0 is closed under limits, and such ob jects are called C -acyclic . ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 4 1.2. Closure prop erties in triangulated categories. F rom no w on, we assume that T is a triangulated category with pro ducts and copro ducts. Motiv ated by top ology , we denote b y Σ the shift op erato r and call it susp ension . Distinguished triangles in T will simply b e called triangles and will b e denoted b y (1.1) X u / / Y v / / Z w / / Σ X, or shortly b y ( u, v , w ). W e sa y that a functor F : T → T pr eserv e s a triangle (1.1 ) if ( F u, F v , F w ) is a triangle. Note that, if this happ ens, then F Σ X ∼ = Σ F X . A full sub categor y S of T will b e called (i) close d under ﬁbr es if X is in S for ev ery triangle (1.1) where Y and Z are in S ; (ii) close d under c oﬁbr es if Z is in S for eve ry tria ngle (1.1) where X and Y are in S ; (iii) close d under extensions if Y is in S for ev ery triangle (1.1) where X and Z are in S ; (iv) trian g ulate d if it is closed under ﬁbres, coﬁbres and extensions. A full sub category of T is called lo c alizing if it is triangulated and closed under copro d- ucts, and c olo c alizing if it is triangulated and closed under pro ducts. If a triangulated sub category S is closed under coun table copro ducts or under coun table pro ducts, then S is automatically closed under retracts; see [37, Lemma 1.4.9] or [46, Prop osition 1 .6 .8]. More generally , a f ull sub category of T will b e called s emilo c alizing if it is closed under copro ducts, coﬁbres, and extensions (hence under retracts and susp ension), but not necess arily under ﬁbres. And a full sub catego r y will b e called semic olo c alizin g if it is closed under pro ducts, ﬁbres, and extensions (therefore under retracts and desusp ension as w ell). Semilo calizing subcat ego ries are also called c o c om plete pr e-ais l e s elsew here [4], [53], and semicolo calizing subcatego r ies are called c omplete pr e-c o aisles —the terms “aisle” and “coaisle” originated in [40]. See also [6] for a related discussion of torsion pairs and t -structures in triangulated categories. A reﬂection L on T will b e called semiexact if the sub category of L -lo cal ob jects is semicolo calizing, and exa c t if it is colo calizing. Dually , a coreﬂection C will b e called semiexact if the sub category of C -colo cal ob jects is semilocalizing and exact if it is lo cal- izing. If L is a semiexact reﬂection with unit l , then, since the class of L -lo cal ob jects is closed under desusp ension, there is a nat ur a l morphism ν X : LX → Σ − 1 L Σ X suc h that ν X ◦ l X = Σ − 1 l Σ X for all X , and hence a natural morphism (1.2) Σ ν X : Σ LX − → L Σ X suc h that Σ ν X ◦ Σ l X = l Σ X . As w e next sho w, if Σ LX ∼ = L Σ X for a giv en ob ject X , then Σ ν X is automatically an isomorphism. Lemma 1.1. Supp ose that L is a semiexact r eﬂe ction. If Σ LX is L -lo c al for a given obje ct X , then Σ ν X is an isomorphism. Pr o of. If Σ L X is L -lo cal, then there is a (unique) morphism h : L Σ X → Σ LX suc h that h ◦ l Σ X = Σ l X . Thu s Σ ν X ◦ h ◦ l Σ X = l Σ X , whic h implies that Σ ν X ◦ h = id, b y the ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 5 univ ersal prop erty of L . Similarly , h ◦ Σ ν X ◦ Σ l X = Σ l X , and hence Σ − 1 h ◦ ν X ◦ l X = l X , from whic h it follows that Σ − 1 h ◦ ν X = id, or h ◦ Σ ν X = id. This prov es that Σ ν X has indeed an in verse .  Theorem 1.2. L et T b e a triangulate d c ate gory. F or a semiexact r eﬂe ction L on T , the fol lowing assertions ar e e quivalent: (i) L is exa c t. (ii) The class o f L -lo c al obj e cts is close d under Σ . (iii) Σ LX ∼ = L Σ X for al l X . (iv) Σ ν X : Σ LX → L Σ X is an isomorphis m for al l X . (v) L pr eserves al l triangles. Pr o of. The equiv alence b et w ee n (i) and (ii) fo llo ws from the deﬁnitions. The fa ct that (ii) ⇒ (iv) is give n b y Lemma 1.1, a nd ob viously (iv) ⇒ (iii) ⇒ (ii) . In order to pro v e that ( ii) ⇒ (v), let ( u, v , w ) b e a triangle, and let C b e a coﬁbre of Lu . Th us w e can c ho ose a morphism ϕ yielding a comm utativ e diagram of triangles X u / / l X   Y v / / l Y   Z w / / ϕ   Σ X − Σ u / / Σ l X   Σ Y Σ l Y   LX Lu / / LY / / C / / Σ LX − Σ Lu / / Σ LY . Since C is a ﬁbre of a morphism b et w een L - lo cal ob jects, it is itself L -lo cal, since L is semiexact. F rom the ﬁv e-lemma it follows that the morphism T ( C , W ) → T ( Z , W ) induced b y ϕ is an isomorphism for ev ery L -lo cal ob ject W , and therefore ϕ is an L -lo calization, so C ∼ = LZ . Then the induced morphisms LY → LZ and LZ → L Σ X (using Σ ν X ) a re equal to Lv and Lw resp ectiv ely , b y the univ ersal prop erty of L . This pro v es that L preserv es ( u, v , w ). Finally , (v) ⇒ (iii), so the argumen t is complete.  There is of course a dua l result for semiexact coreﬂections, with a similar pro of. W e omit the details. Theorem 1.3. L et T b e a triangulate d c ate gory. A r eﬂe ction L on T is semiexact if and only i f L pr eserves triangles X → Y → Z → Σ X wh e r e Z is L -lo c al, and a c or eﬂe ction C on T is sem i e xact if and only i f C pr eserves triangle s X → Y → Z → Σ X in which X is C -c olo c al. Pr o of. W e o nly pro v e the ﬁrst part, as the second pa rt is pro v ed dually . Assume that L is a semiex act reﬂection and let C b e a coﬁbre of l X ◦ ( − Σ − 1 w ). Then there is a commutativ e diagram of t riangles Σ − 1 Z − Σ − 1 w / / X u / / l X   Y v / / ϕ   Z w / / Σ X Σ l X   Σ − 1 Z / / LX / / C / / Z / / Σ LX ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 6 where C is L -local since L is semiexact. Again b y the ﬁv e-lemma, ϕ induces an isomor- phism T ( C , W ) ∼ = T ( Y , W ) f or ev ery L -lo cal ob ject W . Hence C ∼ = LY , and the univers al prop ert y of L implies then that the resulting arrow s LX → LY , LY → Z and Z → L Σ X are Lu , Lv and Lw , as needed. Con v ers ely , let ( u, v , w ) b e a triang le where X and Z are L -lo cal. Then, since L preserv e s this tr ia ngle, w e hav e a comm utativ e diagram Σ − 1 Z − Σ − 1 w / / Σ − 1 l Z   X u / / l X   Y v / / l Y   Z w / / l Z   Σ X Σ l X ◦ Σ ν X   Σ − 1 LZ / / LX Lu / / LY Lv / / LZ Lw / / L Σ X where l X and l Z are isomorphisms, and Σ ν X is also an isomorphism by Lemma 1.1. It then follows that l Y is also an isomorphism and hence Y is L -lo cal. Similar ly , if Y and Z are L -lo cal, then X is L -lo cal. Therefore, the sub category of L -lo cal ob j ects is closed under ﬁbres and extensions, as claimed.  1.3. Orthogonality and semiorthogonalit y. Seve ral kinds of orthogonality can b e considered in a triangulated category . In this article it will b e con v enie n t to use the same nota tion as in [7]. Th us, for a class of ob jects D in a triangulated category T with pro ducts and copro ducts, w e write ⊥ D = { X | T ( X , Σ k D ) = 0 for all D ∈ D a nd k ∈ Z } , D ⊥ = { Y | T (Σ k D , Y ) = 0 for all D ∈ D a nd k ∈ Z } . F or ev ery class of ob jects D , the class ⊥ D is lo calizing and D ⊥ is colo calizing. A lo cal- izing subcategory C is called close d if C = ⊥ D for some D , or equiv a lently if C = ⊥ ( C ⊥ ), and a colo calizing sub category L is called close d if L = D ⊥ for some D , or equiv a lently if L = ( ⊥ L ) ⊥ . F or example, if w e w ork in the homotop y category o f sp ectra and E is a sp ectrum, then the statemen t X ∈ ⊥ E holds if and only if E ∗ ( X ) = 0, where E ∗ is the reduced cohomology theory represen ted by E . Th us, ⊥ E is the class of E ∗ -acyclic sp ectra. (Here and later, w e write ⊥ E instead of ⊥ { E } for simplicit y .) Let us in troduce the follo wing v arian t, whic h w e call semiortho go n ality : L D = { X | T ( X , Σ k D ) = 0 for all D ∈ D a nd k ≤ 0 } , D L = { Y | T (Σ k D , Y ) = 0 for all D ∈ D and k ≥ 0 } . Similarly as ab ov e, for ev ery class of ob j ects D the class L D is semilocalizing, while D L is semicolo calizing. A semilo calizing subcategory C will b e called close d if C = L D for some class of ob j ects D , or equiv a lently if C = L ( C L ). A semicolo calizing sub category L will b e called close d if L = D L for some D , or equiv alently if L = ( L L ) L . Note that, if a class D is preserv ed b y Σ and Σ − 1 , t hen L D = ⊥ D and D L = D ⊥ . Therefore, if a lo calizing sub category C is closed, then it is also closed as a semilo calizing sub category , sinc e C = ⊥ ( C ⊥ ) = L ( C ⊥ ). The dual assertion is of course also true. ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 7 Semiexact reﬂections and semiexact coreﬂections are link ed throug h the following ba sic result, whic h generalizes Lemma 3.1 .6 in [37]; compare also with [6, Proposition 2 .3 ], [7, Lemma 1.2], and [41, Prop osition 4.12.1]. Theorem 1.4. In every triangulate d c a te gory T t her e is a bije ctive c orr esp ondenc e b e- twe en semiexact r eﬂe ctions an d semiexact c or eﬂe ctions such that, if a r e ﬂe ction L is p air e d with a c or eﬂe ction C under this bije ction, then the fol lowing hold: (i) F or every X , the morphisms l X : X → LX and c X : C X → X ﬁt into a triangle C X / / X / / LX / / Σ C X. (ii) The class L of L -lo c al obje cts c oincides with the clas s of C -acyclic s , and the class C of C -c olo c al obje cts c oincides with the class of L -acyclics. (iii) The class C is e qual to L L , and L is e qual to C L . (iv) L is exact if and only if C i s exact. In this c ase, C = ⊥ L and L = C ⊥ . Pr o of. Let L b e a semiexact reﬂection. F or ev e ry X in T , c ho ose a ﬁbre C X of the unit morphism l X : X → LX . Thus , for ev ery X in T w e ha v e a triangle (1.3) C X c X / / X l X / / LX / / Σ C X. If we apply L to (1.3), since LX is L - lo cal, Theorem 1.3 implies that LC X Lc X / / LX Ll X / / LLX / / Σ LC X is a triangle, and hence LC X = 0. F or each morphism f : X → Y , c ho ose a mor phism C f : C X → C Y such that the follo wing diagra m comm utes: Σ − 1 LX / / Σ − 1 Lf   C X c X / / C f   X l X / / f   LX Lf   Σ − 1 LY / / C Y c Y / / Y l Y / / LY . Then C f is unique, since LC X = 0 implies that T ( C X , Σ k LY ) = 0 for k ≤ 0, and there- fore c Y : C Y → Y induces a bijection T ( C X , C Y ) ∼ = T ( C X , Y ). This yields functorialit y of C and nat uralit y of c . Moreo v er, for each X the f ollo wing diagram comm utes: C C X c C X / / C c X   C X l C X / / c X   LC X Lc X   C X c X / / X l X / / LX . Here the fact that LC X = 0 implies that c C X is a n isomorphism. And, since c X induces a bijection T ( C C X , C X ) ∼ = T ( C C X , X ), w e infer that c C X = C c X and therefore C is a coreﬂection. In order to prov e that C is semiexact, w e need to show tha t the class of C - colo cal ob jects is closed under coﬁbres and extensions. F or this, let X → Y → Z b e a t r iangle ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 8 and assume ﬁrst that X a nd Z are C -colo cal. Consider the following diagra m, where all the columns and the central ro w are triangles: C X / /   C Y / /   C Z   X / /   Y / /   Z   LX / / LY / / LZ . Since C X ∼ = X and C Z ∼ = Z , w e ha v e that LX = 0 and LZ = 0. Since L is semiex act, L Σ X = 0 as well. Th is implies that T ( Y , W ) ∼ = T ( Z , W ) for ev ery L -lo cal ob ject W . Hence the morphism LY → LZ is an isomorphism. Therefore LY = 0 and C Y ∼ = Y , as needed. Second, a ssume that X and Y are C -colo cal. Then LX = 0 and LY = 0 and the same argumen t tells us that LZ = 0, so C Z ∼ = Z . P art (ii) follows directly fro m (i). The ﬁrst claim of (iii) is pro v ed a s fo llo ws. Let X ∈ C . Since LX = 0 and the class of L -lo cal ob jects is closed under desusp ension, w e hav e that T ( X , Σ k D ) ∼ = T ( LX , Σ k D ) = 0 for k ≤ 0 and all D ∈ L . This tells us that X ∈ L L . Con v ersely , if X ∈ L L , t hen T ( LX, LX ) ∼ = T ( X , LX ) = 0. Therefore LX = 0, so X ∈ C . The second part is pro v ed dually . The ﬁrst claim of (iv) fo llows b y considering the comm utativ e diagram Σ C X Σ c X / /   Σ X Σ l X / / Σ LX / /   ΣΣ C X   C Σ X c Σ X / / Σ X l Σ X / / L Σ X / / Σ C Σ X , and using Theorem 1 .2 (and its dual). The rest is prov ed with the same argumen ts as in part (iii).  In the homotopy category of sp ectra, ev ery f -lo calization functor L f in the sense of [11] or [19] is a reﬂection, and cellularizations Cell A are coreﬂections. Classes of f -lo cal sp ectra are closed under ﬁbres, but not under coﬁbres nor extensions, in general. Dually , A -cellular classes are closed under coﬁbres. Nulliﬁcation functors P A (i.e., f -lo calizations where f : A → 0, suc h as P ostnik o v sections) a re semiex act reﬂections. Homological lo calizations of sp ectra (o r , more generally , nu lliﬁcations P A where Σ A ≃ A ) are exact reﬂections. One prov es as in [16] that the k ernel of a n ulliﬁcation P A is precisely the closure under extens ions of the imag e of Cell A . 1.4. T orsion pairs and t -structures. F or a class of ob jects D in a triangulated category T with pro ducts and copro ducts, w e denote by lo c( D ) the smallest lo calizing sub category of T t ha t con tains D ; that is, the in ters ection of all the lo calizing sub categories of T that con tain D . W e use the terms colo c( D ), slo c( D ), and scolo c ( D ) ana logously , and w e say ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 9 that eac h of these is g e n er ate d b y D . If D consists of only one ob ject, then we sa y that the resp ectiv e classe s are singly gener ate d . Note that if D = { D i } i ∈ I is a set (not a prop er class), then lo c( D ) = lo c  ` i ∈ I D i  and colo c ( D ) = colo c  Q i ∈ I D i  , and similarly with slo c ( D ) and scoloc( D ). Th us, in the presence of pro ducts and copro d- ucts, “g enerated b y a set” and “singly generated” mean the same thing. It is imp o r tan t to relate classes generated by D in this sense with the corresp onding closures of D under orthogona lity or semiorthogonalit y . Although this seems to b e diﬃcult in g eneral, it follo ws from Theorem 1 .4 t ha t reﬂectiv e colo calizing or semicolocalizing sub categories are closed, and coreﬂectiv e lo calizing or semilo calizing subcategor ies are also closed. This has the f o llo wing consequence. Prop osition 1.5. L et D b e any class of obje c ts in a triangulate d c ate gory with pr o ducts and c opr o ducts. (i) If scolo c( D ) is r eﬂe ctive, then scolo c( D ) = ( L D ) L , and if slo c( D ) is c or eﬂe ctive, then sloc ( D ) = L ( D L ) . (ii) If colo c( D ) is r eﬂe ctive, then colo c( D ) = ( ⊥ D ) ⊥ , and if lo c( D ) is c or eﬂe ctive, then lo c( D ) = ⊥ ( D ⊥ ) . Pr o of. W e only prov e the ﬁrst claim, as the others follo w similarly . Since D ⊆ scolo c( D ), w e ha v e ( L D ) L ⊆ ( L scolo c( D )) L . If scolo c( D ) is reﬂectiv e, then it follo ws fr o m part (iii) of Theorem 1 .4 that scolo c( D ) is closed. Hence, ( L D ) L ⊆ scolo c( D ). The rev erse inclusion follo ws from the fact that ( L D ) L is a semicolo calizing sub category containing D , and scolo c( D ) is minimal with t his prop ert y .  W e remark that, fo r ev ery class D , w e ha v e D L = slo c( D ) L , a nd similarly with left semiorthogonalit y o r orthogonality in either side. T o pro v e this assertion, o nly the in- clusion D L ⊆ slo c( D ) L needs to b e c hec k ed, and this is done as follo ws. Since L ( D L ) is semilo calizing a nd con tains D , it also con tains slo c( D ). Hence, (1.4) D L = ( L ( D L )) L ⊆ slo c ( D ) L , as claimed. Prop osition 1.6. L et D b e any class of obje cts i n a triangulate d c ate gory T . Supp o s e that for e ac h X ∈ T ther e is a triangle C X → X → LX → Σ C X wher e C X ∈ slo c( D ) and LX ∈ slo c( D ) L . Then C deﬁnes a semiexact c o r eﬂe ction onto slo c( D ) . Pr o of. If Y is an y ob ject in slo c( D ), then T ( Y , LX ) = 0 and T ( Y , Σ − 1 LX ) = 0. Hence, the morphism C X → X induces a bijection T ( Y , C X ) ∼ = T ( Y , X ), so C is a coreﬂection on to slo c( D ), hence semiexact.  This fact will be used in Section 3. There is of course a dual result, and there are corresp onding facts fo r exact coreﬂections and exact reﬂections; cf. [4 6, Theorem 9.1.13]. Under the hy p otheses of Prop osition 1.6, the classes slo c( D ) and slo c( D ) L form a torsion p air as deﬁned in [6, I.2.1]. This yields the follo wing fact, whic h also relates these not io ns with t -structures; see [4, § 1 ] as we ll. ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 10 Theorem 1.7. In every triangulate d c a te gory T t her e is a bije ctive c orr esp ondenc e b e- twe en the fol lowin g classes: (i) R eﬂe ctive semic olo c alizing sub c ate gories. (ii) Cor eﬂe ctive sem ilo c alizing sub c ate gories. (iii) T orsio n p ai rs . (iv) t -s tructur es. Pr o of. The bijectiv e corresp ondence b et w e en (i) a nd ( ii) has b een established in Theo- rem 1.4. The bijectiv e corresp ondence b et w ee n torsion pairs and t -structures is pro v ed in [6, I.2.13]. If C is a coreﬂectiv e semilo calizing sub category , then, by Prop osition 1.6, ( C , C L ) is a torsion pair. Con v ersely , if ( X , Y ) is a torsion pair, then [6, I.2.3] tells us that X is coreﬂectiv e and semilo calizing, while Y is reﬂectiv e and semic olo calizing.  1.5. T ensor t r iangulated categories. T o conclude this in troductory section, let T b e a tensor triangulate d c ate gory , in t he sense of [5], [37], [54]. More precisely , we assume that T ha s a closed symmetric monoidal structure with a unit ob ject S , tensor pro duct denoted b y ∧ and in ternal ho m F ( − , − ), compatible with the triangulated structure and suc h that T ( X , F ( Y , Z )) ∼ = T ( X ∧ Y , Z ) naturally in all v ariables; cf. [37, A.2.1]. Then a full sub category C of T is called an ide al if E ∧ X is in C for eve ry X in C and all E in T , and a full sub category L is called a c o ide al if F ( E , X ) is in L for ev ery X in L and all E in T . A lo c alizing ide al is a lo calizing sub category that is a lso an ideal, and similarly in the dual case. In t he homotopy category of sp ectra, all lo calizing sub categories ar e ideals and all colo- calizing subcategories are coideals . As shown in [37, Lemma 1.4.6], the same happ ens in an y mo no gen i c stable homotop y category (i.e., suc h that the unit of the monoidal struc- ture is a small generator). In [37], for stable homotop y categories, the terms lo c alization and c olo c a l i z ation we re used in a more restrictive sense tha n in the presen t article. Th us an exact r eﬂection L w as called a lo calization in [37, Deﬁnition 3.1.1] if LX = 0 for an ob ject X implies that L ( E ∧ X ) = 0 for ev ery E ; in other w ords, if the class of L -acyclic ob jects is a lo calizing ideal. Dually , an exact coreﬂection C w as called a colo calization if C X = 0 implies that C ( F ( E , X )) = 0 for ev ery E , i.e., if the class of C -acyclic ob jects is a colo calizing coideal. F or eac h class o f ob jects D , t he class o f those X suc h that F ( X, D ) = 0 for all D ∈ D is a lo calizing ideal, and the class of those Y suc h that F ( D , Y ) = 0 for all D ∈ D is a colo calizing coideal. If C is a lo calizing ideal, then C ⊥ is a colo calizing coideal, and, if L is a colo calizing coideal, then ⊥ L is a lo calizing ideal. Thus , under the bijectiv e corresp ondence g iv en by Theorem 1.4, lo calizations in the sense of [37] are also paired with colo calizations. 2. Reflect ive colocalizing s ubca te gories Bac kground on lo cally presen table and access ible categories can b e f ound in [2] and [45]. The basic deﬁnitions are as follow s. Let λ b e a regular cardinal. A nonempt y small category is λ -ﬁlter e d if, give n an y set o f o b jects { A i | i ∈ I } where | I | < λ , there is an ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 11 ob ject A and a morphism A i → A for eac h i ∈ I , and, moreov er, giv en a n y set of parallel arro ws b et w een any tw o ob jects { ϕ j : B → C | j ∈ J } where | J | < λ , there is a mo r phism ψ : C → D such that ψ ◦ ϕ j is the same morphism for all j ∈ J . Let K b e an y catego ry . A diag ram D : I → K where I is a λ -ﬁltered small category is called a λ -ﬁlter e d diagr am , and, if D has a colimit, then colim I D is called a λ -ﬁlter e d c oli m it . An ob ject X of K is λ -pr esentable if the functor K ( X, − ) preserv es λ -ﬁltered colimits. The cat ego ry K is λ -ac c essible if all λ -ﬁltered colimits exis t in K and there is a set S of λ - presen table ob jects suc h tha t ev ery o b ject of K is a λ - ﬁltered colimit o f ob jects from S . It is called ac c essible if it is λ -accessible fo r some λ . A co complete a ccessible category is called lo c al ly pr esentable . Th us, t he passage f r om lo cally presen table to accessible catego r ies amoun ts to w eak- ening the assumption of co completeness by imp o sing only that enough colimits exist. As explained in [2, § 2.1], using directed colimits instead of ﬁltered colimits in the deﬁnitions leads to the same concepts of accessib ilit y and local prese n tabilit y . As explained in [2, 6.3], V opˇ enka’s principle is equiv alen t to the statemen t that, given any family o f obje cts X s of an ac c essible c ate gory ind e xe d by the class of a l l or dinals, ther e is a morphism X s → X t for some or dina ls s < t . A f unctor γ : K → T b etw een tw o categories will b e called essential ly surje ctive on sour c es if, for ev ery ob ject X and ev ery collection of morphisms { f i : X → X i | i ∈ I } in T (where I is an y discrete category , p ossibly a prop er class), there is an ob ject K and a collection of morphisms { g i : K → K i | i ∈ I } in K together with isomorphisms h : γ K → X and h i : γ K i → X i for all i rendering the follo wing diagram commutativ e: (2.1) γ K h ∼ =   γ g i / / γ K i h i ∼ =   X f i / / X i . F or example, if K is a mo del category [36] and γ : K → Ho( K ) is the canonical functor on to the corresp onding homotopy category (which w e assume to b e the identit y on ob jects), then γ is essen tially surjectiv e o n sources —and also on sinks; here “sources” and “sinks” are mean t as in [1]. Indeed, give n a collection { f i : X → X i | i ∈ I } in Ho( K ), w e can c ho ose a coﬁbran t replacemen t q : K → X and a ﬁbran t replacemen t r i : X i → K i for eac h i . Then w e can pic k a morphism g i : K → K i for eac h i suc h that the zig-zag X K q o o g i / / K i X i r i o o represen ts f i . Then the c hoice s h = γ q and h i = ( γ r i ) − 1 render (2.1) comm utat ive. Theorem 2.1. L et T b e a c ate gory with pr o ducts and supp ose given a functor γ : K → T wher e K is ac c essible and γ is essential ly surje c tive on sour c es. I f V opˇ enka’s principl e holds, then every ful l sub c ate g ory L ⊆ T clo se d under pr o ducts is we akly r eﬂe ctive. ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 12 Pr o of. W rite L as the union of an ascending c hain of full sub categories indexed by the ordinals, L = [ i ∈ Ord L i , where eac h L i is the closure under pro ducts of a small sub category A i . F or each ob ject X of T , let X i b e the pr o duct of t he co domains of all morphisms from X to ob jects of A i , and let f i : X → X i b e the induced morphism. Then ev ery morphism from X to some ob ject of L i factors through f i and hence f i is a w eak reﬂection of X on to L i . No w, as in [2, 6.26 ], in order to pro v e that L is w eakly reﬂectiv e, it suﬃces to ﬁnd an ordinal i suc h that, for all j ≥ i , the morphism f j can b e factor ized as f j = ϕ ij ◦ f i for some ϕ ij : X i → X j . In other w ords, ( X ↓ L )( f i , f j ) 6 = ∅ for all j ≥ i , where ( X ↓ L ) denotes the comma category of L under X . Supp ose t he con trary . Then there are ordinals i 0 < i 1 < · · · < i s < · · · , where s ranges o v er all the ordinals, suc h that (2.2) ( X ↓ L )( f i s , f i t ) = ∅ if s < t . Since γ is essen tially surjectiv e on sources, there is a morphism g i : K → K i in K for each ordinal i , and there are isomorphisms h : γ K → X a nd h i : γ K i → X i suc h that f i ◦ h = h i ◦ γ g i for all i . Then ( K ↓ K )( g i s , g i t ) = ∅ if s < t , since, if there is a mo r phism G : K i s → K i t with G ◦ g i s = g i t , t hen F = h i t ◦ γ G ◦ ( h i s ) − 1 satisﬁes F ◦ f i s = f i t , con tradicting (2.2). Since the category ( K ↓ K ) is accessible by [2, 2.44], this is incompatible with V op ˇ enk a’s principle, according to [2, 6.3].  W e next sho w that the existence of a w eak reﬂection implies the existence o f a reﬂection under a ssumptions that do not require any further input from large-cardinal theory . W e sa y that ide mp otents split in a category T if for ev ery mor phism e : A → A suc h that e ◦ e = e t here are morphisms f : A → B and g : B → A suc h that e = g ◦ f a nd f ◦ g = id. This is aut o matic in a category with co equalizers, since f can b e chose n to b e a co equalizer of e and the identit y , and g is determined b y the unive rsal prop erty of the co equalizer. It also holds in other imp ortan t cases; for instance, by [46, Prop osition 1.6.8], idemp oten ts split in an y triangulated categor y with coun table coproducts or coun table pro ducts. W e will need the follo wing result, whic h, as p ointed out to us by Chorn y , can b e derive d from [32, VII.2 8 H]. A self-contained pro of is giv en here f o r the sake of completeness. Recall that a we ak limi t of a diagram D : I → T , where T is a ny category and I is a small category , is an ob ject X of T together with a natura l transformation ν : X → D (where X is seen as a constant functor, so ν is a cone to D ) such that an y other na tural transformation Y → D with Y in T factorizes thro ug h ν , not necessarily in a unique w a y . W eak colimits are deﬁned dually . Theorem 2.2. L et T b e a c ate gory with pr o ducts wher e idemp otents split. L et L b e a we a k ly r eﬂe ctive s ub c ate gory of T close d unde r r etr acts, and assume that e v ery p air of p ar al lel arr ows in L has a we ak e qualizer that lies in L . Then L is r eﬂe ctive. ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 13 Pr o of. Note ﬁrst that, by [2, R emark 4.5(3)], since L is w eakly reﬂectiv e and closed under retracts, it is also closed under pro ducts. Recall also that reﬂectiv e or weak ly reﬂectiv e sub categories a r e tacitly a ssumed to b e full. Let A b e an y ob ject of T and let r 0 : A → A 0 b e a w eak reﬂection o f A on to L . Let I denote the set of a ll pairs o f morphisms ( f , g ) : A 0 ⇒ A 0 suc h that f ◦ r 0 = g ◦ r 0 , and let u 1 : A 1 → A 0 b e a w eak equalizer of the pair ( Q i ∈ I f i , Q i ∈ I g i ) : A 0 ⇒ Q i ∈ I A 0 . By h ypo thesis, we ma y c ho ose u 1 in L . Since u 1 is a w eak equalizer, there is a morphism r 1 : A → A 1 suc h that u 1 ◦ r 1 = r 0 . Moreo v er, since r 0 is a w eak reﬂection and A 1 is in L , t here is a morphism t 1 : A 0 → A 1 suc h that t 1 ◦ r 0 = r 1 . Then ( u 1 ◦ t 1 , id) ∈ I and hence u 1 ◦ t 1 ◦ u 1 = u 1 . It follow s that t 1 ◦ u 1 is idemp oten t and hence it splits. That is, there are morphisms u 2 : A 2 → A 1 and t 2 : A 1 → A 2 suc h that u 2 ◦ t 2 = t 1 ◦ u 1 and t 2 ◦ u 2 = id. W e next pro v e that, if w e pic k r 2 = t 2 ◦ r 1 , then r 2 is a reﬂection of A on to L . First of all, A 2 is a retract of A 1 and hence A 2 is in L . Second, f r o m the equality r 0 = u 1 ◦ u 2 ◦ r 2 it follo ws that r 2 is a weak reﬂection o f A onto L . No w, giv en a morphism f : A → X with X in L , since r 2 is a w eak reﬂection, there is a morphism g : A 2 → X suc h that g ◦ r 2 = f . Supp ose that there is another h : A 2 → X with h ◦ r 2 = g ◦ r 2 . Let w : B → A 2 b e a w eak equalizer o f g and h with B in L . Then, as w e next sho w, w has a right in v ers e, so the equalit y g ◦ w = h ◦ w implies that g = h , as needed. In order to pro v e that w has a righ t inv erse, note that, since h ◦ r 2 = g ◦ r 2 , there is a morphism t : A → B with w ◦ t = r 2 . Since r 0 is a we ak reﬂection of A onto L , there is a morphism s : A 0 → B suc h that s ◦ r 0 = t . Now u 1 ◦ u 2 ◦ w ◦ s ◦ r 0 = r 0 ; hence ( u 1 ◦ u 2 ◦ w ◦ s, id) ∈ I , from whic h it follows that u 1 ◦ u 2 ◦ w ◦ s ◦ u 1 = u 1 . Finally , note that t 2 ◦ t 1 ◦ u 1 ◦ u 2 = t 2 ◦ u 2 ◦ t 2 ◦ u 2 = id and therefore w ◦ s ◦ u 1 = t 2 ◦ t 1 ◦ u 1 = t 2 , so w ◦ s ◦ u 1 ◦ u 2 = id, as claimed.  Corollary 2.3. Every we akly r e ﬂe ctive sub c ate gory close d under r e tr acts and ﬁbr es in a triangulate d c ate gory with pr o ducts is r eﬂe ctive. Pr o of. This is implied by Theorem 2 .2, since a ﬁbre of f − g is a w eak equalizer of t w o giv en parallel arrows f and g , and idemp otents split in a triangulated category if coun table pro ducts exist, according to [46 , Remark 1.6.9].  Dually , ev ery we akly coreﬂectiv e sub category closed under retra cts and coﬁbres in a triangulated category T with coproducts is coreﬂectiv e. Neeman pro v ed this fact in [47, Prop osition 1.4] for thic k sub categories, without assuming the exis tence of copro ducts in T , but imp osing that idempoten ts split in T . Putting together Theorem 2.1 and Theorem 2 .2, w e state the main result of this section. A model category K is called stable [51, 2.1.1] if it is p oin ted (i.e., the unique map from the initial ob j ect to the terminal ob ject is a n isomorphism) and the susp ension and loo p op erators are in v ers e equiv alences on t he homotop y category Ho( K ). It then follows t hat Ho( K ) is tr iangulated, where the triangles come from ﬁbre or coﬁbre sequences in K (see [36, 6.2.6]), and has pro ducts and copro ducts o v er arbitrary index sets, coming from those of K . In fact, Ho( K ) has w eak limits and colimits, but it is neither complete ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 14 nor co complete in g eneral; see [37, 2.2]. Quillen equiv alenc es of stable mo del categories preserv e ﬁbre and coﬁbre sequences and hence the triangulated structure of Ho( K ). Theorem 2.4. L et K b e a l o c al l y pr esentable c ate gory with a stable mo del c ate gory struc- tur e. I f V opˇ enka’s princ iple ho lds, then every ful l sub c ate gory L of Ho( K ) clo s e d under pr o ducts and ﬁbr es is r eﬂe ctive. If L is semic olo c alizing, then the r eﬂe ction is semiexact. If L is c olo c alizing, then the r eﬂe ction is exact. Pr o of. The cano nical functor γ : K → Ho( K ) is essen tially surjectiv e on sources and sinks. Hence γ satisﬁes the assumptions of Theorem 2.1, from whic h it follow s that L is w eakly reﬂectiv e. Closure of L under retra cts follows from the Eilen b erg swindle, as in [37, Lemma 1.4.9]. Then Corollary 2.3 implies that L is in fact reﬂectiv e. The other statemen ts hold b y the deﬁnitions of the terms in v olv ed.  W e do not kno w if the assumption that L b e closed under ﬁbres is necessary for t he v alidity of Theorem 2.4. W e note how eve r that an imp ortant kind of reﬂectiv e sub cate- gories, namely classes of f - lo cal ob jects in the sense of [19] or [33] in homoto py categories of suitable mo del categories, a re closed under ﬁbres. Under the assumptions of Theorem 2.4, if Ho( K ) is tensor triangulated and a giv en colo calizing subcatego ry L ⊆ Ho( K ) is a coideal, then the exact reﬂection L giv en b y Theorem 2 .4 is automatically a lo calizatio n in the sense of [37], that is, the class of L -acyclic ob j ects is then a lo calizing ideal. Corollary 2.5. L et K b e a lo c al ly pr ese n table stable mo del c ate gory. I f V opˇ enka’s prin- ciple holds, then every close d semilo c alizing sub c ate gory of Ho( K ) is c or eﬂe ctive. Pr o of. Let C b e a closed semilo calizing subcategor y of Ho ( K ). Then C L is a semic olo- calizing sub category , whic h is reﬂectiv e by Theorem 2.4. Hence L ( C L ) is coreﬂectiv e b y Theorem 1.4, and it is equal to C since C is closed b y assumption.  As o bserv e d in Subsection 1.3, if a lo calizing sub category C is closed, then it is also closed if view ed as a semilocalizing subcatego r y . Hence, the statemen t of Corollary 2.5 is also true for closed lo calizing subcategor ies. It w ould b e v ery in teresting to ha v e a coun terexample (if there is one) to the statemen t of Theorem 2.4 under some set-theoretical assumption incompatible with V opˇ enk a’s prin- ciple. W e next give a partial resu lt in this direction, based on [15] and [20 ], whic h sho ws that Theorem 2.1 cannot b e prov ed in ZFC. This result answ ers the second part of Op en Problem 5 from [2, p. 2 96]. Prop osition 2.6. Assumi n g the nonex i s tenc e of me asur able c ar dinals, ther e is a f ul l sub c ate gory of the c ate gory of ab elian gr oups which i s clo se d under pr o ducts and r etr acts but not we akly r eﬂe ctive. Pr o of. Let C b e the closure of the class of groups Z κ / Z <κ under pro ducts and retracts, where κ runs o v er all cardinals, and Z κ denotes a pro duct o f copies of t he integers indexed b y κ while Z <κ denotes the subgroup of sequence s whose supp ort (i.e., the set of nonzero ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 15 en tries) has cardinalit y smaller than κ . Assume that w : Z → A is a w eak reﬂection of Z on to C . Then there is a retraction Y i ∈ I Z κ i / Z <κ i r − → A for some set of cardinals { κ i } i ∈ I . Choo se a regular cardinal λ bigger than the sum Σ i ∈ I κ i . Let d : Z → Z λ b e the diago nal and p : Z λ → Z λ / Z <λ the pro jection. Since w is a w eak reﬂection, there is a homomorphism f : A → Z λ / Z <λ with f ◦ w = p ◦ d . F ollo wing [15, Lemma 6.1 ], there is a homomorphism g : A → Z λ suc h that f = p ◦ g . Since the image of d is not contained in Z <λ , w e hav e f 6 = 0 and th us g 6 = 0. Since r is an epimorphism, g ◦ r 6 = 0. No w, since Z λ maps on to Q i ∈ I Z κ i , there is a nonzero homomorphism h : Z λ → Z λ whic h v anishes o n the direct sum ⊕ i<λ Z , since it factors through Q i ∈ I Z κ i / Z <κ i . Hence, by comp osing h with a suitable pro jection, w e o btain a nonzero homomorphism Z λ → Z whic h v anishe s on the direct sum ⊕ i<λ Z . Ac cording to [26, 94.4], this fact implies the existence of measu rable cardinals. This contradiction prov es the statemen t.  Corollary 2 .7. Assuming the nonexistenc e of m e asur able c a r dinals, ther e is a ful l sub- c ate gory of the hom otopy c ate gory of sp e ctr a which is close d under pr o ducts and r etr acts but not we akly r eﬂe ctive. Pr o of. Consider the full em b edding H of the category of ab elian g roups into the homotop y category of sp ectra giv en b y assigning to eac h ab elian group A an Eilenberg–Mac Lane sp ectrum H A represen ting ordinary cohomology with co eﬃcien ts in A . Since H preserv es pro ducts and its image is closed under retracts, it sends the class C considered in the pro of of Prop osition 2.6 to a class H C of spectra closed under pro ducts and retracts. This class H C is not w eakly reﬂectiv e, since the ab ov e argument sho w s tha t H Z do es not a dmit a w eak reﬂection on to H C if there are no measurable cardinals.  3. Coreflective localizing subca tegories In Section 2 w e pro v ed that, if K is a lo cally presen table categor y with a stable mo del category structure, then V op ˇ enk a’s principle implies tha t all colo calizing sub categories (in fact, all full subcatego r ies closed under pro ducts and ﬁbres) of Ho( K ) are reﬂectiv e. Our purp ose in this sec tion is to study if V op ˇ enk a’s principle a lso implies that all lo calizing o r semilo calizing sub categories of Ho( K ) are coreﬂectiv e . By Corollary 2.5, this is equiv alen t to asking if they are close d. W e pro vide an aﬃrmativ e answ er b y assuming that K b e coﬁbrantly generated [36, Def- inition 2.1.17], in additio n to b eing lo cally presen table. Recall that a coﬁbrantly generated mo del category whose underlying category is lo cally presen table is called c om binatorial . It was sho wn in [2 3] and [24] tha t a mo del category is com binatorial if and only if it is Quillen equiv alen t to a lo calization of some category of diagrams of simplicial sets with resp ect to a set of morphisms. Hence, examples ab ound. When K is combinatorial, w e not only prov e that semilo calizing sub categories o f Ho( K ) ar e coreﬂectiv e, but w e moreo v er sho w that they are singly generated. ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 16 Although, for the v alidit y o f our argumen ts, w e need that K b e simplicial [30, II.3 ], it is not neces sary to imp ose this as a restriction, due to the follo wing fact. Prop osition 3.1. Every ( s table) c ombinatorial mo del c ate gory is Quil len e quivalent to a (stable) simplicial c om binatorial m o del c ate gory. Pr o of. F o r a small catego ry C , denote b y U + C , as in [25], the category of functors fro m C op to the category sSets ∗ of p oin ted simplicial sets. According t o [23, Corollary 6 .4] and [25, Prop osition 5.2], for ev ery com binatorial mo del category K there is a small category C suc h that K is Quillen equiv alen t t o the left Bousﬁeld lo calizat io n of U + C with resp ect to a certain set of mor phisms. The category U + C is com binatorial, p ointed, simplicial and left prop er, and so is an y of its lo calizations. Since Quillen equiv alences prese rv e the suspension and lo op functors, ev ery p oin ted mo del catego r y whic h is Quillen equiv alent to a stable one is itself stable.  One crucial prop erty of com binatorial mo del categories that w e will use in this section is the follo wing. F or ev ery com binatorial mo del category K there is a regular cardinal λ suc h that, if X : I → K and Y : I → K are diagrams where I is a small λ -ﬁltered category , and a morphism of dia g rams f : X → Y is giv e n suc h that f i : X i → Y i is a w eak equiv alence for eac h i ∈ I , then the induced map colim I X → colim I Y is also a weak equiv alence. F or a pro of of this fact, see [24, Prop osition 2.3]. Another feature of combinatorial mo del categories is that, if K is combinatorial and I is an y small category , then the pr o j e ctive mo del structur e (in whic h w eak equiv alences and ﬁbrations are ob ject wise) and the inje c tive mo del structur e (in whic h w eak equiv a- lences and coﬁbra t io ns are ob ject wis e) exist o n the diagr am category K I ; see [42, Prop osi- tion A.2.8.2]. In fact, as sho wn in [33, Theorem 11.6.1], for the exis tence o f the pro jective mo del structure it is enough that K b e coﬁbran tly generated. If K I is equipped with the pro jectiv e mo del structure, then the constan t functor K → K I is righ t Quillen and therefore its left adjoint colim I : K I → K is left Quillen, so it preserv es coﬁbrations, trivial coﬁbrations, and w eak equiv alenc es b etw een coﬁbran t diagrams [30, I I.8.9]. Hence, its total left deriv ed functor ho colim I exists. Since we will need to use explicit form ulas to compute homotopy colimits, w e recall, b efore going further, a n um b er of basic facts ab o ut homotopy colimits in mo del categories. Our main sources a re [12], [28], [30], [33], [34], [36], [52]. F or simplicit y , we restrict our discussion to p ointed simplicial mo del categories, whic h is suﬃcien t for o ur purp oses. The unp oin ted case w ould b e treated analogously . 3.1. A r eview of homotop y c o limits. Let K b e a p ointed simplicial mo del category . Let ∗ b e the initial and terminal ob ject, and let ⊗ denote the tensoring of K o v er p ointe d simplicial sets. F or eac h simplicial set W , w e denote b y W + its union with a disjoint basep oin t. Let ∆ denote the category whose ob jects are ﬁnite ordered sets [ n ] = (0 , 1 , . . . , n ) for n ≥ 0, and whose morphisms are nondecreasin g functions. Let ∆[ n ] b e the simplicial set whose set of k -simplices is the set of morphisms [ k ] → [ n ] in ∆ , and denote by ∆ + : ∆ → sSets ∗ the functor that sends [ n ] to ∆[ n ] + . If X is a coﬁbran t ob ject in K , ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 17 then X ⊗ ∂ ∆[1] + → X ⊗ ∆[1] + is a coﬁbra t io n yielding a cylinder for X , and hence Σ X ≃ X ⊗ S 1 ; cf. [36 , 6.1.1] The r e alization | B | of a simplicial ob ject B : ∆ op → K is the co equalizer of the tw o morphisms (3.1) a [ m ] → [ n ] B n ⊗ ∆[ m ] + / / / / a [ n ] B n ⊗ ∆[ n ] + induced by B n → B m and ∆[ m ] + → ∆[ n ] + , resp ectiv ely , fo r each morphism [ m ] → [ n ] in ∆ ; see [30, VI I.3.1]. Using co end nota tion [44, IX.6], this can b e written as | B | = Z n B n ⊗ ∆[ n ] + = B ⊗ ∆ op ∆ + . Supp ose giv en functors X : I → K and W : I op → sSets ∗ , where W will b e called a weight . The (tw o - sided) b ar c onstruction B ( W, I , X ) ∈ K ∆ op is the simplicial ob ject with (3.2) B ( W , I , X ) n = a i n → ··· → i 0 X i n ⊗ W i 0 , whose k th face map omits i k using the iden tit y on X i n ⊗ W i 0 if 0 < k < n , and using W i 0 → W i 1 if k = 0 a nd X i n → X i n − 1 if k = n . D egeneracies are giv en b y insertions o f the iden tit y . If we c ho ose as w eigh t the constant diagr a m S at the 0 t h sphere S 0 , then w e denote B I X = B ( S, I , X ) and call it a simplicia l r eplac ement of X . The (p ointed) homotopy c olim it of a functor X : I → K is deﬁned as (3.3) ho colim I X = | B I X | . It follow s that homotop y colimits comm ute; that is, giv e n X : I × J → K , ho colim I ho colim J X ∼ = ho colim I × J X ∼ = ho colim J ho colim I X . F rom (3 .3) and (3.1) one obtains the Bousﬁeld–Kan formula [12, XII.2.1], [33, 18 .1.2], as follows. Let N ( i ↓ I ) op b e the nerv e of the category ( i ↓ I ) op for eac h i ∈ I . Th us , N ( i ↓ I ) op + is the realization of the simplicial space ∆ op → sSets ∗ that consists in degree n of a copro duct of copies of S 0 indexed b y the set of sequences i → i n → · · · → i 0 of morphisms in I . Since co equalizers comm ute, w e hav e ho colim I X ∼ = co eq " a i → j X i ⊗ N ( j ↓ I ) op + / / / / a i X i ⊗ N ( i ↓ I ) op + # = X ⊗ I N ( − ↓ I ) op + . (W e note that ( I ↓ i ) w as used in [1 2] instead of ( i ↓ I ) op .) In other w ords, ho colim I X is a w eigh ted colimit of X with w eigh t N ( − ↓ I ) op + : I op → sSets ∗ . F or simplicial o b jects ∆ op → K , the n th skeleton sk n is the comp osite of the tr uncatio n functor K ∆ op → K ∆ op n with its left adjoint, where ∆ n is the full sub category of ∆ with ob jects { [0] , . . . , [ n ] } ; see [30, VI I.1.3]. Th us (sk n B ) m ∼ = B m if m ≤ n , and hence B ∼ = colim n sk n B for ev ery simplicial ob ject B . ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 18 The n th latching obje ct of B : ∆ op → K is deﬁned as L n B = (sk n − 1 B ) n for eac h n . As explained in [30, VI I.2], the category K ∆ op of simplicial o b jects in K admits a mo del structure (called R e e dy mo del structur e ) where weak equiv alences are ob jectw ise and coﬁ- brations are morphisms f : X → Y suc h that L n Y ` L n X X n → Y n is a coﬁbration for all n . Th us an ob ject B is Reedy coﬁbran t if and only if the natural morphisms L n B → B n are coﬁbrations in K f o r all n . If B is Reedy coﬁbrant, then each sk eleton sk n B is also Reedy coﬁbrant, and the inclusions sk n − 1 B ֒ → sk n B are Reedy coﬁbrations; see e.g. [8, Prop o sition 6.5]. As sho wn in [30, VI I.3.6], the realization functor is left Quillen if K ∆ op is equipp ed with the Reedy mo del structure. The follo wing consequence is crucial. Lemma 3.2. L et K b e a p ointe d simp l i c i al mo del c ate g o ry and let I b e smal l. (a) If a diagr am X : I → K is obje ctwise c oﬁbr ant, then B I X is R e e dy c oﬁ b r ant. (b) If f : X → Y is an obje c twise we ak e quivalenc e in K I and the diagr ams X and Y ar e obje ctwise c oﬁ b r ant, then the induc e d mo rp hism ho colim I X → ho colim I Y is a we ak e q uiva lenc e of c oﬁbr ant obje c ts. Pr o of. Let B = B I X . Thu s B n = ` i n → ··· → i 0 X i n for all n ≥ 0, and w e ma y write (3.4) B n = L n B ` Z n B , where L n B includes the “degenerate” summands of B n , i.e., those lab elled b y se quences i n → · · · → i 0 where some arrow is an iden tit y , and Z n B collects the rest. Then the inclusion L n B → B n is a coproduct of the identit y L n B → L n B and ∗ → Z n B , whic h is a coﬁbratio n sinc e X take s coﬁbran t v alues. This prov es part (a). Then part (b) follo ws from the fact that realization is left Q uillen, since B I f : B I X → B I Y is a w eak equiv alence b et w een Reedy coﬁbra n t ob jects.  Th us, if deﬁne d a s in (3.3), the homotopy colimit is o nly homotop y inv ar ian t on obje ct- wise c oﬁbr ant diagrams. F or this reason, it is often con v enie n t t o “correct” ho colim I b y comp osing it with a coﬁbrant replacemen t functor in K , as in [52, Deﬁnition 8.2]. The fundamen tal fact that, if made homotopy in v ariant, ho colim I yields a total left deriv ed functor o f colim I is explained as follows. F or each dia g ram X : I → K there is a natural morphism (3.5) ho colim I X − → colim I X , since colim I X is the co equalizer of t he t w o face morphisms ` i → j X i ⇒ ` i X i ; that is, the morphism (3.5) tak es the form (3.6) X ⊗ I N ( − ↓ I ) op + − → X ⊗ I S. This morphism is a w eak equiv alence only in some cases; cf. [12, XI I.2]. F or instance, it is so if I has a terminal ob j ect. More imp ortan tly , (3.6) is a w eak equiv a lence if X is c oﬁ br ant in the pro jectiv e mo del structure of K I . T o sho w this, use the fact, pro v ed in [28, Theorem 3.2], that ( − ) ⊗ I ( − ) is a left Quillen functor in tw o v ariables if the pro jectiv e mo del structure exists and is c hosen on K I and t he injectiv e mo del structure is considered in sSets I op ∗ . Accordingly , if X is a pro jectiv ely coﬁbrant diagram, then ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 19 X ⊗ I ( − ) prese rv es w eak equiv a lences b et w een (ob jectwis e) coﬁbran t ob j ects, so (3 .6) is indeed a w eak equiv alence. It is also true, as sho wn in [28, Theorem 3.3 ], that ( − ) ⊗ I ( − ) is left Quillen in tw o v ariables if the pro j ective mo del structure is considered in sSets I op ∗ and the injectiv e mo del structure exists and is c hosen on K I . Th us, since N ( − ↓ I ) op + → S is a pro j ective ly coﬁbran t approximation in sSets I op ∗ , the Bousﬁeld–Kan form ula displa ys in fact ho colim I as a left deriv e d functor of colim I , provid ed that w e restrict it to ob ject wis e coﬁbrant diagrams (i.e., coﬁbran t in the injectiv e mo del structure). F or some purp oses it is useful to consider the following functoria l pro jectiv ely coﬁbrant replacemen t of a g iv en diagram X : I → K . A ssume that X ta kes coﬁbrant v alues (or comp ose it with a coﬁbrant replacemen t functor in K otherwise). Consider the functor B ( I ↓− ) X : I × ∆ op → K giv en by ( B ( I ↓− ) X )( j, [ n ]) = ( B ( I ↓ j ) ( X ◦ U j )) n = a i n → ··· → i 0 → j X i n , where U j : ( I ↓ j ) → I sends eac h arrow i → j to i , and let e X = | B ( I ↓− ) X | . Th us, (3.7) e X j = | B ( I ↓ j ) ( X ◦ U j ) | = ho colim ( I ↓ j ) ( X ◦ U j ) for a ll j ∈ I . Since ( I ↓ j ) has a terminal ob ject for eac h j , the natural morphism e X → X is an ob ject wise w eak equiv alence. Using the fact that realization is a left adjoint and hence comm utes with colimits, one obtains a canonical isomorphism (3.8) colim I e X = colim j e X j = colim j ho colim ( I ↓ j ) ( X ◦ U j ) = colim j | B ( I ↓ j ) ( X ◦ U j ) | ∼ = | colim j B ( I ↓ j ) ( X ◦ U j ) | ∼ = | B I X | = ho colim I X . In order to pro v e that the diagr a m e X is indeed pr o jectiv ely coﬁbran t, view B ( I ↓− ) X as an ob ject in ( K I ) ∆ op and c hec k that it is Reedy coﬁbrant if the pro jectiv e mo del structure is c hosen in K I , similarly as in part (a) of Lemma 3.2. Although pro j ectiv ely coﬁbran t diagrams are not easy to c haracterize in general, w e note the follo wing w ell-kno wn sp ecial case for subsequen t reference. Lemma 3.3. L et λ b e an inﬁ nite or dinal and let K b e a mo del c ate gory. Supp ose that, for an obje ctwise c oﬁbr ant diagr am X : λ → K , e ach morphism X i → X i +1 with i < λ is a c oﬁbr ation and the induc e d morphism colim i<α X i → X α is also a c oﬁbr ation for every limit or di n al α < λ . Then the diagr am X is pr oje ctively c oﬁbr ant in K λ . Pr o of. F o r eac h ob jectw ise trivial ﬁbration A → B in K λ and eac h morphism X → B , the existence of a lifting X → A fo llo ws by transﬁnite induction.  F or an ob jectwis e coﬁbran t diagram X : I → K , the homotop y colimit ho colim I X can b e ﬁltered as fo llo ws. Let B = B I X , and denote F n = | sk n B | . Since B is Reedy coﬁbrant, the Bousﬁel d –Kan m a p ho colim ∆ op B = B ⊗ ∆ op N ( − ↓ ∆ op ) op + − → B ⊗ ∆ op ∆ + = | B | ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 20 is a weak equiv alence; cf. [12, XI I.3.4 ], [33, 18.7.1]. Since homotopy colimits comm ute, ho colim I X = | B | ≃ ho colim ∆ op B ≃ ho colim ∆ op ho colim n sk n B (3.9) ∼ = ho colim n ho colim ∆ op sk n B ≃ ho colim n F n . This equiv alence, whic h w as our main goal in this subs ection, is relev an t in the con text of triangulated categories, since it allows us to replace a homotop y colimit indexed by a n arbitrary small cat ego ry by another one indexed by a coun tably inﬁnite ordinal, whic h ﬁts in to a w ell-kno wn triangle in v olving coun table copro ducts and the shift map, as in [46, Deﬁnition 1.6.4]. 3.2. Singly generated semilocalizing sub categories are coreﬂectiv e. The ﬁltration displa y ed in (3.9 ) of a homotopy colimit was used in [9], [10] to sho w that the class of acyclics of an y homolo g y theory o n sp ectra is closed under homotop y colimits, as a k ey ingredien t o f the pro of of the existence of ho mo lo gical lo calizations. The v alidit y of t he same argumen t for lo calizing sub categories o f stable homotop y categories w as suggested in [3 7, Remark 2.2.5]. A similar argument in deriv ed categories of Grothendiec k categories w as used fo r ﬁltered homotop y colimits in [3, Theorem 3.1 ]. W e generalize it as follows. Prop osition 3.4. L et K b e a stable simplicial mo del c ate gory and let γ : K → Ho( K ) denote the c an o n ic al functor. L et C b e a semilo c alizi n g sub c ate gory of Ho( K ) . If a diagr am X : I → K is obje ctwise c oﬁbr ant and γ X i ∈ C for al l i ∈ I , then γ ho colim I X ∈ C . Pr o of. Let B = B I X b e the simplicial replacemen t of X , as in (3.3), a nd let F n = | sk n B | . As explained in [30, VI I.3 .8 ] or [31 , 5 .2], since r ealizat io n comm utes with colimits, there is a natura l pushout diagram (3.10) ( B n ⊗ ∂ ∆[ n ] + ) ` ( L n B ⊗ ∂ ∆[ n ] + ) ( L n B ⊗ ∆[ n ] + ) / /   B n ⊗ ∆[ n ] +   F n − 1 / / F n . According to part (a) of Lemma 3.2, since the diagram X is ob j ect wis e coﬁbrant, B is Reedy coﬁbrant. Hence, b y Quillen’s SM7 axiom for a simplicial mo del category [30, I I.3.12], the upper arrow in ( 3.10) is a coﬁbration. Therefore, the coﬁbre F n /F n − 1 is isomorphic to the coﬁbre of the upp er arrow in (3.10), whic h is isomorphic to Z n B ⊗ S n if w e write, as in (3.4), B n = L n B ` Z n B , where Z n B contains the nondegenerate summands of B n . Hence, the sequenc e γ F n − 1 / / γ F n / / Σ n γ Z n B is part of a triangle in Ho( K ). Since Z n B is a copro duct of o b j ects X i with i ∈ I , it follo ws inductiv ely that γ F n ∈ C for all n . Since F n − 1 → F n is a coﬁbration b et w ee n coﬁbrant ob jects for ev ery n , Lemma 3 .3 implies that γ ho colim n F n ∼ = γ colim n F n , ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 21 and γ colim n F n is a coﬁbre of a morphism ` n γ F n → ` n γ F n in Ho( K ), na mely the diﬀerence b et w ee n the iden tit y and the shift ma p, from whic h it f o llo ws that γ ho colim n F n is in C , b ecause C is semilo calizing. Since, a s observ ed in (3.9), ho colim n F n ≃ ho colim I X , the claim is pro v ed.  Corollary 3.5. If K is a p ointe d simplicial c ombinatorial mo del c a te gory and we den o te by γ : K → Ho( K ) the c anon ic al functor, then ther e is a r e gular c ar dinal λ such that: (a) F o r every λ -ﬁlter e d obje ctwise c oﬁbr ant diagr am X : I → K , the natur al morphism ho colim I X → colim I X is a we ak e quivalenc e. (b) If K is stable and C is a semilo c alizing sub c ate gory of Ho( K ) , then, for every λ -ﬁlter e d diagr am X : I → K with γ X i ∈ C for al l i , we have γ colim I X ∈ C . Pr o of. By [2 4, Prop osition 7.3], for a com binatorial category K there is a regular cardinal λ suc h that λ -ﬁltered colimits of we ak equiv alences are we ak equiv alenc es. Let X : I → K b e an ob jectw ise coﬁbrant diagram where I is λ -ﬁltered. Let e X → X b e the ob ject wis e w eak equiv alence deﬁned in (3.7). Then, b y our c hoice of λ , the induced morphism colim I e X − → colim I X is a weak equiv alence. Since colim I e X ∼ = ho colim I X by (3.8), part (a) is prov ed. No w let X : I → K b e any diagram where I is λ - ﬁltered, and let Q b e a coﬁbrant replacemen t functor in K . F rom our c hoice of λ w e infer that colim I X is w eakly equiv alen t to colim I QX and hence to ho colim I QX , b y part (a). Therefore, if K is stable, then for ev ery semilo calizing sub category C of Ho( K ) it follows fro m Prop o sition 3.4 that γ colim I X ∈ C if γ X i ∈ C for all i ∈ I .  W e emphasize that the cardinal λ in the statemen t of Coro lla ry 3.5 dep ends only on K , not on the sub category C . The follo wing is another useful pro p ert y of triangulat ed categories with mo dels. A sp e- cial case is discussed in [37, Remark 2 .2 .8]. ( The assumption that K b e simplicial is not really necessary here nor in Corollary 3.5, since homotop y colimits can be used, with t he same basic prop erties, in all mo del categories; see [33, Chapter 19].) Lemma 3.6. L et K b e a stable simplicial mo del c ate gory and denote by γ : K → Ho( K ) the c anonic a l functor. L et I b e any smal l c ate gory such that the pr o j e ctive mo del structu r e exists on K I . Supp os e given morphisms of obje ctwise c oﬁbr ant diagr ams X → Y → Z i n K I such that γ X → γ Y → γ Z is p art of a triangle in Ho( K I ) . Then γ ho colim I X / / γ ho colim I Y / / γ ho colim I Z is p art of a triangle in Ho( K ) . Pr o of. Since colim I is left Quillen if the pro jectiv e mo del structure is considered on K I , its total left deriv ed functor preserv es triangles, as sho wn in [36, Prop osition 6.4 .1]. Alternativ ely , this result follows from the fact that homo t o p y colimits comm ute, since, b y assumption, Z is w eakly equiv a lent to the homotopy coﬁbre in K I of the giv en mor- phism X → Y , i.e., the homotop y pushout of ∗ ← X → Y , and ho colim I is homotop y in v ariant on ob jectwis e coﬁbran t diagrams.  ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 22 Sp ecial cases or v arian ts of the next result ha v e b een described in [4, Theorem 3.4] for deriv ed categories of G rothendiec k categories; in [6, Prop osition I I I.2.6] for compactly generated to r sion pairs; in [37, Prop osition 2.3.1] for alg ebraic stable homotop y categories; in [39, Theorem 3.1] for deriv ators; and in [4 3 , Prop osition 16.1] for stable ∞ -categories. The core of the argument w as ﬁrst used b y Bousﬁeld in [9]. W e note that, if the dual statemen t could b e pro v ed without larg e-cardinal assumptions, namely that singly generated colo calizing sub categories are reﬂectiv e in ZF C, this w ould imply the existence of cohomolo g ical lo calizations o f sp ectra in ZFC, a long-standing unsolv ed pro blem. Theorem 3.7. If K is a stable c ombinatorial mo del c ate gory, then every singly gener ate d semilo c alizing sub c ate gory of Ho( K ) is c or eﬂe ctive. Pr o of. By Prop osition 3.1, w e may assume that K is simplicial. Let γ : K → Ho( K ) denote the canonical functor. Let C b e a semilo calizing sub category of Ho( K ) and supp ose that C = slo c( A ) fo r some ob ject A . Pic k, fo r eac h n ≥ 0, a coﬁbran t ob ject B n in K suc h that γ B n ∼ = Σ n A , and c ho ose a regular cardinal λ and a ﬁbrant replacemen t functor R in K suc h that: (i) B n is λ -presen table for ev ery n ≥ 0; (ii) all λ -ﬁltered colimits of w eak equiv alenc es are w eak equiv alences; (iii) the functor R preserv es λ -ﬁltered colimits. This is p ossible according to [24, Prop osition 2.3 ] and [2 4, Prop osition 7.3], due to the assumption that K is com binatorial. In order to construct a coreﬂection onto C , w e pro ceed similarly as in [9 , Prop osition 1.5 ] or a s in the pro of of [37, Prop osition 2 .3.17]. F or an y ob ject X o f K —whic h w e assume ﬁbran t and coﬁbrant—, tak e Y 0 = X a nd let W 0 b e a copro duct of copies of B n for n ≥ 0 indexed b y all morphisms in K ( B n , Y 0 ). Let u 0 : W 0 → Y 0 b e given b y f : B n → Y 0 on the summand corresp onding to f . Next, let Y 1 b e the homotopy coﬁbre of u 0 . More precisely , fa ctor u 0 in to a coﬁbration e u 0 : W 0 → e Y 0 follo w ed by a trivial ﬁbratio n φ 0 : e Y 0 → Y 0 ; let Y ′ 1 b e the pushout of e u 0 and W 0 → ∗ , and let Y ′ 1 → Y ′′ 1 b e a trivial coﬁbration with Y ′′ 1 ﬁbran t. Since Y 0 is coﬁbran t, there is a left in v ers e Y 0 → e Y 0 to φ 0 and hence a morphism Y 0 → Y ′′ 1 , whic h w e factor again into a coﬁbrat io n v 0 : Y 0 → Y 1 follo w ed by a trivial ﬁbration Y 1 → Y ′′ 1 . Thu s it follo ws from our c hoices that Y 1 is b oth ﬁbran t and coﬁbrant, and W 0 u 0 / / Y 0 v 0 / / Y 1 yields a triangle in Ho( K ), since γ W 0 γ u 0 / / γ Y 0 γ v 0 / / γ Y 1 is isomorphic to γ W 0 γ e u 0 / / γ e Y 0 / / γ Y ′ 1 , whic h is a canonical tria ng le. ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 23 No w rep eat the pro cess with Y 1 in the place of Y 0 . In this wa y w e construct inductiv ely , for ev ery or dina l i , a sequence (3.11) W i u i / / Y i v i / / Y i +1 yielding a triangle in Ho( K ) , where Y i +1 is ﬁbran t and coﬁbran t, v i is a coﬁbration, and W i is a copro duct of copies of B n for n ≥ 0 , to gether with a morphism w i +1 : X → Y i +1 suc h that w i +1 = v i ◦ w i (with w 1 = v 0 ). If α is a limit ordinal, ta ke Z α = colim i<α Y i , and let Z α → Y α b e a trivial coﬁbration with Y α ﬁbran t. Since eve ry (p ossibly tra nsﬁnite) comp osition of coﬁbratio ns is a coﬁbration, the morphism X → Z α giv en b y w i for i < α is a coﬁbratio n, and hence the comp osite w α : X → Y α is also a coﬁbration. Let Y : λ → K be the diagra m give n b y the ob jects Y i and the maps v i for i < λ . Then Y is coﬁbran t in K λ , b y Lemma 3 .3, and the constant diagram X : λ → K at the ob ject X is also coﬁbran t in K λ . L et F b e the homotopy pullbac k of the map X → Y giv en b y the morphisms w i and the trivial map ∗ → Y in K λ . Th us, γ F / / γ X / / γ Y is part of a triangle in Ho( K λ ), since K λ is stable. Let Q b e a coﬁbran t r eplacemen t functor in K , and let QF be the comp osite of Q and F . Thus QF is ob ject wise coﬁbrant and, by Lemma 3.6, (3.12) γ ho colim i<α QF i / / γ X / / γ ho colim i<α Y i is part o f a triangle for eac h limit ordinal α ≤ λ . By the octahedral axiom in Ho( K ) a nd (3.11), there is a tria ngle γ QF i / / γ QF i +1 / / γ W i for eac h o rdinal i . Therefore, it follo ws from tra nsﬁnite induction that γ QF i ∈ slo c ( A ) for all ordinals i , since eac h γ W i is constructed from A b y means of susp ensions and copro ducts. If α is a limit ordinal, then γ Y α ∼ = γ colim i<α Y i ∼ = γ ho colim i<α Y i b ecause Y is coﬁbran t, and it t hen follo ws f r om (3.1 2) that γ QF α ∼ = γ ho colim i<α QF i , whic h is in slo c( A ) by Prop osition 3.4. Let C X = colim i<λ QF i and let LX = colim i<λ Y i , and note that the natural morphisms ho colim i<λ QF i → C X and ho colim i<λ Y i → LX are w eak equiv alences, by part (a) of Corollary 3.5 and b y our c hoice of λ . By Lemma 3.6, the sequenc e γ C X / / γ X / / γ LX is part o f a triangle in Ho( K ). By Prop osition 3 .4, γ C X ∈ slo c( A ). Again by our c hoice of λ , w e hav e RLX ∼ = colim i<λ RY i . Now ev ery f : Σ n A → γ LX in Ho( K ) can b e lifted to a morphism e f : B n → RLX in K , as RLX is ﬁbran t. Once more b y our c hoice of λ , this mo r phism e f f actors through RY k for some k < λ , since B n is λ -presen table. Since (3.11) yields a triangle fo r all i , the comp osite Σ n A / / γ RY k / / γ RY k +1 ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 24 is zero. This implies that f : Σ n A → γ LX is zero. Therefore, γ LX ∈ A L , and, b y (1.4), A L = slo c( A ) L . This prov es that C is a coreﬂection onto C , using Prop osition 1.6.  W e note that the reﬂection L obtained in the previous pro of is a n ulliﬁc ation P A in the sense of [11] and [19], and the sub cat ego ry C is th us the closure under extensions of the class of A -cellular ob jects. W e ha v e giv en the argumen t in full detail to stress the fact that it w orks for semilo c al- izing sub categories. It then also w orks for lo calizing sub categories, since, if C is g ener- ated b y an ob j ect A a s a lo calizing sub category , then it is generated by ` n ≤ 0 Σ n A as a semilo calizing sub category . How e v er, in the case of a lo calizing sub catego r y , t here is an alternativ e, m uc h shorter pro of of Theorem 3.7 whic h do es no t require the existence of mo dels. Instead, it is based on Bro wn represen tabilit y . A similar argumen t can b e fo und in [41, Theorem 7.2.1]. Prop osition 3.8. L et T b e a w e l l-gener a te d triangulate d c ate go ry with c opr o ducts. Then every singly gener ate d lo c alizin g sub c ate go ry of T is c or e ﬂe ctive. Pr o of. By [46, Prop osition 8.4.2 ], the category T satisﬁes Bro wn represen tabilit y , and T = ∪ α T α , i.e., ev ery ob ject of T is α -compact for some inﬁnite cardinal α . Let C b e a lo calizing subcatego ry of T generated b y some ob ject A . Then A ∈ T α for some inﬁnite cardinal α . Hence, it follo ws from [46, Corollary 4.4.3] that the V erdier quotien t category T / C has small hom-sets. The existence of a coreﬂection on to C amoun ts to the existence of a right adjoin t to the inclusion C ֒ → T , and this is equiv alen t to the existence of a righ t adjoin t to the functor F : T → T / C (see [46, Prop osition 9.1.1 8 ]). Since T / C has small hom-sets, a right adjoin t G : T / C → T can be deﬁned as follo ws. If X is a n y ob ject of T / C , then GX is obta ined b y Bro wn repres en tabilit y , namely ( T / C )( F ( − ) , X ) ∼ = T ( − , GX ).  Recall from [50, Prop osition 6.10] that, if K is a stable combin atorial mo del category , then Ho( K ) is indeed w ell generated. 3.3. Semilo calizing sub categories are singly generated. W e remark that the wa y in whic h V op ˇ enk a’s principle is used in Theorem 3.9 b elo w is diﬀeren t f r o m the wa y in whic h it was used in Section 2. What w e need here is the fa ct that, b y [2 , Theorem 6.6 and Corollary 6.18], if V opˇ enka’s princi p le holds, then every ful l sub c ate go ry of a lo c al ly pr e- sentable c ate gory close d under λ -ﬁlter e d c olimits for some r e g ular c ar dinal λ is ac c essib l e . The following argumen t was used similarly in [13, Lemma 1.3] and [17 , Lemma 1.3]. Theorem 3.9. L et K b e a stable c ombi natorial mo del c ate gory. If V opˇ enka’s principle holds, then every semilo c alizin g sub c ate gory of Ho( K ) is singly gener ate d and c or eﬂe ctive. Pr o of. First replace K with a Quillen equiv alent stable simplicial combinatorial mo del category , whic h is p ossible according to Prop osition 3.1. L et C b e a semilo calizing sub- category of T = Ho( K ). W rite it as the union of an ascending c hain of full sub catego r ies C = [ i ∈ Ord C i , ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 25 indexed b y the ordinals, where f o r each i there is an ob j ect A i ∈ C suc h that C i = slo c( A i ). Then, by Theorem 3.7, each C i is coreﬂectiv e. Consider the corresponding classes S i = γ − 1 ( C i ), where γ : K → Ho( K ) is the canonical functor. These form an ascending chain of full sub categories o f K . L et S = ∪ i ∈ Ord S i = γ − 1 ( C ). By Corollary 3.5, there is a regular cardinal λ suc h that eac h S i is closed under λ -ﬁltered colimits, and so is S . Since K is lo cally presen table, V op ˇ enk a ’s principle implies that S is acces sible [2, The- orem 6.6 and Corollary 6.18]. Hence, there is a regular cardinal µ , whic h we ma y choo se bigger than λ , and a set X of µ -presen table ob jects in S suc h that ev e ry ob ject of S is a µ -ﬁltered colimit of ob jects from X . Since X is a set, w e hav e X ⊆ S k for some ordinal k . Hence, ev ery ob ject of S is a µ -ﬁltered colimit of ob jects from S k . But the class S k is closed under µ -ﬁltered colimits, since eve ry µ -ﬁltered colimit is also λ -ﬁltered. Therefore, S k = S , that is, the c hain {S i | i ∈ Ord } ev entually stabilizes. Then {C i | i ∈ Ord } also stabilize s, since C i = γ ( S i ) for all i . This pro v es that C = C k for some k , whic h is singly generated and coreﬂectiv e.  Under the assumptions of Theorem 3.9 , eve ry lo calizing sub cat ego ry C is also singly generated, since we may infer from Theorem 3.9 that C = slo c( A ) for some ob ject A , and then C = lo c( A ) as well. It also f o llo ws that, under the assumptions of Theorem 3.9, all semilo calizing sub cate- gories (a nd all lo calizing sub categories) a re closed. If w e assume, in additio n, that T is tensor triangulat ed, and apply Theorem 3.9 to a lo calizing ideal, then the corresp onding coreﬂection C is a colo calizatio n in the sense of [37]; that is, if X is suc h that C X = 0, then C ( F ( E , X )) = 0 for ev ery ob ject E in T . Hence, the question ask ed aft er [37, Lemma 3.6.4 ] of whether all lo calizing ideals are closed ha s an aﬃrmat ive answ er in tensor t riangulated cat ego ries with com binatorial mo dels, ass uming V op ˇ enk a’s principle. 4. Nullity class e s and cohomological B ous fie ld classes It follo ws from Theorem 1.4, Theorem 2.4 a nd Theorem 3.9 that, if V op ˇ enk a’s prin- ciple holds, then in ev ery tr ia ngulated catego r y T with com binatorial mo dels there is a bijectiv e corresp ondence b et w een lo calizing subcategories and colo calizing sub categories. This answ ers aﬃrmativ e ly [48, Problem 7.3 ] under the assumptions made here. In fact, under the same assumptions, there is also a bijectiv e corresp ondence b etw een semilo calizing sub catego r ies and semicolo calizing sub categories. Hence, w e ha v e: Corollary 4 .1. Under V opˇ enka’s principle, every semilo c alizing sub c ate g ory o f a trian- gulate d c ate gory with c ombinatorial mo dels is p art of a t -structur e, and the same happ ens for every semic olo c alizin g sub c ate g ory. Pr o of. As stated in Theorem 1.7, ev ery reﬂectiv e semicolo calizing sub category yields a t -structure, and so do es ev ery coreﬂectiv e semilo calizing sub category . Theorem 2.4 ensures reﬂectivit y of all semicolo calizing sub categories and Theorem 3.9 ensures coreﬂectivit y of all semilo calizing sub categories, under the assumptions made.  ARE ALL LOCALIZIN G SUBCA TEGORIES COR EFLECTIVE? 26 Another consequence of our results is the following. Theorem 4.2. L et T b e a trian g ulate d c ate gory with c ombinatoria l mo dels. Assuming V opˇ enka’s principle, ev ery semic olo c alizing sub c ate gory of T is e qual to E L for some obje ct E and ev ery c olo c alizing sub c ate gory is e qual to E ⊥ for some E . Pr o of. Let L b e a semicolo calizing sub category of T . Theorem 2.4 ensures that L is r eﬂec- tiv e and hence L = ( L L ) L , by Prop osition 1.5. No w consider L L , whic h is a semilo calizing sub category , hence singly generated by Theorem 3.9. That is, L L = slo c( E ) for some E . Consequen tly , L = ( L L ) L = slo c( E ) L = E L b y (1.4), whic h prov es our ﬁrst claim. W e argue in t he same w ay for a colo calizing sub category .  Semicolo calizing sub categories o f the form E L for some ob ject E are called n ul li ty classes , since E L consists of ob jects X that are E -nul l , in the sense that T (Σ k E , X ) = 0 for k ≥ 0 (this terminology is consisten t with [16] or [19], but slightly diﬀers from that used in [53]). Th us, the f ollo wing corollary is a rew ording o f Theorem 4.2 . Corollary 4.3. Assuming V opˇ enka’s p ri n ciple, every semic olo c alizing sub c ate go ry of a triangulate d c ate gory with c omb i natorial mo dels is a nul lity class. It was show n in [53] that there is a prop er class o f distinct nullit y classes E L in the deriv ed catego r y of Z or in the ho mo t op y category of sp ectra. Ho w ev er, it is unkno wn if there is a prop er class or only a set of distinct classe s of the form E ⊥ . The same problem is op en f o r classes of the form ⊥ E . A localizing sub catego ry o f the form ⊥ E f or some ob ject E is called a c oh o m olo gi c al Bousﬁeld class ; cf. [35]. It follow s from Corollary 2.5 that cohomolog ical Bo usﬁeld classes of sp ectra are coreﬂectiv e under V opˇ enk a’s principle —this w as ﬁrst pro v ed in [13], [15 ]. How ev er, w e do not know if eve ry lo calizing sub category of sp ectra is a cohomolo g ical Bousﬁeld class. Indeed, w e could not prov e that colo calizing subcategories are singly generated, not ev en under V op ˇ enk a ’s principle a nd in the presence of combinatorial mo dels. As we next explain, there seems to b e a reason for this. 4.1. T orsion t heories in ab elian c ategories. In an ab elian category , the analogue of a semilo calizing subcategory is a full sub cat ego ry closed under colimits and extensions (this is usually called a torsion class ), a nd the a nalogue of a semicolo calizing sub category is a full sub category closed under limits and extensions (called a torsion-fr e e class ). In w ell-p o w ered ab elian catego r ies, torsion classes are coreﬂectiv e a nd torsion-free classes are reﬂectiv e; see [18]. A torsion class closed under subgroups is called her e ditary . These corresp ond to the lo calizing sub categories. Hereditary torsion classes of mo dules ov er a ring a re singly generated and t heir orthogonal torsion-free classes ar e also singly generated; see [22 ]. In the non-hereditary case, the situation is mor e in triguing. On one hand, under V opˇ enk a’s principle, ev ery tor sion class of ab elian groups is singly generated. 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Are all localizing subcategories of stable homotopy categories coreflective?

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