ON FIBRANT OBJECTS IN MODEL CATEGORIES

Transcription

1 Theory and Applcatons o Categores, ol. 33, No. 3, 2018, pp ON FIBRANT OBJECTS IN MODEL CATEGORIES ALERY ISAE Abstract. In ths paper, we study propertes o maps between brant objects n model categores. We gve a characterzaton o weak equvalences between brant objects. I every object o a model category s brant, then we gve a smple descrpton o a set o generatng cobratons. We show that to construct such a model structure t s enough to check some relatvely smple condtons. 1. Introducton The ramework o model categores, ntroduced n [7], s an mportant tool n homotopy theory. In ths paper, we study propertes o maps between brant objects. We wll show that weak equvalence between brant objects have a smple characterzaton n terms o generatng cobratons (Proposton 3.5), whch s smlar to the characterzaton descrbed n [9]. In partcular, every object o a model category s brant, then we get a complete characterzaton o weak equvalences. We wll prove several propertes o model categores usng ths characterzaton. For example, we wll show that a rght Qullen unctor U : D C relects weak equvalence between brant objects and only F (C) generates the class o cobratons o D n a certan sense, where F s a let adjont to U and C s the class o cobratons o C (Proposton 3.11). Ths proposton s useul snce ths condton on rght Qullen unctors appears n the characterzaton o Qullen equvalences. I we x the class o cobratons, then the class o weak equvalences or whch there exsts model structure n whch all objects are brant s unque t exsts, so ths rases a natural queston: when does such a model structure exst? We prove several necessary and sucent condtons or ths to be true. Examples o applcatons o these constructons are model structures on strct ω-categores (whch was constructed n [5]) and topologcal spaces. Examples o new model structures constructed by means o results o ths paper wll be gven n [4]. I a model structure n whch all objects are brant exsts, then the class o weak equvalences o ths model structure s the smallest class o weak equvalences among model structures wth ths class o cobratons. Thus a model category n whch all objects are brant s let-determnant (as dened n [8]). But the converse s not true. For example, the category o smplcal sets (as well as every Grothendeck topos [1]) Receved by the edtors and, n nal orm, Transmtted by Stephen Lack. Publshed on Mathematcs Subject Classcaton: 55U35. Key words and phrases: Qullen model structures, brant objects. c alery Isaev, Permsson to copy or prvate use granted. 43

2 44 ALERY ISAE carres a let-determnant model structure wth monomorphsms as cobratons, but a model structure on ths category n whch all objects are brant and wth ths class o cobratons does not exst. To see ths, consder a cylnder C or the termnal object. Then C [0] C does not have a map rom [1] whch maps aces o ths smplex to ponts o C whch were not amalgamated. Thus C [0] C cannot be brant. Thus the class o model structures n whch all objects are brant s much narrower than the class o let-determnant model structures. On the other hand model categores n whch all objects are brant have propertes not shared by let-determnant ones. Oten a set o generatng trval cobratons s dened usng cardnalty bounds on domans and codomans o maps. For example, let Bouseld localzatons are oten constructed n ths way. In general, there s no useul explct descrpton o generatng trval cobratons, but all objects o a model category are brant, then we can gve a smple explct constructon o a set o generatng trval cobratons (Proposton 3.6). Also, as we already noted, model categores n whch all objects are brant have a smple characterzaton o weak equvalences n terms o generatng cobratons. Let us dscuss the man obstacles that we ace when we try to construct such model structure n whch all objects are brant. Snce we have an explct characterzaton o weak equvalences n terms o cobratons, we just need to check that the classes o weak equvalences, cobratons and bratons satsy the requred condtons. The rst problem s to prove that ths class o weak equvalences satses the 2-out-o-3 property. The most dcult part s to show that : Y and g : Y Z are maps such that and g are weak equvalences, then g s also a weak equvalence. For example, t s one o the man obstacles n [5] and [4]. Theorem 4.3 mples that we can prove the 2-out-o-3 property or weak equvalences, then to construct the model structure, we just need to prove that the maps n J I -cell are weak equvalences, where J I s the set o generatng cobratons dened n secton 4. A smlar problem occurs when we try to construct a transerred model structure. I F : C D s a let adjont unctor and C s a cobrantly generated model category, then to construct a model structure on D, we just need to prove that maps n F (J)-cell are weak equvalences, where J s a set o generatng trval cobratons o C. Ths condton holds we can construct unctoral path objects or brant objects. Smlar result holds n our case too (Corollary 4.6). Also, analogous result holds we there are cylnder objects nstead o path objects (Corollary 4.9). Our man result gves a necessary and sucent condton or the exstence o a model structure that satses some condtons. Thus we may compare our theorem to other analogous crteron that exst n the lterature. We already mentoned the transerred model structures. Another such result s proved n [10], whch generalzes a theorem o Csnsk proved n [1]. Frst o all, the results o Olschok apples to model categores n whch all objects are cobrant and our results apples to model categores n whch all objects are brant. Snce there are not many model categores n whch both o these condtons hold, the two results rarely can be appled to construct the same model structure. The results o Olschok use Je Smth s theorem, and hence apples only to

3 ON FIBRANT OBJECTS IN MODEL CATEGORIES 45 locally presentable categores. Our results apples to all bcomplete categores and gve a model structure wth explctly descrbed set o generatng trval cobratons. Fnally, the class o weak equvalences s completely determned by the class o cobratons n our settng whle the theorems o Olschok gve us some reedom n choosng the class o weak equvalences. The paper s organzed as ollows. In secton 2, we recall basc propertes o model categores and establsh the notaton that we use n ths paper. In secton 3, we prove several results that are related to brant objects, bratons and trval cobratons between them. Frst o all, we gve a characterzaton o weak equvalences between brant objects. Then we prove that the brant objects are determned by a set S o trval cobratons, then trval cobratons and bratons wth brant codomans are determned by a set o generatng cobratons whch s explctly dened n terms o S. Fnally, we prove a necessary and sucent condton or a rght Qullen unctor to relect weak equvalences between brant objects. In secton 4, we demonstrate a method o constructng model structures n whch all objects are brant. We prove several necessary and sucent condtons or such a model structure to exst. 2. Prelmnares In ths secton, we recall several standard denton and propertes related to model categores. Most o the propostons are ether standard or a varaton o such propostons and wll be gven ether wthout a proo or wth a short proo. The denton that we use are also mostly standard. We want to apply these dentons n categores wthout model structure, so we need a slghtly more general dentons o cylnder and path objects. Let C be a category and let be an object o C. Then a cylnder object or s an object C( ) together wth maps γ 0, γ 1 : C( ). I : U s a morphsm o C, then a relatve cylnder object or C s a cylnder object (C U ( ), γ 0, γ 1 ) or such that γ 0 = γ 1. I C has the ntal object 0, then a relatve cylnder object or 0 s just a cylnder object or. A morphsm o cylnder objects C( 1 ) and C( 2 ) s a par o maps : 1 2 and C() : C( 1 ) C( 2 ) whch commute wth γ 0 and γ 1. Note that we do not requre that there exsts a map s : C( ) such that s γ 0 = s γ 1 = d. We also do not requre maps γ 0, γ 1 to be weak equvalences n any sense. A (let) homotopy (wth respect to C( )) between maps, g : s a map h : C( ) such that h γ 0 = and h γ 1 = g. Maps are homotopc there s a homotopy between them. Maps are homotopc relatve to : U (wth respect to C U ( )) there s a homotopy wth respect to C U ( ) between them. Note that maps are homotopc relatve to only = g. I maps and g are homotopc, then we wrte g, and they are homotopc relatve to, then we wrte g. Note that s relexve and only there exsts a map s : C U ( ) such that s γ 0 = s γ 1 = d. In ths case we wll say that C U ( ) s relexve. Relaton r s relexve and only there exsts a map t : P () such that p 0 t = p 1 t = d. In ths case we wll say that P () s relexve. I P () s relexve, then we say that a rght

4 46 ALERY ISAE homotopy h : P () between, g : s constant on : U h = t. In ths case, we wrte r g and say that and g are homotopc relatve to. Let C be a category and let be an object o C. A path object or s an object P () together wth maps p 0, p 1 : P (). A (rght) homotopy between morphsms, g : s a map h : P () such that p 0 h = and p 1 h = g. We say that and g are rght homotopc and wrte r g there exsts a rght homotopy h : P () between them. A morphsm o path objects P () and P (Y ) s a par o maps : Y and P () : P () P (Y ) whch commute wth p 0 and p 1. Let, Y be objects o a category C and R some relaton on the set C(, Y ). Gven two morphsms : U and g : Y, we say that has the let ltng property (LLP) up to R wth respect to g, and g has the rght ltng property (RLP) up to R wth respect to or every commutatve square o the orm U u h v there s a dotted arrow h : such that h = u and (g h) R v. We say a map has the rght ltng property up to R wth respect to an object t has ths property wth respect to the map 0. Gven a morphsm and a set o morphsms I, the map has the let (rght) ltng property up to R wth respect to I t has ths property wth respect to all morphsms n I. Note that a map has the rght (let) ltng property t has ths property up to the equalty relaton. Oten, we need to talk about maps whch have RLP up to wth respect to some map. In ths case, we wll say that a map has RLP up to relatve homotopy wth respect to. Let R be the maxmal relaton on the set C(, Y ), that s or every 1, 2 : Y, we have 1 R 2. We wll say that g : Y s pure wth respect to : U g has RLP up to R wth respect to. Thus g s pure wth respect to and only every square as above has a lt n whch the top trangle commutes but the lower need not. The noton o pure morphsm s (ormally) smlar to the concept o λ-pure morphsm, used n the theory o accessble categores. We lst a ew elementary propertes o pure morphsms n the ollowng proposton: 2.1. Proposton. The ollowng holds n every category C: R 1. I g has RLP up to some relaton wth respect to, then g s pure wth respect to. 2. Pure maps are closed under composton. 3. I : Y and g : Y Z are maps such that g s pure, then s also pure. 4. Every splt monomorphsm s pure wth respect to all maps. 5. I a map s pure wth respect to tsel, then t s a splt monomorphsm. g Y,

5 ON FIBRANT OBJECTS IN MODEL CATEGORIES 47 Model categores were ntroduced n [7]. For an ntroducton to the theory o model categores see [2, 3]. We wll denote the sets o cobratons, bratons and weak equvalences o a model category by C, F, and W, respectvely. Let C be a model category. Then there s a relexve relatve cylnder object C U ( ) or every map : U o C. There s also a relexve path object P () or every object o C. The ollowng proposton s standard: 2.2. Proposton. I : U s a cobraton and s a brant object, then maps, g : are let homotopc relatve to and only they are rght homotopc relatve to. Ths homotopy relaton s an equvalence relaton. The dentons o a deormaton retract and a strong deormaton retract are standard. A map : Y s an ncluson o a deormaton retract there s a map g : Y such that g = d and g d Y. A map : Y s an ncluson o a strong deormaton retract the homotopy s relatve to. The ollowng lemmas are slght generalzatons o standard propertes o model categores Lemma. [Homotopy extenson property] Let C be a category, and let : U be a morphsm o C. Suppose that or an object o C, there exsts a path object p 0, p 1 : P () such that p 0 has RLP wth respect to. Let : U C, u : U, and v : be maps, and let h : U P () be a homotopy between v and u. Then there exsts a map v : and a homotopy h : P () between v and v such that h = h. Proo. Let h : U P () be a homotopy between v and u. Consder the ollowng dagram: U h P () By assumpton, we have a lt h : P (). Then we can dene v as p 1 h. Ths lemma can be llustrated as ollows: v p 0 U u U u r v v r v I we have a dagram on the let, then we can nd a map v such that the dagram on the rght commutes. Moreover, we restrct the homotopy between v and v on U, then we get the orgnal homotopy between v and u. Let r be the relexve transtve closure o r. Then the prevous lemma also holds or r n place o r :

6 48 ALERY ISAE 2.4. Lemma. Let C be a category that satses condtons o the prevous lemma. Let : U C, u : U, and v : be maps, and let h 1,..., h n : U P () be a sequence o homotopes such that p 1 h j = p 0 h j+1 or every 1 j < n, p 0 h 1 = v and p 1 h n = u. Then there exsts a map v : and a sequence o homotopes h 1,..., h n : P () such that p 1 h j = p 0 h j+1 or every 1 j < n, p 0 h 1 = v and p 1 h n = v and h j = h j or every 1 j n. Proo. Apply the prevous lemma n tmes. Let C be a category and let I be a class o morphsms o C. Then we dene I-nj to be the class o morphsms o C that has RLP wth respect to I, I-co to be the class o morphsms o C that has LLP wth respect to I-nj, and I-cell to be the class o transnte compostons o pushouts o elements o I. Elements o I-cell are called relatve I-cell complexes. Every relatve I-cell complex belongs to I-co. We say that a set I o maps o a cocomplete category C permts the small object argument the domans o maps n I are small relatve to I-cell. The ollowng proposton s standard and appears n ths orm n [3] Proposton. Suppose that C s a complete and cocomplete category, W s a class o morphsms o C, and I, J are sets o morphsms o C. Then C s a cobrantly generated model category wth I as the set o generatng cobratons, J as the set o generatng trval cobratons, and W as the class o weak equvalences and only the ollowng condtons are satsed: (A1) I and J permt the small object argument. (A2) W has the 2-out-o-3 property and s closed under retracts. (A3) I-nj W. (A4) J-cell W I-co. (A5) Ether J-nj W I-nj or I-co W J-co. 3. Propertes o model categores In ths secton, we wll prove varous propertes o model categores that are related to brant objects. In partcular, we wll gve a characterzaton o weak equvalences between brant objects Weak equvalences between brant objects. The ollowng propostons gve useul characterzatons o trval cobratons and weak equvalences between brant objects. The characterzaton or weak equvalences s smlar to the one descrbed n [9].

7 ON FIBRANT OBJECTS IN MODEL CATEGORIES Proposton. Let C be a model category. A cobraton between brant objects s a weak equvalence and only t s an ncluson o a strong deormaton retract. Ths characterzaton o trval cobratons s probably well-known. Smlar propostons are proved n [2], but we could not nd a reerence or ths property, so we nclude a proo or the sake o convenence. Proo. Every ncluson o a deormaton retract becomes an somorphsm n the homotopy category and every such map s a weak equvalence. To prove the converse, let us show that every weak equvalence wth a brant doman s pure wth respect to cobratons. Ths ollows rom the act that every weak equvalence can be actored nto a trval cobrant and a trval braton. Every trval braton s pure wth respect to cobratons, and every trval cobraton : Y wth a brant doman s a splt monomorphsm snce we have a lt n the dagram below, hence t s pure wth respect to any map. d Y Now, let : Y be a trval cobraton between brant objects. Snce s brant, has a retracton g : Y. By the 2-out-o-3 property, g s a weak equvalence. Snce Y s brant, g s pure wth respect to cobratons. Consder the ollowng dagram: Y Y [ g,d Y ] Y [γ 0,γ 1 ] C (Y ) Snce g s pure wth respect to cobratons, we have a lt, whch gves us a homotopy between g and d Y. Now, we need to prove a techncal lemma, whch we ormulate n a general orm snce we wll need t later Lemma. Let C be a ntely cocomplete category, and let : U and g : Y Z be maps o C. Let C U ( ) be a relatve cylnder object or. Suppose that or every A {Y, Z}, there exsts a path object p 0, p 1 : P (A) A, whch satsy the ollowng condtons: 1. For every A {Y, Z}, p 0 : P (A) A has RLP wth respect to and t s pure wth respect to [γ 0, γ 1 ] : U C U ( ). 2. There exsts a morphsm o path objects (g, P (g)) : P (Y ) P (Z). 3. Ether p 1 : P (Z) Z has RLP wth respect to or there exsts a map s : P (Z) P (Z) such that p 0 s = p 1 and p 1 s = p 0. Let : Y be a map o C such that has RLP up to r wth respect to U, and g has RLP wth respect to up to. Then g also has RLP wth respect to up to. g s g

8 50 ALERY ISAE Proo. Suppose that we have a commutatve square as below. Then there exsts a map u x : U such that u x r u whch means that there s a sequence o homotopes h 1,..., h n : U P (Y ) or some n such that p 0 h 1 = u x, p 1 h n = u, and p 1 h j = p 0 h j+1 or every 1 j < n. U u x u v r Y Z g We can dene a sequence o homotopes h 1 u,..., h n u between g u x and v. I p 1 has RLP wth respect to, then let h j u = P (g) h j ; otherwse let h j u = s s P (g) h j. By Lemma 2.4, there exsts a map v z : Z and a sequence o homotopes h 1 3,..., h n 3 : P (Z) between v z and v such that h j 3 = h j u. Indeed, p 1 has RLP wth respect to, then we can apply Lemma 2.4 to p 1, p 0 : P (Z) Z. I we dened h j u as s s P (g) h j, then we can apply Lemma 2.4 to s P (g) h j to get a sequence o homotopes h 1 4,..., h n 4 between v and v z. Then we can dene h j 3 as s h j 4. By assumpton on g, there exsts a map v x : such that g v x and v z are relatvely homotopc. Note that h 1,..., h n s a sequence o homotopes between u x = v x and u. Thus, by Lemma 2.4, we have a map v y : Y and a sequence o homotopes h 1 y,..., h n y : P (Y ) between v x and v y such that h j y = h j. In partcular, v y = u. Thus we only need to prove that g v y and v are homotopc relatve to. I h j u = P (g) h j, then let h j 1 = P (g) h j y. I h j u = s s P (g) h j, then let h j 1 = s s P (g) h j y. Then h j 1 = h j u. Thus we have a sequence o maps [h j 1, h j 3] : U P (Z). Now, let us show that p 1 h j 1 p 1 h j 3 whenever p 0 h j 1 p 0 h j 3. Indeed, h 0 s a relatve homotopy between p 0 h j 1 and p 0 h j 3, then consder the ollowng dagram: U [h j 1,hj 3 ] P (Z) [γ 0,γ 1 ] C U ( ) h 0 Z p 0 Snce p 0 s pure wth respect to [γ 0, γ 1 ], we have a lt q : C U ( ) P (Z). Then p 1 q s a relatve homotopy between p 1 h j 1 and p 1 h j 3. Fnally, to complete the proo we need to show that g v y and v are relatvely homotopc. Snce g v y = p 1 h n 1 and v = p 1 h n 3, the prevous observaton mples that t s enough to prove that p 0 h 1 1 and p 0 h 1 3 are relatvely homotopc. But p 0 h 1 1 = g v x and p 0 h 1 3 = v z and these maps are relatvely homotopc by denton o v x.

9 ON FIBRANT OBJECTS IN MODEL CATEGORIES 51 I I s a class o maps o a category C, then let us denote by J I the class o maps γ 0 : C U ( ) or each U I. Ths notaton s slghtly abusve snce the class J I depends on the choce o cylnder objects C U ( ). I the cylnder objects are not speced explctly n some statement, then t holds or any choce o these objects. Now, we wll show that I generates the class o cobratons o a model category n whch all objects are brant, then a map s a weak equvalence and only t has RLP up to relatve homotopy wth respect to I. Thus the ollowng proposton mples that a map o such a model category s a weak equvalence and belongs to J I -nj, then t s a trval braton. Ths mples that J I can be used as a set o generatng trval cobratons t permts the small object argument (see Proposton 3.6) Proposton. Let C be a category, and let I be a class o maps o C. I : Y J I -nj has RLP up to relatve homotopy wth respect to I, then I-nj. Proo. Suppose we have a commutatve square as below. We need to nd a lt such that both trangles commute. U I u g v Y By assumpton, we have a lt g : together wth a relatve homotopy h : C U ( ) Y between g and v. Snce has RLP wth respect to γ 0, we have a lt h : C U ( ) n the ollowng dagram: γ 0 C U ( ) h Then h γ 1 s a requred lt n the orgnal square. Indeed, the top trangle commutes because h γ 1 = h γ 0 = g = u and the bottom trangle commutes because h γ 1 = h γ 1 = v. Now we assume that C s a model category and C U ( ) s a correct relatve cylnder object, that s t s a actorzaton o the map [d, d ] : U nto a cobraton [γ 0, γ 1 ] : U C U ( ) ollowed by a weak equvalence C U ( ). The ollowng propostons gves a characterzaton o weak equvalences between brant objects Proposton. Let C be a model category, and let I be a class o maps o C whch generates the class o cobratons (that s, the class o cobratons equals to I-co). Let : Y be a morphsm o C such that and Y are brant. Then the ollowng condtons are equvalent: 1. s a weak equvalence. 2. has RLP up to relatve homotopy wth respect to cobratons. g h Y.

10 52 ALERY ISAE 3. has RLP up to relatve homotopy wth respect to I. Proo. (1 2) Consder the ollowng commutatve square n whch c s a cobraton: U c Y Factor nto a trval cobraton : Z ollowed by a trval braton p : Z Y. Snce c s a cobraton and p s a trval braton, we have a lt q : Z. Snce s a trval cobraton between brant objects, by Proposton 3.2, t has a retracton r : Z such that r d Z. c U q r Z p Y Z Then r q s a requred lt. (2 3) Obvous. (3 1) Factor nto a trval cobraton : Z ollowed by a braton g : Z Y. Let us prove that has RLP up to r wth respect to every object. By Proposton 3.2, there exsts a map g : Z and a homotopy g d Z. By Proposton 2.2, we have a rght homotopy h : Z P (Z) between g and d Z. For every morphsm u : U Z we can dene a lt u : U Y as g u. Then h u s a homotopy between u and u. Snce Z and Y are brant, condtons o Lemma 3.3 are satsed. Hence, g has RLP up to relatve homotopy wth respect to I. Snce J I conssts o trval cobratons, by Proposton 3.4, g s a trval braton. Thus s a weak equvalence by the 2-out-o-3 property. Oten the class o cobratons s generated by a much smaller class I. Thus the last condton n the prevous proposton gves us a useul characterzaton o weak equvalences between brant objects whch s easy to very n practce. In partcular, every object n a model category s brant, then ths proposton gves us a complete characterzaton o weak equvalences, whch we wll use n the next secton. The equvalence 1 3 s well-known or topologcal spaces. A varaton o ths proposton s proved n [9], where a map between brant objects s proved to be a weak equvalence and only t satses a slghtly stronger verson o the second condton o Proposton 3.5. Let C be a model category n whch every object s brant. I C has a set o generatng cobratons I whch satses a mld addtonal hypothess, then usng the characterzaton o weak equvalence that we gave n Proposton 3.5, we can construct a set o generatng

11 ON FIBRANT OBJECTS IN MODEL CATEGORIES 53 trval cobratons or ths model structure. Ths shows that the model category s cobrantly generated n ths case, and the set o generatng trval cobratons has a smple explct descrpton n terms o generatng cobratons Proposton. Let C be a model category n whch every object s brant. Suppose that the class o cobratons s generated by a set I such that the domans and the codomans o maps n I are small relatve to I-cell. Then the model structure s cobrantly generated. Proo. By the small object argument, there exsts a relexve relatve cylnder object C U ( ) such that [γ 0, γ 1 ] : U C U ( ) s a cobraton and the map C U ( ) s a trval braton. We prove that J I s a set o generatng trval cobratons. Let us check the condtons o Proposton 2.5: (A1) The set I permts the small object argument by assumpton. By [3, Proposton ], the codomans o maps n I are small relatve to I-co. Snce maps n J I are cobratons (see (A4) below) and the domans o maps n J I are the codomans o maps n I, the set J I also permts the small object argument. (A2) The class o weak equvalences s closed under retracts and satses 2-out-o-3 snce C s a model category. (A3) Snce I generates cobratons, the class I-nj conssts o trval bratons. (A4) A map γ 0 : C U ( ) n J I s the composton o maps U C U ( ), where the rst map s a pushout o a cobraton U and the second map s a cobraton by assumpton. Moreover, the map C U ( ) s a weak equvalence; hence, γ 0 s also a weak equvalence by 2-out-o-3. (A5) By Proposton 3.4 and Proposton 3.5, J I -nj W, then I-nj Trval cobratons and bratons wth brant codomans. I some o the objects o a model category are not brant, then sometmes we can characterze brant objects as those whch have the RLP wth respect to some set o trval cobratons S whch s consderably smaller than a set o generatng cobratons. For example, the Joyal model structure on smplcal sets has weak Kan complexes as brant objects, whch are smplcal sets that have RLP wth respect to nner horns. But nner horns do not generate the whole class o trval cobratons. Another example s the category o marked smplcal sets constructed n [6, Proposton ]. Fnally, [2, Lemma ] mples that a let Bouseld localzaton o a model category also has an explct descrpton o a set o maps characterzng brant objects provded the orgnal model category has such a descrpton. Let S be a set o trval cobratons such that an object s brant and only t has the RLP wth respect to S. We should not expect that there s always a smple explct

12 54 ALERY ISAE descrpton o a set o generatng trval cobratons n terms o S. But we can show that the class (J I S)-co contans all trval cobratons wth brant codomans. Moreover, a map wth a brant codoman s a braton and only t has the RLP wth respect to ths set. For smplcal sets wth the Joyal model structure, (a slghtly stronger verson o) ths result was proved n [6, Corollary ] (where t s attrbuted to Joyal) Proposton. Let C be a model category. Let I be a set o generatng cobratons, and let S be a set o trval cobratons such that every object that has the RLP wth respect to S s brant. Suppose that the domans and the codomans o maps n I and the domans o maps n S are small relatve to I-cell. Then a map wth a brant codoman s a trval cobraton and only t belongs to (J I S)-co. A map wth a brant codoman s a braton and only t belongs to (J I S)-nj. Proo. Maps n S are trval cobratons by assumpton and maps n J I are trval cobratons by (A4) n the proo o Proposton 3.6. Thus maps n (J I S)-co are trval cobratons and bratons belong to (J I S)-nj. Let : Z be a trval cobraton such that Z s brant. Factor nto a map g : Y (J I S)-cell ollowed by a map h : Y Z (J I S)-nj. g Y Z By the 2-out-o-3 property, h s a weak equvalence. Snce t s a weak equvalence between brant objects, by Proposton 3.5, t has RLP up to relatve homotopy wth respect to I. By Proposton 3.4, I-nj. Hence, we have a lt Z Y as shown n the dagram above. Thus s a retract o g and t belongs to (J I S)-co. Let : Y be a map n (J I S)-nj such that Y s brant. To prove that t s a braton, we need to show that t has RLP wth respect to every trval cobraton. Let : U be a trval cobraton, and let u : U and v : Y be maps such that the obvous square commutes. Factor v nto a map Z S-cell ollowed by a map Z Y S-nj. u U Z h Z Y Snce Z s brant and the map U Z s a trval cobraton, t belongs to (J I S)-co. Snce belongs to (J I S)-nj, we have a lt Z n the dagram above. Thus has RLP wth respect to.

13 ON FIBRANT OBJECTS IN MODEL CATEGORIES Qullen equvalences. Let U : D C be a rght Qullen unctor wth let adjont F : C D. Then F U s a Qullen equvalence and only U relects weak equvalences between brant objects and or every cobrant, map η UF () U(r F ()) URF () s a weak equvalence, where η s the unt o the adjuncton and r F () s a brant replacement or F (). In ths subsecton, we gve an equvalent condton or the rst part o ths characterzaton, whch s easer to very sometmes. We prove that U relects weak equvalences between brant objects and only F (C C ) generates C D n a certan sense, where C C and C D are classes o cobratons o C and D respectvely. To descrbe ths characterzaton, we need to ntroduce a bt o notaton. Ths notaton s smlar to the usual notaton related to ltng propertes, but we work wth ltng propertes up to a homotopy relaton. Let I be a class o cobratons o some model category C. Then we dene I-nj h as the set o maps o C between brant objects whch have RLP up to relatve homotopy wth respect to I. We dene I-co h as the set o cobratons o C whch have LLP up to relatve homotopy wth respect to I-nj h. We wll call elements o I-co h weak I-cobratons. As usual, we have the ollowng propertes: I I-co h, I-co h -nj h = I-nj h, I-co h -co h = I-co h, I-nj I-nj h, and I J, then J-nj h I-nj h and I-co h J-co h. Proposton 3.5 mples that C-nj h = W and I-co = C, then I-nj h = W and I-co h = C. The ollowng lemma s analogous to [3, Lemma 2.1.8]: Lemma. Let U : D C be a rght Qullen unctor wth let adjont F : C D. Then the ollowng are true: 1. I s a cobraton n C and g s a map between brant objects n D, then F () has LLP up to relatve homotopy wth respect to g and only has ths property wth respect to U(g). 2. U(F I-nj h ) I-nj h or every class o cobratons I. 3. F (I-co h ) F I-co h or every class o cobratons I. Proo. The proo s dentcal to the proo o the analogous lemma or ordnary ltng propertes. 1. Snce F s a let Qullen unctor, t preserves relatve cylnder objects, and hence also relatons or every cobraton. Now, the statement holds by adjuncton as usual. 2. Let g F I-nj h and I. Then g has RLP up to F () wth respect to F (), and by (1), U(g) has RLP up to wth respect to. Thus U(g) I-nj h, as requred. 3. Let I-co h and g F I-nj h. Then by (2), U(g) has RLP up to wth respect to. By (1), g has RLP up to F () wth respect to F (). Thus F () F I-co h.

14 56 ALERY ISAE Now, we can prove the characterzaton o Qullen equvalences. Sometmes ths condton s easer to very. For example, F (C C )-co contans cobratons o D, then ths condton holds mmedately Proposton. Let U : D C be a rght Qullen unctor wth a let adjont F : C D. Then U relects weak equvalences between brant objects and only C D F (C C )-co h. Proo. Let : Y be a map o D such that and Y are brant. Then U() s a weak equvalence and only U() C C -nj h by Proposton 3.5. But U() C C -nj h and only F (C C )-nj h by the prevous lemma. Thus we only need to prove that F (C C )-nj h W D and only C D F (C C )-co h. But ths ollows rom the acts that W D = C D -nj h and C D -co h = C D whch hold by Proposton Corollary. A Qullen adjuncton F U s a Qullen equvalence and only the map URF () s a weak equvalence or every cobrant and C D F (C C )-co h Propertes o weak I-cobratons. Just as ordnary cobratons, the class o weak I-cobratons s closed under retracts as we wll see n the proposton below. But t s also closed under weaker noton o retracts that we now ntroduce. Let : Y and g : Z W be maps o some model category. Then we say that s a weak retract o g there exsts a dagram o the ollowng orm: g Z Y W Y where the composton o the top row s the dentty morphsm and the composton o the bottom row s a trval cobraton. In order to use Corollary 3.12, we need to be able to construct weak I-cobratons. The ollowng proposton gves us several useul closure propertes Proposton. For every model category C and every class o cobratons I n t, the class o weak I-cobratons s closed under pushouts, transnte compostons, and weak retracts. Proo. Let : U be a weak I-cobraton. Let : U be a pushout o, and let : Y be a map n I-nj h. Then consder the ollowng dagram: U U Y Snce I-co h, there exst a map v : and a homotopy h : P (Y ) between v and Y constant on. Then by the unversal property o pushouts, there

15 ON FIBRANT OBJECTS IN MODEL CATEGORIES 57 exst maps v : and h : P (Y ) such that v s v, U v s U, h P (Y ) s h, and h s the constant homotopy: U U v r Y Thus has LLP up to wth respect to. Now, let us show that weak I-cobratons are closed under transnte compostons. Let : λ C be a λ-sequence such that the maps x α : α α+1 are weak I- cobratons. Consder the ollowng commutatve square: u 0 Y λ v Z where Y and Z are brant and I-nj h. We construct partal unctons g and h rom λ + 1 to the class o morphsms o C such that g α : α Y and h α : α P (Z) by transnte recurson on α. Let g 0 = u, and let h 0 be the constant homotopy on u. The maps g α+1 and h α+1 are dened whenever g α and h α are dened and h α s a homotopy between g α and v x α,λ (where x α,λ s the map α λ ). By the homotopy extenson property or rght homotopes n model categores, there exst a map v : α+1 Z such that v x α = g α and a homotopy h : α+1 P (Z) between v and v x α+1,λ such that h x α = h α : g α α Y x α v α+1 v x α+1,λ Snce x α s a weak I-cobraton, there exst a map g α+1 : α+1 Y such that g α+1 x α = g α and a homotopy h : C α ( α+1 ) Z between g α+1 and v. Consder the ollowng dagram: h α+1 P (Z) γ 1 C α ( α+1 ) h Z h,v x α+1,λ s Z Z Snce γ 1 s a trval cobraton and P (Z) Z Z s a braton, there exsts a lt h n ths dagram. We dene h α+1 as h γ 0. Note that h α+1 s a homotopy between g α+1 and v x α+1,λ. Moreover, h α+1 x α = h γ 0 x α = h γ 1 x α = h x α = h α.

16 58 ALERY ISAE I α s a lmt ordnal, the maps g α and h α are dened and only g β+1 x β = g β and h β+1 x β = h β or every β < α. I ths holds, then the maps {g β } β<α and {h β } β<α determne cones or α λ C, so we can dene g α and h α by the unversal property o colmts. It s easy to see by nducton on α that maps g α and h α are always dened and commute wth the maps x α. In partcular, we have a map g λ : λ Y such that g λ x 0,λ equals to u. Moreover, the map h λ : λ P (Z) s a relatve homotopy between g λ and v. Fnally, let us prove that I-cobratons are closed under weak retracts. Suppose that : U s a weak retract o a weak I-cobraton : Y. Then there exst maps Y and Y such that Y s a trval cobraton. Let g : Z W I-nj h, u : U Z, and v : W be maps such that v = g u. Snce W s brant, v actors through Y. Thus we have the ollowng dagram: U U u Z Y W Snce s a weak I-cobraton, there exsts a lt Y Z such that Y Z g W s relatvely homotopc to Y W. Then Y Z s the requred lt n the orgnal square. We can also prove the ollowng standard proposton: Proposton. Let I be a set o maps n a model category admttng the small object argument. Then every weak I-cobraton s a weak retract o a relatve I-cell complex. Proo. Let : Y be a weak I-cobraton. Let r Y : Y R(Y ) be a brant replacement or Y. Factor r Y nto a relatve I-cell complex Z ollowed by a map Z R(Y ) I-nj. Snce Z R(Y ) s a weak equvalence between brant objects, we have a lt n the ollowng dagram: g Z Y r Y R(Y ) Thus s a weak retract o Z. 4. Exstence o model structures In ths secton, we wll gve necessary and sucent condtons or the exstence o a model structure n whch all objects are brant. Throughout ths secton let C be a xed complete and cocomplete category and I a set o maps o C such that the domans and the codomans o maps n I are small relatve to I-cell. Suppose that or every

17 ON FIBRANT OBJECTS IN MODEL CATEGORIES 59 map : U n I, there exsts a relexve relatve cylnder object C U ( ) such that [γ 0, γ 1 ] : U C U ( ) I-co. Note that such a cylnder object always exsts by the small object argument, but sometmes we can choose another object whch s more convenent to work wth. Let J I = { γ 0 : C U ( ) : U I }, and let W I be the set o maps whch have RLP up to relatve homotopy wth respect to I. We wll consder the ollowng condtons: For every composable J I -cell and g, g W I, then g W I (*) For every composable J I -cell W I and g, g W I, then g W I (* ) Frst, let us prove a general lemma about splt monomorphsms. It seems that t should be well-known, but we could not nd a proo o ths result n the lterature, so we nclude t here or the sake o convenence Lemma. Splt monomorphsms are closed under pushouts, retracts, and transnte compostons. Proo. The only nontrval statement s the last part. To prove that splt monomorphsms are closed under transnte compostons, consder a λ-sequence : λ C such that the maps x α : α α+1 are splt monomorphsms. We construct a partal uncton r rom λ+1 to the class o morphsms o C such that r α : α 0 by transnte recurson on α. Let r 0 = d 0. Let r α+1 = r α x α whenever r α s dened, where x α : α+1 α s a splttng o x α. Fnally, α s a lmt ordnal and r β+1 x β = r β or every β < α, then the maps r β determne a cone or α λ C. Snce α s a colmt o ths dagram, we can dene r α by the unversal property these condtons hold; otherwse, r α s undened. It s easy to see by nducton on α that maps r α are always dened and commute wth the maps x α. In partcular, we have a retracton r λ : λ 0 o the map 0 λ. Now, we can prove the man techncal lemma o ths secton: 4.2. Lemma. I condton (*) holds, then the ollowng are true: 1. Every weak equvalence actors nto a map n J I -cell ollowed by a map n I-nj. 2. Every weak equvalence has RLP up to relatve homotopy wth respect to I-co. 3. For every : Y and g : Y Z, g W I and g W I, then W I. 4. For every : Y and g : Y Z, W I and g W I, then g W I. 5. J I -cell W I.

18 60 ALERY ISAE Proo. Let : Z be a weak equvalence. Factor nto maps : Y J I -cell and g : Y Z J I -nj. By assumpton, g W I. By Proposton 3.4, g I-nj. Ths proves (1). Ths mples that weak equvalences are pure wth respect to cobratons. Indeed, maps n I-nj are pure wth respect to cobratons and maps n J I -cell are pure wth respect to all maps snce they are splt monomorphsms. The last statement ollows rom Lemma 4.1 snce the maps γ 0 : C U ( ) are splt monomorphsms by assumpton. Now, let us prove (2). Snce every weak equvalence actors nto a map n J I -cell ollowed by a map n I-nj, to prove that every weak equvalence has RLP up to wth respect to every cobraton : U, t s enough to show that every map n J I -cell has ths property. Let : Y be a map n J I -cell. It has a retracton g : Y whch s a weak equvalence by condton (*). Hence, g s pure wth respect to cobratons. Let u : U and v : Y be maps such that the obvous square commutes. Consder the ollowng dagram: U [γ 0,γ 1 ] C U ( ) [ g v,v] Snce g s pure wth respect to cobratons, we have a relatve homotopy between g v and v. Thus g v s a requred lt n the orgnal square: g v s U u g v v Now, let us prove (3). Let : Y and g : Y Z be maps such that g W I and g W I. Consder the ollowng dagram: Y U u Y g v Y g Z Snce g W I, we have a lt q : and a homotopy h : C U ( ) Z between g q and g v. Consder the ollowng dagram: U [ q,v] Y [γ 0,γ 1 ] C U ( ) h g Z

19 ON FIBRANT OBJECTS IN MODEL CATEGORIES 61 Snce g s pure wth respect to cobratons, we have a lt h : C U ( ) Y whch gves us a homotopy between q and v. Now, let us show that every : Y J I -cell s a weak equvalence. Let g : Y be a retracton o. Snce g = d, the retracton g s a weak equvalence by (*). By (3), s a weak equvalence too. Fnally, let us prove that weak equvalences are closed under compostons. To do ths, t s enough to show that relaton s transtve. Let h 0 : C U ( ) be a homotopy between : and :, and let h 1 : C U ( ) be a homotopy between : and :. Then consder the ollowng dagram: γ 1 C U ( ) γ 0 C U ( ) p 1 Z s p 0 γ 1 C U ( ) Snce p 0 J I -cell, q s a weak equvalence. Consder the ollowng dagram: q U [p 0 γ 0,p 1 γ 1 ] Z [γ 0,γ 1 ] C U ( ) C U ( ) Snce q s pure wth respect to cobratons, we have a lt h : C U ( ) Z. Then [h 0, h 1 ] h s a homotopy between and Theorem. Let C be a bcomplete category, and let I be a set o maps o C such that the domans and the codomans o maps n I are small relatve to I-cell. Suppose that or every map : U n I, there exsts a relexve relatve cylnder object C U ( ) such that [γ 0, γ 1 ] : U C U ( ) I-co. Then there exsts a model structure on C wth I-co as the class o cobratons and J I -co as the class o trval cobratons and only condton (* ) holds. Proo. Frst, suppose that such a model structure on C exsts. Then every object s brant (snce J I -co are splt monomorphsms by Lemma 4.1) and C U ( ) s a correct cylnder object. By Proposton 3.5, W I s the class o weak equvalences. Hence, condton (* ) holds. Now, suppose that condton (* ) holds. Let us very the condtons o Proposton 2.5: (A1) Snce the domans o maps n J I are the codomans o maps n I, the classes I and J I permt the small object argument by assumpton. q

20 62 ALERY ISAE (A2) The closure o W I under retracts s obvous. One part o the 2-out-o-3 property holds by assumpton and the other two parts ollow rom Lemma 4.2. (A3) Snce C U ( ) s relexve, a map has RLP wth respect to every map n I, t also has RLP up to relatve homotopy wth respect to these maps. (A4) Maps n J I are cobratons by denton. Maps n J I -cell are weak equvalences by Lemma 4.2. (A5) Proposton 3.4 mples that a map s a weak equvalence and has RLP wth respect to J I -nj, then t has RLP wth respect to I. I maps n I satsy some mld addtonal assumptons, then we can smply the condton n Theorem 4.3: 4.4. Proposton. Suppose that the domans o maps n I are cobrant. Then condtons (*) and (* ) are equvalent. Proo. Condton (* ) obvously mples condton (*). Let us prove the converse. Let : Y and g : Y Z be maps such that W I and g W I. By Lemma 4.2, we can actor nto maps : J I -cell and g : Y I-nj. Snce J I -cell and g g = g W I, condton (*) mples that g g W I. Consder the ollowng dagram: U u Y u v Snce U s cobrant, we have a lt u : U. Snce g g W I, we have a lt v : such that g g v v. Then g v s a requred lt n the orgnal square. Thus the man problem s to very condton (*). There are a ew ways to do ths, but the dea s the same: we need to assume that there exsts some noton o homotopy on sets o maps whch satses some condtons. There are two standard ways to do ths: usng path and cylnder objects. Now, we present ths constructons. Note that we use very weak notons o path and cylnder objects (see secton 2) Proposton. Condton (*) holds and only or every object, there exsts a path object P () such that the ollowng condtons hold: 1. For every : Y, there exsts a morphsm o path objects (, P ()) : P () P (Y ), Z g g

21 ON FIBRANT OBJECTS IN MODEL CATEGORIES Ether p 1 has RLP wth respect to I or there exsts a map s : P () P () such that p 0 s = p 1 and p 1 s = p p 0 has RLP wth respect to I. 4. Ether path objects are relexve and maps p 0, p 1 : P () have RLP wth respect to J I or maps n J I -cell have RLP up to r wth respect to the domans o maps n I. Proo. I condton (*) holds, then we can dene path objects as usual usng a actorzaton o the dagonal nto maps P () J I -cell and P () J I -nj. The second and the thrd condtons ollow rom Lemma 4.2 and Proposton 3.4. The rst condton s obvous snce P () can be constructed as a lt n the ollowng dagram: Y P (Y ) P () Y Y The last condton s also obvous snce the path object s relexve and p 0, p 1 J I -nj by constructon. Let us prove the converse. I the second opton o (4) holds, then condton (*) holds by Lemma 3.3. Thus we only need to prove that the rst opton o (4) mples the second. Indeed, let : Y be a map n J I -cell, and let g : Y be ts retracton. Consder the ollowng dagram: Y t g,d Y P (Y ) Y Y p 0,p 1 We have a lt h : Y P (Y ) whch gves us a rght homotopy between g and d Y. Now, or every u : U Y, we can dene a lt u = g u : U and a homotopy h u between u and u. Thus has RLP up to r wth respect to any object Corollary. Let C be a category and let I be a set o maps o C such that the domans o maps n I are cobrant and the condtons o Theorem 4.3 are satsed. Then there exsts a model structure on C wth I-co as the class o cobratons and J I -co as the class o trval cobratons and only there exst path objects that satsy the condtons o Proposton Example. An example o a model category dened n ths way s a olk model structure on the category o ω-categores whch was constructed n [5]. The condtons o Proposton 4.5 ollow rom the results o [5], but some o them are not needed or ths proposton. Thus the constructon o ths model structure can be somewhat smpled usng the general results o ths secton.

22 64 ALERY ISAE Instead o path objects, we could try to use cylnder objects to very condton (*). The advantage o ths approach s that we do not need to dene a cylnder object or every object o the category, only or objects that are domans and codomans o generatng cobratons. The dsadvantage s that we stll need to very that maps n J-cell has RLP up to wth respect to the domans o generatng cobratons. Recall that a cylnder object or an object s any object C() together wth maps γ 0, γ 1 : C(). We do not requre C() to be a proper cylnder object. That s, maps C() may not be weak equvalences and a map C() may not exst at all Proposton. Suppose that, or every object whch s ether the doman or the codoman o a map n I, there exsts a cylnder object C() such that the ollowng condtons hold: 1. For every : U I, there exsts a morphsm o cylnder objects (, C()) : C(U) C( ). 2. There exsts a map s : C() C() such that s 0 = 1, s 1 = 0, and C() s = s C(). 3. These cylnder objects satsy the homotopy extenson property. That s, : U C, u : U and v : are maps, and h : C(U) s a homotopy between v and u, then there exsts a map v : and a homotopy h : C( ) between v and v such that h = h C(). 4. Maps n J I -cell have RLP up to wth respect to the domans o maps n I. 5. For every : U I, we have a lt p n the ollowng dagram: U [γ 0,γ 1 ] C U ( ) p T where T = C U ( ) ( U ) (C( ) C(U) C( )) s the pushout o maps [γ 0, γ 1 ] : U C U ( ) and γ 0 γ0 γ 0 : U C( ) C(U) C( ), and : U T s the composte U γ 1 γ1 γ 1 C( ) C(U) C( ) T. Then condton (*) holds. I the domans o maps n I are cobrant, then the converse s true. Proo. Assume that condtons (1)-(5) hold. Then the proo o (*) s smlar to the proo o Lemma 3.3. Suppose that we have a commutatve square as below, where J I -cell

23 ON FIBRANT OBJECTS IN MODEL CATEGORIES 65 and g W I. By condton (4), there exsts a map u x : U and a sequence o homotopes h 1,..., h n : C(U) Y between u x and u. U u x u v Then we have a sequence o homotopes g h 1 s,..., g h n s between v and g u x. By the homotopy extenson property, there exsts a map v z : Z and a sequence o homotopes h 1 3,..., h n 3 : C( ) Z between v and v z such that h j 3 C() = g h j s. Snce g W I, there exsts a map v x : and a homotopy h 2 : C U ( ) Z between g v x and v z. Note that h 1 s s,..., h n s s s a sequence o homotopes between u x = v x and u. Thus, by the homotopy extenson property, we have a map v y : Y and a sequence o homotopes h 1 y,..., h n y : C( ) Y between v x and v y such that h j y C() = h j s s. In partcular, v y = u. Thus we only need to prove that g v y and v are homotopc relatve to. I we let h j 1 = g h j y, then h j 1 C() = g h j s s = h j 3 s C(). Thus we have a sequence o maps [h j 1, h j 3 s] : C( ) C(U) C( ) Z. I [h j 1, h j 3 s] (γ 0 γ0 γ 0 ) : U Z extends to C U ( ), then [h j 1, h j 3 s] (γ 1 γ1 γ 1 ) also extends to C U ( ). Indeed, [h j 1, h j 3 s] (γ 0 γ0 γ 0 ) = h 0 [γ 0, γ 1 ] or some h 0 : C U ( ) Z, then there s a map q : T Z constructed by the unversal property o the pushout T. Then q p : C U ( ) Z s an extenson o [h j 1, h j 3 s] (γ 1 γ1 γ 1 ), where p : C U ( ) T s the map rom condton (5). Fnally, note that h 2 s an extenson o [g v x, v z ] = [h 1 1, h 1 3 s] (γ 0 γ0 γ 0 ). It ollows that we have an extenson o [h n 1, h n 3 s] (γ 1 γ1 γ 1 ) = [g v y, v], whch denes a relatve homotopy between g v y and v. Now, let us assume that the domans o maps n I are cobrant and condton (*) holds. Then Theorem 4.3 and Proposton 4.4 mply that we have a structure o a model category. Thus the condtons (1)-(4) are obvous. Let us prove condton (5). Consder the ollowng dagram: Y Z g U C U ( ) U γ 1 γ1 γ 1 γ 0 γ0 γ 0 C( ) C(U) C( ) T U C U ( ) C U ( ) C U ( )

24 66 ALERY ISAE Snce the map γ 0 γ0 γ 0 : U C( ) C(U) C( ) s a trval cobraton, the map C U ( ) T s also a trval cobraton. It ollows that the map T C U ( ) s a weak equvalence, whch mples that t has the RLP wth respect to cobratons up to a relatve homotopy. Snce the horzontal map U T n the dagram s, we have a map p : C U ( ) T such that U C U ( ) p T equals to Corollary. Let C be a category and let I be a set o maps o C such that the domans o maps n I are cobrant and the condtons o Theorem 4.3 are satsed. Then there exsts a model structure on C wth I-co as the class o cobratons and J I -co as the class o trval cobratons and only there exst cylnder objects that satsy the condtons o Proposton Example. An example o a model category dened n ths way s the usual model structure on topologcal spaces. I we dene C() as the usual cylnder [0, 1], then the condtons o Proposton 4.8 are easy to very drectly. Reerences [1] Dens-Charles Csnsk, Thores homotopques dans les topos, Journal o Pure and Appled Algebra 174 (2002), no. 1, [2] P.S. Hrschhorn, Model categores and ther localzatons, Mathematcal Surveys and Monographs, Amer Mathematcal Socety, [3] M. Hovey, Model categores, Mathematcal Surveys and Monographs, Amercan Mathematcal Socety, [4]. Isaev, Model structures on categores o models o type theores, (2016), arv: [5] Yves Laont, Franços Métayer, and Krzyszto Worytkewcz, A olk model structure on omega-cat, Advances n Mathematcs 224 (2010), no. 3, [6] Jacob Lure, Hgher topos theory, Annals o mathematcs studes, Prnceton Unversty Press, Prnceton, N.J., Oxord, [7] Danel G. Qullen, Homotopcal algebra, Lecture Notes n Mathematcs, vol. 43, Sprnger-erlag, [8] J. Roscký and W. Tholen, Let-determned model categores and unversal homotopy theores, Transactons o the Amercan Mathematcal Socety 355 (2003), [9] R. M. ogt, The HELP-Lemma And Its Converse In Qullen Model Categores, (2010), arv: [10] Marc Olschok, Let determned model structures or locally presentable categores, Appled Categorcal Structures 19 (2011), no. 6, Kushelevskaya doroga, 3k12, kv. 172, Sant Petersburg, Russa, Emal: valery.saev@gmal.com Ths artcle may be accessed at