模型结构介绍 张继平（北京大学） 章璞（上海交通大学） 2021年12月3日 有限群的表示 ,有限维Hopf n投射 范畴，加法范砖的复形范畴，微分分次代数范畴，拓扑空间范畴，都有自然的（闭）模型结构模型 范畴中的思想方法概括、发展、并影响了某些数学领域中的思想方法.例如，V.Voevodsky获 Fields奖的工作N,现在，模型结构在诸如表示论、高阶K-理论、同伦论、代数拓扑、理论 计算机科学中的同伦类型论中，都有重要的应用，也成为前沿的研究对象 Daniel gray ouillen(19402011)1964年以偏微分方程方面的工作获哈佛大学博士学位 不久，受D.Km的影响，他研究代数拓扑和代数学.1967年他引入模型范畴并以此研究同伦 论：1971年他用有限群模表示论证明拓扑学中的As猜想：1972年他建立高阶K理论， 中引入正合范畴：1976年他证明Sc心猜想：主理想整环上的n元多项式环上的有限生成投 射模都是自由模(A.A.Suslin也独立地得到这一结果)，1978年他获Fields奖 文献[Q]清晰简洁，是这一理论很好很快的入门书.和可是这方面的专著文献 旧3是有帮助的综述性文章 这个讲义的初稿基于我们2010年关于余挠对和正合范畴的笔记手稿.此次由于系列讲座 的推动，添加Quillen模型结构方面的内容.在这个短课中，我们将介绍（闭）模型结构的基本 概今和性质，说明模型结构与闭模型结构之间的关系，证明Ab范畴上相容的闭模型结构与 ,二元组之间 对应的名定理。从 一对遗传、完备、相容的余挠对构造HO 组.介绍正合范畴及其基本性质：给出Frobenins范畴上的自然的（闭）模型结构.我们也将从
.(0 ‹U² (ÆåÆ) Ÿ‚ (˛°œåÆ) 2021 c 12 3 F âÆ˛(4) .(⁄(4) .âÆߥ D. Quillen 3 [Q1] •⁄\ (èÎÑ [Q2]). kÅ+L´âÆ,kÅë Hopf ìÍ âÆ,g\ìÍ âÆ, Dz Gorenstein › âÆ, \{âÆE/âÆ,á©©gìÍâÆ,ˇ¿òmâÆ, —kg,(4) .(. . âÆ•géê{V)!u–!øKè , ÍÆ+ç•géê{.~X, V. Voevodsky º Fields ¯Ûä [V]. y3ß .(3ÃXL´ÿ!p K-nÿ!”‘ÿ!ì͡¿!nÿ OéÅâÆ•”‘a.ÿ•ß—káA^ßè§èc˜ÔƒÈñ. Daniel Gray Quillen (1940-2011) 1964 c±†á©êßê°ÛäºMÃåÆƨƆ. ÿ»,… D. Kan Kè, ¶Ôƒì͡¿⁄ìÍÆ. 1967 c¶⁄\ .âÆø±dÔƒ”‘ ÿ¶1971c¶^kÅ+ L´ÿy²ˇ¿Æ• Adams flé¶1972 c¶Ô·p K-nÿߟ •⁄\‹âƶ1976c¶y² Serre flé: ÃnéDz n ıë™Ç˛kÅ)§› —¥gd (A. A. Suslin è’·/˘ò(J). 1978c¶º Fields ¯. ©z [Q1]òfl{',¥˘ònÿÈ–ÈØ\Ä÷. [H1] ⁄ [Hir] ¥˘ê°;Õ.©z [H3] ¥kêœn„5©Ÿ. ˘á˘¬–vƒu·Ç 2010 c'u{LÈ⁄‹âÆ)PÃv.dgduX˘å ̃, V\ Quillen .(ê°SN. 3˘á·ë•ß·ÇÚ0 (4) .(ƒ Vg⁄5üß`² .(Ü4 .(Ém'X. y² Abel âÆ˛ÉN4 .(Ü Hovey n|ÉmòòÈAÕ¶½n; lòÈ¢D!!ÉN{LÈE Hovey n |. 0 ‹âÆ9Ÿƒ 5ü¶â— Frobenius âÆ˛g,(4) .(. ·ÇèÚl 1
Abel范畴上的余挠对出发，在适当的条件下，得到其复形范畴上的两个余挠对，进而得到复 形范畴上的一个Hovey王元组，从而得到复形范畴上的一个相容的闭模型结构.时间所限， 我们未涉及重要的同伦理论.附录中可以查到需要反复用到的拉回和推出方面的主要结论. 感谢周远扬教授邀请我们做这方面的系列讲座.感谢李志伟教授的讨论并帮忙校对；感谢 高楠教授指出打印错误. 目录 1模型结构 2闭模型结构 3Abel范畴上相容的闭模型结构与Hovey三元组 4Hovey三元组的一种构造 5正合范畴 6 Frobenius范畴上的()模型结构 7复形范畴上的诱导Hovey三元组 8附录：拉回和推出 参考文献 2
Abel âÆ˛{LÈ—uß3·^áeߟE/âÆ˛¸á{LÈ, ? E /âÆ˛òá Hovey n|ß l E/âÆ˛òáÉN4 .(. ûm§Åß ·Çô9ᔑnÿ.N¹•å±IááE^.£⁄Ì—ê°Ãá(ÿ. a ±«û·Çâ˘ê°X˘å.a oìï«?ÿøêaÈ; a pô«ç—ã<Üÿ. 8¹ §1 .( §2 4 .( §3 Abel âÆ˛ÉN4 .(Ü Hovey n| §4 Hovey n|ò´E §5 ‹âÆ §6 Frobenius âÆ˛(4) .( §7 E/âÆ˛p Hovey n| §8 N¹: .£⁄Ì— Ωz 2
S1模型结构 本节我们将介绍模型结构的概念和初步性质。 $1.1 Model structures 1.1([Q1])A model structure on a category M is a triple(Cofib(M),Fib(M).Weq(M) of classes of morphisms,where the morphisms in the three classes are called cofibrations (usually denoted by),fibrations(usually denoted by),and weak equivalences,respectively,satisfying the following conditions (M1)-(M5): (Ml）)(提升性)Given a commutative square wherei∈Cofb(MW）andp∈Fib(MW,if eitheri∈WeqM)orp∈Weq(M),then there exists a dotted arrows:B such that a=si,b=ps. (M2)(分解性)Any morphismf:X-一→has the followingtwo factorization (i)f=pi,where iE Cofib(M)n Weq(M),pE Fib(M); (ii)f=p'i',where i'E Cofib(M),p'E Fib(M)n Weq(M). (M3)Both Fib(M)and Cofib(M)are closed under compositions.Isomorphisms are both fibrations and cofibrations. Fibrations are closed under pullback.i.e.,given a pullback square with pE Fib(M),then p'E Fib(M). Cofibrations are closed under pushout,i.e.,given a pushout square 。1 (1.2 1￡： withi∈Cofib(M),then∈Cofib(M). (M4)For any pullback square as (1.1),if pE Fib(M)nWeq(M),then p'Weq(M). For any pushout square as(1.2),if iCofib(M)nWeq(M),then'Weq(M)
§1 .( !·ÇÚ0 .(Vg⁄–⁄5ü. §1.1 Model structures ½¬ 1.1 ( [Q1]) A model structure on a category M is a triple (Cofib(M), Fib(M), Weq(M)) of classes of morphisms, where the morphisms in the three classes are called cofibrations (usually denoted by ,→), fibrations (usually denoted by ), and weak equivalences, respectively, satisfying the following conditions (M1) - (M5): (M1) (J,5) Given a commutative square A a / _ i X p B b / s > Y where i ∈ Cofib(M) and p ∈ Fib(M), if either i ∈ Weq(M) or p ∈ Weq(M), then there exists a dotted arrow s : B −→ X such that a = si, b = ps. (M2) (©)5) Any morphism f : X −→ Y has the following two factorizations. (i) f = pi, where i ∈ Cofib(M) ∩ Weq(M), p ∈ Fib(M); (ii) f = p 0 i 0 , where i 0 ∈ Cofib(M), p 0 ∈ Fib(M) ∩ Weq(M). (M3) Both Fib(M) and Cofib(M) are closed under compositions. Isomorphisms are both fibrations and cofibrations. Fibrations are closed under pullback, i.e., given a pullback square • / p 0 • p • /• (1.1) with p ∈ Fib(M), then p 0 ∈ Fib(M). Cofibrations are closed under pushout, i.e., given a pushout square • i / • • i 0 /• (1.2) with i ∈ Cofib(M), then i 0 ∈ Cofib(M). (M4) For any pullback square as (1.1), if p ∈ Fib(M) ∩ Weq(M), then p 0 ∈ Weq(M). For any pushout square as (1.2), if i ∈ Cofib(M) ∩ Weq(M), then i 0 ∈ Weq(M). 3
(M5)Lety be morphisms in M.If two of the three morphisms f,9.gf are weak equivalences,then so is the third.(这通常称为弱等价的“二推三"性质.特别地，弱等 价的合成还是弱等价) Any isomorphism is a weak equivalence 将Cofib(M)nWeq(M)中的态射称为平几余纤维：将Fib(M)nWeq(M)中的态射称 为平凡纤维， 模型结构的定义并不假定拉回和推出的存在性：而只是说，如果（平凡）纤维的拉回存在 则拉回后得到的态射仍是（平凡）纤维：如果（平凡）余纤维的推出存在则推出后得到的态射仍 是（平凡）余纤维 1.2 ([Q1])A category M endowed with a model structure is called a model category,if (MO)Mis closed under finite projective and inductive limits. 模型范畴M中的条件(M0)保证了在M中，初对象(initialobject)、终对象(termina object,or,final object)*、核(kernel)、余核(cokernel)、有限个对象的余积(coproduct小、有 限个对象的积(product)、拉回(pullback)、推出(pushout),都是存在的. 为方便起见，以下在本讲义中，总假定所考虑的范畴有零对象0.从而，初对象0和终对 象*均存在，且0=0=* $1.2 Terminologies (1)Let M be a category (i)Let f and g be morphisms in M.The morphism f is said to be a retract,of g. if there exists two in the morphism category Mor(M) such that=Idx,p=Idy (Let iand p be morphisms of M.We say that i has the lef lifting pro erty(LLP)with respecttop,andphas(RLP)with respect to,provided that for any commutative square
(M5) Let X f −→ Y g −→ Z be morphisms in M. If two of the three morphisms f, g, gf are weak equivalences, then so is the third. (˘œ~°èf d“Ìn”5ü. AO/, f d‹§Ñ¥f d.) Any isomorphism is a weak equivalence. Ú Cofib(M) ∩ Weq(M) •°è ²Ö{në¶Ú Fib(M) ∩ Weq(M) •° è ²Önë. .(½¬øÿb½.£⁄Ì—35¶ ê¥`, XJ (²Ö) në.£3 K.£E¥(²Ö)në¶XJ (²Ö) {nëÌ—3KÌ—E ¥(²Ö){në. ½¬ 1.2 ( [Q1]) A category M endowed with a model structure is called a model category, if (M0) M is closed under finite projective and inductive limits. .âÆ M •^á (M0) y 3 M •, –Èñ (initial object) ∅!™Èñ (terminal object, or, final object) ?!ÿ (kernel)!{ÿ (cokernel)!kÅáÈñ{» (coproduct)!k ÅáÈñ» (product)!.£ (pullback)!Ì— (pushout), —¥3. èêBÂÑ, ±e3 ˘¬•, ob½§ƒâÆk"Èñ 0. l , –Èñ ∅ ⁄™È ñ ? ˛3, Ö ∅ = 0 = ?. §1.2 Terminologies (1) Let M be a category. (i) Let f and g be morphisms in M. The morphism f is said to be a retract, †£ of g, if there exists two morphisms ϕ : f −→ g and ψ : g −→ f in the morphism category Mor(M), such that ψϕ = Idf . That is, there exists the following commutative diagram X ϕ1 / f X0 ψ1 / g X f Y ϕ2 /Y 0 ψ2 /Y such that ψ1ϕ1 = IdX, ψ2ϕ2 = IdY . (ii) Let i and p be morphisms of M. We say that i has the left lifting property (LLP) with respect to p, and p has the right lifting property (RLP) with respect to i, provided that for any commutative square A a / i X p B b / s > Y 4
there exists a morphism s:B such that a=si.b=ps. morphisms,denote by(R)the class of morphisms which have the LLP (lef lifting property)with respect to every morphism inR. For a classL of morphisms,denote byr(L)the class of morphisms which have the RLP (right lifting property)with respect to every morphism in L. (2)Let (Cofib(M),Fib(M),Weq(M)be a model structure on categoryM.Then ()提升性()可以重新表述成 Cofib(M)n Weq(M)E I(Fib(M));Cofib(M)I(Fib(M)n Weq(M)); Fib(M)n Weq(M)Er(Cofib(M)):Fib(M)Er(Cofib(M)nWeq(M)) (i)An object X is called a cofibrant object,,余纤维对象，if0-→X is a cofibration. 用M。或C表示所有余纤维对象作成的类， An object Y is called a fibrant object,,纤维对象，ifY-→0 is a fibration. 用M或F表示所有纤维对象作成的类 An object W is called atrivial防ect,平凡对象，if0-→W is a weak equivalence.由弱等 价的二推三性质即知：W是平凡对象当且仅当W-一→0是弱等价， 用M:或W表示所有平凡对象作成的类， 于是，得到三个对象类，或三个全子范畴：C,F,W, 习题设(Coib(M),Fib(M),Weq(M)是范睛M上的一个模型结构，A和B是两个 平凡对象.则任意态射∫：A-一→B都是弱等价. 51.3弱等价的分解 引理1.3在一个模型结构中，弱等价恰是平凡余纤维与平凡纤维的合成，即 Weq(M)=(Fib(M)nWeq(M))-(Cofib(M)nWeq(M)) 证明由弱等价的合成还是弱等价（参见M5)即知 (Fib(M)）nWeq(M)·(Cofb(M)nWeq(M)sWeq(M). Weq(M)S(Fib(M)n Weq(M)).(Cofib(M)n Weq(M))
there exists a morphism s : B −→ X such that a = si, b = ps. For a class R of morphisms, denote by l(R) the class of morphisms which have the LLP (left lifting property) with respect to every morphism in R. For a class L of morphisms, denote by r(L) the class of morphisms which have the RLP (right lifting property) with respect to every morphism in L. (2) Let (Cofib(M), Fib(M), Weq(M)) be a model structure on category M. Then (i) J,5 (M1) å±#L„§ Cofib(M) ∩ Weq(M) ⊆ l(Fib(M)); Cofib(M) ⊆ l(Fib(M) ∩ Weq(M)); Fib(M) ∩ Weq(M) ⊆ r(Cofib(M)); Fib(M) ⊆ r(Cofib(M) ∩ Weq(M)). (ii) An object X is called a cofibrant object, {nëÈñ, if 0 −→ X is a cofibration. ^ Mc ½ C L´§k{nëÈñä§a. An object Y is called a fibrant objectßnëÈñ, if Y −→ 0 is a fibration. ^ Mf ½ F L´§knëÈñä§a. An object W is called a trivial objectß²ÖÈñ, if 0 −→ W is a weak equivalence. df d“Ìn”5ü=µW ¥²ÖÈñÖ= W −→ 0 ¥f d. ^ Mt ½ W L´§k²ÖÈñä§a. u¥, náÈña, ½ná fâÆ: Cß F, W. SK (Cofib(M), Fib(M), Weq(M)) ¥âÆ M ˛òá .(, A ⁄ B ¥¸á ²ÖÈñ. K?ø f : A −→ B —¥f d. §1.3 f d©) ⁄n 1.3 3òá .(•, f dT¥²Ö{nëܲÖn닧, = Weq(M) = (Fib(M) ∩ Weq(M)) · (Cofib(M) ∩ Weq(M)). y² df d‹§Ñ¥f d (ÎÑ (M5)) = (Fib(M) ∩ Weq(M)) · (Cofib(M) ∩ Weq(M)) ⊆ Weq(M). áÉ, w ∈ Weq(M). d (M2) k©) w = pi, Ÿ• i ∈ Cofib(M)∩Weq(M), p ∈ Fib(M). 2d (M5) (=ßf d“Ìn”5ü) p ∈ Weq(M), p ∈ Fib(M) ∩ Weq(M). u¥ Weq(M) ⊆ (Fib(M) ∩ Weq(M)) · (Cofib(M) ∩ Weq(M)). 5
$1.4态射分解的函子性 事实14设范M有三个态射是，记为Coi(M,Fi(M,We(M)满足)和M2 则2)中的分解具有函子性，即，若有交换图 则 ()若f=pi,g=,其中,/E Fib(M),，t∈Cofib(M)n Weq(M,则存在s使得 si=ip.p's=up. (②)若f=g助，g=g了，其中g,g∈Fib(M0nWeq(M).,∈Coib(M),则存在t使得 tj=j'p q't=vq. 证明只证（②）.结论()的证明完全类似.考虑交换图： 由于j∈Cofb(M),g∈Fib(M)nWeq(M),故由(M)知存在t使得与='g,t=g.口 6
§1.4 ©)ºf5 Ø¢ 1.4 âÆ M knáa, Pè Cofib(M), Fib(M), Weq(M)ߘv (M1) ⁄ (M2). K (M2) •©)‰kºf5, =, ekÜ„ • f / ϕ • ψ • g /• K (1) e f = pi, g = p 0 i 0 , Ÿ• p, p 0 ∈ Fib(M), i, i 0 ∈ Cofib(M) ∩ Weq(M), K3 s ¶ si = i 0ϕ, p0 s = ψp. • f / i ϕ • ψ • s p ?? • g / i 0 • • p 0 ? (2) e f = qj, g = q 0 j 0 , Ÿ• q, q 0 ∈ Fib(M) ∩ Weq(M), j, j 0 ∈ Cofib(M), K3 t ¶ tj = j 0ϕ, q 0 t = ψq. • f / j ϕ • ψ • t q ? • g / j 0 • • q 0 ? y² êy (2). (ÿ (1) y² aq. ƒÜ„: • j 0ϕ / j • q 0 • ψq / t ? • du j ∈ Cofib(M), q 0 ∈ Fib(M) ∩ Weq(M), d (M1) 3 t ¶ tj = j 0ϕ, q0 t = ψq. 6
S2闭模型结构 Quillen也引入闭模型结构，它优于模型结构：其中三个态射类中的任意一个态射类由其 余两个态射类唯一确定.本节我们将介绍闭模型结构的概念、基本性质，特别地，我们要说明 闭模型结构是模型结构，并给出一个模型结构是闭的充分必要条件。 $21闭模型结构 2.1 ([Q2],p.233)A closed model structure on a category M is a triple (Cofib(M). Fib(M),Weq(M))of classes of morphisms,where the morphisms in the three classes are (CIl)）(弱等价的s二推三"性质)LetX三y_9,Z be morphisms in M.If two of the nces,then so is the third.特别地，弱等价的合成还是弱 (C2)(仁个态射类均对retract封闭)Iffisaretract ofg,andgis acofibration(6 ibration weak equivalence).then so isf. (CM3)=(Ml)(提升性)Given a commutative square wherei∈Cofib(and p∈Fib(M),if eitheri∈Weq(M)orp∈Weq(M),then there exists a morphism s:BX such that a =si,b=ps. (CM4)=(M2)(分解性)Any morphism f:X-→Y has two factorizations: (i)f=pi,where iE Cofib(M)n Weq(M).pE Fib(M): (ii)f=p'i',where i'E Cofib(M).p'E Fib(M)n Weg(M). 2.([Q2)A category Mendowed with a closed model structure is called a closed model ategory,if (CMO)=(MO)Mis closed under finite projective and inductive limits 请注意，现在不少文献中的模型结构（范畴）就是指闭模型结构（范畴） s22例子 这个理论的一个特点：每一个例子就是一条重要定理.因此，立即给出（闭）模型范畴的例 子是困难的.知道Frobenius范畴有自然的模型结构以后，会得到很多例子
§2 4 .( Quillen è⁄\4 .(, ß`u .(: Ÿ•náa•?øòáadŸ {¸áaçò(½. !·ÇÚ0 4 .(Vg!ƒ 5ü. AO/ß·Çá`² 4 .(¥ .(,øâ—òá .(¥4ø©7á^á. §2.1 4 .( ½¬ 2.1 ( [Q2], p.233) A closed model structure on a category M is a triple (Cofib(M), Fib(M), Weq(M)) of classes of morphisms, where the morphisms in the three classes are respectively called cofibrations (usually denoted by ,→), fibrations (usually denoted by ), and weak equivalences, satisfying the following conditions (CM1) - (CM4): (CM1) (f d“Ìn”5ü) Let X f −→ Y g −→ Z be morphisms in M. If two of the morphisms f, g, gf are weak equivalences, then so is the third. AO/, f d‹§Ñ¥f d. (CM2) (náa˛È retract µ4) If f is a retract of g, and g is a cofibration (fibration, weak equivalence), then so is f. (CM3)=(M1) (J,5) Given a commutative square A a / _ i X p B b / s > Y where i ∈ Cofib(M) and p ∈ Fib(M), if either i ∈ Weq(M) or p ∈ Weq(M), then there exists a morphism s : B −→ X such that a = si, b = ps. (CM4)=(M2) (©)5) Any morphism f : X −→ Y has two factorizations: (i) f = pi, where i ∈ Cofib(M) ∩ Weq(M), p ∈ Fib(M); (ii) f = p 0 i 0 , where i 0 ∈ Cofib(M), p 0 ∈ Fib(M) ∩ Weq(M). ½¬ 2.2 ( [Q2]) A category M endowed with a closed model structure is called a closed model category, if (CM0)=(M0) M is closed under finite projective and inductive limits. û5ø, y3ÿ©z• .((âÆ)“¥ç4 .((âÆ). §2.2 ~f ˘ánÿòáA:µzòá~f“¥ò^á½n. œd, ·=â—(4) .âÆ~ f¥(J. Frobenius âÆkg, .(±ß¨Èı~f. 7
按惯例，我们先指出一些例子，但除(1)以外，它们但是证明很长且困难的定理。 ()易知：如果(Cofb(M,Fb(M),Weq(M)是范畴M上的（闭）模型结构（相应地 范畴)，则(Fib(M),Cofb(M),Weq(M)是反范睛Mp上的（闭）模型结构（相应地，范畴）， (②)设A是有足够多投射对象的Abl范畴.则A上的下有界复形范畴C+(4)是闭模型 范畴，其中的余纤维是复形的单态射且其余核是投射对象的复形，纤维是复形的满态射，弱等 价是复形的拟同构，即诱导出上同调对象之间同构的复形态射.参见[QL,Chapter I,§小.这 在1，Theorem2.3.1川中被推广至无界复形范畴上.也参见例7.l5. (③)加法范畴A上的复形短正合列0-→X二y二Z一→0称为链可裂短 正合列，如果0-→Xn二ymg二Zn-→0是可裂短正合列，Yn∈Z,即存在同枸 7:X"⊕Zm-一→ym使得下图交换 →0 0→mP →0 其等价描述参见引理58.此时链映射称为链可裂单射，链映射9称为缝可裂满射.注意 在链可裂短正合列中，作为复形.Y·未必同构于X·由Z. 加法范畴A的复形范畴C(4,C+(A),C-(4),C(A),都是闭模型范畴，其中 余纤维是链可裂单射，纤维是链可裂满射，弱等价是同伦等价（即同伦范畴中的同构） 这与（②）中的模型结构是不同的.也参见命题6.9 (④设R是环，存在非自由模的授射模则下有界复形范畴c+(RMo)是模型范畴但非 闭模型范畴，其中的余纤维是复形的单态射且其余核是自由R模的复形，纤维是复形的满态 射，弱等价是复形的拟同构.参见[QL,Chapter,Is5. (⑤)拓扑空间和连续映射作成的范畴是闭模型范畴，其中的弱等价是弱同伦等价，纤维 是Se纤维，余纤维是对所有平凡纤维（即Scre纤维且弱同伦等价）有左提升性质的连续 映射.参见[QL,Chapter II,s3,Theorem. (6)微分分次代数范畴是闭模型范畴，其中的弱等价是拟同构，纤维是g代数的满同态， 余纤维是对所有平凡纤维有左提升性质的dg代数同态.参见[GM,V3 Theorem] s2.3基本性质 在本小节中，若无特殊声明，总假设(Cob( (M)是范畴M上的一个 闭模型结构，不再每次陈述。我们 将看到 ,闭模型 最重要的性质是 态射类Coib(M) Fib(M),Weq(M)中的任意两个态射类唯一地确定第三个态射类，这是“闭”的含义， 比较模型结构和闭模型结构的定义，立即看出：范畴M上的模型结构(Cob(M),ib(M) Weq(M)是闭模型结构当且仅当它满足(CM2),即三个态射类Cofib(M),Fib(M),Weq(M) 均对retract封闭
U.~, ·Çkç—ò ~f, ÿ (1) ± , ßÇ¥y²ÈÖ(J½n. (1) ¥µXJ (Cofib(M), Fib(M), Weq(M)) ¥âÆ M ˛(4) .( (ÉA/ß âÆ), K (Fib(M), Cofib(M), Weq(M)) ¥áâÆ Mop ˛(4) .( (ÉA/ßâÆ). (2) A ¥kv ı›Èñ Abel âÆ. K A ˛ek.E/âÆ C +(A) ¥4 . âÆ, Ÿ•{në¥E/¸ÖŸ{ÿ¥›ÈñE/ßnë¥E/˜ßf d¥E/[”, =p—˛”NÈñÉm”E/. ÎÑ [Q1, Chapter I, § 1]. ˘ 3 [H1, Theorem 2.3.11] •Ì2ñÃ.E/âÆ˛. èÎÑ~ 7.15. (3) \{âÆ A ˛E/·‹ 0 −→ X• f • −−→ Y • g • −→ Z • −→ 0 °èÛå· ‹, XJ 0 −→ Xn f n −−→ Y n g n −−→ Z n −→ 0 ¥å·‹, ∀ n ∈ Z, =3” γ : Xn ⊕ Z n −→ Y n ¶e„Ü 0 /Xn ( 1 0) /Xn ⊕ Z n (0,1) / γ Z n /0 0 /Xn f n /Y n g n /Z n /0. Ÿ d£„ÎÑ ⁄n 5.8. dûÛN f • °èÛå¸, ÛN g • °èÛå˜. 5ø 3Ûå·‹•, äèE/, Y • ô7”u X• ⊕ Z • . \{âÆ A E/âÆ C b (A), C +(A), C −(A), C(A), —¥4 .âÆ, Ÿ• {në¥Ûå¸ßnë¥Ûå˜, f d¥”‘ d (=”‘âÆ•”). ˘Ü (2) • .(¥ÿ”. èÎÑ·K 6.9. (4) R ¥Ç, 3ögd › . Kek.E/âÆ C +(R-Mod) ¥ .âÆö 4 .âÆ, Ÿ•{në¥E/¸ÖŸ{ÿ¥gd R- E/ßnë¥E/˜ ßf d¥E/[”. ÎÑ [Q1, Chapter I, § 5]. (5) ˇ¿òm⁄ÎYNä§âÆ¥4 .âÆ, Ÿ•f d¥f”‘ dßnë ¥ Serre nëß{në¥È§k²Önë (= Serre nëÖf”‘ d)kÜJ,5üÎY N. ÎÑ [Q1, Chapter II, § 3, Theorem 1]. (6) á©©gìÍâÆ¥4 .âÆ, Ÿ•f d¥[”, në¥ dg ì͘”ß {në¥È§k²ÖnëkÜJ,5ü dg ìÍ”. ÎÑ [GM, V.3, Theorem]. §2.3 ƒ 5ü 3 !•, eÃAœ(², ob (Cofib(M), Fib(M), Weq(M)) ¥âÆ M ˛òá 4 .(,ÿ2zgù„. ·ÇÚw, 4 .(Åá5ü¥náa Cofib(M), Fib(M), Weq(M) •?ø¸áaçò/(½1náa. ˘¥ “4” ¹¬. ' .(⁄4 .(½¬, ·=w—: âÆ M ˛ .( (Cofib(M), Fib(M), Weq(M)) ¥4 .(Ö=ߘv (CM2), =náa Cofib(M), Fib(M), Weq(M) ˛È retract µ4. 8
然而，要说明闭模型结构是模型结构，还需要若干准备，主要是验证(3)和QM4 命题2.30 ne has (1)Cofib(M)=I(Fib(M)nWeq(M)).i.e.,cofibrations are ecisely those morphism (2)Cofib(M)Weq(M)=I(Fib(M)),i.e.,trivial cofibrations are precisely those mor phisms which have LLP with respect toall fibrations. (3)Fib(M)=r(Cofib(M)nWeq(M)),i.e.,fibrations ar recisely those morphism which have RLP(the right lifting property)with respect to all trivial cofibrations. (4)Fib(M)Weq(M)=r(Cofib(M,ie,trivial cofibrations are precisely those mo phisms which have RLP with respect to all cofibrations (5)Weq(M)=r(Cofib(M)).I(Fib(M)). (1)The axiom (CM3)shows that Cofib(M)(Fib(M)Weq(M)). Conversely,for an arbitrary morphism f E /(Fib(M)n Weq(M)),by (CM4)one has a factorization f=p'i,where E Cofib(M)and p'E Fib(M)n Weq(M).So one gets a commutative square By the assumption fl(Fib(M)nWeq(M)).f has LLP with respect to p',i.e..there exists a morphism r such that the two triangles commutes.Since r=Id,it follows that f is a retract of i,as the following commutative diagram shows Since'is a cofibration,it follows from (CM2)that f is a cofibration,ie..fCofib(M).This proves Cofib(M)=I(Fib(M)Weq(M). (2)The axiom (CM3)shows that Cofib(M)Weq(M)I(Fib(M)) Conversely,for an arbitrary morphismf(Fib(M)),by (CM)one has a factorization f=pi,whereiCofib(M)Weq(M)andp Fib(M).So one gets a commutative square 9
, , á`²4 .(¥ .(,ÑIáeZO, Ãá¥y (M3) ⁄ (M4). ·K 2.3 One has (1) Cofib(M) = l(Fib(M) ∩ Weq(M)), i.e., cofibrations are precisely those morphisms which have LLP (the left lifting property) with respect to all trivial fibrations. (2) Cofib(M) ∩ Weq(M) = l(Fib(M)), i.e., trivial cofibrations are precisely those morphisms which have LLP with respect to all fibrations. (3) Fib(M) = r(Cofib(M)∩Weq(M)), i.e., fibrations are precisely those morphisms which have RLP (the right lifting property) with respect to all trivial cofibrations. (4) Fib(M) ∩ Weq(M) = r(Cofib(M)), i.e., trivial cofibrations are precisely those morphisms which have RLP with respect to all cofibrations. (5) Weq(M) = r(Cofib(M)) · l(Fib(M)). y² (1) The axiom (CM3) shows that Cofib(M) ⊆ l(Fib(M) ∩ Weq(M)). Conversely, for an arbitrary morphism f ∈ l(Fib(M) ∩ Weq(M)), by (CM4) one has a factorization f = p 0 i 0 , where i 0 ∈ Cofib(M) and p 0 ∈ Fib(M) ∩ Weq(M). So one gets a commutative square • i 0 / f=p 0 i 0 • p 0 • x ? • By the assumption f ∈ l(Fib(M) ∩ Weq(M)), f has LLP with respect to p 0 , i.e., there exists a morphism x such that the two triangles commutes. Since p 0x = Id, it follows that f is a retract of i 0 , as the following commutative diagram shows • f • i 0 • f • x /• p 0 /• Since i 0 is a cofibration, it follows from (CM2) that f is a cofibration, i.e., f ∈ Cofib(M). This proves Cofib(M) = l(Fib(M) ∩ Weq(M)). (2) The axiom (CM3) shows that Cofib(M) ∩ Weq(M) ⊆ l(Fib(M)). Conversely, for an arbitrary morphism f ∈ l(Fib(M)), by (CM4) one has a factorization f = pi, where i ∈ Cofib(M) ∩ Weq(M) and p ∈ Fib(M). So one gets a commutative square • i / f=pi • p • x ? • 9
By the assumption f(Fib(M)).has LLP with respect to p,i.e.,there exists a morphism x such that the two triangles commutes.So f is a retract of i,as the following commutative diagram shows Sinceiis a trivial cofibration,it follows from (CM2)that fisa trivial cofration, Cofib(M)Weq(M).This shows Cofib(M)Weq(M)=I(Fib(M)). 结论(3)和(4)的证明是类似的，留作习题。 (5)由引理1.3的证明即知Weq(M)=(Fib(M)nWeq(M)·(Cofb(M)nWeq(M).再 由（②）和（④即得。 引理2.4 One has (1)The classes Fib(M)and Fib(M)nWeq(M)are closed under compositions. (2)The class Fib(M)is closed under pullback. (3)The class Fib(M)Weq(M)is closed under pullback (1)The classes Cofib(M)and Cofib(M)Weq(M)are closed under compositions (2)The class Cofib(M)is closed under pushout (3)The class Cofib(M)Weq(M)is closed under pushout 证明只证(1)，(2)，(3).其余结论留作习题。 ()设··二·是M中（两个可合成）的态射，其中p和均是纤维.要证p也是 纤维.由命题2.3(3知，只要证明pp对任意平凡余纤维i有右提升性质.为此，假设有态射ā 和b满足bi=(pp)a.则i=p(pm).参见下图.因为p是纤维，由提升性即知存在态射x使 得m=i,b=z 因为卫是纤维，再对交换方块m=i使用提升性即知存在态射x使得a一ti,工=m.从 而a=xi,b=x=(p)江.这就证明了印对任意平凡余纤维有右提升性质，从而由命题 2.3(3)知pp∈r(Cofib(M)n Weq(M0)=Fib(M0. 如果p和d还都是弱等价，则p也是弱等价（参见(CM1),故pp∈Fib(M)nWeq(M)
By the assumption f ∈ l(Fib(M)), f has LLP with respect to p, i.e., there exists a morphism x such that the two triangles commutes. So f is a retract of i, as the following commutative diagram shows. • f • i • f • x /• p /• Since i is a trivial cofibration, it follows from (CM2) that f is a trivial cofibration, i.e., f ∈ Cofib(M) ∩ Weq(M). This shows Cofib(M) ∩ Weq(M) = l(Fib(M)). (ÿ (3) ⁄ (4) y²¥aq, 3äSK. (5) d⁄n 1.3y²= Weq(M) = (Fib(M) ∩ Weq(M)) · (Cofib(M) ∩ Weq(M)). 2 d (2) ⁄ (4) =. ⁄n 2.4 One has (1) The classes Fib(M) and Fib(M) ∩ Weq(M) are closed under compositions. (2) The class Fib(M) is closed under pullback. (3) The class Fib(M) ∩ Weq(M) is closed under pullback. (10 ) The classes Cofib(M) and Cofib(M) ∩ Weq(M) are closed under compositions. (20 ) The class Cofib(M) is closed under pushout. (30 ) The class Cofib(M) ∩ Weq(M) is closed under pushout. y² êy (1), (2), (3). Ÿ{(ÿ3äSK. (1) • p −→ • p 0 −→ • ¥ M •(¸á勧), Ÿ• p ⁄ p 0 ˛¥në. áy p 0p è¥ në. d·K 2.3(3), êáy² p 0p È?ø²Ö{në i kmJ,5ü. èd, bk a ⁄ b ˜v bi = (p 0p)a. K bi = p 0 (pa). ÎÑe„. œè p ¥në, dJ,5=3 x ¶ pa = xi, b = p 0x. • p A pa / a 8 i • p 0 B b / x 8 x 0 @ • œè p 0 ¥nëß2ÈÜê¨ pa = xi ¶^J,5=3 x 0 ¶ a = x 0 i, x = px0 . l a = x 0 i, b = p 0x = (p 0p)x 0 . ˘“y² p 0p È?ø²Ö{nëkmJ,5ü, l d·K 2.3(3) p 0p ∈ r(Cofib(M) ∩ Weq(M)) = Fib(M). XJ p ⁄ p 0 Ñ—¥f d, K p 0p è¥f d (ÎÑ (CM1)), p 0p ∈ Fib(M)∩Weq(M). 10
