$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