(②)Let be a pullback square,and p a fibration.We want to prove that f is also a fibration.Again by Proposition 2.3(3),it suffices to prove that fr(Cofib(M)nWeq(M)).Assume that we have a commutative square with i Cofib(M)Weq(M),we want to find a morphism such thati=s andf=t. Since p E Fib(M)=r(Cofib(M)nWeq(M)),there exists a morphism y such that yi=as and pybt,as the following commutative diagram shows. Since py=bt,by the universal property of pullback,there exists a unique morphism such that fr=t and ar=y,as the following commutative diagram shows: Consider the following commutative diagram with bti=pyi =pas. Since axi yi as fs=ti, fri=ti
(2) Let • a / f • p • b /• be a pullback square, and p a fibration. We want to prove that f is also a fibration. Again by Proposition 2.3(3), it suffices to prove that f ∈ r(Cofib(M) ∩ Weq(M)). Assume that we have a commutative square • s / i • f • t / x ? • with i ∈ Cofib(M) ∩ Weq(M), we want to find a morphism x such that xi = s and fx = t. Since p ∈ Fib(M) = r(Cofib(M) ∩ Weq(M)), there exists a morphism y such that yi = as and py = bt, as the following commutative diagram shows. • s / i • a / f • p • t / y 7 • b /• Since py = bt, by the universal property of pullback, there exists a unique morphism x such that fx = t and ax = y, as the following commutative diagram shows: • y t & x • a / f • p • b /• Consider the following commutative diagram • as ti & xi s • a / f • p • b /• with bti = pyi = pas. Since as = as, fs = ti, axi = yi = as, fxi = ti. 11
By the universal property of a pullback,we have s=ri.Hence ri=s and fz=t.This completes the proof. 结论（③）与(2)的证明类似.留作习题 2.5 Isomorphisms are in Cofib(M)nFib(M)n Weq(M). 证明以下证明的工具是使用命题2.3.Let fbe an isomorphism.From the following com- tative diagram PEF:b(M) }6 we know that fE1(Fib(M))Cofib(M)n Weq(M). From the following commutative diagram Cofib(M)nWeq[M)3i we know that fEr(Cofib(M)Weq(M))=Fib(M). ThusfE Cofib(M)nFib(M)nWeq(M). 52.4模型结构与闭模型结构的关系 对范畴M上的模型结构(Coib(M),Fib(M),Wq(M),方便起见，将如下三个条件合 起来称为公理(M6: (a)Cofib(M)=1(Fib(M)nWeq(M),即，余纤维恰好是对所有的平凡纤维都具有左提 升性质的态射。 (b)Fib(M)=r(Cofb(M)nWeq(M),即，纤维恰好是对所有的平凡余纤维都具有右提 升性质的态射. (c)Weg(M)=r(Cofib(M)).I(Fib(M)) 特别地，公理(M6)成立显示三个态射类Cob(M),Fib(M),Weq(M)中任意一个态射 类由其余两个态射类唯一确定· 将上述(a),)以及如下两个条件(@和(@)合起来称为公理M7): (d)Cofb(M)nWeg(M)=l(Fib(M),即，平凡余纤维恰好是对所有的纤维都具有左提 升性质的态射 ()Fib(M)Wea(M)=r(Cofb(M),即，平凡纤维恰好是对所有的余纤维都具有右提 升性质的态射
By the universal property of a pullback, we have s = xi. Hence xi = s and fx = t. This completes the proof. (ÿ (3) Ü (2)y²aq. 3äSK. ⁄n 2.5 Isomorphisms are in Cofib(M) ∩ Fib(M) ∩ Weq(M). y² ±ey²Û‰¥¶^·K 2.3. Let f be an isomorphism. From the following commutative diagram • a / f • p∈Fib(M) • b / af−1 7 • we know that f ∈ l(Fib(M)) = Cofib(M) ∩ Weq(M). From the following commutative diagram • a / Cofib(M)∩Weq(M)3i • f • b / f −1 b 7 • we know that f ∈ r(Cofib(M) ∩ Weq(M)) = Fib(M). Thus f ∈ Cofib(M) ∩ Fib(M) ∩ Weq(M). §2.4 .(Ü4 .('X ÈâÆ M ˛ .( (Cofib(M), Fib(M), Weq(M)), êBÂÑ, ÚXená^á‹ Â5°è˙n (M6): (a) Cofib(M) = l(Fib(M) ∩ Weq(M)), =, {nëT–¥È§k²Önë—‰kÜJ ,5ü. (b) Fib(M) = r(Cofib(M) ∩ Weq(M)), =, nëT–¥È§k²Ö{në—‰kmJ ,5ü. (c) Weq(M) = r(Cofib(M)) · l(Fib(M)). AO/, ˙n (M6) §·w´náa Cofib(M), Fib(M), Weq(M) •?øòá adŸ{¸áaçò(½. Ú˛„ (a), (b) ±9Xe¸á^á (d) ⁄ (e) ‹Â5°è˙n (M7): (d) Cofib(M) ∩ Weq(M) = l(Fib(M)), =, ²Ö{nëT–¥È§knë—‰kÜJ ,5ü. (e) Fib(M) ∩ Weq(M) = r(Cofib(M)), =, ²ÖnëT–¥È§k{në—‰kmJ ,5ü. 12
引理2.6设(Cofib(M),Fib(M),Weq(M)是范咋M上的模型结构，且满足公理(QM7).则 ()余纤维和平几余纤维均对retract封闭. (1)纤维和平凡纤维均对retract封闭. 证明只证()，结论(1)的证明类似，留作习题.设态射f是余纤维g的一个retract,即，有 交换图 X =ldy.要证∫也是余纤维.由性质(a).只要证f∈libM)n →X X 特别地，有交换长方块(b2g-a心.因为g是余纤维，由提升性即得s使得sg-a,s b2.于是有g=8y2:Y-→A满足 s'f=sp2!=sgo1=avrio=a,ps'psp2 =bap2=b. 因此f∈l(Fib(M)nWeq(M)=Cofb(M),即，f是余纤维. 若g还是平凡余纤维，则利用性质（④）类似地可证∫也是平凡余纤维. 为了方便于以后（指后面的定理3.13）的应用，下述引理中的条件看上去似乎比较复杂，但 这些条件要比模型范畴的要求弱。 引理2.7设范骑M有态射类Co6b(M),Fib(M),Weq(M),满足提升性(Q),分解性(M2 和弱等价（指Weq(M)中的态射）的“二推三”性质.若M满足下列条件之一，则弱等价对 缩回封闭. (1)设平凡纤维（即Fb(M)nWq(M)中的态射）的拉回存在，且平几纤维对拉回封闭， 且平几余纤维（即Cofb(M)nWeq(M)中的态射）对缩回封闭. (②)设平凡余纤维的推出存在，且平凡余纤维对推出封闭，且平凡纤维对缩回封闭. 证明设M满足上述条件().设态射∫是弱等价g的一个缩回.要证f也是弱等价.由假 设，有交换图
⁄n 2.6 (Cofib(M), Fib(M), Weq(M)) ¥âÆ M ˛ .(,Ö˜v˙n (M7). K (1) {në⁄²Ö{në˛È retract µ4. (10 ) në⁄²Önë˛È retract µ4. y² êy (1), (ÿ (10 ) y²aq, 3äSK. f ¥{në g òá retract, =ßk Ü„ X ϕ1 / f X0 ψ1 / g X f Y ϕ2 /Y 0 ψ2 /Y Ÿ• ψ1ϕ1 = IdX, ψ2ϕ2 = IdY . áy f è¥{në. d5ü (a), êáy f ∈ l(Fib(M) ∩ Weq(M)). kÜê¨ bf = pa, Ÿ• p ∈ Fib(M) ∩ Weq(M). ÎÑe„.KXeÜ„ X ϕ1 / f X0 ψ1 / g X f a /A p Y ϕ2 /Y 0 ψ2 / s 7 Y b /B AO/ßkÜê¨ (bψ2)g = p(aψ1). œè g ¥{në, dJ,5= s ¶ sg = aψ1, ps = bψ2. u¥k s 0 = sϕ2 : Y −→ A ˜v s 0 f = sϕ2f = sgϕ1 = aψ1ϕ1 = a, ps0 = psϕ2 = bψ2ϕ2 = b. œd f ∈ l(Fib(M) ∩ Weq(M)) = Cofib(M), =ßf ¥{në. e g Ñ¥²Ö{në, K|^5ü (d) aq/åy f 襲Ö{në. è êBu±(ç°½n 3.13)A^, e„⁄n•^áw˛q'E,, ˘ ^áá' .âÆá¶f. ⁄n 2.7 âÆ M ka Cofib(M), Fib(M),Weq(M), ˜vJ,5 (M1), ©)5 (M2) ⁄f d (ç Weq(M) •) /Ìn05ü. e M ˜ve^áÉò, Kf dÈ †£µ4. (1) ²Önë (= Fib(M) ∩ Weq(M) •).£3ßÖ²ÖnëÈ.£µ4, Ö²Ö{në (= Cofib(M) ∩ Weq(M) •) Ȇ£µ4. (2) ²Ö{nëÌ—3ßÖ²Ö{nëÈÌ—µ4, Ö²ÖnëȆ£µ4. y² M ˜v˛„^á (1). f ¥f d g òᆣ. áy f è¥f d. db , kÜ„ X ϕ1 / f X0 ψ1 / g X f Y ϕ2 /Y 0 ψ2 /Y 13
其中1p1=dx,22=1dy.根据(M2,有如下分解f=p， 其中1是余纤维，卫是平凡纤维，由弱等价对合成封闭的题设，只要证明i是平凡余纤维.为此， 考虑平凡纤维D：Z-Y和：y-Y的拉回方块（由颗设，这总存在）由平凡纤维对 .由2P2p=n,知存在唯一的态射 又由2g=f1=pi1,知存在唯一的态射j使得g射=g,j=i1.即有如下交换图 Z- 由刚=9以及g和g均是弱等价知方是弱等价（弱等价的“二推三”性质） 又由2-∫=i,知存在唯一的态射r:X-一Z使得下图交换： 注意到态射p:X-→Z和态射j1:X-→Z均满足r的性质，即 qpi=papi=paf. qjp1=991=p2, vi=iv=i
Ÿ• ψ1ϕ1 = IdX, ψ2ϕ2 = IdY . ä‚ (M2), f kXe©) f = pi, X f / i % Y Z p 9 Ÿ• i ¥{në, p ¥²Önë. df dÈ‹§µ4K, êáy² i ¥²Ö{në. èd, ƒ²Önë p : Z −→ Y ⁄ ψ2 : Y 0 −→ Y .£ê¨ (dKߢo3). d²ÖnëÈ .£µ4K, ²Önë q Ü ψ ¶ ψ2q = pψ. d ψ2ϕ2p = p, 3çò ϕ : Z −→ Z 0 , ¶ qϕ = ϕ2p, ψϕ = IdZ. =kXeÜ„: Z = ϕ2p % ϕ Z 0 ψ / q Z p Y 0 ψ2 /Y qd ψ2g = fψ1 = piψ1, 3çò j ¶ qj = g, ψj = iψ1. =kXeÜ„: X0 iψ1 g % j Z 0 ψ / q Z p Y 0 ψ2 /Y d qj = g ±9 g ⁄ q ˛¥f d j ¥f d (f d“Ìn”5ü). qd ψ2ϕ2f = f = pi, 3çò r : X −→ Z 0 ¶e„Ü: X i ϕ2f % r Z 0 ψ / q Z p Y 0 ψ2 /Y 5ø ϕi : X −→ Z 0 ⁄ jϕ1 : X −→ Z 0 ˛˜v r 5ü, = qϕi = ϕ2pi = ϕ2f, ψϕi = i; qjϕ1 = gϕ1 = ϕ2f, ψjϕ1 = iψ1ϕ1 = i. 14
由r的唯一性知oi=j91.于是即有如下交换图 XXX 其中g=1z.于是i是j的缩回.我们希望能证明j是平凡余纤维，但目前不知道j是否 是余纤维，必须作进一步的考虑。 将方分解成j=可分，其中才是余纤维，可是平凡纤维，即 X- 入H 因为)和可均是弱等价，故才也是弱等价，从而才是平凡余纤维」 由p1=1=i,即下图中的方块交换 因为是平凡纤维，：是余纤维，由()知存在：Z一H使得1=，=.于 是有如下交换图： X1X'1、X 其中p中=p=z.于是i是平凡余纤维子的缩回.因此，由平凡余纤维对缩回封闭的 题设知i是平凡余纤维， 下述定理给出了模型结构与闭模型结构的关系（其中（②）中()与()的等价性在[Q) 中是用同伦范畴的理论得到的，这里是与李志伟讨论直接得到的；而()在[Q则中并未提及)， 定理2.8(D.Quillen,.1967)()但闭模型结构是模型结枸.从而，闭模型范咋是模型范哈 (②设(Coi,M,WM)是范M上的模型结构.知平纤维的拉回 存在，或者平几余纤维的推出存在，则下迷等价 15
d r çò5 ϕi = jϕ1. u¥=kXeÜ„: X ϕ1 / i X0 ψ1 / j X i Z ϕ / p Z 0 ψ / q Z p Y ϕ2 /Y 0 ψ2 /Y Ÿ• ψϕ = IdZ. u¥ i ¥ j †£. ·ÇF"Uy² j ¥²Ö{në, 8cÿ j ¥ƒ ¥{në, 7Lä?ò⁄ƒ. Ú j ©)§ j = p 0 j 0 , Ÿ• j 0 ¥{në, p 0 ¥²Önë, = X0 j / j 0 % Z 0 H p 0 9 œè j ⁄ p 0 ˛¥f d, j 0 è¥f d, l j 0 ¥²Ö{në. d p 0 j 0ϕ1 = jϕ1 = ϕi, =e„•ê¨Ü X j 0ϕ1 / i H p 0 Z ϕ / ϕ 0 > Z 0 œè p 0 ¥²Önë, i ¥{në, d (M1) 3 ϕ 0 : Z −→ H ¶ ϕ 0 i = j 0ϕ1, p 0ϕ 0 = ϕ. u ¥kXeÜ„: X ϕ1 / i X0 ψ1 / j 0 X i Z ϕ 0 / p H ψp0 / qp0 Z p Y ϕ2 /Y 0 ψ2 /Y Ÿ• ψp0ϕ 0 = ψϕ = IdZ. u¥ i ¥²Ö{në j 0 †£. œd, d²Ö{nëȆ£µ4 K i ¥²Ö{në. e„½nâ— .(Ü4 .('X (Ÿ• (2) • (i) Ü (ii) d53 [Q1] •¥^”‘âÆnÿ,˘p¥Üoìï?ÿÜ; (iii) 3 [Q1] •øôJ9). ½n 2.8 (D. Quillen, 1967) (1) 4 .(¥ .(. l , 4 .âÆ¥ .âÆ. (2) (Cofib(M), Fib(M), Weq(M)) ¥âÆ M ˛ .(. XJ²Önë.£ 3ß½ˆ²Ö{nëÌ—3ßKe„ d 15