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