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. !·ÇÚ04.(Vg!ƒ5ü. 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) (fd“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/, fd‹§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,.(±￾ߨÈı~f. 7