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 ¥fd. ^ Mt ½ W L´§k²ÖÈñä§a. u¥, náÈña, ½náfâÆ: Cß F, W. SK  (Cofib(M), Fib(M), Weq(M)) ¥âÆ M ˛òá.(, A ⁄ B ¥¸á ²ÖÈñ. K?ø f : A −→ B —¥fd. §1.3 fd©) ⁄n 1.3 3òá.(•, fdT¥²Ö{nëܲÖn닧, = Weq(M) = (Fib(M) ∩ Weq(M)) · (Cofib(M) ∩ Weq(M)). y² dfd‹§Ñ¥fd (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) (=ßfd“Ìn”5ü) p ∈ Weq(M),  p ∈ Fib(M) ∩ Weq(M). u¥ Weq(M) ⊆ (Fib(M) ∩ Weq(M)) · (Cofib(M) ∩ Weq(M)). 5