By Hans Joachim Baues
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Then M, with cofibrations and weak equivalences as in M is a cofibration category. Dually Mf with fibrations and weak equivalences as in M is afibration category. Moreover, if M is proper (M, cof, we) is a cofibration category and (M, fib, we) is a fibration category. Proof: (C3) follows directly from (M2). We now prove (C4). For any object X we have by (M2) a factorization X >-^--* R -> e of X -+ e. We claim that R is a fibrant model. In fact, for each trivial cofibration R >-=> Q we have by (M 1) 16 I Axioms and examples the commutative diagram Q` )e This shows that each e-fibrant object in M is a fibrant model.
The algebra is augmented by e = po. We clearly have Q T(V) = V. For a map a: V -+ W of degree 0 let T(a):T(V)-> T(W) be given by T(a)(x t Ox Qx xn) = ax t 0 px axn. Then QT (a) = a.
The proofs below can be generalized without difficulty. Compare the result of Quillen (1967) that ChainR is a model category. Let ChainR be the full subcategory of ChainR consisting of chain complexes which are bounded below. 4) Proposition. 2) the category ChainR is a cofibration category for which all objects are fibrant. 12) below. 5) (V (D V')n = V" p+ V;,. The direct sum of chain complexes, V O+ V', satisfies in addition d(x + y) _ dx + dy for xeV, yeV'. = (@ Vi®X Wj. i+j=n R Here we assume that Vi is a right R-module and that Wj is a left R-module for i, 6 The category of chain complexes 41 j eZ.
Algebraic Homotopy by Hans Joachim Baues