 By Robert M.F. Moss, Charles B. Thomas

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We are going to construct Ωσ : G(σ) → E(σ) by induction on the dimension of σ. Let dim(σ) = 0. Since Ψσ : F (σ) → G(σ) is an (acyclic) coﬁbration in M, by the lifting axiom, there exists h : G(σ) → E(σ) which makes the following diagram 13. F unb (K, M) AS A MODEL CATEGORY 33 commutative: G E(σ) xY x h xx Ψσ Φσ xx x  x  G B(σ) G(σ) F (σ) Deﬁne Ωσ := h. Let us assume that the appropriate morphisms Ωσ : G(σ) → E(σ) have been constructed for all the simplices σ ∈ K such that dim(σ) < n. Let dim(τ ) = n.

In order to capture those properties of absolute coﬁbrations which are local, we introduce now the notion of a relative coﬁbration. 1. Definition. Let f : L → K be a map and Ψ : F → G be a natural transformation in F unbf (L, M). For any simplex σ : ∆[n] → L, pull-back Ψ along σ ∂∆[n] → ∆[n] → L, take colimits, and deﬁne: MΨ (σ) := colim colim∆[n] F o colim∂∆[n] F colim∂∆[n] Ψ G colim∂∆[n] G (cf. Section 12). • We say that Ψ : F → G is an (acyclic) f -coﬁbration if, for any simplex σ ∈ L such that f (σ) is non-degenerate in K, the morphism MΨ (σ) → G(σ) is an (acyclic) coﬁbration in M.

1. 2. Remark. 1). A natural transformation Ψ ∈ F unbf (L, M) is a weak equivalence, a ﬁbration, or a coﬁbration if and only if, for any simplex σ : ∆[n] → L, its pull-back σ ∗ Ψ is so in F unbf ◦σ (∆[n], M). 1. Consider the reduction L −→ red(f ) → K of f . 1. 3. Remark. Let f : L → K be a map and L −→ red(f ) → K be its reduction. 1 relies on the fact that the pair of adjoint functors: F unbf (L, M) o (fred )k (fred ) ∗ G F unb red(f ), M is a Quillen equivalence (see ) of model categories.