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By Virgil Snyder, Charles Herschel Sisam

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Let dim(K) = n. , K fits into a push-out square:  G ∆[n] ∂∆[n]  σ   N  GK where dim(N ) < n. 6 (2) to show that colimK Ψ is an (acyclic) cofibration. The pull-back of Ψ along N → L is an (acyclic) cofibration, and thus by the inductive assumption so is colimN Ψ. Since the map ∂σ : ∂∆[n] → K can send non-degenerate simplices to degenerate ones, the pull-back of Ψ along ∂∆[n] → K is not a cofibration. Therefore we can not apply the inductive assumption directly to argue that colim∂∆[n] Ψ is an (acyclic) cofibration.

17. 2. We want to warn the reader, who could be tempted to skip the end of the chapter, that it contains a very fundamental tool for the study of bounded diagrams: the reduction process (cf. Section 18). 5). In fact this construction preserves more properties. 1. 1. Definition. Let f : L → K be a map of spaces and F : L → C be a functor. We say that F is f -bounded if, for any simplex σ ∈ L such that f (σ) = si ξ 17. 2). A simplex σ ∈ L is called f -non-degenerate if f (σ) is non-degenerate in K.

The pull-back of Ψ along σ is an (acyclic) (f ◦ σ)-cofibration. 3, the map Ψ σ colim∆[n] F = F (σ) −→ G(σ) = colim∆[n] G is an (acyclic) cofibration. Proof of 3. If Ψ is an acyclic f -cofibration in F unbf (L, M), then in particular it is an f -cofibration. Now part 1 of the corollary shows that, for any simplex σ ∈ L, Ψσ is a weak equivalence. Assume that Ψ is an f -cofibration and, for any σ ∈ L, Ψσ is a weak equivalence. For any simplex σ : ∆[n] → L, consider the following commutative diagram associated with σ: colim∂∆[n] F  F (σ) @A G colim∂∆[n] G vvv vvv vvv v8  G MΨ (σ) G G(σ) BCy Ψσ To prove the second implication, we have to show that MΨ (σ) → G(σ) is a weak equivalence for any σ ∈ L.

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