By Mitrinovic D.S., Ulcar J., Anderssen R.S.

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2 (2)). 2. Proposition. For any map f : L → K, the following triangle commutes: fk F un(L, C) ss s colimsL s sss 6 C G F un(K, C) u uu colimK u u uz uu Proof. Let F : L → C be a diagram. 2 (1), colimL F = colimcolimK df F = colimK colimdf F = colimK f k F . 3. Proposition. Let f : L → K be a map of spaces. Then the pull-back process f ∗ : F un(K, C) → F un(L, C) is right adjoint to the left Kan extension functor f k : F un(L, C) → F un(K, C). 4. Corollary. The left Kan extension f k : F un(L, C) → F un(K, C) commutes with colimits.

Assume that σ contains the vertex i. Let k be such that lk = i. We consider two cases. , lk+1 = i+1. In this case (di ◦ si )(σ) = (lm > · · · > lk+2 > i = i > · · · > l0 ) ∈ (∆[n + 1])m , and hence (di ◦ si )(σ) = sk dk+1 σ. Since si (σ) ∈ ∆[n] is of the form sk τ , the morphisms F (dk ) : F (dk σ) → F (σ) and F (dk+1 ) : F (dk+1 σ) → F (σ) are isomorphisms (F is si -bounded). We deﬁne s∗i F (σ) → F (σ) to be the composite: s∗i F (σ) = F (di ◦ si )(σ) = F (sk dk+1 σ) F (sk ) G F (dk+1 σ) F (dk+1 ) G F (σ) It is clear that this composite is an isomorphism as F (sk ) is so (F is a bounded diagram).

Let f : L → K be a map and F : L → M be an f -bounded diagram. Then F is f -coﬁbrant if and only if, for any simplex σ : ∆[n] → L such that f (σ) is non-degenerate in K, the morphism colim∂∆[n] F → F (σ) is a coﬁbration in M. The most signiﬁcant aspect of being a relative coﬁbration is that this property can be checked locally. This is the key feature that absolute coﬁbrations are missing. Relative coﬁbrations have been introduced to enlarge the class of absolute coﬁbrations so the notion of coﬁbrancy would become local.