By Wojciech Chacholski, Jerome Scherer

During this paper we enhance homotopy theoretical equipment for learning diagrams. particularly we clarify tips to build homotopy colimits and boundaries in an arbitrary version classification. the most important proposal we introduce is that of a version approximation. A version approximation of a class $\mathcal{C}$ with a given type of vulnerable equivalences is a version type $\mathcal{M}$ including a couple of adjoint functors $\mathcal{M} \rightleftarrows \mathcal{C}$ which fulfill definite houses. Our key outcome says that if $\mathcal{C}$ admits a version approximation then so does the functor class $Fun(I, \mathcal{C})$. From the homotopy theoretical viewpoint different types with version approximations have comparable homes to these of version categories.They admit homotopy different types (localizations with appreciate to susceptible equivalences). in addition they can be utilized to build derived functors via taking the analogs of fibrant and cofibrant replacements. a class with susceptible equivalences could have a number of valuable version approximations. We make the most of this risk and in every one scenario pick out one who fits our wishes. during this means we end up the entire basic homes of the homotopy colimit and restrict: Fubini Theorem (the homotopy colimit - respectively restrict- commutes with itself), Thomason's theorem approximately diagrams listed through Grothendieck structures, and cofinality statements. because the version approximations we current right here encompass yes functors 'indexed through spaces', the most important position in all our arguments is performed through the geometric nature of the indexing different types.

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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.