Quillen lemma

As an application of the gluing lemma, let us prove the so-called Quillen lemma, which turned out to be a useful tool in Combinatorial Algebraic Topology 

Let ip P Q be an order-preserving map such that for any x (E Q the complex is contractible. Then the induced map between the simpli- 

Proof of the fact. For x e A we have A x = Q x- Therefore A A x) = A Q x) is contractible. The conditions of the Quillen lemma are thereby satisfied, and the fact follows by induction, since A iQj.  

Homotopy colimits is a subject of several books and surveys. A classical source is provided by [BoK72] see also [Vo73]. Probably the first use of homotopy colimits in combinatorics can be found in [ZZ93]. Our treatment of the general theory in this chapter has an introductory character. We recommend [WZZ99] for a more in-depth approach. 

Our treatment is self-contained with the exception of the standard Theorem 15.18, whose proof can be found in many textbooks, see, for example, [Bre93, Chapter I, Section 12]. 

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The Quillen lemma originally appeared in [Qu73] and [Qu78], with both sources being well worth a further look. 

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