Homotopy colimit

Dually, homotopy colimit is defined and denoted by hocolim, if T has small coproducts. 

Now consider the general case for a. As the functors in question on F changes coproducts to products, the map in question is a quasi-isomorphism if F is a direct sum of complexes bounded above with equivariant cohomology groups. Indeed, a direct product of quasi-isomorphisms of complexes of PM(A i) is again quasi-isomorphic. In particular, the lemma holds if F is a homotopy colimit of objects of As any object F of Dem X,) is the... 

Unlike the theory of homotopy colimits the theory of homotopy limits for simplicial sheaves on sites is different from the corresponding theory for simplicial sets because the analog of I.emma 1.20 does not hold for infinite homotopy limits. As a result holim functor may not preserve weak equivalences even between systems of pointwise fibrant objects unless the objects arc actually fibrant. An example of such a situation for an inhnite product is given below. A more sophisticated example is given in 1.30. 

Example 15.9. Let us see a few examples of homotopy colimits involving some of the diagrams that have previously appeared in the text. 

The homotopy colimit of the diagram in Example 15.2(4) consists of two spaces X, Y, and a cylinder connecting the two copies of X n F inside X and Y with each other see the last space in Figure 15.3. 

For the diagram from Example 15.9(1), we see that pb is the canonical projection of the homotopy colimit to whereas p/ is the quotient map that collapses each string along the attached cylinder to a point. 

It is now time to formalize an important property of homotopy colimits their flexibility with respect to homotopy type. It turns out that if the topological spaces in a diagram are replaced by homotopy equivalent ones in a coherent way (as a diagram map), then the homotopy type of the homotopy colimit... 

It is not difficult to see that the pair (X, A) is NDR. Therefore it is enough to show that the inclusion map A X is a homotopy equivalence. As we have said, A deformation retracts onto hocolimX>i, which in turn, by the construction of homotopy colimit, deformation retracts onto On the other hand, again by the construction of homotopy colimit, the whole space X deformation retracts onto the mapping cylinder of the map X n) An —> Bn-Since X(n) is a homotopy equivalence, we conclude that A is... 

Remark 15.13. It may be worthwhile to explicitly mention the following special case when all the spaces v) in the diagram are contractible, then we have a diagram map from V to the point diagram defined in Example 15.9(0). It follows by Theorem 15.12 that the homotopy colimit of our diagram is homotopy equivalent to the base trisp, with the homotopy equivalence given by Pfc. 

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. 

WZZ99] V. Welker, G.M. Ziegler, R. Zivaljevic, Homotopy colimits - comparison lemmas for combinatorial applications, J. Reine Angew. Math. 509 (1999), pp. 117-149. 

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