# Homotopy colimits preserve weak equivalences

It is well known that homotopy colimits of diagrams are constructed so that if one has weak equivalences between all objects of two diagrams (under the same indexing category) the induced map between the two homotopy colimits is as well a weak equivalence. However, I have only found proofs working in abstract model categories where annoying cofibrancy assumptions have to be made. I am looking for a nice geometric proof in the category of Topological spaces, or, for that matter, in the category of CW complexes, omitting these. Thanks in advance!

• If you use use the definition of homotopy colimit as the Kan extension (to the localization) of the limit functor, the homotopy colimit automatically preserves weak equivalences. I don't know what cofibrancy assumptions you're talking about. – user40276 Jun 4 '15 at 11:33
• It really depends on what you mean by homotopy colimit. – Zhen Lin Jun 4 '15 at 13:09
• I would like to assume the definition which firstly obtains a simplicial replacement of the diagram, and then applies a certain coequalizer functor to realize the geometric intuition which thinks of the maps in the diagram as edges in simplices and associates mapping cylinders with them. Specifically, I am referring to definition 4.5 in pages.uoregon.edu/ddugger/hocolim.pdf, which uses the notion of a simplicial replacement introduced on page 15. – TheHumanHighway Jun 4 '15 at 13:21
• In the link that you cite, there is a proof of what you want just after the definition of homotopy colimit (and a reference for a general proof). Maybe you want some intuition about the cofibrant replacement functor. If you know about sheaves, it's analogous to the situation where you substitute a sheaf by its injective resolution. In the case of good topological spaces, it's just the cylinder object. – user40276 Jun 5 '15 at 3:45