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Is geometric realization $|\cdot|:\mathbf{Top}^{\mathbf{\Delta}^{\textrm{op}}}\rightarrow \mathbf{Top}$ a left Quillen functor? If so, under what model structure on $\mathbf{Top}^{\mathbf{\Delta}^{\textrm{op}}}$? I would guess the Reedy model structure.

A reference would be ideal.


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Shouldn't it be "left" instead of "right"? – Tom Goodwillie Mar 23 '11 at 0:05
yeah left. oops, thanks! – Alan Wilder Mar 23 '11 at 5:51
up vote 3 down vote accepted

The results on homotopy invariance of the geometric realization of simplicial spaces go back at least to May's The Geometry of Iterated Loop Spaces, although he doesn't explicitly mention model categories. It is indeed true that the geometric realization of simplicial spaces is a left Quillen functor with respect to the Reedy model structure. One reference is Proposition VII.3.6 of Simplicial Homotopy Theory by Goerss and Jardine. There is also a survey of related results in nLab.

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