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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.

Thanks

edited Mar 23 2011 at 5:52

asked Mar 22 2011 at 23:47

Alan Wilder
449●2●8

	Shouldn't it be "left" instead of "right"? – Tom Goodwillie Mar 23 2011 at 0:05
	yeah left. oops, thanks! – Alan Wilder Mar 23 2011 at 5:51

Karol Szumiło · Accepted Answer · 2011-03-23 10:57:50Z UTC

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.

geometric realization on $\mathbf{sTop}$

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