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



geometric realization on $\mathbf{sTop}$
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](http://ncatlab.org/nlab/show/geometric+realization+of+simplicial+topological+spaces).


