MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.


share|cite|improve this question
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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.