The original question I had was:

If I have a sequence of simplicial spaces

$$A\to B\to C$$

which is degree-wise a homotopy fibration, under which conditions is the geometric realization also a homotopy fibration?

I bet there are tons of results on this. I have found the following theorem published by Anderson:


If $X\to Y$ is a map of simplicial spaces such that $\pi_0(f)$ is a Kan-fibration, and if the higher groupoids $\Pi_\infty(X)$ and $\Pi_\infty(Y)$ are fully fibrant, then for any map $g:Y'\to Y$ of simplicial spaces, if $X'$ is the homotopy theoretic fiber product of $Y'$ with $X$ over $Y$, $|X'|$ is the homotopy theoretic fiber product of $|Y'|$ with $|X|$ over $|Y|$.

Now the theorem answers the question, by letting $X=\ast$ and $Y=C$. The condition on $\pi_0(f)$ becomes then something easy, but I am having trouble understanding the motivation behind the $\Pi_\infty(Y)$ condition. In fact I have quite a lot of structure on the simplicial spaces in question and I doubt that it is even prudent to work with the definition itself. Can anybody enlighten me?

For me $C=Y$ is itself in every degree the classifying space of a category and moreover a group-like H-space.

Edit: I just remembered a different result by Waldhausen (Algebraic K-theory of generalized free products, Lemma 5.2)

Let $$A\to B\to C$$ be a sequence of bisimplicial sets such that that the composition is constant. Suppose geometric realization in one direction gives homotopy fibrations with connected base. Then the realization is a homotopy fibration.

So I guess the questions are:

  1. Does $C$ being the realization of a connected simplicial set imply that $\Pi_\infty(C)$ is fully fibrant?
  2. Can I say something if the base is not connected, but a group-like H-space, which implies that connected components are homotopy equivalent?
  • $\begingroup$ Could you add a reference for Anderson's paper? $\endgroup$ – Fernando Muro Sep 30 '13 at 15:49
  • $\begingroup$ Sure, it's 'fibrations and geometric realizations' Bulletin of the AMS, Vol 84 No 5, Sept 1978 $\endgroup$ – Simon Markett Sep 30 '13 at 16:08
  • $\begingroup$ I'd have to look at the Anderson paper to be sure, but I suspect his funny $\Pi_\infty$ condition is related to something called the "$\pi_*$-Kan condition", which was introduced by Bousfield and Friedlander as a hypothesis for results of exactly the one you quote from Anderson. I believe there is a discussion of this in the Goerss-Jardine book on simplicial sets. $\endgroup$ – Charles Rezk Oct 1 '13 at 13:13
  • $\begingroup$ Also look at mathoverflow.net/questions/18926/… $\endgroup$ – Charles Rezk Oct 1 '13 at 13:14

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.