TITLE: A fibrant-objects structure on Top
QUESTION [7 upvotes]: (Sorry for the crossposting, but I'm really interested in this question).
One can define (Paragraph 1.5, page 10) a fibrant-object structure on a suitable cartesian closed category of topological spaces $\bf Top$, called the $\pi_0$-fibrant structure:

A $\pi_0$-equivalence is a map inducing a bijection at the level of $\pi_0$
A $\pi_0$-fibration is a continuous map $p\colon E\to B$ having the RLP with respect to the map $\{0\}\to [0,1]$ including the 0:
[I'm not able to reproduce the diagram, the TeX engine seems not to accept the "array" environment]

Every property defining a fibrant structure can be easily shown in the way you see.
Now I'm interested in extending this. The natural definition for a $\pi_n$-equivalence is a map $A\to B$ inducing isomorphisms $\pi_i(A)\to \pi_i(B)$ for all $0\le i\le n$.

What should a $\pi_n$-fibration be in order to define a fibrant structure $\pi_n\text{-}\bf Top$ for all $n\in\mathbb N$?
What if we "go to the limit" (and can it be done?) $\varinjlim_n \big(\pi_n\text{-}\bf Top\big)$ of these fibrant structures? Do we recover a known fibrant structure, obtained forgetting cofibrations and mutual lifting properties of a suitable model structure, on $\bf Top$?

REPLY [2 votes]: There was a mistake in an earlier version of the paper that you mention. If you define  $\pi_0$-fibrations and $\pi_0$-equivalences the way that you did, they do not give the structure of a category of fibrant objects on $\bf Top$. The reason is that then the acyclic fibrations will not be closed under pullbacks, as they should be by (FW1) loc. cit.
A counterexample is the map $R\to S^1$ of the real line to the circle that gives the universal cover ($t\to \exp(2\pi it)$). This is a $\pi_0$-acyclic fibration. Take the pullback of this map with respect to $*\to S^1$. The pullback is the fiber of the first map at the point, which is $Z$ (integers). The map from $Z$ to the point $*$ is not a $\pi_0$-equivalence.
I remarked this to Otgonbayar Uuye and he fixed this in the new version. The main conclusion he wanted to draw from it, namely, that Schochet fibrations and homotopy equivalences form a category of fibrant objects on $C^*$-algebras, still holds. The point is to use the standard (Quillen) model structure on $\bf Top$.
As for larger $n$, we can begin with the Quillen model structure on $\bf Top$, and take its left Bousfield Localization with respect to the map $S^{ n+1}\to *$. We obtain a new model structure on $\bf Top$ (so restricting to the fibrant objects we get a fibrant objects structure), with the weak equivalences being the $\pi_n$-equivalences.
Note, however, that there are less fibrations in this model structure then in the usual one. In other words, any $\pi_n$-fibration would also be a Serre fibration. For e.g., the fibrant objects in the localized model structure would be (not all spaces, but only) spaces $X$ such that $\pi_i(X)=0$ for $i> n$.