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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:

  1. A $\pi_0$-equivalence is a map inducing a bijection at the level of $\pi_0$
  2. 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: $$ \begin{array}{ccc} \{0\} & \to & E \\ \downarrow &\nearrow_{\exists \alpha}& \downarrow p\\ [0,1] &\to_{\forall\gamma} & B \end{array} $$ 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 $A\to B$ indcuing 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 $\bf Top$?

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I add as a comment what I tried to do: "having the RLP with respect to $\{0\}\to [0,1]$" can be generalized in "having the RLP with respect to any of the $i+2$ inclusions $\triangle_i\to \triangle_{i+1}$, $\triangle_i$ being the $i$-simplex. I strongly suspect I'm asking too much in some ways, and not enough in others. – Fosco Loregian Nov 10 '12 at 11:07
Probably this should be asked at mathoverflow ;). – Martin Brandenburg Nov 30 '12 at 10:58

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