(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:

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: [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 $\bf Top$?