(Sorry for the [crossposting](https://math.stackexchange.com/questions/234108/a-fibrant-objects-structure-on-bf-top), but I'm really interested in this question).


One can [define](http://arxiv.org/abs/1011.2926) (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 $\bf Top$?
> 
> 
> 



A fibrant-objects structure on Top
Your question was addressed in the following paper:


Carmen Elvira-Donazar and Luis-Javier Hernandez-Patricio. [Closed model categories for the $n$-type of spaces and simplicial sets](http://www.sci-prew.inf.ua/v118/1/S0305004100073485.pdf). Math. Proc. Camb. Phil. Soc. (1995), 118, 93.


Allow me to define an $n$-fibration by quoting from the introduction: Let $I^p$ be the $p$-dimensional unit cube, $V^{p-1}$ be the union of all faces of $I^p$ except for $I^p\times \{1\}$ and $\partial I^p$ the boundary of $I^p$. A map $f$ is an $n$-fibration if it has the right lifting property with respect to $V^{p-1}\to I^p$ (for $0 < p \leq n+1$) and with respect to $V^{n+1}\to \partial I^{n+2}$.


With this definition, and your notion of an $n$-equivalence they prove that $Top$ (meaning a suitable cartesian-closed version) is a model category. So you can forget all mention of cofibrations and get the fibrant-object structure you wanted. The proof proceeds by way of Simplicial Sets, so if you read that paper you'll probably learn loads more about $n$-fibrations. For instance, Corollary 2.1 says trivial $n$-fibrations are exactly maps which have the RLP with respect to $\partial I^p\to I^p$ for $0\leq p\leq n+1$.


It is not difficult to see from the description of $n$-equivalences and $n$-fibrations that in the limit as $n\to \infty$ you get the usual model structure on $Top$. I should mention that [this paper of Golasinski and Goncalves](http://www.emis.de/journals/BBMS/Bulletin/bul972/golasinski.pdf) credits this model structure to Tim Porter and J.L. Hernandez via *Categorical models of $n$-types for procrossed complexes and $\mathcal{J}\_n$-prospaces* from the 1990 Barcelona Conference on Algebraic Topology. But I couldn't find an online copy of that, so I went with the reference above instead.


Note that the dual question to your question (declaring $X\to Y$ to be an $n$-equivalence if $\pi\_k(X)\to \pi\_k(Y)$ is an isomorphism for all $k>n$) has also been answered, and again there is a model structure. Here is a reference:


J. Ignacio Extremiana Aldana, L. Javier Hernández Paricio, and M. Teresa Rivas Rodríguez. [A closed model category for ($n-1$)-connected spaces](http://www.ams.org/journals/proc/1996-124-11/S0002-9939-96-03606-4/). Proc. Amer. Math. Soc. 124 (1996), 3545-3553 




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EDIT (April 1, 2013):


I recently learned of another paper in this vein by the same authors, thanks to a comment of Fernando Muro over at [this MO thread](https://mathoverflow.net/questions/126199/top-model-structures/126206#126206). Here is the reference:


J. Ignacio Extremiana Aldana , L. Javier Hernández Paricio , M. Teresa Rivas Rodríguez. [Closed Model Categories For [n,m]-Types](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.50.594) (1997)


This combines the two types of truncation mentioned above to get a model structure on [n,m]-types (truncated by $n$ below and $m$ above) whose homotopy category is equivalent to the category of $n$-reduced CW complexes with dimension $\leq m+1$ and $m$-homotopy classes of maps. It actually does even more, because it gives a different model structure whose homotopy category is equivalent to the homotopy category of $(n-1)$-connected, $(m+1)$-coconnected CW complexes.


