A fibrant-objects structure on Top

Question

(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 $\bf Top$?

Answer 1

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. 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 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. 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. Here is the reference:

J. Ignacio Extremiana Aldana , L. Javier Hernández Paricio , M. Teresa Rivas Rodríguez. Closed Model Categories For [n,m]-Types (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.

Answer 2

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 $i> n$.

Answer 3

This might be a naive answer but here is a suggestion for the definition of $\pi_n$-fibrations: maps having the RLP with respect to the the map $\Delta^k\to\Delta^k\times I$ for any $k\leq n$.

In the limit you will get the "obvious" fibrant-object structure on $Top$ which comes from its usual model structure (recall that the full subcategory of fibrant objects in any model category is a category of fibrant objects.. and that all objects are fibrants in $Top$).