Return to Answer

There was a mistake in an earlier version of [the paper][2] 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 $i> n$. [2]: http://arxiv.org/abs/1011.2926

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 $i> n$.

There was a mistake in an earlier version of [the paper][2] 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 an 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 $i> n$. [2]: http://arxiv.org/abs/1011.2926

There was a mistake in an earlier version of [the paper][2] 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 $i> n$. [2]: http://arxiv.org/abs/1011.2926

There was a mistake in an earlier version of [the paper][2] 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 an 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 $\bf Top$ (and, in particular, a fibrant objects structure), with the weak equivalences being the $\pi_n$-equivalences (that is, maps $A\to B$ inducing isomorphisms $\pi_i(A)\to \pi_i(B)$ for all $0\le i\le n$).

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 $i> n$. [2]: http://arxiv.org/abs/1011.2926

There was a mistake in an earlier version of [the paper][2] 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 an 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 $i> n$. [2]: http://arxiv.org/abs/1011.2926