I'm reading the Hirschhorn's book model categories and their localization and I have a question about frames and resolutions.

Following the book (definition 16.6.1) a cosimplicial frame on an object $X$ in a model category $\mathcal{M}$ is a cosimplicial object $A^{*}$ in $\mathcal{M}^{\Delta}$ such that

1)$A^{*}$ is weakly equivalent (with respect to the Reedy model structure on $\mathcal{M}^{\Delta}$) to $ccX$, the constant cosimplicial object in $\mathcal{M}^{\Delta}$, and the induced map at the degree $0$ $A^{0}\to X$ is an isomorphism in $\mathcal{M}$, and

2) If $X$ is cofibrant then $A^{*}$ is cofibrant in the Reedy model structure.

On the other hand a cosimplicial resolution of $X$ is a cofibrant replacement (in the Reedy model structure) of $ccX$.

Now for $A^{*}$ a cosimplicial resolution, we define the functor $\mathcal{M}(A,-)\: : \: \mathcal{M}^{op}\to sSet$ via $\mathcal{M}(A, Y)_{n}:=Hom(A^{n},Y)$. By proposition 16.5.2 we know that this functor satisfies the following: for any Reedy cofibration $i\: : \: A^{*}\to B^{*}$ between cosimplicial resolutions and any fibration $p\: : \:Y\to Y$ the map

$$(P)\quad\mathcal{M}(B, Y)\to \mathcal{M}(A, Y)\times_{\mathcal{M}(A, Z)}\mathcal{M}(B, Z)$$

is a fibration which is trivial if either $p$ or $i$ is trivial.

My questions

1) If $A^{*}$ is only a cosimplicial object, the functor $\mathcal{M}(A,-)\: : \: \mathcal{M}^{op}\to sSet$ doesn't satisfy (P) in general (e.g example 16.4.7). Assume that $A^{*}$ is a cosimplicial frame (not necessarily cofibrant). Does the functor $\mathcal{M}(A,-)$ satisfy (P)?

2) I ask this because it seems to me that the above property is used in the proof of Theorem 19.3.1, unless I'm wrong?

Some comments:

We can look at this question as: What are the minimal conditions on $A^{*}$ and $B^{*}$ such that $(P)$ is true?

I look at the proof of Theorem 16.5.2. There are some observations.

1) Very strange: If $A^{*}$ and $B^{*}$ are simply cosimplicial objects, either $p$ a trivial fibration or $i$ is a trivial cofibration then (P) is true and the resulting map is a trivial fibration (this is proposition 16.4.6).

2) Now assume that neither $p$ or $i$ are weak equivalences. Note that the proof of theorem 16.5.2 comes from proposition 16.4.12, and the proof of this is a consequence of lemma 16.4.11

So the question is: let $A^{*}$ be a cosimplicial object in $\mathcal{M}$, under which conditions all the inclusions $\Lambda[n,k]\to \Delta[n]$, for $n\geq 1$, $n\geq k\geq 0$, induce trivial cofibrations $$ (Q1)\quad A\otimes \Lambda[n,k]\to A\otimes \Delta[n]? $$ Lemma 16.4.11 show that if $A^{*}$ is a cosimplicial resolution, then (Q1) is true.

3) The proof of Lemma 16.4.11 is an induction over $n$. The fact that $A^{*}$ is a resolution give a proof for the case $n=1$ (the first step). This step is equivalent to: $$(Q2)\quad A^{1}\text{ is a cylinder object on }A^{0}\text{ where }d_{0},d_{1}\text{ are cofibrations. }$$ On the other hand the higher induction steps don't depend by $A^{*}$.

4) Conclusion: (Q1) is true if $A^{*}$ and $B^{*}$ satisfy (Q2). Then (P) is true if $A^{*}$ and $B^{*}$ satisfy (Q2).

Question: Let $A^{*}$ be a cosimplicial frames, does $A^{*}$ satisfy (Q1)?


1 Answer 1


Consider the Hovey's definition of cosimplicial frame. A cosimplicial frame $\tilde{X}^{*}$ on an object $X\in \mathcal{M}$ is a factorization $p^{*}(X)\to \tilde{X}^{*}\to ccX$ of the fold map $p^{*}(X)\to ccX$, where the first map is a cofibration and the second is a weak equivalence that is an isomorphism in degree zero. Here $$p^{*}(X)_{n}:=\coprod _{[n]}X$$ is the $n+1$ fold product and the cosimplicial structure comes from the simplex category $\Delta$. Now these cosimplicial frames satisfy that any inclusions $\partial\Delta[n]\to\Delta[n]$ induces a cofibration $$\tilde{X}^{\partial \Delta[n]}\to \tilde{X}^{\Delta[n]}.$$ This property doesn't work in the Hirschhorn's definition of simplicial frame: e.g let $\mathcal{M}$ be a simplicial model category, then for any $X\in\mathcal{M}$ the functor $X\otimes-$ is a cosimplicial frame in the sense of Hirshhorn but not in the sense of Howey.

But using the Hovey's definition it is possible prove (Q2). I don't know how to prove (Q2) using only the Hirschhorn's definition.


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