6
$\begingroup$

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)?

$\endgroup$

1 Answer 1

1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.