Contractibility of the category of cosimplicial resolutions

Let $$γ:C→M\gamma : \mathcal{C} \to \mathcal{M}$$ be a functor and define a cosimplicial resoultion of $$γ\gamma$$ as a functor $$Γ:C→MΔ\Gamma: \mathcal{C} \to \mathcal{M}^{\Delta}$$ such that

• $$ΓC\Gamma C$$ is Reedy cofibrant for every $$C∈CC \in \mathcal{C}$$
• for every $$CC$$ there is a natural weak equivalence $$w(C):ΓC∼→c∗γCw(C):\Gamma C \xrightarrow{\sim} c^* \gamma C$$

We can define a category $$R=coRes(γ)\mathcal{R}=\text{coRes}(\gamma)$$ where the morphisms are natural transformations $$η:Γ1→Γ2\eta:\Gamma_1 \to \Gamma_2$$ such that for all $$CC$$ the obvious triangles commute i.e. we have $$w2(C)∘ηC=w1(C)w_2(C) \circ \eta_C = w_1(C)$$ for all $$C.C.$$

I would like to understand why this category, as is well known, is
contractible.

Since I do not understand anything of the proof I found in the text I consulted, I am trying to prove it by myself in the following way:

• A resolution exists because for every $$C,C,$$ we can find a cofibrant object $$XCX_C$$ in $$MΔ\mathcal{M}^{\Delta}$$ and a weak equivalence $$XC∼→c∗γCX_C \xrightarrow{\sim} c^*\gamma C$$ and this defines a functor $$X(C)=XCX(C)=X_C$$ by functorial factorization.
• For every $$Γ∈R,\Gamma \in \mathcal{R},$$ by functorial factoriazion there is a morphism $$X→Γ.X \to \Gamma.$$
• If I call weak equivalence in $$R\mathcal{R}$$ a map $$η\eta$$ such that $$ηC\eta_C$$ is a weak equivalence in the Reedy model structure in $$MΔ\mathcal{M}^{\Delta}$$ for all $$C,C,$$ then given any map of resolutions $$η:Γ1→Γ2,\eta:\Gamma_1 \to \Gamma_2,$$ by commutativity of the triangle we have that $$η\eta$$ is a weak equivalence under this defintion.
• Now, my naive intuition is that the contractibility of $$R\mathcal{R}$$ should follow from the fact that if we formally invert all morphisms in $$R=coRes(γ)\mathcal{R}=\text{coRes}(\gamma)$$, the resulting localization $$R[R−1]\mathcal{R}[\mathcal{R}^{-1}]$$ is a simply connected groupoid, hence contractible.
• I put on $$R\mathcal{R}$$ the equivalence relation given by identifying all parallel morphisms, which is a congruence. In this way, all morphisms become invertible in the quotient so that I can call $$R/∼=R[R−1]\mathcal{R}/{\sim}=\mathcal{R}[\mathcal{R}^{-1}]$$ and I have the quotient functor $$q:R→R[R−1].q:\mathcal{R}\to \mathcal{R}[\mathcal{R}^{-1}].$$
• For every $$Γ,\Gamma,$$ the arrow category $$Γ↓q\Gamma \downarrow q$$ is contractible having initial object, so I conclude by Quillen’s theorem A.

Is this proof reasonable?

Edit The last bullet point is wrong because when I pass to the comma category I lose the initial object.

Also, apparently we cannot just pass to the quotient without using some extra propery of $$R\mathcal{R}$$: if it were possible to apply the reasoning I wanted to make, it would imply that any category with an object $$XX$$ such that $$Hom(X,A)≠∅\text{Hom}(X,A) \neq \emptyset$$ and $$Hom(A,X)≠∅\text{Hom}(A,X) \neq \emptyset$$ for all $$AA$$ would become contractible. And I just found counterexamples to this fact in this other question.

I still wonder if by using some more property of $$R\mathcal{R}$$, for example the fact that the maps I am inverting were all weak equivalences in some model structure, we can still deduce the contractibility of $$R\mathcal{R}$$ from that of $$R[R−1]\mathcal{R}[\mathcal{R}^{-1}]$$ along the quotient functor in this case.

Answer

Since you have functorial factorisations you should exploit that to the hilt.

If $$M\mathcal{M}$$ is a model category with functorial factorisations then the category $$cM\mathbf{c}\mathcal{M}$$ of cosimplicial objects in $$M\mathcal{M}$$, with the Reedy model structure, is also a model category with functorial factorisations. There is an obvious fully faithful embedding $$M→cM\mathcal{M} \to \mathbf{c} \mathcal{M}$$, so we may as well just forget about cosimplicial objects and just prove the following claim:

For every model category $$M\mathcal{M}$$ with functorial factorisations and every diagram $$F:C→MF: \mathcal{C} \to \mathcal{M}$$, the full subcategory $$Q(F)\mathcal{Q} (F)$$ of the over-category $$[C,M]/F[\mathcal{C}, \mathcal{M}]_{/ F}$$ spanned by the componentwise cofibrant replacements of $$FF$$ is contractible.

Indeed, let $$Q:M→MQ : \mathcal{M} \to \mathcal{M}$$ be a functor and let $$p:Q⇒idMp : Q \Rightarrow \textrm{id}_\mathcal{M}$$ be a natural transformation such that, for every object $$MM$$ in $$M\mathcal{M}$$, $$QMQ M$$ is a cofibrant object in $$M\mathcal{M}$$ and $$pM:QM→Mp_M : Q M \to M$$ is a weak equivalence in $$M\mathcal{M}$$. Such $$QQ$$ and $$pp$$ exist because $$M\mathcal{M}$$ has functorial factorisations. Then, for every natural transformation $$α:F′⇒F\alpha : F' \Rightarrow F$$ and every object $$CC$$ in $$C\mathcal{C}$$, we have the following commutative square in $$M\mathcal{M}$$:
$$\require{AMScd} \begin{CD} Q F’ C @>{p_{F’ C}}>> F’ C \\ @V{Q \alpha_C}VV @VV{\alpha_C}V \\ Q F C @>>{p_{F C}}> F C \end{CD}\require{AMScd} \begin{CD} Q F' C @>{p_{F' C}}>> F' C \\ @V{Q \alpha_C}VV @VV{\alpha_C}V \\ Q F C @>>{p_{F C}}> F C \end{CD}$$
This is all natural in $$CC$$, so we actually have a commutative square in $$[\mathcal{C}, \mathcal{M}][\mathcal{C}, \mathcal{M}]$$, hence a zigzag $$(Q F, p F) \leftarrow (Q F’, \alpha \bullet p F’) \rightarrow (F’, \alpha)(Q F, p F) \leftarrow (Q F', \alpha \bullet p F') \rightarrow (F', \alpha)$$ in the overcategory $$[\mathcal{C}, \mathcal{M}]_{/ F}[\mathcal{C}, \mathcal{M}]_{/ F}$$. But $$(Q F, p F)(Q F, p F)$$ is a componentwise cofibrant replacement of $$FF$$, and this is natural in $$F’F'$$, so we have a zigzag of natural transformations connecting the identity functor on $$\mathcal{Q} (F)\mathcal{Q} (F)$$ and a constant functor. Therefore $$\mathcal{Q} (F)\mathcal{Q} (F)$$ is contractible.

If you are geometrically inclined, you may think of the above proof as constructing a deformation retract of $$\mathcal{Q} (F)\mathcal{Q} (F)$$ to a point. Of course, any space with a deformation retract to a point is contractible. The gist of the argument is widely applicable and can be used in contexts where one does not have a model structure per se – this, I think, is the point of Part II of Homotopy limit functors on model categories and homotopical categories.

Attribution
Source : Link , Question Author : giuseppe , Answer Author : Zhen Lin