Do weak equivalences in a model category admit a calculus of fractions? Suppose it is given a model category $M$, $W$ being the set of its weak equivalences. One can define the localization of $M$ along $W$, denoted by $W^{-1}M$. In order to work with $W^{-1}M$, it would be of great help if $W$ admitted a calculus of fractions. So, my question is: is that true in general, or perhaps under some appropriate assumptions? I suspect the answer in general is "no", but I really hope it is "yes under appropriate assumptions": I'm currently working on dg-categories and craving for a "simple" description of the category $\mathrm{Ho}(\mathbf{dg\text{-}cat})$ of small dg-categories localized along quasi-equivalences.
Thanks in advance for any answer!
 A: At the end I'll give you a reference to a counterexample in the paper of Meier and Ozornova. But before that, I want to talk in the other direction: besides the 3-arrow calculus that Charles Rezk mentions, there's another sense in which $W$ "almost" admits a calulus of fractions. It's a result in the same paper.
Recall the functor $\mathrm{Ex}: \mathsf{sSet} \to \mathsf{sSet}$, which is right adjoint to barycentric subdivision. There is a canonical natural transformation $1 \Rightarrow \mathrm{Ex}$, allowing to define maps $\mathrm{Ex}^n \to \mathrm{Ex}^{n+1}$ for every $n$, and most famously the colimit $\mathrm{Ex}^\infty$ was shown by Kan to be a fibrant replacement functor for the Kan-Quillen model structure on $\mathsf{sSet}$; in particular $\mathrm{Ex}^\infty X$ is a Kan complex for every simplicial set $X$ (I don't know if this extends further -- is $\mathrm{Ex}^\infty f$ a Kan fibration for every simplicial map $f$? Is $\mathrm{Ex}F$ injective-fibrant for every diagram $F$?). But moreover, a functor $F: \mathcal{C} \to \mathcal{D}$ between categories is fibrant in the Thomason model structure on $\mathsf{Cat}$ iff $\mathrm{Ex}^2 F$ is a Kan fibration (where we identify a category with its nerve).
All of this is just to motivate looking at $\mathrm{Ex}^2 \mathcal{C}$ for a category $\mathcal{C}$. Meier and Ozornova show that if $W$ is the category of weak equivalences in a model category (actually, only a substantially weaker structure called a "partial model category" is necessary), then $\mathrm{Ex}^2 W$ is a Kan complex (i.e. $W$ is Thomason-fibrant).
Why am I telling you this? Well, Meier and Ozornova also show something surprisingly (to me) clean: 


Theorem: If $\mathcal{C}$ is a category, then $\mathrm{Ex}^1 \mathcal{C}$ is a Kan complex if and only if $\mathcal{C}$ admits a left calculus of fractions.


It follows that $\mathcal{C}$ admits a two-sided calculus of fractions if and only if both $\mathrm{Ex}^1 \mathcal{C}$ and $\mathrm{Ex}^1 \mathcal{C}^\mathrm{op}$ are Kan complexes. Meier and Ozornova also suggest (Question 2 at the end) that there might be notions of ``$n+1$-arrow calculus" giving clean characterizations of $\mathrm{Ex}^n \mathcal{C}$ being a Kan complex -- which in particular would recover the result that the weak equivalences in a model category admit a 3-arrow calculus, since they are Kan after applying $\mathrm{Ex}^2$.
Finally, Meier and Ozornova provide examples answering your actual question in the negative. Finite sets and injections (Example 5.1) form a partial model category whose weak equivalences do not admit a left calculus of fractions. More pertinently (Example 5.3), $\mathsf{Cat}$ with the Thomason model structure is itself a model category which does not admit a calculus of left fractions.
 Note, though, that the weak equivalences in a left (resp. right) proper model category admits a left resp. right) caculus of fractions. 
A: I'm writing this answer to consolidate and make precise some of the claims found in the comments to this question and the other MO question that David White linked to.
Short answer: you have to restrict to fibrant (or, dually, cofibrant) objects, and then quotient by the appropriate homotopies before obtaining a calculus of fractions.
Let $M$ be a model category, and $W$ its weak equivalences. Let $M_f$ denote the full subcategory of fibrant objects.
Then $M_f$ is a category of fibrant objects, with fibrations and weak equivalences coming from $M$. In any category of fibrant objects, we can define homotopies between maps$^\dagger$. Let $M_f / \sim $ be the category obtained by quotienting out the homotopies. 
Then (the image of) $W$ admits a calculus of fractions (i.e. is a multiplicative system) in $M_f / \sim$, and further $\text{Ho}(M) \cong \text{Ho}(M_f) \cong \text{Ho}(M_f/ \sim)$. This is shown in the ncatlab entry I linked.
In many model categories (including $\textbf{dg-cat}$ and $\textbf{Ch}(R)$ with the projective model structure), all objects are fibrant, so $M_f = M$, and thus $M/\sim$ admits a calculus of fractions. But I have not found a reference that shows that $M/\sim$ always admits a calculus of fractions. If this is in fact true, it would be great if someone could provide a reference!
$^\dagger$ Note that the notion of right homotopy is the same in model categories as well as categories of fibrant objects, but (as far as I can tell) the notion of homotopy does not appear to be the same.
