TITLE: A category with weak equivalences which is not a model category
QUESTION [9 upvotes]: I'm only considering complete and cocomplete categories. A pair $(\mathfrak{X} , \mathfrak{W}) $ is, by definition, a category with weak equivalences if  $ \mathfrak{X} $ is a category and $ \mathfrak{W} $ is a subcategory satisfying the $2$ out of $3$ axiom.
I was wondering if there are nice examples of categories with weak equivalences for which there is no model structure. 
Thank you

REPLY [7 votes]: Here is another (more combinatorial) answer.  Let $\kappa$ be any (nice) cardinal, and let $\mathcal{A}$ be the union of the full subcategory of $\mathbf{Set}$ consisting of all sets of cardinality less than $\kappa$ and the subcategory of all isomorphisms in $\mathbf{Set}$.  $\mathcal{A}$ is closed under retracts and 2-of-3, but it is NOT the subcategory of weak equivalences of a model structure on $\mathbf{Set}$.  We can see this just by listing all of the (nine) model structures on $\mathbf{Set}$, and noting that the subcategories of weak equivalences are

all morphisms
all isomorphisms
all morphisms between nonempty sets (plus the identity on $\emptyset$)

(See Tom Goodwillie's answer to https://mathoverflow.net/questions/29653.) In particular, model categories can determine whether you're empty or nonempty, but they can't differentiate between different set sizes.