There is an example of a <i>model category</i> structure on a preorder of families of sets, defined as follows: $X\leq Y$ iff every element $x\in X$ is bounded $x\leq y_x$ by some element of $y_x\in Y$;
families $X$ and $Y$ are weakly equivalent iff $X\leq Y$ and for every $y\in Y$ there is $x\in X$ such that $y$ and $x$ differ by finitely many elements; cofibrant objects are families of countable sets. 

The example is also quite easy, however, it is set theoretic in nature. 
You may find the definitions in

<a href="http://arxiv.org/abs/1102.5562">A homotopy theory for set theory, I</a>

<a href="http://arxiv.org/abs/1006.4647">A homotopy approach to set theory</a>