Background
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Model categories are an axiomization of the machinery underlying the study of topological spaces up to homotopy equivalence. They consist of a category $C$, together with three distinguished classes of morphism: Weak Equivalences, Fibrations, and Cofibrations. There are then a series of axioms this structure must satisfy, to guarantee that the classes behave analogously to the topological maps they are named after. The axioms can be found [here](http://en.wikipedia.org/wiki/Model_category).


(As far as I know...) The main practical advantage of this machinery is that it gives a rather concrete realization of the localization category $C/\sim$ where the Weak Equivalences have been inverted, which generalizes the homotopy category of topological spaces. The main conceptual advantage is that it is a first step towards formalizing the concept of "a category enriched over topological spaces".


A discussion of examples and intuition can be found at [this question](https://mathoverflow.net/questions/2185/how-to-think-about-model-categories).


The Question
------------


The examples found in the answers to [Ilya's question](https://mathoverflow.net/questions/2185/how-to-think-about-model-categories), as well as in the introductory papers I have read, all have a model category structure that could be expected. They are all examples along the lines of topological spaces, derived categories, or simplicial objects, which are all conceptually rooted in homotopy theory and so their model structures aren't really surprising.


I am hoping for an example or two which would elicit disbelief from someone who just learned the axioms for a model category. Along the lines of someone who just learned what a category being briefly skeptical that any poset defines a category, or that '$n$-cobordisms' defines a category.



What are surprising examples of Model Categories?
Here is an [example](http://arxiv.org/abs/0906.4087) that surprised me at some time in the past.
Bisson and Tsemo introduce a nontrivial model structure on the topos of directed graphs.
Here a directed graph is simply a $4$-tuple $(V,E,s,t)$ where an arc $e \in E$ starts at $s(e) \in V$ and ends at $t(e) \in V$. Fibrations are maps that induce a surjection on the set of outgoing arcs of each vertex, cofibrations are embeddings obtained by attaching a bunch of trees, and weak equivalences are maps that induce a bijection on the sets of cycles.


In this model structure fibrant objects are graphs without sinks and
cofibrant objects are graphs with exactly one incoming arc for every vertex.
Cofibrant replacement replaces a graph by the disjoint union of its cycles
with the obvious morphism into the original graph.


We have a chain of inclusions of categories $A\to B \to C\to D$, where $D$ is the topos of directed graphs, $C$ is the full subcategory of $D$ consisting of all graphs with exactly
one incoming arc for each vertex, $B$ is the full subcategory of $C$ consisting of all graphs with exactly one outgoing arc for each vertex, and $A$ is the full subcategory of $B$ consisting of all graphs such that $s=t$.


Each functor is a part of a Quillen adjunction and total left and right derived
functors compute nontrivial information about graphs under consideration.


Two finite graphs are homotopy equivalent iff they are isospectral iff their zeta-functions coincide.


