What are surprising examples of Model Categories?

Question

Background

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.

(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.

The Question

The examples found in the answers to Ilya's question, 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.

Answer 1

Here is an example 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.

Answer 2

The category of sets admits precisely nine model category structures, no more no less.

I learned this fact from Tom Goodwillie's comments on a different MO question. It always shocks people when I mention it to them, so I guess it is surprising. I am not sure which is more surprising, that you can actually compute all the model structures or that there are exactly nine of them. Working out the details is such a fun exercise that I don't want to spoil the fun here.

Answer 3

There is a preprint (arXiv) by Finnur Larusson explaining a model structure on equivalence relations. From the abstract:

We give a detailed exposition of the homotopy theory of equivalence relations, perhaps the simplest nontrivial example of a model structure.

Answer 4

There is a series of paper by Philippe Gaucher on the arxiv that deal with model categories in the context of theoretical computer science. E.g.:

• Abstract homotopical methods for theoretical computer science (0707.1449)

The purpose of this paper is to collect the homotopical methods used in the development of the theory of flows initialized by author's paper ``A model category for the homotopy theory of concurrency''.

• A model category for the homotopy theory of concurrency (math/0308054)

We construct a cofibrantly generated model structure on the category of flows such that any flow is fibrant and such that two cofibrant flows are homotopy equivalent for this model structure if and only if they are S-homotopy equivalent. This result provides an interpretation of the notion of S-homotopy equivalence in the framework of model categories.

I guess it is just because of my ignorance, but to me this was unexpected.

Answer 5

(more detail on the answer by mmm: Gavrilovich in http://arxiv.org/abs/1006.4647 and then further works of Gavrilovich and Hasson http://arxiv.org/abs/1102.5562 and then Gavrilovich, Hasson and Kaplan http://arxiv.org/abs/1111.3489 explore in depth connections with pcf theory (a part of set theory, one could say) - in particular, they recover Shelah's covering number $cov(\lambda, \aleph_1,\aleph_1, 2)$)... How surprising it is depends (I guess) on how much you are used to "detect" homotopic content in areas where it was seemingly not present.

Answer 6

Here is an example of a poset which is a model category. The construction is set-theoretic and mentions Continuum Hypothesis.

Answer 7

There is Krause and Nikolaus' model structure on the category of group presentations whose homotopy category is the category of groups.