What are surprising examples of Model Categories? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T06:17:14Z http://mathoverflow.net/feeds/question/18336 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/18336/what-are-surprising-examples-of-model-categories What are surprising examples of Model Categories? Greg Muller 2010-03-16T04:36:08Z 2013-05-26T22:41:56Z <h2>Background</h2> <p>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 <a href="http://en.wikipedia.org/wiki/Model_category" rel="nofollow">here</a>.</p> <p>(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".</p> <p>A discussion of examples and intuition can be found at <a href="http://mathoverflow.net/questions/2185/how-to-think-about-model-categories" rel="nofollow">this question</a>.</p> <h2>The Question</h2> <p>The examples found in the answers to <a href="http://mathoverflow.net/questions/2185/how-to-think-about-model-categories" rel="nofollow">Ilya's question</a>, 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.</p> <p>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.</p> http://mathoverflow.net/questions/18336/what-are-surprising-examples-of-model-categories/18337#18337 Answer by Dmitri Pavlov for What are surprising examples of Model Categories? Dmitri Pavlov 2010-03-16T05:10:11Z 2012-03-12T22:50:20Z <p>Here is an <a href="http://arxiv.org/abs/0906.4087" rel="nofollow">example</a> 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.</p> <p>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.</p> <p>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 $s=t$.</p> <p>Each functor is a part of a Quillen adjunction and total left and right derived functors compute nontrivial information about graphs under consideration.</p> <p>Two finite graphs are homotopy equivalent iff they are isospectral iff their zeta-functions coincide.</p> http://mathoverflow.net/questions/18336/what-are-surprising-examples-of-model-categories/18350#18350 Answer by Henning Arnór Úlfarsson for What are surprising examples of Model Categories? Henning Arnór Úlfarsson 2010-03-16T10:11:55Z 2010-03-16T10:11:55Z <p>There is a preprint (<a href="http://arxiv.org/abs/math/0611344v2" rel="nofollow">arXiv</a>) by Finnur Larusson explaining a model structure on equivalence relations. From the abstract:</p> <blockquote> <p>We give a detailed exposition of the homotopy theory of equivalence relations, perhaps the simplest nontrivial example of a model structure.</p> </blockquote> http://mathoverflow.net/questions/18336/what-are-surprising-examples-of-model-categories/18388#18388 Answer by philip314 for What are surprising examples of Model Categories? philip314 2010-03-16T15:40:03Z 2010-03-17T10:21:59Z <p>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.:</p> <ul> <li>Abstract homotopical methods for theoretical computer science (<a href="http://arxiv.org/abs/0707.1449" rel="nofollow">0707.1449</a>)</li> </ul> <blockquote> <p>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''.</p> </blockquote> <ul> <li>A model category for the homotopy theory of concurrency (<a href="http://arxiv.org/abs/math/0308054" rel="nofollow">math/0308054</a>)</li> </ul> <blockquote> <p>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.</p> </blockquote> <p>I guess it is just because of my ignorance, but to me this was unexpected.</p> http://mathoverflow.net/questions/18336/what-are-surprising-examples-of-model-categories/23237#23237 Answer by mmm for What are surprising examples of Model Categories? mmm 2010-05-02T07:06:49Z 2013-05-26T22:41:56Z <p><a href="http://mishap.sdf.org/by:gavrilovich/what:work-in-progress/blat.pdf" rel="nofollow">Here</a> is an example of a poset which is a model category. The construction is set-theoretic and mentions Continuum Hypothesis.</p> http://mathoverflow.net/questions/18336/what-are-surprising-examples-of-model-categories/90947#90947 Answer by Andrés Villaveces for What are surprising examples of Model Categories? Andrés Villaveces 2012-03-11T22:57:51Z 2012-03-11T22:57:51Z <p>(more detail on the answer by mmm: Gavrilovich in <a href="http://arxiv.org/abs/1006.4647" rel="nofollow">http://arxiv.org/abs/1006.4647</a> and then further works of Gavrilovich and Hasson <a href="http://arxiv.org/abs/1102.5562" rel="nofollow">http://arxiv.org/abs/1102.5562</a> and then Gavrilovich, Hasson and Kaplan <a href="http://arxiv.org/abs/1111.3489" rel="nofollow">http://arxiv.org/abs/1111.3489</a> 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 <em>surprising</em> it is depends (I guess) on how much you are used to "detect" homotopic content in areas where it was seemingly not present.</p> http://mathoverflow.net/questions/18336/what-are-surprising-examples-of-model-categories/91010#91010 Answer by Chris Schommer-Pries for What are surprising examples of Model Categories? Chris Schommer-Pries 2012-03-12T19:07:56Z 2012-03-12T19:07:56Z <p><strong>The category of sets admits precisely nine model category structures, no more no less.</strong> </p> <p>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 <em>all</em> 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. </p>