A Model Structure on Symmetric Monoidal Categories - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T10:34:09Z http://mathoverflow.net/feeds/question/17569 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/17569/a-model-structure-on-symmetric-monoidal-categories A Model Structure on Symmetric Monoidal Categories Eric Finster 2010-03-09T03:21:08Z 2010-03-18T22:13:59Z <p>The recent article found <a href="http://arxiv.org/abs/1002.3622" rel="nofollow">here</a> revisits Thomason's proof that symmetric monoidal categories model all connective spectra, but stops short of showing that there is a full closed model structure on this category (as does, it seems, Thomason's original paper.) Is there such a thing?</p> <p>My guess is some lifting similar to how the model structure on small categories is derived would work, but I'm not sure if there are any complications.</p> http://mathoverflow.net/questions/17569/a-model-structure-on-symmetric-monoidal-categories/18673#18673 Answer by Tony Elmendorf for A Model Structure on Symmetric Monoidal Categories Tony Elmendorf 2010-03-18T22:13:59Z 2010-03-18T22:13:59Z <p>One basic problem is that the category of symmetric monoidal categories isn't complete. Its completion, in a basic sense, is the category of multicategories, on which it seems reasonable to conjecture there is a model category structure whose homotopy category "is" the connective part of stable homotopy -- we hope to prove this soon. See Elmendorf and Mandell, "Permutative categories, multicategories, and algebraic K-theory", which just appeared in Algebraic and Geometric Topology.</p>