# A Model Structure on Symmetric Monoidal Categories

The recent article found here 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?

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.

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Elmendorf's conjecture below has a corresponding result in terms of dendroidal sets which is proved here: arxiv.org/abs/1203.6891 –  Peter Arndt Apr 2 '12 at 13:23

## 1 Answer

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.

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Yes, shortly after writing this it had occurred to me that I wasn't sure if the the category had all the required limits and colimits and this might be an obstruction. The idea of using multicategories is intriguing. I'll be sure to have a look at your paper. –  Eric Finster Mar 20 '10 at 23:45
What's the state of this work in progress? Isn't the category of multicategories just the category of (non-symmetric) colored operads? Does the recent work of Marcy Robertson get the required model structure? –  David White Aug 10 '12 at 18:41