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Let me preface this by saying that I don't know much category theory.

I am running into a situation where I have a double category and additionally there is a multiplication. Moreover, choosing either the vertical or the horizontal arrows makes my thing a symmetric monoidal category. Has this structure been studied somewhere?

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  • $\begingroup$ As far as I understand this (among other things) will be treated in Gaitsgory and Rozenblyum's upcoming book. $\endgroup$ – Adam Gal Apr 25 '14 at 13:21
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Symmetric Monoidal and Cartesian Double Categories as a Semantic Framework for Tile Logic by Roberto Bruni, José Meseguer, Ugo Montanari

Mathematical Structures in Computer Science / Volume 12 / Issue 01 / February 2002, pp 53-90 DOI: http://dx.doi.org/10.1017/S0960129501003462, Published online: 26 February 2002

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There is a preprint by Mike Shulman, Constructing Symmetric Monoidal Bicategories, which seems to treat the problem of constructing a symmetric monoidal bicategory from a symmetric monoidal double category. You can find it on arXiv, here: http://arxiv.org/abs/1004.0993

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