# symmetric monoidal double categories?

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?

• As far as I understand this (among other things) will be treated in Gaitsgory and Rozenblyum's upcoming book. Apr 25 '14 at 13:21

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

• These symmetric monoidal double categories (which are the same ones mentioned in John's answer) use only one direction (say, vertical) for the coherence isomorphisms. But the main result is that if the double category has sufficient companion pairs then these coherences can be lifted to horizontal ones that are coherent up to isomorphism as in a symmetric monoidal bicategory. The OP didn't describe his particular double category, but it could be that this is the reason that choosing "either" vertical or horizontal directions makes a SMC. Aug 15 '20 at 1:12

Kenny Courser has written a nice thesis on symmetric monoidal double categories and their applications:

• Kenny Courser, Open Systems: A Double Categorical Perspective, Ph.D. thesis, U. C. Riverside, 2020. Available at arXiv:2008.02394.

Also, Mike Shulman's paper mentioned above has a new installment, which explains symmetric monoidal double categories, the maps between them, and the maps between those - and how to turn these into symmetric monoidal double categories, and maps between them, and maps between those:

• L. W. Hansen and M. Shulman, Constructing symmetric monoidal bicategories functorially. Available at arXiv:1901.09240.

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