This is a report on ongoing work aimed towards giving an abstract reconstruction of the Boardman--Vogt tensor product of symmetric multicategories, as well as an extension of this to multiprofunctors due to Dwyer and Hess.
In this talk, I hope to cover the basic abstract setting, which is that of a so-called normal oplax monoidal double category. This is a bit less than what is usually meant by a monoidal double category, but is exactly what one gets in the leading example, which for us, is the double category of coloured symmetric sequences. The plan is to explain how such structure arises abstractly, in terms of a presentation of symmetric sequences as a Kleisli double category.
This is joint work with Nicola Gambino and Christina Vasilakopoulou.