Given two algebraic theories S and T, their commuting tensor product is a new algebraic theory in which the operations of S and T are required to commute with each other. The analogue of this operation for symmetric operads is the famous Boardman-Vogt tensor product. A unified account of these commuting tensor products was given by Garner and Lopez-Franco.
Here, we extend their analysis in two respects. First, we generalise it so as to make it applicable to the many-object case, recovering as special case the Boardman-Vogt tensor product of symmetric multicategories, subsuming work of Elmendorf and Mandell. Secondly, we investigate how the commuting tensor product acts on bimodules, generalising work of Dwyer and Hess.
This is joint work with Richard Garner and Christina Vasilakopoulou.
