The Matrix Product of Coloured Symmetric Sequences

In 2008, Maia and Méndez defined the operation of arithmetic product of species of structures, extending the calculus of species of structures introduced by Joyal in the ‘80s. In 2014, as part of their work on the Boardman-Vogt tensor product of bimodules, Dwyer and Hess rediscovered independently this operation and studied it in the context of symmetric sequences and named it matrix multiplication.

In this talk, based on joint work in progress with Richard Garner and Christina Vasilakopoulou, we extend the matrix multiplication from symmetric sequences to coloured symmetric sequences and show that it determines an oplax monoidal structure on the the bicategory of coloured symmetric sequences. In order to do this, we establish general results on lifting monoidal structures to Kleisli double categories. This approach allows us to attack and solve the difficult problem of verifying the coherence conditions for a monoidal bicategory in an efficient way.