Premonoidal and Kleisli double categories

Abstract

We give a double categorical version of the recently introduced notion of premonoidal bicategories. We introduce two sorts of funny products on double categories granting them closed funny monoidal structures. We classify binoidal structures and non-Cartesian monoidal products for double categories according to the versions of multimaps used in their defining multicategories. We characterize strict and semi-strict premonoidal double categories as (pseudo)monoids in the funny monoidal (2-)category of double categories. We prove that a premonoidal double category D is purely central if and only if its binoidal structure is given by a pseudodouble quasi-functor (a multimap for a multicategory in the style of Gray) if and only if it admits a monoidal structure. For such D, we introduce pure center and show that the monoidal structure on D extends to it. We also discuss one-sided and general center double categories. Exploiting the companion-lifting properties of vertical structures in a double category into their horizontal counterparts, we prove a series of further results simplifying proofs for the corresponding bicategorical findings. We introduce vertical strengths on vertical double monads and horizontal strengths on horizontal double monads and prove that the former induce the latter. We show that vertical strengths induce actions of the induced horizontally monoidal double category on the corresponding Kleisli double category of the induced horizontal double monad. We prove that there is a 1-1 correspondence between horizontal strengths and extensions of the canonical action of the double category on itself. Finally, we show that for a bistrong vertical double monad, the corresponding Kleisli double category is premonoidal.