Skew monoidal structures on actegories
Pavla Procházková
TL;DR
The paper addresses how to construct skew monoidal structures on actegories from strong actions by leveraging a left adjoint to $J_*$, yielding a tensor $A \lhd B = J_!A * B$ and associator/unitors built via adjoint transposition. It proves that the resulting adjunction $J_! \dashv J_*$ is monoidal, with $J_*$ being lax monoidal and, when $J_!$ is strong monoidal, invertibility of the key constraint maps, connecting to warpings and opmonoidal monads. The framework unifies existing skew monoidal examples (e.g., Kan-extension warpings, bialgebroid-related tensors) and provides mechanisms to transfer braidings and closedness from the acting category to the induced skew structure. It also analyzes braidings for induced skew monoidal actegories and gives sufficient conditions for left/right closedness, enriching the theory and its applications to categorical structures arising from actions.
Abstract
We present a construction of skew monoidal structures from strong actions. We prove that the existence of a certain adjoint allows one to equip the actegory with a skew monoidal structure and that this adjunction becomes monoidal. This construction provides a unifying framework for the description of several examples of skew monoidal categories. We also demonstrate how braidings on the original monoidal category of a given action induce braidings on the resulting skew monoidal structure on the actegory and describe sufficient conditions for closedness of the resulting skew monoidal structure.