Skew monoidal structures on actegories

Pavla Procházková

Masaryk University



Abstract: Skew monoidal categories are a generalisation of monoidal categories, where we omit the requirement for invertibility of constraint morphisms and fix a certain combination of orientations of those. An early example of this structure was described by Altenkirch Chapman and Uustalu in 2010 in the paper Monads Need Not Be Endofunctors. The name skew monoidal category was introduced in 2012 by Szlachányi, who studied structures induced by bialgebroids.

In my Master’s thesis, I have been studying how monoidal actions of categories, where we have the existence of a certain adjoint functor, induce a skew monoidal structure on the actegory (i. e. that acted-upon category). A large class of examples of skew monoidal categories can be described in this way, including the example of Altenkirch, Chapman and Uustalu and examples involving bialgebras, which are special cases of Szalchányi’s motivating bialgebroid examples.

In this talk, I want to explore some properties of skew monoidal categories induced by this construction. The adjunction involved in the construction can be shown to be monoidal with respect to the original tensor of the acting category and the newly defined skew-tensor on the actegory. This observation is closely related e. g. to the correspondence between relative monads and certain ordinary monads described by Altenkirch, Chapman, and Uustalu.

I also plan to discuss how braidings on the acting monoidal category induce braidings on the induced skew monoidal category. In addition, I will touch on what are some sufficient conditions for the induced structures to be closed.