Skew monoidal structures on categories of algebras

There is a long tradition of constructing monoidal or closed structure on the category of algebras for a monad that is assumed to be commutative, monoidal, cartesian closed,or similar. In each case, one builds a tensor product classifying bilinear maps using a coequalizer. This approach, initiated by Linton’s description of the construction, has been studied by Kock, Guitart and others, while Seal has recently examined the monoidal case in some detail. In this talk we explore these ideas for skew monoidal categories, viz. suitably directed versions of monoidal categories in which the structuralmaps are not assumed to be invertible. Under standard conditions I will show that, for a strong monad T on a skew monoidal category, the category of T-algebras acquires a skew monoidal structure with a tensor product classifying left-linear maps. I will then characterise the monoids for this left-linear monoidal structure as precisely the T-monoids of Fioreet al., and give two constructions for free monoids in skew monoidalcategories.