Skew-monoidal reflection and lifting theorems

Abstract: The Day Reflection Theorem gives conditions under which a reflective subcategory of a closed monoidal category can be equipped with a closed monoidal structure in such a way that the reflection adjunction becomes a monoidal adjunction. We adapt this result to skew monoidal categories. The beauty of this variant is further evidence that the direction choices involved in the skew notion are important for organizing, and adding depth to, certain mathematical phenomena. We also provide conditions under which a skew monoidal structure can be lifted to the category of Eilenberg-Moore algebras for a comonad.