We study Kan extensions in three weakenings of the Eilenberg-Moore double category associated to a double monad, that was introduced by Grandis and Paré. To be precise, given a normal oplax double monad T on a double category K, we consider the double categories consisting of pseudo T-algebras, `weak' vertical T-morphisms, horizontal T-morphisms and T-cells, where `weak' means either `lax', `colax' or `pseudo'. Denoting these double categories by Alg_w(T), where w = l, c or ps accordingly, our main result gives, in each of these cases, conditions ensuring that (pointwise) Kan extensions can be lifted along the forgetful double functor Alg_w(T) --> K. As an application we recover and generalise a result by Getzler, on the lifting of pointwise left Kan extensions along symmetric monoidal enriched functors. As an application of Getzler's result we prove, in suitable symmetric monoidal categories, the existence of bicommutative Hopf monoids that are freely generated by cocommutative comonoids.
Keywords: double monad, algebraic Kan extension, free bicommutative Hopf monoid
2010 MSC: 18D05, 18C15, 18A40, 16T05
Theory and Applications of Categories, Vol. 30, 2015, No. 5, pp 86-146.