Algebraic Kan extensions in double categories · arXivDesk
1406.6994Jun 26, 201461 pages. Changes in v2 include a slightly sharpened statement of the main theorem (5.7) as well as a new remark (5.8). This is the final version, as it appears in TAC
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
Nearby in the stack
-cells, where `weak' means either `lax', `colax' or `pseudo'. Denoting these double categories by
AlgwT
, 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
AlgwT→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.