Algebraic Kan extensions in double categories

Abstract

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 $\mathcal{K}$, 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 ${\mathsf{Alg}}_{w}T$, 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 ${\mathsf{Alg}}_{w}T\to \mathcal{K}$. 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.

61 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

References in corpus (3)

• Operads revisited
• On pointwise Kan extensions in double categories
• Algebraic weighted colimits

Cited by in corpus (7)

• Regular patterns, substitudes, Feynman categories and operads
• A double-dimensional approach to formal category theory
• On pointwise Kan extensions in double categories
• A Criterion for Kan Extensions of Lax Monoidal Functors
• A categorical approach to the maximum theorem
• Left Kan extensions that are algebraic over colax-idempotent 2-monads
• Cartesian double theories: A double-categorical framework for categorical doctrines