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Algebraic Kan extensions in double categories

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Seerp Roald Koudenburg

We study Kan extensions in three weakenings of the Eilenberg-Moore
double category associated to a double monad, that was introduced by
Grandis and ParĂ©. 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.

Published 2015-02-04.

http://www.tac.mta.ca/tac/volumes/30/5/30-05.pdf

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