two preprints on Kan extensions in/for double categories

two preprints on Kan extensions in/for double categories

[Marco Grandis]

This is to announce two preprints, which take on our study of weak
double categories and deal with Kan extensions. The first two parts,
dealing with limits and adjoints, respectively, where published in
"Cahiers" (1999, 2004) and can also be found on my server.

M. Grandis

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M. Grandis - R. Pare,
Kan extensions in double categories (On weak double categories, Part
III),
Dip. Mat. Univ. Genova, Preprint 553 (2007).

http://www.dima.unige.it/~grandis/Dbl3.pdf   (ps)

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---, Lax Kan extensions for double categories (On weak double
categories, Part IV),
Dip. Mat. Univ. Genova, Preprint 554 (2007).

http://www.dima.unige.it/~grandis/Dbl4.pdf   (ps)

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Abstracts.

Part III. This paper deals with Kan extensions in a weak double
category. Absolute Kan extensions are closely related to the
orthogonal adjunctions introduced in a previous paper. The pointwise
case is treated by introducing internal comma objects, which can be
defined in an arbitrary double category.

Part IV. Right Kan extensions for weak double categories extend
double limits and other constructions, called vertical companions and
vertical adjoints, studied in previous papers. We prove that these
particular cases are sufficient to construct all pointwise unitary
lax right Kan extensions, along those lax double functors which
satisfy a Conduche type condition. Double categories 'based on
profunctors' are complete, i.e. have all such constructions, while
the double category of commutative squares on a complete category is
not, in general.